Lesson 8

In this lesson, students will be finding percent increase and percent decrease by multiplying by an appropriate factor. For example, to find a 36% increase in a quantity, we can multiply that quantity by 1.36. This warm-up prompts them to think about the scale factor needed to multiply one number to get another.

Display one problem at a time. Give students 30 seconds of quiet think time per problem and ask them to give a signal when they have an answer and a strategy. Allow students to share their answers with a partner and note any discrepancies. Follow with a whole-class discussion.

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
• “Do you agree or disagree? Why?”

In this activity, students are calculating percent increase and decrease in the context of interest and depreciation. They may have encountered these contexts before this unit, but be sure that everyone understands the basic idea of interest and depreciation before students begin work.

Students have enough experience to solve these problems using various strategies and representations. Look for students writing different but correct expressions in terms of \(x\).

Remind students what interest is, give examples of when interest is applied (savings accounts, credit cards, etc.), and discuss depreciation value of an item.

Students in groups of 2. Give students 3–5 minutes of quiet work time, followed by partner then whole-class discussion.

If students have a hard time organizing their information, suggest that they use a table.

Students might struggle to write an expression in terms of \(x\). Have students describe in words how they calculated the previous problems with numbers and use that to come up with an expression.

Select students with different ways of writing correct expressions in terms of \(x\) to share their expression. Connect both problems back to the previous work with the distributive property by asking students:

• How else can the first problem be written? (\(x + 0.06x\) or \((1 + 0.06)x\) or \(1.06x\))
• How else can the second problem be written? (\(x - 0.15x\) or \((1 - 0.15)x\) or \(0.85x\))

In this activity, students match equations that represent a percent increase situation to the situations they represent.

Arrange students in groups of 2. 2 minutes of quiet think time followed by 2 minutes of partner discussion. 

Ask one or more students to share which equation they matched with each situation, and resolve any discrepancies. Once the matches are agreed upon, ask students how they would solve the equation to find the amount without actually solving the equations.

The purpose of this activity is for students to use equations to represent situations of percent increase and decrease. Additionally, students identify the original and new amount in the double number lines to reinforce what they learned in earlier lessons (that the original amount pertains to 100%).

As students work on the task, look for students who created various equations for the last question.

Show students the double number line from the activity in the previous lesson.

Help them make connections to the first problem. Tell them that they can go back to the previous lesson to see the double number lines for the others if that helps.

Arrange students in groups of 2. Give 5–8 minutes of quiet work time. After 5 minutes allow students to work with a partner or to continue to work alone.

Students may continue to struggle to recognize the original amount and new amount with the proper percentages on the double number line. Remind them that the original amount always corresponds to 100%.

Select students to share the values they identified as original amount and the new amount for a few problems. Discuss how 100% always corresponds to the original value and when there is an increase in the value the new value corresponds to a percentage greater than the original 100%.

Select students to share the different equations they came up with. Discuss how the distributive property is useful for finding the percentage that corresponds with the new value instead of the percentage of the change.

Discuss how solving problems about percent change may require either multiplying or dividing numbers. It can be confusing, but it helps to first express the relationship as an equation and then think about how you can find the unknown number. Looking at the examples below, the first two require multiplication, but the others require division.

Using the structure A% of B is C:

• \((1.5) \boldcdot 12 = c\)
• \((0.80) \boldcdot 15 = c\)
• \(a \boldcdot (1,200) = 1,080\)
• \(a \boldcdot (1.50) = 1.75\)
• \((0.75) \boldcdot b = 12\)
• \((1.25) \boldcdot b = 61, 600\)