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NUMBER SENSE AND NUMERATION

REPRESENTING NUMBER

1. WRITTEN FORM Examples: i. One ii. Twelve iii. Six hundred eight iv. Seven thousand eight three v. Seven hundredths vi. Four hundred thousandths vii. Thirty two thousand, four hundred ninety six viii. Nine hundred five billion, six hundred fifteen million, two hundred eight thousand, five hundred eight.

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2. STANDARD FORM Information:  Beginning with the smallest whole number (the ones), there are THREE digits in each grouping of numbers and then a space before the next set of digits. For the largest place value, there may be one, two or three digits.  Zero is a digit which represents a place value unless it is at the beginning of a whole number, and then it just a place holder Examples: i. 32 469 ii. 905 615 208 508

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3. EXPANDED FORM Information:  Numbers written in standard form can be expanded out to show the value of each digit  When you do this, you what each digit in the number actually represents numerically (its place value)

2 6 7 5 9 8 3

TENS

ONES

MILLIONS

HUNDREDS THOUSANDS

OF THOUSANDS

TENS OF THOUSANDS TENS OF

HUDREDS Examples: i. 8 765 = 8 000 + 700 + 60 + 5 ii. 943 567 832.23 = 900 000 000 + 40 000 000 + 3 000 000 + 5 00 000 + 60 000 + 7 000 + 800 + 30 + 2 + 0.2 + 0.03 4

PLACE VALUE Information: our number system has ten digits. They are: 0, 1, 2 , 3, 4, 5, 6, 7, 8, and 9

How those digits are arranged makes a number. It is the POSITION of the digit in the number that gives the digit a value.

The further to the right a digit is in a number) the larger the value of the digit is.

The further to the left a digit is (including in the portion) the smaller the value of the digit is.

Example: 88 888.88 Even though all of the digits are 8, each place value means the 8s are worth different values. The underlined 8 is in the ten thousand place value position. It represents 8 groups of 10 000 or 80 000. The boxed digit 8 is in the hundredths place value position. It represents 8 groups of a hundredth, or .08 5

EXPONENTS 2 EXPONENT

POWER 5

The exponent shows how many times you must multiple the base number times itself. Examples: 52 means 5 x5, which equals 25 53 means 5 x 5 x 5, which equals 125 54 means 5 x 5 x 5 x 5 = 625

6 Notice how the exponent POWERS OF TEN and the number of zeros There is NO power of one! in the number are the same! MATH IS ALL 102 = 10 x 10 = 100 ABOUT PATTERNS 103 = 10 x 10 x 10 = 1 000 104 = 10 x 10 x 10 x 10 = 10 000 105 = 10 x 10 x 10 x 10 x 10 = 100 000 106 = 10 x 10 x 10 x 10 x 10 x 10 = 1 000 000 107 = 10 x 10 x 10 x 10 x 10 x 10 x 10 = 10 000 000 108 = 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 100 000 000 109 = 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 1 000 000 000 10 10 = 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 10 000 000 000

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EXPANDED FORM USING POWERS OF TEN Since we have looked at Powers of ten, the next step is to show numbers in expanded form using powers of ten! Example: 45 678  The digit 4 represents 4 groups of 10 000, or 40 000.  We know that 104 is 10 000  We can show 40 000 as 4 x 104

 The digit 5 represents 5 groups of 1 000 or 5 000  We know that 103 is 1 000  We can show 5 000 as 5 x 103

So, in expanded for, using Power of Ten:

45 678 = 4 x 104 + 5 x 103 + 6 x 102 + 70 + 8.

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DIVISIBILITY RULES

Divisible by…? The Strategy Example 2 Must end in an even number 252, 756, 928, 4 354 (o, 2 , 4, 6, 8) 3 Add all digits together until they form 76 825 either a single digit or a digit you 7+6+8+2+5 = 28 recognize as being divisible by 3. If the sum of the digits is divisible by three, 2+8 = 10 then so is the number 10 is NOT divisible by 3, so neither is 76 825.

4 Look at the last 2 digits. If they are 9 6 5 7 57 is not divisible by 2, then so is the number. divisible by 4, so neither is 9 657. 5 Any number that ends in a zero or a five 6 890 is divisible by 5 is divisible by 5 because is ends in a zero. 6 Both rules for 2 and 3 apply 3 246 – last digit is even, so divisible by 2 and sum of digits is 15 which is divisible by 3, so it is divisible by 6 9 Sum of digits is divisible by 9 48 456 4+8+4+5+6 = 27 27 is divisible by 9, so so is 48 456 10 Any number that ends in a zero is 56 090 – is divisible by divisible by 10 10

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MULTIPLES Think of your multiplication tables… Multiples of a number are the product of that number times any other number. So, for example, the first 4 multiples of 2 are: 2 x 1 = 2 2 x 2 = 4 2 x 3 = 6 2 x 4 = 8 The first 6 multiples of 3 are: 3, 6, 9, 12, 15, 18, but 300 and 252 are also multiples of 3 because they can be divided by 3 with no remainder. 10

FACTORS Factors are one of two or more numbers that, when multiplied together, produce a give product.

So, for example, the factors of 6 are 2 and 3 because 2 x 3 = 6.

Sometimes we are asked to find ALL of the factors of a number. That means all of the possible combinations of numbers that we could multiply together to give a specific product.

EXAMPLE: Because ALL numbers can be made by multiplying the number by 1, we do NOT 36 include 1 as a factor of a number All of the factors of 36 are: (1 x 36) 2 x 18 3 x 12 4 x 9

6 x 6

The factors of the number 36 are: 2, 3, 4, 6, 9, 12, 18 and 36. 11

FACT FAMILIES – INVERSE OPERATIONS

You will be very surprised how often you find that fact families are helpful as you progress through intermediate and high school math! You learned about fact families in primary grades, but guess what? You apply (use) this knowledge in all different forms of more challenging math! Fact families show opposite or INVERSE operations. Here are two examples of a fact families. 5 + 8 = 13 4 + 3 = 12 8 + 5 = 13 3 + 4 = 12 13 – 8 = 5 12 ÷ 3 = 4

13 – 5 = 8 12 ÷ 4 = 3 Fact families show how numbers are related. They teach us that we can use the inverse operation to help us. The inverse operation of addition is subtraction and the inverse operation of multiplication is division. 12

OPERATIONS IN MATH There are 4 basic operations in math. The answers you get by using different operations have different names! The answer to addition is called the

SUM

The answer to subtraction is called the DIFFERENCE

The answer to multiplication is called the PRODUCT The answer to division is called the

QUOTIENT

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“FRIENDLY” NUMBERS Okay, well, numbers are not really either friendly or unfriendly, but some numbers are a bit easier to work with than others, so we call them, “friendly numbers.” Examples of friendly numbers include any multiple of 10, because with a bit of practise, these numbers (because they end in zeroes) are easy to add, subtract, multiply and divide. What is one person’s idea of a friendly number may differ from another’s, but usually, the following are considered, “friendly:” 2, 5, 10, 20, 25, 50, 100, 1 000

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INTEGERS – POSITIVE AND NEGATIVE NUMBERS An integer is a WHOLE number that is not a fraction. Our number system is INFINITE which means that it never ends. You can ALWAYS add to or take away from a number. Numbers that are BELOW (LESS THAN) zero are called NEGATIVE integers (or numbers!). Numbers that are ABOVE (GREATER THAN) zero are called POSITIVE integers (or numbers!).

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

We encounter negative numbers all the time in our real lives. In the winter, in Ontario, the temperature is often below zero! We would show 25° below zero as -25°celcius. If you are below sea level, then this is measured as below zero.

This is below sea level!

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PRIME AND COMPOSITE NUMBERS

We describe whole numbers in different ways. One of the ways is by looking at how many factors a whole number has. PRIME NUMBERS Some numbers only have 2 factors, one and the number itself. These numbers are called PRIME NUMBERS. Think about a glass plate made of a number. If you throw it up and it can only break into 2 sections, (1 and the number itself), it is PRIME. Another way to think about this is to describe PRIME NUMBERS as only having TWO .

The first 20 PRIME NUMBERS are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67 and 71.

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PRIME FACTORIZATION (FACTOR TREES) We often have to find the smallest possible prime multiples of a number. To do this we make a FACTOR TREE. We can use any multiples of the target number to begin. So, look at the 36. We can find many sets of multiples to, “smash it up.” Let’s use 4 x 9 as our example 36 36 1 x 36 4 x 9 2 x 18 2 x 2 x 3 x 3 3 x 12 4 x 9 Because 2 and 3 are the smallest PRIME numbers when 36 is 6 x 6 broken down, they are the PRIME FACTORS of 36 17

Let’s try another one. Here are the PRIME FACTORS of 210 210 10 x 21 2 x 5 x 7 x 3 So the prime factors of 210 are: 2, 3, 5 & 7 SHOWING PRIME FACTORS IN EXPONENTIAL FORM (Gr. 7/8)

To simplify the 432 bottom row we 2 x 216 could show it as 24 x 33 2 x 2 x 108 2 x 2 x 2 x 54 2 x 2 x 2 x 3 x 18 2 x 2 x 2 x 3 x 3 x 6 2 x 2 x 2 x 3 x 3 x 2 x 3 18

USEFUL PREFIXES TO HELP UP RECOGNIZE NUMBER VALUES (USUALLY EITHER LATIN OR GREEK BASED Prefix Meaning uni 1 bi 2 tri 3 quad 4 penta 5 hexa 6 hepta 7 octa 8 nona 9 deca 10 19

GREATEST COMMON FACTOR (GCF) & LOWEST COMMON MULTIPLES (LCM)

Often we look at two numbers and see what might be common (shared) between them. GREATEST COMMON FACTOR (GCF) Example: 12 20

3 x 4 2 x 10 3 x 2 x 2 2 x 2 x 5

Both 12 and 20 share two sets of 2 (I have shown one set circles in red and one set circled in yellow). Bring ONE of each set down and show it as a multiplication problem

2 x 2 = 4 The GCF of 12 and 20 is 4 Notice that you have “left-over,” factors that are NOT shared by 12 and 20. I have shown them boxed in green. Bring down the product of the shared factors (4) and multiply that number by any of the left over (not shared) factors. 4 x 3 x 5 4 x 3 x 5 = 60 The LCM of 12 and 20 is 60 To find the next common multiple, double the lowest one (60 + 60 = 120). To find the next one, add another 60 (120 + 60 = 180), keep adding the LCM to find each subsequent multiple. 20

COMPOSITE NUMBERS Some numbers have more than only 2 factors. These numbers are called COMPOSITE NUMBERS. Think again about the glass plate made of a number. If you throw it up and it can break in two lots of ways, it is COMPOSITE NUMBER. Another way to think about this is to describe COMPOSITE NUMBERS as having more than TWO DIVISORS.

The first 20 COMPOSITE NUMBERS are:

4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33…

The number 1 is NEITHER prime nor composite!

In music, a COMPOSITION is a made up of many different notes. In math, a

COMPOSITE number is made up of more than just 2 numbers.