Lesson 9

Students perform mental calculations by applying strategies involving the distributive property.

Display one problem at a time. Give students 30 seconds of quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Once students have had a chance to share a few different ways of reasoning about this product, focus on explanations using the distributive property like \(5 \boldcdot 98 = 5 \boldcdot (90 + 8)\) or \(5 \boldcdot (100 - 2)\), then writing out the two products from distributing. Remind students of the distributive property, and let them know they will spend the next few lessons working with it.

The purpose of this activity is to remind students of rectangle diagrams they worked with in a previous unit to represent multiplication. It is also to introduce the convention that for example the expression \(6 \boldcdot 3 + 2\) equals 20. If we want the sum to be carried out before the product, we would need to use parentheses like \(6 \boldcdot (3+2)\).

Allow students 10 minutes of quiet work time, followed by a whole-class discussion.

Students may conclude that \(6 \boldcdot 3 + 2\) represents the area of the rectangle. This is a good opportunity to introduce a convention. When we have multiplication and addition in the same expression, it is the convention that the multiplication is done first. So \(6 \boldcdot 3 + 2\) equals \(18 + 2\), or 20, so it doesn’t represent the area of the rectangle, which we know to be 30 square units. If you want the addition to be done first, you need to use parentheses. Therefore, \(6 (3+2)\) does represent the area of the large rectangle. Remind students that “next to” implies multiplication.

By using the rectangle, we can tell that \(6 (3+2)=6 \boldcdot 3 + 6 \boldcdot 2\). This is another example of two expressions that are equivalent because of the distributive property.

This is for practice going back and forth with the distributive property using numbers, but also to make the point that invoking the distributive property can help you do computations in your head. Some expressions that would be difficult to brute force become simpler after using the distributive property to write an equivalent expression.

Note that there is more than one way to rewrite the last row. For example, \(24-16\) could also be written as \(2(12-8)\) where \((12-8)\) is the difference of two terms. A term is a single number or variable, or variables and numbers multiplied together. This is fine, since there is no reason to insist that students use the greatest common factor at this time. Students should recognize that there is more than one factor that would work, and that the resulting expressions are equivalent. In a subsequent unit students will explicitly study the idea of a greatest common factor.

Give students the following setup: Suppose your business makes 15 items for $17 each and sells them for $20 each. How would you find your profit? One way is to write \(15\boldcdot 20 - 15\boldcdot 17\). But that’s a lot of calculations! An easier way to get the answer is to write \(15(20-17)\), which is easier to figure out in your head.

Allow students 10 minutes of quiet work time, followed by a whole-class discussion.

Students might understand how to expand an expression with parentheses but struggle with how to approach a sum. Encourage students to think about the rectangle diagrams they have seen and draw a diagram of a partitioned rectangle. Ask students what the sum represents and help them to see that it can represent the sum of the areas of the two smaller rectangles. Remind students that the rectangles have the same width, and ask what that width might have been to produce the two areas, what factor the two areas have in common. Then have them consider the other factors (the lengths) that would produce those products for the areas. 

Invite students to share their strategies and reasoning. Include students who used diagrams of partitioned rectangles. Ask if they noticed any interesting patterns, or if they want to share some examples of their own of calculations that can be made simpler by using the distributive property to write an equivalent expression.