7th Grade Math Expressions 2015-11-17 www.njctl.org

Table of Contents Mathematical Expressions Click on a topic to 
go to that section. Order of Operations [This object is a pull tab] Teacher Notes Vocabulary Words are bolded in the presentation. The text box the word is in is then linked to the page at the end of the presentation with the word defined on it. The Distributive Property Like Terms Translating Words Into Expressions Evaluating Expressions Glossary & Standards

Mathematical Expressions Return to Table of Contents

Expressions Algebra extends the tools of arithmetic, which were developed to work with numbers, so they can be used to solve real world problems. This requires first translating words from your everyday language (i.e. English, Spanish, French) into mathematical expressions. Then those expressions can be operated on with the tools originally developed for arithmetic.

An Expression may contain: numbers, variables, mathematical operations Expressions An Expression may contain: numbers, variables, mathematical operations Example: 4x + 2 is an algebraic expression.

Circle the terms of this expression. What is a Term? Terms of an expression are the parts of the expression which are separated by addition or subtraction. Circle the terms of this expression. Example: 4x + 2 Circle the terms and then click to check. There are two terms: 4x; 2

What is a Constant? A constant is a fixed value, a number on its own, whose value does not change. A constant may either be positive or negative. Example: 4x + 2 Circle the constant and then click to check. In this expression 2 is the constant.

What is a Variable? A variable is any letter or symbol that represents a changeable or unknown value. Example: 4x + 2 Circle the variable and then click to check. In this expression x is the variable.

What is a Coefficient? A coefficient is a number multiplied by a variable. It is located in front of the variable. Example: 4x + 2 Circle the coefficient and then click to check. In this expression 4 is the coefficient.

If a variable contains no visible coefficient, the coefficient is 1. [This object is a pull tab] Math Practice MP6: Attend to precision. Continuously emphasize that the coefficient of 1 or -1 exists when when only the variable (or the negative variable) is given. If a variable contains no visible coefficient, the coefficient is 1. Example 1: x + 7 is the same as (1)x + 7 Example 2: -x + 7 is the same as (-1)x + 7

In 2x - 12, the variable is "x". 1 True False True Answer [This object is a pull tab] Answer True True False

In 6y + 20, the variable is "y". 2 True False True Answer [This object is a pull tab] Answer True True False

In 3x + 4, the coefficient is 3. [This object is a pull tab] Answer True True False

In 9x + 2, the coefficient is 2. 4 In 9x + 2, the coefficient is 2. [This object is a pull tab] Answer False True False

What is the constant in 7x - 3? 5 What is the constant in 7x - 3? [This object is a pull tab] Answer D A 7 B x C 3 D - 3

What is the coefficient in - x + 3? 6 What is the coefficient in - x + 3? [This object is a pull tab] Answer C A none B 1 C -1 D 3

7 x has a coefficient. True True False Answer [This object is a pull tab] Answer True True False

19 has a coefficient. 8 True False False Answer [This object is a pull tab] Answer False True False

Order of Operations Return to Table 
of Contents

Mathematics has its grammar, just like any language. Order of Operations Mathematics has its grammar, just like any language. Grammar provides the rules that allow us to write down ideas so that a reader can understand them. A critical set of those rules is called the order of operations.

Order of Operations The order of operations allows us to read an expression and interpret it as intended. It lets us understand what the author meant. For instance, the below expression could mean many different things without an agreed upon order of operations. How would you evaluate this expression? (5-8)(5)(3)-42÷2+8÷4+(3-2)

Use Parentheses Parentheses will make your life much easier. Each time you do an operation, keep the result in parentheses until you use it for the next operation. You'll be able to read your own work, and avoid mistakes. When you're done, read each step you did and you should be able to check your work. Also, when you substitute a value into an expression, put it in parentheses first...that'll save you a lot of trouble.

Order of Operations (5-8)(5)(3)-42÷2+8÷4+(3-2) Do all operations in parentheses first. (-3)(5)(3)-42÷2+8÷4+(1) Then, do all exponents and roots. (-3)(5)(3)-(16)÷2+8÷4+1 Then, do all multiplication and division. (-45)-(8)+(2)+1 Then, do all addition and subtraction. -50

Order of Operations One acronym used for the order of operations is PEMDAS which stands for: Parentheses Exponents/Roots Multiplication/Division Addition/Subtraction This order helps you read an expression...but it also helps you write expressions that others can read. Since parentheses are always done first, you can always eliminate confusion by putting parentheses around what you want to be done first. They may not be needed, but they don't ever hurt.

-7 + (-3)[5 - (-2)] Order of Operations Let's simplify this step by step... What should you do first? 5 - (-2) = 5 + 2 = 7 What should you do next? (-3)(7) = -21 What is your last step? -7 + (-21) = -28 -7 + (-3)[5 - (-2)] click to reveal click to reveal click to reveal

Order of Operations Let's simplify this step by step... What should you do first? What should you do second? Click to Reveal Click to Reveal

Order of Operations Let's simplify this step by step... What should you do third? What should you do last? Click to Reveal Click to Reveal

-12÷3(-4) 9 Simplify the expression. 16 Answer [This object is a pull tab] Answer 16 9 Simplify the expression. -12÷3(-4)

Simplify the expression. [-1 - (-5)] + [7(3 - 8)] 10 Simplify the expression. [-1 - (-5)] + [7(3 - 8)] [This object is a pull tab] Answer -31

Simplify the expression. 40 - (-5)(-9)(2) 11 Simplify the expression. 40 - (-5)(-9)(2) [This object is a pull tab] Answer -50

Simplify the expression. 5.8 - 6.3 + 2.5 12 Simplify the expression. 5.8 - 6.3 + 2.5 [This object is a pull tab] Answer 2

Simplify the expression. -3(-4.7)(5-3.2) 13 Simplify the expression. -3(-4.7)(5-3.2) [This object is a pull tab] Answer 25.38

Simplify the expression. 14 Simplify the expression. [This object is a pull tab] Answer -4

Complete the first step of simplifying. What is your answer? 15 Complete the first step of simplifying. What is your answer? [This object is a pull tab] Answer -12.4 - 6[4.1 - (-5.3)] [3.2 + (-15.6)] - 6[4.1 - (-5.3)]

Complete the next step of simplifying. What is your answer? 16 Complete the next step of simplifying. What is your answer? [This object is a pull tab] Answer -12.4 - 6[9.4] [3.2 + (-15.6)] - 6[4.1 - (-5.3)] click to reveal 
step from 
previous slide -12.4 - 6[4.1 - (-5.3)]

Complete the next step of simplifying. What is your answer? 17 Complete the next step of simplifying. What is your answer? [This object is a pull tab] Answer -12.4 - 56.4 [3.2 + (-15.6)] - 6[4.1 - (-5.3)] click to reveal steps from previous slides -12.4 - 6[4.1 - (-5.3)] -12.4 - 6[9.4]

Complete the next step of simplifying. What is your answer? 18 Complete the next step of simplifying. What is your answer? [This object is a pull tab] Answer -68.8 [3.2 + (-15.6)] - 6[4.1 - (-5.3)] click to reveal steps from previous slides -12.4 - 6[4.1 - (-5.3)] -12.4 - 6[9.4] -12.4 - 56.4

Simplify the expression. 19 Simplify the expression. [This object is a pull tab] Answer 1.76

Simplify the expression. 20 Simplify the expression. [This object is a pull tab] Answer 40 3 4

Simplify the expression 21 Simplify the expression [This object is a pull tab] Answer -45

Simplify the expression 22 Simplify the expression [This object is a pull tab] Answer 27

Simplify the expression (-4.75)(3) - (-8.3) 23 Simplify the expression (-4.75)(3) - (-8.3) [This object is a pull tab] Answer -5.95

Order of Operations -62.75 Solve this one in your groups. Answer [This object is a pull tab] Answer -62.75 Solve this one in your groups.

Order of Operations -21.25 How about this one? Answer [This object is a pull tab] Answer -21.25 How about this one?

Simplify the expression 24 Simplify the expression [This object is a pull tab] Answer 20.8

Simplify the expression [(-3.2)(2) + (-5)(4)][4.5 + (-1.2)] 25 Simplify the expression [(-3.2)(2) + (-5)(4)][4.5 + (-1.2)] [This object is a pull tab] Answer -87.12

Simplify the expression 26 Simplify the expression [This object is a pull tab] Answer -23.2

Simplify the expression 27 Simplify the expression [This object is a pull tab] Answer 23

Simplify the expression 28 Simplify the expression [This object is a pull tab] Answer -20.50

Evaluate the expression (9 - 13)2 ÷ 2(3 - 1) + 9 ∙ 8 - (5 + 6) 29 Evaluate the expression (9 - 13)2 ÷ 2(3 - 1) + 9 ∙ 8 - (5 + 6) [This object is a pull tab] Answer 65

Evaluate the expression 7 ∙ 9 − (7 − 4)3 ÷ 9 + (10 − 12) 30 Evaluate the expression 7 ∙ 9 − (7 − 4)3 ÷ 9 + (10 − 12) [This object is a pull tab] Answer 58

Evaluate the expression (7 + 3)2 ÷ 25 + 4 ∙ 2 - (7 + 8) 31 Evaluate the expression (7 + 3)2 ÷ 25 + 4 ∙ 2 - (7 + 8) [This object is a pull tab] Answer -3

Order of Operations and Fractions The simplest way to work with fraction is to imagine that the numerator and the denominator are each in their own set of parentheses. Before you divide the numerator by the denominator, you must have them both in simplest form. And, then you must be very careful about what you can do with them.

Order of Operations and Fractions For instance, a common error is shown below: I CANNOT divide the top and the bottom by x to get: Rather, I have to think of the denominator (1+x) as being in parentheses. Until I can simplify that further (which I can't) this is the simplest form. x 1+x x 1 1+1 x (1+x) x 1+x

Order of Operations and Fractions How would you evaluate this expression? (4)(3)-32÷5+6÷2+(5-8) 7-8

Order of Operations (4)(3)-32÷5+6÷2+(5-8) 7-8 First, recognize that terms in a denominator act like they are in parentheses. Then, do all operations in parentheses first. (Keep all results in parentheses until the next operation.) Then, do all exponents and roots. (4)(3)-32÷5+6÷2+(5-8) (7-8) (4)(3)-32÷5+6÷2+(-3) (-1) (4)(3)-(9)÷5+6÷2+(-3) (-1)

Order of Operations (4)(3)-9÷5+6÷2+(-3) (-1) Then, all multiplication and division Then, do all addition and subtraction. Then, divide the numerator by the denominator. (12)-(1.8)+(3)+(-3) (-1) (10.2) (-1) (-10.2)

Simplify the expression. 32 Simplify the expression. [This object is a pull tab] Answer -4

Evaluate the expression 3(5 − 3)3 + 5(7 + 5) − 9 2 ∙ 5 + 5 33 Evaluate the expression 3(5 − 3)3 + 5(7 + 5) − 9 2 ∙ 5 + 5 [This object is a pull tab] Answer 5

Evaluate the expression 2(9 − 4)2 + 8 ∙ 6 − 3 3 ∙ 42 + 2 34 Evaluate the expression 2(9 − 4)2 + 8 ∙ 6 − 3 3 ∙ 42 + 2 [This object is a pull tab] Answer 1.9 = 19 10

Evaluate the expression −4(2 − 8)2 + 7(−3) + 15 5(25 − 12) 35 Evaluate the expression −4(2 − 8)2 + 7(−3) + 15 5(25 − 12) [This object is a pull tab] Answer −1.5 = − 3 2

36 Select the correct number from each group of numbers to complete the equation. [This object is a pull tab] Answer A 2 H -3/4 _____ _____ A 2 E 2 B -2 F -2 C 3/4 G 4/3 D -4/3 H -3/4 From PARCC EOY sample test non-calculator #6

B, Step 2 has the first error. 37 Chris made at least one error as she found the value of this expression. Identify the step in which Chris made her first error. After identifying the step with the first error, write the corrected steps and find the final answer. [This object is a pull tab] Answer B, Step 2 has the first error. A Step 1: 2(-20) + 3(-25) + 5(20) + 4(50) B Step 2: (3 + 2)(-20 + -25) + (5 + 4)(20 + 50) C Step 3: 5(-45) + 9(70) D Step 4: -225 + 630 E Step 5: 405 From PARCC PBA sample test calculator #5

The Distributive Property Return to Table 
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This slide & the next one address MP4. Area Model [This object is a pull tab] Math Practice This slide & the next one address MP4. Ask: What connections do you see between the distributive property and the area of a rectangle? (MP4) Write an expression for the area of a rectangle whose width is 4 and whose length is x + 2 4 x 2

Area Model 4 x 2 You can think of this as being two rectangles. One has an area of (4)(x) and the other has an area of (4)(2) An expression for the total area would be 4x + 8 Or as one large rectangle of area (4)(x+2).

Distributive Property Finding the area of each rectangle demonstrates the distributive property. 4(x + 2) 4(x) + 4(2) 4x + 8 The 4 is distributed to each term of the sum (x + 2). [This object is a pull tab] Math Practice The next 2 slides address MP1, MP5 & MP6. Ask: What information do you know? (MP1) What does the distributive property mean? (MP6) Can you do this mentally? (MP5)

Distributive Property Now you try: 6(x + 4) =   5(x + 7) = [This object is a pull tab] Answer 6x + 24 5x + 35

Write an expression equivalent to: Distributive Property Write an expression equivalent to: [This object is a pull tab] Answer 2(x - 1) 2x - 2 4(x - 8) 4x - 32 2(x - 1) 4(x - 8)

Distributive Property a(b + c) = ab + ac Example: 2(x + 3) = 2x + 6 (b + c)a = ba + ca Example: (x + 7)3 = 3x + 21 a(b - c) = ab - ac Example: 5(x - 2) = 5x - 10 (b - c)a = ba - ca Example: (x - 3)6 = 6x - 18

Distributive Property The Distributive Property can be used to eliminate parentheses, so you can then combine like terms. For example:  3(4x - 6)     3(4x) - 3(6)     12 x - 18 

Distributive Property The Distributive Property can be used to eliminate parentheses, so you can then combine like terms. For example: -2(x + 3) -2(x) + -2(3) -2x + -6 -2x - 6  

Distributive Property The Distributive Property can be used to eliminate parentheses, so you can then combine like terms. For example: -3(4x - 6) -3(4x) - -3(6) -12x - -18 -12x + 18

Simplify 4(7x + 5) using the distributive property. 38 Simplify 4(7x + 5) using the distributive property. [This object is a pull tab] Answer C A 7x + 20 B 28x + 5 C 28x + 20

Simplify -6(2x + 4) using the distributive property. 39 Simplify -6(2x + 4) using the distributive property. [This object is a pull tab] Answer D A 12x + 4 B -12x + 24 C 12x - 4 D -12x - 24

Simplify -3(5m - 8) using the distributive property. 40 Simplify -3(5m - 8) using the distributive property. [This object is a pull tab] Answer B A -35m - 8 B -15m + 24 C 15m - 24 D -15m - 24

Distributing a Negative Sign A negative sign outside of the parentheses represents a multiplication by (-1). For example:   -(3x + 4) (-1)(3x + 4) (-1)(3x) + (-1)(4) -3x - 4

Use the Distributive Property to simplify the expression. -(6x - 7) 41 Use the Distributive Property to simplify the expression. -(6x - 7) [This object is a pull tab] Answer A A -6x + 7 B -6x - 7 C 6x - 7 D 6x + 7

Use the Distributive Property to simplify the expression. -(-x - 9) 42 Use the Distributive Property to simplify the expression. -(-x - 9) [This object is a pull tab] Answer D A -x + 9 B x - 9 C -x - 9 D x + 9

Use the Distributive Property to simplify the expression. -(2x + 5) 43 Use the Distributive Property to simplify the expression. -(2x + 5) [This object is a pull tab] Answer B A -2x + 5 B -2x - 5 C 2x - 5 D 2x + 5

Use the Distributive Property to simplify the expression. -(-5x + 3) 44 Use the Distributive Property to simplify the expression. -(-5x + 3) [This object is a pull tab] Answer C A -5x + 3 B -5x - 3 C 5x - 3 D 5x + 3

45 ) 4(x + 6) is the same as 4 + 4(6). False True False Answer [This object is a pull tab] Answer False True False

46 Use the distributive property to rewrite the expression without parentheses. 2(x + 5) [This object is a pull tab] Answer B A 2x + 5 B 2x + 10 C x + 10 D 7x

47 Use the distributive property to rewrite the expression without parentheses. 3(x - 6) [This object is a pull tab] Answer B A 3x - 6 B 3x - 18 C x - 18 D 15x

48 Use the distributive property to rewrite the expression without parentheses. -4(x - 9) [This object is a pull tab] Answer C A -4x - 36 B 4x - 36 C -4x + 36 D 32x

49 Use the distributive property to rewrite the expression without parentheses. -(4x - 2) [This object is a pull tab] Answer C A -4x - 2 B 4x - 2 C -4x + 2 D 4x + 2

50 Use the distributive property to rewrite the 
expression without parentheses. 0.6(3.1x + 17) [This object is a pull tab] Answer C A 186x + 102 B 1.86x + 17 C 1.86x + 10.2 D .631x + .617

51 Use the distributive property to rewrite the 
expression without parentheses. 0.5(10x - 15) [This object is a pull tab] Answer C A 5x - 75 B 10x - 7.5 C 5x - 7.5 D 5x - 15

52 Use the distributive property to rewrite the 
expression without parentheses. 1.3(6x + 49) [This object is a pull tab] Answer B A 78x + 637 B 7.8x + 63.7 C 7.8x + 49 D 1.36x + 1.349

Like Terms Return to Table of Contents

Like Terms Like Terms: Terms in an expression that have the same 
variable(s) raised to the same power Like Terms 6x and 2x 5y and 8y 4x2 and 7x2 NOT Like Terms 6x and x2 5y and 8 4x2 and x4

Identify all of the terms like 5y. 53 Identify all of the terms like 5y. [This object is a pull tab] Answer C, D, E A 5 B 4y2 C 18y D 8y E -1y

Identify all of the terms like 8x. 54 Identify all of the terms like 8x. [This object is a pull tab] Answer A, E A 5x B 4x2 C 8y D 8 E -10x

Identify all of the terms like 8xy. 55 Identify all of the terms like 8xy. [This object is a pull tab] Answer C, E A 5x B 4x2y C 3xy D 8y E -10xy

Identify all of the terms like 2y. 56 Identify all of the terms like 2y. [This object is a pull tab] Answer A, C, E A 51y B 2w C 3y D 2x E -10y

Identify all of the terms like 14x2. 57 Identify all of the terms like 14x2. [This object is a pull tab] Answer B, E A 5x B 2x2 C 3y2 D 2x E -10x2

Identify all of the terms like 0.75x5. 58 Identify all of the terms like 0.75x5. [This object is a pull tab] Answer B, E A 75x B 75x5 C 3y2 D 2x E -10x5

Identify all of the terms like 2 3 x 59 Identify all of the terms like [This object is a pull tab] Answer A, D A 5x B 2x2 C 3y2 D 2x E -10x2

Identify all of the terms like 1 4 x2 60 Identify all of the terms like [This object is a pull tab] Answer C, D A 5x B 2x C 3x2 D 2x2 E -10x

Combining Like Terms This slide addresses MP2 & MP6 [This object is a pull tab] Math Practice This slide addresses MP2 & MP6 Ask: What do like terms mean? (MP6) When we combine them, what changes mathematically? How do they change? (MP2) Emphasize the addition/subtraction of the coefficients w/ the degree of the variable remaining the same. (MP6) Simplify by combining like terms 6x + 3x (6 + 3)x 9x  Notice when combining like terms you add/subtract the coefficients 
but the variable remains the same.

Simplify by combining like terms [This object is a pull tab] Math Practice MP6: Attend to precision. Emphasize the addition/subtraction of the coefficients w/ the degree of the variable remaining the same. Simplify by combining like terms 4 + 5(x + 3) 4 + 5(x) + 5(3) 4 + 5x + 15 5x + 19  Notice when combining like terms you add/subtract the coefficients 
but the variable remains the same.

Simplify by combining like terms 7y - 4y (7 - 4)y 3y Notice when combining like terms you add/subtract the coefficients 
but the variable remains the same. [This object is a pull tab] Math Practice MP6: Attend to precision. Emphasize the addition/subtraction of the coefficients w/ the degree of the variable remaining the same.

Simplify the expression 8x + 9x. 61 Simplify the expression 8x + 9x. [This object is a pull tab] Answer B A x B 17x C -x D cannot be simplified

Simplify the expression 7y - 5y. 62 Simplify the expression 7y - 5y. [This object is a pull tab] Answer A A 2y B 12y C -2y D cannot be simplified

Simplify the expression 6 + 2x + 12x. 63 Simplify the expression 6 + 2x + 12x. A 6 + 10x [This object is a pull tab] Answer C B 20x C 6 + 14x D cannot be simplified

Simplify the expression 7x + 7y. 64 Simplify the expression 7x + 7y. [This object is a pull tab] Answer C A 14xy B 14x C 14y D cannot be simplified

The next 10 slides address MP3 [This object is a pull tab] Math Practice The next 10 slides address MP3 Ask: Does the answer seem reasonable? Why or why not? How can you prove that the answer is true/false? Teachers: Use the Math Practice tab to assist with questioning on the next 10 slides

65 ) 8x + 3x is the same as 11x. True True False Answer [This object is a pull tab] Answer True True False

66 ) 7x + 7y is the same as 14xy. False True False Answer [This object is a pull tab] Answer False True False

67 ) 4x + 4x is the same as 8x2. False True False Answer [This object is a pull tab] Answer False True False

68 ) -12y + 4y is the same as -8y. True True False Answer [This object is a pull tab] Answer True True False

69 ) -3 + y + 5 is the same as 2y. False True False Answer [This object is a pull tab] Answer False True False

70 ) -3y + 5y is the same as 2y. True True False Answer [This object is a pull tab] Answer True True False

) 7x - 3(x - 4) is the same as 4x +12. 71 ) 7x - 3(x - 4) is the same as 4x +12. [This object is a pull tab] Answer True True False

72 ) 7 + 5(x + 2) is the same as 5x + 9. False True False Answer [This object is a pull tab] Answer False True False

73 ) 4 + 6(x - 3) is the same as 6x -14. True True False Answer [This object is a pull tab] Answer True True False

) 3x + 2y + 4x + 12 is the same as 9xy + 12. 74 ) 3x + 2y + 4x + 12 is the same as 9xy + 12. True [This object is a pull tab] Answer False False

Which expression represents the perimeter of the home plate? 75 The lengths of the sides of home plate in 
baseball are represented by the expressions in 
the accompanying figure. Which expression represents the perimeter of the home plate? [This object is a pull tab] Answer D yz y x A 5xyz B 2x + 2yz C 2x + 3yz D 2x + 2y + yz

Find an expression for the perimeter of the octagon. 76 Find an expression for the perimeter of the octagon. [This object is a pull tab] Answer B A x +24 x x+2 x+3 7 B 6x + 24 C 24x D 30x

Translating Words Into Expressions Return to Table 
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Translating Between Words and Expressions [This object is a pull tab] Math Practice This entire lesson addresses MP1 & MP2 & MP4. Ask: What are the key words in the problem/question? (MP1) How could you represent the problem with symbols & numbers? (MP2) What connections do you see between this word problem and the phrases in the previous examples? (MP4) Key to solving algebra problems is translating words into mathematical expressions. The two steps to doing this are: Taking English words and converting them to mathematical words. Taking mathematical words and converting them into mathematical symbols. We're going to practice the second of these skills first, and then the first...and then combine them.

List words that indicate addition. [This object is a pull tab] Answer Sum Total Add Plus Increased by More Than Altogether

List words that indicate subtraction. [This object is a pull tab] Answer Minus Difference Take away  Less than  Subtract Decreased by Less Fewer than Subtracted from

List words that indicate multiplication. [This object is a pull tab] Answer Product Times Of Twice... Double... Multiplied by

List words that indicate division. [This object is a pull tab] Answer Divided by Quotient of Half... Fractions Divisible by Divisibility

Less and Less Than MP6: Attend to precision. [This object is a pull tab] Math Practice MP6: Attend to precision. Make sure that students know the order of the numbers when "less than" (and "more than" w/ addition) appear. Additional Q's to help: What number is _ less than _? How did you get your answer? (MP2) - Note: Fill in the blanks with any numbers. Less and Less Than Be aware of the difference between "less" and "less than". For example: "Eight less three" and "three less than eight" are equivalent expressions, so what is the difference in wording? Eight less three: 8 - 3 Three less than eight: 8 - 3 When you see "less than", take the second number minus the 
first number.

Reverse the Order As a rule of thumb, if you see the words "than" or "from" it means you have to reverse the order of the two numbers or variables when you write the expression. Examples: 8 less than b means b - 8 3 more than x means x + 3 x less than 2 means 2 - x

Multiplication The many ways to represent multiplication. How do you represent "three times a"? (3)(a)  3(a)   3 a  3a The preferred representation is 3a. When a variable is being multiplied by a number, the number (coefficient) is always written in front of the variable. The following are not allowed:  3xa ...  The multiplication sign looks like another variable  a3 ... The number is always written in front of the variable

How do you represent "b divided by 12"? Representation of Division How do you represent "b divided by 12"? b ÷ 12 b ∕ 12 b 12

Sort the words by operation. [This object is a pull tab] Answer The example on this slide addresses MP6. Quotient Product Sum Total Ratio Difference Less Than More Fraction Multiply Per Multiply Ratio Sum Total Less Than Quotient Difference More Per Fraction Product

Translate the Words into Algebraic Expressions Using the Red Characters . 1 2 3 4 5 6 7 8 9 + - ÷ j [This object is a pull tab] Answer * Note numbers and symbols are infinitely cloned. 3j 8 ÷ j 7 - j j + 5 j - 4 Three times j Eight divided by j j less than 7 5 more than j 4 less than j

Write the Expression 23 + m The sum of twenty-three and m Answer [This object is a pull tab] Answer 23 + m The sum of twenty-three and m

Write the Expression 4k The product of four and k Answer [This object is a pull tab] Answer 4k The product of four and k

Write the Expression d - 24 Twenty-four less than d Answer [This object is a pull tab] Answer d - 24 Twenty-four less than d

**Remember, sometimes you need to use parentheses for a quantity.** Write the Expression [This object is a pull tab] Answer 4(8-j) **Remember, sometimes you need to use 

parentheses for a quantity.** Four times the difference of eight and j

Write the Expression 7w 12 The product of seven and w, divided by 12 [This object is a pull tab] Answer 7w 12 The product of seven and w, divided by 12

Write the Expression (6+p)2 The square of the sum of six and p Answer [This object is a pull tab] Answer (6+p)2 The square of the sum of six and p

77 The sum of 100 and h B A 100 B 100 + h C 100 - h D 100 + h h 200 [This object is a pull 
tab] Answer B A 100 h B 100 + h C 100 - h D 100 + h 200

The quotient of 200 and the quantity of p times 7 78 The quotient of 200 and the quantity of p times 7 [This object is a pull tab] Answer A A 200 7p B 200 - (7p) C 200 ÷ 7p D 7p 200

Thirty five multiplied by the quantity r less 45 79 Thirty five multiplied by the quantity r less 45 [This object is a pull tab] Answer D A 35r - 45 B 35(45) - r C 35(45 - r) D 35(r - 45)

80 a less than 27 A 27 - a B a 27 C a - 27 D 27 + a A Answer [This object is a pull tab] Answer A A 27 - a B a 27 C a - 27 D 27 + a

Translating English Sentences to Mathematical Sentences Now, we know how to translate a mathematical sentence in words to a mathematical expression in symbols. Next, we need to practice translating from English sentences to mathematical sentences. Then, we can translate from English sentences to mathematical expressions.

Translating From English Sentences Write a mathematical sentence, in words, for the below. Then translate that into a mathematical expression. The total amount of money my friends have, if each of my seven friends has x dollars. click for mathematical sentence 7 multiplied by x click for mathematical expression 7x

Translating From English Sentences Write a mathematical sentence, in words, for the below. Then translate that into a mathematical expression. My age if I am x years older than my 12 year old brother click for mathematical sentence 12 added to x click for mathematical expression x + 12

Translating From English Sentences Write a mathematical sentence, in words, for the below. Then translate that into a mathematical expression. How many apples each person gets if starting with 15 apples, 5 are eaten and the rest are divided equally by 2 friends. click for mathematical sentence The total of 15 minus 5 divided by 2 click for mathematical expression (15-5)/2

Translating From English Sentences Write a mathematical sentence, in words, for the below. Then translate that into a mathematical expression. My speed if I travel d meters in s seconds click for mathematical sentence d divided by s click for mathematical expression d/s

Translating From English Sentences Write a mathematical sentence, in words, for the below. Then translate that into a mathematical expression. How much money I make if I earn r dollars per hour and work for 28 hours click for mathematical sentence r multiplied by 28 click for mathematical expression 28r

Translating From English Sentences Write a mathematical sentence, in words, for the below. Then translate that into a mathematical expression. My height if I am 6 inches less than twice the height of my sister, who is h inches tall click for mathematical sentence 6 less than two times h click for mathematical expression 2h - 6

81 The total number of jellybeans if Mary had 5 jellybeans for each of 4 friends. [This object is a pull tab] Answer C A 5 + 4 B 5 - 4 C 5 x 4 D 5 ÷ 4

If n + 4 represents an odd integer, the next larger 82 If n + 4 represents an odd integer, the next larger odd integer is represented by [This object is a pull tab] Answer D A n + 2 B n + 3 C n + 5 D n + 6 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

83 Jenny earns $15 an hour waitressing plus $150 in tips 
on a Friday night. What expression represents her 
total earnings? [This object is a pull tab] Answer C A 150 - 15h B h 150 C 15h + 150 D 15 + h

84 Bob's age if he is 2 years less than double the age of his brother who is z years old? Answer C A 2z + 2 B z 2 C 2z - 2 D z - 2

Variables When choosing a variable, there are some letters that 
are often avoided: l, i, t, o, O, s, S Why might these letters be avoided? It is best to avoid using letters that might be confused for numbers or operations. In the case above (1, +, 0, 5) Click

85 Bob has x dollars. Mary has 4 more dollars 
than Bob. Write an expression for Mary's 
money. [This object is a pull tab] Answer C A 4x B x - 4 C x + 4 D 4x + 4

86 The width of the rectangle is five inches less 
than its length. The length is x inches. Write an 
expression for the width. [This object is a pull tab] Answer B A 5 - x B x - 5 C 5x D x + 5

87 Frank is 6 inches taller than his younger brother, Pete. Pete's height is P. Write an expression for Frank's height. [This object is a pull tab] Answer B A 6P B P + 6 C P - 6 D 6

Write an expression for the dog's weight. 88 A dog weighs three pounds more than twice the weight of a cat, whose weight is c pounds. Write an expression for the dog's weight. [This object is a pull tab] Answer A A 2c + 3 B 3c + 2 C 2c + 3c D 3c

89 Write an expression for Mark's test grade, given that he scored 5 less than Sam who earned a score of x. [This object is a pull tab] Answer B A 5 - x B x - 5 C 5x D 5

90 Tim ate four more cookies than Alice. Bob ate twice as many cookies as Tim. If x represents the number of cookies Alice ate, which expression represents the number of cookies Bob ate? [This object is a pull tab] Answer C A 2 + (x + 4) B 2x + 4 C 2(x + 4) D 4(x + 2) From the New York State Education Department. Office of Assessment Policy, Development and Administration. 

Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

Evaluating Expressions Return to Table 
of Contents

Evaluating Expressions [This object is a pull tab] Math Practice The next 5 slides address MP7 Ask: What do you know about the Order of Operations that can apply to this situation? When evaluating algebraic expressions, the process is fairly straight forward. 1. Write the expression. 2. Substitute in the value of the variable (in parentheses). 3. Simplify/Evaluate the expression.

Evaluate (4n + 6)2 for n = 1 Write: Substitute: Simplify: (4n + 6)2 (4(1) + 6)2 (4 + 6)2 (10)2 100

Evaluate 4(n + 6)2 for n = 2 Write: Substitute: Simplify: 4(n + 6)2 4((2) + 6)2 4(8)2 4(64) 256

Evaluate (4n + 6)2 for n = -1 Write: Substitute: Simplify: (4n + 6)2 (4(-1) + 6)2 ((-4) + 6)2 (2)2 4

12x + 23 Let x = 8, then use the magic looking glass 

to reveal the correct value of the expression 104 106 108 116 114 118 128 130

4x + 2x3 Let x = 2, then use the magic looking glass 
to reveal the correct value of the expression 800 20 72 24 114 130 118 128

91 Evaluate 3h + 2 for h = 3 3(3) + 2 11 Answer [This object is a pull tab] Answer 3(3) + 2 11

92 Evaluate 2(x + 2)2 for x = -10 2(-10 + 2)2 2(-8)2 Answer 128 [This object is a pull tab] Answer 2(-10 + 2)2 2(-8)2 128

93 Evaluate 2x2 for x = 3 2(3)2 2(9) 18 Answer [This object is a pull tab] Answer 2(3)2 2(9) 18

94 Evaluate 4p - 3 for p = 20 4(20) - 3 80 - 3 77 Answer [This object is a pull tab] Answer 4(20) - 3 80 - 3 77

95 Evaluate 3x + 17 when x = -13 Answer 3(-13) + 17 -39 + 17 -22 [This object is a pull tab] Answer 3(-13) + 17 -39 + 17 -22

96 Evaluate 3a for a = -12 9 3(-12) 9 -36 -4 Answer [This object is a pull tab] Answer 3(-12) 9 -36 -4

c a 97 Evaluate 4a + for a = 8, c = -2 8 4(8) + -2 32 + (-4) 28 Answer [This object is a pull tab] Answer 4(8) + 32 + (-4) 28 8 -2

98 If t = -3, then 3t2 + 5t + 6 equals A -36 B -6 C 6 D 18 [This object is a pull tab] Answer 3(-3)2 + 5(-3) + 6 27 + (-15) + 6 D: 18 A -36 B -6 C 6 D 18 From the New York State Education Department. Office of Assessment Policy, Development and Administration. Internet. Available from www.nysedregents.org/IntegratedAlgebra; accessed 17, June, 2011.

Evaluate 3x + 2y for x = 5 and y = 1 2 99 Evaluate 3x + 2y for x = 5 and y = [This object is a pull tab] Answer 3(5) + 2( ) 15 + 1 16 1 2

Evaluate 8x + y - 10 for x = and y = 50 4 100 Evaluate 8x + y - 10 for x = and y = 50 [This object is a pull tab] Answer 8( ) + 50 - 10 2 + 50 - 10 42 1 4

[This object is a pull tab] Teacher Notes Vocabulary Words are bolded in the presentation. The text box the word is in is then linked to the page at the end of the presentation with the word defined on it. Glossary & Standards Return to Table of Contents

Coefficient The number multiplied by the variable and is located in front of the variable. 1x + 7 Tricky! 7 3 5 When not present, the coefficient is assumed to be 1. 4x + 2 These are not coefficients. These are constants! - 1x2 +18 Back to Instruction

These are not constants. These are coefficients! A fixed number whose value does not change. It is either positive or negative. Tricky! 8 4 110 7x 3z 3y 4x + 2 π 69 1/2 These are not constants. These are coefficients! 7 0.45 Back to Instruction

The Distributive Property A property that allows you to multiply all the terms on the inside of a set of parenthesis by a term on the outside of the parenthesis. 3(2 + 4) = a(b + c) = ab + ac (3)(2) + (3)(4) = 3(x + 4) = 48 a(b + c) = ab + ac a(b - c) = ab - ac 6 + 12 = 18 (3)(x) + (3)(4) = 48 3(2 - 4) = 3x + 12 = 48 (3)(2) - (3)(4) = 3x = 36 6 - 12 = -6 x = 12 Back to Instruction

An expression contains: number, variables, and at least one operation. Remember! 7x = 21 4x + 2 11 = 3y + 2 7x "7 times x" "7 divided by x" 7 x 11 - 1 = 3z + 1 Back to Instruction

Like Terms Terms in an expression that have the same variable raised to the same power. 3x 5x3 x3 27x3 x NOT LIKE TERMS! 5x2 1/2x 5 -2x3 15.7x 1/4x3 5x -5x3 -2.3x 5x 5x4 2.7x3 Back to Instruction

Any letter or symbol that represents a changeable or unknown value. Variable Any letter or symbol that represents a changeable or unknown value. any letter towards end of alphabet! 4x + 2 l, i, t, o, O, s, S x y z u v Back to Instruction

Standards for Mathematical Practices MP1 Make sense of problems and persevere in solving them. MP2 Reason abstractly and quantitatively. MP3 Construct viable arguments and critique the reasoning of others. MP4 Model with mathematics. MP5 Use appropriate tools strategically. MP6 Attend to precision. MP7 Look for and make use of structure. MP8 Look for and express regularity in repeated reasoning. Click on each standard to bring 
you to an example of how to 
meet this standard within the unit.

This slide & the next one address MP4. Area Model [This object is a pull tab] Math Practice This slide & the next one address MP4. Ask: What connections do you see between the distributive property and the area of a rectangle? (MP4) Write an expression for the area of a rectangle whose width is 4 and whose length is x + 2 4 x 2

Order of Operations The order of operations allows us to read an expression and interpret it as intended. It lets us understand what the author meant. For instance, the below expression could mean many different things without an agreed upon order of operations. How would you evaluate this expression? (5-8)(5)(3)-42÷2+8÷4+(3-2)

Order of Operations Mathematics has its grammar, just like any language. Grammar provides the rules that allow us to write down ideas so that a reader can understand them. A critical set of those rules is called the order of operations.