Multi-Subject CST Mathematics Preparation Competency 0001- Number Systems and Quantity 1.1 Number Systems and Vector and Matrix Quantities
February 2016
© NYC Teaching Fellows 2014 Agenda
Introduction to Competency
Content Review and Sample Problems
Additional Practice
/ 2 Introduction to Competency 0001- NUMBER AND QUANTITY
Performance Expectations
The New York State Grade 7-12 Multi-Subject teacher • Demonstrates knowledge of the properties of numbers, number systems, and how number systems are extended. • Demonstrates understanding of real and complex numbers and understands the complex number system as an extension of the real number system. • Applies properties of complex numbers, works accurately with real numbers, and uses them to solve mathematical and real-world problems. • Works with vectors and matrices. • Has a deep understanding of ratios and proportional relationships, and applies connections between multiplication and division and ratios and rates. • Analyze relationships between ratios and fractions, solves problems involving ratios and rates, and demonstrates the ability to work accurately with ratios and proportional relationships.
New York State Education Department / 3 Introduction to Competency 0001- Number Systems and Vector and Matrix Quantities
1.1 Number Systems and Vector and Matrix Quantities
• Applies properties of signed rational numbers, ordering, and absolute value of rational numbers • Solves mathematical and real-world problems involving the four basic operations with rational numbers • Applies and extends understanding of arithmetic and the order of operations to algebraic expressions, equations, and inequalities • Uses rational approximations of irrational numbers (e.g., comparing the size of irrational numbers, locating irrational numbers on the number line) • Knows properties of repeating decimal expansions and converts between repeating decimal expansions and rational numbers • Performs operations with numbers expressed in scientific notation • Works with radicals and exponents and rewrites expressions involving radicals and rational exponents using the properties of exponents
New York State Education Department / 4 Introduction to Competency 0001- Number Systems and Vector and Matrix Quantities
1.1 Number Systems and Vector and Matrix Quantities
• Uses units as a way to understand problems and to guide the solution of multistep problems, and chooses and interprets units consistently in formulas • Performs arithmetic operations with complex numbers and represents complex numbers and their operations on the complex plane • Uses complex numbers to solve quadratic equations with real coefficients that have complex solutions • Represents and models with vector quantities and performs operations on vectors both algebraically and graphically • Performs basic operations on matrices and uses matrices in applications • Demonstrates knowledge of how to analyze and interpret assessment data to inform and plan instruction that engages and challenges all students to meet or exceed the NYCCLS related to number systems and vector and matrix quantities
New York State Education Department / 5 Agenda
Introduction to Competency
Content Review and Sample Problems
Additional Practice
/ 6 Number Systems: Applies properties of signed rational numbers, ordering, and absolute value of rational numbers
Signed rational numbers
• What is a rational number? o Any number that can be expressed as a ratio or two integers. Basically, a number that can be written in fractional form. o Watch - Understanding Rational Numbers • What is a signed rational number? o A positive or negative number that can be written as a fraction.
Practice- Identify rational numbers
New York State Education Department / 7 Number Systems: Applies properties of signed rational numbers, ordering, and absolute value of rational numbers
Ordering Rational Numbers
• How to order signed rational numbers o To order rational numbers, first write them in the same form and then order the numbers. For example, convert all numbers to fractions or decimals. o Watch- Ordering Rational Numbers
Practice- Ordering rational numbers
New York State Education Department / 8 Number Systems: Applies properties of signed rational numbers, ordering, and absolute value of rational numbers
Absolute Value
• What is absolute value? o Absolute value describes the distance of a number on the number line from zero without considering which direction from zero the number lies. o The absolute value of a number is never zero. o The symbol for absolute value is two straight lines surrounding the number.
o Watch- Find absolute value o Read- Absolute Value
Practice-Absolute value of rational numbers
New York State Education Department / 9 Number Systems: Applies properties of signed rational numbers, ordering, and absolute value of rational numbers
Operations with Rational Numbers • Adding integers o Rule- If signs are the same, ADD and keep the signs. If signs are different (one positive and one negative) SUBRTRACT the values and use the sign of the number with the largest absolute value o 4 +18 = 22 o -5 + -6 = -11 o Watch- Adding integers with the same signs o Watch- Adding integers with different signs o Watch- Adding fractions with different signs • Subtracting Integers o Rule- Keep the first number the same, Change subtraction sign to addition, Change the sign of the last number to it’s opposite. Then use addition rules. o OR add the additive inverse o 3 – (-9) 3 + 9 = 12 o -8 – 7 -8 + -7 = -15 o Watch- Subtracing Integers o Watch- Subtracting fractions Practice- Add and subtract integers
New York State Education Department / 10 Number Systems: Applies properties of signed rational numbers, ordering, and absolute value of rational numbers
Operations with Rational Numbers • Multiplying Integers o Rule- If signs are the same the product is positive. If signs are different (one positive and one negative) the product is negative o -4 x -3 = 12 o 8 x (-2) = - 16
• Dividing Integers o Rule- If signs are the same the quotient is positive. If signs are different (one positive and one negative) the quotient is negative o -8 ÷ (-2) = 4 o -15 ÷ 3 = -5
• Watch- Multiplying and dividing integers
• Practice- Multiply and divide integers
New York State Education Department / 11 Number Systems: Solves mathematical and real-world problems involving the four basic operations with rational numbers
Applying operations with rational numbers to mathematical situations (expressions and equations) • Simplify an expression by distributing a negative -3(x + 6) Distribute the -3 to both terms in the parentheses -3(x) + -3(6) Then multiply using integer operation rules -3x - 18 o Watch- Distributive property with integers Practice- Simplify expressions involving integers
• Evaluate expressions with integers Evaluate the expression -2x + 6y if x = -5 and y = -4 Substitute -5 for x and -4 for y and then solve using order of operations -2(-5) + 6(-4) Now multiply 10 – 24 Subtract using integer operations -14 o Watch- Evaluate variable expressions with integers Practice- Evaluate variable expressions involving integers
New York State Education Department / 12 Number Systems: Solves mathematical and real-world problems involving the four basic operations with rational numbers
Applying operations with rational numbers to real-world problems • Calculations with rational numbers are used when recording investment transactions. o Deposits are added to an account balance; money is deposited into the account. o Withdraws are subtracted from an account balance; money is taken out of the account. o Gains are added to an account balance; they are positive returns on the investment. o Losses are subtracted from an account balance; they are negative returns. • Temperature is another way to work with rational numbers o Watch -Use addition and subtraction to solve real-world problems involving decimals o Watch- Negative number word problems
Practice- Integer addition and subtraction word problems
New York State Education Department / 13 Number Systems: Applies and extends understanding of arithmetic and the order of operations to algebraic expressions, equations, and inequalities Order of operations • Order of operations is a rules that defines which procedures to perform first in a given mathematical expression. o Parentheses- Perform operations in the parentheses first o Exponents- Perform any work with exponents or radicals o Multiplication and Division- working from left to right, do all multiplication and division o Addition and Subtraction- - working from left to right, do all addition and subtraction -8 2 2 + 7 ( -4 + 1) Add in parentheses first -8 2 2 + 7 (-3) Exponents -8 4 + 7 (-3) Multiplication -32 + -21 Addition -53 • Watch- Order of operations with integers
Practice- Order of operations with integers
New York State Education Department / 14 Number Systems: Applies and extends understanding of arithmetic and the order of operations to algebraic expressions, equations, and inequalities Equations • Solving equations with rational numbers is the same as solving basic equations. o Solving an equation means getting the variable alone on one side of the equation to find it’s value. o To get the variable alone, you use inverse operations to undo what has been done to the variable. o Addition and subtraction are inverse operations. o Whatever you do to one side of the equation, you must also do the other side to maintain the equality.
x + 5.7 = 2.5 Subtract 5.7 from each side x + 5.7 – 5.7 = 2.5 – 5.7 by using rules for subtraction x = -3.2
• Watch- Solving equations with integers • Watch- Solving equations with fractions and decimals • Practice- Solving equations
New York State Education Department / 15 Number Systems: Applies and extends understanding of arithmetic and the order of operations to algebraic expressions, equations, and inequalities
Inequalities • Solving inequalities with integers is just like solving equations EXCEPT when multiplying or dividing by a negative, you SWITCH the inequality sign. -4x > 12 -4x > 12 Divide by – 4 on both sides of the inequality -4 -4 x < -3 Remember to switch the inequality sign
• Watch-Solving inequalities with integers • Practice- Solving inequalities with integers
New York State Education Department / 16 Number Systems: Uses rational approximations of irrational numbers (e.g., comparing the size of irrational numbers, locating irrational numbers on the number line)
Irrational Numbers • What is an irrational number? o An irrational number is a number that CANNOT be expressed as a fraction. o Non-repeating, non-terminating decimals, non-perfects square roots, and π
• Watch- Identifying irrational numbers • Practice- Sorting rational and irrational numbers
New York State Education Department / 17 Number Systems: Uses rational approximations of irrational numbers (e.g., comparing the size of irrational numbers, locating irrational numbers on the number line) Comparing irrational numbers o To compare irrational numbers, each number needs to be converted to the same form. Typically a decimal approximation is best.
• Rounding non-terminating and non-repeating decimals o Best to round to the thousandth place – 0.42372345 ≈ 0.424 • Decimal approximation for π ≈ 3.14 • Converting non-perfect square roots to decimals- o Watch-Approximating non perfect square roots o Watch- Ordering irrational numbers
• Practice- Estimating square roots • Practice- Estimating square roots 2
New York State Education Department / 18 Number Systems: Uses rational approximations of irrational numbers (e.g., comparing the size of irrational numbers, locating irrational numbers on the number line) Locate irrational numbers on the number line • Since the decimal form of an irrational number is an approximate value, we can approximate where the values appear on the number line.
• Watch- Locating non-perfect square roots on the number line • Practice- Estimating square roots
New York State Education Department / 19 Number Systems: Knows properties of repeating decimal expansions and converts between repeating decimal expansions and rational numbers
Converting repeating decimals to fractions • To convert a repeating decimal to a fraction, 1. Take the digits that repeat and make those digits the numerator 2. The number of digits that repeat denote the number of 9’s that become the denominator.
• 0.456456456…. ≈ 456 999
• Watch- Repeating decimals to fractions
• Practice- Repeating decimals to fractions • Practice- Repeating decimals to fractions 2
New York State Education Department / 20 Number Systems: Performs operations with numbers expressed in scientific notation
Scientific Notation • What is scientific notation? • Scientific notation is a way of expressing really large and small numbers by using power of ten.
• 0.0000000056 is written as 5.6 x 10-9 • 4,920,000,000 is written as 4.92 x 109
• Watch- Scientific Notation • Watch- Orders of magnitude
• Practice- Scientific notation
New York State Education Department / 21 Number Systems: Performs operations with numbers expressed in scientific notation
Operations with scientific notation • Multiply numbers in scientific notation o First, multiply the coefficients. o Next, use laws of exponents to add the exponents for the powers of 10. o Lastly, convert to scientific notation if necessary. • Watch- Multiply numbers in scientific notation • Watch- Multiply numbers in scientific notation 2 • Practice- Multiply numbers in scientific notation
• Divide numbers in scientific notation o First, divide the coefficients. o Next, use laws of exponents to subtract the exponents for the powers of 10. o Lastly, convert to scientific notation if necessary. • Watch- Divide numbers in scientific notation • Watch- Divide numbers in scientific notation 2 • Practice- Divide numbers in scientific notation
New York State Education Department / 22 Number Systems: Works with radicals and exponents involving radicals and rational exponents using the properties of exponents
Evaluate expressions using exponents • If multiplying two powers with a like base, then add the exponents 푥푎 푥푏 = 푥푎+푏 o Watch- Multiplying expressions with exponents o Watch- Multiplying expressions with exponents 2 o Practice- Multiplication with exponents
푥푎 • If dividing two powers with a like base, then subtract the exponents = 푥푎−푏 푥푏 o Watch- Dividing expressions with exponents part 1 o Watch- Multiplying expressions with exponents part 2 o Practice- Divide with exponents
• If raising a power to a power, then multiply the exponents (푥푎)푏 = 푥푎푏 o Watch- Raising a power to a power o Practice- Raising a power to a power
New York State Education Department / 23 Number Systems: Works with radicals and exponents involving radicals and rational exponents using the properties of exponents
Evaluate expressions using exponents • Negative exponents o Expressions with a negative exponent are written as fractions 1 . 푥−푎 = 푥푎
• Watch-Negative exponents
• Practice- Negative exponents
New York State Education Department / 24 Number Systems: Works with radicals and exponents involving radicals and rational exponents using the properties of exponents
Evaluate expressions using exponents • Negative exponents o Expressions with a negative exponent are written as fractions (express as the reciprocal) 1 푥−푎 = 푥푎 o Watch-Negative exponents o Practice- Negative exponents
• Integers raised to rational exponents 푚 푛 푛 푛 First, 푥 푛 = 푥푚 or equivalently 푥푚 = ( 푥)푚
Then, if n is odd 푛 푥푛 = 푥
But, if n is even 푛 푥푛 = 푥 o Watch- Integers raised to fractional exponents Part 1 o Watch- Integers with fractional exponents Part 2 o Practice- Negative exponents
New York State Education Department / 25 Number Systems: Performs arithmetic operations with complex numbers and represents complex numbers and their operations on the complex plane
Complex numbers • What is a complex number? o A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and I is the imaginary unit. o Watch- Imaginary and complex numbers Part 1 o Watch- Imaginary and complex numbers Part 2 o Practice- Introduction to complex numbers
• Plotting complex numbers on the complex plane o Complex Numbers can be represented on what we call the Complex Plane. o The Complex Plane is the same thing as your typical x, y-plane except… . ‘x-axis’ becomes the ‘real axis’ . ‘y-axis’ becomes the ‘imaginary axis’
New York State Education Department / 26 Number Systems: Performs arithmetic operations with complex numbers and represents complex numbers and their operations on the complex plane
Complex numbers o Example: To graph -1 + 3i, you would go one unit to the left and three units up
o Watch- Plotting on the Complex Plane o Practice- Plotting on the Complex Plane
New York State Education Department / 27 Number Systems: Performs arithmetic operations with complex numbers and represents complex numbers and their operations on the complex plane
Complex numbers • Adding and subtracting complex numbers o When adding, or subtracting, two complex numbers, you simply treat i as a variable and combine like terms o Watch- Adding Complex Number Video o Watch- Subtracting Complex Number Video o Practice- Adding and Subtracting Complex Numbers
• Multiplying two complex numbers o When multiplying two complex numbers, the two complex numbers as binomials and then use distributive property and don’t forget that i2 = -1 . Example: (2 – 3i) (-1 + 4i) = -2 + 8i + 3i – 12i2 = -2 + 8i + 3i – 12(-1) = -2 + 8i + 3i + 12 = 10 + 11i o Watch- Multiplying Complex Numbers Part 1 o Watch- Multiplying Complex Numbers Part 2 o Practice- Multiplying Complex Numbers
New York State Education Department / 28 Number Systems: Performs arithmetic operations with complex numbers and represents complex numbers and their operations on the complex plane
Complex numbers • Dividing Complex Numbers o To divide complex numbers, you must multiply by the conjugate. o To find the conjugate of a complex number all you have to do is change the sign between the two terms in the denominator. o Distribute in both the numerator and denominator to remove the parenthesis.
o Watch- Dividing Complex Numbers o Practice- Dividing Complex Numbers
New York State Education Department / 29 Number Systems: Uses complex numbers to solve quadratic equations with real coefficients that have complex solutions
Square Roots of Negative Numbers • Quadratic equations with a negative discriminant have no real number solution. o However, if we extend our number system to allow complex numbers, quadratic equations will always have a solution o Since the solution to a quadratic equation involves the square root of the discriminant, we need to look at square roots of negative numbers. o Example: −9 = +3푖 표푟 − 3푖
o Watch- Imaginary roots of negative numbers
• Negative square roots and the Quadratic Equation o Solve the Quadratic Equation like normal, except remember how to find the square root of negative numbers.
o Watch- Quadratic equation and complex numbers
New York State Education Department / 30 Number Systems: Represents and models with vector quantities and performs operation on vectors both algebraically and graphically
Vectors • What is a vector? o Has both magnitude and direction. o The length of the line shows its magnitude and the arrowhead points in the direction. o Watch- Understanding vectors o Practice- Identifying magnitude of vectors
Operations with Vectors • Adding vectors o We can add vectors by graphing them and joining them head-to-tail or adding the coordinates o Watch- Adding Vectors • Subtracting vectors o First, reverse the direction of the vector we want to subtract. o Then add them as usual o Practice- Adding and Subtracting vectors
New York State Education Department / 31 Number Systems: Represents and models with vector quantities and performs operation on vectors both algebraically and graphically
Multiplying Vectors by a Scalar • What is a scalar? o A scalar is just an ordinary number. o A vector is often written in bold, so we know it is not a scalar. o Example: kb is actually the scalar k times the vector b
• When we multiply a vector by a scalar it is called “scaling” a vector, because we change how big or small the vector is o Watch- Multiplying vectors by a scalar o Practice- Multiply a vector by a scalar
New York State Education Department / 32 Number Systems: Performs basic operations on matrices and uses matrices in applications Matrices • What are matrices? o A Matrix is a two-dimensional arrangement of numbers. o A Matrix is described by the number of values found in the rows and columns. o Example: This is a 2 by 3 matrix o Matrices are just ways to represent numbers. o Watch- Understanding matrices o Practice- Matrix dimensions
• Adding and subtracting matrices o To add or subtract matrices: perform the operation to the numbers in the matching position. o Matrices must be the same size o Watch- Matrix addition and subtraction o Practice- Adding and Subtracting matricies o Practice- Adding and Subtracting matricies practice 2
New York State Education Department / 33 Number Systems: Performs basic operations on matrices and uses matrices in applications
Matrices • Multiplying matrices o We can multiply a matrix by some value (constant). o Each value in the matrix is multiplied by the constant.
o Watch- Multiplying matrices o Practice- Multiplying matrices
• Applications with matrices o Matrices are used in representing the real-world data’s like the traits of people’s population, habits, etc.
o Watch- Applications with matrices o Practice- Applications of matrices
New York State Education Department / 34 Agenda
Introduction to Competency
Content Review and Sample Problems
Additional Practice
/ 35 Additional Practice Problems Number and Quantity
Exponents: For any real x, which of the following is equivalent to x6 + x5 + x4?’
A. x4(x3 + x + 1) B. x4(x2 + 1) C. x5(x2 + x + 1) D. x4(x2 + x + 1) E. x3(x2 + x + 1)
New York State Education Department / 36 Additional Practice Problems Number and Quantity
Exponents Solution D
The GCF is x4 so once factored out, the solution is x4(x2 + x + 1). Remember, when multiplying variables, we add the exponents.
New York State Education Department / 37 Additional Practice Problems Number and Quantity
Fractions: Jessica gets her favorite shade of purple paint by mixing 1/3 cup of blue paint with 1/2 cup of red paint. How many cups of blue paint does Jessica need to make 20 cups of her favorite purple paint? A. 8 B. 12 C. 10 D. 6
New York State Education Department / 38 Additional Practice Problems Number and Quantity
Fractions Solution A
8 cups of blue paint. One batch of purple paint contains 1/3 cup of blue paint and 1/2 cup of red paint. This will make a total of 1/2 + 1/3 = 5/6 of a cup of purple paint. In order to make 20 cups of purple paint, we need 20 ÷ 5/6 = 24 batches. Each batch has 1/3 of a cup of blue paint so 24 batches will contain 24 × 1/3 = 8 cups of blue paint. Each batch has 1/2 cup of red paint so 24 batches will contain 24 × 1/2 = 12 cups of red paint.
New York State Education Department / 39 Additional Practice Problems Number and Quantity
Solve Problems involving the four basic operations:
The average of the integers 24, 6, 12, x and y is 11. What is the value of the sum x + y? A. 11 B. 17 C. 13 D. 15 E. 9
New York State Education Department / 40 Additional Practice Problems Number and Quantity
Solving Problems Including the Four Basic Operations: Solution C
The average of the 5 numbers is (24 + 6 + 12 + x + y)/5. (24 + 6 + 12 + x + y)/5 = 11 24 + 6 + 12 + x + y = 11 · 5 24 + 6 + 12 + x + y = 55 x + y = 13
New York State Education Department / 41 Additional Practice Problems Number and Quantity
Simplifying Fractions with Variables:
Simplify the following,
A. 16z + 4 B. 16z C. z D. 2z + 2 E. 12z
New York State Education Department / 42 Additional Practice Problems Number and Quantity
Simplifying Fractions with Variables Solution: E
(16z3/2z2) can be simplified to 8z and (4z5/z4) can be simplified to 4z so we are then left with 8z + 4z which will give us 12z.
New York State Education Department / 43 Additional Practice Problems Number and Quantity
Operations with Numbers in Scientific Notation
Simplify. 2.1×10-5 + 0.038×10-3 + 0.41×10-4 =
A. 1E-4 B. 1E-5 C. 1E-3 D. 2.1E-4 E. 2.1 E-5
New York State Education Department / 44 Additional Practice Problems Number and Quantity
Operations with Numbers in Scientific Notation Solution: A
First we want to rewrite all of our numbers in scientific notation 2.1×10-5 = 2.1×10-5 0.038×10-3 = 3.8×10-5 0.41×10-4 = 4.1×10-5
Now we can rewrite our problem. 2.1×10-5 + 3.8×10-5 + 4.1×10-5
Since all of our terms are raised to the negative fifth power, we can factor out the “times ten to the negative fifth” and add 2.1 + 3.8 +4.1 = (2.1 + 3.8 + 4.1)×10-5= 10×10-5
To rewrite 10×10-5in scientific notation we would have 1x10-4 which is the same as 1E-4.
New York State Education Department / 45 Additional Practice Problems Number and Quantity
Simplifying Complex Numbers
Rewrite the complex number 푧 = −16 + 4푖6 in a standard form z = a + bi
A. z = 4 - 4i B. z = 4 + 4i C. z = -4 - 4i D. z = -4 + 4i
New York State Education Department / 46 Additional Practice Problems Number and Quantity
Simplifying Complex Numbers Solution: D First we can simplify −16 to 4i
Then we can rewrite 4푖6 as 4(푖2)3 which will give us 4(-1)3 we can then simplify 4(-1)3 further to -4
When we write this is standard form we will have -4 + 4i
New York State Education Department / 47 Additional Practice Problems Number and Quantity
Adding Complex Numbers
Calculate z = z1 + z2: 2 z1 = 3i - 2i + 1 2 z2 = -i - i + 5
a) z = 4 - 3i b) z = 3 - i c) z = -2 + i d) z = -1 - 3i
New York State Education Department / 48 Additional Practice Problems Number and Quantity
Adding Complex Numbers Solution: A
2 2 z = z1 + z2 = 3i - 2i + 1 + (-i - i + 5) = 2i2 - 3i + 6 Combine like terms = 2(-1) - 3i + 6 Change i2 to -1 = -2 - 3i + 6 Multiply 2(-1) = 4 - 3i Combine like terms
New York State Education Department / 49 Additional Practice Problems Number and Quantity
Setting up a Matrix
2 2 z = z1 + z2 = 3i - 2i + 1 + (-i - i + 5) = 2i2 - 3i
New York State Education Department / 50