4.1.1 • Contrasting Two Relationships by Reasoning
Warm Up
A customer at a bagel shop is buying 13 bagels. The store worker says, “That would be $16.25.” Jaime, Pedro, and Hannah, who are in the shop, all think it is a mistake.
- Jaime says, “Shouldn’t the total be $13.25?”
- Pedro says, “I think it should be $13.00.”
- Hannah says, “No, I think it should be $11.25.”
Consider how the store worker, Jaime, Pedro, and Hannah could all be right.
In the following activity, you will be asked to use the pricing information displayed at the bagel shop to determine who is correct about the pricing for 13 bagels.
Classroom Connection
Log into student.desmos.com using the information provided by your teacher to complete the activity.
1. Create a table of values using the pricing information from the bagel shop’s sign to identify the different possibilities for calculating the price of 13 bagels.
| Number of bagels | Store worker’s price calculation (in dollars) |
Jaime’s price calculation (in dollars) |
Pedro’s price calculation (in dollars) |
Hannah’s price calculation (in dollars) |
|---|---|---|---|---|
| 1 | ||||
| 2 | ||||
| 3 | ||||
| 4 | ||||
| 5 | ||||
| 6 | ||||
| 7 | ||||
| 8 | ||||
| 9 | ||||
| 10 | ||||
| 11 | ||||
| 12 | ||||
| 13 |
| Number of bagels | Shopkeeper’s price | Jaime’s price | Pedro’s price | Hannah’s price |
|---|---|---|---|---|
| 1 | 1.25 | |||
| 2 | 2.50 | |||
| 3 | 3.75 | |||
| 4 | 5.00 | |||
| 5 | 6.25 | |||
| 6 | 7.50 | 6.00 | ||
| 7 | 8.75 | 7.25 | ||
| 8 | 10.00 | 8.50 | ||
| 9 | 11.25 | 9.75 | 8.00 | |
| 10 | 12.50 | 11.00 | 9.25 | |
| 11 | 13.75 | 12.25 | 10.50 | |
| 12 | 15.00 | 12.00 | 11.75 | 10.00 |
| 13 | 16.25 | 13.25 | 13.00 | 11.25 |
2. Use the table you created in the previous question to explain how each person in the scenario is correct.
a. Using the table you created in question 1, explain why the store worker thinks the total for 13 bagels should cost $16.25.
Use the sentence stem: The store worker thinks the price for 13 bagels should be $16.25 because …
The store worker thinks the price for 13 bagels should be $16.25 because she multiplied 13 times 1.25. She charged the customer for each bagel individually.
b. Using the table you created in question 1, explain why Jaime thinks the total for 13 bagels should cost $13.25.
Use the sentence stem: Jaime thinks the price for 13 bagels should be $13.25 because …
Jaime thinks the price for 13 bagels should be $13.25 because that is the cost of buying two sets of six bagels and then one more bagel.
c. Using the table you created in question 1, explain why Pedro thinks the total for 13 bagels should cost $13.00.
Use the sentence stem: Pedro thinks the price for 13 bagels should be $13.00 because …
Pedro thinks the price for 13 bagels should be $13.00 because he added the costs of buying 10 bagels and then 3 more bagels.
d. Using the table you created in question 1, explain why Hannah thinks the total for 13 bagels should cost $11.25.
Use the sentence stem: Hannah thinks the price for 13 bagels should be $11.25 because …
Hannah thinks the price for 13 bagels should be $11.25 because she noticed that a dozen bagels could be purchased for $10. She only needs one more bagel for $1.25 to have 13 bagels. That means her total cost is $11.25.
3. Create a table of values using the pricing information from the bagel shop’s sign to identify the best price for each number of bagels bought.
| Number of bagels | Best price (in dollars) |
|---|---|
| 1 | |
| 2 | |
| 3 | |
| 4 | |
| 5 | |
| 6 | |
| 7 | |
| 8 | |
| 9 | |
| 10 | |
| 11 | |
| 12 | |
| 13 |
| Number of bagels | Best price (in dollars) |
|---|---|
| 1 | 1.25 |
| 2 | 2.50 |
| 3 | 3.75 |
| 4 | 5.00 |
| 5 | 6.25 |
| 6 | 6.00 |
| 7 | 7.25 |
| 8 | 8.50 |
| 9 | 8.00 |
| 10 | 9.25 |
| 11 | 10.50 |
| 12 | 10.00 |
| 13 | 11.25 |
4. How might a price list like this benefit the owner of the bagel shop?
Your answer may vary, but here are some samples:
- A price list like this benefits the owner because it encourages customers to buy more bagels since they will think they may be getting a better deal.
- A price list like this benefits the owner because it rewards customers who buy larger quantities with cheaper prices.
5. As a customer, do you like items to be priced like this or not? Explain your reasoning.
Your answer may vary, but here are some samples:
- I do not like price lists like this because it would be easy to be overcharged for my purchases.
- I do not mind price lists like this because I can get a better deal for 12 bagels compared to buying them at the individual price.
In this activity, because there was only one best price for a particular number of bagels, the best price is a function of the number of bagels bought.
A function assigns one output to each input.