Lesson 7

The purpose of this warm-up is to elicit the idea that percent increases and decreases can be represented with double number lines, which will be useful when students use double number lines in a later activity. While students may notice and wonder many things about these images, recognizing the original amount, represented by 100%, is an important discussion point.

Arrange students in groups of 2. Tell students that they will look at an image, and their job is to think of at least one thing they notice and at least one thing they wonder. Display the image for all to see. Ask students to give a signal when they have noticed or wondered about something. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After each response, ask the class if they agree or disagree and to explain alternative ways of thinking, referring back to the images each time. If the original quantity, represented by 100%, does not come up during the conversation, ask students to discuss this idea.

The purpose of this activity is for students to use double number line diagrams to represent situations of percent increase and decrease. Additionally, students identify the original and new amount in the double number lines to reinforce that the original amount pertains to 100%.

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

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:

• \((1.5) \boldcdot 12 = c\)
• \((0.80) \boldcdot 15 = c\)
• \(a \boldcdot (1,\!200) = 1,\!080\)
• \(a \boldcdot (1.25) = 1.50\)
• \((0.75) \boldcdot b = 12\)
• \((1.25) \boldcdot b = 6,\!600\)

The purpose of this activity is to help students understand that percent increase should be interpreted as a percent of the original or starting amount. Students are presented with two different methods to reason about and decide with which one they agree (MP3).

As students work on the activity, look for students who agree with Clare and Priya, these students should be asked to share during the whole-class discussion. The focus of the discussion should be on why Priya's number line is the correct one and how the term percent increase affects the original or starting amount (it goes over 100%).

Give students 1 minute of quiet think time followed by partner and then whole-class discussion.

If students do not see why Clare's number line is incorrect, ask them to think about the original amount of ounces the juice box comes with. If the packaging claims to be more than what it started with should it be more or less than the original amount (100%)?

Ask selected students to share their reasoning. Ask students, When the packaging says “20% more juice,” that means 20% more than what? (The amount of juice in the original packaging.) Explain that phrases like “percent more” or “percent less” are expressing the percent of some original amount. So, if we use a double number line strategy, it makes sense to associate the original or starting amount with 100%.

The purpose of this activity is for students to encounter a situation where the quantity given is not the whole amount, but rather is the amount after a decrease. In this case, they are given the amount after a 10% decrease. They should make the connection from previous lessons that the amount given is 90% of the whole.

Give students 3–5 minutes of quiet work time followed with whole-class discussion.

If time is limited pick the second problem to talk about during the discussion.

For the percent decrease problem, students may calculate 10% of 234, getting a change of 23.4 turtles and an original number of 257.4 turtles. Remind them that the percent decrease describes the change as a percentage of the original value, not as a percentage of the new value. If needed, prompt students to represent the situation using a double number line, placing 234 to the left by 10% of the quantity they want to find, which should be associated with 100%.

Ask students:

• How are the two problems the same? How are they different?
• What information is given in the first problem that is not given in the second? Second and not the first?
• In which problem were you given the quantity that represented 100%?

Connect the previous discussion about the original value being 100% by asking:

• For the percent increase question, what percentage value were you given? (students might answer 10% here but that is not actually the case)
• How did you use that to figure out the original number of green sea turtles?