Lesson 12

The purpose of this warm-up is for students to review how to solve for percentage change and represent these situations with expressions. Analyzing the structure of equivalent expressions for the same situation helps students see how the quantities in it are related. Since there are many equivalent expressions to represent how to find percentage change in a situation, encourage students to look for relationships between the expressions.

Arrange students in groups of 2. Give students 1 minute of quiet work time followed by 2 minutes to compare their responses with their partner. During the partner discussion, tell students to discuss the expressions they have in common, ones they don’t and then try to come to an agreement on the correct expressions that represent the price of the item after the discount. Follow with a whole-class discussion. 

Some students may choose expressions that represent the discount itself instead of the price of the item after the discount. Ask those students to refer back to the situation to identify which piece of the problem the expression they chose finds. If students are still unclear, it may be helpful to give students a price for \(x\) such as $10 and ask students if 20% of $10 makes sense as the new price of the item after the discount and then what piece of the problem they found. 

Ask students to indicate whether each expression represents the price of the item after the discount. If all students agree on an expression, ask 1 or 2 students to explain their reasoning and move to the next expression. Record and display their responses for all to see. If there is a disagreement on an expression, ask students to explain their reasoning for both choices and come to an agreement.

After the class had agreed on the four expressions that represent the price of the item after the discount, record them as a list and display them all to see. Ask students to discuss any connections they see between the expressions to show they are equivalent. 

The first three questions help students recall how tape diagrams can be used to represent a percentage increase situation and draw connections to an equation representing percent increase. Monitor for different approaches in the first three questions (see solutions for different approaches).

The last question is review of previous work in this unit. This fourth question can be used for additional practice, but it can be safely skipped if time is short.

Keep students in the same groups. Tell students to work on the first three questions and pause for discussion. Give 5 minutes of quiet work time and time to share their responses with a partner, followed by a whole-class discussion. If time permits, the last question can be used as more practice on work from earlier in the unit.

If students bring up that the diagram represents 120% or \(\frac65\), or if they refer to each equal part as 20% or \(\frac15\), ask what whole the fraction or percent refers to. They should understand that the whole is the amount from Day 2, \(d+5\).

Select groups with different approaches to share their responses to the first three questions. 

This activity offers four word problems. Depending on time constraints, you may have all students complete all four problems or assign a different problem to each group. The problems increase in difficulty. It is suggested that students create a visual display of one of the problems and do a gallery walk or presentation, but if time is short you may choose to just have students work in their workbooks or devices.

Keep students in the same groups. Either instruct students to complete all four problems or assign one problem to each group. If opting to have students do presentations or a gallery walk, distribute tools for making a visual display.

Give students 5–6 minutes quiet work time and partner discussion followed by a whole-class discussion or gallery walk.

If students created a visual display and you opt to conduct a gallery walk, ask students to post their solutions. Distribute sticky notes and ask students to read others’ solutions, using the sticky notes to leave questions or comments. Give students a moment to review any questions or comments left on their display.

Invite any students who chose to draw a diagram to share; have the class agree or disagree with their diagrams and suggest any revisions. Next, invite students who did not try to draw a diagram to share strategies. Ask students about any difficulties they had creating the expressions and equations. Highlight equivalent expressions that represent the same quantity and different strategies for solving equations.