Lesson 21

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.

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.  

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. 

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. 

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.

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 \(\text-3(4-9x)\):

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

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. 

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. 

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.

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.”