Lesson 22

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.

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.

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. 

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. 

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. 

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.

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. 

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.”

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

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.

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