What combination of coins could I have?
I have 45 cents. I have an odd number of coins. What combination of coins could I have?
Hint: An odd number can be divided into 2 equal groups with 1 left over.
- Think about the value of each coin. How can you use this information to help you make a combination of 45 cents?
- Build one combination of 45 cents. How can you use different coins to build another combination?
- How many ways can you find to make 45 cents?
- Imagine I had no quarters or pennies. How would that change your solution?
- Imagine I had $1.45 in coins. What combinations of coins could I have?
In this task, students consider combinations of 45 cents while dealing with the constraint that the total number of coins must be odd. Students may engage in a range of mathematical skills as they investigate this task, including counting, adding, subtracting, early multiplication, and place value.
There are multiple correct combinations of coins. Students might come up with a single combination or multiple combinations that would satisfy the parameters of this problem. They may begin by counting by 10s and 1s to get to 45 and then check their solution to make sure the total number of coins is odd. Students may try to build the combination with the fewest coins, starting with a larger coin amount (such as the quarter). They may also engage in skip-counting by 5s until they reach 45.
Various apps can be used to represent the context of this problem.
- In the Money Pieces app, students can build combinations with visual representations using either the money pieces (as shown here) or a combination of the various coins (as shown here). They may build one combination, count and label the total number of coins and then build a different combination. The app allows for students to easily represent their combinations with coin representations, providing a strong connection between actual coins and images of coins.
- Students could also use the Number Line app to represent the problem. They might start at 0 and make jumps to match coin amounts, such as jumping by 10 to represent a dime or 5 to represent a nickel (as shown here). Because jumps of 1 are not always preferable, this app may push students to think about larger coin amounts to take fewer jumps.
- In the Number Pieces app, students may be more likely to think of the coin amounts in groups of 10s and 1s (or dimes and pennies). Students could represent the problem by using base-ten pieces to represent dimes or two nickels. They can use the ones pieces to represent pennies or group them together in sets of 5 to represent a nickel (as shown here.)
Students could solve this problem by finding a single combination of 45 cents. Encourage them to consider other combinations of 45 cents. What other combination of coins would work? How many different combinations can you find?
To push students’ thinking further, invite them to consider: Are some coins easier to use to make combinations? Why or why not? How many different combinations of 45 cents can you make using one quarter? Did you find all of the combinations? How do you know?