GENERATING EQUIVALENT EXPRESSIONS ENGAGE NY- LESSONS 1 & 2

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4 GENERATING EQUIVALENT EXPRESSIONS ENGAGE NY- LESSONS 1 & 2

5 FIND THE PRODUCT: x 4.2 x -2 x

6 CALCULATE:

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11 CORE ENRICHMENT Interim Analysis Paper Pencil

12 The cost (c) to hire a dog trainer is directly proportional to the amount of time (h) in hours spent training the dog. Sue spent $660 on 12 hours of obedience training for her dog Muffin. a. Write an equation representing the proportional relationship between c and h. b. Jon wants to use the same dog trainer for his dog Tex, but he only has $440 to spend. How many hours of training can he get for Tex? Explain your reasoning.

13 The cost (c) to hire a dog trainer is directly proportional to the amount of time (h) in hours spent training the dog. Sue spent $660 on 12 hours of obedience training for her dog Muffin. a. Write an equation representing the proportional relationship between c and h.

14 b. Jon wants to use the same dog trainer for his dog Tex, but he only has $440 to spend. How many hours of training can he get for Tex? Explain your reasoning. $ Time (hrs)

15 Engage NY- Lesson 1: Generating Equivalent Expressions Any order, any grouping property: Combines the Associative Property & the Commutative Property of Multiplication. Associative Property: Doesn t matter how you group the numbers, but whatever numbers are grouped, perform that operation first.

16 Engage NY- Lesson 1: Generating Equivalent Expressions Any order, any grouping property: Combines the Associative Property & the Commutative Property of Multiplication. Commutative Property: Order doesn t matter. =

17 Coefficient of the Term: The number next to a variable that represents the amount of each variable. Variables with no numbers represent an understood 1. Engage NY- Lesson 1: Generating Equivalent Expressions Expression: No equal sign Variable: Letters in an expression Equivalent Expressions: 2 expressions that are equal Equal sign in between both expressions Term: Each summand of an expression 3 terms

18 SUPPLIES New Seats Agenda- meet me at the chalkboard Pencil After Agenda new top dog- who won pride points? Homework: page 266 all Notebook

19 Variable: a symbol (such as a letter) that represents a number, it is a placeholder for a number. Numerical Expression: a number, or it is any combination of sums, differences, products, or divisions of numbers that evaluates to a number, but does NOT have an equal sign. Statements such as 3 + or 3 0 are not numerical expressions because neither represents a point on the number line. Value of a Numerical Expression: the number found by evaluating the expression is a numerical expression, and its value is 5.

20 Expression : a numerical expression, or it is the result of replacing some (or all) of the numbers in a numerical expression with variables. We can start out with a numerical expression, such as , and replace some of the numbers with letters to get 1 3 x + y z. We can build such expressions from scratch, as in x + x(y z), and note that if numbers were placed in the expression for the variables x, y, and z, the result would be a numerical expression.

21 Equivalent Expressions: Two expressions are equivalent if both expressions evaluate to the same number for every substitution of numbers into all the letters in both expressions. This description becomes clearer through lots of examples and linking to the associative, commutative, and distributive properties. An Expression in Expanded Form : An expression that is written as sums (and/or differences) of products whose factors are numbers, variables, or variables raised to whole number powers is said to be in expanded form. A single number, variable, or a single product of numbers and/or variables is also considered to be in expanded form. An Expression in Standard Form : An expression that is in expanded form where all like-terms have been collected is said to be in standard form, also means simplified.

22 Term (description): Each summand of an expression in expanded form is called a term. For example, the expression 2x + 3x + 5 consists of 3 terms: 2x, 3x, and 5. Coefficient of the Term (description): The number found by multiplying just the numbers in a term together. For example, given the product 2 x 4, its equivalent term is 8x. The number 8 is called the coefficient of the term 8x. An Expression in Standard Form: An expression in expanded form with all its like terms collected is said to be in standard form. For example, 2x + 3x + 5 is an expression written in expanded form; however, to be written in standard form, the like terms 2x and 3x must be combined. The equivalent expression 5x + 5 is written in standard form.

23 Engage NY- Lesson 1: Generating Equivalent Expressions Rewrite 5x + 3x in expanded form Rewrite 5x - 3x in expanded form

24 ANY ORDER, ANY GROUPING PROPERTY WITH ADDITION Rewrite 5x + 3x in expanded form Rewrite 5x - 3x in expanded form

25 Engage NY- Lesson 1: Generating Equivalent Expressions 3a + 2b + 6c 3(2) + 2(5) + 6(-3) a + 2a 4a 9a 4a 5a 5(2) 10 Generate Equivalent Expressions: a = 2, b = 5, c = -3 SHOW ALL WORK 1. Substitute values into variables using parenthesis to represent multiplication. 2. Multiply each term with signs 3. Answer Combine Like Terms a = 2, b = 5, c = -3 SHOW ALL WORK 1. Collect all terms that are the same. 2. Substitute values into variables using parenthesis to represent multiplication. 3. Multiply each term with signs 4. Answer

26 HOMEWORK: PINK PACKET PAGES 3-4 ALL

27 SUPPLIES: 1. GRADING PEN- YOU WILL WRITE IN THE CORRECT ANSWERS ON YOUR HW WITH A GRADING PEN. 2. PINK PACKET 3. HW: PAGES 3 & 4 4. NOTEBOOK

28 Engage NY- Lesson 2: Generating Equivalent Expressions Find the sum of 2x + 1 and 5x and justify your steps.

29 FIND THE SUM OF 2X + 1 AND 5X & JUSTIFY YOUR STEPS (2x + 1) + 5x: Original Expression 2x + 5x + 1: Commutative Property of addition (2x + 5x) + 1: Associative Property of addition 7x + 1: Combined like terms The ORANGE steps can be combined!

30 2X X = 7X + 1 Why was the associative property AND the commutative properties both used? Did the use of these properties change the value of the expression? How can we confirm that the expressions (2x + 1) + 5x and 7x + 1 are equivalent expressions?

31 Engage NY- Lesson 2: Generating Equivalent Expressions Find the sum of (-3a + 2) + (5a 3) and justify your steps.

32 FIND THE SUM & JUSTIFY YOUR STEPS (-3a + 2) + (5a 3) -3a a + (-3): Additive Inverse (add the opposite) -3a + 5a (-3): Commutative Property 2a + (-1): Combine like terms 2a 1: Adding the inverse & simplify expression

33 Engage NY- Lesson 2: Generating Equivalent Expressions Find the product of 2x and 3 and justify your steps.

34 ANY ORDER, ANY GROUPING WITH MULTIPLICATION 2x 3 Original Expression (x 3) Associative Property of Multiplication (any grouping) 2 (3 x) Commutative Property of Multiplication (any order) 6x Simplify using multiplication If a product of factors is being multiplied, the any order, any grouping property allows us to multiply those factors in any order by grouping them together in any way.

35 Engage NY- Lesson 2: Generating Equivalent Expressions 3 2x 3 2 x 6x 12x 13x -1x = -x Use any order, any grouping to find equivalent expressions. Variables with a 1 in front are just written with a variable. The 1 or -1 is understood. -x = 5 (-1 -x) = (-1 5) x = -5 To get rid of the x, multiply both sides by -1 bc negative x negative = positive. -(4x) = -4x When there is a negative on the outside of ( ), that means multiply by -1

36 Engage NY- Lesson 2: Generating Equivalent Expressions 0x = 0 Zero x s = ZERO = 0 3 ⅓ = 1 Additive Inverse: Adding the opposite number to equal ZERO. Multiplicative Inverse: Multiplying by the reciprocal to equal ONE.

37 Engage NY- Lesson 2: Generating Equivalent Expressions (2x +3y 4) Combining expressions vertically: -(5x + 2) When lining up vertically, make sure each term is lined up above or below its mate with its sign. (2x + 3y 4) +( 5x 2) -3x + 3y 6 Find the additive inverse- and change ALL signs.

38 Engage NY- Lesson 2: Generating Equivalent Expressions The opposite of a sum is the sum of its opposites

39 HOMEWORK: PACKET PAGES 5-7

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41 COMPLETE PACKET PAGES 8-10