What’s Worth $1?
What coin collections are worth exactly $1?
Tammy has 8 coins with a total value of $1. What coins might she have? Can you find more than one possible solution?
- $1 is worth 100¢. How can you add quarters (25¢), dimes (10¢), nickels (5¢), and pennies (1¢) to get a sum of 100¢? You can also use half dollars (50¢).
- Are there coins you can trade for others so that you end up with exactly 8 coins?
- Can you find all the possible solutions?
- Tammy says that it is impossible to use exactly 94 coins to make $1. Do you agree? Why or why not? Can you find another number of coins that it’s impossible to make $1 with?
This task asks students to find collections of 8 coins with a total value of $1. Solving this problem requires them to trade coins of equivalent value, determine multiple unknown addends when given the sum, and work flexibly with addends within 100.
There are multiple strategies that students may use to solve this problem. They might start by making a coin collection that is worth $1 but has an incorrect number of coins and then trade coins of equal value until they have exactly 8 coins. For example, they might start with 4 quarters (4 coins total), trade 2 of the quarters for 5 dimes (7 coins total), and so on. You could also see students make a coin collection that is close to the problem's specifications (e.g., 6 coins with a total value of 70¢), then make adjustments to the collection until they have exactly 8 coins with a total of 100¢. Many students might discover that it is useful to start building their collections using coins of greater value, then add coins of increasingly lesser value.
Various apps may be used to represent the context of this problem.
- The Money Pieces app allows students to switch between standard coins and money pieces that are sized proportionally to their value. Students could start with a large $1 money piece (a 10 x 10 array) and then place smaller coin pieces on top until the $1 piece is completely covered with 8 coin pieces (as shown in this work-in-progress here). Alternatively, students could work with the standard coins, as shown in this work-in-progress here. One feature of the Money Pieces app is the Trade button, allowing students to swap coins or money pieces for equivalent amounts.
- In the Number Line app, students might use 8 jumps of 50, 25, 10, 5, or 1 to make a total of 100 (as shown in this work-in-progress here).
As students develop solutions, encourage them to include expressions or equations to represent their thinking (including multiplication for Grade 3 students). Ask questions such as these: How did you decide to start with those coins? I see that you found one way to use 8 coins to make $1. How can you trade coins from that collection to find a different solution to the problem? How many different solutions do you think that there are? Do you think it's possible to have pennies in your final coin collection? Why or why not?