Lesson 8

Students worked with the distributive property with variables in grade 6 and with numbers in earlier grades. In order to understand the two ways of solving an equation of the form \(p(x+q)=r\) in the upcoming lessons, it is helpful to have some fluency with the distributive property. 

Arrange students in groups of 2. Give 3 minutes of quiet work time and then invite students to share their responses with their partner, followed by a whole-class discussion.

Focus specifically on why 1 and 3 are equivalent to lead into the next activity. You may also recall the warm-ups from prior lessons and ask if \(2(x+3)\) is equal to 10, or other numbers, how much is \(x+3\)?

Ask students to write another expression that is equivalent to \(2(x+3)\) (look for \(2x+6\)).

This activity continues the work of using a balanced hanger to develop strategies for solving equations. Students are presented with a balanced hanger and are asked to explain why each of two different equations could represent it. They are then asked to find the unknown weight. Note that no particular solution method is prescribed. Give students a chance to come up with a reasonable approach, and then use the synthesis to draw connections between the diagram and each of the two equations. Students notice the structure of equations and diagrams and find correspondences between them and between solution strategies.

Keep students in the same groups. Give 5–10 minutes of quiet work time and time to share their responses with a partner, followed by a whole-class discussion.

Have one student present who did \(7=x+3\) first, and another student present who subtracted 6 first. If no one mentions one of these approaches, demonstrate it. Show how the hanger supports either approach. The finished work might look like this for the first equation:

\(\begin {align} 14&=2(x+3) \\ 7&=x+3 \\ \end{align}\)

\(\begin {align} 7&=x+3 \\ 4&=x \\ \end{align}\)

For the second equation, rearrange the right side of the hanger, first, so that 2 \(x\)’s are on the top and 6 units of weight are on the bottom. Then, cross off 6 from each side and divide each side by 2. Show this side by side with “doing the same thing to each side” of the equation.

The first question is straightforward since each diagram uses a different letter, but it’s there to make sure students start with \(p(x+q)=r\). If some want to rewrite as \(px+pq=r\) first, that’s great. We want some to do that but others to divide both sides by \(p\) first. Monitor for students who take each approach.

Keep students in the same groups. Give 5–10 minutes of quiet work time and time to share their responses with a partner, followed by a whole-class discussion.

Select one hanger for which one student divided by \(p\) first and another student distributed \(p\) first. Display the two solution methods side by side, along with the hanger.