Question
Lesson Activity
Understanding Multiplication of Rational Numbers
This activity will help you meet these educational goals:
Mathematical Practices—You will make sense of problems and solve them, reason abstractly and quantitatively, and use mathematics to model real-world situations.
Directions
Read the instructions for this self-checked activity. Type in your response to each question, and check your answers. At the end of the activity, write a brief evaluation of your work.
Activity
Quiana took out a loan to pay for a new car. Initially, she owed the lender $15,234.68. She has repaid $247.43 of the loan each month for the past 5 months. What is the net change to the loan from Quiana’s perspective over the past 5 months?
Complete the steps below to help Quiana calculate the loan amount that she’s repaid to the lender.
Part A
Write a rational number to express the initial loan amount as it relates to Quiana’s bank account.
Space used (includes formatting): 27 / 30000
Part B
What is the sign of the rational number you wrote in part A for Quiana’s loan? What does the sign represent?
Space used (includes formatting): 0 / 30000
Part C
Write a rational number that represents the amount of the monthly payment from Quiana’s perspective.
Space used (includes formatting): 0 / 30000
Part D
Quiana made five equal montly payments from her bank account. Which rational number shows something being removed five times?
Space used (includes formatting): 0 / 30000
Part E
Now, write an expression using the two terms from parts C and D to represent the net change to Quiana’s loan as a result of her payments.
Space used (includes formatting): 0 / 30000
Part F
One way to simplify expressions with negative terms is to distribute the -1s. Rewrite the expression from Part E, factoring out a -1 from any negative value.
Space used (includes formatting): 0 / 30000
Part G
Next, rewrite the expression by grouping the -1s and grouping the numbers that are not -1s.
Space used (includes formatting): 0 / 30000
Part H
Now find the product of the values that are not -1s and rewrite the expression using this product.
Space used (includes formatting): 0 / 30000
Part I
Before we move on, what does multiplying a number by -1 mean?
Space used (includes formatting): 0 / 30000
Part J
When simplifying the expression in part H, how many times will the sign change on the positive term? Will the final expression be positive or negative?
Space used (includes formatting): 0 / 30000
Part K
Simplify the expression in part H. Following the five payments, what is the net change in the loan amount from Quiana’s viewpoint?
Space used (includes formatting): 0 / 30000
Self-Evaluation
How did you do? Rate your work on a scale of 1 to 5, with 5 as the highest score. Then write a brief evaluation of your work below. Note what you learned and what challenged you.
Space used (includes formatting): 0 / 30000
Understanding Multiplication of Rational Numbers
This activity will help you meet these educational goals:
Mathematical Practices—You will make sense of problems and solve them, reason abstractly and quantitatively, and use mathematics to model real-world situations.
Directions
Read the instructions for this self-checked activity. Type in your response to each question, and check your answers. At the end of the activity, write a brief evaluation of your work.
Activity
Quiana took out a loan to pay for a new car. Initially, she owed the lender $15,234.68. She has repaid $247.43 of the loan each month for the past 5 months. What is the net change to the loan from Quiana’s perspective over the past 5 months?
Complete the steps below to help Quiana calculate the loan amount that she’s repaid to the lender.
Part A
Write a rational number to express the initial loan amount as it relates to Quiana’s bank account.
Space used (includes formatting): 27 / 30000
Part B
What is the sign of the rational number you wrote in part A for Quiana’s loan? What does the sign represent?
Space used (includes formatting): 0 / 30000
Part C
Write a rational number that represents the amount of the monthly payment from Quiana’s perspective.
Space used (includes formatting): 0 / 30000
Part D
Quiana made five equal montly payments from her bank account. Which rational number shows something being removed five times?
Space used (includes formatting): 0 / 30000
Part E
Now, write an expression using the two terms from parts C and D to represent the net change to Quiana’s loan as a result of her payments.
Space used (includes formatting): 0 / 30000
Part F
One way to simplify expressions with negative terms is to distribute the -1s. Rewrite the expression from Part E, factoring out a -1 from any negative value.
Space used (includes formatting): 0 / 30000
Part G
Next, rewrite the expression by grouping the -1s and grouping the numbers that are not -1s.
Space used (includes formatting): 0 / 30000
Part H
Now find the product of the values that are not -1s and rewrite the expression using this product.
Space used (includes formatting): 0 / 30000
Part I
Before we move on, what does multiplying a number by -1 mean?
Space used (includes formatting): 0 / 30000
Part J
When simplifying the expression in part H, how many times will the sign change on the positive term? Will the final expression be positive or negative?
Space used (includes formatting): 0 / 30000
Part K
Simplify the expression in part H. Following the five payments, what is the net change in the loan amount from Quiana’s viewpoint?
Space used (includes formatting): 0 / 30000
Self-Evaluation
How did you do? Rate your work on a scale of 1 to 5, with 5 as the highest score. Then write a brief evaluation of your work below. Note what you learned and what challenged you.
Space used (includes formatting): 0 / 30000
Answers
GPT-4o mini
Sure, let's work through each part of the activity!
### Activity
#### Part A
**Write a rational number to express the initial loan amount as it relates to Quiana’s bank account.**
The initial loan amount can be expressed as a negative rational number since it represents money owed. Thus, the initial loan amount is:
-15,234.68
---
#### Part B
**What is the sign of the rational number you wrote in part A for Quiana’s loan? What does the sign represent?**
The sign of the rational number is negative (-). This negative sign represents that Quiana has a liability or debt – she owes this amount of money to the lender.
---
#### Part C
**Write a rational number that represents the amount of the monthly payment from Quiana’s perspective.**
The amount of the monthly payment that Quiana makes is:
-247.43
---
#### Part D
**Quiana made five equal monthly payments from her bank account. Which rational number shows something being removed five times?**
To show the removal of the monthly payment five times, we can multiply the monthly payment by 5:
-5 * 247.43
This means she is removing:
- (5 * 247.43) or -1237.15 (the total amount repaid after 5 months).
---
#### Part E
**Now, write an expression using the two terms from parts C and D to represent the net change to Quiana’s loan as a result of her payments.**
The net change in the loan from Quiana's perspective after 5 months of payments is:
-15,234.68 + (- (5 * 247.43))
or
-15,234.68 - 1237.15
---
#### Part F
**One way to simplify expressions with negative terms is to distribute the -1s. Rewrite the expression from Part E, factoring out a -1 from any negative value.**
Factoring out a -1, we get:
-(15,234.68 + 1237.15)
---
#### Part G
**Next, rewrite the expression by grouping the -1s and grouping the numbers that are not -1s.**
This would be:
-(15,234.68 + 1237.15)
---
#### Part H
**Now find the product of the values that are not -1s and rewrite the expression using this product.**
Calculate:
15,234.68 + 1237.15 = 16,471.83
Now express the equation:
-16,471.83
---
#### Part I
**Before we move on, what does multiplying a number by -1 mean?**
Multiplying a number by -1 means changing its sign. If the number is positive, it becomes negative, and if it is negative, it becomes positive.
---
#### Part J
**When simplifying the expression in part H, how many times will the sign change on the positive term? Will the final expression be positive or negative?**
The sign will change once when factoring out the -1. Since the original expression is negative, the final expression will remain negative.
---
#### Part K
**Simplify the expression in part H. Following the five payments, what is the net change in the loan amount from Quiana’s viewpoint?**
The simplified expression is:
-16,471.83
Thus, the net change to the loan following the five payments is a decrease of $16,471.83 from the original loan amount.
---
### Self-Evaluation
**How did you do? Rate your work on a scale of 1 to 5, with 5 as the highest score. Then write a brief evaluation of your work below. Note what you learned and what challenged you.**
I would rate my work a 5. I clearly understood the process of representing the loan and payments as rational numbers and successfully simplified the expressions. I learned how important it is to keep track of both positive and negative values in a financial context, which can be challenging at times. Understanding how to work with negative numbers is crucial for modeling real-world financial situations accurately.
### Activity
#### Part A
**Write a rational number to express the initial loan amount as it relates to Quiana’s bank account.**
The initial loan amount can be expressed as a negative rational number since it represents money owed. Thus, the initial loan amount is:
-15,234.68
---
#### Part B
**What is the sign of the rational number you wrote in part A for Quiana’s loan? What does the sign represent?**
The sign of the rational number is negative (-). This negative sign represents that Quiana has a liability or debt – she owes this amount of money to the lender.
---
#### Part C
**Write a rational number that represents the amount of the monthly payment from Quiana’s perspective.**
The amount of the monthly payment that Quiana makes is:
-247.43
---
#### Part D
**Quiana made five equal monthly payments from her bank account. Which rational number shows something being removed five times?**
To show the removal of the monthly payment five times, we can multiply the monthly payment by 5:
-5 * 247.43
This means she is removing:
- (5 * 247.43) or -1237.15 (the total amount repaid after 5 months).
---
#### Part E
**Now, write an expression using the two terms from parts C and D to represent the net change to Quiana’s loan as a result of her payments.**
The net change in the loan from Quiana's perspective after 5 months of payments is:
-15,234.68 + (- (5 * 247.43))
or
-15,234.68 - 1237.15
---
#### Part F
**One way to simplify expressions with negative terms is to distribute the -1s. Rewrite the expression from Part E, factoring out a -1 from any negative value.**
Factoring out a -1, we get:
-(15,234.68 + 1237.15)
---
#### Part G
**Next, rewrite the expression by grouping the -1s and grouping the numbers that are not -1s.**
This would be:
-(15,234.68 + 1237.15)
---
#### Part H
**Now find the product of the values that are not -1s and rewrite the expression using this product.**
Calculate:
15,234.68 + 1237.15 = 16,471.83
Now express the equation:
-16,471.83
---
#### Part I
**Before we move on, what does multiplying a number by -1 mean?**
Multiplying a number by -1 means changing its sign. If the number is positive, it becomes negative, and if it is negative, it becomes positive.
---
#### Part J
**When simplifying the expression in part H, how many times will the sign change on the positive term? Will the final expression be positive or negative?**
The sign will change once when factoring out the -1. Since the original expression is negative, the final expression will remain negative.
---
#### Part K
**Simplify the expression in part H. Following the five payments, what is the net change in the loan amount from Quiana’s viewpoint?**
The simplified expression is:
-16,471.83
Thus, the net change to the loan following the five payments is a decrease of $16,471.83 from the original loan amount.
---
### Self-Evaluation
**How did you do? Rate your work on a scale of 1 to 5, with 5 as the highest score. Then write a brief evaluation of your work below. Note what you learned and what challenged you.**
I would rate my work a 5. I clearly understood the process of representing the loan and payments as rational numbers and successfully simplified the expressions. I learned how important it is to keep track of both positive and negative values in a financial context, which can be challenging at times. Understanding how to work with negative numbers is crucial for modeling real-world financial situations accurately.