Lesson 16

The purpose of this task is to help students develop (or remind them of) shortcuts for computing percentages of a number and percentages off a number. Students may construct tape diagrams or double number line diagrams to help them make sense of these problems. Monitor for these representations so that they can be connected to shortcuts presented in the activity synthesis.

Provide access to calculators.

If any students used a representation like a tape diagram or a number line to reason, present their work alongside the work of a student who used more efficient methods. For example, you could have a tape diagram where the whole tape is labeled “200,” one piece is labeled 35% and the other 65%. Students may reason along the lines “I know 10% of 200 is 20, which means 5% of 200 is 10, which means 35% is 70.” But place these representations alongside numerical expressions like \(0.35(200)\) and \(0.65(200)\). Both knowing efficient ways to compute these values, but also why they make sense, are important.

To prepare for the next activity, make sure to mention that 8.5% of 140 can be computed by multiplying 140 by 0.085.

The purpose of this task is to introduce the idea of interest, and show through example how the balance is different if someone earns interest on their interest, versus someone who withdraws the interest every year. The activity asks a question without much scaffolding, offering an opportunity for students to make sense of the problem and persevere in solving (MP1).

In addition to “interest,” students may need help interpreting terms like “deposit” and “withdrawal.” Use the launch to gauge how much time to spend on helping students understand the terms associated with bank accounts.

Monitor for different ways students organize their work.

Ask students if they have ever heard of earning interest on a bank account. After one or more students share their understanding, clarify that for some types of bank accounts, the bank gives you a percentage of the amount in the account every year. For example, if an account gives 2% interest, and you deposit $100 in the account, you would earn an additional $2 from the bank after a year. The reason banks do this is because the money in your account doesn’t just sit there, but the bank uses it to accomplish other things, like loaning to other customers.

The interest that the bank deposits in your account is now yours. If you leave this money alone, you also earn interest on the new money. So if you left your $102 alone, after a year, you would earn 2% of that amount in interest. 2% of 102 is 2.04, so you would earn $2.04, and your new balance would be $104.04.

Provide access to calculators.

Select students to share their work who organized it clearly. For example, the information and calculations could be organized in tables, like this: 

year	0	1	2	3	4	5
Kiran's account balance	1000	1000	1000	1000	1000	1000
amount Kiran has withdrawn	0	75	150	225	300	375
Jada's account balance	1000	1075	1155.63	1242.30	1335.47	1435.63

The important point to draw out is that Jada earned interest on her interest, and Kiran did not. Questions for discussion:

• “From their original deposit of $1000, how much did Kiran have at the end of year 2?” ($1150.)
• “How much did Jada have at the end of year 2?” ($1155.83.)
• “Why did Jada end up with slightly more money in year 2? “She earned 7.5% on $1075, since she left her interest in the bank. But Kiran only earned 7.5% on $1000, since he withdrew the interest from the bank.”

In this practice activity, students have an opportunity to solve a problem where a percentage change is applied repeatedly. After students have had a chance to make a visual display, conduct a gallery walk or invite one group for each problem to present their solution. When students explain why they agree or disagree with a character’s statement, they are justifying their reasoning and critiquing the reasoning of others (MP3).

Arrange students in groups of 2–3 and distribute tools for making a visual display. Either assign one problem to each group or allow them to choose a problem. Encourage students to make a rough draft of their solution before making their visual display, and make sure that their work is organized, clear, and easy to understand.

Much of the discussion will occur within the groups. If time permits, consider conducting a gallery walk. Post each group’s visual display around the room, and give every student a few sticky notes to add questions or comments to the displays.

Alternatively, invite one group for each problem to present their work to the class. To keep other students involved, ask:

• “Can you restate or summarize this group’s approach?”
• “Think about this group’s solution. What do you agree or disagree with?”
• “What are some choices the group made that made their display easy to understand? What are some things they could improve?”