Over and Over
These materials, when encountered before Algebra 1, Unit 5, Lesson 16 support success in that lesson.
16.1: The Phone for the Lowest Price (10 minutes)
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.
A phone is on sale. Which sale results in the lowest price? Explain your reasoning.
- A $200 phone that is 35% off.
- A $200 phone for sale at 60% of its original price.
- A $150 phone for sale at 87.5% of its original price.
- A $140 phone that is 8.5% off.
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.
16.2: Two Saving Methods (15 minutes)
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.
Two people open bank accounts and deposit $1,000 each. Both bank accounts earn 7.5% interest every year.
- Kiran leaves $1,000 alone, but withdraws the interest each year.
- Jada leaves the interest in the bank account.
How much money does each person have after 5 years? Explain.
Select students to share their work who organized it clearly. For example, the information and calculations could be organized in tables, like this:
|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.”
16.3: Wild Sales (20 minutes)
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.
- Tyler really wanted a $200 jacket he saw in a shop window on his way to school every day. On Monday he noticed a sign on the jacket that said, “20% off each day!” The fine print said, “Price is reduced by 20% today, and will be reduced by an additional 20% of the previous price every day until the jacket is sold.” Tyler wondered if the jacket would be free on Friday. Diego does not think it will be free. Who do you agree with, and why?
- Elena saw that a local store was having a 50% off everything storewide sale. She had a coupon for 50% off the final price of any item at that store, and the coupon said, “Combine with any other offers!” She said, “I think I can get anything I want for free!” Priya said, “Not for free, but for 75% off, I think.” Who do you agree with, and why?
- Andre and Clare are saving money. They each started with $20. Each week for 3 weeks, Andre increased his savings by 25%. Clare didn’t increase her savings for 2 weeks, but in the third week, she increased her savings by 75% to keep up with Andre. Andre claims he raised more money. Clare says they are even. Who do you agree with, and why?
- Jada and Lin are saving money. They each started with $50. Each week for 4 weeks, Jada increased her savings by 15%. After 2 weeks, Lin increases her savings by 30%, and after another 2 weeks, by 30% again. Jada says that they both increased their savings by 60% and have the same amount. Lin is not sure. Do they have the same amount of savings? If not, who has more savings? Explain your reasoning.
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?”