Identifying Equivalent Expressions Teachers: PLEASE look at the notes section on each slide for additional important information. Lesson 4.1.3 6.EE.3 6.EE.4

Yesterday… We learned how to evaluate an algebraic expression using a skill called substitution. You can use some of the words that the students came up with when explaining step 1: plug in, fill in, take the place of…

Yesterday… To evaluate an algebraic expression when you are given the value of a variable: Step 1: Substitute (or replace) the variable with the given number. Step 2: Solve the expression You can use some of the words that the students came up with when explaining step 1: plug in, fill in, take the place of… Example Evaluate 2y + 3 when y = 4 2(4) +3 8 + 3 11

Today… We are going to use these skills to determine if two algebraic expressions are equivalent expressions. We will start with an exploration to see if you can discover on your own what makes two expressions equivalent.

Explore & Discover

For each value that you plugged in… Exploration (Set – A) Evaluate each expression for the given values. Have the students work the exploration problems for Set-A. Make sure they record their answers on their exploration sheet (provided in lesson plan). Review the answers, then click mouse to display first exploration question. Answer: YES For each value that you plugged in… Did the answer for Expression A and Expression B calculate to be the same? Yes or No

For each value that you plugged in… Exploration (Set – B) Evaluate each expression for the given values. Have the students work the exploration problems for Set-B. Make sure they record their answers on their exploration sheet (provided in lesson plan). Review the answers, then click mouse to display next exploration question. Answer: YES For each value that you plugged in… Did the answer for Expression A and Expression B calculate to be the same? Yes or No

For each value that you plugged in… Exploration (Set – C) Evaluate each expression for the given values. Have the students work the exploration problems for Set-C. Make sure they record their answers on their exploration sheet (provided in lesson plan). Review the answers, then click mouse to display first exploration question. Answer: NO… When we plugged in 5, we got different answers. For each value that you plugged in… Did the answer for Expression A and Expression B calculate to be the same? Yes or No

Exploration Set-A Set-B Set-C Answer: Set-A and Set-B contain equivalent expressions. For any value that we plugged in, Expression A and Expression B calculated to be the same. On Set-C, that was not the case. Equivalent expressions are expressions that are the same (or equal), even though they may look a little different. TWO of our SETS contain expressions that are equivalent expressions. Which two sets do you think they are? Explain why you think these two sets contain equivalent expressions and why the remaining set does not?

Equivalent Expressions Finding Equivalent Expressions

How do I know if Two Expressions are Equivalent? Method 1: Substitute Values Substitute several values into each expression as we did in the Explore Activity. If they calculate to be the same number in every situation then they are equivalent expressions. Method 2: Simplify & Match You can apply the properties of operations to simplify each expression. If the simplified versions of both expressions are identical, then you can say that they are equivalent expressions. Properties of Operations - Distributive property, commutative property, associative property…

Let’s take a look at a problem where the Simplify & Match Method would be a good method to use to determine if expressions are equivalent…

EXAMPLE - Simplify & Match Method Which expression is equivalent to 7(x + 4)? A. x + 11 B. x + 28 C. 7x + 4 D. 7x + 28 We need to simplify the expression 7(x+4) using the distributive property in the question. Answer choices A, B, C, and D are already in simplest form. Answer: D First Ask Yourself… Is there anything that we need to simplify in the question or answer choices to generate identical expressions? If yes, then… Apply the appropriate property and match.

You need to recognize that BOTH methods will work for every single problem, but in some cases one method may be easier to use than the other. Let’s take some problems and use both methods to determine if the expressions are equivalent.

Guided Practice #1 Are the expressions 3x + 5 and 3(x+5) equivalent expressions? Use both methods to justify your answer. Method 1 Substitute Values Method 2 Simplify & Match Method 1: Substitute Values Use several values to test to see if 3x + 5 and 3(x+5) are equivalent. Answer: The expressions are not equivalent expressions Method 2: Simplify and Match 3(x+5) simplifies to 3x + 15 when the distributive property is applied. Answer: 3x+15 and 3x+5 are not identical so they are not equivalent expressions.

Guided Practice #2 Are the expressions 2y + 5 and 5 + 2y equivalent expressions? Use both methods to justify your answer. Method 1 Substitute Values Method 2 Simplify & Match Method 1: Substitute Values Use several values to test to see if 2y+5 and 5+2y are equivalent. Answer: They expressions are equivalent expressions. Method 2: Simplify and Match Use the commutative property to swap the order of the terms on the second expression. Answers: 2y+5 and 2y+5 are identical so the expressions are equivalent expressions.

Guided Practice #3 Are the expressions 3x and (x)(x)(x) equivalent expressions? Use both methods to justify your answer. Substitute Values Method Simplify & Match Method Method 1: Substitute Values Use several values to test to see if 3x and (x)(x)(x) are equivalent. Answer: They are not equivalent expressions. Method 2: Simplify and Match (x)(x)(x) can be written as x³. Answers: 3x and x³ are not identical so they are not equivalent expressions.

Equivalent Expression Finding More Than One Equivalent Expression

WHY do students seem to miss the problem more than one correct answer? Why, Why, Why! WHY do students seem to miss the problem when asked to select more than one correct answer? Select ALL that Apply

Because they take more effort Why, Why, Why! Because they take more effort Lack of proper technique when solving the problem! AND

ALL STUDENTS There is NO easy way… and… there is NO getting around this. If there is an expression in the question and it is not in simplest form, then REWRITE it. Next, you MUST go through EACH answer choice and either eliminate it or select it. Yes… that will require you to study and possibly simplify every single solitary answer choice looking for identical matches!

Example of a Proper Technique Select ALL of the expressions that are equivalent to 10 (x + 6). First… simplify the expression in the question by distributing 10x +60 60 + 10x 10x + 60 10x + 6 10x + 50 + 10 5(2x + 12) 2(5x + 6) Already Simplified - Match (Commutative Property of Addition) Already Simplified - Match Already Simplified - Does NOT Match Must Simplify to 10x + 60 - Match Must Simplify to 10x + 60 (Distributive Property) - Match Must Simplify to 10x + 12 (Distributive Property) - Does NOT Match

Now you can try these next 5 problems…

You Try #1 Which expression is equivalent to 2(6x+3)? A. 12x + 3 B. 6x + 6 C. 3(4x + 2) D. 12(x + 3) Can you think of another option that would make this a “Select All” type question? We need to simplify the expression 2(6x+3) using the distributive property in the question. 12x + 6 Answer choices A, B, are already in simplest form, but Do Not Match. Answer choice C and D need to be simplified using the distributive. Answer: C Use Simplify & Match… Is there anything that we need to simplify in the question or answer choices to generate identical expressions?

Use Simplify & Match Method… You Try #2 Which expressions are equivalent to 9w + 6? SELECT ALL that apply. 6 + 9w 9w + 4 + 5 9(w + 6) 3(3w + 2) 3(3w + 6) The expression in the question is already simplified. 9w + 6 * 6 + 9w Already simplified. Match (Commutative Property of Addition) 9w + 4 + 5 Must simplify. 9w + 9 Does Not Match 9(w+6) Must simplify. 9w + 54 Does Not Match * 3(3w + 2) Must simplify. 9w + 6 Match 3(3w + 6) Must simplify. 9w + 18 Does Not Match Use Simplify & Match Method… Is there anything that we need to simplify in the question or answer choices to generate identical expressions?

Use Simplify & Match Method… You Try #3 Select ALL the expressions that are equivalent to 4(3 + 5x). 12 + 5x 20x + 12 12 + 20x 6 + 6 + 5x 2(6x + 10) 4(5x + 3) The expression in the question must be simplified (Distributive Property). 12 + 20x 12 + 5x Already simplified Does Not Match * 20x + 12 Already simplified Match (Commutative Property of Addition) * 12 + 20x Already simplified Match 6 + 6 + 5x Must simplify 12 + 5x Does Not Match 2(6x + 10) Must simplify 12x + 20 Does Not Match * 4(5x + 3) Must simplify 20x + 12 (Distributive Property/Commutative Property) Match Use Simplify & Match Method… Is there anything that we need to simplify in the question or answer choices to generate identical expressions?

Do any other values work? You Try #4 Determine if the expressions below are equivalent by substituting in values for x. Use the following values: 0, 1, 2, 5 x² + 2 2x + 1 + 1 x² + 2 2x+1+1 0² + 2 = 2 2(0) + 1+1 = 2 1² + 2 = 3 2(1) + 1+1 = 4 2² + 2 = 6 2(2) + 1+1 = 6 5² + 2 = 27 2(5) + 1+1 = 12 Not Equivalent Do any other values work?

You Try #5 Determine if the expressions below are equivalent expressions by substituting 3 values of your choice for m. 14m + 16m 30m 2m(7 + 8) All 3 are equivalent because they calculate to the same number no mater which value is substituted in.

Let’s Get Creative! Write two equivalent expressions following the rules below. Each expression: Must have at least one variable Must have two different operations Must have at least two numbers Prove they are equivalent using substitution One example could be: 2(x + 3) and x + 3 + x + 3 This activity can be done as a team by giving each team a poster paper and having them explain their methods. These can then be displayed around the room or in the hall as they will be unique and higher order.

Are v – 18 and w – 18 equivalent expressions? Why or Why not? Closure Challenge Are v – 18 and w – 18 equivalent expressions? Why or Why not? No because different letters would represent different numbers. They would not calculate/simplify to the same answer.