Lesson 7

The purpose of this warm-up is to rekindle anything students remember about percentages and representations they use to reason about them.

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.

Invite students to share different representations and ways of reasoning. Record student strategies and nonchalantly write an equation for each in the process.

Students perform repeated calculations and then generalize with an algebraic expression (MP8). The purpose of this activity is to help students see that any basic percentage problem like “\(n\) percent of this is that” can be represented with an equation in the form \(px=q\).

Using any insights from the warm-up as an example, remind students of any efficient method they know to compute a percentage. For example, 25% of 16 can be computed using \(\frac{25}{100} \boldcdot 16.\)

Arrange students in groups of 2. Give 5–10 minutes of quiet work time and time to share responses with a partner, followed by a whole-class discussion.

Students might not understand that we are trying to find the whole that we know an amount is a certain percent of. Encourage them to draw a tape diagram or a double number line to visualize the relationship between the three quantities.

The purpose of this discussion is for students to see how writing and solving an equation can be an efficient way to solve a problem about percentages. In the course of the discussion, they should see three equations written and solved. If any students used representations like tape diagrams or double number lines to reason about the problem, it can be advantageous to display these alongside the equations so that students can make connections between strategies they understand well and the more abstract strategy of writing and solving an equation. 

In this activity, students are asked to write an equation but are not given a letter to use. This is an opportunity to explain to students that when they decide to use a letter to represent something, they need to state what the letter represents.

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

The focus of the discussion should be the selection of a variable to represent an unknown quantity. Invite students to share how they decided where to use a variable, what it represented in the story, what letter they used and why. Ask why it is important to state what the letter represents and where they made that statement in their solutions.