8.1: Concert Tickets (10 minutes)
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.
Priya wants to buy three tickets for a concert. She has earned $135 and each ticket costs $50. She borrows the rest of the money she needs from a bank and buys the tickets.
- How can you represent the amount of money that Priya has after buying the tickets?
- How much more money will Priya need to earn to pay back the money she borrowed from the bank?
- How much money will she have after she pays back the money she borrowed from the bank?
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.
8.2: Cafeteria Food Debt (10 minutes)
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.
Supports accessibility for: Organization; Attention
At the beginning of the month Kiran had $24 in his school cafeteria account. Use a variable to represent the unknown quantity in each transaction below and write an equation to represent it. Then, represent each transaction on a number line. What is the unknown quantity in each case?
In the first week he spent $16 on lunches. How much was in his account then?
Then he deposited some more money and his account balance was $28. How much did he deposit?
Then he spent $34 on lunches the next week. How much was in his account then?
Then he deposited enough money to pay off his debt to the cafeteria. How much did he deposit?
- Explain why it makes sense to use a negative number to represent Kiran's account balance when he owes money.
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
Design Principle(s): Optimize output (for explanation)
8.3: Bank Statement (10 minutes)
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.
Supports accessibility for: Memory; Conceptual processing
Design Principle(s): Cultivate conversation; Support sense-making
Here is a bank statement.
If we put withdrawals and deposits in the same column, how can they be represented?
Andre withdraws $40 to buy a music player. What is his new balance?
- If Andre deposits $100 in this account, will he still be in debt? How do you know?
Are you ready for more?
The national debt of a country is the total amount of money the government of that country owes. Imagine everyone in the United States was asked to help pay off the national debt. How much would each person have to pay?
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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.
Main learning points:
- We can use positive numbers to represent payments into a bank account (deposits) and negative numbers to represent money taken out of an account (withdrawals).
- We can also use a negative balance to represent debt (owing money).
- We can use the additive inverse to quickly find how much money is needed to reach a balance of zero.
- What words do we use to mean "money added into" or "money taken out of" an account?
- How can we represent owing money?
- Why does it make sense to use negative numbers to represent debt?
- How can we tell how much money is needed to pay off a debt?
8.4: Cool-down - Buying a Bike (5 minutes)
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Student Lesson Summary
Banks use positive numbers to represent money that gets put into an account and negative numbers to represent money that gets taken out of an account. When you put money into an account, it is called a deposit. When you take money out of an account, it is called a withdrawal.
People also use negative numbers to represent debt. If you take out more money from your account than you put in, then you owe the bank money, and your account balance will be a negative number to represent that debt. For example, if you have \$200 in your bank account, and then you write a check for \$300, you will owe the bank \$100 and your account balance will be -\$100.
|starting balance||deposits and withdrawals||new balance|
|0||50||\(0 + 50\)|
|50||150||\(50 + 150\)|
|200||-300||\(200 + (\text-300)\)|
In general, you can find a new account balance by adding the value of the deposit or withdrawal to it. You can also tell quickly how much money is needed to repay a debt using the fact that to get to zero from a negative value you need to add its opposite.