Lesson 8

This activity is to prepare students for the warm-up in the associated Algebra 1 lesson, where they will need to explain why a diagram can be represented by two different expressions.

• “How would the values of the two expressions compare if \(x\) is 2?” (They would each equal 56.)
• “If \(x\) is 10?” (They would each equal 112.)
• “If \(x\) is 1,000?” (They would each equal 7,042.)
• “We call the two expressions ‘equivalent expressions.’ What does that mean? How would you define it?” (Equivalent expressions have the same value for all values of the variables.)

Highlight that the two expressions \(42+7x\) and \(7(6+x)\) are equivalent expressions. This means they have the same value for any value of \(x\). Even though we can’t check all possible values of \(x\), we know they are equivalent because they represent the same area, and because they are equivalent by the distributive property.

The work in this activity is similar to the work in the associated Algebra 1 lesson, except that it is more scaffolded, and it focuses on operations with numbers instead of with variables.

Display the two rectangles from the first part of the activity. Ask students, “What do you notice and wonder about the two rectangles?” Students might notice that they appear to be identical, and that the side lengths of B are equal to the side lengths of A. They might wonder why B is partitioned into smaller rectangles with the dashed lines.

Ask students to make sense of the different ways to express the area of the last rectangle. In the course of this discussion, ensure they see work that looks like:

\((10+6)(10+1)\)

\(10 \boldcdot 10 + 10 \boldcdot 6 + 10 \boldcdot 1 + 6 \boldcdot 1\)

\(100 + 60 + 10 + 6\)

Notice that the first 10 is multiplied by both 10 and 1, then the 6 is multiplied by both 10 and 1. These four products are added. This is an example of the distributive property.

These practice exercises offer an opportunity to practice rewriting expressions using the distributive property. There may be more work here than can reasonably completed in the available time. If that is the case, consider assigning a subset of items that would be most beneficial for your students to practice.

Note that one of the questions gives an expression like \(3x+12\) for the area of a rectangle, and asks for a possible length and width. While 3 and \(x+4\) may be the most likely response, another example of a valid response would be \(\frac15\) and \(15x+60\).

The intention is for students to reason using the distributive property or by drawing diagrams, so restrict access to calculators unless it’s necessary to access the activity.

Arrange students in groups of 2–4. Encourage students to check in with their group as they work.

The purpose of this lesson was to remind students of a way to understand the distributive property using expressions for the area of a rectangle. Possible questions for discussion:

• “What are two different ways to think about the area of the rectangles we’ve seen in this lesson?” (You can add the areas of the individual pieces, or you can multiply the entire length by the width.)
• “What is an example of two expressions that are equivalent because of the distributive property?” (Answers vary.)
• “How would you use a diagram to explain to someone who was absent today why \(a(b+c)=ab+ac\) for any value of \(a\), \(b\), and \(c\)?” (Use a rectangle with one side labeled \(a\) and the other \(b+c\). The rectangle is subdivided into two regions with areas \(ab\) and \(ac\). These total to the area of the entire rectangle, which is also length times width, or \(a(b+c)\).)