Gr 9 Maths.Pdf

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Table of Contents

Page Lessons Topic Number 1 Introduction to Maths 8-12 2 Introduction to Maths - Roman Numerals 12-16 3 Number Systems 16-23 4 Number Systems 16-23 5 Factors and Multiples 23-26 6 Worksheet - Factors 26 7 Worksheet - Factors 26 8 Squares 26-31 9 Worksheet - Squares 31 10 Cubes 31-39 11 Cubes 31-39 12 Worksheet - Cubes 40 13 Number Patterns and Sequences 40-43 14 Worksheet - Number Patterns 44 15 Worksheet - Number Patterns 44 16 Rational and Irrational Numbers 44-57 17 Rational and Irrational Numbers 44-57 18 Worksheet - Classifying Numbers 57 19 Worksheet - Classifying Numbers 57 20 Negative Numbers 57-68 21 Negative Numbers 57-68 22 Two Dimensional Shapes and Measurement 68-69 23 Theorem of Pythagoras 70-76 24 Theorem of Pythagoras 70-76

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Page 2 of 362 25 Worksheet - Theorem of Pythagoras 76 26 Worksheet - Theorem of Pythagoras 76 27 Parallelograms and Trapeziums 76-84 28 Worksheet - Parallelograms and Trapeziums 84 Area and Perimeter of Regular and Irregular 29 Polygons 84-83 30 Worksheet - Area 83 31 Worksheet - Perimeter 83 32 Area and Perimeter of a Circle 83-99 33 Area and Perimeter of a Circle 83-99 34 Worksheet - Area of a Circle 99 35 Worksheet - Perimeter of a Circle 99 36 Notation and Powers of 10 100-106 37 Division of Powers 106-109 38 Worksheet - Powers 109 39 Scientific Notation 109-112 40 Worksheet - Scientific Notation 112 41 Square Roots 113-117 42 Multiplication and Division of Exponents 117-120 43 Multiplication and Division of Exponents 117-120 44 Addition and Subtraction of Polynomials 120-125 45 Addition and Subtraction of Polynomials 120-125 46 Worksheet - Polynomials 125 47 Transformations 125-135 48 Transformations 125-135 49 Worksheet - Nets 135 50 Congruent Triangles in Polygons 135-142

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Page 3 of 362 51 Congruent Triangles in Polygons 135-142 52 Activity - Create Your Own Pattern 142 53 Enlargement 142-148 54 Congruency and Similarity 148-157 55 Congruency and Similarity 148-157 56 Worksheet - Similarity and Congruency 158 57 Types of Transformations 15-163 58 General Discussion 164-169 59 General Discussion 164-169 60 General Discussion 164-169 61 Equivalent Expressions 170-171 62 Worksheet - Equivalent Expression 171 63 Worksheet - Equivalent Expression 171 64 Products and Factors 171-176 65 Factorization 176-181 66 Factorization 176-181 67 Worksheet - Factorization 181 68 Worksheet - Factorization 181 69 Worksheet - Factorization 181 70 Difference of Two Squares 181-184 71 Worksheet - Difference of Two Squares 185 72 Worksheet - Difference of Two Squares 185 73 Algebraic Fractions 185-191 74 Algebraic Fractions 185-191 75 Worksheet - Algebraic Fractions 191 76 Worksheet - Algebraic Fractions 191 77 Mathematical Relationships 191-192

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Page 4 of 362 78 Conjectures 193-195 79 Functions 195-207 80 Functions 195-207 81 Functions 195-207 82 Types of Functions and Gradients 208-214 83 Types of Functions and Gradients 208-214 84 Inverse Operations 215-216 85 Inverse Functions 216-224 86 Inverse Functions 216-224 87 Solving Equations Using Inverse Operations 224-227 88 Worksheet - Expressions and Equations 227 89 Worksheet - Expressions and Equations 227 90 Points to Remember When Solving Equations 227-228 91 Equations with Variables on Both Sides 228-229 92 Worksheet - Working with Variables 229 93 Worksheet - Working with Variables 229 94 Equations with Brackets 229-231 95 Worksheet - Equations with Brackets 231 96 Worksheet - Equations with Brackets 231 97 Equations with Fractions 231-236 98 Quadratic Equations 236-241 99 Quadratic Equations 236-241 100 Quadratic Equations 236-241 101 Solving Problems with Algebraic Models 242-244 102 Solving Equations by Trial and Improvement 244-245 103 Solid Geometry 245-246 104 Polyhedra and Non Polyhedra 246-247

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Page 5 of 362 105 Volume of a Pyramid 247-249 106 Volume of a Pyramid 247-249 107 Volume of a Cone 249-251 108 Combining Cubes 251-254 109 Ratios 254-260 110 Ratios 254-260 111 Rate 260-261 112 Worksheet - Ratios and Rate 262 113 Worksheet - Ratios and Rate 262 114 Proportion 262-269 115 Proportion 262-269 116 Gradient 269-273 117 Worksheet - Gradient 273 118 Worksheet - Gradient 273 119 Worksheet - Gradient 273 120 Volume and Capacity 273-277 121 Volume and Capacity 273-277 Calculating the Height and Radius of a Cylinder 122 Given the Volume 277-282 123 Volumes of Prisms 283-287 124 Worksheet - Volume of Prisms 288 Calculating the Surface Area of a Cylinder and a 125 Right Prism 288-301 Calculating the Surface Area of a Cylinder and a 126 Right Prism 288-301 Calculating the Surface Area of a Cylinder and a 127 Right Prism 288-301 128 Calculating the Surface Area of a Cylinder and a 288-301

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Page 6 of 362 Right Prism 129 Worksheet - Surface Area 301 130 Worksheet - Surface Area 301 131 Worksheet - Surface Area 301 132 Statistical Graphs or Charts - Bar Graphs 301-302 133 Line Graphs and Pictograms 302-304 134 Pie Charts 304-306 135 Frequency Diagrams and Polygons 306-313 136 Frequency Diagrams and Polygons 306-313 137 Measures of Central Tendency (Mean) 313 138 Measures of Central Tendency (Median) 314 139 Measures of Central Tendency (Mode) 314-318 140 Worksheet - Measures of Central Tendency 319 141 Worksheet - Measures of Central Tendency 319 142 Measures of Dispersion 319-322 143 Dealing with Bivariate Data 322-327 144 Collecting Data 327-331 145 Collecting Data 327-331 146 Activity - Making a Questionnaire 331 147 Simple Interest 331-333 148 Worksheet - Simple Interest 334 149 Worksheet - Simple Interest 334 150 Hire Purchase Loan 334-335 151 Worksheet - Hire Purchase Loan 336 152 Compound Interest 336-342 153 Compound Interest 336-342 154 Worksheet - Compound Interest 343

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Page 7 of 362 155 Inflation Rates 343 156 Depreciation Rates 343-346 157 Worksheet - Depreciation Rates 346 158 Exchange Rates 346-347 159 Worksheet - Exchange Rates (Q1 - 7) 347 160 Worksheet - Exchange Rates (Q8 - 15) 347 161 Worksheet - Exchange Rates (Q16 - 20) 347 162 Commission and Rentals 347-348 163 Worksheet - Calculating Commission 348 164 Worksheet - Calculating Commission 348 165 Similarity 348-352 166 Worksheet - Similarity 352 167 Worksheet - Similarity 352 168 Congruency 352-355 169 Worksheet - Congruency 355 170 Worksheet - Congruency 355 171 Worksheet - Congruent Triangles 355 172 Worksheet - Congruent Triangles 355 173 Congruent Angles 355-356 174 The Probability Scale 356-361 175 Worksheet - Probability 361 176 Worksheet - Probability 361 177 Revision 361 178 Revision 361 179 Revision 361 180 Revision 361 181 Revision 361

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Lesson 1: Introduction to Maths

GLOSSARY

Symmetry The quality of being made up of exactly similar parts facing each other or around an axis. Probability It is a ratio of the number of ways an event can occur to the total number of possible outcomes. Factor A number that will divide into another number exactly. Multiple The multiple of a number is the product of the number and any other whole number. (2,4,6,8 are multiples of 2) Prime numbers Prime numbers are integers that are greater than 1 and are only divisible by them and 1. Composite Number Composite number has at least one other factor aside from its own. A composite number cannot be a prime number. Square root Square a number, you multiply it by itself. The square root of a number is the value of the number when multiplied by itself, gives you the original number. For instance 12 squared is 144, the square root of 144 is 12. Exponent Exponent means a little number in the upper corner of the number that indicates the number of times you multiply the base number (like in 3 to the 4th, 3 would be the base) times itself. 3 to the 4th means 3x3x3x3. 3 to the 4th equal 81. Oddly, any number to the power of 0 equals 1. Even 0 to the power of 0 equals one. That is what the term exponents mean.

Scientific Notation Scientific notation, also known as standard form or as exponential notation, is a way of writing numbers that accommodates values too large or small to be conveniently written in standard decimal notation. ... Variable An element, feature, or factor that is liable to vary or change.

Mathematics equips children with a uniquely powerful set of tools to understand and change the world. These tools include logical reasoning, problem-solving skills, and the ability to think in abstract ways. As such, mathematics is a creative discipline. It can stimulate moments of happiness and wonder when a child solves a problem for the first time, discovers a more efficient solution to a problem or suddenly sees hidden connections.

Throughout history, mathematics has shaped the way we view the world. The early study of astronomy demanded the expansion of our understanding of mathematics and made possible such realizations as the size and weight of the earth, our distance from the sun, the fact that we revolve around it, and other discoveries that allowed us to move forward in our body of knowledge without which we would not have any of our modern marvels of technology.

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Page 9 of 362 Mathematics remains as important today. Many life stages and skills require a solid grasp of mathematics, from entering university to balancing a household budget, applying for a home loan, or assessing a possible business opportunity. When children eventually leave education and seek out a career, they will inevitably need to call upon the mathematical skills and strategies they have learnt at school. They will soon realize that many careers require a solid understanding of maths. Doctors, lawyers, accountants and other professionals use maths on a daily basis, as do builders, plumbers, engineers and managers. Math’s is a critical skill for many professions and opens a world of opportunity for children.

Mathematics in Everyday Life It is sometimes difficult for students to appreciate the importance of Mathematics. They often find the subject boring and hard to understand. With this project, we will hopefully help our students realize that Mathematics is not just a subject on their timetable but a tool they use in their everyday life!

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Page 10 of 362 Math is everywhere and yet, we may not recognize it because it does not look like the math we did in school. Math in the world around us sometimes seems invisible. However, math is present in our world all the time--in the workplace, in our homes, and in life in general. When you buy a car, follow a recipe, or decorate your home, you are using math principles.

Percentages are used in our everyday life and we may not even realize it!

Car Logos Most manufacturers use symmetry of some kind in designing their logos. For example, Audi uses four intersecting circles in a line. This pattern has one line of reflection symmetry.

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On a basic level, you need to be able to count, multiply, subtract and divide. Mathematics is around us. It is present in different forms whenever we pick up the phone, manage the money, travel to some place, play soccer, meet new friends; unintentionally in all these things mathematics is involved. There are illustrations that testify the presence of mathematics in everything that we are doing.

Cooking: the idea of proportion For a Chocolate cake: 5 eggs,3/4 cup of sugar, 1/2 cup of vegetable oil,

Bank: savings and credit With some good understanding of simple and compound interest, you can manage the way your money grows.

Chance to win in lottery: Probability The mathematical concept that deals with the chance of winning a lottery game is probability...

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Area

Geometry in clothing Symmetry in the nature

Symmetry in the nature: a flower

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Lesson 2: Introduction to Maths –Roman Numerals

The Roman Number System

The Romans used several different systems for writing numbers. Sometimes they wrote numbers like this: I II III IV V and other times they used the Greek numbers. Roman people did not always write numbers the same way, either - people knew what you meant even if you did it a little differently. Here is a table showing all of the Roman numerals.

I 1 L 50 II 2 C 100 III 3 D 500 IV (or IIII) 4 M 1000 V 5 VI 6 VII 7 VIII 8 IX (or VIIII) 9 X 10

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Using the calculator

Counting and Calculating Mechanical Devices

Abacus – The first known calculating device

The abacus was a simple wooden box with beads strung, which are moved towards the mid-bar to perform calculation. You bring the beads near the bar and count to get result. It is a manual process. Thus, Abacus is essentially a memory aid rather than a calculating device. An Abacus is divided into two parts – heaven, the upper deck and earth, the lower deck- divided by a mid-bar. There are two beads in each string on heaven and 5 beads on earth. The value of each bead on heaven is considered 5 and on earth 1. Therefore, if you pull one heaven bead and 3-earth bead near the mid-bar, it represented the number 8.

Abacus is a Latin word that has its origins in the Greek words abax or abakon (meaning “table” or “tablet”)

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Page 16 of 362 Napier’s bones Abacus is about ancient past. When we look upon the modern history, it is 1614 when John Napier invented Logarithm – a branch of mathematics to multiply and divide extremely large or small numbers. This is considered a principal invention of Napier. John Napier publicly propounded the method of logarithms in 1614; it is a set of rods (11 rods in a set). Numbers are carved on each rod and can be used to perform multiplication, division with the help of logarithm. These rods were made up of bones, must be the reason for the name. Calculation is performed by properly aligning the proper rods against each other and by inspection.

Lesson 3-4: Number Systems

The Egyptian Number System Here are some of the symbols for powers of ten that were used in the Egyptian number system below:

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Page 17 of 362 This hieroglyphic numeration was a written version of a concrete counting system using material objects. To represent a number, the sign for each decimal order was repeated as many times as necessary. To make it easier to read the repeated signs they were placed in groups of two, three, or four and arranged vertically.

Example 1:

In writing the numbers, the largest decimal order would be written first. The numbers were written from right to left.

Example 2:

Below are some examples from tomb inscriptions

Addition and Subtraction The techniques used by the Egyptians for these are essentially the same as those used by modern mathematicians today. The Egyptians added by combining symbols. They would combine all the units ( ) together, then all of the tens ( ) together, then all of the hundreds ( ), etc. If the scribe had more

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Page 18 of 362 than ten units ( ), he would replace those ten units by . He would continue to do this until the number of units left was less than ten. This process was continued for the tens, replacing ten tens with , etc.

For example, if the scribe wanted to add 456 and 265, his problem would look like this

The scribe would then combine all like symbols to get something like the following

He would then replace the eleven units ( ) with a unit ( ) and a ten ( ). He would then have one unit and twelve tens. The twelve tens would be replaced by two tens and one one-hundred. When he was finished, he would have 721, which he would write as

Subtraction was done much the same way as we do it except that when one has to borrow, it is done with writing ten symbols instead of a single one.

Multiplication Egyptians method of multiplication is clever, but can take longer than the modern day method. This is how they would have multiplied 5 by 29

When multiplying they would began with the number they were multiplying by 29 and double it for each line. Then they went back and picked out the numbers in the first column that added up to the first number (5). They used the distributive property of multiplication over addition.

29(5) = 29(1 + 4) = 29 + 116 = 145

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Page 19 of 362 Division The way they did division was similar to their multiplication. For the problem 98/7, they thought of this problem as 7 times some number equals 98. Again, the problem was worked in columns.

This time the numbers in the right-hand column are marked which sum to 98 then the corresponding numbers in the left-hand column are summed to get the quotient. Therefore, the answer is 14. 98 = 14 + 28 + 56 = 7(2 + 4 + 8) = 7*14

 Natural numbers Natural numbers are what you use when you are counting one to one objects. You may be counting pennies, buttons, or cookies. When you start using 1,2,3,4 and so on, you are using the counting numbers or to give them a proper title, you are using the natural numbers.

The numbers 1, 2, 3, 4…. Are known as the natural numbers. We put these numbers into “{ }” to indicate that they from a set. So we call the set {1; 2; 3; …} the set of natural numbers.

The symbol used to represent the set of natural numbers is IN = {1; 2; 3; 4; …}.

It can be represented on the number line as:

 Whole numbers With the inclusion of 0, a new set of numbers is formed.

Whole numbers are easy to remember. They're not fractions, they're not decimals, and they’re simply whole numbers. The only thing that makes them different from natural numbers is that we include the zero when we are referring to whole numbers. However, some mathematicians will also include the zero in natural numbers and I am not going to argue the point. I will accept both if a reasonable argument are presented. Whole numbers are 1, 2, 3, 4, and so on.

The set = {0; 1; 2; 3; ..} is called the set of whole numbers or counting numbers and may be represented on the number line as:

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 Integers Integers can be whole numbers or they can be whole numbers with a negative signs in front of them.

Individuals often refer to integers as the positive and negative whole numbers including zero. Integers are -4, -3, -2, -1, 0, 1, 2, 3, and 4 and so on. And the symbol used to represent integers is Z.

Z= {-3; -2; -1; 0; 1; 2; 3 ..} and may be represented on the number line as:

Here is your cheat sheet to help you remember what to do with positive and negative numbers (integers) with adding, subtracting, multiplying and dividing.

1. Positive + Positive = Positive: 5 + 4 = 9 Negative + Negative = Negative: (- 7) + (- 2) = - 9

Sum of a negative and a positive number: Use the sign of the larger number and subtract

(- 7) + 4 = -3 6 + (-9) = - 3 (- 3) + 7 = 4 5 + (-3) = 2

2. Subtracting Rules:

Negative - Positive = Negative: (- 5) - 3 = -5 + (-3) = -8 Positive - Negative = Positive + Positive = Positive: 5 - (-3) = 5 + 3 = 8 Negative - Negative = Negative + Positive = Use the sign of the larger number and subtract (Change double negatives to a positive) (-5) - (-3) = ( -5) + 3 = -2 (-3) - ( -5) = (-3) + 5 = 2

3. Multiplying Rules:

Positive x Positive = Positive: 3 x 2 = 6 Negative x Negative = Positive: (-2) x (-8) = 16

Negative x Positive = Negative: (-3) x 4 = -12 Positive x Negative = Negative: 3 x (-4) = -12

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Page 21 of 362 4. Dividing Rules:

Positive ÷ Positive = Positive: 12 ÷ 3 = 4 Negative ÷ Negative = Positive: (-12) ÷ (-3) = 4 Negative ÷ Positive = Negative: (-12) ÷ 3 = -4 Positive ÷ Negative = Negative: 12 ÷ (-3) = -4

 Rational Numbers Rational numbers have integers AND fractions (common and improper fractions) AND decimals.

Now you can see that numbers can belong to more than one classification group. Rational numbers can also have repeating decimals, which you will see be written like this: 0.54444444... which simply means it repeats forever, sometimes you will see a line drawn over the decimal place which means it repeats forever, instead of having a ...., the final number will have a line drawn above it.

- A rational number is any number that can be written in the form where a and b are

integers and . If the number is a decimal, it will be either terminating or recurring.

- Recurring decimals are decimals that have a repeating pattern of numbers. Examples of recurring decimals are: a) 0,333 … b) 0,454 545 …

Examples of terminating decimals are decimals that have an end. a) 3,52 - Rational numbers include all natural numbers, whole numbers and integers. Examples of rational numbers are:

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- The symbol used to represent the set of rational numbers is Q and irational numbers is Q’. Examples of irrational and rational numbers is:

Activity 1: Natural, Irrational, Whole, Rational Numbers and Integers

Please answer the following questions.

1. List three examples of: a) Irrational numbers b) Integers c) Whole numbers d) Natural numbers e) Rational numbers

2. Give an example of a Roman numeral. 3. Shannon and 4 of her friends decided to have a cupcake sale at school to raise money for abused victims. After the sale, there were 23 cub cakes left. Shannon and her four friends decide to divide the cupcakes evenly amongst themselves. a) How many whole cupcakes will each person get?

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Page 23 of 362 b) How many cupcakes will be left over? c) In order to share the remaining whole cupcakes evenly, how many pieces should each cupcake be cut into, and how many will each of the 5 friends get? 4. Find a rational number between 0,123 and 0,124. 5. Write down a rational number close to, but bigger than, 1.

Lesson 5: Factors and Multiples

In the problem 3 x 4 = 12, 3 and 4 are factors and 12 is the product.

A factor is One of two or more expressions that are multiplied together to get a product. Factoring is like taking a number apart. It means to express a number as the product of its factors. Factors are either composite numbers or prime numbers (except that 0 and 1 are neither prime nor composite).

The number 12 is a multiple of 3, because it can be divided evenly by 3.

Multiple is A multiple of a number is the product of that number and any other whole number. Zero is a multiple of every number. 3 x 4 = 12

3 and 4 are both factors of 12

12 is a multiple of both 3 and 4.

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A factor is simply a number that is multiplied to get a product. Factoring a number means taking the number apart to find its factors--it's like multiplying in reverse. Here are lists of all the factors of 16, 20, and 45.

16 --> 1, 2, 4, 8, 16

20 --> 1, 2, 4, 5, 10, 20

45 --> 1, 3, 5, 9, 15, 45 12 --> 12, 24, 36, 48, 60 . . .

5 --> 5, 10, 15, 20, 25 . . .

7 --> 7, 14, 21, 28, 35 . . .

Factors are either composite numbers or prime numbers. A prime number has only two factors, one and itself, so it cannot be divided evenly by any other numbers. Here's a list of prime numbers up to 100. You can see that none of these numbers can be factored any further.

PRIME NUMBERS to 100 2,3,5,7,11,13,17,19,23,29,31,37,41,43, 47,53,59,61,67,71,73,79,83,89,97

A composite number is any number that has more than two factors. Here's a list of composite numbers up to 20. You can see that they can all be factored further. For example, 4 equals 2 times 2, 6 equals 3 times 2, 8 equals 4 times 2, and so forth.

By the way, zero and one are considered neither prime nor composite numbers-they're in a class by themselves!

COMPOSITE NUMBERS up to 20 4,6,8,9,10,12,14,15,16,18,20

You can write any composite number as a product of prime factors. This is called prime factorization.

To find the prime factors of a number, you divide the number by the smallest possible prime number and work up the list of prime numbers until the result is itself a prime number. Let's use this method to find the prime factors of 168. Since 168 are even, we start by dividing it by the smallest prime number, 2. 168 divided by 2 is 84.

84 divided by 2 is 42. 42 divided by 2 is 21. Since 21 are not divisible by 2, we try dividing by 3, the next biggest prime number. We find that 21 divided by 3 equals 7, and 7 is a prime number. We know 168 are now fully factored. We simply list the divisors to write the factors of 168.

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Page 25 of 362 168 ÷ 2 = 84 84 ÷ 2 = 42 42 ÷ 2 = 21 21 ÷ 3 = 7 Prime number

Prime factors = 2 × 2 × 2 × 3 × 7

To check the answer, multiply these factors and make sure they equal 168. Here are the prime factors of the composite numbers between 1 and 20.

4 = 2 × 2 6 = 3 × 2 8 = 2 × 2 × 2 9 = 3 × 3 10 = 5 × 2 12 = 3 × 2 × 2 14 = 7 × 2 15 = 5 × 3 16 = 2 × 2 × 2 × 2 18 = 3 × 3 × 2 20 = 5 × 2 × 2

Example:

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Activity 2: Factors and Multiples

Enter the missing number for each prime factorization in the empty space. 1. 5 x 3 x 3 x ____ = 135 2. 13 x 11 x ____ = 2145 3. 7 x 5 x 11 x ____ = 1155 4. 7 x 7 x 13 x ____ = 4459 5. 5 x 5 x 7 x ____ = 350 6. 13 x 5 x 7 x ____ = 2275

State if the statements given are true or false: 1. 20 is a multiple of 10 2. 2 is a factor of every natural number 3. 12 is a multiple of 1 4. 6 is a factor of 38 5. 22 is a multiple of 10 6. 5 is a multiple of 35 7. 1 is a multiple of every natural number

Lesson 6-7: Worksheets

Please check your calendar for information on this lessons.

Lesson 8: Squares

In mathematics, a square number, sometimes also called a perfect square, is an integer that is the square of an integer; in other words, it is the product of some integer with itself. So, for example, 9 is a square number, since it can be written as 3 × 3.

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The usual notation for the formula for the square of a number n is not the product n × n, but the equivalent exponentiation n2, usually pronounced as "n squared". The name square number comes from the name of the shape. This is because a square with side length n has area n2.

Square numbers are non-negative. Another way of saying that a (non-negative) number is a square number is that its square root is again an integer. For example, √9 = 3, so 3 is the square root of 9 because 3 x 3 = 9.

A positive integer that has no perfect square divisors except 1 is called square-free. For a non-negative integer n, the nth square number is n2, with 02 = 0 being the zero the square. The concept of square can be extended to some other number systems. If rational numbers are included, then a square is the ratio of two square integers, and, conversely, the ratio of two square integers is a square (e.g., 4/9 = (2/3)2).

Starting with 1, there are square numbers up to and including m, where the expression represents the floor of the number x.

Examples The squares (sequence A000290 in OEIS) smaller than 602 are:

02 = 0 12 = 1 22 = 4 32 = 9 42 = 16 52 = 25 62 = 36 72 = 49 82 = 64 92 = 81 102 = 100 112 = 121 122 = 144 132 = 169 142 = 196 152 = 225 162 = 256 172 = 289 182 = 324 192 = 361 202 = 400 212 = 441 222 = 484 232 = 529 242 = 576 252 = 625 262 = 676 272 = 729 282 = 784 292 = 841 302 = 900 312 = 961 322 = 1024 332 = 1089 342 = 1156 352 = 1225 362 = 1296 372 = 1369 382 = 1444 392 = 1521 402 = 1600 412 = 1681 422 = 1764 432 = 1849 442 = 1936 452 = 2025 462 = 2116 472 = 2209 482 = 2304 492 = 2401 502 = 2500 512 = 2601 522 = 2704 532 = 2809 542 = 2916 552 = 3025 562 = 3136 572 = 3249 582 = 3364 592 = 3481

The difference between any perfect square and its predecessor is given by the identity .

Equivalently, it is possible to count up square numbers by adding together the last square, the last square's root, and the current root, that is, .

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Page 28 of 362 Properties The number m is a square number if and only if one can arrange m points in a square: m = 12 = 1

m = 22 = 4

m = 32 = 9

m = 42 = 16

m = 52 = 25

The expression for the nth square number is n2. This is also equal to the sum of the first n odd numbers as can be seen in the above pictures, where a square results from the previous one by adding an odd number of points (shown in magenta). The formula follows:

So for example, 52 = 25 = 1 + 3 + 5 + 7 + 9.

There are several recursive methods for computing square numbers. For example, the nth square number can be computed from the previous square by .

Alternatively, the nth square number can be calculated from the previous two by doubling the (n − 1)-th square, subtracting the (n − 2)-th square number, and adding 2, because n2 = 2(n − 1)2 − (n − 2)2 + 2.

For example, 2 × 52 − 42 + 2 = 2 × 25 − 16 + 2 = 50 − 16 + 2 = 36 = 62.

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Page 29 of 362 A square number is also the sum of two consecutive triangular numbers. The sum of two consecutive square numbers is a centered square number. Every odd square is also a centered octagonal number.

Another property of a square number is that it has an odd number of divisors, while other numbers have an even number of divisors. An integer root is the only divisor that pairs up with itself to yield the square number, while other divisors come in pairs.

Lagrange's four-square theorem states that any positive integer can be written as the sum of four or fewer perfect squares. Three squares are not sufficient for numbers of the form 4k (8m + 7). A positive integer can be represented as a sum of two squares precisely if its prime factorization contains no odd powers of primes of the form 4k + 3. This is generalized by Waring's problem.

A square number can end only with digits 0, 1, 4, 6, 9, or 25 in base 10, as follows: 1. If the last digit of a number is 0, its square ends in an even number of 0s (so at least 00) and the digits preceding the ending 0s must also form a square. 2. If the last digit of a number is 1 or 9, its square ends in 1 and the number formed by its preceding digits must be divisible by four. 3. If the last digit of a number is 2 or 8, its square ends in 4 and the preceding digit must be even. 4. If the last digit of a number is 3 or 7, its square ends in 9 and the number formed by its preceding digits must be divisible by four. 5. If the last digit of a number is 4 or 6, its square ends in 6 and the preceding digit must be odd. 6. If the last digit of a number is 5, its square ends in 25 and the preceding digits must be 0, 2, 06, or 56.

In base 16, a square number can end only with 0, 1, 4 or 9 and  in case 0, only 0, 1, 4, 9 can precede it,  in case 4, only even numbers can precede it.

In general, if a prime p divides a square number m then the square of p must also divide m; if p fails to divide m∕p, then m is definitely not square. Repeating the divisions of the previous sentence, one concludes that every prime must divide a given perfect square an even number of times (including possibly 0 times). Thus, the number m is a square number if and only if, in its canonical representation, all exponents are even.

Squarity testing can be used as alternative way in factorization of large numbers. Instead of testing for divisibility, test for squarity: for given m and some number k, if k2 − m is the square of an integer n then k − n divides m. (This is an application of the factorization of a difference of two squares.) For example, 1002 − 9991 is the square of 3, so consequently 100 − 3 divides 9991. This test is deterministic for odd divisors in the range from k − n to k + n where k covers some range of natural numbers k ≥ √m.

A square number cannot be a perfect number.

The sum of the series of power numbers

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Page 30 of 362

can also be represented by the formula

The first terms of this series (the square pyramidal numbers) are: 0, 1, 5, 14, 30, 55, 91, 140, 204, 285, 385, 506, 650, 819, 1015, 1240, 1496, 1785, 2109, 2470, 2870, 3311, 3795, 4324, 4900, 5525, 6201... (sequence A000330 in OEIS).

All fourth powers, sixth powers, eighth powers and so on are perfect squares.

Special cases  If the number is of the form m5 where m represents the preceding digits, its square is n25 where n = m × (m + 1) and represents digits before 25. For example the square of 65 can be calculated by n = 6 × (6 + 1) = 42 which makes the square equal to 4225.

 If the number is of the form m0 where m represents the preceding digits, its square is n00 where n = m2. For example the square of 70 is 4900.

 If the number has two digits and is of the form 5m where m represents the units digit, its square is AABB where AA = 25 + m and BB = m2. Example: To calculate the square of 57, 25 + 7 = 32 and 72 = 49, which means 572 = 3249.

Odd and even square numbers Squares of even numbers are even (and in fact divisible by 4), since (2n)2 = 4n2.

Squares of odd numbers are odd, since (2n + 1)2 = 4(n2 + n) + 1.

It follows that square roots of even square numbers are even, and square roots of odd square numbers are odd.

As all even square numbers are divisible by 4, the even numbers of the form 4n + 2 are not square numbers.

As all odd square numbers are of the form 4n + 1, the odd numbers of the form 4n + 3 are not square numbers.

Squares of odd numbers are of the form 8n + 1, since (2n + 1)2 = 4n(n + 1) + 1 and n(n + 1) is an even number.

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Activity no 3: Squares

1. Evaluate the following without using a calculator:

2. The number 49 is one less than twice 25. In other words 49 = 2 x -1. a) Find other square numbers that are one less (or perhaps one more) than twice another square number. b) Now find square numbers that are: (i) One less or more than three times another square number.

Lesson 9: Worksheet

Please check your calendar for information on this lesson.

Lesson 10: Cubes

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Page 32 of 362 What is a cube number? A cube number (or a cube) is a number you can write as a product of three equal factors of natural numbers.

Formula: k=a*a*a=a³ (k and a stand for integers.)

On the other hand, a cube number results by multiplying an integer by itself three times. Formula: a*a*a=a³=k (a and k stand for integers.)

The same factor is called the base.

After this, a negative number like (-2)³= -8 or a fraction number like (2/3)³=8/27 are suspended.

If it is appropriate, the number 0 is also a cubic number.

These are the first 100 cube numbers.

You can illustrate the name cube number by the following drawings.

If a is the side of a cube, the volume is a³.

The picture pairs make a 3-D view possible.

Cube Root It is easy to find a cube number. It is more difficult, to find the base of a cube number.

This procedure is called extracting the cube root of n.

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The cube root of a natural number can be written as

1 Computation with the calculator In former times you found cube numbers by ready-reckoners, today there are calculators.

You act with the TI30 like this: After entering the number you press the keys yx , (1/3) and finally = .

Example: [2299968] -- [yx] -- [(1/3)] -- [=] . You get 22999681/3 =132. It is even simpler to give in 2299968^(1/3) into the searching field of Google or Bing. The cube root appears after pressing the enter key.

2 Determination by nested intervals 22999681/3 The number must be between 100 and 200 (100³=1000000 and 200³=4000000). It must be between 130 and 140 (130³=2197000 and 140³=2744000). It must be near 130 and 8 is on the ones place. Then 132 comes into consideration. Result: 22999681/3=132

3 Determination by factorization 22999681/3 You write the number in factors and develop the cube number this way. 2299968 = 8*287496 = 8*27*10648 = 8*27*8*1331=8*27*8*11*121=8*8*27*11³=(2*2*3*11)³=132. Result: 22999681/3 = 132.

Cube Roots of Negative Numbers? There is (-2)³=-8. You could think that then (-8)1/3=-2 is. In former times this notation was common in schools, too. Today you demand that a1/3 is only defined for a>0 or a=0. The term (-8)1/3 it’s not allowed. Thus you avoid conflicts as the following calculations show.

There is (-8)1/3= (-8)2/6= [(-8)2]1/6= (64)1/6= (26)1/6=2. That means that (-8)1/3 would be ambiguous. You would get two results depending on the way of calculation.

This is possible. The equation (-2)³=-8 is changed to - (81/3) =-2. Generally x³=a leads to x=-(-a) 1/3 for a<0.

Sequences and Series Series of the cube numbers There is a series to each sequence and this is the sequence of the partial sums. For cube numbers this is the sequence sn= 1³+2³+3³+...+n³. 1³+2³+3³+...+n³= [n (n+1)/2]² is given.

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Page 34 of 362 Proof by mathematical induction >The statement holds for n=1: s1= [1*(1+1)/2]²=1. >Assume that the formula holds for all natural numbers n: sn= [n (n+1)/2]². >You must show that then the formula also holds for n+1: sn+1 = [(n+1) (n+2)/2]².

Calculation: Convert the product [(n+1) (n+2)] ² into a sum at first.

[(n+1)(n+2)]²=(n²+3n+2)²=n4+(3n+2)²+2n²(3n+2)=n4+(9n²+12n+4)+(6n³+4n²)=n4+6n³+12n²+12n+4

There is sn+1 = sn+(n+1)³ = [n(n+1)/2]²+(n+1)³ = [n²(n+1)²+4(n+1)³]/4 = [n²(n²+2n+1)+(4n³+12n²+12n+4)]/4 = (n4+6n³+13n²+12n+4)/4 = [(n+1)(n+2)]²/4 = [(n+1)(n+2)/2]², qed..

The formula 1³+2³+3³+...+n³= [n (n+1)/2]² contains the series of the natural numbers, because the sum can be written as 1+2+3+ … +n=n (n+1)/2.

Thus there is the formula 1³+2³+3³+...+n³ = (1+2+3+...+n) ², which is illustrated below.

Interpretation as an arithmetic series of third order The sequence of the cube numbers is also an arithmetic series of third order. A figure follows to explain this.

In every new row you find the difference of two numbers a row higher.

The feature is that you reach a constant difference 6 after three steps. There are formulas for these sequences, so that you can prove the formula 1³+2³+3³+...+n³= [n (n+1)/2]² in another way.

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Page 35 of 362 Series of odd numbers There is 1³=1, 2³=3+5, 3³=7+9+11, 4³=13+15+17+19 ... and 1³+2³+3³+4³+... = 1+3+5+7+9+11+13+15+17+19+...

Series of the reciprocal cube numbers There is another series of cube numbers, the one of the reciprocal cube numbers: 1/1³+1/2³+1/3³+...+1/n³.

It is convergent. You find the limit, the Riemann's zeta-function for 3, on the Wikipedia page 1729 (number).

Series of the centred hexagonal numbers The sequence of the centred hexagonal numbers is 1, 7, 19, 37, 61, 91, 127, 169, 217, 271, 331...

The formula is an=3n²-3n+1. There is 1, 1+7=8, 1+7+19=27, 1+7+19+37=64... The series to the sequence of the centred hexagonal numbers is the sequence of the cube numbers.

Calculation for proving:

The formulas 1²+2²+3³+...+n²= (1/6) n (n+1) (2n+1) and 1+2+3+...+n= (1/2) n (n+1) are used.

Sequence of the centred cube numbers The sequence of the centred cube numbers is 1, 9, 35, 91, 189, 341, 559, 855, 1241...

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Page 36 of 362 The picture pairs make a 3-D view possible. The formula is an=2n³-3n²+3n-1.

If you add two consecutive cube numbers, the centred cube numbers develop.

This is given by 2n³-3n²+3n-1=n³+ (n -1)³.

Atomium in Brussels, a centred cube

It stands 102 meters (335 ft) tall. There are nine steel spheres connected by tubes. Photos from July 2011.

Sequence of the perfect numbers T.L. Heath (1861-1940) proved that every even perfect number - except 6 - is the sum of 2(n-1)/2 cube numbers,

For example: 28=1³+3³, 496=1³+3³+5³+7³, 8128=1³+3³+5³+7³+9³+11³+13³+15³.

Waring's Problem The English mathematician Eduard Waring (1734-1798) maintained the following statement among others.

"Every natural number is either a cube number or the sum of 2,3,4,5,6,7,8 or 9 cube numbers." (2), page 37ff.

That means that 9 is a smallest number.

It can be more than 9 as the following sum of 180³ with 64 (!) cubic numbers shows.

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Page 37 of 362 180³ = 6³+7³+8³+...+67³+68³+69³ (1). Already 4 summands will do, 180=1³+3³+3³+5³.

The first numbers 1=1³ 2=1³+1³ 3=1³+1³+1³ 4=1³+1³+1³+1³ 5=1³+1³+1³+1³+1³ 6=1³+1³+1³+1³+1³+1³ 7=1³+1³+1³+1³+1³+1³+1³ 15=2³+1³+1³+1³+1³+1³+1³+1³ 23=2³+2³+1³+1³+1³+1³+1³+1³+1³

Dissection of the numbers <= 100, found by a simple computer program

The smallest number of the summands is determined.

>The numbers 2, 9, 16, 28, 35, 54, 65, 72, 91 have sums with 2 cubes at least: 2, 9, 16, 28, 35, 54, 65, 72, and 91 >Sums with 3 cubes: 3, 10, 17, 24, 29, 36, 43, 55, 62, 66, 73, 80, 81, 92, 99 >Sums with 4 cubes: 4, 11, 18, 25, 30, 32, 37, 44, 51, 56, 63, 67, 70, 74, 82, 88, 89, 93, 100 >Sums with 5 cubes: 5, 12, 19, 26, 31, 33, 38, 40, 45, 52, 57, 59, 68, 71, 75, 78, 83, 90, 94, 96, 97 >Sums with 6 cubes: 6, 13, 20, 27, 34, 39, 41, 46, 48, 53, 58, 60, 69, 76, 79, 84, 86, 95, 98 >Sums with 7 cubes: 7, 14, 21, 42, 47, 49, 61, 77, 85, 87 >Sums with 8 cubes: 15, 22, 50

(All numbers are 15, 22, 50, 114, 167, 175, 186, 212, 231, 238, 303, 364, 420, 428, 454.)

>Sums with 9 cubes for 23, (and 239 only) >8, 27, 64 are the cube numbers < 100.

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Page 38 of 362 Sum of cubes The equation x²+y²=z² leads to the Pythagorean triples The more general equation xn+yn=zn has no solution for n>2. If you cannot write a cube number as a sum of two cubes, you can look for sums with three or more cubes.

A nice example is 3³+4³+5³=6³, particularly, because there are four consecutive numbers.

Special Cube Numbers Square numbers among the cubes

There are cubes, which also are squares.

You can construct them step by step by squaring cube numbers.

23 leads to 26=64, 33 to 36=729, 4³ to 46=4 096, ...

The next cube numbers and at the same time square numbers up to 1 million are 15 625, 46 656, 117 649, 262 144 und 531 441. I must not forget 1.

Palindromes among the cube numbers

343=7³ 131=11³, 1030301=101³, 1003003001=1001³, ...

1367631=111³, 1030607060301=10101³, 1003006007006003001=1001001³, ...

1334996994331=11011³, 1331399339931331=110011³, ...

1033394994933301=1011010³, ...

It is noticed that all bases are also palindromes.

A cube number is the third power of its digit sum. 512=8³=(5+1+2)³ 4913=17³=(4+9+1+3)³ 5832=18³=(5+8+3+2)³ 17576=26³=(1+7+5+7+6)³ 19683=17³=(1+9+6+8+3)³

Terms with equal digits 3³+7³=37*(3+7) 4³+8³=48*(4+8) 14³+7³=147*(14+7) 14³+8³=148*(14+8)

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Page 39 of 362 Cube numbers are written in the digits 1 to 9. No digit is twice or more. 125*438976=380³ 8*24137569=578³ 8*32461759=628³

Two numbers with common features The bases form an arithmetic progression 180³ = 6³+7³+8³+...+67³+68³+69³ 540³ = 34³+35³+ ... +158³ 2856³ = 213³+214³+ ... +555³ 5544³ = 406³+407³+ ... +917³ 16834³ = 1134³+1135³+ ... +2133³ 3990³ = 290³+293³+ ... +935³ 29880³ = 2108³+2111³+ ... +4292³ 408³ = 149³+256³+363³ 440³ = 230³+243³+265³+269³+282³ 1155³ = 435³+506³+577³+648³+719³+790³ 2128³ = 553³+710³+867³+1024³+1181³+1338³+1475³ 168³ = 28³+41³+54³+67³+80³+93³+106³+119³ 64085³ =935³+5868³+10801³+15734³+20667³+25600³+30533³+35466³+40399³+45332³ 495³ = 15³+52³+89³+126³+163³+200³+237³+274³+311³+248³

Activity no 4: Cubes

1. Calculate the following without using a calculator:

2. Calculate the following also without using a calculator:

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Lesson 12: Worksheet

Please check your calendar for information on this lesson.

Lesson 13: Number Patterns and Sequences

Number Sequences are lists of numbers that form a relationship with each other and are connected in some way. They follow a pattern which you have to identify to extend the list of numbers or fill in the blanks.

Each number in the sequence is called a 'term' - so in the sequence 5, 6, 7, 8 ... 5 is the 1st term, 6 is the 2nd term, 7 is the 3rd term and so on.

Every sequence has a 'rule', once you discover the rule, you can use it to work out the missing numbers. For the sequence 1,2,3,4,5 ... the rule would be 'plus 1', so the next number would be 5 + 1 = 6, next is 6 +1 = 7 and so on.

There are several types of sequences and we will look in depth at the common types. Odd and Even odd end in 1,3,5,7,9 - even end in 2,4,6,8,0)

6, 8, 10, 12, 14, _ , _ ... This is a sequence of even numbers.

What are the next two numbers in the sequence ..... 16 and 18

5, 7, 9, 11, 13, _ , _ ... This is a sequence of odd numbers.

What are the next two numbers in the sequence ..... 15 and 17

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Page 41 of 362 Sequences can also go in reverse 22, 20, 18, 16, 14, _ , _ - rule is 'even numbers decreasing'

Arithmetic Sequence An Arithmetic Sequence either adds or subtracts a value to the previous term.

1, 5, 9, 13, 17, 21, 25, _ , _ ... Rule is 'add 4'

A useful strategy for dealing with sequences is to write them down, giving enough space to draw and label the hops from one term to another.

The rule is 'add 5' Sometimes you will be asked to find a term at the beginning or in the middle of the sequence

The rule is 'add 4' and you will have to work backwards to find the missing term

Geometric Sequence A Geometric Sequence is when the previous term is multiplied or divided by a value.

2,4,8,16,32,64, _ , _ ... Rule is 'multiply by 2' or double

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Page 42 of 362 Changing Number This is where the value changes with each step

Repeated Pattern This is where the pattern is repeated after a certain number of steps

Triangular Numbers

Square Numbers

Cube Numbers

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Page 43 of 362 Fibonacci Numbers

Rule is: The number is found by adding the two previous terms. (Term 5 is found by adding term 3 to term 4 ... 1 + 2 = 3)

Activity 5: Number sequences

1. Complete the number sequences. What is the rule for each sequence??

Example: 3, 6, 9, 12, 15, 18, 21, 24. Rule: Add 3

1. 8, 12, 16, _, _, _ Rule: 2. 19, 28, 37, _, _, _ Rule: 3. 33, 27, 21, _, _, _ Rule: 4. 13, 25, 37, _, _, _ Rule: 5. 100, 96, 92, _, _, _ Rule:

2. Can I find missing numbers in a sequence? Use a comma to separate numbers. Explain your predictions for the missing numbers.

Example: 54, 56, 58, 60, 62, 64, 66, 68 the rule is ADD 2 85, 82, 79, 76, 73, 70, 67, 64 the rule is SUBTRACT 3 a) 15, 17, __, __, 23, 25, __, __ The rule is b) __, 33, 38, __, 48, __, __, 63 The rule is c) __, __, 26, __, __, 56, __, __ The rule is d) 91, __, 81, __, __, 66, __, __ The rule is e) __, 62, __, __, __, __, __, 2 The rule is f) __, 85, __, __, 79, __, 75, __ The rule is g) __, 75, 78, 81, __, __, __, __ The rule is h) 34, __, __, __, 50, __, __, 62 The rule is i) __, __, __, __, 28, __, 18, 13 The rule is j) __, __, __, 87, __, __, __, 123 The rule is k) __, 84, __, __, 66, __, __, 48 The rule is l) 13, __, 27, 34, __, __, __, __ The rule is

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Lesson 14-15: Worksheets

Please check your calendar for information on this lessons.

Lesson 16-17: Rational and Irrational Numbers

Rational Numbers A rational number is a number that can be written as a ratio. That means it can be written as a fraction, in which both the numerator (the number on top) and the denominator (the number on the bottom) are whole numbers.  The number 8 is a rational number because it can be written as the fraction 8/1.  Likewise, 3/4 is a rational number because it can be written as a fraction.  Even a big, clunky fraction like 7,324,908/56,003,492 is rational, simply because it can be written as a fraction.

Every whole number is a rational number, because any whole number can be written as a fraction. For example, 4 can be written as 4/1, 65 can be written as 65/1, and 3,867 can be written as 3,867/1.

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Page 45 of 362 Here are some more examples:

Number As a Fraction Rational? 5 5/1 Yes 1.75 7/4 Yes .001 1/1000 Yes 0.111 1/9 Yes √2 (square root of 2) ? NO ! Oops! The square root of 2 cannot be written as a simple fraction! And there are many more such numbers, and because they are not rational they are called Irrational.

Formal Definition of Rational Number More formally we would say: A rational number is a number that can be in the form p/q where p and q are integers and q is not equal to zero.

So, a rational number is: p / q Where q is not zero

Examples: p q p / q =

1 1 1/1 1

1 2 1/2 0.5

55 100 55/100 0.55

1 1000 1/1000 0.001

253 10 253/10 25.3

7 0 7/0 No! “q” can’t be zero!

Using Rational Numbers

How to add, subtract, multiply and divide rational numbers A rational number is a number that can be written as a simple fraction (i.e. as a ratio).

Examples: Number As a Fraction 5 5/1

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Page 46 of 362 1.75 7/4 .001 1/1000 0.111... 1/9

In general ... So, a rational number looks like this: p / q But q cannot be zero, as that would be dividing by zero.

How to Add, Subtract, Multiply and Divide If the rational number is something simple like 3, or 0.001, then just use mental arithmetic, or your calculator! But if it is still in the form p / q, then read on to find how to handle it.

A rational number is a fraction, so you could also refer to:  Adding Fractions,  Subtracting Fractions, ½  Multiplying Fractions and  Dividing Fractions But here I will be showing you those operations in a more Algebra- like way. You might also like to read Fractions in Algebra.

Adding Fractions

There are 3 Simple Steps to add fractions:  Step 1: Make sure the bottom numbers (the denominators) are the same  Step 2: Add the top numbers (the numerators), put the answer over the denominator.  Step 3: Simplify the fraction (if needed). Example 1: 1 1 + 4 4 Step 1. The bottom numbers (the denominators) are already the same. Go straight to step 2.

Step 2. Add the top numbers and put the answer over the same denominator: 1 1 1 + 1 2 + = = 4 4 4 4

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Step 3. Simplify the fraction: 2 1 = 4 2

In picture form it looks like this: 1/ 1/ + 1/ = 2/ = 2 4 4 4

2 1 ... and do you see how /4 is simpler as /2 ? (see Equivalent Fractions.)

Example 2: 1 1 + 3 6

Step 1: The bottom numbers are different. See how the slices are different sizes? 1/ + 1/ = ? 3 6

We need to make them the same before we can continue, because we can't add them like that.

The number "6" is twice as big as "3", so to make the bottom numbers the same we can multiply the top and bottom of the first fraction by 2, like this:

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Page 48 of 362 × 2

1 2 = 3 6

× 2

Important: you multiply both top and bottom by the same amount, to keep the value of the fraction the same

Now the fractions have the same bottom number ("6"), and our question looks like this: 2/ + 1/ 6 6

The bottom numbers are now the same, so we can go to step 2.

Step 2: Add the top numbers and put them over the same denominator: 2 1 2 + 1 3 + = = 6 6 6 6 In picture form it looks like this: 2/ + 1/ = 3/ 6 6 6

Step 3: Simplify the fraction:

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Page 49 of 362 3 1 = 6 2 In picture form the whole answer looks like this: 2 1 3 1 /6 + /6 = /6 = /2

Subtracting Fractions

You might like to read Adding Fractions first. There are 3 simple steps to subtract fractions  Step 1. Make sure the bottom numbers (the denominators) are the same  Step 2. Subtract the top numbers (the numerators). Put the answer over the same denominator.  Step 3. Simplify the fraction.

Example 1: 3 1 – 4 4 Step 1. The bottom numbers are already the same. Go straight to step 2.

Step 2. Subtract the top numbers and put the answer over the same denominator: 3 1 3 – 1 2 – = = 4 4 4 4 Step 3. Simplify the fraction: 2 1 = 4 2 (If you are unsure of the last step see Equivalent Fractions.) Example 2: 1 1 – 2 6

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Step 1. The bottom numbers are different. See how the slices are different sizes? We need to make them the same before we can continue, because we can't subtract them like this:

1/ − 1/ = ? 2 6

1 To make the bottom numbers the same, multiply the top and bottom of the first fraction ( /2) by 3 like this: × 3

1 3 = 2 6

× 3 And now our question looks like this: 3/ − 1/ 6 6

The bottom numbers (the denominators) are the same, so we can go to step 2. Step 2. Subtract the top numbers and put the answer over the same denominator: 3 1 3 – 1 2 – = = 6 6 6 6 In picture form it looks like this: 3/ − 1/ = 2/ 6 6 6

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Page 51 of 362

Step 3. Simplify the fraction: 2 1 = 6 3

Multiplying Fractions

Multiply the tops, multiply the bottoms.

There are 3 simple steps to multiply fractions 1. Multiply the top numbers (the numerators). 2. Multiply the bottom numbers (the denominators). 3. Simplify the fraction if needed.

Example: 1 2 × 2 5

Step 1. Multiply the top numbers: 1 1 2 × 2 × = 2 = 2 5

Step 2. Multiply the bottom numbers: 1 1 2 × 2 2 × = = 2 2 5 × 10 5

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Page 52 of 362

Step 3. Simplify the fraction: 2 1 = 10 5

With Pizza Here you can see it with pizza ...

... and do you see how two-tenths is simpler as one-fifth?

Dividing Fractions

Turn the second fraction upside down, then multiply. There are 3 Simple Steps to Divide Fractions: Step 1. Turn the second fraction (the one you want to divide by) upside-down (this is now a reciprocal). Step 2. Multiply the first fraction by that reciprocal Step 3. Simplify the fraction (if needed)

Example: 1 1 ÷ 2 6

Step 1. Turn the second fraction upside-down (it becomes a reciprocal): 1 6 becomes 6 1

Step 2. Multiply the first fraction by that reciprocal: 1 6 1 × 6 6 × = = 2 1 2 × 1 2

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Page 53 of 362 Step 3. Simplify the fraction: 6 = 3 2

History of Irrational Numbers Apparently Hippasus (one of Pythagoras' students) discovered irrational numbers when trying to represent the square root of 2 as a fraction (using geometry, it is thought). Instead he proved you couldn't write the square root of 2 as a fraction and so it was irrational.

However Pythagoras could not accept the existence of irrational numbers, because he believed that all numbers had perfect values. But he could not disprove Hippasus' "irrational numbers" and so Hippasus was thrown overboard and drowned!

Irrational Numbers All numbers that are not rational are considered irrational. An irrational number can be written as a decimal, but not as a fraction.

An irrational number has endless non-repeating digits to the right of the decimal point. Here are some irrational numbers:

π = 3.141592… = 1.414213…

Although irrational numbers are not often used in daily life, they do exist on the number line. In fact, between 0 and 1 on the number line, there are an infinite number of irrational numbers!

An Irrational Number is a real number that cannot be written as a simple fraction. Irrational means not Rational.

Examples:

Rational Numbers

OK. A Rational Number can be written as a Ratio of two integers (i.e. a simple fraction). Example: 1.5 is rational, because it can be written as the ratio 3/2 Example: 7 is rational, because it can be written as the ratio 7/1 Example 0.333... (3 repeating) is also rational, because it can be written as the ratio 1/3

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Page 54 of 362 Irrational Numbers But some numbers cannot be written as a ratio of two integers ...

...they are called Irrational Numbers.

It is irrational because it cannot be written as a ratio (or fraction), not because it is crazy!

Example: π (Pi) is a famous irrational number.

π = 3.1415926535897932384626433832795 (and more...) You cannot write down a simple fraction that equals Pi.

22 The popular approximation of /7 = 3.1428571428571... is close but not accurate. Another clue is that the decimal goes on forever without repeating.

Rational vs. Irrational So you can tell if it is Rational or Irrational by trying to write the number as a simple fraction.

Example: 9.5 can be written as a simple fraction like this: 19 9.5 = /2 So it is a rational number (and so is not irrational) Here are some more examples: Rational or Number As a Fraction Irrational? 1.75 7/4 Rational .001 1/1000 Rational √2 ? Irrational ! (square root of 2)

Square Root of 2 Let's look at the square root of 2 more closely.

If you draw a square (of size "1"), what is the distance across the diagonal?

The answer is the square root of 2, which is 1.4142135623730950...(etc.)

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But it is not a number like 3, or five-thirds, or anything like that ...... in fact you cannot write the square root of 2 using a ratio of two numbers ... I explain why on the Is It Irrational? page, ... and so we know it is an irrational number

Famous Irrational Numbers

Pi is a famous irrational number. People have calculated Pi to over one million decimal places and still there is no pattern. The first few digits look like this: 3.1415926535897932384626433832795 (and more ...)

The number e (Euler's Number) is another famous irrational number. People have also calculated e to lots of decimal places without any pattern showing. The first few digits look like this: 2.7182818284590452353602874713527 (and more ...)

The Golden Ratio is an irrational number. The first few digits look like this: 1.61803398874989484820... (and more ...)

Many square roots, cube roots, etc. are also irrational numbers. Examples: √3 1.7320508075688772935274463415059 (etc.) √99 9.9498743710661995473447982100121 (etc.)

But √4 = 2 (rational), and √9 = 3 (rational) ...... so not all roots are irrational.

Note on Multiplying Irrational Numbers Have a look at this:  π × π = π2 is irrational  But √2 × √2 = 2 is rational  So be careful ... multiplying irrational numbers can result in a rational number!

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Activity 6: Using Rational Numbers

(A) (B) (C) (D)

(A) 4 (B) (C) (D)

Activity 7: Irrational Numbers

1. Which one of the following is NOT irrational?

(A) (B) (C) (D)

2. Which one of the following is NOT irrational?

(A) (B) (C) (D)

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Page 57 of 362 3. Answer the following question below.

A, B, C and D are rectangles.

The length of the diagonal of which rectangle is not irrational?

(A) A (B) B (C) C (D) D

Lesson 18-19: Worksheets

Please check your calendar for information on this lessons.

Lesson 20-21: Negative Numbers

In this section we will see how our number system can be extended to include negative numbers.

Look at the number line below:

The numbers to the left of zero are negative and the numbers to the right of zero are positive. If we have two numbers, we say that one is less than the other if it is further left on the number line.

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Page 58 of 362 For example, "2 is less than 7" because 2 is further left. This can also be written as "2 < 7". We say that one number is greater than another if it is further right on the number line. For example, "3 is greater than -5" because 3 is further right. This can also be written as "3 > - 5".

When you first learned your numbers, way back in elementary school, you started with the counting numbers: 1, 2, 3, 4, 5, 6, and so on. Your number line looked something like this:

Later on, you learned about zero, fractions, decimals, square roots, and other types of numbers, so your number line started looking something like this:

Addition, multiplication, and division always made sense — as long as you didn't try to divide by zero — but sometimes subtraction didn't work. If you had "9 – 5", you got 4:

...but what if you had "5 – 9"? You just couldn't do this subtraction, because there wasn't enough "space" on the number line to go back nine units:

You can solve this "space" problem by using negative numbers. The "whole" numbers start at zero and count off to the right; these are the positive integers. The negative integers start at zero and count off to the left:

Note the arrowhead on the far right end of the number line above. That arrow tells you the direction in which the numbers are getting bigger. In particular, that arrow also tells you that the negatives are getting smaller as they move off to the left. That is, –5 is smaller than – 4. This might seem a bit weird at first, but that's okay; negatives take some getting used to. Let's look at a few inequalities, to practice your understanding. Refer to the number line above, as necessary.

Complete the following inequality: 3 _____ 6 Look at the number line: Since 6 is to the right of 3, then 6 is larger, so the correct inequality is: 3 < 6

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Page 59 of 362  Complete the following inequality: –3 _____ 6 Look at the number line: Every positive number is to the right of every negative number, so the correct inequality is: –3 < 6

 Complete the following inequality: –3 _____ –6 Look at the number line: Since –6 is to the left of –3, then –3, being further to the right, is actually the larger number. So the correct inequality is: –3 > –6

 Complete the following inequality: 0 _____ 1 Zero is less than any positive number, so: 0 < 1

 Complete the following inequality: 0 _____ –1 Zero is greater than any negative number, so: 0 > –1

Adding and Subtracting Negative Numbers How do you deal with adding and subtracting negatives? The process works similarly to adding and subtracting positive numbers. If you are adding a negative, this is pretty much the same as subtracting a positive, if you view "adding a negative" as adding to the left.

Let's return to the first example from the previous page: "9 – 5" can also be written as "9 + (–5)". Graphically, it would be drawn as "an arrow from zero to nine, and then a 'negative' arrow five units long":

...and you get "9 + (–5) = 4". Now look back at that subtraction you couldn't do: 5 – 9. Because you now have negative numbers off to the left of zero, you also now have the "space" to complete this subtraction. View the subtraction as adding a negative 9; that is, draw an arrow from zero to five, and then a "negative" arrow nine units long:

...or, which is the same thing:

Then 5 – 9 = 5 + (–9) = –4.

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Page 60 of 362 Of course, this method of counting off your answer on a number line won't work so well if you're dealing with larger numbers. For instance, think about doing "465 – 739". You certainly don't want to use a number line for this. You know that the answer to "465 – 739" has to be negative, (because "minus 739" will take you somewhere to the left of zero), but how do you figure out which negative number is the answer?

Look again at "5 – 9". You know now that the answer will be negative, because you're subtracting a bigger number than you started with (nine is bigger than five). The easiest way of dealing with this is to do the subtraction "normally" (with the smaller number being subtracted from the larger number), and then put a "minus" sign on the answer: 9 – 5 = 4, so 5 – 9 = – 4. This works the same way for bigger numbers (and is much simpler than trying to draw the picture): since 739 – 465 = 274, then 465 – 739 = –274.

Adding two negative numbers is easy: you're just adding two "negative" arrows, so it's just like "regular" addition, but in the opposite direction. For instance, 4 + 6 = 10, and –4 – 6 = –4 + (–6) = –10.

But what about when you have lots of both positive and negative numbers?

 Simplify 18 – (–16) – 3 – (–5) + 2 Probably the simplest thing to do is convert everything to addition, group the positives together and the negatives together, combine, and simplify. It looks like this: 18 – (–16) – 3 – (–5) + 2 = 18 + 16 – 3 + 5 + 2 = 18 + 16 + (–3) + 5 + 2 = 18 + 16 + 5 + 2 + (–3) = 41 + (–3) = 41 – 3 = 38 "Whoa! Wait a minute!" you say. "How do you go from ' – (–16)' to ' + 16' in your first step?" This is actually a fairly important concept, and, if you're asking, I'm assuming that your teacher's explanation didn't make much sense to you. So I won't give you a "proper" mathematical explanation of this "the minus of a minus is a plus" rule. Instead, here's a mental picture that I ran across in an algebra newsgroup:

Imagine that you're cooking some kind of stew, but not on a stove. You control the temperature of the stew with magic cubes. These cubes come in two types: hot cubes and cold cubes.

If you add a hot cube (add a positive number), the temperature goes up. If you add a cold cube (add a negative number), the temperature goes down. If you remove a hot cube (subtract a positive number), the temperature goes down. And if you remove a cold cube (subtract a negative number), the temperature goes UP! That is, subtracting a negative is the same as adding a positive. Now suppose you have some double cubes and some triple cubes. If you add three double-hot cubes (add three-times-positive-two), the temperature goes up by six. And if you remove two triple-cold cubes (subtract two-times-negative-three), you get the same result. That is, –2(–3) = + 6.

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Page 61 of 362 There's another analogy that I've been seeing recently. Letting "good" be "positive" and "bad" be "negative", you could say:

good things happening to good people: a good thing good things happening to bad people: a bad thing bad things happening to good people: a bad thing bad things happening to bad people: a good thing

The above isn't a technical explanation or proof, but I hope it makes the "minus of a minus is a plus" and "minus times minus is plus" rules seem a bit more reasonable. Let's look at a few more examples:

 Simplify –43 – (–19) – 21 + 25. –43 – (–19) – 21 + 25 = –43 + 19 – 21 + 25 = (–43) + 19 + (–21) + 25 = (–43) + (–21) + 19 + 25 = (–64) + 44 = 44 + (–64)

Technically, I can only move the numbers around as I did in the steps above after I have converted everything to addition. I cannot reverse a subtraction, only an addition. In practical terms, this means that I can only move the numbers around if I move their signs with them. If I move only the numbers and not their signs, I will have changed the value and will end up with the wrong answer. Continuing...

44 + (–64) = 44 – 64 Since 64 – 44 = 20, then 44 – 64 = –20.

 Simplify 84 + (–99) + 44 – (–18) – 43. 84 + (–99) + 44 – (–18) – 43 = 84 + (–99) + 44 + 18 + (–43) = 84 + 44 + 18 + (–99) + (–43) = 146 + (–142) = 146 – 142 = 4

Multiplying and Dividing Negative Numbers

Turning from addition and subtraction, how do you do multiplication and division with negatives? Actually, we've already covered the hard part: you already know the "sign" rules:

plus times plus is plus (adding many hot cubes raises the temperature) minus times plus is minus (removing many hot cubes reduces the temperature) plus times minus is minus (adding many cold cubes reduces the temperature) minus times minus is plus (removing many cold cubes raises the temperature)

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Page 62 of 362 The sign rules work the same way for division; just replace "times" with "divided by". Here are a couple examples of the rules in division:

(Remember that fractions are just another form of division!)

You may notice people "cancelling off" minus signs. They are taking advantage of the fact that "minus times minus is plus". For instance, suppose you have (–2)(–3)(–4). Any two negatives, when multiplied together, become one positive. So pick any two of the multiplied (or divided) negatives, and "cancel" their signs:  Simplify (–2)(–3)(–4). (–2)(–3)(–4) = (–2)(–3)(–4) = (+6)(–4) = –24

If you're given a long multiplication with negatives, just cancel off "minus" signs in pairs: Simplify (–1)(–2)(–1)(–3)(–4)(–2)(–1). (–1)(–2)(–1)(–3)(–4)(–2)(–1) = (–1)(–2)(–1)(–3)(–4)(–2)(–1) = (+1)(+2)(–1)(–3)(–4)(–2)(–1) = (1)(2)(–1)(–3)(–4)(–2)(–1) = (1)(2)(+1)(+3)(–4)(–2)(–1) = (1)(2)(1)(3)(–4)(–2)(–1) = (1)(2)(1)(3)(+4)(+2)(–1) = (1)(2)(1)(3)(4)(2)(–1) = (2)(3)(4)(2)(–1) = 48(–1) = –48

Here's another example:

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Page 63 of 362 Negatives through parentheses The major difficulty that people have with negatives is in dealing with parentheses; particularly, in taking a negative through parentheses. The usual situation is something like this:

–3(x + 4)

If you had "3(x + 4)", you would know to "distribute" the 3 "over" the parentheses:

3(x + 4) = 3(x) + 3(4) = 3x + 12

The same rules apply when you're dealing with negatives. If you have trouble keeping track, use little arrows:

 Simplify 3(x – 5). 3(x – 5) = 3(x) + 3(–5) = 3x – 15

 Simplify –2(x – 3). –2(x – 3) = –2(x) – 2(–3) = –2x + 2(+3) = –2x + 6

The other trouble, related to the previous one, is with subtracting a parentheses. You can keep track of the subtraction sign by converting the subtraction to a multiplication by negative one:  Simplify 4 – (2 + x).

Don't be afraid to write in the little "1" and draw in the little arrows. You should do whatever you need to do to keep your work straight and get the right answer.

 Simplify 6 – (3x – 4[1 – x]).

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Page 64 of 362 1 (x – 2)  Simplify /3 – /3.

Note that I converted from subtracting a fraction to adding a negative one times a fraction. It is very easy to "lose" the minus when you're adding messy polynomial fractions like this. The most common mistake is to put the minus on the x and forget to take it through to the –2. Take particular care with fractions!

Exponents and Negative Numbers Now you can move on to exponents, using the cancelation-of-minus-signs property of multiplication. For instance, (3)2 = (3)(3) = 9. In the same way:  Simplify (–3)2 (–3)2 = (–3)(–3) = (+3)(+3) = 9

Note the difference between the above exercise and the following:

 Simplify –32 –32 = –(3)(3) = (–1)(9) = –9

In the second exercise, the square (the "to the power 2") was only on the 3; it was not on the minus sign. Those parentheses make all the difference in the world! Be careful with them, especially when you are entering expressions into software. Different software may treat the same expression very differently, as one researcher has demonstrated very thoroughly.

 Simplify (–3)3 (–3)3 = (–3)(–3)(–3) = (+3)(+3)(–3) = (9)(–3) = –27

 Simplify (–3)4 (–3)4 = (–3)(–3)(–3)(–3) = (+3)(+3)(–3)(–3)

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Page 65 of 362 = (+3)(+3)(+3)(+3) = (9)(9) = 81

 Simplify (–3)5 (–3)5 = (–3)(–3)(–3)(–3)(–3) = (+3)(+3)(–3)(–3)(–3) = (+3)(+3)(+3)(+3)(–3) = (9)(9)(–3) = –243

Note the pattern: A negative number taken to an even power gives a positive result (because the pairs of negatives cancel), and a negative number taken to an odd power gives a negative result (because, after cancelling, there will be one minus sign left over). So if they give you an exercise containing something slightly ridiculous like (–1)1001, you know that the answer will either be +1 or –1, and, since 1001 is odd, then the answer must be –1.

You can also do negatives with roots, but only if you're careful. You can do , because there is a number that squares to 16. That is,

...because 42 = 16. But what about ? Can you square anything and have it come up negative? No! So you cannot take the square root (or the fourth root, or the sixth root, or the eighth root, or any other even root) of a negative number. On the other hand, you can do cube roots of negative numbers. For instance:

...because (–2)3 = –8. For the same reason, you can take any odd root (third root, fifth root, seventh root, etc.) of a negative number.

Activity 8: Negative numbers

Using the number line above, decide whether there should be a less than (<) or greater than (>) symbol between each pair of numbers below.

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Page 66 of 362 (a) -5 ____ 4 (b) 3 ____ 7 (c) -6 ____ -9 (d) 2 ____ -2

Activity 9: Negative numbers

Work out the answers to the questions below and fill in the answers in the blank space. Look at the temperature scale below and work out the temperature:

(a) 3º C warmer than -1º C? ____ ºC (b) 6º C colder than -3º C? ____ ºC (c) 5º C warmer than -5º C? ____ ºC (d) 8º C warmer than -7º C? ____ ºC (e) 8º C warmer than -7º C? ____ ºC

Activity 10: Negative numbers

These questions are similar to those in Activity 9 but they are about numbers, not temperatures. Work out the correct number.

(a) 3 more than -2? ____ (b) 5 more than -7? ____ (c) 5 less than -4? ____ (d) 5 more than -20? ____ (e) 12 less than 10? ____ (f) 6 less than 1? ____ (g) 6 more than -10? ____ (h) 16 less than 3? ____ (i) 6 more than 5? ____ (j) 20 more than -8? ____

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Page 67 of 362

Activity 11: Negative numbers

In this question you need to decide whether one number is greater or less than another number. Choose the < or > symbol

For example, if the question said 4 __ - 2 then you would put in the > symbol, because 4 > - 2.

(a) 4 ____ 2 (b) -6 ____ -2 (c) -3 ____ 4 (d) 2 ____ -4 (e) -6 ____ -7 (f) -6 ____ -5 (g) 0 ____ 1 (h) -1 ____ 0

Activity 12: Negative numbers

In this question you need to decide whether each statement is true or false.

For example, the statement 4 > - 3 is TRUE, but the statement - 4 > - 3 is FALSE.

(a) 6 > 7 ______(b) 5 > - 6 ______(c) -6 < - 7 ______(d) - 1 > 0 ______(e) - 3 < 2 ______(f) - 7 < 6 ______(g) - 4 > -3 ______(h) - 5 < - 2 ______

Activity 13: Negative numbers

In this question you need to give any integer which could go in the blank space.

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For example, if the question said 4 > ___ > - 2 then any of the integers 3, 2, 1, 0 or -1 could fit.

(a) 5 < ____ < 7 (b) - 5 < ____ < - 3 (c) - 3 > ____ > - 7 (d) - 6 < ____ < 0 (e) - 1 < ____ < 2

Lesson 22: Two Dimensional Shapes and Measurement

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Page 69 of 362

Below you will find revision of work you did in Grade 8

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Page 70 of 362 Lesson 23-24: Theorem of Pythagoras

Pythagoras' Theorem

Years ago, a man named Pythagoras found an amazing fact about triangles: If the triangle had a right angle (90°) ...... and you made a square on each of the three sides, then the biggest square had the exact same area as the other two squares put together!

It is called "Pythagoras' Theorem" and can be written in one short equation: a2 + b2 = c2

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Page 71 of 362 Note:  c is the longest side of the triangle  a and b are the other two sides

Definition The longest side of the triangle is called the "hypotenuse", so the formal definition is: In a right angled triangle: the square of the hypotenuse is equal to the sum of the squares of the other two sides. Sure ... ? Let's see if it really works using an example.

Example: A "3,4,5" triangle has a right angle in it.

Let's check if the areas are the same: 32 + 42 = 52 Calculating this becomes: 9 + 16 = 25 It works ... like Magic!

Why Is This Useful? If we know the lengths of two sides of a right angled triangle, we can find the length of the third side. (But remember it only works on right angled triangles!)

How Do I Use it? Write it down as an equation:

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Page 72 of 362

a2 + b2 = c2

Now you can use algebra to find any missing value, as in the following examples: Example: Solve this triangle. a2 + b2 = c2 52 + 122 = c2 25 + 144 = c2 169 = c2 c2 = 169

c = √169 c = 13 You can also read about Squares and Square Roots to find out why √169 = 13 Example: Solve this triangle. a2 + b2 = c2 92 + b2 = 152 81 + b2 = 225 Take 81 from both sides: b2 = 144

b = √144 b = 12 Example: What is the diagonal distance across a square of size 1?

a2 + b2 = c2 12 + 12 = c2 1 + 1 = c2 2 = c2 c2 = 2 c = √2 = 1.4142...

It works the other way around, too: when the three sides of a triangle make a2 + b2 = c2, then the triangle is right angled.

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Page 73 of 362 Example: Does this triangle have a Right Angle?

Does a2 + b2 = c2 ?  a2 + b2 = 102 + 242 = 100 + 576 = 676  c2 = 262 = 676 They are equal, so ... Yes, it does have a Right Angle!

Example: Does an 8, 15, 16 triangle have a Right Angle? Does 82 + 152 = 162 ?  82 + 152 = 64 + 225 = 289,  but 162 = 256 So, NO, it does not have a Right Angle

Example: Does this triangle have a Right Angle?

Does a2 + b2 = c2 ? Does (√3)2 + (√5)2 = (√8)2 ? Does 3 + 5 = 8 ?

Yes, it does! So this is a right-angled triangle

Activity 14: Theorem of Pythagoras

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Page 74 of 362

Solve this triangle. And choose the correct answer.

(A) c = 5 (B) c = 25 (C) c = √527 (D) c = 31

Activity 15: Theorem of Pythagoras

Solve this triangle and choose the correct answer below.

(A) a = 5 (B) a = √35 (C) a = √135 (D) a = √377

Activity 16: Theorem of Pythagoras

What is the length of the diagonal of a rectangle of length 3 and width 2?

(A) √5 (B) √13 (C) 5 (D) 6

Activity 17: Theorem of Pythagoras

Which one of the following triangles is NOT a right triangle?

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Page 75 of 362

(A) (B)

Activity 18: Theorem of Pythagoras

What is the length of the side x?

(A) x = 5 (B) x = √34 (C) x = 2√11 (D) x = 2√61

Activity 19: Theorem of Pythagoras

Only one of these triangles is really a right triangle. Which one?

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Page 76 of 362

(A) A (B) B (C) C (D) D

Lesson 25-26: Worksheets

Please check your calendar for information on this lessons.

Lesson 27: Parallelograms and Trapeziums

Parallelograms

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Page 77 of 362 A parallelogram is a quadrilateral in which both pairs of opposite sides are parallel. Special relationships are a reality between the measures of consecutive angles, opposite angles and opposite sides of a parallelogram. The square is the most obvious parallelogram, because it has 2 sets of parallel sides.

Facts about a Parallelogram (1) The degree measure of the four angles of a parallelogram add up to 360 degrees. Remember that all quadrilaterals (4 sided figures) have angles which add up to 360 degrees. Here's a sample:

Then: a + b + c + d = 360 degrees

(2) The degree measure of any two consecutive angles add up to 180 degrees. In parallelogram ABCD:

angle a + angle b = 180 degrees angle b + angle c = 180 degrees

angle c + angle d = 180 degrees

angle a + angle d = 180 degrees

(3) Opposite angles have the same measure in terms of degrees.

In parallelogram ABCD:

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angle a = angle c angle b = angle d

Sides of a Parallelogram In parallelogram ABCD:

(1) Opposite sides are parallel:

side AD || side BC side AB || side CD NOTE: The symbol || means parallel. (2) Opposite sides have the same lengths:

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side AD = side BC side AB = side CD

Diagonals of a Parallelogram The diagonals of a parallelogram divide the parallelogram into two side-by-side triangles. As shown in the picture below, diagonal AC forms equal alternate interior angles with each pair of parallel sides. We can also see that there are two triangles in the picture below. Triangle 1 is congruent to triangle 2 by ASA (Angle- Side-Angle) Method.

Where did the two triangles come from? They were formed by diagonal AC.

I should also note that diagonals of a parallelogram bisect each other as shown in the picture below.

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AE = EC DE = EB where E is the midpoint of BOTH diagonals.

Sample:

In parallelogram WXYZ, the measure of angle X = 4a - 40 and the measure of angle Z = 2a - 8. Find the measure of angle W?

Solution:

(1) Find the value of a.

Since angles X and Z are opposite angles of parallelogram WXYZ, they have the same measure. We equate the terms and solve for a.

4a - 40 = 2a - 8 4a - 2a = 40 - 8 2a = 32 a = 32/2 a = 16

(2) Since a = 16, we now find the measure of angle X by substituting 16 for a in 4a - 40. angle X = 4a - 40 angle X = 4(16) - 40 angle X = 64 - 40 angle X = 24

(3) We also know that consecutive angles of a parallelogram are supplementary (add up to 180 degrees) and so, this fact is used to find the measure of angle W. angle W = 180 - angle X angle W = 180 - 24 angle W = 156

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Page 81 of 362 Conjecture: The area of a trapezium is half the sum of the parallel sides multiplied by the height.

Justification: Area of trapezium ABCD

= area ABCD + area BCD

= AB•h + CD•h

= h (AB + CD)

Example:

1. Determine the areas of each of the shapes

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Page 82 of 362 Solution

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Activity 20: Parallelograms and trapeziums

Determine the area of the following shapes below.

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Lesson 28: Worksheet

Please check your calendar for information on this lesson.

Lesson 29: Area and Perimeter of Regular and Irregular Polygons

A polygon is a plane shape with straight sides. Is it a Polygon? Polygons are 2-dimensional shapes. They are made of straight lines, and the shape is "closed" (all the lines connect up).

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Polygon Not a Polygon Not a Polygon (straight sides) (has a curve) (open, not closed)

Polygon comes from Greek. Poly- means "many" and -gon means "angle".

Types of Polygons Simple or Complex A simple polygon has only one boundary, and it doesn't cross over itself. A complex polygon intersects itself! Many rules about polygons don't work when it is complex.

Simple Polygon Complex Polygon (this one's a Pentagon) (also a Pentagon)

Concave or Convex A convex polygon has no angles pointing inwards. More precisely, no internal angles can be more than 180°. If there are any internal angles greater than 180° then it is concave. (Think: concave has a "cave" in it)

Convex Concave

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Regular or Irregular If all angles are equal and all sides are equal, then it is regular, otherwise it is irregular

Regular Irregular

More Examples

Complex Polygon (a "star polygon", in Concave Octagon Irregular Hexagon this case, a pentagram)

Names of Polygons

If it is a Regular Polygon...

Name Sides Shape Interior Angle

Triangle (or Trigon) 3 60°

Quadrilateral (or Tetragon) 4 90°

Pentagon 5 108°

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Hexagon 6 120°

Heptagon (or Septagon) 7 128.571°

Octagon 8 135°

Nonagon (or Enneagon) 9 140°

Decagon 10 144°

Hendecagon (or Undecagon) 11 147.273°

Dodecagon 12 150°

Triskaidecagon 13 152.308°

Tetrakaidecagon 14 154.286°

Pentadecagon 15 156°

Hexakaidecagon 16 157.5°

Heptadecagon 17 158.824°

Octakaidecagon 18 160°

Enneadecagon 19 161.053°

Icosagon 20 162°

Triacontagon 30 168°

Tetracontagon 40 171°

Pentacontagon 50 172.8°

Hexacontagon 60 174°

Heptacontagon 70 174.857°

Octacontagon 80 175.5°

Enneacontagon 90 176°

Hectagon 100 176.4°

Chiliagon 1,000 179.64°

Myriagon 10,000 179.964°

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Page 88 of 362 Megagon 1,000,000 ~180°

Googolgon 10100 ~180°

n-gon n (n-2) × 180° / n

For polygons with 13 or more sides, it is OK (and easier) to write "13-gon", "14-gon" ... "100-gon", etc.

Area of Irregular Polygons Introduction I just thought I would share with you a clever technique I once used to find the area of general polygons.

The polygon could be regular (all angles are equal and all sides are equal) or irregular

Regular Irregular

The Example Polygon Let's use this polygon as an example:

Coordinates The first step is to turn each vertex (corner) into a coordinate, like on a graph:

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Area Under One Line Segment Now, for each line segment, work out the area down to the x-axis.

So, how do we calculate each area?

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Average the two heights, then multiply by the width Example: For the shape highlighted above, we take the two heights (the "y" coordinates 2.28 and 4.71) and work out the average height: (2.28+4.71)/2 = 3.495

Work out the width (the difference between the "x" coordinates 2.66 and 0.72) 2.66-0.72 = 1.94

The area under that line segment is width×height 1.94 × 3.495 = 6.7803 Add Them All Up Now add them all up! But the trick is to add when they go forwards (positive width), and subtract when they go backwards (negative width).

If you always go clockwise around the polygon, and always subtract the second "x" coordinate from the first, it works out naturally, like this: From To

x y x y Avg Height Width (+/-) Area (+/-) 0.72 2.28 2.66 4.71 3.495 1.94 6.7803 2.66 4.71 5 3.5 4.105 2.34 9.6057 5 3.5 3.63 2.52 3.01 -1.37 -4.1237 3.63 2.52 4 1.6 2.06 0.37 0.7622 4 1.6 1.9 1 1.3 -2.1 -2.7300 1.9 1 0.72 2.28 1.64 -1.18 -1.9352 Total: 8.3593

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Page 91 of 362 And It looks like this:

So that's it! The area is 8.3593

Activity 21: Area of an irregular polygon

Use the area of an irregular polygon method to find the area of the polygon.

(A) 18 (B) 19.5 (C) 22 (D) 31.5

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Activity 22: Irregular polygon method

Use the area of an irregular polygon method to find the area of the polygon.

(A) 34 (B) 37.5 (C) 39 (D) 42.5

Lesson 30-31: Worksheets

Please check your calendar for information on this lessons.

Lesson 32-33: Area and Perimeter of a Circle

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 The point on the circle are all equidistant from a central point called the centre.  The straight-line segment from the centre to the circle is the radius, r, of the circle.  A diameter, d, is the straight-line segment going through the centre with its end points on the circle.

 An arc of a circle is a part of the circle subtended by chord.  A sector of a circle is the region enclosed between an arc and two radii.  A segment of a circle is that area or portion of a circle between a chord and an arc of the circle.  A semi-circle is half the circle or the arc subtended by the diameter.  A chord is a straight line segment with its end points on the circle. If it goes through the centre then it is the diameter.  A secant is a straight line that cuts the circle at two points.  A tangent line meets the circle at one point only.

Perimeter of a Circle

Perimeter is the distance around a closed figure and is typically measured in millimetres (mm), centimetres (cm), metres (m) and kilometres (km). These units are related as follows:

10 mm = 1 cm 100 cm = 1 m 1000 m = 1 km

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Page 94 of 362 The word 'perimeter' is also sometimes used instead of circumference.

If we know the radius Given the radius of a circle, the circumference or perimeter can be calculated using the formula below:

Perimeter (P) = 2 · π · R where: R is the radius of the circle π is Pi, approximately 3.142

If we know the diameter If we know the diameter of a circle, the circumference can be found using the formula

Perimeter (P) = π · D where: D is the diameter of the circle π is Pi, approximately 3.142

If we know the area If we know the area of a circle, the circumference can be found using the formula:

Perimeter (P) = √(4 · π · A ) where: A is the area of the circle π is Pi, approximately 3.142

Example 1: A circular flower-bed has a radius of 9 m. Find the perimeter/circumference of the flower-bed.

Solution: P = 2 · π · R P = 2 · 3.1416 · 9 P = 56.5487 cm

So, the perimeter/circumference of the flower-bed is 56.5487 m.

Example 2: Find the perimeter of the given circle whose diameter is 4.4 cm.

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Solution: Given that: Diameter of the circle (D) = 4.4 cm. We know the formula to find the perimeter of the circle if the diameter is given, namely π· D. Substitute the diameter 4.4 and Pi value as 3.14 in the above formula. Perimeter = (3.14)(4.4) = 13.82 Therefore 13.82 cm is the perimeter of the given circle.

Example 3: If the radius is 11.7 cm. Find perimeters (circumference) of the circle. Solution: Given that: Radius (r) = 11.7cm Perimeter (circumference) of circle P = 2 π r Substitute the r value in the formula, we have: P = 2 x 3.14 x 11.7 P = 79.56 cm Thus, the perimeter of the circle is 79.56cm

Example 4: Find the perimeter and area of the circle, if the radius of the circle is 8cm. Solution: We have given the radius, which is 8cm. So, by using the formula of the perimeter of the circle, we have: P = 2πr P = 2×3.14×8 P = 50.24 cm And for the area of the circle:- A = π r2 A = 3.14×(8)2 A = 200.96cm2

Example 5: The wheel of a bullock cart has a radius of 6 m. If the wheel rotates once how much distance does the cart move?

Solution: If the wheel rotates once, the cart will move by a distance equal to the perimeter of the wheel. Step 1:

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Page 96 of 362 P = 2πr P = 2× 3.14× 6 = 37.68 m

Thus, the bullock cart moves 37.68 m in one revolution of the wheel.

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Page 98 of 362

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Page 99 of 362

Activity 23: Perimeter of a Circle

Identify the parts of the circle labeled (a) to (f).

Activity 23: Perimeter of a Circle

For each of the shapes find the: (2) Perimeter (3) Area

Lesson 34-35: Worksheets

Please check your calendar for information on this lessons.

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Lesson 36: Notation and Powers of 10

Note: Index and Power mean the same things as Exponent)

The exponent (or index or power) of a number says how many times to use the number in a multiplication.

102 means 10 × 10 = 100 (It says 10 is used 2 times in the multiplication)

Example: 103 = 10 × 10 × 10 = 1,000  In words: 103 could be called "10 to the third power", "10 to the power 3" or simply "10 cubed"

Example: 104 = 10 × 10 × 10 × 10 = 10,000  In words: 104 could be called "10 to the fourth power", "10 to the power 4" or "10 to the 4" You can multiply any number by itself as many times as you want using this notation (see Exponents), but powers of 10 have a special use...

Powers of 10 "Powers of 10" is a very useful way of writing down large or small numbers. Instead of having lots of zeros, you show how many powers of 10 you need to make that many zeros

Example: 5,000 = 5 × 1,000 = 5 × 103  5 thousand is 5 times a thousand. And a thousand is 103. So 5 times 103 = 5,000  Can you see that 103 is a handy way of making 3 zeros?

Scientists and Engineers (who often use very big or very small numbers) like to write numbers this way.

Example: The Mass of the Sun The Sun has a Mass of 1.988 × 1030 kg. It would be too hard for scientists to write 1,988,000,000,000,000,000,000,000,000,000 kg (And very easy to make a mistake counting the zeros!)

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Page 101 of 362 Example: A Light Year (the distance light travels in one year)

It is easier to use 9.461 × 1015 meters, rather than 9,461,000,000,000,000 meters

Exponents In the table below, the number 2 is written as a factor repeatedly. The product of factors is also displayed in this table

Factors Product of Factors Description 2 x 2 = 4 2 is a factor 2 times 2 x 2 x 2 = 8 2 is a factor 3 times 2 x 2 x 2 x 2 = 16 2 is a factor 4 times 2 x 2 x 2 x 2 x 2 = 32 2 is a factor 5 times 2 x 2 x 2 x 2 x 2 x 2 = 64 2 is a factor 6 times 2 x 2 x 2 x 2 x 2 x 2 x 2 = 128 2 is a factor 7 times 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 256 2 is a factor 8 times

Writing 2 as a factor one million times would be a very time-consuming and tedious task. A better way to approach this is to use exponents. Exponential notation is an easier way to write a number as a product of many factors.

BaseExponent The exponent tells us how many times the base is used as a factor.

For example, to write 2 as a factor one million times, the base is 2, and the exponent is 1,000,000. We write this number in exponential form as follows:

2 1,000,000 read as two raised to the millionth power

Example 1: Write 2 x 2 x 2 x 2 x 2 using exponents, then read your answer aloud. 2 x 2 x 2 x 2 x 2 = 2 raised to the fifth

Solution: 25 power

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Let us take another look at the table from above to see how exponents work.

Exponential Factor Standard Form Form Form 22 = 2 x 2 = 4 23 = 2 x 2 x 2 = 8 24 = 2 x 2 x 2 x 2 = 16 25 = 2 x 2 x 2 x 2 x 2 = 32 26 = 2 x 2 x 2 x 2 x 2 x 2 = 64 27 = 2 x 2 x 2 x 2 x 2 x 2 x 2 = 128 28 = 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 = 256

So far we have only examined numbers with a base of 2. Let's look at some examples of writing exponents where the base is a number other than 2.

Example 2: Write 3 x 3 x 3 x 3 using exponents, then read your answer aloud. Solution: 3 x 3 x 3 x 3 = 34 3 raised to the fourth power

Example 3: Write 6 x 6 x 6 x 6 x 6 using exponents, then read your answer aloud. Solution: 6 x 6 x 6 x 6 x 6 = 65 6 raised to the fifth power

Example 4: Write 8 x 8 x 8 x 8 x 8 x 8 x 8 using exponents, then read your answer aloud. Solution: 8 x 8 x 8 x 8 x 8 x 8 x 8 = 87 8 raised to the seventh power

Example 5: Write 103, 36, and 18 in factor form and in standard form.

Solution: Exponential Factor Standard Form Form Form 103 10 x 10 x 10 1,000 36 3 x 3 x 3 x 3 x 3 x 3 729 18 1 x 1 x 1 x 1 x 1 x 1 x 1 x 1 1

The following rules apply to numbers with exponents of 0, 1, 2 and 3:

Rule Example Any number (except 0) raised to the zero 1490 = 1

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Page 103 of 362 power is equal to 1. Any number raised to the first power is 81 = 8 always equal to itself. If a number is raised to the second power, 32 is read as three we say it is squared. squared If a number is raised to the third power, we 43 is read as four say it is cubed. cubed

Summary: Whole numbers can be expressed in standard form, in factor form and in exponential form. Exponential notation makes it easier to write a number as a factor repeatedly. A number written in exponential form is a base raised to an exponent. The exponent tells us how many times the base is used as a factor.

Negative Exponents Negative? What could be the opposite of multiplying? Dividing! A negative exponent means how many times to divide one by the number. Example: 8-1 = 1 ÷ 8 = 0.125 You can have many divides: Example: 5-3 = 1 ÷ 5 ÷ 5 ÷ 5 = 0.008 But that can be done an easier way: 5-3 could also be calculated like: 1 ÷ (5 × 5 × 5) = 1/53 = 1/125 = 0.008

In General That last example showed an easier way to handle negative exponents:  Calculate the positive exponent (an)  Then take the Reciprocal (i.e. 1/an)

More Examples: Negative Exponent Reciprocal of Positive Exponent Answer 4-2 = 1 / 42 = 1/16 = 0.0625 10-3 = 1 / 103 = 1/1,000=0.001 (-2)-3 = 1 / (-2)3 = 1/(-8) = -0.125

What if the Exponent is 1, or 0? 1 If the exponent is 1, then you just have the number itself (example 91 = 9) 0 If the exponent is 0, then you get 1 (example 90 = 1) But what about 00 ? It could be either 1 or 0, and so people say it is "indeterminate".

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Page 104 of 362 It All Makes Sense My favorite method is to start with "1" and then multiply or divide as many times as the exponent says, then you will get the right answer, for example:

Example: Powers of 5 .. etc.. 52 1 × 5 × 5 25 51 1 × 5 5 50 1 1 5-1 1 ÷ 5 0.2 5-2 1 ÷ 5 ÷ 5 0.04

.. etc..

If you look at that table, you will see that positive, zero or negative exponents are really part of the same (fairly simple) pattern.

Be Careful About Grouping To avoid confusion, use parentheses () in cases like this:

(-2)2 = (-2) × (-2) = 4 -22 = -(22) = - (2 × 2) = -4

(ab)2 = ab × ab ab2 = a × (b)2 = a × b × b

Activity 25: Exponents

What is the value of 63 ?

(A) 18 (B) 216 (C) 729 (D) 1,296

Activity 26: Exponents

What is the value of 34 ? (A) 12 (B) 27 (C) 34 (D) 81

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Activity 27: Exponents

What is the value of 2-4?

(A) -8 (B) 0.125 (C) 0.0625 (D) 16

Activity 28: Exponents

What is the value of 5-2?

(A) -25 (B) -10 (C) 0.04 (D) 0.1

Activity 29: Exponents

What is the value of (-3)5?

(A) -243 (B) -125 (C) 125 (D) 243

Activity 30: Exponents

What is the value of (-2)-5? (A) -0.03125 (B) 0.03125 (C) 10 (D) 32

Activity 31: Exponents

What is the value of (-5)-3?

(A) (B) (C) (D)

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Activity 32: Exponents

What is the value of (-10)0? (A) -10 (B) 1 (C) 10 (D) zero

Activity 33: Exponents

What is the value of 52 + 62 ? (A) 22 (B) 51 (C) 61 (D) 121

Activity 34: Exponents

What is the value of 22 - 2-2 ? (A) 0 (B) 3.75 (C) 4 (D) 4.25

Lesson 37: Division of Powers

 We know that: = 1 000

 We can write this using power notation as: = =

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 So, in general the rule is: = , where a 0

 Using the rule, we see that: =  Written out, it looks like this:

So,

This gives us another rule:

Therefore, in general:

a = 1, where a 0

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Page 109 of 362

Activity 35: For this activity please check your calendar for related worksheet

Lesson 38: Worksheets

Please check your calendar for information on this lesson.

Lesson 39: Scientific Notation

Scientific Notation (also called Standard Form in Britain) is a special way of writing numbers that makes it easier to use big and small numbers.

Example: 102 = 100, so 700 = 7 × 105 7 × 102 is "Scientific Notation"

Example: 4,900,000,000 1,000,000,000 = 109 , so 4,900,000,000 = 4.9 × 109 in "Scientific Notation"

The number is written in two parts:  Just the digits (with the decimal point placed after the first digit), followed by  × 10 to a power that puts the decimal point where it should be (i.e. it shows how many places to move the decimal point).

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In this example, 5326.6 is written as 5.3266 × 103, because 5326.6 = 5.3266 × 1000 = 5326.6 × 103

How to Do it

To figure out the power of 10, think "how many places do I move the decimal point?" If the number is 10 or greater, the decimal point has to move to the left, and the power of 10 will be positive.

If the number is smaller than 1, the decimal point has to move to the right, so the power of 10 will be negative:

Example: 0.0055 would be written as 5.5 × 10-3

Because 0.0055 = 5.5 × 0.001 = 5.5 × 10-3 Example: 3.2 would be written as 3.2 × 100

We didn't have to move the decimal point at all, so the power is 100 But it is now in Scientific Notation

Check After putting the number in Scientific Notation, just check that:  The "digits" part is between 1 and 10 (it can be 1, but never 10)  The "power" part shows exactly how many places to move the decimal point

Why Use It? Because it makes it easier when you are dealing with very big or very small numbers, which are common in Scientific and Engineering work.

Example: it is easier to write (and read) 1.3 × 10-9 than 0.0000000013 It can also make calculations easier, as in this example:

Example: a tiny space inside a computer chip has been measured to be 0.00000256m wide, 0.00000014m long and 0.000275m high.

What is its volume? Let's first convert the three lengths into scientific notation:  width: 0.000 002 56m = 2.56×10-6

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Page 111 of 362  length: 0.000 000 14m = 1.4×10-7  height: 0.000 275m = 2.75×10-4

Then multiply the digits together (ignoring the ×10s): 2.56 × 1.4 × 2.75 = 9.856 2.57 Last, multiply the ×10s: 10-6 × 10-7 × 10-4 = 10-17 (this was easy: I just added -6, -4 and -7 together) The result is 9.856×10-17 m3

It is used a lot in Science:

Example: Suns, Moons and Planets The Sun has a Mass of 1.988 × 1030 kg. It would be too hard for scientists to have to write 1,988,000,000,000,000,000,000,000,000,000 kg

Activity 36: Scientific Notation Write 3.56 × 1011 as an ordinary number

(A) 3,560,000,000,000 (B) 356,000,000,000

Activity 37: Scientific Notation Write 7.085 × 10-14 as an ordinary number

(A)0.000 000 000 007 085 (B) 0.000 000 000 000 708 5

Activity 38: Scientific Notation The speed of light in a vacuum is 299 792 458 m/s What is this written in Scientific notation?

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(A) 0.299 792 458 × 109 m/s (B) 29.979 245 8 × 107 m/s

Activity 39: Scientific Notation Photocopy paper is packaged in reams (500 sheets). The thickness of the pack is 41 mm. What is the thickness of one sheet of paper written in Scientific Notation using meters?

(A) 8.2 × 10-2 m (B) 8.2 × 10-3 m

Activity 40: Scientific Notation The mass of the Moon is 73,000,000,000,000,000,000,000 kg What is this written in Scientific notation?

(A) 7.3 × 1021 kg (B) 7.3 × 1022 kg

Activity 41: Scientific Notation The rest mass of an electron is 0.000 000 000 000 000 000 000 000 000 000 910 938 kg What is this written in Scientific notation?

(A) 0.910938 × 10-32 kg (B) 91.0938 × 10-34 kg

Lesson 40: Worksheets

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Page 113 of 362 Lesson 41: Square Roots

Squares and Square Roots First learn about Squares, then Square Roots are easy.

How to Square A Number To square a number, just multiply it by itself ...

Example: What is 3 squared?

3 Squared = = 3 × 3 = 9

"Squared" is often written as a little 2 like this:

This says "4 Squared equals 16" (the little 2 says the number appears twice in multiplying)

Squares From 12 to 62 1 Squared = 12 = 1 × 1 = 1 2 Squared = 22 = 2 × 2 = 4 3 Squared = 32 = 3 × 3 = 9 4 Squared = 42 = 4 × 4 = 16 5 Squared = 52 = 5 × 5 = 25 6 Squared = 62 = 6 × 6 = 36

You can also find the squares on the Multiplication Table:

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Negative Numbers You can also square negative numbers.

Example: What happens when you square (-5) ? Answer: (-5) × (-5) = 25 (because a negative times a negative gives a positive) When you square a negative number you get a positive result. Just the same as if you had squared a positive number:

Note: if someone says "minus 5 squared" do you:  Square the 5, then do the minus?  Or do you square (-5) ? You get different answers: Square 5, then do the minus: Square (-5): -(5×5) = -25 (-5)×(-5) = +25

Always make it clear what you mean, and that is what the "( )" are for.

Square Roots A square root goes the other way:

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Page 115 of 362 3 squared is 9, so a square root of 9 is 3

A square root of a number is ...... a value that can be multiplied by itself to give the original number.

A square root of 9 is ...... 3, because when 3 is multiplied by itself you get 9.

It is like asking: What can I multiply by itself to get this?

To help you remember think of the root of a tree: "I know the tree, but what is the root that produced it?" In this case the tree is "9", and the root is "3".

Here are some more squares and square roots:

The Square Root Symbol

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This is the special symbol that means "square root", it is sort of like a tick, and actually started hundreds of years ago as a dot with a flick upwards.

It is called the radical, and always makes math look important!

You can use it like this:

you would say "square root of 9 equals 3"

Example: What is √25? Well, we just happen to know that 25 = 5 × 5, so if you multiply 5 by itself (5 × 5) you will get 25. So the answer is: √25 = 5

Example: What is √36 ? Answer: 6 × 6 = 36, so √36 = 6

Perfect Squares The perfect squares are the squares of the whole numbers:

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 etc. Perfect 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 ... Squares:

Try to remember at least the first 10 of those.

Calculating Square Roots It is easy to work out the square root of a perfect square, but it is really hard to work out other square roots.

Example: what is √10? Well, 3 × 3 = 9 and 4 × 4 = 16, so we can guess the answer is between 3 and 4.  Let's try 3.5: 3.5 × 3.5 = 12.25  Let's try 3.2: 3.2 × 3.2 = 10.24  Let's try 3.1: 3.1 × 3.1 = 9.61  ...

Getting closer to 10, but it will take a long time to get a good answer!

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At this point, I get out my calculator and it says: 3.1622776601683793319988935444327 But the digits just go on and on, without any pattern. So even the calculator's answer is only an approximation!

Activity 42: Square roots

Calculate the square roots without the use of a calculator. Then use a calculator to calculate the answers and compare these to your first answers.

a) b) c) d) e)

Lesson 42-43: Multiplication and Division of Exponents

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Page 118 of 362 Variables with Exponents How to Multiply and Divide them What is a Variable with an Exponent?

A Variable is a symbol for a number we don't know yet.

It is usually a letter like x or y.

An exponent (such as the 2 in x2) says how many times to use the variable in a multiplication.

Example: y2 = yy (yy means y multiplied by y, because in Algebra putting two letters next to each other means to multiply them) Likewise z3 = zzz and x5 = xxxxx

Exponents of 1 and 0 Exponent of 1 If the exponent is 1, then you just have the variable itself (example x1 = x) We usually don't write the "1", but it sometimes helps to remember that x is also x1

Exponent of 0 If the exponent is 0, then you are not multiplying by anything and the answer is just "1" (example y0 = 1)

Multiplying Variables with Exponents So, how do you multiply this: (y2)(y3)

We know that y2 = yy, and y3 = yyy so let us write out all the multiplies: y2 y3 = yyyyy

That is 5 "y"s multiplied together, so the new exponent must be 5: y2 y3 = y5

But why count the "y"s when the exponents already tell us how many? The exponents tell us that there are two "y"s multiplied by 3 "y"s for a total of 5 "y"s: y2 y3 = y2+3 = y5

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So, the simplest method is to just add the exponents! (Note: this is one of the Laws of Exponents) (Note: I used "·" to mean multiply. In Algebra we don't like to use "×" because it looks too much like the letter "x")

Negative Exponents Mean Dividing! 1 1 1 x-1 = x-2 = x-3 = x x2 x3

Get familiar with this idea, it is very important and useful!

Dividing y3 So, how do you do this? y2

yyy If we write out all the multiplies we get: yy

We can remove any matching "y"s that are both top and bottom y (because y/y = 1), so we are left with: So 3 "y"s above the line get reduced by 2 "y"s below the line, leaving only 1 "y" : y3 yyy = = y3-2 = y1 = y y2 yy

OR, you could have done it like this: y3 = y3y-2 = y3-2 = y1 = y y2

So ... just subtract the exponents of the variables you are dividing by! The "z"s got completely cancelled out! (Which makes sense, because z2/z2 = 1) You can see what is going on if you write down all the multiplies, then "cross out" the variables that are both top and bottom: x3 y z2 xxx y zz xx xx x2 = = = = x y2 z2 x yy zz y y y

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Page 120 of 362 But once again, why count the variables, when the exponents tell you how many? Once you get confident you can do the whole thing quite quickly "in place" like this:

Activity 43: Exponents

Simplify (2x5y2)(3y3)

(A) 6x5y6 (B) 6xy10 (C) 5x5y5 (D) 6x5y5

Activity 44: Exponents

The length of an electronic component is 0.000 126 5 mm Write this in Engineering Notation.

(A) 126.5 × 10-6 mm (B) 12.65 × 10-5 mm (C) 1.265 × 10-4 mm (D) 0.126 5 × 10-3 mm

Activity 45: Exponents

The radius of the sun is 695 500 km What is its approximate volume written in Scientific notation?

(A) 1.405 × 1018 km3 (B) 1,405 × 1015 km3 (C) 607.6 × 1010 km3 (D) 6.076 × 1012 km3

Lesson 44-45: Addition and Subtraction of Polynomials

Polynomials A polynomial is made up of terms that are only added, subtracted or multiplied. polynomial looks like this:

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example of a polynomial this one has 3 terms

Polynomial comes from poly- (meaning "many") and -nomial (in this case meaning "term") ... so it says "many terms"

A polynomial can have: constants (like 3, -20, or ½) variables (like x and y) exponents (like the 2 in y2), but only 0, 1, 2, 3, ... etc. and they can be combined using: + addition, - subtraction, and × multiplication ... but not division!

Those rules keeps polynomials simple, so they are easy to work with!

Polynomial or Not?

These are polynomials:  3x  x - 2 2 7  -6y - ( /9)x  3xyz + 3xy2z - 0.1xz - 200y + 0.5  512v5+ 99w5

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Page 122 of 362  1

(Yes, even "1" is a polynomial, it has one term which just happens to be a constant).

And these are not polynomials  2/(x+2) is not, because dividing is not allowed  1/x is not  3xy-2 is not, because the exponent is "-2" (exponents can only be 0,1,2,...)  √x is not, because the exponent is "½" (see fractional exponents)

But these are allowed:  x/2 is allowed, because it is also (½)x (the coefficient is ½, or 0.5)  also 3x/8 for the same reason (the coefficient is 3/8, or 0.375)  √2 is allowed, because it is a constant (= 1.4142...etc.)

Monomial, Binomial, Trinomial There are special names for polynomials with 1, 2 or 3 terms:

How do you remember the names? Think cycles!

(There is also quadrinomial (4 terms) and quintinomial (5 terms), but those names are not often used)

Can Have Lots and Lots of Terms Polynomials can have as many terms as needed, but not an infinite number of terms.

Variables Polynomials can have no variable

Example: 21 is a polynomial. It has just one term, which is a constant. Or one variable Example: x4-2x2+x has three terms, but only one variable (x)

Or two or more variables

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Page 123 of 362 Example: xy4-5x2z has two terms, and three variables (x, y and z)

What is Special about Polynomials? Because of the strict definition, polynomials are easy to work with. For example we know that:

 If you add polynomials you get a polynomial  If you multiply polynomials you get a polynomial So you can do lots of additions and multiplications, and still have a polynomial as the result. Also, polynomials of one variable are easy to graph, as they have smooth and continuous lines.

Example: x4-2x2+x

See how nice and smooth the curve is?

Degree The degree of a polynomial with only one variable is the largest exponent of that variable. Example: The Degree is 3 (the largest exponent of x) .

Standard Form The Standard Form for writing a polynomial is to put the terms with the highest degree first.

Example: Put this in Standard Form: 3x2 - 7 + 4x3 + x6 The highest degree is 6, so that goes first, then 3, 2 and then the constant last: x6 + 4x3 + 3x2 - 7

You don't have to use Standard Form, but it helps. Adding and Subtracting Polynomials A polynomial looks like this:

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example of a polynomial this one has 3 terms

To add polynomials you simply add any like terms together... so what is a like term?

Like Terms Like Terms are terms whose variables (and their exponents such as the 2 in x2) are the same. In other words, terms that is "like" each other. Note: the coefficients (the numbers you multiply by, such as "5" in 5 xs) can be different.

Example: 7x x -2x πx

Are all like terms because the variables are all x Example: (1/3)xy2 -2xy2 6xy2 xy2/2

Are all like terms because the variables are all xy2

Adding Polynomials Two Steps:  Place like terms together  Add the like terms

Example: Add 2x2 + 6x + 5 and 3x2 - 2x - 1

Start with: 2x2 + 6x + 5 + 3x2 - 2x - 1

Place like terms together: 2x2 + 3x2 + 6x - 2x + 5 - 1

Add the like terms: (2+3)x2 + (6-2)x + (5-1) = 5x2 + 4x + 4

Adding Several Polynomials You can add several polynomials together like that. Example: Add (2x2 + 6y + 3xy), (3x2 - 5xy - x) and (6xy + 5) Line them up in columns and add: 2x2 + 6y + 3xy 3x2 - 5xy – x

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Page 125 of 362 6xy + 5 5x2 + 6y + 4xy - x + 5 Using columns helps you to match the correct terms together in a complicated sum.

Activity 46: Polynomials (Please go to calendar to do this activity)

Lesson 46: Worksheet

Please check your calendar for information on this lesson.

Lesson 47-48: Transformations

The three main Transformations are:

Rotation Turn!

Reflection Flip!

Translation Slide!

After any of those transformations (turn, flip or slide), the shape still has the same size, area, angles and line lengths.

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Page 126 of 362 If one shape can become another using Turns, Flips and/or Slides, then the two shapes are called Congruent.

Resizing The other important Transformation is Resizing (also called dilation, contraction, compression, enlargement or even expansion). The shape becomes bigger or smaller:

Resizing

If you have to Resize to make one shape become another then the shapes are not Congruent, but they are Similar.

Congruent or Similar So, if one shape can become another using transformation, the two shapes might be Congruent or just Similar If you ... Then the shapes are ...

... only Rotate, Reflect and/or Congruent Translate

... need to Resize Similar

Translation In Geometry, "Translation" simply means Moving ...... without rotating, resizing or anything else, just moving. To Translate a shape:

Every point of the shape must move:  the same distance  in the same direction.

Writing it Down Sometimes we just want to write down the translation, without showing it on a graph.

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Example: if we want to say that the shape gets moved 30 Units in the "X" direction, and 40 Units in the "Y" direction, we can write:

This says "all the x and y coordinates will become x+30 and y+40"

Reflection Reflections are everywhere ... in mirrors, glass, and here in a lake. ... what do you notice ?

Every point is the same distance from the central line ! ... and ... The reflection has the same size as the original image The central line is called the Mirror Line ...

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Page 128 of 362 Can A Mirror Line Be Vertical? Yes. Here my dog "Flame" shows a Vertical Mirror Line (with a bit of photo magic)

In fact Mirror Lines can be in any direction. Imagine turning the photo at the top in different directions ...... the reflected image is always the same size, it just faces the other way:

A reflection is a flip over a line

How Do I Do It Myself? Just approach it step-by-step. For each corner of the shape:

1. Measure from the point to the 2. Measure the same distance 3. Then connect the mirror line (must hit the mirror again on the other side and place a new dots up! line at a right angle) dot.

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Labels It is common to label each corner with letters, and to use a little dash (called a Prime) to mark each corner of the reflected image.

Here the original is ABC and the reflected image is A'B'C'

Some Tricks

X-Axis If the mirror line is the x-axis, just change each (x,y) into (x,-y)

Y-Axis If the mirror line is the y-axis, just change each (x,y) into (-x,y)

Fold the Paper And if all else fails, just fold your sheet of paper along the mirror line and then hold it up to the light !

Rotation "Rotation" means turning around a centre: The distance from the centre to any point on the shape stays the same. Every point makes a circle around the centre.

Here a triangle is rotated around the point marked with a "+"

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Activity 47: Rotation

Please answer the following question.

When this 'L'-shape is rotated about the origin (0,0) by 90° anticlockwise (counter clockwise), which one of these would it look like?

A B

C D

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Activity 48: Rotation

Please answer the following question.

When this triangle is: * rotated about the point (1, 1) by 90° clockwise, and * then rotated about (1, 1) by another 180°, which one of these would it look like?

A B

C D

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Activity 49: Reflection

Please answer the following question.

The rectangle is reflected in the line y = 4.

Which one of the following shows the correct image?

A B

C D

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Activity 50: Reflection

Please answer the following question.

The L-shape A′B′C′D′′F′ is the image of the L-shape ABCDEF after reflection in which of the following lines?

A The y axis B The line x = -3 C The line y = x D The line y = -x

Activity 51: Translation

Please answer the following question.

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The rectangle is translated 3 units in the positive x direction and 1 unit in the negative y direction.

Which one of the following shows the correct image?

A B

C D

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Activity 52: Translation

Please answer the following question.

Rectangle A′B′C′D′ is the image of rectangle ABCD after which of the following translations?

A 4 units in the negative x direction and 5 units in the negative y direction B 5 units in the negative x direction and 4 units in the negative y direction C 4 units in the positive x direction and 5 units in the positive y direction D 5 units in the positive x direction and 4 units in the positive y direction

Lesson 49: Worksheet

Please check your calendar for information on this lesson.

Lesson 50-51: Congruent Triangles in Polygons

Congruent Triangles Triangles that have exactly the same size and shape are called congruent triangles. The symbol for congruent is ≅. Two triangles are congruent when the three sides and the three angles of one triangle have the same measurements as three sides and three angles of another triangle.

The triangles in Figure 1 are congruent triangles.

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Figure 1 Congruet triangles

Corresponding parts The parts of the two triangles that have the same measurements (congruent) are referred to as corresponding parts.

This means that Corresponding Parts of Congruent Triangles are Congruent (CPCTC). Congruent triangles are named by listing their vertices in corresponding orders. In Figure

1 , Δ BAT ≅ Δ ICE. Example 1: If Δ PQR ≅ Δ STU which parts must have equal measurements?

These parts are equal because corresponding parts of congruent triangles are congruent.

Tests for congruence

To show that two triangles are congruent, it is not necessary to show that all six pairs of corresponding parts are equal. The following postulates and theorems are the most common methods for proving that triangles are congruent (or equal).

Postulate 13 (SSS Postulate): If each side of one triangle is congruent to the corresponding side of another triangle, then the triangles are congruent (Figure 2).

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Figure 2 The corresponding sides (SSS) of the two triangles are all congruent

Postulate 14 (SAS Postulate): If two sides and the angle between them in one triangle are congruent to the corresponding parts in another triangle, then the triangles are congruent (Figure 3).

Figure 3 two sides and the included angle (SAS) of one triangle are congruent to the corresponding part of the other trangle.

Postulate 15(ASA Postulate): If two angles and the side between them in one triangle are congruent to the corresponding parts in another triangle, then the triangles are congruent (Figure 4).

Figure 4 Two angles and their common side (ASA) in one triangle are congruent to the corresponding parts of the other triangle.

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Page 138 of 362 Theorem 28 (AAS Theorem): If two angles and a side not between them in one triangle are congruent to the corresponding parts in another triangle, then the triangles are congruent (Figure 5).

Figure 5 Two angles and the side opposite one of these angles (AAS) in one triangle are congruent to the corresponding parts of the other triangle.

Postulate 16 (HL Postulate): If the hypotenuse and leg of one right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent (Figure 6).

Figure 6 The hypotenuse and one leg (HL) of the first right triangle are congruent to the corresponding parts of the second right triangle.

Theorem 29 (HA Theorem): If the hypotenuse and an acute angle of one right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent (Figure 7).

Figure 7 The hypotenuse and an acute angle (HA) of the first right triangle are congruent to the corresponding parts of the second right triangle.

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Theorem 30 (LL Theorem): If the legs of one right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent (Figure 8 ).

Figure 8 The legs (LL) of the first right triangle are congruent to the corresponding parts of the second right triangle.

Theorem 31 (LA Theorem): If one leg and an acute angle of one right triangle are congruent to the corresponding parts of another right triangle, then the triangles are congruent (Figure 9 ).

Figure 9 One leg and an acute angle (LA) of the first right triangle are congruent to the corresponding parts of the second right triangle.

Example 2: Based on the markings in Figure 10 , complete the congruence statement Δ ABC ≅Δ .

Figure 10 Congruent triangles.

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Δ YXZ, because A corresponds to Y, B corresponds to X, and C corresponds, to Z.

Example 3: By what method would each of the triangles in Figures 11 (a) through 11 (i) be proven congruent?

Figure 11 Methods of proving pairs of triangles congruent.

 (a) SAS.  (b) None. There is no AAA method.  (c) HL.  (d) AAS.  (e) SSS. The third pair of congruent sides is the side that is shared by the two triangles.  (f) SAS or LL.  (g) LL or SAS.  (h) HA or AAS.  (i) None. There is no SSA method.

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Example 4: Name the additional equal corresponding part(s) needed to prove the triangles in Figures 12 (a) through 12 (f) congruent by the indicated postulate or theorem.

Figure 12 Additional information needed to prove pairs of triangles congruent.

 (a) BC = EF or AB = DE ( but notAC = DF because these two sides lie between the equal angles).  (b) GI = JL.  (c) MO = POandNO = RO.  (d) TU = WXandSU = VX.  (e) m ∠ T = m ∠ E andm ∠TOW = m ∠ EON.  (f) IX = EN or SX = TN (but not IS = ET because they are hypotenuses).

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Activity 53: Congruent triangles in polygons For this activity please check your calendar for the related worksheet

Activity 54: Congruent triangles in polygons

Transformations of polygons are used extensively in Islamic architecture.

For each of the examples below:

1. Discuss the transformation in the design.

A. B.

Lesson 52: Worksheet

Please check your calendar for information on this lesson.

Lesson 53: Enlargement

Scale factor enlargement Shapes B, C and D are all enlargements of shape A. However, they are all different sizes and in different positions on the screen.

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When describing enlargements you must state their size and their position.

Scale factor The size of an enlargement is described by its scale factor.

For example, a scale factor of 2 means that the new shape is twice the size of the original. A scale factor of 3 means that the new shape is three times the size of the original.

In each of the diagrams, the smaller polygon has been enlarged through the point O. Each side in the enlargement is the same multiple of the corresponding side of the smaller polygon. This multiple is called the scale factor of the enlargement.

So if the sides on the enlarged figure are twice as long as those in the original figure, this means that the scale factor is 2.

Centre of enlargement The position of the enlarged shape is described by the centre of enlargement (0).

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Example To enlarge the triangle with a scale factor of 2 and centre of enlargement 0, you should do the following:

Enlarging a triangle with a scale factor of 2

A line is drawn from point O through point A of a triangle

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Page 145 of 362 Extent lines form point O

Points A1, B1 and C1 are connected to form an enlarged triangle

A line is drawn from point O through point A of a triangle

For a scale factor of 2: 0A' = 2 × 0A 0B' = 2 × 0B 0C' = 2 × 0C

For a scale factor of 3: 0A' = 3 × 0A 0B' = 3 × 0B 0C' = 3 × 0C

Sometimes the centre of enlargement lies on or within the original shape.

Fractional scale factor You already know that the size of an enlargement is described by its scale factor.

For example, a scale factor of 2 means that the new shape is twice the size of the original. A scale factor of 3 means that the new shape is three times the size of the original.

1 It therefore follows that a scale factor of /2 means that the new shape is half the size of the original.

Example 1 To enlarge the triangle with a scale factor of /2 and centre of enlargement 0, do the following:

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Page 146 of 362 Enlarging a triangle with a scale factor of one half

A right angled triangle

Enlarging a triangle with a scale factor of one half

A line is drawn from point O to point A of the triangle

Enlarging a triangle with a scale factor of one half

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Extend lines from point O to B and C, indicate new scale factor points

Enlarging a triangle with a scale factor of one half

Connect new scale factor points to create smaller triangle

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Activity 55: Enlargement

For this activity please check your calendar for related worksheet

Activity 56: Enlarging a triangle with a scale factor of one half

What is the scale factor of enlargement in this diagram?

Lesson 54-55: Congruency and Similarities

Similar

Two shapes are Similar if the only difference is size (and possibly the need to turn or flip one around).

Resizing is the Key If one shape can become another using Resizing (also called dilation, contraction, compression, enlargement or even expansion), then the shapes are Similar:

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These Shapes are Similar!

There may be Turns, Flips or Slides, Too! Sometimes it can be hard to see if two shapes are Similar, because you may need to turn, flip or slide one shape as well as resizing it.

Rotation Turn!

Reflection Flip!

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Translation Slide!

Examples These shapes are all Similar:

Resized Resized and Reflected Resized and Rotated

Why is it Useful? When two shapes are similar, then:  corresponding angles are equal, and  the lines are in proportion.

This can make life a lot easier when solving geometry puzzles, as in this example:

Example: What is the missing length here?

Notice that the red triangle has the same angles as the main triangle ...... they both have one right angle, and a shared angle in the left corner

In fact you could flip over the red triangle, rotate it a little, then resize it and it would fit exactly

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Page 151 of 362 on top of the main triangle. So they are similar triangles.

So the line lengths will be in proportion, and we can calculate: ? = 80 × (130/127) = 81.9

(No fancy calculations, just common sense!)

Congruent If one shape can become another using Turns, Flips and/or Slides, then the two shapes are called Congruent:

Rotation Turn!

Reflection Flip!

Translation Slide!

After any of those transformations (turn, flip or slide), the shape still has the same size, area, angles and line lengths.

Examples These shapes are all Congruent:

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Rotated Reflected and Moved Reflected and Rotated

Congruent or Similar? The two shapes need to be the same size to be congruent. When you need to resize one shape to make it the same as the other, the shapes are called Similar.

If you ... Then the shapes are ...

... only Rotate, Reflect and/or Translate Congruent

... also need to Resize Similar

Congruent? Why such a funny word that basically means "equal"? Probably because they would only be "equal" if laid on top of each other. Anyway it comes from Latin congruere, "to agree". So the shapes "agree"

Activity 57: Similar

How many triangles similar to this equilateral triangle are there in the following diagram?

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Page 153 of 362 A 10 B 16 C 26 D 27

Activity 58: Similar

How many rectangles similar to this rectangle are there in the following diagram?

A 16 B 20 C 24 D 25

Activity 59: Similar

These two quadrilaterals are similar.

What is the value of x (the length of B'C') ?

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A B 5 C 6 D

Activity 60: Similar

In the diagram:

. AB is parallel to ED, . the points B, C and E lie on a straight line, and . the points A, C and D also lie on a straight line.

What is the value of e (the length of CD) ?

A 14.5 B 16.5 C 17.5 D 18

Activity 61: Similar

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The two trapezoids are similar. What is the value of x?

A 2 B 2.5 C 3 D 3.125

Activity 62: Congruent

Which shape is not congruent to the other three?

A A B B C C D D

Activity 63: Congruent

How many triangles congruent to this equilateral triangle are there in the following diagram?

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A 10 B 16 C 26 D 27

Activity 64: Congruent

How many congruent trapezoids are there in the following diagram?

A 8 B 12 C 16 D 21

Activity 65: Congruent

How many parallelograms congruent to this parallelogram

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are there in the following diagram?

A 12 B 16 C 18 D 24

Activity 66: Similar

How many rectangles congruent to this rectangle

are there in the following diagram?

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Lesson 56: Worksheet

Please check your calendar for information on this lesson.

Lesson 57: Types of Transformations

Isometry A transformation that preserves congruence. In other words, a transformation in which the image and pre-image have the same side lengths and angle measurements. The following transformations maintain their mathematical congruence. o Translations (a translation is considered a 'direct isometry' because it not only maintains congruence, but it also, unlike reflections and rotations, preserves its orientation. o Rotations o Reflections o Dilations

On the other hand, a dilation is not an isometry because its image is not congruent with its pre-image.

Translate a point A translation is the same as sliding/shifting an object. The notation for translate is T(+a,+b)--where a and b represent how much you slide in the x and the y directions, respectively. For instance, look at the picture below.

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Reflections: How to reflect a point Reflect point across x axis, y axis and other lines.

A reflection is a kind of transformation. It is basically a ‘flip’ of a shape over the line of reflection.

Very often reflections are performed using coordinate notation as they all are on this page. The coordinates allow us to easily describe the image and its pre-image.

Examples of the most common types of reflections in math

 Reflection in the x-axis. A reflection in the x-axis can be seen in the picture below in which A is reflected to its image A'. The general rule for a reflection in the x-axis: (A,B) (A, −B)

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 Reflection in the y-axis. A reflection in the y-axis can be seen in the picture below in which A is reflected to its image A'. The general rule for a reflection in the y-axis: (A,B) (−A, B)

Reflection in the line y = x.  A reflection in the line y = x can be seen in the picture below in which A is reflected to its image A'. The general rule for a reflection in the y-axis: (A, B) (B, A)

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Rotations of points, shapes In math, a rotation lives up to its name!

To rotate an object you need a centre of ration and how much you want to rotate it. By convention, positive rotations go counter clockwise, and negative rotations go clockwise.

 "Centre" is the 'centre of rotation.' This is the point around which you are performing your mathematical rotation.

 "Degrees" stands for how many degrees you should rotate. A positive number usually by convention means counter clockwise.

Very often rotations are performed using coordinate notation as they all are on this page. The coordinates allow us to easily describe the image and its pre-image.

Examples of the most common rotations

 Rotation by 90° about the origin: R(origin, 90°) A rotation by 90° about the origin can be seen in the picture below in which A is rotated to its image A'. The general rule for a rotation by 90° about the origin is (A,B) (-B, A)

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 2) Rotation by 180° about the origin: R(origin, 180°) A rotation by 180° about the origin can be seen in the picture below in which A is rotated to its image A'. The general rule for a rotation by 180° about the origin is (A,B) (-A, -B)

 3) Rotation by 270° about the origin: R(origin, 270°) A rotation by 270° about the origin can be seen in the picture below in which A is rotated to its image A'. The general rule for a rotation by 270° about the origin is (A,B) (B, -A)

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Activity 67: Types of transformation

Reflection

Translation

Rotation

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Lesson 58-60: General Discussion

Terms with the same powers of the variables are said to be like terms.

An expression can be simplified by adding or subtracting like terms:

It is not possible to simplify an expression into an equivalent from by adding or subtracting unlike terms:

Example Add the polynomials:

Solution

Remember only to add or subtract only the LIKE terms!

It is important to put brackets around the second term. Can you see why?

Example Subtract

Solution

A polynomial is an algebraic expression in which the exponent of the variables is natural numbers. The variables are not used in the denominators of the terms.

For example,

But

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The degree of the polynomial is the highest exponent of the variable.

A binomial expression is an expression consisting of two terms.

Expressions To solve a problem we need to find the value of an unknown quantity or quantities. The unknown quantity or quantities are represented by letters that are known as pro-numerals.

An expression is a combination of mathematical terms using operations such as addition, subtraction, multiplication, division, brackets, powers or roots.

For example, 2x + 3y is an expression involving addition where 2x and 3y are called the terms of the expression. Terms often include pro-numerals but may also be a number called a constant term.

Terms with the same pronominal or pro-numerals are called like terms.

Terms that do not have the same pro-numerals are called unlike terms.

Addition and Subtraction Only like terms can be added (or subtracted) to simplify an expression involving addition (or subtraction).

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Page 166 of 362 Example 1

Solution:

Product of Binomial Expressions The product of two binomial expressions is called a binomial product.

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Page 167 of 362

This can be split up into two parts as follows:

Algebraic Method

Setting out: Often, we set out the solution as follows:

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Example 5

Solution:

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Activity 68: Polynomials

Add the polynomials

Activity 69: Subtract

Activity 70: Simplify

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Lesson 61: Equivalent Expressions

Definition of Equivalent Expression  Two algebraic expressions are said to be equivalent if their values obtained by substituting the values of the variables are same.

More about Equivalent Expression  To symbolize equivalent expressions an equality (=) sign is used.

Examples of Equivalent Expression  3(x + 3) and 3x + 9 are equivalent expressions, because the value of both the expressions remains same for any value of x. For instance, for x = 4, 3(x + 3) = 3(4 + 3) = 21 and 3(x + 9) = 3 × 4 + 9( x + 3) = 21.  The expressions 6(x2 + y + 2) and 6x2 + 6y + 12 are equivalent expressions and can also be written as 6(x2 + y + 2) = 6x2 + 6y + 12.

Solved Example on Equivalent Expression Choose an expression that is equivalent to the expression 2n + 7(3 + n).

Choices: A. 9n + 21 B. -9n + 21 C. -9n – 21 D. n + 21

Correct Answer: A

Solution: Step 1: 2n + 7(3 + n) [Original expression.] Step 2: = 2n + 7(3) + 7(n) [Use the distributive property.] Step 3: = 2n + 21 + 7n [Multiply.] Step 4: = 2n + 7n + 21 [Use the commutative property.] Step 5: = 9n + 21 [Combine like terms.]

Activity 71: Equivalent expressions

Which expressions are equivalent to 4b ? A b+2(b+2b) B 3b+b C 2(2b)

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Activity 72: Equivalent expressions Which expressions are equivalent to 2(4f + 2g)? A 8f + 4g B 8f + 2g C 4(2f + g) D 4f + 4f + 4g

Lesson 62-63: Worksheets

Please check your calendar for information on this lessons.

Lesson 64: Products and Factors

Multiplication involves at least two numbers. The two numbers are multiplied together to give the product. The numbers multiplied are called the factors of the product number. Example: 4 x 5 is 20. Factors are numbers being multiplied.

In a multiplication problem, the numbers that are multiplied are called as factors and multiplying these factors is called as product. Factor means taking a number way from the product. The reverse of multiplication is called as factor. For example: Multiplying 4 and 2 we get 8 as a product. Here 4 and 2 is called as two factors. It can be divided into two division facts: 8 ÷ 2 = 4, and 8 ÷ 4 = 2. So, 8 are divisible by both 4 and 2. The factors of 8 are 4 and 2.

Numerical factors are numbers used as factors.

Example: Here, 20 is the product and 4, 5 are the factors of 20. 4 and 5 exactly divides 20.

Literal factors are alphabets or letters being used in place of or represent numbers used for multiplication.

Example: (m) (n) = 24; here, 'm' and 'n' are literal factors. Interchanging the factors does not change the product value (commutative law of multiplication). Interchanging of factors may be used to (a) simplify the multiplication (b) re-arrange the factors in a preferred order (c) check the product value

Factors and Products Rules There are some rules for factors and products.  Finding factor is a method of breaking down the number into smaller ones.  The factor of a number exactly divides that number.

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Page 172 of 362  All factors of a number are less than or equal to the number.  Each number is a factor of itself.  1 is a factor of all numbers.

Thus, a product is the multiple of all its factors. These are some of the properties of factors and products.

Divisibility Rules Divisibility rules help to find the factors of a number. Finding factors of a number means, we start dividing the number to get exact divisors.

 Products are opposites of factors.  Products of a number are obtained by multiplying that number with other numbers.  20 is the product of 5 with 4. 25 is the product of 5 with 5.  Product of a number is always greater than or equal to the number. 20 > 5.

Every number is the product of itself and 1.

Factors and Products Examples Given below are some of the examples on factors and products.

Solved Examples Question 1: Find the factors of 60.

Solution: Applying divisibility test, 60 is divisible by 10

60 = 6 x 10

60 = 3 x 2 x 2 x 5

Factors of 60 are 1, 2, 3, 5, 60.

Question 2: Find the factors of 24

Solution: Factor rainbows are a method of listing the factors of a number and the factor pairs that multiply to give the number are joined by curves.

Factors of 24 = 1 x 24 = 2 x 12 = 4 x 6= 8 x 3 Factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24.

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Question 3: Find the factors of 120

Solution: Factor tree is another method of representing factors.

120 = 12 x 10 = 4 x 3 x 2 x 5 = 2 x 2 x 3 x 2 x 5

We break down 120 to simple factors using the divisibility test. Then, each factor is broken down to more simple numbers.

The factors of 120 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60,120.

 Expressions that are equal for all values of the variables are called equivalent expressions or identities.

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Page 174 of 362  We indicate identities by means of the equals sign (=) or the equivalence or identity signs.

 Changing the expression to an equivalent form helps us to use the expression more efficiently.

Area of rectangles A and B = area of rectangle A + area of rectangle B

x(y + z) = xy + xz

 The identity in the box is known as the distributive law.  We say that multiplication is distributive over addition.  When working from left to right, we are finding a product.  When working from right to left, we are factorising.

Study the following examples.

Example Simplify x(2x – 3)

Solution

Example Simplify

Solution

Example  Multiplication is commutative, that is ab = ba. We can see this in: 3 x 2 = 2 x 3  So, 3x(2x – 5) = (2x – 5)3x

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Simplify 2x(x – 3) + (x + 7)2x

Solution

Example

Solution

Activity 73: Products and factors

Find the products. Simplify if possible. a. 3x(x – 5) b. (x + 2)4x c. -3x(x – 2) d. (x – 2) – 3x e. (x – 2) (3x) f. -3x – (x – 2)

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Page 176 of 362 g. 5a(2 + b) + (2 + b)5a h. 2x – (x + 3) – x i. Ab + a(b – 3) – (5a + 2ab) + 7a

Activity 74: Products and factors

Please answer the following question. 3(x + 2)x – x(x – 4)

Activity 75: Products and factors

Please answer the following question.

Mr. Schutte buys apples for his horses. 30 apples arrive separately and in two large baskets. The delivery note explains that each basket contains five packets and two boxes. There are eight apples in each packet and 15 in each box. Does Mr. Schutte have enough apples for 120 horses?

Lesson 65-66: Factorization

Factorisation is the opposite process of expanding brackets. For example, expanding brackets would require to be written as . Factorisation would be to start with and to end up with .

The two expressions and are equivalent; they have the same value for all values of x.

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Page 177 of 362 In previous grades, we factorised by taking out a common factor and using difference of squares.

Common factors Factorising based on common factors relies on there being factors common to all the terms. For example, can be factorised as follows:

Example: Factorising using a switch around in brackets Question Factorise: .

Answer Use a “switch around” strategy to find the common factor. Notice that

We will use the distributive law from right to left and write terms as factors.

Let’s look at the following examples.

Example Factorise 2x(x + 1) – 3(x +1)

Solution 2x(x + 1) – 3(x + 1) = (x + 1) (2x – 3)

 The HCF is (x +1)

Example Factorise

Solution

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Page 178 of 362  The HCF is

Example Factorise

Solution = 4ab x 3a + 4ab x 4 = 4ab(3a + 4)

 The HCF is 4ab

Example Factorise ab + a

Solution ab + a = ab + a.1 = a(b +a)

 The common factor in both terms is a

Note: It is always useful to check the factorization by multiplying the factors, so a(b + 1) = ab + a.

Example Factorise 5a + 30

Solution What is the highest common factor of 5a and 30?

5a + 30 = 5 x a + 5 x 6 = 5(a + 6)

 The HCF is 5 We want to multiply two binomials: (a + b)(c + d). This result can be illustrated using areas of rectangles.

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The total area of all four rectangles is (a + b)(c + d)

So, (a+ b)(c + d) = ac + ad + bc + bd

Using the result obtained, study the examples:

Example Simplify (x – 5)(x – 3) Solution

Example Simplify (x – 2)(x + 1)

Solution (x – 2)(x + 1) = x.x + x.1 – 2.x – 2.1

Notice that the product of two binomials can be obtained by multiplying each of the terms in the first factor by each of the terms in the second factor:

Where ac is the product of the First terms, ad, the product of the Outer terms,

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Page 180 of 362 bc, the product of the Inner terms, bd, the product of the Last terms. OR

Examine the products: First terms + Outer terms + Inner terms + Last terms

Notice that the products of the outer terms (-3x) and the inner terms (2x) are like terms, that simplify when added: -3x + 2x = - x.

This leads to a useful order in which to multiply out (FOIL):

Study the examples and find the products that follow.

Use as few steps as possible.

Example Simplify (x + 3)(x + 7)

Solution

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Example Simplify (x – 10)(x + 4)

Solution

Example Simplify (2x – 3)(3x – 1)

Solution

Activity 76: Factorization

Factorise each expression. 1. 2. 3.

Lesson 67-69: Worksheets

Please check your calendar for information on this lessons.

Lesson 70: Difference of Two Squares

We have seen that

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Page 182 of 362 Therefore

For example, can be written as which is a difference of two squares. Therefore, the factors of are and .

To spot a difference of two squares, look for expressions:  consisting of two terms;  with terms that have different signs (one positive, one negative);  with each term a perfect square.

For example: ; ; .

Example: The difference of two squares Question Factorise: . Answer Take out the common factor

Factorise the difference of two squares

Let’s look at some more differences of two squars. (Notice the subtraction sign between the terms.)

You may remember seeing expressions like this one when you worked with multiplying algebraic expressions. Do you remember ...

If you remember this fact, then you already know that:

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Example: Factor: x2 - 9 Both x2 and 9 are perfect squares. Since subtraction is occurring between these squares, this expression is the difference of two squares. These answers could What times itself will give x2 ? The answer is x. also be negative values, What times itself will give 9 ? The answer is 3. but positive values will make our work easier. The factors are (x + 3) and (x - 3). Answer: (x + 3) (x - 3) or (x - 3) (x + 3) (order is not important) Example:

Factor 4y2 - 36y6 There is a common factor of 4y2 that can be factored out first in this problem, to make the problem easier. 4y2 (1 - 9y4)

In the factor (1 - 9y4), 1 and 9y4 are perfect squares (their coefficients are perfect squares and their exponents are even numbers). Since subtraction is occurring between these squares, this expression is the difference of two squares.

What times itself will give 1? The answer is 1. What times itself will give 9y4 ? The answer is 3y2 . The factors are (1 + 3y2) and (1 - 3y2).

Answer: 4y2 (1 + 3y2) (1 - 3y2) or 4y2 (1 - 3y2) (1 + 3y2)

If you did not see the common factor, you can begin with observing the perfect squares. Both 4y2 and 36y6 are perfect squares (their coefficients are perfect squares and their exponents are even numbers). Since subtraction is occurring between these squares, this expression is the difference of two squares.

What times itself will give 4y2 ? The answer is 2y.

What times itself will give 36y6 ? The answer is 6y3 .

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Page 184 of 362 The factors are (2y + 6y3) and (2y - 6y3).

Answer: (2y + 6y3) (2y - 6y3) or (2y - 6y3) (2y + 6y3)

These answers can be further factored as each contains a common factor of 2y: 2y (1 + 3y2) • 2y (1 - 3y2) = 4y2 (1 + 3y2) (1 - 3y2)

Activity 77: Difference of two squares

Factorise each expression.

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Lesson 71-72: Worksheets

Please check your calendar for information on this lessons.

Lesson 73-74: Algebraic Fractions

To do algebraic fractions, you need to remember how to do numerical fractions.

Some reminders:

 When you multiply by 1, the value does not change.  1 is the identity element for multiplication and division.

 To divide by a fraction, invert it and multiply

  Add or subtract 0 and the value remains the same. 0 is the identity element for addition and subtraction.  Any number multiplied by 0 gives 0, in other words, a x 0 = 0



Study the following examples.

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Look for the HCF of the numerator and denominator. The largest factor of 12 and 8 is 4. Why do you think we picked the lowest exponent for the variable factors?

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 We can cancel out only if there is a multiplication or division sing between terms, For example



You can add, subtract, multiply and divide fractions in algebra in the same way that you do in simple arithmetic.

Adding Fractions To add fractions there is a simple rule:

(You can see why this works on the Common Denominator page).

Example: x + y = (x)(5) + (2)(y) = 5x+2y

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Page 188 of 362 2 5 (2)(5) 10

Example: x + 4 x - 3 (x+4)(4) + (3)(x-3) 4x+16 + 3x-9 7x+7 + = = = 3 4 (3)(4) 12 12

Subtracting Fractions Subtracting fractions is very similar to adding, except that the + is now -

Example: x + 2 x (x+2)(x-2) - (x)(x) x2-22 - x2 -4 - = = = x x - 2 x(x-2) x2 - 2x x2 - 2x

Multiplying Fractions Multiplying fractions is the easiest one of all, just multiply the tops together, and the bottoms together:

Example: 3x x (3x)(x) 3x2 x2 × = = = x-2 3 3(x-2) 3(x-2) x-2

Dividing Fractions

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Page 189 of 362 To divide fractions, first "flip" the fraction you want to divide by, then use the same method as for multiplying:

Example: 3y2 y 3y2 2 (3y2)(2) 6y2 6y ÷ = × = = = x+1 2 x+1 y (x+1)(y) (x+1)(y) x+1

 The denominators of fractions must be the same in order to add or subtract fractions.  If the denominators are not the same, the fractions must be changed to equivalent fractions. They must all have the same denominators.  To do this, find the lowest common denominator (LCD).  To change to an equivalent fraction, both the numerator and the denominator must be multiplied by the same number. Then the fraction would then have been multiplied by 1, which does not change its value.

Example

Solution

The lowest common denominator is the lowest common multiple of 3 and 5. So the LCD = 15.

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Activity 78: Algebraic fractions

What is the least common denominator for the fractions

A 12 B 27 C 45 D 135

Activity 79: Algebraic fractions

What is the least common denominator for the fractions

A 24 B 36 C 48 D 72

Activity 80: Algebraic fractions

A B C D

Activity 81: Algebraic fractions

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A B C D

Activity 82: Algebraic fractions

A B 1 C D

Lesson 75-76: Worksheets

Please check your calendar for information on this lesson.

Lesson 77: Mathematical Relationships

Relationships between input and output variables can be represented in many different ways.

We can use:

 Flow diagrams  Tables  Symbols or a formula (this is using algebraic language)  Descriptions in words  Graphs

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Page 193 of 362 Lesson 78: Conjunctures

Definition of Conjecture  Conjecture is a statement that is believed to be true but not yet proved.

Examples of Conjecture  The statement "Sum of the measures of the interior angles in any triangle is 180°" is a conjecture.  Here is another such conjecture: "If two parallel lines are cut by a transversal, the corresponding angles are congruent."

Solved Example on Conjecture Sam proposed that "3n + 1 yields a prime number for any even number n." and called it a conjecture. He explained his conjecture using the numbers 2, 4, and 6. He got 7, 13, and 19. Is his proposition a conjecture?

Choices: A. Yes B. No Correct Answer: B

Solution: Step 1: A conjecture is a statement that is believed to be true but not yet proved or disproved. Step 2: Sam's proposition is not a conjecture, because for n = 8, 3n + 1 gives 25, which is not a prime number.

Related Terms for Conjecture  Statement  Inductive Reasoning  Proof

Braam and Martin are asked to investigate the digits of the multiples of 9 and then to make a conjecture. 1 x 9 = 9 2 x 9 = 18 3 x 9 = 27 4 x 9 = 36 5 x 9 = 45 6 x 9 = 54

Braam’s conjecture: “If you add the digits, you will always get 9.”

Martin disagree: “That will only work up to a point. If you take 11 x 9 = 99, adding the digits gives you 18!”

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Page 194 of 362 The case above illustrates that when we make a conjecture, it is important not to check it against the given data only. You must also check your conjecture beyond the given data so that you can justify your statement for all cases. You must make sure that your conjecture works for all possible cases. If we look at data over the precipitation in a city for 29 out of 30 days and see that, it has been raining every single day it would be a good guess that it will be raining the 30th day as well. A conjecture is an educated guess that is bases on known information.

Example If we were given information about the quantity and formation of section 1, 2 and 3 of stars our conjecture would be as follows.

This method to use a number of examples to arrive at a plausible generalization or prediction could also be called inductive reasoning.

If our conjecture would turn out to be false, it is called a counterexample.

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Page 195 of 362

Activity 83: Conjecture

Make a conjecture for the following figure

Lesson 79-81: Worksheets

Please check your calendar for information on this lesson.

Lesson 82-83: Functions

 You have seen that there is a relationship between the input value and the output value.  The output value depends on the input value. We refer to the input value as the independent variable and the output value as the dependent variable.

When we work with functions in Mathematics, we can replace the phrase ‘depends on’ with the phrase ‘is a function of’.

Example of functions in real life:  A baby’s mass depends on its age  A person’s fitness depends on the number of hours of exercise done.

Thus, we can also say:  A baby’s mass is a function of its age  A person’s fitness is a function of the number of hours of exercise done.

What is a Function? A function relates an input to an output.

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It is like a machine that has an input and an output. And the output is related somehow to the input.

"f(x) = ... " is the classic way of writing a function. f(x) And there are other ways, as you will see!

Input, Relationship, Output I will show you many ways to think about functions, but there will always be three main parts:  The input  The relationship  The output

Example: "Multiply by 2" is a very simple function. Here are the three parts: Input Relationship Output 0 × 2 0 1 × 2 2 7 × 2 14 10 × 2 20 ......

For an input of 50, what would be the output?

Some Examples of Functions  x2 (squaring) is a function  x3+1 is also a function  Sine, Cosine and Tangent are functions used in trigonometry  and there are lots more! But we are not going to look at specific functions ...... instead; we will look at the general idea of a function.

Names First, it is useful to give a function a name.

The most common name is "f", but you can have other names like "g" ... or even "marmalade" if you want.

But let us use "f":

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You would say "f of x equals x squared" what goes into the function is put inside parentheses () after the name of the function:

So f(x) shows you the function is called "f", and "x" goes in And you will often see what a function does with the input: f(x) = x2 shows you that function "f" takes "x" and squares it.

Example: with f(x) = x2:  an input of 4  becomes an output of 16. In fact we can write f(4) = 16.

The "x" is Just a Place-Holder! Don't get too concerned about "x", it is just there to show you where the input goes and what happens to it.

It could be anything! So this function: f(x) = 1 - x + x2

Would be the same function if I wrote:  f(q) = 1 - q + q2  h(A) = 1 - A + A2  w(θ) = 1 - θ + θ2 It is just there so you know where to put the values: f(2) = 1 - 2 + 22 = 3

Sometimes There is No Function Name Sometimes a function has no name, and you might just see something like: y = x2

But there is still:  an input (x)  a relationship (squaring)  and an output (y)

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Page 198 of 362 Relating At the top I said that a function was like a machine. But a function doesn't really have belts or cogs or any moving parts - and it doesn't actually destroy what you put into it! A function relates an input to an output.

Saying "f(4) = 16" is like saying 4 is somehow related to 16. Or 4 → 16

Example: this tree grows 20 cm every year, so the height of the tree is related to its age using the function h: h(age) = age × 20 So, if the age is 10 years, the height is: h(10) = 10 × 20 = 200 cm

Here are some example values: age h(age) = age × 20 0 0 1 20 3.2 64 15 300 ......

What Types of Things Do Functions Process? "Numbers" seems an obvious answer, but ...... which numbers? For example, the tree-height function h(age) = age×20 makes no sense for an age

less than zero. ... it could also be letters ("A"→"B"), or ID codes ("A6309"→"Pass") or stranger things.

So we need something more powerful, and that is where sets come in: A set is a collection of things. Here are some examples: Set of even numbers: {..., -4, -2, 0, 2, 4, ...} Set of clothes: {"hat”, “shirt",...}

Set of prime numbers: {2, 3, 5, 7, 11, 13, 17, ...}

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Page 199 of 362 Positive multiples of 3 that are less than 10: {3, 6, 9}

Each individual thing in the set (such as "4" or "hat") is called a member, or element. So, a function takes elements of a set, and gives back elements of a set.

A Function is Special But a function has special rules:  It must work for every possible input value  And you can only have one relationship for each input value

This can be said in one definition:

Formal Definition of a Function A function relates each element of a set with exactly one element of another set (possibly the same set).

The Two Important Things! 1. "...each element..." means that every element in X is related to some element in Y. We say that the function covers X (relates every element of it). (But some elements of Y might not be related to at all, which is fine.) 2. "...exactly one..." means that a function is single valued. It will not give back 2 or more results for the same input. So "f(2) = 7 or 9" is not right!

(one-to-many) (many-to-one) This is NOT OK in a function But this is OK in a function

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Page 200 of 362 If a relationship does not follow those two rules then it is not a function ... it would still be a relationship, just not a function.

Example: The relationship x → x2

Could also be written as a table: X: x Y: x2 3 9 1 1 0 0 4 16 -4 16 ......

It is a function, because:  Every element in X is related to Y  No element in X has two or more relationships So it follows the rules. (Notice how both 4 and -4 relate to 16, which is allowed.)

Example: This relationship is not a function:

It is a relationship, but it is not a function, for these reasons:  Value "3" in X has no relation in Y

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Page 201 of 362  Value "4" in X has no relation in Y  Value "5" is related to more than one value in Y

(But the fact that "6" in Y is not related to does not matter)

Vertical Line Test On a graph, the idea of single valued means that no vertical line would ever cross more than one value. If it crosses more than once it is still a valid curve, but it would not be a function.

Infinitely Many My examples have just a few values, but functions usually work on sets with infinitely many elements. Example: y = x3  The input set "X" is all Real Numbers  The output set "Y" is also all the Real Numbers

I cannot show you ALL the values, so I just give a few as an example:

X: x Y: x3 -2 -8 -0.1 -0.001 0 0 1.1 1.331 3 27 and so on... and so on...

Domain, Codomain and Range In our examples above  the set "X" is called the Domain,  the set "Y" is called the Codomain, and  the set of elements that get pointed to in Y (the actual values produced by the function) is called the Range.

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Page 202 of 362 We have a special page on Domain, Range and Codomain if you want to know more.

So Many Names! Functions have been used in mathematics for a very long time, and lots of different names and ways of writing functions have come about.

Here are some common terms you should get familiar with:

Example: with z = 2u3:  "u" could be called the "independent variable"  "z" could be called the "dependent variable" (it depends on the value of u)

Example: with f(4) = 16:  "4" could be called the "argument"  "16" could be called the "value of the function"

Ordered Pairs I said I would show you many ways to think about functions, and here is another way: You can write the input and output of a function as an "ordered pair", such as (4,16). They are called ordered pairs because the input always comes first, and the output second: (input, output) So it looks like this: ( x, f(x) )

Example: (4,16) means that the function takes in "4" and gives out "16" Set of Ordered Pairs A function can then be defined as a set of ordered pairs:

Example: {(2,4), (3,5), (7,3)} is a function that says "2 is related to 4", "3 is related to 5" and "7 is related 3". Also, notice that:  the domain is {2,3,7} (the input values)  and the range is {4,5,3} (the output values)

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Page 203 of 362 But the function has to be single valued, so we also say

"if it contains (a, b) and (a, c), then b must equal c"

Which is just a way of saying that an input of "a" cannot produce two different results.

Example: {(2,4), (2,5), (7,3)} is not a function because {2,4} and {2,5} means that 2 could be related to 4 or 5. In other words it is not a function because it is not single valued

A Benefit of Ordered Pairs We can graph them...... because they are also coordinates! So a set of coordinates is also a function (if they follow the rules above, that is)

A Function Can be in Pieces You can create functions that behave differently depending on the input value

Example: A function with two pieces:  when x is less than 0, it gives 5,  when x is 0 or more it gives x2

Here are some example values: x y -3 5 -1 5 0 0 2 4 4 16 ......

Read more at Piecewise Functions.

Explicit vs Implicit Before I finish, I would like to mention the terms "explicit" and "implicit". "Explicit" is when the function shows you how to go directly from x to y, such as:

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Page 204 of 362 y = x3 - 3 When you know x, you can find y That is the classic y = f(x) style.

"Implicit" is when it is not given directly such as: x2 - 3xy + y3 = 0 When you know x, how do you find y? It may be hard (or impossible!) to go directly from x to y. "Implicit" comes from "implied", in other words shown indirectly.

Graphing  The Function Grapher can only handle explicit functions,  The Equation Grapher can handle both types (but takes a little longer, and sometimes gets it wrong).

Conclusion  a function relates inputs to outputs  a function takes elements from a set (the domain) and relates them to elements in a set (the codomain).  all the outputs (the actual values related to) are together called the range  a function is a special type of relation where: o every element in the domain is included, and o any input produces only one output (not this or that)  an input and its matching output are together called an ordered pair  so a function can also be seen as a set of ordered pairs

Activity 84: Functions f(x) = - 2 x 2 + 6 x - 3 find f(- 2).

A -10 B -4 C -23 D 14

Activity 85: Functions

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Page 205 of 362 Does the equation y 2 + x = 1 represent a function y in terms of x?

A. y = + √(1 - x) or y = - √(1 - x) B. y=8 C. y=(1 - x) 2

Activity 86: Functions Work out what “a” is: h(x) = 3x2 + ax – 1

A. a = 0.2 B. a= 1 C. a = 0.5 D. a = -6

Activity 87: Functions evaluate the function:

h(x) = x2 + 2 for x = −3

A -11 B -5 C 12 D 15

Activity 88: Functions

Which one of the following is not a function?

(A)

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(B)

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(C)

(D)

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Page 208 of 362

Lesson 82-83: Types of Functions and Gradients

If the function value changes by a constant amount when the input variable is increased by 1, then this is called a linear function.

The values in the second row of the table are called the function values.

The function represented in the table is linear. This is because the function values increase by a constant amount of 2 when the input variable is increased by 1.

We also say that the gradient of the function is 2.

Graphically, the function is represented by a straight line.

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Page 209 of 362 Note:

We are assuming that the quotient, is constant no matter what the change in x is.

For example: if we take the points (-1; -1) and (2; 5), then:

The function represented in the tableis not linear. This is because the function values increase by varying amounts when the input variable is increased by 1.

The value of the quotient, is not constant.

The gradient of the function varies.

A linear function has a constant gradient. A non-linear function has a varying gradient.

The formula or algebraic rule for a linear function is often written as: y = mx + c or y = ax + c.

Let us look at an example:

a linear function

The function rule gives the relationship between the independent (x) and dependent variable (y).

In this case it is: “Multiply the input number by 3 and add 1.”

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Page 210 of 362 In symbolic form it is: y = 3x + 1.

- The constant difference between the values is 3.

The constant difference is the same as the coefficient of the x-term in the formula.

- If the input value is increased by 1 and there is a constant difference in the output value, then this constant difference is the gradient.

So in the formula of the linear function, the coefficient of the x-term gives the gradient of the linear function.

Linear Function Definition of Linear Function  A function that can be graphically represented in the Cartesian coordinate plane by a straight line is called a Linear Function.

More about Linear Function  A linear function is a first degree polynomial of the form, F(x) = m x + c, where m and c are constants and x is a real variable.  The constant m is called slope and c is called y-intercept.

Examples of Linear Function  y = 3x + 5 is a linear function.  The graph of the function y = 2x is shown below. This is a linear function since the points fit onto a straight line.

Solved Example on Linear Function Identify the graph that represents a linear function.

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Choices: A. Graph 1 B. Graph 3 C. Graph 4 D. Graph 2 Correct Answer: C

Solution: Step 1: The graph of a linear function is a straight line. Step 2: Graph 4 is a straight line. Step 3: So, Graph 4 represents a linear function.

Nonlinear Functions Linear functions are functions where x is raised only to the first power. On graphs, linear functions are always straight lines. y = mx + b 3x + 5y - 10 = 0 y = 88x are all examples of linear equations.

The graphs of nonlinear functions are not straight lines.

In this topic, we will be working with nonlinear functions with the form y = ax2 + b and y = ax3 b where a and b are integers.

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Quadratic functions: y = ax2 + b

The graph of the function y = ax2 + b will look like a "U". This "U" shape graph is called a parabola. When a is positive, then the parabola opens up. When a is negative, then the parabola opens down. The highest or lowest point of parabolas is called the vertex. b determines where the vertex is on the graph. When b=0, the vertex is on the origin (0,0). When b = h where h is an integer, the vertex is on the point (0, h).

In this graph, the vertex is the lowest point. b = 0 because the vertex is on the origin. Use the point (2,12) to find a. y = ax 2 12 = a(2) 2 12 = 4a 3 = a The equation is y = 3x.

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In this graph, the vertex is the highest point. b = -5 because the vertex is on (0, -5). Use the point (1, -7) to find a. y = ax2 - 5 -7 = a(1)2 - 5 -7 = a - 5 -2 = a The equation is y = -2x - 5.

Cubic functions: y = ax3 + b The graph of a cubic function has this shape

b = 0 when the point of transition (from an upwards curve to a downwards curve) is on the origin (0,0). This is an example of y = ax3 where a is negative. Use the point (1, -2) to find a. y = ax3 -2 = a(1)3 -2 = a The equation is y = -2x3.

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b = -5 because the point of transition is on (0, -5). Use the point (1, -2) to find a. y = ax 3 - 5 -2 = a(1) 3 - 5 -2 = a - 5 3 = a The equation is y = 3x3 - 5.

Activity 89: Linear Function

Given f(x) = 2x • 2, what is the value of f(•5)?

A f(-5)=-15 B f(-5)= 32 C f(-5)= -6 D f(-5)= -12

Activity 90: Linear Function

Given g(x) = •5x • 3, what is the value of g(x) = 7?

A g(•2) = 7 B g(-2) = -9 C g(-2) = 12 D g(-2) = -22

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Lesson 84: Inverse Operations

Inverse Operation

The operation that reverses the effect of another operation. Example: Addition and subtraction are inverse operations

Another Example: Multiplication and division are inverse operations.

Operation

A mathematical process.

The most common are add, subtract, multiply and divide (+, -, ×, ÷ ).

But there are many more, such as squaring, square root, etc.

If it isn't a number it is probably an operation.

Example: In 25 + 6 = 31, the operation is add

Inverse Inverse means the opposite in effect. The reverse of.

The Inverse of Adding is Subtracting

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Page 216 of 362 Adding moves you one way, subtracting moves you the opposite way. Example: 20 + 9 = 29 can be reversed by 29 - 9 = 20 (back to where we started)

And the other way around: Example: 15 - 3 = 12 can be reversed by 12 + 3 = 15 (back to where we started)

The Inverse of Multiplying is Dividing Multiplying can be "undone" by dividing. Example: 5 × 9 = 45 can be reversed by 45/9 = 5

It works the other way around too, dividing can be undone my multiplying. Example: 10 / 2 = 5 can be reversed by 5 × 2 = 10

But Not With 0 You can't divide by 0, so don't try! Example: 5 × 0 = 0 cannot be reversed by 0/0 = ???

Lesson 85-86: Inverse Functions

Inverse Functions An inverse function goes in the opposite direction!

Let us start with an example:

Here we have the function f(x) = 2x+3, written as a flow diagram:

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The Inverse Function just goes the other way:

So the inverse of: 2x+3 is: (y-3)/2

The inverse is usually shown by putting a little "-1" after the function name, like this: f-1(y)

We say "f inverse of y"

So, the inverse of f(x) = 2x+3 is written: f-1(y) = (y-3)/2 (I also used y instead of x to show that we are using a different value.)

Back to Where We Started The cool thing about the inverse is that it should give you back the original value:

If the function f turns the apple into a banana, Then the inverse function f-1 turns the banana back to the apple

Example: Using the formulas from above, we can start with x=4: f(4) = 2×4+3 = 11

We can then use the inverse on the 11: f-1(11) = (11-3)/2 = 4

And we magically get 4 back again!

We can write that in one line: f-1( f(4) ) = 4

"f inverse of f of 4 equals 4"

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So applying a function f and then its inverse f-1 gives us the original value back again: f-1( f(x) ) = x

We could also have put the functions in the other order and it still works: f( f-1(x) ) = x

Example: Start with: f-1(11) = (11-3)/2 = 4

And then: f(4) = 2×4+3 = 11

So we can say: f( f-1(11) ) = 11 "f of f inverse of 11 equals 11"

Solve Using Algebra You can work out the inverse using Algebra. Put "y" for "f(x)" and solve for x:

The function: f(x) = 2x+3 Put "y" for "f(x)": y = 2x+3 Subtract 3 from both sides: y-3 = 2x Divide both sides by 2: (y-3)/2 = x Swap sides: x = (y-3)/2

Solution (put "f-1(y)" for "x") : f-1(y) = (y-3)/2 This method works well for more difficult inverses.

Fahrenheit to Celsius A useful example is converting between Fahrenheit and Celsius: 5 To convert Fahrenheit to Celsius: f(F) = (F - 32) x /9 -1 9 The Inverse Function (Celsius back to Fahrenheit) is: f (C) = (C × /5) + 32 For You: see if you can do the steps to create that inverse!

Inverses of Common Functions It has been easy so far, because we know the inverse of Multiply is Divide, and the inverse of Add is Subtract, but what about other functions? Here is a list to help you:

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Page 219 of 362 Inverses Careful!

<=>

<=> Don't divide by zero

<=> x and y not zero

<=> x and y ≥ 0

n not zero

<=> or (different rules when n is odd, even, negative or positive)

<=> y > 0

<=> y and a > 0

<=> -π/2 to +π/2

<=> 0 to π

<=> -π/2 to +π/2

(Note: you can read more about Inverse Sine, Cosine and Tangent.)

Careful! Did you see the "Careful!" column above? That is because some inverses work only with certain values.

Example: Square and Square Root If you square a negative number, and then do the inverse this happens: Square: (-2)2 = 4 Inverse (Square Root): √(4) = 2

But we didn't get the original value back! We got 2 instead of -2. Our fault for not being careful! So the square function (as it stands) does not have an inverse

But we can fix that!

Restrict the Domain (the values that can go into a function).

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Example: (continued) Just make sure you don't use negative numbers. In other words, restrict it to x ≥ 0 and then you can have an inverse.

So we have this situation:  x2 does not have an inverse  but {x2 | x ≥ 0 } (which says "x squared such that x is greater than or equal to zero" using set- builder notation) does have an inverse.

Why Would There Be No Inverse? Let us see graphically what is going on here: To be able to have an inverse you need unique values.

Just think ... if there are two or more x-values for one y-value, how do you know which one to choose when going back?.

No Inverse Inverse is Possible

When a y-value has more than When there is a unique y-value for one x-value, how do you know every x-value you can always "go which x-value to go back to? back" from y to x.

So we have this idea of "a unique y-value for every x-value", and it actually has a name. It is called "Injective" or "One-to-one":

If a function is "One-to-one" (Injective) it has an inverse.

Domain and Range So what is all this talk about "Restricting the Domain"?

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In its simplest form the domain is all the values that go into a function (and the range is all the values that come out).

As it stands the function above does not have an inverse.

But you could restrict the domain so there is a unique y for every x ...

... and now you can have an inverse:

Note also:  The function f(x) goes from the domain to the range,  The inverse function f-1(y) goes from the range back to the domain.

Or...

You could plot them both in terms of x ... so it is now f-1(x), not f-1(y).

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f(x) and f-1(x) are like mirror images (flipped about the diagonal). In other words: The graph of f(x) and f-1(x) are symmetric across the line y=x

Example: Square and Square Root (continued)

First, we restrict the Domain to x ≥ 0:  {x2 | x ≥ 0 } "x squared such that x is greater than or equal to zero"  {√x | x ≥ 0 } "square root of x such that x is greater than or equal to zero"

And you can see they are "mirror images" of each other about the diagonal y=x.

Note: we could have restricted the domain to x ≤ 0 and the inverse would then be f-1(x) = -√x:  {x2 | x ≤ 0 }  {-√x | x ≥ 0 } Which are inverses, too.

Not Always Solvable! It is sometimes not possible to find an Inverse of a Function. Example: f(x) = x/2 + sin(x)

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Page 223 of 362 We cannot work out the inverse of this, because we cannot solve for "x": y = x/2 + sin(x) y ... ? = x

Notes on Notation Even though we write f-1(x), the "-1" is not an exponent (or power): f-1(x) ...is different to... f(x)-1 Inverse of the function f(x)-1 = 1/f(x)

f (the Reciprocal)

Summary  The inverse of f(x) is f-1(y)  You can find an inverse by reversing the "flow diagram"  Or you can find an inverse by using Algebra: o Put "y" for "f(x)", and o Solve for x  You may need to restrict the domain for the function to have an inverse

Activity 91: Inverse functions

Find the inverse function of f.: f(x) = 3x - 2

Activity 92: Inverse functions

Find the inverse function of f: f(x) = -x 2 + 2 , x >= 0

Activity 93: Inverse functions What is the inverse of the function f:

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Page 224 of 362 f(x) = x 2 - 2x , x >= 1

f -1(x) = 1 + sqrt(x + 1)

Activity 94: Inverse functions

What is the inverse of the function f:

f(x) = 2 / x

Activity 95: Inverse functions

What is the inverse of the function f:

f(x) = (x + 1) / (x - 1)

Lesson 87: Solving Equations using Inverse Operations

Reversing Operations The goal in solving an equation is to get the variable by itself on one side of the equation and a number on the other side of the equation.

To isolate the variable, we must reverse the operations acting on the variable. We do this by performing the inverse of each operation on both sides of the equation. Performing the same operation on both sides of an equation does not change the validity of the equation, or the value of the variable that satisfies it.

Reversing Multiple Operations When more than one operation acts on a variable in an algebraic equation, apply the reverse of the order of operations to reverse the operations. Here is the order in which you should reverse operations: 1. Reverse addition and subtraction (by subtracting and adding) outside parentheses.

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Page 225 of 362 2. Reverse multiplication and division (by dividing and multiplying) outside parentheses. 3. Remove (outermost) parentheses, and reverse the operations in order according to these three steps.

Be sure to check your answer! The value of the variable, when plugged in for the variable, should make the equation true.

Example 1: Solve for x : 5x + 9 = 44

1. Reverse addition: 5x + 9 - 9 = 44 - 9 5x = 35

2. Reverse multiplication: = x = 7 3. No parentheses.

Check: 5(7) + 9 = 44 Thus, x = 7 .

Example 2: Solve for y : 3( - 1) = 15

1. No addition or subtraction outside the parentheses 2. Reverse multiplication: 3( -1)÷3 = 15÷3 ( - 1) = 5 3. Within parentheses:

1. Reverse subtraction: - 1 + 1 = 5 + 1 = 6 2. Reverse division: ×4 = 6×4 y = 24 3. No parentheses.

Check: 3( - 1) = 15 Thus, y = 24 .

Example 3: Solve for z : 4(3(z - 11) + 6) = 48

1. No addition or subtraction.

2. Reverse multiplication: = (3(z - 11) + 6) = 12 3. Within parentheses:

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Page 226 of 362 1. Reverse addition: 3(z - 11) + 6 - 6 = 12 - 6 3(z - 11) = 6 2. Reverse multiplication: = (z - 11) = 2 3. Within parentheses 1. Reverse subtraction: z - 11 + 11 = 2 + 11 z = 13 2. No multiplication or division. 3. No parentheses.

Check: 4(3(13 - 11) + 6) = 48 Thus, z = 13 .

Sometimes, the equation will not start out simplified. If this is the case, simplify the equation before reversing the operations.

Example 4: Solve for x : 6x - 5 - 2x + 3 - 2 = 4

First, simplify the equation by combining like terms: 4x - 4 = 4 1. Reverse subtraction: 4x - 4 + 4 = 4 + 4 4x = 8 2. Reverse multiplication: = x = 2 3. No parentheses.

Check: 6(2) + 5 - 2(2) - 3 + 2 = 12 - 5 - 4 + 3 - 2 = 4 Thus, x = 2

Activity 96: Expressions and Equations

Problem : Solve for x : + 4 = 7

Activity 97: Expressions and Equations

Problem : Solve for y : 4(y - 3) - 30 = 2

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Page 227 of 362

Activity 98: Expressions and Equations

Problem : Solve for z : 8 =

Activity 99: Expressions and Equations

Problem : Solve for p : 5 - 2p = 1

Lesson 88-89: Worksheets

Please check your calendar for information on this lessons.

Lesson 90: Points to Remember when Solving Equations

- Isolate the variable. - To undo an operation, apply the inverse operation. - Do the same to both sides. - Keep equal signs under each other. - Use going from line to line. - Check your solution by substituting into the original equation.

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Page 228 of 362

Activity 100: Solve equations by using inverse operations

Solve: 1. x + 2=3 2. x - 2 =3 3. 2x =8 4. x/ 2 = 8

Lesson 91: Equations with Variables on Both Sides

Solving Equations with a Variable on Both Sides Sometimes, the unknown quantity will appear on both sides of an equation. This is where the properties learned in 5.1 and 5.2 come in handy. A quantity with a variable can be treated just like a quantity without variables -- a quantity with a variable follows all the rules learned in the last two sections. For example, we can add a quantity with a variable to both sides without changing the equation or the values that make it true:

15 - x = 4x 15 - x + x = 4x + x 15 + 0x = 5x 15 = 5x 3 = x x = 3

Similarly, we can subtract a term with a variable from both sides of the equation: 5x = 6 + 2x 5x - 2x = 6 + 2x - 2x 3x = 6 + 0x 3x = 6 x = 2

After simplifying, the first step in solving an equation with a variable on both sides is to get the variable on one side. This is done by reversing the addition or subtraction of one of the terms with the variable. In other words, we must add to both sides or subtract from both sides one of the quantities that contains the variable. It is generally easier to add or to subtract the smaller quantity from the larger quantity, so we are working with positive coefficients, but either way works. Once the variable is on one side only, we can proceed using inverse operations, as in 4.1 and 4.2.

Example 1. Solve for x : 3x + 2x = 12 - x  Simplify: 5x = 12 - x

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Page 229 of 362  Get the variable on one side: o 5x + x = 12 - x + x o 6x = 12  Solve using inverse operations: o = o x = 2  Check: 3(2) + 2(2) = 12 - 2 ? Yes!

Example 2. Solve for y : 5y - 3 = 3y + 5  The equation is already simplified.  Get the variable on one side: o 5y - 3 - 3y = 3y + 5 - 3y o 5y - 3y - 3 = 3y - 3y + 5 o 2y - 3 = 5  Solve using inverse operations: o 2y - 3 + 3 = 5 + 3 o 2y = 8 o = o y = 4  Check: 5(4) - 3 = 3(4) + 5 ? Yes!

Activity 101: Equations with variables on both sides

1. Problem : Solve for x : 16 - x = 7x 2. Problem : Solve for x : 4x + 12 = 6(x - 1) 3. Problem : Solve for y : 5 + 2y = 8y 4. Problem : Solve for l : 3(l - 2) = 2l + 1 5. Problem : Solve for m : 17 + 3m = - (5 - m)

Lesson 92-93: Worksheets

Please check your calendar for information on this lessons.

Lesson 94: Equations with Brackets

Before equations with brackets can be solved, the brackets must be removed.

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Equations Containing Brackets To solve the equation containing brackets, we may proceed as follows:  Remove the brackets by using the Distributive Law.  Collect the pro-numeral terms on the left-hand side of the equation and the numerical terms on the right-hand side of the equation by doing the same thing to both sides of the equation.

Example

Solution:

Example

Solution:

Example Solve for x: 3 (x – 4) + 5 = x + 3

Solution: 3 (x – 4) + 5 = x + 3 (Use the Distributive Law and remove the brackets.) 3 x – 12 + 5 = x + 3

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Page 231 of 362 3 x – 7 = x + 3

3x – x - 7 = x – x + 3 (Group the variables on the left-hand side. Use the inverse to Remove x from the right-hand side.) 2x – 7 = 3

2x – 7 + 7 = 3 + 7 (Isolate the x-term by adding 7 to both sides.)

x = 5

Check: LHS = 3(5 – 4) + 5 = 3(1) + 5 = 8

RHS = 5 + 3 = 8

LHS = RHS x = 5 is a solution.

Activity 102: Equations with brackets

Solve the equations by removing the brackets first.

1. (x – 2) + 4 = 2x 2. 2(x + 1) = 3(x + 1) 3. x + 1 = -(x + 3) 4. 1 + (x – 1) = 2x +3 5. 4(3p + 4) – 6p = 46 6. 6(p – 1) – p = - 2(p + 1) + p – 1 7. 2x + 3(x – 2) = 4

Lesson 95-96: Worksheets

Please check your calendar for information on this lessons.

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Page 232 of 362

Lesson 97: Equations with Fractions

TO SOLVE AN EQUATION WITH fractions, we transform it into an equation without fractions -- which we know how to solve. The technique is called clearing of fractions. Example 1. Solve for x: x x − 2 + = 6. 3 5 Solution. Clear of fractions as follows: Multiply both sides of the equation -- every term -- by the LCM of denominators. Every denominator will then cancel. We will then have an equation without fractions.

The LCM of 3 and 5 is 15. Therefore, multiply every term on both sides by 15: x x − 2 15· + 15· = 15· 6 3 5 Each denominator will now cancel into 15 -- that is the point -- and we have the following simple equation that has been "cleared" of fractions:

5x + 3(x − 2) = 90.

It is easily solved as follows:

5x + 3x − 6 = 90

8x = 90 + 6

96 x = 8

= 12.

We say "multiply" both sides of the equation, yet we take advantage of the fact that the order in which we multiply or divide does not matter. Therefore we divide the LCM by each denominator first, and in that way clear of fractions.

Example 2. Clear of fractions and solve for x: x 5x 1 − = 2 6 9

Solution. The LCM of 2, 6, and 9 is 18. Multiply each term by 18 -- and cancel.

9x − 15x = 2. It should not be necessary to actually write 18. The student should simply look at x , and see that 2 will go into 18 nine (9) times. That term

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Page 233 of 362 2 therefore becomes 9x.

5x Next, look at , and see that 6 will to into 18 three (3) times. That 6 term therefore becomes 3· −5x = −15x. 1 Finally, look at , and see that 9 will to into 18 two (2) times. That 9 term therefore becomes 2 · 1 = 2. Here is the cleared equation, followed by its solution: 9x − 15x = 2

−6x = 2

2 x = −6

1 x = − 3

In the following problems, clear of fractions and solve for x: x x Problem 1. − = 3 2 5

The LCM is 10. Here is the cleared equation and its solution:

5x − 2x = 30

3x = 30

x = 10.

On solving any equation with fractions, the very next line you write -- 5x − 2x = 30 -- should have no fractions. x 1 x Problem 2. = + 6 12 8

The LCM is 24. Here is the cleared equation and its solution:

4x = 2 + 3x

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Page 234 of 362 4x − 3x = 2

x = 2

x x x − 2 + = Problem 3. 3 2 5

The LCM is 30. Here is the cleared equation and its solution:

6(x − 2) + 10x = 15x

6x − 12 + 10x = 15x

16x − 15x = 12

12. x =

x − 1 x = 4 7 Problem 4.

The LCM is 28. Here is the cleared equation and its solution:

7(x − 1) = 4x

7x − 7 = 4x

7x − 4x = 7

3x = 7

7 x = 3

We see in this Problem that when a single fraction is equal to a single fraction, then the equation can be cleared by "cross-multiplying." If a c = , b d then ad = bc.

x − 5 Problem 5. x − 3 = 2 3

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Page 235 of 362

Here is the cleared equation and its solution:

2(x − 3) = 3(x − 5)

2x − 6 = 3x – 15

2x − 3x = − 15 + 6

−x = −9

x = 9

x − 3 x + 1 = x − 1 x + 2 Problem 6.

Here is the cleared equation and its solution:

(x − 3)(x + 2) = (x − 1)(x + 1)

x² −x − 6 = x² − 1

−x = −1 + 6

−x = 5

x = −5

Activity 103: Equations with fractions

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Page 236 of 362

Lesson 98-100: Quadratic Equations

An example of a Quadratic Equation:

The name Quadratic comes from "quad" meaning square, because the variable gets squared (like x2).

It is also called an "Equation of Degree 2" (because of the "2" on the x) The Standard Form of a Quadratic Equation looks like this:

 a, b and c are known values. a can't be 0.  "x" is the variable or unknown (you don't know it yet).

Here are some more examples: In this one a=2, b=5 and c=3

This one is a little more tricky: 2  Where is a? In fact a=1, as we don't usually write "1x "  b = -3  And where is c? Well, c=0, so is not shown.

Oops! This one is not a quadratic equation, because it is missing

x2 (in other words a=0, and that means it can't be quadratic)

Hidden Quadratic Equations! So the "Standard Form" of a Quadratic Equation is ax2 + bx + c = 0

But sometimes a quadratic equation doesn't look like that! For example:

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Page 237 of 362 In disguise → In Standard Form a, b and c x2 = 3x -1 Move all terms to left hand side x2 - 3x + 1 = 0 a=1, b=-3, c=1 Expand (undo the brackets), 2(w2 - 2w) = 5 2w2 - 4w - 5 = 0 a=2, b=-4, c=-5 and move 5 to left z(z-1) = 3 Expand, and move 3 to left z2 - z - 3 = 0 a=1, b=-1, c=-3 5 + 1/x - 1/x2 = 0 Multiply by x2 5x2 + x - 1 = 0 a=5, b=1, c=-1

Have a Play With It I have a "Quadratic Equation Explorer" so you can see:  the graph it makes, and  the solutions (called "roots").

How To Solve It? The "solutions" to the Quadratic Equation are where it is equal to zero. There are usually 2 solutions (as shown in the graph above).

They are also called "roots", or sometimes "zeros"

There are 3 ways to find the solutions: 1. You can Factor the Quadratic (find what to multiply to make the Quadratic Equation) 2. You can Complete the Square, or 3. You can use the special Quadratic Formula:

Just plug in the values of a, b and c, and do the calculations.

We will look at this method in more detail now.

About the Quadratic Formula Plus/Minus First of all what is that plus/minus thing that looks like ± ?

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Page 238 of 362 The ± means there are TWO answers:

Here you see why you can get two answers:

But sometimes you don't get two real answers, and the "Discriminant" shows why ...

Discriminant Do you see b2 - 4ac in the formula above? It is called the Discriminant, because it can "discriminate" between the possible types of answer:  when b2 - 4ac is positive, you get two Real solutions  when it is zero you get just ONE real solution (both answers are the same)  when it is negative you get two Complex solutions

I will explain about Complex solutions after you have seen how to use the formula.

Using the Quadratic Formula Just put the values of a, b and c into the Quadratic Formula, and do the calculations. Example: Solve 5x² + 6x + 1 = 0 Coefficients are: a = 5, b = 6, c = 1

Quadratic Formula: x = [ -b ± √(b2-4ac) ] / 2a

Put in a, b and c: x = [ -6 ± √(62-4×5×1) ] / (2×5)

Solve: x = [ -6 ± √(36-20) ]/10 x = [ -6 ± √(16) ]/10

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Page 239 of 362 x = ( -6 ± 4 )/10 x = -0.2 or -1

Answer: x = -0.2 or x = -1

And you can see them on this graph.

Check -0.2: 5×(-0.2)² + 6×(-0.2) + 1 = 5×(0.04) + 6×(-0.2) + 1 = 0.2 -1.2 + 1 = 0 Check -1: 5×(-1)² + 6×(-1) + 1 = 5×(1) + 6×(-1) + 1 = 5 - 6 + 1 = 0

Remembering The Formula I don't know of an easy way to remember the Quadratic Formula, but a kind reader suggested singing it to "Pop Goes the Weasel": "x equals minus b "All around the mulberry bush ♫ ♫ plus or minus the square root The monkey chased the weasel of b-squared minus four a c The monkey thought 'twas all in fun all over two a" Pop! goes the weasel" Try singing it a few times and it will get stuck in your head!

Complex Solutions? When the Discriminant (the value b2 - 4ac) is negative you get Complex solutions ... what does that mean? It means your answer will include Imaginary Numbers. Wow!

Example: Solve 5x² + 2x + 1 = 0 Coefficients are: a = 5, b = 2, c = 1

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Note that The Discriminant is negative: b2 - 4ac = 22 - 4×5×1 = -16

Use the Quadratic Formula: x = [ -2 ± √(-16) ] / 10

The square root of -16 is 4i (i is √-1, read Imaginary Numbers to find out more)

So: x = ( -2 ± 4i )/10

Answer: x = -0.2 ± 0.4i

The graph does not cross the x-axis. That is why we ended up with complex numbers.

In some ways it is actually easier ... you don't have to calculate the solutions, just leave it as -0.2 ± 0.4i.

Summary  Quadratic Equation in Standard Form: ax2 + bx + c = 0  Quadratic Equations can be factored  Quadratic Formula: x = [ -b ± √(b2-4ac) ] / 2a  When the Discriminant (b2-4ac) is: o positive, there are 2 real solutions o zero, there is one real solution o negative, there are 2 complex solutions

Activity 104: Quadratic Equations

Solve the quadratic equation 6x2 + 7x - 3 = 0

A x = 1/3 or -1.5 B x = -1/3 or 1.5 C x = -1/6 or 3 D x = 1/6 or -3

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Activity 105: Quadratic Equations

Solve the equation 10x – 1/x = 3

A 1/2 B 35 C 0.4 D 1/4

Activity 106: Quadratic Equations

Solve the quadratic equation 4x2 + 3x - 27 = 0

A B

C D

Activity 107: Quadratic Equations

Solve the quadratic equation 15x2 - 26x - 21 = 0

A B

C D

Activity 108: Quadratic Equations

Solve the quadratic equation 21x2 - 12x + 1 = 0

A 0.074 or 0.645 B 0.101 or 0.470 C -0.645 or 0.074 D -0.470 or 0.101

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Page 242 of 362 Lesson 101: Solving Problems with Algebraic Models

- Until now, you have learnt various methods of solving equations. - In this section, you will be given problems to solve using that knowledge. - The first step will be to change the description of the problem that is given in words into a diagram, expression or equation involving a variable. - This algebraic equation is an example of a mathematical model. - The equation then has to be solved. - Lastly, the mathematical solution of the model has to be checked back against the original context as described in words to see if it makes sense. - The modelling process can be represented by the diagram:

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Activity 109: Algebraic equation

For each question, form an algebraic equation and then solve it to find the answer. Remember to translate your answers back into the context and to check them.

1. A mother is three times as old as her daughter. Six years ago, the mother's age was six times that of her daughter. How old are they now? 2. A coin collection amounting to $25 consists of nickels and dimes. There are 3 times as many nickels and dimes. There are 3 times as many nickels as dimes. How many coins of each kind are there? 3. Taylor is five times as old as Spenser. The sum of their ages is eighteen. How old are Taylor and Spencer?

Lesson 102: Solving Equations by Trial and Improvement

Trial and Improvement is a method of solving equations when you can't do it by normal algebraic methods. It's normally a 3 mark question so understanding it is vital if your aiming for a grade C in maths! You will need a calculator to answer these questions.

Example Question a) Show that the equation x3 - 6x + 1 = 0 has a solution between 2 and 3. b) Solve this equation correct to 1 decimal place.

Solution a) To show that x3 - 6x + 1 = 0 has a solution between 2 and 3 we need to substitute x = 2 and x = 3 into the equation.

When x = 2 we get: (2)3 - 6(2) + 1 = 8 - 12 + 1 = -3. When x = 3 we get: (3)3 - 6(3) + 1 = 27 - 6(3) + 1 = 10.

Now notice when x = 2 we get an answer less than 0 and when x = 3 we get an answer greater than 0. This means there must be a value of x between 2 and 3 which is equal to 0. b) We now need to find this solution correct to 1 decimal place. To do this we need to draw a table to test different values of x. Since we know the answer is between 2 and 3 it makes sense to start with x = 2.5.

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Page 245 of 362 x x3 - 6x + 1 Comment

2 -3 Too small 3 10 Too big 2.5 1.625 Too big 2.3 -0.633 Too small 2.4 0.424 Too big

Now since 2.3 was too small and 2.4 was too big we know the solution is between these values. To find our answer to 1 decimal place we have to try one more value in the middle of these, when x = 2.35.

When x = 2.35 we get: (2.35)3 - 6(2.35) + 1 = -0.122125. Finally since 2.35 is too small then we can say 2.3 must also be too small. Hence x = 2.4 to 1 decimal place.

Activity 110: Solving equations by trail and improvement

1. The equation x3 + x = 15 has a solution between 2 and 3. Find this solution correct to one decimal place. x =

2. The equation x3 - 2x + 6 = 0 has a solution between -3 and -2. Find this solution correct to one decimal place. x =

Lesson 103: Solid Geometry

Solid Geometry

Solid Geometry is the geometry of three-dimensional space, the kind of space we live in ...

Three Dimensions

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It is called three-dimensional, or 3D because there are three dimensions: width, depth and height.

Simple Shapes Let us start with some of the simplest shapes:

 Cube  Cuboid  Volume of a Cuboid

Properties Solids have properties (special things about them), such as:

 volume (think of how much water it could hold)  surface area (think of the area you would have to paint)

Lesson 104: Polyhedra and Non Polyhedra

There are two main types of solids, "Polyhedra", and "Non-Polyhedra":

Polyhedra :

(they must have flat Platonic Solids faces)

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Page 247 of 362

Prisms

Pyramids

Non-Polyhedra:

(if any surface is not Sphere Torus

flat)

Cylinder Cone

Lesson 105: Volume of a Pyramid

Look carefully at the pyramid shown below. The volume of a pyramid can be computed as shown Pyramid:

Volume = (B × h)/3 B is the area of the base

H is the height

The base of the pyramid can be a rectangle, a triangle, or a square. Compute the area of the base accordingly

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Page 248 of 362

Volume of a square pyramid

Example #1:

A square pyramid has a height of 9 meters. If a side of the base measures 4 meters, what is the volume of the pyramid?

Since the base is a square, area of the base = 4 × 4 = 16 m2

Volume of the pyramid = (B × h)/3 = (16 × 9)/3 = 144/3 = 48 m3

Volume of a rectangular pyramid

Example #2:

A rectangular pyramid has a height of 10 meters. If the sides of the base measure 3 meters and 5 meters, what is the volume of the pyramid?

Since the base is a rectangle, area of the base = 3 × 5 = 15 m2

Volume of the pyramid = (B × h)/3 = (15 × 10)/3 = 150/3 = 50 m3

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Page 249 of 362 Volume of a triangular pyramid

Example: #3

A triangular pyramid has a height of 8 meters. If the triangle has a base of 4 meters and a height of 3 meters, what is the volume of the pyramid?

Notice that here, you are dealing with two different heights. Avoid mixing the height of the pyramid with the height of the triangle

Since the base is a triangle, area of the base = (b × h)/2 = (4 × 3)/2 = 12/2 = 6 m2

Volume of the pyramid = (B × h)/3 = (6 × 8)/3 = 48/3 = 16 m3

Activity 111: Volume of a pyramid

1. The base of a right pyramid 10 √7 feet high is a triangle whose sides are 9 feet, 11feet and 16 feet. Find the volume of the pyramid. 2. The base of right pyramid is a triangle whose sides are 28 cm, 25cm and 17 cm. If the volume of the pyramid be 11120 cubic cm, find its height. 3. The base of a right pyramid is a square of 40 cm and its slant height is 25 cm. If the value of cube is equal to the volume of the pyramid find the length of a side of the cube. 4. The base of a right pyramid is a square of side 40 cm and length of an edge through the vertex is 5√41 cm. If the volume of a cube is equal to the volume of the pyramid, then find length of the side of the cube.

Lesson 107: Volume of a Cone

The volume of a cone is 1/3(Area of Base)(height) = 1/3 π r2 h

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Page 250 of 362 Given the radius and h, the volume of a cone can be found by using the formula:

Formula: Vcone = 1/3 × b × h

b is the area of the base of the cone. Since the base is a circle, area of the base = pi × r2

2 Thus, the formula is Vcone = 1/3 × pi × r × h

Use pi = 3.14

Example #1:

Calculate the volume if r = 2 cm and h = 3 cm

2 Vcone = 1/3 × 3.14 × 2 × 3

Vcone = 1/3 × 3.14 × 4 × 3

Vcone = 1/3 × 3.14 × 12

Vcone = 1/3 × 37.68

Vcone = 1/3 × 37.68/1

Vcone = (1 × 37.68)/(3 × 1)

Vcone = 37.68/3

3 Vcone = 12.56 cm

Example #2:

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Page 251 of 362 Calculate the volume if r = 4 cm and h = 2 cm

2 Vcone = 1/3 × 3.14 × 4 × 2

Vcone = 1/3 × 3.14 × 16 × 2

Vcone = 1/3 × 3.14 × 32

Vcone = 1/3 × 100.48

Vcone = 1/3 × 100.48/1

Vcone = (1 × 100.48)/(3 × 1)

Vcone = 100.48/3

3 Vcone = 33.49 cm

Activity 112: Volume of a cone

1. Find the volume of a cone of radius 3 cm and height 6 cm ? 2. Find the volume of a cone of radius 10 cm and height 15 cm ?

Lesson 108: Combing Cubes

What is a Soma Cube?

The seven pieces of cubes are named Soma cubes. Altogether you have 1x3 + 6x4 = 27 separate cube lets.

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Page 252 of 362 Main Problem

The main problem of the Soma "research" is to assemble the seven Soma pieces or 27 cubes let’s to make a 3x3x3 cube. The chance of solving this three-dimensional puzzle is good, because there are 240 possibilities to put the cube together, not counting symmetries. If you try to solve the puzzle for the first time, you need approximately 15 min. You have a better chance if you start with the three-dimensional pieces 5,6,7......

Three solutions 1 2 3 Solution 1 was my first solution, which I kept in my mind. Solution 2 is easy to remember: You start with the three 3D pieces 5,6, and ... 7. Piece 4 follows...... Solution 3 is one of the few solutions without piece 7 forming a corner. All 240 solutions

Positions of the Soma Pieces 3 and 2 Piece 2 and 3 contain a 1x1x3 bar. The shapes of the Soma cubes result in the following statements. Piece 2 forms 0,1 or 2 corners. Piece 3 forms either 0 or 2 corners.

Now there are two possibilities: (1) Piece 3 forms no corners. The other six pieces form at most 2+5 corners. This case is not possible, because you don't have eight corners. (2) Piece 3 forms two corners. The other 6 pieces form 6 corners. Therefore piece 2 must form at least one corner.

Results:

...... Piece 3 forms two corners. Soma piece 2 must always form a corner and must not lie in the middle. This can be the first steps to prove that there are 240 solution.

Figures of Soma Cubes

It is interesting to look for new 27-cubelet-shapes. This is one example, a "car".

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More well-known figures are sofa, bed, bathtub, gate, gravestone, and tower.

There are innumerable figures of Soma cubes. This is proved by the following thoughts about designing new ones. All Soma cubes have 27 cube lets. If you give a mat of 5x4 = 20 cube lets, there are [20 ... above 7] = 20!/13!/7! = 77520 places for the remaining seven cube lets, so 77520 figures. ... There is an example on the left. If you form a bar with five cube lets, there are two left. So you get [15 above 2] =15!/13!/2! = 105 figures. There is an example on the left. If you demand ... symmetry, you only get 18 cube figures. You can see them as top view on the ...... right and in perspective below. ...

Two figures are insoluble.

Magnification Problem Can you imitate a soma cube with magnification? It is a productive problem with pentominos and tetra cubes, but not with Soma cubes.

Only the cube with three cube lets is possible. You need the rest of the Soma ... cubes to build it with double magnification. ...

Making of Soma Cubes If you want to play with Soma cubes you have to construct them by hand. The simplest way is to buy a length of wood which is square in cross-section, cut it into cubes and glue the cubes together.

Another method is gluing dice. The best idea is to use a two component glue, because it needs time to harden. Then you can form the Soma pieces without having to hurry. A cheap but difficult method is constructing them from a sheet of paper. You have to design the base of every Soma cube and fold it.

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Loops cube (27=4+4+4+5+5+5), designed and made by René Dawir

Order in the solution: 564321 ...

Activity 113: Combining cubes

Please answer the following questions.

Use the 1 cm x 1cm x 1 cm cubes. Cubes must be placed squarely on each other and not like this picture below.

1. How many different shapes can be made from four identical cubes? 2. How many different shapes can be made from four identical cubes?

Lesson 109-110: Ratios

A ratio is a mathematical way of comparing two quantities.

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Page 255 of 362 Some everyday uses of ratio are:  “Six out of every eight students in my class watched the Word Series”,  “I drove 55 miles per hour on the expressway, and people were speeding by me”,  “The recipe for cookies contained one cup of sugar for every two cups of flour”

If the ratio compares two quantities with different units that cannot be converted to a common unit, it is called a rate.

For Example:  55 miles per hour is a rate.

Whether you call the comparison a ratio or rate, it can be written as a fraction, as a comparison using the word to, or as a comparison using a colon.

You can set up a ratio whenever you are comparing two numbers.

A ratio shows the relative sizes of two or more values.

There are 3 blue squares to 1 yellow square

Ratios can be shown in different ways:

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Page 256 of 362 Example:

There is 1 boy and 3 girls, so you can write the ratio as:

1:3 (for every one boy there are 3 girls)

1/4 are boys and 3/4 are girls

0.25 are boys (by dividing 1 by 4)

25% are boys (0.25 as a percentage)

Using Ratios The trick with ratios is to always multiply the numbers in the ratio by the same value.

Example: 4 : 5 is the same as 4×2 : 5×2 = 8 : 10

Scaling

 The ratio of the Indian Flag is 2:3, that means that for every 2 (inches, centimeters, whatever) of height there should be 3 of width.

 If you made this flag 20 cm high, it should be 30 cm wide.

 If you made this flag 40 inches high, it should be 60 inches wide (which is still in the ratio 2:3)

If you want to draw a horse at 1/10th the normal size, you need to multiply all sizes by 1/10th.

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Example: This horse in real life is 1500mm high and 2000 mm long, so the ratio of its height to length is 1500 : 2000

What is that ratio when you draw it?

Answer: 1500 : 2000 = 1500×1/10 : 2000×1/10 = 150 : 200

You can pick any reduction/enlargement you want that way.

Big Foot?

Allie measured her foot and it was 21cm long, and then she measured her Mother’s foot, and it was 24cm long.

“I must have big feet my foot is nearly as long as my Mom’s!”

But then she thought to measure how tall she and her Mom were, and found she was 134cm tall, and her Mom was 153cm.

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In a table this was:

To work out the Ratio she simply divided her Mom's measurements by hers:

"Oh!" she said, "the Ratios are nearly the same" "So my foot is only as big as it should be for my height, and is not really too big."

A "Concrete" Example Concrete is made by mixing cement, sand, stones and water.

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Page 259 of 362 A typical mix of cement, sand and stones is written as a ratio, such as 1:2:6. You can multiply all values by the same amount and you will still have the same ratio. 10:20:60 is the same as 1:2:6 So if you used 10 buckets of cement, you should use 20 of sand and 60 of stones.

Example: If you have just put 12 buckets of stones into a wheelbarrow, how much cement and how much sand should you add to make a 1:2:6 mix?

Let us lay it out in a table to make it clearer:

You can see that you have 12 buckets of stones but the ratio says 6.

That is OK, you simply have twice as many stones as the number in the ratio ... so you need twice as much of everything to keep the ratio.

Here is the solution:

And the ratio 2:4:12 is the same as 1:2:6 (because they show the same relative sizes)

Why are they the same ratio? In the 1:2:6 ratio there is 3 times more Stones as Sand (6 vs 2), and in the 2:4:12 ratio there is also 3 times more Stones as Sand (12 vs 4) ... similarly there is twice as much Sand as Cement in both ratios.

That is the good thing about ratios. You can make the amounts bigger or smaller and so long as the relative sizes are the same then the ratio is the same.

So the answer is: add 2 buckets of Cement and 4 buckets of Sand. (You will also need water and a lot of stirring....)

Activity 114: Ratio

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Page 260 of 362 In the following diagram

What is the ratio of orange squares to white squares?

 11:5  5:11  5:16  16:5

Lesson 111: Rate

A rate is used to compare quantities that have different units, for example, kilometers and hours. A rat tells us how much there is of one quantity for every unit of another quantity. Speed is a rate. For example, it can show how many kilometers were travelled in every one hour.

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Activity 115: Rates

Use a calculator where necessary and write your answers correct to tow decimal places.

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Page 262 of 362 Lesson 112-113: Worksheets

Please check your calendar for information on this lessons.

Lesson 114-115: Proportion

Numbers are said to increase or decrease in proportion if they have a constant quotient or a constant product.

Two quantities in direct proportion have a constant quotient.

Number of hours 1 2 4 6 7

Total cost of hire R25 R50 R100 R150 R175

In the table above, when any amount in the top row is divided by the corresponding amount in the bottom row, the quotient obtained is a constant. In this case it is When any amount in the bottom row is divided by the corresponding amount in the top row, the quotient obtained is also a constant. Now it is 25.

Two quantities in inverse proportion have a constant product.

When an amount in the top row is multiplied by the corresponding amount in the bottom row, there is a constant product of 2 500.

 Quantities that are in direct proportion have a constant quotient. Quantities that are in inverse proportion have a constant product.  If two quantities do not have a constant product or quatient, then they are not in proportion.

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Page 263 of 362 A proportion is a statement that two ratios (fractions) are equal. The statement 7/14 = 14/16 is an example of a proportion. In other words, it means “7 is to 8 as 14 is to 16.

To Solve a Proportion  Set up a proportion, making sure that the fraction on the left side of the equal sign is set up in the same order as the fraction on the right side.  Set up the method of solution by setting the missing value equal to the cross product of the diagonal of the two given numbers divided by the remaining number.  Solve by multiplying the cross product and then dividing.

In a proportion you find that when you multiply the diagonal numbers, the results are equal. This fact can help you solve many math problems involving proportional relationships. Cross multiplication is the multiplication of the numerator of the first ratio by the denominator of the second ratio and the multiplication of the denominator of the first ratio by the numerator of the second ratio.

Example:1

7 14 ----- = ----- 8 16

7 × 16 = 112 and 8 × 14 = 112 show that the cross products are equal. Notice that if the product 112 is divided by one factor, the result is the other factor.

For instance, 112 ÷ 7 = 16 because 7 × 16 = 112.

Example:2 21 3 --- = --- 70 10

21 * 10 = 70 * 3 210 = 210

If one term of a proportion is not known, cross multiplication can be used to find the value of that term. x 3 -- = -- 70 10

x * 10 = 70 * 3

10x = 210

10x 210

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Page 264 of 362 --- = --- 10 10 x = 21

When solving a problem that has a ratio, proportions are often a lot easier to work with. There are a few ways to solve these problems.

Method 1: Cross multiplication. This is where you’ll multiply two proportions in a criss-cross to help solve the equation. Check out the examples below.

Method 2: Multiply the first proportion in order to make equivalent fractions.

The really important part for both methods: compare apples to apples and oranges to oranges. If you’re comparing girls to boys on one side of the equal sign, you must also compare girls to boys on the other side (not boys to girls). The units for both numerators must match and the units for both denominators must match.

Check out this baffling baking situation: Your great-grandma’s chocolate bourbon ball cookie recipe calls for 3 cups of flour, so you will have to make a small batch. You need to match the proportion of flour to brown sugar in the recipe (great- grandma wouldn’t have it any other way).

How much brown sugar do you need?  The recipe calls for a ratio of 3 cups flour to 6 tablespoons brown sugar.

 That is a ratio of , or . (It’s ok to set up ratios where the measurements are different, like cups and tablespoons.)  You only have 2 cups of flour, but the ratio of flour to brown sugar needs to stay constant.  The unknown quantity, the amount of brown sugar we need, will be represented with the

variable, x. So .  As you can see the units of the numerators match and the units of the denominators match, with cups of flour on top and tablespoons of brown sugar on the bottom for both.

We can solve this two different ways:

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Directly Proportional and Inversely Proportional

Directly proportional: as one amount increases, another amount increases at the same rate.

The symbol for "directly proportional" is ∝ ∝ (Don't confuse it with the symbol for infinity ∞)

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Page 266 of 362 Example: you are paid $20 an hour How much you earn is directly proportional to how many hours you work Work more hours, get more pay; in direct proportion. This could be written: Earnings ∝ Hours worked  If you work 2 hours you get paid $40  If you work 3 hours you get paid $60  etc ...

Constant of Proportionality The "constant of proportionality" is the value that relates the two amounts

Example: you are paid $20 an hour (continued) The constant of proportionality is 20 because:

Earnings = 20 × Hours worked This can be written: y = kx Where k is the constant of proportionality

Example: y is directly proportional to x, and when x=3 then y=15.

What is the constant of proportionality? They are directly proportional, so: y = kx Put in what we know (y=15 and x=3): 15 = k × 3 Solve (by dividing both sides by 3): 15/3 = k × 3/3 5 = k × 1 k = 5 The constant of proportionality is 5: y = 5x When you know the constant of proportionality you can then answer other questions

Example: (continued) What is the value of y when x = 9? y = 5 × 9 = 45

What is the value of x when y = 2? 2 = 5x x = 2/5 = 0.4

Inversely Proportional Inversely Proportional: when one value decreases at the same rate that the other increases.

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Example: speed and travel time Speed and travel time are Inversely Proportional because the faster you go the shorter the time.  As speed goes up, travel time goes down  And as speed goes down, travel time goes up This: y is inversely proportional to x

Is the same thing as: y is directly proportional to 1/x

k Which can be written: y = x

Example: Four people can paint a fence in 3 hours. How long will it take six people to paint it? (Assume everyone works at the same rate) It is an Inverse Proportion:  As the number of people goes up, the painting time goes down.  As the number of people goes down, the painting time goes up. We can use: t = k/n Where:  t = number of hours  k = constant of proportionality  n = number of people

We know that t = 3 when n = 4 3 = k/4 3 × 4 = k × 4 / 4 12 = k k = 12

So now we know: t = 12/n

And when n = 6: t = 12/6 = 2 hours

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So 6 people will take 2 hours to paint the fence.

How many people are needed to complete the job in half an hour? ½ = 12/n n = 12 / ½ = 24 So it needs 24 people to complete the job in half an hour. (Assuming they don't all get in each other's way!)

Proportional to ... It is also possible to be proportional to a square, a cube, an exponential, or other function! Example: Proportional to x2

A stone is dropped from the top of a high tower. The distance it falls is proportional to the square of the time of fall. The stone falls 19.6 m after 2 seconds, how far does it fall after 3 seconds?

We can use: d = kt2 Where:  d is the distance fallen and  t is the time of fall

When d = 19.6 then t = 2 19.6 = k × 22 19.6 = 4k k = 4.9 So now we know: d = 4.9t2 And when t = 3: d = 4.9 × 32 d = 44.1 So it has fallen 44.1 m after 3 seconds.

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Page 269 of 362

Activity 116: Proportion

1. It takes 4 men 6 hours to repair a road. How long will it take 8 men to do the job if they work at the same rate?

Lesson 116: Gradient

Gradient (Slope) of a Straight Line The Gradient (also called Slope) of a straight line shows how steep a straight line is.

The rate of change is the amount by which the output variable (y) changes when the input variable (x) is increased by 1.

The gradient of a line actually represents the rate of change.

Calculate The method to calculate the Gradient is: Divide the change in height by the change in horizontal distance

Change in Y Gradient = Change in X

Examples:

3 The Gradient of this line = = 1 3

So the Gradient is equal to 1

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4 Gradient = = 2 2

(The line is steeper, and so the Gradient is larger)

3 Gradient = = 0.6 5

(The line is less steep, and so the Gradient is smaller)

Positive or Negative? Important:  Starting from the left end of the line and going across to the right is positive (but going across to the left is negative).  Up is positive, and down is negative

-4 Gradient = = –2 2

That line goes down as you move along, so it has a negative Gradient.

Straight Across

Gradient = 0 = 0

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Page 271 of 362 5

A line that goes straight across (Horizontal) has a Gradient of zero.

Straight Up and Down

3 Gradient = = undefined 0

That last one is a bit tricky ... you can't divide by zero, so a "straight up and down" (Vertical) line's Gradient is "undefined".

Rise and Run Sometimes the horizontal change is called "run", and the vertical change is called "rise" or "fall":

They are just different words, none of the calculations change

Activity 117: Gradient

Question 1 What is the gradient (slope) of this line:

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A -2 B C D 2

Question 2 What is the gradient (slope) of this line:

A B C D Question 3 What is the gradient (slope) of this line:

A B C D

Question 4 What is the gradient (slope) of this line:

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A -3 B C D 3

Question 5 What is the gradient (slope) of this line:

A -4 B C 4 D undefined

Lesson 117-119: Worksheets

Please check your calendar for information on this lessons.

Lesson 120-121: Volume and Capacity

Volume The volume of a solid is the amount of space it occupies or we can say volume is the amount of space occupied by an object.

The unit for measuring small volumes is the cubic centimetre (cm³). This is the amount of space occupied by a cube with edges of 1 cm.

The unit for measuring larger volumes is the cubic metre (m³).

The general, the volume of any right prism or cylinder can be found using this formula:

Volume = area of base x height

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Capacity Capacity is the amount of liquid a container can hold when it is full.

Cube

A cube is a solid with 6 square faces and 12 equal edges. Volume of cube = 4 x 4 x 4 = 64 cm3

Cuboid

A cuboid is a solid with 6 faces which are all rectangles.

Volume of cuboid = Length x Breadth x Height = 12 x 8 x 5

= 480 cm3

Length of cuboid = Volume Breadth x Height = 480 8 x 5

= 12cm

Breadth of cuboid = Volume Length x Height

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Page 275 of 362 = 480 12 x 5

= 8 cm

Height of cuboid = Volume Length x Breadth = 480_ 12 x 8

= 5 cm

Example 1 A rectangular tank 35 cm by 30 cm by 20 cm contains water to a height of 15 cm. Find the volume of water in the tank. (Give your answer in litres)

Volume of water in the tank = 35 x 30 x 15 = 15 750 cm3 = 15.75 litres Example 2 A container has a square base of 8 cm. What is the height of the box if its volume is 384 cm3?

Area of square base = 8 x 8

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Page 276 of 362 = 64 cm2

Height of container = Volume Base area = 384 64 = 6 cm

- Volume is measured in cubic milimetres (mm³), cubic centimetres (cm³) and cubic metres (m³). - Capacity is the interior volume of a container and is measured in units of millilitres (ml) and liters (L).

Here are the conversions you will need to know:

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Activity 118: Volume and capacity

Convert the following into liters and milliliters:

1. 2000 ml 2. 7000 ml 3. 5300 ml 4. 6450 ml 5. 9325 ml 6. 21238 ml 7. 150595 ml 8. 6921 ml 9. 2525 ml

The volume of a cube of side length 2 m is: A) 8 m³ B) 6 m3 C) 8 m

Lesson 122: Calculating the Height and Radius of a Cylinder given the Volume

When working with litres and centimetres in the same equation, first convert the litres to cubic centimetres before inserting values into the formula.

Example The cylinder in the figure below has a radius of 50 cm and a volume of 800 L. Calculate the height of this cylinder correct to the nearest whole number.

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Solution

Example The cylinder given below has a volume of 25 000 L and a height of 500 cm. Determine the radius of the cylinder correct to the nearest whole number.

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Page 279 of 362 Solution

A good rule of thumb to remember is: - When converting from a larger unit to a smaller unit, multiply by the appropriate factor. There will be more of the smaller unit than the original larger unit. - When converting from a smaller unit to a larger unit, divide by the appropriate factor. There will be less of the larger unit than the original smaller unit.

Example The following triangular right prism is given.

The triangular face has sides of 10m, 10 m and 12m. the length is 20 m. Determine the volume of this container.

Solution The triangular face from the base of the prism. Therefore we are going to need the area of this face before determining the colume.

In any isosceles triangle, the height from the intersection of the equal sides bisects the base.

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Example Deon wishes to collect the rainwater from the roof of his house for use during dry periods. His local hardware store has two tanks from which to choose. The one has a circular base of diameter 800 mm and height 1 600mm. The other has a square base of side 800 mm and height 1 300 mm.

a) Which container can store the most water?

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Page 281 of 362 b) How many litres will each container hold? Give your answer correct to the nearest whole number. Solution Since question b) asks for the number of litres, it may be wise to work in centimetres.

To convert mm to cm, we are going from a smaller unit to a larger unit. So we divide by 10.

Activity 119: Calculate the volume

1. Calculate the volume of each of the solids below correct to one decimal place. The measurements are all given in centimetres.

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2. A typical soft-drink can has a raduis of 33 mm and a height of 110 mm.

Tinned fish comes in a can of diameter 100 mm and height 40 mm.

Give your answers correct to the nearest ml.

a. Calculate the volume of the soft drink can. b. Calculate the volume of the can of tinned fish. c. Would the contents of the soft-drink can fit into the fish tin? d. Can you give a good reason why cooldrinks are not solid in cans that have the same shape as the fish tins?

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Lesson 123: Volumes of Prisms

Prisms A prism has flat sides and the same cross section all along its length!

A cross section is the shape you get when cutting straight across an object.

The cross section of this object is a triangle... .. it has the same cross section all along its length ...... and so it's a triangular prism.

Try drawing a shape on a piece of paper (using straight lines!), Then imagine it extending up from the sheet of paper, - that's a prism !

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No Curves! A prism is a polyhedron, which means the cross section will be a polygon (a straight-edged figure) ... so all sides will be flat!

No curved sides.

For example, a cylinder is not a prism, because it has curved sides.

These are all Prisms:

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Regular and Irregular Prisms All the previous examples are Regular Prisms; because the cross section is regular (in other words it is a shape with equal edge lengths, and equal angles.)

Here is an example of an Irregular Prism:

Volume of a Prism The Volume of a prism is simply the area of one end times the length of the prism

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Volume = Area × Length

Example: What is the volume of a prism whose ends have an area of 25 in2 and which is 12 in long?

Answer: Volume = 25 in2 × 12 in = 300 in3

(Note: we have an Area Calculation Tool)

Other Things to Know

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Page 287 of 362 When calculating the volume of irregular prisms, two approaches can be used: 1. You can add the volumes of the prisms that were used to build the object For example:

2. Or you can imagine the regular prism that was cut away to form the irregular prism.

Here is another example:

Activity 120: Volumes prisms

Calculate the volume of each prism

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Lesson 124: Worksheet

Please check your calendar for information on this lesson.

Lesson 125-128: Calculating the Surface of a Cylinder and a Right Prism

A is a polyhedron with two congruent faces, called that lie in parallel planes. The other faces, called are parallelograms formed by connecting the corresponding vertices of the bases. The segments connecting these vertices are lateral edges.

The altitude or height of a prism is the perpendicular distance between its bases. In each lateral edge is perpendicular to both bases Prisms that have lateral edges that are not perpendicular to the bases are oblique prisms. The length of the oblique lateral edges is the slant height of the prism.

Prisms are classified by the shapes of their bases. For example, the figures above show one rectangular prism and one triangular prism. The surface area of a polyhedron is the sum of the areas of its faces. The lateral area of a polyhedron is the sum of the area of its lateral faces.

Example Find the surface of a right rectangular prism with a height of 8 inches, a length of 3 inches, and a width of 5 inches.

Solution Begin by sketching the prism, as shown below. The prism has 6 faces, two of each of the following:

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Right prisms and cylinders

Definition 1: Right prism A right prism is a geometric solid that has a polygon as its base and vertical sides perpendicular to the base. The base and top surface are the same shape and size. It is called a “right” prism because the angles between the base and sides are right angles.

A triangular prism has a triangle as its base, a rectangular prism has a rectangle as its base, and a cube is a rectangular prism with all its sides of equal length. A cylinder is another type of right prism which has a circle as its base. Examples of right prisms are given below: a rectangular prism, a cube, a triangular prism and a cylinder.

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Figure 1

Surface area of prisms and cylinders

Definition 2: Surface area Surface area is the total area of the exposed or outer surfaces of a prism.

This is easier to understand if we imagine the prism to be a cardboard box that we can unfold. A solid that is unfolded like this is called a net. When a prism is unfolded into a net, we can clearly see each of its faces. In order to calculate the surface area of the prism, we can then simply calculate the area of each face, and add them all together.

For example, when a triangular prism is unfolded into a net, we can see that it has two faces that are triangles and three faces that are rectangles. To calculate the surface area of the prism, we find the area of each triangle and each rectangle, and add them together.

In the case of a cylinder the top and bottom faces are circles and the curved surface flattens into a rectangle with a length that is equal to the circumference of the circular base. To calculate the surface area we therefore find the area of the two circles and the rectangle and add them together.

Below are examples of right prisms that have been unfolded into nets:

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A rectangular prism unfolded into a net is made up of six rectangles.

A cube unfolded into a net is made up of six identical squares.

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Page 292 of 362

A triangular prism unfolded into a net is made up of two triangles and three rectangles. The sum of the lengths of the rectangles is equal to the perimeter of the triangles.

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Page 293 of 362 A cylinder unfolded into a net is made up of two identical circles and a rectangle with a length equal to the circumference of the circles.

Example 1: Finding the surface area of a rectangular prism

Question Find the surface area of the following rectangular prism:

Answer Sketch and label the net of the prism

Find the areas of the different shapes in the net

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Page 294 of 362 Find the sum of the areas of the faces cm .

Write the final answer The surface area of the rectangular prism is 160 cm2.

Example 2: Finding the surface area of a triangular prism Question Find the surface area of the following triangular prism:

Answer Sketch and label the net of the prism

Find the area of the different shapes in the net To find the area of the rectangle, we need to calculate its length, which is equal to the perimeter of the triangles.

To find the perimeter of the triangle, we have to first find the length of its sides using the theorem of Pythagoras:

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Find the sum of the areas of the faces

Write the final answer The surface area of the triangular prism is 240 cm2.

Example 3: Finding the surface area of a cylindrical prism Question Find the surface area of the following cylinder (correct to 1 decimal place):

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Page 296 of 362 Answer

Sketch and label the net of the prism

Find the area of the different shapes in the net

Write the final answer The surface area of the cylinder is 2513,28 cm2.

Volume of prisms and cylinders Definition 3: Volume Volume is the three dimensional space occupied by an object, or the contents of an object. It is measured in cubic units.

The volume of a right prism is simply calculated by multiplying the area of the base of a solid by the height of the solid.

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Page 297 of 362 Calculating volume

Rectangular prism

(8)

Triangular prism

(9)

Cylinder

(10)

Table 1

Example 4: Finding the volume of a cube

Question Find the volume of the following cube:

Answer Find the area of the base

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Page 298 of 362 ( Multiply the area of the base by the height of the solid to find the volume

( Write the final answer The volume of the cube is .

Example 5: Finding the volume of a triangular prism Question Find the volume of the triangular prism:

Answer Find the area of the base

( Multiply the area of the base by the height of the solid to find the volume

(

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Page 299 of 362

Write the final answer The volume of the triangular prism is .

Example 6: Finding the volume of a cylindrical prism

Question Find the volume of the following cylinder (correct to 1 decimal place):

Answer Find the area of the base

Multiply the area of the base by the height of the solid to find the volume

Write the final answer The volume of the cylinder is .

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Page 300 of 362

Activity 121: Calculating the surface area of a cylinder and a right prism

Problem 1: Calculate the surface area of the following prisms:

Problem 2: If a litre of paint covers an area of 2 m2, how much paint does a painter need to cover: 1. a rectangular swimming pool with dimensions 4 m m m (the inside walls and floor only);

2. the inside walls and floor of a circular reservoir with diameter 4 m and height 2,5 m.

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Page 301 of 362

Activity 122: Calculating the surface area of a cylinder and a right prism

Problem 1: Calculate the volumes of the following prisms (correct to 1 decimal place):

Lesson 129-131: Worksheets

Please check your calendar for information on this lesson.

Lesson 132: Statistical Graph or Charts

When you have collected and recorded your data, you can represent it in a diagram. Depending on the results, you can use a frequency diagram, pie chart, line graph, pictogram, frequency diagram, frequency polygon or a scatter diagram.

Bar charts Use a bar chart to compare two or more values with a small set of results . Drawing a bar chart In a bar chart, the height of the bar shows the frequency of the result. As the height of bar represents frequency, label the vertical axis 'Frequency'. The labelling of the horizontal axis depends on what is being represented by the bar chart.

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Page 302 of 362 Example:

Leon conducts a survey to find the number of people in each of the cars arriving at his school gate between 8.30am and 9.00am. His results are shown in the bar chart below:

a) How many cars contained 1 person? b) How many cars contained more than 3 people? c) c) Why there are only a small number of cars containing 1 person?

Answers: a) 8 cars contained 1 person b) 14 cars contained more than 3 people (10 + 4 = 14) c) Most cars would be driven by parents bringing their children to school, only a few would contain just a teacher or a sixth former.

Lesson 133: Line Graphs and Pictograms

Line graphs A line graph is often used to show a trend over a number of days or hours. It is plotted as a series of points, which are then joined with straight lines. The ends of the line graph do not have to join to the axes.

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Page 303 of 362 Reading line graphs

Example This line graph shows the midday temperature over a period of 7 days.

You can see at a glance that the temperature was at its highest on Monday and that it started to fall in the middle of the week before rising again at the end of the week.

Question a) What was the lowest temperature and on what day did it occur? b) On what day was the midday temperature 26°C?

Answer: a) The lowest temperature was 19°C and it occurred on Thursday. b) The midday temperature was 26°C on Tuesday.

Pictograms Pictograms use pictures to represent data. To make sense, a pictogram must always have a key.

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Page 304 of 362 Reading pictograms

Example This pictogram shows the number of pizzas eaten by four friends in the past month:

The key tells you that one pizza on the pictogram represents 4 pizzas eaten, so Alan ate 4 + 2 = 6 pizzas.

Question a) Who ate the most pizzas? b) How many pizzas did Bob eat? c) What was the total number of pizzas eaten by the four friends?

Answer: a) Chris ate the most pizzas. 1 b) Alfie ate 9 pizzas. Remember that each pizza on the pictogram represents 4 pizzas, so /4 circle represents 1 pizza. 1 1 c) There are a total of 11 /2 circles in the pictogram. One circle represents 4 pizzas, so 11 /2 × 4 = 46 pizzas

Lesson 134: Pie Charts

Pie charts Pie charts use different-sized sectors of a circle to represent data.

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Page 305 of 362 Reading pie charts Question This pie chart shows the results of a survey that was carried out to find out how students travel to school.

a) What is the most common method of travel? b) What fraction of the students travel to school by car? c) If 6 students travel by car, how many people took part in the survey?

Answer a) The most common method of travel is bus as this has the largest sector on the pie chart. 1 b) /4 of the students travel by car. 1 c) 6 students travel by car, and this is /4 of the total. Therefore, 24 people were questioned for the survey.

Constructing pie charts To construct a pie chart you need to work out the fraction of the total that the sector represents. You can then convert this to an angle and draw the sector on the chart.

Constructing pie charts - using a table

Example The table below shows the grades achieved by 30 pupils in their end-of-year exam.

To show this information in a pie chart, take the following steps: 1. Work out the total number of pupils: 7 + 11 + 6 + 4 + 2 = 30 2. To work out the angle of each segment, work out the fraction of the total that got each grade. 7 Start with A grades: /30

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Page 306 of 362 7 3. There are 360° in a full turn, so to work out the angle, multiply the fraction by 360: /30 × 360 = 84° The grade A sector has an angle of 84° 4. Repeat this process to find the angle of the segments for the other grades 5. Once you have calculated the angles of the segments, construct the pie chart

Question Copy and complete the following table, then use the data to construct a pie chart.

Answer

Lesson 135: Frequency Diagram and Polygons

Frequency diagrams and polygons This frequency diagram shows the heights of 200 people:

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You can construct a frequency polygon by joining the midpoints of the tops of the bars. Frequency polygons are particularly useful for comparing different sets of data on the same diagram.

Constructing a frequency polygon

Diagram showing data

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Page 308 of 362

Midpoints are marked on each bar and joined together

Scatter diagrams* Scatter diagrams show the relationship between two sets of variables. By looking at the diagram you can see whether there is a link between variables. Where there is a link it is called correlation.

Reading scatter diagrams

Example The English and Maths results of ten classmates are shown in the table below:

To see whether there is a correlation between the Maths and English marks, you could plot a scatter diagram.

The Maths mark is on the horizontal scale and the corresponding English mark on the vertical scale.

Bill's Maths mark was 60 and his English mark was 65, so his results are represented by the orange point at coordinates (60, 65).

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Page 309 of 362 Sample scatter diagram

The diagram shows maths and English marks

You can see that all the points representing Maths marks and English marks lie approximately along a straight line. This shows that there is a correlation between these two variables.

The table below shows the correlations that you can deduct from different patterns of scatter.

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Page 310 of 362 Scatter diagrams Line of best fit The 'line of best fit' goes roughly through the middle of all the scatter points on a graph. The closer the points are to the line of best fit the stronger the correlation is.

Example 10 pupils in a school study French and German at GCSE. Their marks for a recent test are recorded in the table below:

You can draw a scatter diagram to represent these marks. As Pete was absent on the day of the German test you do not have enough information to mark his score.

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Page 311 of 362 Looking at this scatter diagram there is strong positive correlation between the marks in French and the marks in German, so you can draw a line of best fit to show that trend.

Pete scored 70 in French, so using the line of his best fit, you can estimate that his mark in German would have been 72.

A line of best fit can only be drawn if there is strong positive or negative correlation. The line of best fit does not have to go through the origin. The line of best fit shows the trend, but it is only approximate and any readings taken from it will be estimations.

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Activity 123: Statistical graphs or charts

Please answer the following questions.

1. A supermarket chain sold 3600 packets of sausages last month. The pie chart shows the different flavours. a) How many packets of vegetarian sausages were sold?

2. Would you expect there to be positive correlation, negative correlation, or no correlation between the following pairs of variables:

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a) Temperature and sales of ice-cream b) Height and intelligence

Lesson 137: Measures of Central Tendency (Mean)

Measures of central tendency are numbers that describe what is average or typical of the distribution of data. There are three main measures of central tendency: mean, median, and mode.

Mean The mean is the arithmetic average of a set of given numbers.

Definition: In statistics, the mean is the mathematical average of a set of numbers. The average is calculated by adding up two or more scores and dividing the total by the number of scores.

Consider the following number set: 2, 4, 6, 9, 12. The average is calculated in the following manner: 2 + 4 + 6 + 9 + 12 = 33 / 5 = 6.6. So the average of the number set is 6.6.

The mean is the best-known and most widely used measure of central tendency. It is what most people call the "average." It is used to describe the distribution of interval-ratio variables such as age, income, and education.

Definition: Scale of measurement refers to how variables are measured. There are four different scales of measurement:  Nominal: Classifies variables simply in terms of their names and the categories cannot be ranked. The variable “religion” with the response categories “Christian,” “Jewish,” “Muslim,” etc. is an example of a nominal scale of measurement.  Ordinal: Contains non-numeric categories than can be ranked, such as “low,” “medium,” and “high.”  Interval: Contains categories in which the actual distances, or intervals, between categories can be compared. For example, we can say that the difference between ages 20 and 25 is the same as the difference between ages 50 and 55.  Ratio: Like the interval-scale variable, however it has a non-arbitrary zero value.

The mean is calculated by adding up all the scores and dividing the result by the number of scores in the distribution. For example, if five families have 0, 2, 2, 3, and 5 children respectively, the mean number of children is (0 + 2 + 2 + 3 + 5)/5 = 12/5 = 2.4. This means that the five households each have an average of 2.4 children.

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Lesson 138: Measures of Central Tendency (Median)

Median The median is the middle score in a set of given numbers.

Definition: The median is the score located at the centre of a distribution. Consider this set of numbers: 2, 3, 6, 8, 10. The median of this number distribution is 6. For distributions with an even number of scores, take the average of the two middle scores to find the median. The median represents the exact middle of a distribution so that half of the cases are above it and half are below it. It is the middle case in a distribution when the scores are arranged in order from lowest to highest.

For example, let’s suppose we have the following list of numbers: 5, 7, 10, 43, 2, 69, 31, 6, 22. First, we must arrange the numbers in order from lowest to highest. The result is this: 2, 5, 6, 7, 10, 22, 31, 43, 69. The median is 10 because it is the exact middle number. There are four numbers below 10 and four numbers above 10.

What happens if we have an even number of cases in our distribution? If we add the number 87 to the end of our list of numbers above, we have 10 total numbers in our distribution, so there is no single middle number. In this case, you average the scores for the two middle numbers. In our new list, the two middle numbers are 10 and 22. So, we take the average of those two numbers: (10 + 22) /2 = 16. Our median is now 16.

Lesson 139: Measures of Central Tendency (Mode)

Mode The mode is the most frequently occurring score in a set of given numbers.

Definition: The mode is the most frequently occurring score in a distribution. Consider the following number distribution of 2, 3, 6, 3, 7, 5, 1, 2, 3, 9. The mode of these numbers would be 3, since three is the most frequently occurring number.

The mode is the category or score with the highest frequency in the distribution. In other words, it is the most common score, or the score that appears the highest number of times in a distribution. For example, let’s say we are looking at pets owned by 100 families, with the distribution looking like this:

Animal Number of families who own it Dog 60 Cat 35 Fish 17

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Page 315 of 362 Hamster 13 Snake 3

The mode here is "dog" since more families own a dog than any other animal. The mode is always the category or score, not the frequency of that score. For instance, in the above example, the mode is "dog," not 60, which is the number of times dog appears.

Some distributions do not have a mode at all. That is, each category has the same frequency. Other distributions might have more than one mode. For example, when a distribution has two scores or categories with the same highest frequency, it is often referred to as "bimodal."

The mode is the only measure of central tendency that can be used with nominal variables.

Example: The data set shows the heights in cm, of 50 South African Grade 9 learners chosen at random.

a) Decide into which class intervals you will group the data. b) Draw up a tally table of the heights using your intervals. c) Construct a histogram from the tally table. d) Determine: (i) the actual arithmetic mean of the heights. Use your calculator. (ii) an estimate of the mean using th frequency table (iii) the mode (iv) the modal class (v) the median (vi) an estimate of the median from the histogram.

Solution: a) Highest – lowest = 185 – 130 = 55. Generally it is advised to have between 8 and 14 intervals. Here it is convenient to choose an interval or class width of 5 and to start from 130. This gives ust the following intervals: 130-134; 135-139; 140-144; 145-149; 150-154; 155-159; 160-164; 165-169; 170-174; 175-179; 180-184; 185-189.

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Page 316 of 362

b) c) The histogram is on the right. Here are some notes on the histogram:

 130, 131, 132, 133 and 134 are in the first interval.  The class intervals are plotted on the horizontal axis. The frequency is plotted on the vertical axis.  A histogram involves a continuous variable.

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Page 317 of 362  The height measurement is in cm in this case.  In a histogram, the columns touch each other.  When the frequency is 0, the column has zero height.  It is important to work with the area of the histogram.

d) (i) Actual mean

(ii) Estimated mean

Notes:  is the symbol used for the actual mean.  is the symbo used for the mean as estimated from the frequency table.  To answer (iii) and (v) it is best to first do a stem-and-leaf plot.

(iii) Mode = 150 cm. This is the value appearing most often.

(iv) The modal class is 170-174. This interval has the highest frequency.

(v) The median = 163 cm. The median lies midway between the 25th and 26th entry in the ordered stem-and-leaf plot. Both of these values are 163 cm so the median is 163 cm.

(vi) To estimate the median from the histogram, we need to estimate the value at which the area of the histogram is divided into two equal parts. The thick line on thehistogram does so. This happens three quarters of the way into the interval 160-164. This is at abour 163 cm.

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Page 318 of 362 You may have noticed that the:  Mode is not in the modal class  Estimates of the mean and median we obtained from the histogram are good.

Activity 124: Measures of central tendency

Please answer the following questions.

1. Calculate the mean of the test marks for the class using this frequency table. The test marks are out of 10.

Marks 0 1 2 3 4 5 6 7 8 9 10

Frequency 0 1 2 2 3 4 6 4 4 3 1

2. The table give the frequency distribution of IQs (Intelligence Quotients) in a Grade 9 class.

IQ 110 111 112 113 114 115 116 117 118 119 120 121

f 1 1 2 3 5 6 5 4 2 2 0 1

a. What is the modal IQ? b. Calculate the mean IQ for the class. c. What is the median IQ for the class?

3. This table gives rainfall data collected from 74 weather stations grouped into nine classes.

Rainfall in cm Classes 10-29 30-49 50-69 70-89 90-109 110-29 130-149 150-169 170-189

Number 33 8 17 12 15 10 5 2 2 of stations

a. What is the modal class? b. Calculate an estimate of the: (i) mean (ii) median.

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Page 319 of 362

Lesson 140-141: Worksheets

Please check your calendar for information on this lesson.

Lesson 142: Measure of Dispersion

The only measure of dispersion you should already know about is range.

Range = highest value – lowest value.

Range The range is the simplest measure of dispersion. The range can be thought of in two ways. 1. As a quantity: the difference between the highest and lowest scores in a distribution. "The range of scores on the exam was 32." 2. As an interval; the lowest and highest scores may be reported as the range. "The range was 62 to 94," which would be written (62, 94).

The Range of a Distribution Find the range in the following sets of data:

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Quartile Finder Find the quartile scores for the following distribution. (See instructions appearing below the histogram).

The divisions you have just performed illustrate quartile scores. Two other percentile scores commonly used to describe the dispersion in a distribution are decile and quintile scores which divide cases into equal sized subsets of tenths (10%) and fifths (20%), respectively. In theory, percentile scores divide a distribution into 100 equal sized groups. In practice this may not be possible because the number of cases may be under 100.

A box plot is an effective visual representation of both central tendency and dispersion. It simultaneously shows the 25th, 50th (median), and 75th percentile scores, along with the minimum and maximum scores. The "box" of the box plot shows the middle or "most typical" 50% of the values, while the "whiskers" of the box plot show the more extreme values. The length of the whiskers indicate visually how extreme the outliers are.

Below is the box plot for the distribution you just separated into quartiles. The boundaries of the box plot's "box" line up with the columns for the quartile scores on the histogram. The box plot displays the median score and shows the range of the distribution as well.

Quartiles are associated with the median: - The first quartile is one quarter of the way through the data set when it has been ordered from lowest to highest. - The second quartile is same as the median. It is halfway through the data set. - The third quartile is three quarters of the way through the data set.

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Page 321 of 362 - The interquartile range = third quartile – first quartile.

The interquartile range gives us an idea of how the data is bunched or dispersed around the median.

Example: The data set shown is the heights of 50 Grade 9 learners. It is the same set of heights we used in the example on measures of central tendency.

For this set of data find: a. the range b. the first, second and third quartiles c. the interquartile range d. any outliers.

Solution: First make an unordered stem-and-leaf plot. Then order it. When it is ordered, it will look like this:

a. The range = 185 – 130 = 55 cm b. We know that there are 50 values in this data set. To calculate the median, or second quartile, we count 25 items up form the lowest height to get to 163. Since this data set has an even number of values, the middle of the data set is actually between the 25th and 26th value. So: (163 + 163) ÷ 2 = 163. This is the second quartile. To either side of the median lie 25 items of data. The first quartile is a quarter of the way through the datat set. So to get the first quartile, we count to the 13th item from the lowest. This gives 155. Similarly, the 3rd quartile lies 13 items up from the median, that is, at 170 cm.

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Page 322 of 362 c. Interquartile range = third quartile – first quartile = 170 – 155 = 15 d. There is a gap from 130 to 140 and from 177 to 185 that does not fit the trend of the other items. Therefore 130 cm and 185 cm are outliers.

Activity 125: Measures of dispersion

For this activity please check your calendar for related worksheet

Lesson 143: Dealing with Bivariate Data

Introduction to bivariate data

Measures of central tendency, variability, and spread summarize a single variable by providing important information about its distribution. Often, more than one variable is collected on each individual. For example, in large health studies of populations it is common to obtain variables such as age, sex, height, weight, blood pressure, and total cholesterol on each individual.

In this chapter we consider bivariate data, which for now consists of two quantitative variables for each individual. Our first interest is in summarizing such data in a way that is analogous to summarizing single variable data.

By way of illustration, let's consider something with which we are all familiar: age. Let’s begin by asking if people tend to marry other people of about the same age. Our experience tells us "yes," but how good is the correspondence? One way to address the question is to look at pairs of ages for a sample of married couples. Table 1 below shows the ages of 10 married couples.

Going across the columns we see that, yes, husbands and wives tend to be of about the same age, with men having a tendency to be slightly older than their wives. This is no big surprise, but at least the data bear out our experiences, which is not always the case.

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Page 323 of 362 The pairs of ages in Table 1 are from a dataset consisting of 282 pairs of spousal ages, too many to make sense of from a table. What we need is a way to summarize the 282 pairs of ages. We know that each variable can be summarized by a histogram (see Figure 1) and by a mean and standard deviation (See Table 2).

Each distribution is fairly skewed with a long right tail. From Table 1 we see that not all husbands are older than their wives and it is important to see that this fact is lost when we separate the variables.

That is, even though we provide summary statistics on each variable, the pairing within couple is lost by separating the variables. We cannot say, for example, based on the means alone what percentage of couples have younger husbands than wives. We have to count across pairs to find this out. Only by maintaining the pairing can meaningful answers be found about couples per se.

Another example of information not available from the separate descriptions of husbands and wives' ages is the mean age of husbands with wives of a certain age. For instance, what is the average age of husbands with 45-year-old wives? Finally, we do not know the relationship between the husband's age and the wife's age.

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Page 324 of 362 We can learn much more by displaying the bivariate data in a graphical form that maintains the pairing. Figure 2 shows a scatter plot of the paired ages. The x-axis represents the age of the husband and the y-axis the age of the wife.

There are two important characteristics of the data revealed by Figure 2. First, it is clear that there is a strong relationship between the husband's age and the wife's age: the older the husband, the older the wife. When one variable (Y) increases with the second variable (X), we say that X and Y have a positive association. Conversely, when y decreases as x increases, we say that they have a negative association.

Second, the points cluster along a straight line. When this occurs, the relationship is called a linear relationship.

Figure 3 shows a scatter plot of Arm Strength and Grip Strength from 149 individuals working in physically demanding jobs including electricians, construction and maintenance workers, and auto mechanics. Not surprisingly, the stronger someone's grip, the stronger their arm tends to be. There is therefore a positive association between these variables.

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Page 325 of 362

Although the points cluster along a line, they are not clustered quite as closely as they are for the scatter plot of spousal age.

Not all scatter plots show linear relationships. Figure 4 shows the results of an experiment conducted by Galileo on projectile motion. In the experiment, Galileo rolled balls down incline and measured how far they travelled as a function of the release height.

It is clear from Figure 4 that the relationship between "Release Height" and "Distance Travelled" is not described well by a straight line: If you drew a line connecting the lowest point and the highest point, all of the remaining points would be above the line. The data are better fit by a parabola.

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Scatter plots that show linear relationships between variables can differ in several ways including the slope of the line about which they cluster and how tightly the points cluster about the line.

Activity 126: Dealing with bivariate data

The table that follows provides data taken from the 2004 World Population Sheet for 16 West African countries listed. Country Gross National Income per Population per square mile

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Page 327 of 362 person per year in US dollars

Benin 1 060 128

Burkina Faso 1 090 128

Cape Verde 4 920 300

Cote d’Ivoire 1 450 136

Gambia 1 660 355

Ghana 2 080 232

Guinea 2 060 97

Guinea-Bissau 680 110

Liberia - 81

Mali 860 28

Mauritania 1 790 8

Niger 800 25

Nigeria 800 385

Senegal 1 540 143

Sierra Leone 500 187

Togo 1 459 253

a. Would it make sense to predict the Gross National Income per person year of a country on the basis of its population per square mile? Give reasons for your answer.

Lesson 144-145: Collecting Data

There are various terms related to taking surveys. Here are the terms you need to know.

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Page 328 of 362 - A census is when the data required has to be collected from all the people concerned. - If it is not necessary or possible to get the data from everyone concerned, you do a survey. For example, in your tuck shop investigation, if your school has hundreds of learners it would be very time-consuming to give out the questionnaire to all of them to analyze all the data collected. - In statistics, the population is the entire group that the data needs to be collected from. For example, in your tuck shop investigation, the population would be all the learners in your school. In a census, you will be collecting the information from the entire population. - A sample is a selection from the population. In a survey, you will be collecting the data from a sample. For example, one class in the school, 20% of the people in your neighborhood and so on. - A representative sample is one that is made up in a similar way as is the total population. For the results of a survey to provide arguments that could be applied to the population, the sample must be representative of the population. For example, if the roll of the school shows that there are twice as many boys as girls in the school, the sample should have twice as many boys as girls. In your tuck-shop investigation, using a class in the school would not provide a good sample. - If a sample is not representative, we say that it is biased. Using only one grade in your tuck shop sample would bias the data to the age group in that grade.

Reasons for working with a sample - It might not be possible to collect the data from the entire population. - Working with the entire population could be too time consuming and expensive. - Even if it is possible to work with the entire population, it might not be necessary to do so. - A sample might be sufficient to provide the data to answer the question. For example, opinion polls, when done using a good representative sample, have been shown to be very accurate.

Random samples In a random sample, every member of the population has an equal chance of being selected for the sample. For the tuck-shop survey, you could use the school register. You could assign a number to each learner on the register, write the numbers on cards and then draw a card at a time until you have a large enough sample.

Some precautions to take when setting up your sample - The population you draw your sample from needs to be well-defined. Using your ‘neighborhood’ as the population is not well-defined. It would be better to define the population more clearly. For example, ’people permanently living within the boundaries of a particular suburb’ is a well- defined population. - A sample must be random. - The proportion of members in the sample should be similar to that of the population. In large randomly selected samples this is usually the case. - If the sample is too small, it will be difficult to draw conclusions about the whole population.

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Page 329 of 362 Conducting a survey - Be clear about the purpose of your survey. You need to decide what you want to know and why you want to know it. - The data must be collected in the same way from each member of the sample. To make sure of this, draw up a questionnaire. - Give the questionnaire to the members of the sample to complete and return. Often this is done through the post or b handing it out. Ask the members to return the questionnaires in a specific way and by a specific time. - When data is collected through interviews, the questions are asked person-to-person. - The interviewer can than help the person being interviewed by explaining the questions on the questionnaire. - If more than one person is going to be doing the interviews, they need to discuss how to go about it together. This ensures that everyone follows the same procedure. - Even when doing interviews, a questionnaire needs to be drawn up. This helps with collecting and analysing the data. - When using interviews in your survey, all the responses from the sample will be available for analysis. This is because the interviewers will keep all the completed questionnaires. - Try out the questionnaire on three or four people first. You can then improve the questionnaire before using it on the sample upon which you have decided.

Guidelines for designing a good questionnaire - Design the questionnaire so that it will be easy to organize, to read off the data and to interpret the data. - Be polite and grateful to the respondents as done in these examples:

- Do not ask for the name of the respondent if the information is sensitive. In this case, make the questionnaire anonymous. You cannot use an interview to collect sensitive data. - Give clear instructions as to how you want the questions answered. - Use definite ways of asking for responses. For example, yes/no, a number, a tick or cross or a small number of alternatives to choose from. If you use alternatives:

 Make sure that the categories do not overlap as they do in this example:

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Page 330 of 362

 List the main categories and have a category for ‘other’. Include a space for the respondent to write in what the ‘other’ is.

 If it is possible to choose more than one category, state clearly how many should be chosen.

 Make sure that all categories are covered.

 It the categories are a list of possible preferences, you could ask the respondent to rank them by filling numbers into boxes.

- Do not ask for information that will not be needed.

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Page 331 of 362 - Avoid asking questins that people may not be willing to answer. For example: “Are you overweight?” - The questions should be clear and concise. Be careful that the question cannot be misinterpreted. For example, the question “When do you buy from the tuch shop?” is not clear. It could have many different responses, such aws: “Never”, “When I have pocket money”, “At first break” or “On Wednesdays”. - Make sure that the questions are neutral and do not favour a particular response. Sometimes the way a question is phrased can actually lead the respondent to answer in a certain way. These are called leading questions. For example, “What is your opinion on the improved service from the tuch shop?” the word “improved” will lead respondents into saying that the service is better. - The questionnaire should be as short as possible while still covering the data that needs to be collected.

Collecting measurement data in an experiment Sometimes, when doing an investigation, measurements need to be taken. Here are some points that we need to remember to make sure we measure correctly and accurately.

- Before you start, make sure you know how to use the measuring instruments. - Use an instrrument that will give you the right degree of accuracy. For example, when measuring heights, a measuring tape in centimetres is sensitive enough. However, when measuring handspans, centimetres would not be enough. We would need to measure in millimetres. So, in this case, we would use a ruler that has milimetres and centimetres. - Make sure to measure from the zero value. For example, when using a ruler, the 0 mark is usually not at the end of the ruler. When measuring mass, make sure the scale is set to zero before starting. - To obtain accurate measurements when using a ruler or tape measure, look directly over the marks. In this way you will avoid the “error of parallax.” - Have a standard way of taking the measurement. If you don’t do the measurements in the same way every time, your investigation will be flawed. - Record data, as you measure it, in a table.

Lesson 146: Worksheets

Please check your calendar for information on this lesson.

Lesson 147: Simple Interest

When money is paid out in the form of a loan, interest is charged for the use of that money. When you invest money with a bank, the bank will use your money and pay the interest to you.

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Page 332 of 362 - Interest is paid only on the amount invested. This amount is also called the capital or the principal amount. - In the case of a loan, interest is charged on the full amount of loan over the period taken to repay the loan.

Definition of 'Simple Interest' A quick method of calculating the interest charge on a loan. Simple interest is determined by multiplying the interest rate by the principal by the number of periods.

A = P(1 + i.n)

Where: - A is the accumulated or total amount. In the case of an investment, this is the amount investe4d plus the interest earned for the use of the money. In the case of a loan, A is the total amount you have to pay back, with interest included.

- P is the loan amount. The amount invested or borrowed.

- i is the interest rate. It is the simple interest rate per annum. Per annum means ‘per year’ and is abbreviated as p.a.

- n is the duration of the loan, using number of periods. It the number of years that the money has been invested or borrowed.

Example: R8 000 is taken out as a loan for five years at a simple interest rate of 9,5% p.a. How much money must be repaid at the end of five years?

Solution: i = 0,095 A = P(1 + i.n) = 8 000(1 + 0,095 x 5) = R11 800

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Page 333 of 362

Activity 127: Simple interest

This activity is based on an advertisement by a financial services provider. We will call this company FSP.

Read the advertisement and then answer the questions that follow.

Dear Ms. Jones,

Do you need to borrow money urgently?

You could have instant cash in your hands tomorrow. You have been selected to receive this offer of a loan from FSP. Whether you want to add value to your home with some renovations, or spoil yourself with a dream holiday, instant cash of up to R25 000 is available to you right now.

Your Loan Options Use this handy installment table to choose the loan that will suit your budget and circumstances. Find the loan amount you need and choose the repayment period that offers you a monthly repayment with which you feel comfortable.

Amount 24 months 36 months 48 months 60 months R4 000 R231 R176 R149 R133 R8 000 R452 R342 R289 R257 R12 000 R656 R491 R409 R361 R16 000 R872 R651 R542 R478 R20 000 R1 087 R812 R676 R596 R25 000 R1 357 R1 012 R842 R472

 Choice of loan term – up to 5 years to repay our loan!  Fixed interest rate – for the full term of your loan!  Cash to use as you choose – for anything that is important to you.

Answer the questions based on the advertisement and the table provided.

1. Ms. Jones takes out a R25 000 loan from FSP over five years. Calculate how much money she will have to pay in total to repay the loan. 2. How much interest does Ms. Jones pay?

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Page 334 of 362 Lesson 148-149: Worksheets

Please check your calendar for information on this lessons.

Lesson 150: Hire-Purchase Loan

A system by which a buyer pays for a thing in regular instalments while enjoying the use of it.

During the repayment period, ownership (title) of the item does not pass to the buyer. Upon the full payment of the loan, the title passes to the buyer. UK term; the usual US term is instalment buying.

One of the most common applications of simple interest is the hire-purchase loan. Goods, such as appliances, are sometimes purchased on a hire-purchase loan that is usually supplied by a bank or some other financial institution. It is also called a hire-purchase agreement because the buyer enters into a contract or agreement with the institution supplying the loan.

The following points apply to most hire-purchase loans: - A certain percentage of the purchase price, usually 10% to 20%, must be paid in cash as a deposit. - The interest is calculated as simple interest on the full amount of the loan over the period that the loan is repaid. - The goods do not belong to the buyer until the loan is fully paid, and can be taken back if the buyer cannot keep up with the payments. The buyer would lose all the money previously paid. - Insurance costs are often included in the loan agreement in case the goods are lost, damaged or stolen,

Example: Jason takes out a hire-purchase loan of R85 000 for furniture that he has bought for his new house. He repays the loan over a period of four years. The interest charged is 14% p.a. simple interest on the full amount of the loan over four years.

a. Calculate how much he will pay over the four years. b. Calculate how much he will have to pay each month assuming that he pays equal amounts every month for four years.

Solution: a. A = P(1 + i.n) = 85 000(1 + 0,14 x 4) = R132 600 b. The loan, together with the interest accrued (accumulated), must be repaid over four years. He pays monthly, and will therefore make 48 payments. Hence, the monthly payments are

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Page 335 of 362 Example: John buys a tumble-dryer and washing machine that cost R12 000 together. He pays 15% in cash and takes out a hire-purchase loan to pay the balance. Included in the loan is the insurance, which is calculated at 2% p.a. of the purchase price. The loan is repaid over three years. The interest on the loan is 14% p.a. simple interest paid on the full amount of the loan over three years. a. Calculate the amount of money that he must pay on insurance over three years. b. How much does he pay in cash? c. Determine the total amount of the loan, including the insurance costs. d. Determine the total amount that must be repaid on the loan including the interest over three years. e. How much must he pay each month, assuming that he makes equal payments every month over three years? f. Determine the total amount paid for the tumble-dryer and washing machine over three years.

Solution: a. 0,02 x 12 000 x 3 = R720 b. 0,15 x 12 000 = R1 800 c. (12 000 – 1 800) + 720 = R10 920 d. 10 920(1 + 0,14 x 3) = R15 506,40 e. 15 506,40 ÷ 36 = R430,73 f. 15 506,40 + 1 800 = R17 306,40

Note: In the exercise that follow you can assume the following, unless otherwise clearly stated:  All loan repayments start one month after the loan has been taken out.  All loan repayments are made in equal monthly instalments.  If the loan is repaid over n years, then there will be 12n monthly payments. These points apply in general practice.

Activity 128: Hire Purchase

1. Mrs. Moodley takes out a loan of R16 000 that is repaid with simple interest of 12% p.a. over a period of four years. Calculate the amount of money that she must pay each month. 2. Mr. Sithole buys camera equipment at a cost of R32 000. He pays 20% in cash and pays the balance using a hire-purchase loan. The loan is repaid over three years with 14% p.a. simple interest. a. Calculate how much money he pays in cash. b. Determine the amount of the loan. c. Determine the amount of money that he will have to repay on the loan, including the interest over the period of three years. d. Calculate his monthly payments.

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Lesson 151: Worksheets

Please check your calendar for information on this lesson.

Lesson 152: Compound Interest

With Compound Interest, you work out the interest for the first period, add it to the total, and then calculate the interest for the next period, and so on ..., like this:

Here are the calculations for a 5 Year Loan at 10%:

As you can see, it is simple to calculate if you take one step at a time. 1. Calculate the Interest (= "Loan at Start" × Interest Rate) 2. Add the Interest to the "Loan at Start" to get the "Loan at End" of the year 3. The "Loan at End" of the year is the "Loan at Start" of the next year

A simple job, with lots of calculations. But there are quicker ways, using some clever mathematics.

Make A Formula Let us make a formula for the above ... just looking at the first year to begin with: $1,000.00 + ($1,000.00 × 10%) = $1,100.00 We can rearrange it like this:

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Page 337 of 362

So, adding 10% interest is the same as multiplying by 1.10

The result is that we can do a year in one step: 1. Multiply the "Loan at Start" by (1 + Interest Rate) to get "Loan at End"

A simple calculation shows you they are the same: this: $1,000 + ($1,000 x 10%) = $1,000 + $100 = $1,100 is the same as: $1,000 × 1.10 = $1,100

Now, here is the magic ...... the same formula works for any year!  We could do the next year like this: $1,100 × 1.10 = $1,210  And then continue to the following year: $1,210 × 1.10 = $1,331  etc...

So it works like this:

In fact we could go straight from the start to Year 5, if we multiply 5 times: $1,000 × 1.10 × 1.10 × 1.10 × 1.10 × 1.10 = $1,610.51 But it is easier to write down a series of multiplies using Exponents (or Powers) like this:

This does all the calculations in the top table in one go.

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Page 338 of 362 The Formula We have been using a real example, but let's be more general by using letters instead of numbers, like this:

(Can you see it is the same? Just with PV = $1,000, r = 0.10, n = 5, and FV = $1,610.51)

Here is is written with "FV" first: FV = PV × (1+r)n where FV = Future Value PV = Present Value r = annual interest rate n = number of periods

This is the basic formula for Compound Interest.

Remember it, because it is very useful.

Examples How about some examples ...... what if the loan went for 15 Years? ... just change the "n" value:

... and what if the loan was for 5 years, but the interest rate was only 6%? Here:

Did you see how we just put the 6% into its place like this:

... and what if the loan was for 20 years at 8%? ... you work it out!

Going "Backwards" to Work Out the Present Value Let's say your goal is to have $2,000 in 5 Years. You can get 10%, so how much should you start with?

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Page 339 of 362

In other words, you know a Future Value, and want to know a Present Value.

We know that multiplying a Present Value (PV) by (1+r)n gives us the Future Value (FV), so we can go backwards by dividing, like this:

So the Formula is: PV = FV / (1+r)n And now we can calculate the answer: PV = $2,000 / (1+0.10)5 = $2,000 / 1.61051 = $1,241.84 In other words, $1,241.84 will grow to $2,000 if you invest it at 10% for 5 years.

Another Example: How much would you need to invest now, to get $10,000 in 10 years at 8% interest rate?

PV = $10,000 / (1+0.08)10 = $10,000 / 2.1589 = $4,631.93 So, $4,631.93 invested at 8% for 10 Years would grow to $10,000

Compounding Periods Compound Interest is not always calculated per year, it could be per month, per day, etc. But if it is not per year it should say so!

Example: you take out a $1,000 loan for 12 months and it says "1% per month", how much do you pay back?

Just use the Future Value formula with "n" being the number of months: FV = PV × (1+r)n = $1,000 × (1.01)12 = $1,000 × 1.12683 = $1,126.83 to pay back And it is also possible to have yearly interest but with several compounding within the year, which is called Periodic Compounding.

Example, 6% interest with "monthly compounding" does not mean 6% per month, it means 0.5% per month (6% divided by 12 months), and would be worked out like this: FV = PV × (1+r/n)n = $1,000 × (1 + 6%/12)12 = $1,000 × (1.005)12 = $1,000 × 1.06168... = $1,061.68 to pay back

This is equal to a 6.168% ($1,000 grew to $1,061.68) for the whole year.

So be careful to understand what is meant!

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Page 340 of 362 APR

Because it is easy for loan ads to be confusing (sometimes on purpose!), the "APR" is often used.

APR means "Annual Percentage Rate" ... it shows how much you will actually be paying for the year (including compounding, fees, etc).

This ad looks like 6.25%, but is really 6.335%

Here are some examples: Example 1: "1% per month" actually works out to be 12.683% APR (if no fees). And: Example 2: "6% interest with monthly compounding" works out to be 6.168% APR (if no fees). If you are shopping around, ask for the APR.

Break Time! So far we have looked at using (1+r)n to go from a Present Value (PV) to a Future Value (FV) and back again, plus some of the tricky things that can happen to a loan.

Now would be a good time to have a break before we look at two more topics:  How to work out the Interest Rate if you know PV, FV and the Number of Periods.  How to work out the Number of Periods if you know PV, FV and the Interest Rate

Working Out The Interest Rate You can calculate the Interest Rate if you know a Present Value, a Future Value and how many Periods.

Example: you have $1,000, and want it to grow to $2,000 in 5 Years, what interest rate do you need? The formula is: r = ( FV / PV )1/n - 1

Note: the little "1/n" is a Fractional Exponent, first calculate 1/n, then use that as the exponent on your calculator. For example 20.2 would be entered as 2, "x^y", 0, ., 2, =

Now we just "plug in" the values to get the result: r = ( $2,000 / $1,000 )1/5 - 1 = ( 2 )0.2 - 1 = 1.1487 - 1 = 0.1487 And 0.1487 as a percentage is 14.87%, So you would need a 14.87% interest rate to turn $1,000 into $2,000 in 5 years.

Another Example: What interest rate would you need to turn $1,000 into $5,000 in 20 Years?

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Page 341 of 362 r = ( $5,000 / $1,000 )1/20 - 1 = ( 5 )0.05 - 1 = 1.0838 - 1 = 0.0838 And 0.0838 as a percentage is 8.38%. So 8.38% will turn $1,000 into $5,000 in 20 Years.

Working Out How Many Periods You can calculate how many Periods if you know a Future Value, a Present Value and the Interest Rate.

Example: you want to know how many periods it will take to turn $1,000 into $2,000 at 10% interest. This is the formula (note: it uses the natural logarithm function ln): n = ln(FV / PV) / ln(1 + r)

The "ln" function should be on a good calculator. You could also use log, just don't mix the two.

Anyway, let's "plug in" the values: n = ln( $2,000 / $1,000 ) / ln( 1 + 0.10 ) = ln(2)/ln(1.10) = 0.69315/0.09531 = 7.27 Magic! It will need 7.27 years to turn $1,000 into $2,000 at 10% interest.

Another Example: How many years to turn $1,000 into $10,000 at 5% interest? n = ln( $10,000 / $1,000 ) / ln( 1 + 0.05 ) = ln(10)/ln(1.05) = 2.3026/0.04879 = 47.19 47 Years! But we are talking about a 10-fold increase, at only 5% interest.

Calculator

I also made a Compound Interest Calculator that uses these formulas.

Summary The basic formula for Compound Interest is: FV = PV (1+r)n To find the Future Value, where:  FV = Future Value,  PV = Present Value,  r = Interest Rate (as a decimal value), and  n = Number of Periods And by rearranging that formula (see Compound Interest Formula Derivation) we can find any value when we know the other three:

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Page 342 of 362

Find the Present Value when you know a Future PV = FV / (1+r)n Value, the Interest Rate and number of Periods.

Find the Interest Rate when you know the Present r = ( FV / PV )1/n - 1 Value, Future Value and number of Periods.

Find the number of Periods when you know the n = ln(FV / PV) / ln(1 + r) Present Value, Future Value and Interest Rate

Annuities So far we have talked about what happens to a value as time goes by ... but what if you have a series of values, like regular loan payments or yearly investments? That is covered in Annuities, coming soon.

Activity 129: Compound Interest

Please answer the following.

Question 1 If the present value of my investment is $1,000 and the rate of interest is 10% compounded annually, what will the value be after 6 years?

A $1,600 B $1,771.56 C $1,790.85 D $1,948.72

Question 2 If the present value of my investment is $1,000 and the rate of interest is 6% compounded annually, what will the value be after 10 years?

A $1,600 B $1,771.56 C $1,790.85 D $1,898.30

Question 3 If the present value of my investment is $2,500 and the rate of interest is 2% compounded annually, what will the value be after 15 years?

A $3,364.67 B $3,306.25 C $3,250 D $3,047.49

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Page 343 of 362

Lesson 154: Worksheet

Please check your calendar for information on this lesson.

Lesson 155: Inflation Rates

Inflation is a continuous increase in the general level of prices over a period of time. The rate of inflation is calculated by comparing the price of goods and services from one year to the next.

Inflation is normally compounded annually and calculated on the previous year’s prices. The compound interest formula can be used to calculate the price of goods over a period of time using an average rate of inflation.

The average rate of inflation in South Africa from 1980 to 1990 was 14,6% p.a. In 1986 inflation reached a high of 18,6% p.a. However the inflation rate dropped in the 1990s and reached a low of 6,9% p.a. in 1998.

The average rate of inflation from 2000 to 2010 was 7,2% p.a.

Activity 130: Inflation rates

Answer the questions below.

1. What factors contribute to high inflation? 2. What are the negative effects on the economy if inflation remains high and uncontrolled? 3. Find out the current rate of inflation, and the price of a new car that you would like to buy. Calculate what the cost of the car will be when you are 25 years old, assuming that inflation continues at its current rate.

Lesson 156: Depreciation Rates

Depreciation is the loss of value of assets, machinery or equipment through age or use. Depreciation can be calculated in a number of different ways.

The two most common methods of calculating depreciation are: 1. Straight-line depreciation. The depreciation is calculated as a percentage of the original value of the asset each year.

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Page 344 of 362 2. Reducing-balance depreciation. The depreciation is calculated as a percentage of the reduced value of the asset each year. - Straight-line depreciation will result in the equipment having no value at all after a period of time. For example, if the depreciation rate is calculated at 20% p.a. on the original value, then the equipment would have no value after five years. - Depreciation on a reducing balance will ensure that equipment always has some value at the end of a certain period.

Example: A company buys office furniture for R500 000. The accountant depreciates the furniture at 20% p.a. on a straight line basis.

a. Determine the value of the furniture at the end of each year for five years. b. Draw a graph showing the change in value of the furniture over five years. Solution: a. End of first year: 500 000 – 0,20 x 500 000 = R400 000 End of second year: 400 000 – 0,20 x 500 000 = R300 000 End of third year: 300 000 – 0,20 x 500 000 = R200 000 End of fourth year: 200 000 – 0,20 x 500 000 = R100 000 End of fifth year: 100 000 – 0,20 x 500 000 = R0

- Note that the computers depreciate by the same amount every year which is 20% of R500 000. - In a straight line depreciation the balance reduces to zero over a period of time.

- This method of depreciation is called straight line depreciation because the value of the assets depreciates as a straight line as shown by the graph over a period of time.

Example: A company buys office furniture for R500 000. The accountant depreciates the furniture at 20% p.a. on a reducing balance over five years.

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Page 345 of 362

a. Determine the value of the furniture at the end of each year for five years. b. Draw a graph showing the change in value of the furniture over five years.

Solution: a. End of first year: 500 000 – 0,20 x 500 000 = R400 000 End of second year: 400 000 – 0,20 x 400 000 = R320 000 End of third year: 320 000 – 0,20 x 320 000 = R256 000 End of fourth year: 256 000 – 0,20 x 256 000 = R204 800 End of fifth year: 204 800 – 0,20 x 204 800 = R163 840

- Note that the furniture depreciates by the smaller amount every year which is 20% of the previous year’s value. It depreciates on a reducing balance. - In reducing balance depreciation the balance never reduces to zero. - It always retains some value.

The formula for depreciation on a reducing balance can be derived in exaactly the same way as the formula for compound interest.

Derive the formula for yourself.

where A = furture value of the equipment P = present value of the equipment n = number of years i = annual rate of depreciation

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Page 346 of 362 the only difference between this formula and the compound interest formula is the minus sign. The reason for the subtraction is that the equipment is depreciating in value and therefore becoming less valuable each year.

Activity 131: Depreciation

For this activity please check your calendar for related worksheet

Lesson 157: Worksheets

Please check your calendar for information on this lesson.

Lesson 158: Exchange Rates

The exchange rate of the South African rand against other currencies plays a major role in the country’s economy, affecting all exports and imports, as well as travel and tourism. Exchange rates constantly change and reflect the strength of the economy against that of other countries.

The most up-to-date rates can be found on the Internet. The exchange rate against the world’s major currencies such as the euro, the US dollar, and the British pound are reported daily on the financial news on television and radio.

Activity 132: Exchange rates

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Page 347 of 362 Use the table of exchange rates to answer the questions.

1. Brenda is going to Spain on holiday and changes to euros. How many euros does she get? 2. Timothy is going to Florida for his holidays. He saves of his pocket money. How many dollars will he get?

Lesson 159-161: Worksheets

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Lesson 162: Commission and Rentals

Commission is money that is paid to a seller for goods, services or property sold.

- Estate agents will charge the owner commission for selling their property. The commission charged is usually a percentage of the property price. - Insurance salespersons are paid commission for selling insurance policies. - Many businesses prefer to pay their sales staff a commission for selling goods, instead of paying a salary. In some cases, a small basic salary is paid and the rest of the salary is made up from commission.

This means that people who work on commission don’t get a fixed amount of money every month like people in other jobs.

Rental is the money charged for the use of property or for the hire of vehicles or goods.

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Page 348 of 362

Activity 133: Commission and rentals

1. Discuss the advantages and disadvantages of working on a commission basis. 2. Why do estate agents charge commission for selling properties? 3. What do estate agents do to sell properties? 4. Find out the commission rates that are charged for selling property.

Lesson 163-164: Worksheets

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Lesson 165: Similarity

Similar

Two shapes are Similar if the only difference is size (and possibly the need to turn or flip one around).

Resizing is the Key If one shape can become another using Resizing (also called dilation, contraction, compression, enlargement or even expansion), then the shapes are Similar:

These Shapes are Similar!

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Page 349 of 362 There may be Turns, Flips or Slides, Too! Sometimes it can be hard to see if two shapes are Similar, because you may need to turn, flip or slide one shape as well as resizing it.

Rotation Turn!

Reflection Flip!

Translation Slide!

Examples These shapes are all Similar:

Resized Resized and Reflected Resized and Rotated

Why is it Useful? When two shapes are similar, then:

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Page 350 of 362  corresponding angles are equal, and  the lines are in proportion.

This can make life a lot easier when solving geometry puzzles, as in this example:

Example: What is the missing length here?

Notice that the red triangle has the same angles as the main triangle ...

... they both have one right angle, and a shared angle in the left corner

In fact you could flip over the red triangle, rotate it a little, then resize it and it would fit exactly on top of the main triangle. So they are similar triangles.

So the line lengths will be in proportion, and we can calculate:

? = 80 × (130/127) = 81.9 (No fancy calculations, just common sense!)

Congruent or Similar? But when you don't need to resize to make the shapes the same, they are called Congruent.

So, if the shapes become the same:

When you ... Then the shapes are ...

... only Rotate, Reflect and/or Translate Congruent

... also need to Resize Similar

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Page 351 of 362

Activity 134: Similar

Question 1

How many triangles similar to this equilateral triangle

are there in the following diagram?

A 10 B 16 C 26 D 27

Question 2

How many rectangles similar to this rectangle

are there in the following diagram?

A16 B 20 C 24 D 25

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Page 352 of 362 Question 3

These two quadrilaterals are similar.

What is the value of x (the length of B'C') ?

A B 5 C 6 D

Lesson 166-167: Worksheets

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Lesson 168: Congruency

If one shape can become another using Turns, Flips and/or Slides, then the shapes are Congruent:

Rotation Turn!

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Page 353 of 362

Reflection Flip!

Translation Slide!

After any of those transformations (turn, flip or slide), the shape still has the same size, area, angles and line lengths.

Examples These shapes are all Congruent:

Rotated Reflected and Moved Reflected and Rotated

Congruent or Similar? The two shapes need to be the same size to be congruent. When you need to resize one shape to make it the same as the other, the shapes are called Similar.

If you ... Then the shapes are ...

... only Rotate, Reflect and/or Translate Congruent

... need to Resize Similar

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Page 354 of 362 Congruent? Why such a funny word that basically means "equal"? Probably because they would only be "equal" if laid on top of each other. Anyway it comes from Latin congruere, "to agree". So the shapes "agree"

Activity 135: Congruent

Question 1

Which shape is not congruent to the other three?

A A B B C C D D

Question 2 How many triangles congruent to this equilateral triangle

are there in the following diagram?

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Page 355 of 362 A 10 B 16 C 26 D 27

Question 3 How many congruent trapezoids are there in the following diagram?

A 8 B 12 C 16 D 21

Lesson 169-172: Worksheets

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Lesson 173: Congruent Angles

Congruent Angles have the same angle (in degrees or radians). That is all.

These angles are congruent.

They don't have to point in the same direction.

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Page 356 of 362 They don't have to be on similar sized lines.

Just the same angle.

Congruent - why such a funny word that basically means "equal"? Probably because they would only be "equal" if laid on top of each other. Anyway it comes from Latin congruere, "to agree". So the angles "agree"

Lesson 174: The Probability Scale

Many events can't be predicted with total certainty. The best we can say is how likely they are to happen, using the idea of probability.

Tossing a Coin When a coin is tossed, there are two possible outcomes:  heads (H) or

 tails (T) We say that the probability of the coin landing H is ½. And the probability of the coin landing T is ½.

Throwing Dice When a single die is thrown, there are six possible outcomes: 1, 2, 3, 4, 5, 6. The probability of any one of them is 1/6.

Probability In general: Number of ways it can happen Probability of an event happening = Total number of outcomes

Example: the chances of rolling a "4" with a die Number of ways it can happen: 1 (there is only 1 face with a "4" on it) Total number of outcomes: 6 (there are 6 faces altogether) 1 So the probability = 6

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Page 357 of 362

Example: there are 5 marbles in a bag: 4 are blue, and 1 is red. What is the probability that a blue marble will be picked? Number of ways it can happen: 4 (there are 4 blues) Total number of outcomes: 5 (there are 5 marbles in total) 4 So the probability = = 0.8 5

Probability Line You can show probability on a Probability Line:

Probability is always between 0 and 1 Probability is the chance that something will happen. It can be shown on a line. The probability of an event occurring is somewhere between impossible and certain.

As well as words we can use numbers (such as fractions or decimals) to show the probability of something happening:  Impossible is zero  Certain is one.

Here are some fractions on the probability line:

We can also show the chance that something will happen:

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Page 358 of 362 a) The sun will rise tomorrow. b) I will not have to learn mathematics at school. c) If I flip a coin it will land heads up. d) Choosing a red ball from a sack with 1 red ball and 3 green balls

Between 0 and 1  The probability of an event will not be less than 0. This is because 0 is impossible (sure that something will not happen).  The probability of an event will not be more than 1. This is because 1 is certain that something will happen.

Probability is Just a Guide Probability does not tell us exactly what will happen, it is just a guide

Example: toss a coin 100 times, how many Heads will come up? Probability says that heads have a ½ chance, so we would expect 50 Heads. But when you actually try it out you might get 48 heads, or 55 heads ... or anything really, but in most cases it will be a number near 50. Learn more at Probability Index.

Words Some words have special meaning in Probability:

Experiment: an action where the result is uncertain. Tossing a coin, throwing dice, seeing what pizza people choose are all examples of experiments.

Sample Space: all the possible outcomes of an experiment Example: choosing a card from a deck There are 52 cards in a deck (not including Jokers) So the Sample Space is all 52 possible cards: {Ace of Hearts, 2 of Hearts, etc... } The Sample Space is made up of Sample Points:

Sample Point: just one of the possible outcomes Example: Deck of Cards  the 5 of Clubs is a sample point  the King of Hearts is a sample point "King" is not a sample point. As there are 4 Kings that is 4 different sample points.

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Event: a single result of an experiment Example Events:  Getting a Tail when tossing a coin is an event  Rolling a "5" is an event. An event can include one or more possible outcomes:  Choosing a "King" from a deck of cards (any of the 4 Kings) is an event  Rolling an "even number" (2, 4 or 6) is also an event

The Sample Space is all possible outcomes. A Sample Point is just one possible outcome.

And an Event can be one or more of the possible outcomes.

Hey, let's use those words, so you get used to them:

Example: Alex decide to see how many times a "double" would come up when throwing 2 dice. Each time Alex throws the 2 dice is an Experiment. It is an Experiment because the result is uncertain.

The Event Alex is looking for is a "double", where both dice have the same number. It is made up of these 6 Sample Points: {1,1} {2,2} {3,3} {4,4} {5,5} and {6,6}

The Sample Space is all possible outcomes (36 Sample Points): {1,1} {1,2} {1,3} {1,4} ... {6,3} {6,4} {6,5} {6,6}

These are Alex's Results: Experiment Is it a Double? {3,4} No {5,1} No {2,2} Yes {6,3} No

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Page 360 of 362 ......

After 100 Experiments, Alex had 19 "double" Events ... is that close to what you would expect?

Activity 136: Probability

Question 1 A die is thrown once. What is the probability that the score is a factor of 6? A 1/6 B 1/2 C 2/3 D 1

Question 2

The diagram shows a spinner made up of a piece of card in the shape of a regular pentagon, with a toothpick pushed through its centre. The five triangles are numbered from 1 to 5.

The spinner is spun until it lands on one of the five edges of the pentagon. What is the probability that the number it lands on is odd? A 1/5 B 2/5 C 1/2 D 3/5

Question 3 Each of the letters of the word MISSISSIPPI are written on separate pieces of paper that are then folded, put in a hat, and mixed thoroughly.

One piece of paper is chosen (without looking) from the hat. What is the probability it is an I? A 4/11 B 2/5 C 1/3 D ¼

Question 4 A card is chosen at random from a deck of 52 playing cards.

There are 4 Queens and 4 Kings in a deck of playing cards.

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What is the probability it is a Queen or a King?

A B C D

Question 5 A fair coin is tossed three times. What is the probability of obtaining one Head and two Tails? (A fair coin is one that is not loaded, so there is an equal chance of it landing Heads up or Tails up.)

A B C D

Lesson 175-181: Worksheets

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