In this activity, students reason about percentages that are not whole numbers. Students use what they know about place value and properties of operations to calculate fractional percentages of numbers, making use of structure (MP7). Similar to a Math Talk, the problems are sequenced so that students can use the result of the previous problem to help answer the next problem.

Monitor for students who:

• Use 1% to find 0.1%.
• Make substitutions with known quantities to help compute unknown quantities.
• Use the result of the previous problem to help answer the next problem.

Invite students to share how they reasoned about the problems. Highlight strategies by students who recognized that they can use 1% of a number to calculate 0.1% of a number and make multiples of that to get, for example, 0.7% of a number.

In this activity, students find the new value after a percent increase that involves a fractional percentage. The first problem asks about an 8% increase, and the second problem asks about a 0.8% increase, so students can use the relationship between these two percentages to help solve the second problem.

Monitor for students who:

• Calculate the percentage first and then add that to the original amount.
• Multiply the original amounts by 1.08 and 1.008, respectively.

Plan to have students present in this order to support moving them from less efficient to more efficient methods.

As students explain why either city could be considered to have grown by more, they are attending to precision (MP6).

The purpose of this discussion is to compare and contrast the 8% increase with the 0.8% increase. First, invite previously selected students to share how they calculated the new populations for each city. Sequence the discussion of the strategies in the order listed in the Activity Narrative. If possible, record and display the students’ work for all to see.

Connect the different responses to the learning goals by asking questions, such as:

• “How are these strategies the same? How are they different?”
• “How does the 8% increase (or 0.8% increase) show up in each method?”
• “Why do the different approaches lead to the same outcome?”

The key takeaway is that the same strategies work for finding 0.8% as for 8%. We can multiply first and then add, or we can add first and then multiply. The multiplier is just different by a factor of 10 (0.008 instead of 0.08).

Next, invite students to share their reasoning about which city grew by more. Either city could be the correct answer depending on whether students are viewing the growth additively or multiplicatively. Make sure that some students support each perspective. If needed, assign some students to support Lin’s statement and others to support Andre’s statement. The goal is for students to articulate more precisely what is meant by the phrase “grew by more” in each interpretation. To involve more students in the conversation, consider asking:

• “Do you agree or disagree? Why?”
• “Who can restate ’s reasoning in a different way?”
• “Does anyone want to add on to ’s reasoning?”