Lesson 8

There are many ways to think about debt, and the way the lender views it differs the way from the borrower does. This warm-up introduces the idea that we can represent a debt with a signed number. From the perspective of the person who owes money, the debt is usually viewed as a negative number. From the perspective of the bank, it may be viewed as a positive number.

Arrange students in groups of 2. Ask students,

"Priya is buying concert tickets for her and her friends with the money she earns at her part-time job. This month, she has earned $135. Can she buy three $50 tickets for a concert?"

Ask students to discuss with a partner for 1 minute.

Explain that sometimes we can borrow money from a bank to buy things we cannot afford at the time, and then pay the money back to the bank in the future. Give students time for partner discussion followed by whole-class discussion.

The most important thing for students to understand is that when representing a debt with a negative number, the additive inverse tells how much money is needed to pay off the debt. If Priya has -$15, then she needs $15 more to raise her balance back to $0.

If no students suggest that we can represent Priya's money using a negative number, introduce the idea.

In this activity, students solve problems about debts that can be represented with addition and subtraction equations. Some problems ask students to calculate the balance after the transaction and some questions ask students to calculate the amount of the transaction, given the starting and ending balances. Students draw number lines to represent each problem.

The series of questions involves a running balance after deposits and withdrawals are made. Students may represent this by drawing a new diagram for each question, or by adding on to the same diagram. Either approach will work. 

The focus in this activity is on writing a new equation to represent each situation, and on creating a diagram to represent the situation. If students struggle, encourage them to think about what they have already learned about adding and subtracting integers, and assure them that they can use that understanding to reason about money. If students might struggle with the computations, consider providing access to a calculator to take the focus off of computation.

Remind students that a deposit is money paid into an account. If students do not read carefully, they may not realize that they are expected to write an equation and create a diagram for each question, and only record a numerical answer. Ensure they understand what they are expected to do before they begin working.

Give students quiet work time followed by whole-class discussion.

Some students may struggle to write an equation for each problem. Prompt them to identify what amount is unknown in each situation.

The most important thing for students to understand is that all the rules they have learned for adding and subtracting signed numbers still work when applied to the context of negative amounts of money.

Review each of the following types of computations and discuss how they apply to the school cafeteria situations:

• Adding numbers with the same sign
• Adding numbers with opposite signs
• Adding opposites makes 0
• Subtracting as addition with a missing addend
• Subtracting as adding the additive inverse

In this activity, students see that withdrawals, in addition to debts, can also be represented using negative numbers. Students continue using addition and subtraction to solve problems about debt. While solving the last problem, students may begin wondering about multiplying and dividing signed numbers, which will be addressed in the next several lessons.

You may wish to ask students to pause after the first question for discussion. The decision about which numbers to represent with positive versus negative values hinges on whether you are thinking from the perspective of the person or the perspective of the account. Point out that the final balance is represented with a negative number to show that the person owes the bank money (this should be brought out in the launch). Therefore, from the perspective of the account, deposits are positive values and withdrawals are negative values. It would be possible to proceed either way, but it will facilitate discussion later if everyone uses the same convention as a result of work on the first question.

As students work, monitor for students who are expressing their reasoning as addition and subtraction equations or expressions.

Display the bank statement image for all to see, without the questions. Ask students to think of two things they notice and two things they wonder. Give students 1 minute of quiet think time. Select a few students to share. Make sure students understand the meaning of deposit and withdrawal.

Give students quiet work time followed by whole-class discussion.

The most important thing for students to understand is that the rules for adding and subtracting signed numbers can help them solve problems about debts.

Select students to share their solutions. Ask students to indicate whether they agree, disagree, or have any clarifying questions.