In this Warm-up, students encounter some algebraic moves they have studied in the past several lessons and explain why these moves are allowed. The moves discussed here are important to understand as students work toward fluency in writing expressions with fewer terms.
Launch
Display one statement at a time. Give students 30 seconds of quiet think time for each statement and ask them to give a signal when they have an explanation. Ask them to share their explanation with a partner and then follow with whole-class discussion.
Activity
None
Explain why each statement is true.
is equivalent to .
Student Response
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Building on Student Thinking
Activity Synthesis
The purpose of this discussion is to refresh students’ memories of algebra moves that will be useful in this lesson. Ask students to share their reasons why each statement is true. Record and display their responses for all to see. Highlight correct use of precise, mathematical language and give students opportunities to revise their responses to be more precise.
Some questions to spark discussion include:
“How does the placement of the parentheses change how the first expression is evaluated?”
“Why is it okay to change the order of the terms, even though there’s subtraction in the expression?”
“Can you draw a diagram to show why is equivalent to ?”
3.2
Activity
Standards Alignment
Building On
Addressing
Building Toward
7.EE.A.1
Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
In this activity, students explain why algebra moves to simplify an expression are valid. In the process, they see two ways to reason about properties to combine like terms. As students explain how they know two expressions are equivalent, they attend to precision of language (MP6).
Students begin by substituting a value for the variable into expressions believed to be equivalent, and discover that the expressions are equal for that value. They then substitute other values and find that one of the expressions has a different value than the others. This primes students to see why applying properties is the only reliable way to decide whether two expressions are equivalent.
Students follow up by expanding the terms of the expression to consider each instance of the variables individually, and uncover the properties applied in each step of writing the expression with fewer terms.
In order to see the moves and see the properties, students need to see a group of terms, for example, as an object that can be rearranged. This is an example of noticing and making use of structure (MP7).
This activity uses the Collect and Display math language routine to advance conversing and reading as students clarify, build on, or make connections to mathematical language.
Launch
Display the first part of the Task Statement for all to see:
Diego and Jada are both trying to write an expression with fewer terms that is equivalent to
Jada thinks is equivalent to the original expression.
Diego thinks is equivalent to the original expression.
Remind students that we can tell whether the expressions are equivalent by substituting some different values for and and evaluating the expressions.
First, ask students to substitute the values and and evaluate the original expression, Jada’s expression, and Diego’s expression. All the expressions come out to 26. Perhaps Jada’s and Diego’s are both equivalent to the original expression?
Then ask students to choose some different values for and and evaluate: the original expression, Jada’s expression, and Diego’s expression. For any other values of and , Jada and Diego's expressions do not evaluate to the same thing. For example, for and , the original is 11, Jada’s expression is 9, and Diego's is 11.
Emphasize the outcome that the value of Diego’s expression matches the original expression’s, but Jada’s does not. This proves that Jada’s expression is not equivalent to the original, but it does not prove that Diego’s expression is equivalent to the original.
Tell students that experimenting with numbers can tell us that two expressions are not equivalent, but it can't prove that two expressions are equivalent. To prove two expressions are equivalent, we need to show they are equal for every value. But it’s impossible to try every value. So, to prove that two expressions are equivalent, we need to reason about the expressions using the properties that we know.
Arrange students in groups of 2. Allow 6–7 minutes quiet work time and partner discussions followed by a whole class discussion.
Use Collect and Display to create a shared reference that captures students’ developing mathematical language. Collect the language that students use to justify why expressions are equivalent. Display words and phrases, such as “rearrange,” “expand,” “combine,” “rewrite,” and “distribute.”
Action and Expression: Develop Expression and Communication. Provide students with alternatives to writing on paper: students can share their learning orally and kinesthetically, using drawings or physical manipulatives such as connecting cubes or sticky notes. Supports accessibility for: Language, Visual-Spatial Processing
Activity Synthesis
Direct students’ attention to the reference created using Collect and Display. Ask students to share their reason each step is equivalent to the last. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases. (For example, the display may have “You can rewrite subtracting as adding so you can rearrange the terms” already on it, which can be updated with the more precise statement “You can rewrite subtraction as adding the opposite so you can use the commutative property.”)
Invite students to justify that the steps taken by Diego do not change the value of the expressions. Emphasize places where he used the distributive property and the commutative property.
Present a simpler expression, like , and ask students how you could rewrite it with fewer terms. Ensure they see it’s equivalent to , whether justified by or by writing .
Consider using this opportunity to explain the meaning of this lesson’s title. Explain that two terms are called “like terms” when they are both numbers or when they both have the same variable. When we add like terms, we call it “combining like terms.” Use these examples to clarify:
are like terms because they both have the variable . We can combine them as .
are like terms because they are both numbers. We can combine them as 8.
are not like terms because one has a variable and one is a number. We cannot combine them.
3.3
Activity
Optional
Standards Alignment
Building On
Addressing
7.EE.A.1
Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
In this activity, students use what they have learned so far to find an unknown term or expression that makes two expressions equivalent. Students have many strategies at their disposal to reason about the unknown term or expression in each equation. For example, for the first equation , students might reason as in the previous activity and write out the sum of 6 's and ? on one side, and a string of 10's on the other side, and reason that 4 's are needed to make the sides equivalent. Alternatively, they might reason with the distributive property, and rewrite the left side as .
When students compare their results with a partner and resolve discrepancies, they have an opportunity to justify their thinking and critique the reasoning of others (MP3).
Launch
Arrange students in groups of 2. Explain that they will use what they have learned so far to find an unknown term or expression that will make two expressions equivalent. Draw students’ attention to the instructions, which tell them to complete the first set of problems, check in with their partner, and then proceed. If desired, consider asking students to pause after the first set for whole-class discussion.
Engagement: Develop Effort and Persistence. Encourage and support opportunities for peer collaboration. When students share their work with a partner, display sentence frames to support conversation, such as “It looks like . . . .” “I noticed _____ so I . . . .” “How do you know . . . ?” or “That could/couldn’t be true because . . . .” Supports accessibility for: Language, Social-Emotional Functioning
Activity
None
Replace each ? with a term or expression in parentheses that will make the expression on the left side of the equation equivalent to the expression on the right side. Check your results for Set A with your partner and work to reach an agreement before moving on to Set B.
Set A
Set B
Activity Synthesis
The purpose of the discussion is to highlight alternative ways of reasoning about equivalent expressions.
Ask students to share their expressions for each problem. Record and display their responses for all to see. After each student shares, ask the class if they agree or disagree.
The following questions, when applicable, can be used as students share:
“Why didn't you combine terms and numbers?” (Rewriting expressions using the properties of multiplication or the distributive property shows why this doesn't result in an equivalent expression.)
“How did you decide on the components of the unknown term or expression?”
“Did you use the commutative property?”
“Did you use the distributive property?”
MLR8 Discussion Supports. Display sentence frames to support whole-class discussion. “I agree/disagree because . . . .”. Advances: Speaking, Conversing
Lesson Synthesis
Share with students, “Today we saw how properties of operations can help us write an equivalent expression with fewer terms.”
To review these concepts, consider asking students:
“What are some ways we can tell that is not equivalent to ?” (We can substitute any value other than 1 for . For example, if then and .)
“How do we know that is equivalent to ? (We can use the distributive property, .)
“What are some ways we could rearrange the terms in the expression and create an equivalent expression?” ()
Student Lesson Summary
There are many ways to write equivalent expressions, and they may look very different from each other. One way to determine if two expressions are equivalent or not is to substitute the same number for the variable in both expressions.
For example, when is 1, the expression equals 4 and the expression equals 7. This means and are not equivalent.
If two expressions are equal when many different values are substituted for the variable, then the expressions may be equivalent—it is impossible to compare the two expressions for all values. To know for sure, we use properties of operations. For example, is equivalent to because:
Standards Alignment
Building On
6.EE.A.4
Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions and are equivalent because they name the same number regardless of which number stands for.
Diego and Jada are both trying to write an expression with fewer terms that is equivalent to:
Jada thinks is equivalent to the original expression.
Diego thinks is equivalent to the original expression.
We can show that expressions are equivalent by writing out all the variables. Explain or show why each expression (after the first row) is equivalent to the one before it.
Here is another way we can rewrite the expressions. Explain or show why each expression (after the first row) is equivalent to the one before it.
Student Response
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Building on Student Thinking
Students may think the expressions are equivalent after finding them equal for and . Remind students that equivalent expressions must be equal for every possible value of the variable.
Students might have trouble describing the moves in the last two questions and justifying why the expressions are equivalent. Encourage students to closely examine the changes from row to row and consider why the moves shown do not change the value of the expression.
