Lesson 22

Combining Like Terms (Part 3)

22.1: Are They Equal? (5 minutes)

Warm-up

The purpose of this activity is to remind students of things they learned in the previous lesson using numerical examples. Look for students who evaluate each expression and students who use reasoning about operations and properties.

Launch

Remind students that working with subtraction can be tricky, and to think of some strategies they have learned in this unit. Encourage students to reason about the expressions without evaluating them. 

Give students 2 minutes of quiet think time followed by whole-class discussion.

Student Facing

Select all expressions that are equal to \(8-12-(6+4)\).

  1. \(8-6-12+4\)
  2. \(8-12-6-4\)
  3. \(8-12+(6+4)\)
  4. \(8-12-6+4\)
  5. \(8-4-12-6\)

Student Response

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Anticipated Misconceptions

Students who selected \(8-6-12+4\) or \(8-12+(6+4)\) might not understand that the subtraction sign outside the parentheses applies to the 4 and that 4 is always to be subtracted in any equivalent expression.

Students who selected \(8-12+(6+4)\) might think the subtraction sign in front of 12 also applies to \((6+4)\) and that the two subtractions become addition. 

Activity Synthesis

For each expression, poll the class for whether the expression is equal to the given expression, or not. For each expression, select a student to explain why it is equal to the given expression or not. If the first student reasoned by evaluating each expression, ask if anyone reasoned without evaluating each expression. 

22.2: X’s and Y’s (15 minutes)

Activity

In this activity students take turns with a partner and work to make sense of writing expressions in equivalent ways. This activity is a step up from the previous lesson because there are more negatives for students to deal with, and each expression contains more than one variable. 

Launch

Arrange students in groups of 2. Tell students that for each expression in column A, one partner finds an equivalent expression in column B and explains why they think it is equivalent. The partner's job is to listen and make sure they agree. If they don't agree, the partners discuss until they come to an agreement. For the next expression in column A, the students swap roles. If necessary, demonstrate this protocol before students start working.

Representation: Internalize Comprehension. Differentiate the degree of difficulty or complexity by beginning with an example with more accessible values. For example, start with an expression with three terms such as “\(6x-(2x+8)\)" and show different forms of equivalent expressions. Highlight connections between expressions by using the same color on equivalent parts of the expression.
Supports accessibility for: Conceptual processing
Listening, Speaking: MLR8 Discussion Supports. Display sentence frames for students to use to describe the reasons for their matches. For example, “I matched expression ___ with expression ___ because . . . .” or “I used the ___ property to help me match expression ___ with expression ___.” Provide a sentence frame for the partner to respond with, such as: “I agree/disagree with this match because . . . .” These sentence frames provide students with language structures that help them to produce explanations, and also to critique their partner’s reasoning.
Design Principle(s): Maximize meta-awareness; Support sense-making

Student Facing

Match each expression in column A with an equivalent expression from column B. Be prepared to explain your reasoning.

A

  1. \((9x+5y) + (3x+7y)\)
  2. \((9x+5y) - (3x+7y)\)
  3. \((9x+5y) - (3x-7y)\)
  4. \(9x-7y + 3x+ 5y\)
  5. \(9x-7y + 3x- 5y\)
  6. \(9x-7y - 3x-5y\)

B

  1. \(12(x+y)\)
  2. \(12(x-y)\)
  3. \(6(x-2y)\)
  4. \(9x+5y+3x-7y\)
  5. \(9x+5y-3x+7y\)
  6. \(9x-3x+5y-7y\)

 

Student Response

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Anticipated Misconceptions

For the second and third rows, some students may not understand that the subtraction sign in front of the parentheses applies to both terms inside that set of parentheses. Some students may get the second row correct, but not realize how the third row relates to the fact that the product of two negative numbers is a positive number. For the last three rows, some students may not recognize the importance of the subtraction sign in front of \(7y\). Prompt them to rewrite the expressions replacing subtraction with adding the inverse. 

Students might write an expression with fewer terms but not recognize an equivalent form because the distributive property has been used to write a sum as a product. For example, \(9x-7y + 3x- 5y\) can be written as \(9x+3x−7y−5y\) or \(12x-12y\), which is equivalent to the expression \(12(x-y)\) in column B. Encourage students to think about writing the column B expressions in a different form and to recall that the distributive property can be applied to either factor or expand an expression. 

Activity Synthesis

Much discussion takes place between partners. Invite students to share how they used properties to generate equivalent expressions and find matches.

  • “Which term(s) does the subtraction sign apply to in each expression? How do you know?”
  • “Were there any expressions from column A that you wrote with fewer terms but were unable to find a match for in column B? If yes, why do you think this happened?”
  • “What were some ways you handled subtraction with parentheses? Without parentheses?”
  • “Describe any difficulties you experienced and how you resolved them.”

22.3: Seeing Structure and Factoring (10 minutes)

Activity

This activity is an opportunity to notice and make use of structure (MP7) in order to apply the distributive property in more sophisticated ways.

Launch

Display the expression \(18-45+27\) and ask students to calculate as quickly as they can. Invite students to explain their strategies. If no student brings it up, ask if the three numbers have anything in common (they are all multiples of 9). One way to quickly compute would be to notice that \(18-45+27\) can be written as \(2\boldcdot 9 -5\boldcdot 9 +3\boldcdot 9\) or \((2-5+3)\boldcdot 9\) which can be quickly calculated as 0. Tell students that noticing common factors in expressions can help us write them with fewer terms or more simply.

Keep students in the same groups. Give them 5 minutes of quiet work time and time to share their expressions with their partner, followed by a whole-class discussion.

Action and Expression: Internalize Executive Functions. To support development of organizational skills, check in with students within the first 2–3 minutes of work time. Look for students who identify common factors or rearrange terms to write the expressions with fewer terms.
Supports accessibility for: Memory; Organization

Student Facing

Write each expression with fewer terms. Show or explain your reasoning.

  1. \(3 \boldcdot 15 + 4 \boldcdot 15 - 5 \boldcdot 15 \)

  2. \(3x + 4x - 5x\)

  3. \(3(x-2) + 4(x-2) - 5(x-2) \)

  4. \(3\left(\frac52x+6\frac12\right) + 4\left(\frac52x+6\frac12\right) - 5\left(\frac52x+6\frac12\right)\)

Student Response

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Activity Synthesis

For each expression, invite a student to share their process for writing it with fewer terms. Highlight the use of the distributive property. 

Speaking: MLR8 Discussion Supports. Provide sentence frames to help students explain their strategies. For example, “I noticed that ______, so I ______.” or “First, I ________ because ________.” When students share their answers with a partner, prompt them to rehearse what they will say when they share with the full group. Rehearsing provides students with additional opportunities to clarify their thinking.
Design Principle(s): Optimize output (for explanation)

Lesson Synthesis

Lesson Synthesis

Ask students to reflect on their work in this unit. They can share their response to one or more of these prompts either in writing or verbally with a partner.

  • “Describe something that you found confusing at first that you now understand well.”
  • “Think of a story problem that you would not have been able to solve before this unit that you can solve now.”
  • “What is a tool or strategy that you learned in this lesson that was particularly useful?”
  • “Describe a common mistake that people make when using the ideas we studied in this unit and how they can avoid that mistake.”
  • “Which is your favorite, and why? The distributive property, rewriting subtraction as adding the opposite, or the commutative property.”

22.4: Cool-down - R's and T's (5 minutes)

Cool-Down

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Student Lesson Summary

Student Facing

Combining like terms is a useful strategy that we will see again and again in our future work with mathematical expressions. It is helpful to review the things we have learned about this important concept.

  • Combining like terms is an application of the distributive property. For example:

\(\begin{gather} 2x+9x\\ (2+9) \boldcdot x \\ 11x\\ \end{gather}\)

  • It often also involves the commutative and associative properties to change the order or grouping of addition. For example:

\(\begin{gather} 2a+3b+4a+5b \\ 2a+4a+3b+5b \\ (2a+4a)+(3b+5b) \\ 6a+8b\\ \end{gather}\)

  • We can't change order or grouping when subtracting; so in order to apply the commutative or associative properties to expressions with subtraction, we need to rewrite subtraction as addition. For example:

\(\begin{gather} 2a-3b-4a-5b \\ 2a+\text-3b+\text-4a+\text-5b\\ 2a + \text-4a + \text-3b + \text-5b\\ \text-2a+\text-8b\\ \text-2a-8b \\ \end{gather}\)

  • Since combining like terms uses properties of operations, it results in expressions that are equivalent.

  • The like terms that are combined do not have to be a single number or variable; they may be longer expressions as well. Terms can be combined in any sum where there is a common factor in all the terms. For example, each term in the expression \(5(x+3)-0.5(x+3)+2(x+3)\) has a factor of \((x+3)\). We can rewrite the expression with fewer terms by using the distributive property: 

\(\begin{gather} 5(x+3)-0.5(x+3)+2(x+3)\\ (5-0.5+2)(x+3)\\ 6.5(x+3)\\ \end{gather}\)