Lesson 21

Combining Like Terms (Part 2)

21.1: True or False? (10 minutes)

Warm-up

In this warm-up, students consider some correct and incorrect numerical examples of distributing when the expression is being subtracted. A main goal of this lesson is to help students to understand how to write equivalent expressions that contain variables. By first looking at numbers only, students have a way to tell if the expressions are equivalent by evaluating each side.

Look for students who evaluate each side and students who reason about operations and properties.

Launch

Tell students that their job is to consider four equations and decide which of them are true.

The amount of numbers and symbols presented relatively close together might present a challenge. It is important that students think of how the different equations compare to each other, but they also need to consider them one at a time. Provide access to index cards, so that, for example, students can cover up questions 2, 3, and 4 while considering question 1.

Arrange students in groups of 2. Give them 3 minutes of quiet work time and time to share their thoughts with a partner, followed by whole-class discussion.

Student Facing

Select all the statements that are true. Be prepared to explain your reasoning.

1. $$4 - 2(3+7)=4-2\boldcdot 3 -2\boldcdot 7$$
2. $$4 - 2(3+7)=4+\text-2\boldcdot 3 +\text-2\boldcdot 7$$
3. $$4 - 2(3+7)=4-2\boldcdot 3 +2\boldcdot 7$$
4. $$4 - 2(3+7)=4-(2\boldcdot 3 +2\boldcdot 7)$$

Student Response

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Activity Synthesis

Ask students to explain why the true statements are true. Select students who reason by evaluating each side, and also students who reason using properties. For example, statement 2 is true because subtracting 2 is the same as adding negative 2, and then the distributive property is applies.

Then, spend some time on why statement 3 is false. First, we can tell its false because when each side is evaluated, we get $$\text-16=12.$$ The order of operations is just a convention, but we need to all follow one convention so that we can communicate mathematically. When the order of operations is followed on the left side, the result of $$2(3+7)$$ is subtracted from 4. However on the right side of statement 3, only the $$2 \boldcdot 3$$ is being subtracted, and the $$2 \boldcdot 7$$ is being added.

21.2: Seeing it Differently (10 minutes)

Activity

In this activity, students encounter typical errors with signed numbers, operations, and properties. They are tasked with identifying which strategies are correct and for those that are not, describing the error that was made.

Launch

Ensure students understand the task: first they decide whether they agree with each person's strategy, but they also need to describe the errors that were made. Give 5 minutes quiet work time followed by a whole-class discussion.

Representation: Internalize Comprehension. Begin by providing students with rules for adding, subtracting, and multiplying signed numbers. Invite students to share their prior knowledge and ideas using simple examples.
Supports accessibility for: Conceptual processing
Speaking, Writing: MLR3 Clarify, Critique, Correct. This activity provides students with the opportunity to improve upon the written work of another by correcting errors and clarifying meaning. Ask students to select one of the errors they notice, and to produce a written explanation, intended for the student who made the error (Noah, Lin, Jada, or Andre), that describes the error that was made, and how to fix it. Give students 3–5 minutes to complete a first draft before they read their writing to a partner. Provide students with prompts they can use to give each other feedback such as “Can you say that another way?” or “Can you try to explain this using an example?” This will provide students with an additional opportunity to produce language related to writing equivalent expressions.
Design Principle(s): Optimize output (for explanation)

Student Facing

Some students are trying to write an expression with fewer terms that is equivalent to $$8-3(4-9x)$$.

Noah says, “I worked the problem from left to right and ended up with $$20 - 45x$$.”

$$\displaystyle 8 - 3(4-9x)$$

$$\displaystyle 5(4 - 9x)$$

$$\displaystyle 20 - 45x$$

Lin says, “I started inside the parentheses and ended up with $$23x$$.”

$$\displaystyle 8 - 3(4-9x)$$

$$\displaystyle 8 - 3(\text-5x)$$

$$\displaystyle 8 + 15x$$

$$\displaystyle 23x$$

Jada says, “I used the distributive property and ended up with $$27x - 4$$.”

$$\displaystyle 8 - 3(4-9x)$$

$$\displaystyle 8 - (12 - 27x)$$

$$\displaystyle 8 - 12 - (\text-27x)$$

$$\displaystyle 27x - 4$$

Andre says, “I also used the distributive property, but I ended up with $$\text-4 - 27x$$.”

$$\displaystyle 8 - 3(4-9x)$$

$$\displaystyle 8 - 12 - 27x$$

$$\displaystyle \text-4 - 27x$$

1. Do you agree with any of them? Explain your reasoning.
2. For each strategy that you disagree with, find and describe the errors.

Student Response

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Student Facing

1. Jada’s neighbor said, “My age is the difference between twice my age in 4 years and twice my age 4 years ago.” How old is Jada’s neighbor?

2. Another neighbor said, “My age is the difference between twice my age in 5 years and and twice my age 5 years ago.” How old is this neighbor?

3. A third neighbor had the same claim for 17 years from now and 17 years ago, and a fourth for 21 years. Determine those neighbors’ ages.

Student Response

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Activity Synthesis

Ask students, “Which way do you see it?” In grade 6, students learned that equivalent expressions meant two expressions had to be equal for any value of the variable. Since each student’s work contains at least two steps, the steps can be used to identify where the error occurs—where the expressions are no longer equal for a value of the variable.

A handy approach is to rewrite subtraction operations as adding the opposite. If desired, demonstrate such an approach for this expression along with a box to organize the work of multiplying $$\text-3(4-9x)$$:

$$\displaystyle 8 - 3(4-9x)$$ $$\displaystyle 8 + (\text-3)(4 + (\text-9x))$$ $$\displaystyle 8 + (\text-3)(4) + (\text-3)(\text-9x)$$ $$\displaystyle 8+(\text-12)+27x$$ $$\displaystyle \text-4+27x$$

21.3: Grouping Differently (15 minutes)

Activity

In this activity students continue the work of generating equivalent expressions as they decide where to place a set of parentheses and explore how that placement affects the expressions.

Launch

Arrange students in groups of 2. Tell students to first complete both questions independently. Then, trade one of their expressions with their partner. The partner’s job is to decide whether the new expression is equivalent to the original or not, and explain how they know.

Student Facing

Diego was taking a math quiz. There was a question on the quiz that had the expression $$8x - 9 - 12x + 5$$. Diego’s teacher told the class there was a typo and the expression was supposed to have one set of parentheses in it.

1. Where could you put parentheses in $$8x - 9 - 12x + 5$$ to make a new expression that is still equivalent to the original expression? How do you know that your new expression is equivalent?
2. Where could you put parentheses in $$8x - 9 - 12x + 5$$ to make a new expression that is not equivalent to the original expression? List as many different answers as you can.

Student Response

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Anticipated Misconceptions

Students may not realize that they can break up a term and place a parentheses, for example, between the 8 and $$x$$ in the term $$8x$$. Clarify that they may place the parentheses anywhere in the expression.

Activity Synthesis

Questions for discussion:

• “How can you write the original expression in different ways with fewer terms?” ($$\text-4x-4$$, $$\text-4(x+1)$$, $$4(-x-1)$$)
• “Where did you place the parentheses to create an equivalent expression?”
• “Share other ways you placed parentheses and the resulting expression with fewer terms.”
Representation: Develop Language and Symbols. Create a display of important terms and vocabulary.  Include the following terms and maintain the display for reference throughout the unit: parenthesis. The display should include examples of how to create equivalent expressions using parentheses by breaking up the components. Be sure to emphasize how the placement of the parentheses affects the expression.
Supports accessibility for: Memory; Language
Speaking, Writing: MLR8 Discussion Supports. When students interpret the new expression they receive, ask students to press their partner for more details in their explanations by challenging an idea, or asking for an example. Listen for, and amplify language students use to describe how the parentheses influence the resulting expression with fewer terms. This will help call students' attention to the types of details and language to look for to determine if an expression is equivalent to another or not.
Design Principle(s): Maximize meta-awareness; Support sense-making

Lesson Synthesis

Lesson Synthesis

Display the expression $$5-2(3x-x).$$ Ask students to think of a mistake someone would be likely to make when trying to write an expression that is equivalent to this one. Select a student to share, and then ask if anyone can think of a different likely mistake. Continue until each of these common errors arises:

• Subtracting 2 from 5 first, resulting in $$3(3x-x)$$
• Distributing positive 2, resulting in $$5-6x-2x$$
• Thinking that $$3x-x$$ is 3, resulting in $$5-2(3)$$

Then, ask students for strategies for preventing these errors. Reliable properties to use are: rewriting subtraction as adding the opposite, the commutative property of multiplication and addition, and the distributive property. Suggest this way of rewriting this example:

$$\displaystyle 5-2(3x-x)$$

$$\displaystyle 5 + \text-2 (3x + \text-x)$$

$$\displaystyle 5+ \text-2 \boldcdot 3x + \text-2 \boldcdot \text-x$$

$$\displaystyle 5 + \text-6x + 2x$$

$$\displaystyle 5+x(\text-6+2)$$

$$\displaystyle 5 + \text-4x \text { or } 5-4x$$

21.4: Cool-down - How Many Are Equivalent? (5 minutes)

Cool-Down

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Student Lesson Summary

Student Facing

Combining like terms allows us to write expressions more simply with fewer terms. But it can sometimes be tricky with long expressions, parentheses, and negatives. It is helpful to think about some common errors that we can be aware of and try to avoid:

• $$6x-x$$ is not equivalent to 6. While it might be tempting to think that subtracting $$x$$ makes the $$x$$ disappear, the expression is really saying take 1 $$x$$ away from 6 $$x$$'s, and the distributive property tells us that $$6x-x$$ is equivalent to $$(6-1)x$$
• $$7-2x$$ is not equivalent to $$5x$$. The expression $$7-2x$$ tells us to double an unknown amount and subtract it from 7. This is not always the same as taking 5 copies of the unknown.
• $$7-4(x+2)$$ is not equivalent to $$3(x+2)$$. The expression tells us to subtract 4 copies of an amount from 7, not to take $$(7-4)$$ copies of the amount.

If we think about the meaning and properties of operations when we take steps to rewrite expressions, we can be sure we are getting equivalent expressions and are not changing their value in the process.