In this activity, students are given a percent increase or percent decrease and the original amount. They use this information to calculate the new amount. As students interpret the situations and choose a strategy for finding the new amount, they are making sense of problems and persevering in solving them (MP1).

Monitor for students who:

• Draw a double number line.
• Create a table of equivalent ratios.
• Calculate the amount of the increase or decrease and then add or subtract it from the original amount.
• Find the percentage that the new amount is of the original amount and multiply the original amount by that.

The key focus is how students make sense of the multi-step problem and choose what steps are needed to calculate the new amount.  

Have students use the double number line diagram if they need help figuring out the value that is 20% more.

The goal of this discussion is to highlight how students made sense of the percent increase and decrease situations and what steps they chose to calculate the new amount. Display 2–3 approaches from previously selected students for all to see. If time allows, invite students to briefly describe their approach. Use Compare and Connect to help students compare, contrast, and connect the different approaches. Here are some questions for discussion:

• “What do the approaches have in common? How are they different?”
• “How does the ‘20% more‘ or ‘20% less‘ show up in each method?”
• “Did anyone solve the problem the same way, but would explain it differently?”

The key takeaway is that “20% more” means the new amount is 120% of the original amount, while “20% less” means the new amount is 80% of the original amount.

Tell students that the change on the shampoo bottle is an example of a percent increase. The discount on the shirt is an example of a percent decrease. It may be mentioned that in both cases, 100% always corresponds to the original amount before the change; however, future activities will address this concept in depth.

In this activity, students match verbal descriptions to tape diagrams and see how the percentage of the increase or decrease determines the percentage that the new amount is of the original amount. For example, a 15% increase corresponds to 115% of the original amount, while a 30% decrease corresponds to 70% of the original amount. Then students create their own tape diagrams to represent two more situations. As students use the tape diagrams to make sense of the increase or decrease, they are making use of structure (MP7).

The purpose of this discussion is for students to explain the connections between the tape diagrams and the verbal descriptions, including where the 75% and 125% come from.

First, invite students to share how they reasoned about the fruit harvest situations and the given diagrams. For each diagram, consider asking, “Why do both of these verbal descriptions match the same diagram?” If possible, record the relevant reasoning on or near the image.

Display these two images for all to see.

Ask students, “What is the same about these two diagrams? What is different?” (They both represent a 25% increase from last year to this year. The first diagram is partitioned into more pieces than the second diagram.) Explain that the tape diagrams that we have worked with in the past were divided into pieces that were all the same size. Here, we see that tape diagrams can be divided into pieces of different sizes, as long as the size of each piece is clearly indicated.

Next, invite students to share the diagrams they drew for the animal population situations. Consider asking:

• “The new number of ducks is what percentage of the old number of ducks? How do you know?”
• “The new number of mosquitos is what percentage of the old number of mosquitos? How do you know?”