What relationships can be found with equal groups and leftovers?
Breanna has a button collection.
- When she puts her buttons in 3 or 7 equal groups, there are 2 buttons left over.
- When she puts the buttons into 2 equal groups, there is 1 button left over.
How many buttons could Breanna have?
- Does Breanna have an even or odd number of buttons? How do you know?
- To start, choose a number between 11 and 20 and see if the clues she gave are true for the number you chose. If not, how can you change the total to make them true?
- Find other possible numbers of buttons that would work with the clues Breanna gave. What pattern do you see?
- For a different set of buttons, Breanna kept the first clue the same and changed just the second clue to say, “When the buttons are in 2 equal groups, there are no buttons left over.” How could you use the work that you have done to help you find the number of buttons in this set?
This task requires students to think about the relationship between multiplication and division. As students move toward dividing larger numbers, their understanding of the relationship between multiplication and division will become essential. This activity may also draw out some conclusions about prime and composite numbers. As students think about which numbers are multiples of 3 and 7, they may notice that numbers that are multiples of both are also multiples of the product of those two numbers. This is unlike a situation with numbers like 4 and 6, which share a common multiple that is not the product of the two, or like 3 and 9, where one number is a multiple of the other.
Students may choose to try only odd numbers because they know that a set with an odd number of objects cannot be split into 2 equal groups. They may use what they know about multiplication to reason that the number is likely 2 more than a multiple of 7 and 3 because it results in a quotient with a remainder (amount left over) of 2 when divided into 3 or 7 groups. They may choose to think about discrete sets and partition a given number of items into equal groups. Students may choose to then organize the total number to make it easier to count. They might think about the total as a linear distance and each equal group as a set of intervals in that distance.
Various apps may be used to represent the context of this problem.
- In the Number Frames app, students may create partitions to sort the counters into 3 equal groups (as shown here) or 7 equal groups (as shown here). They would likely test that number to see if it makes 2 equal groups with 1 left over (as shown here.) Students can then use the frames in the Number Frames app to help them find the total number of counters (as shown here.)
- The Pattern Blocks app can be used to model grouping the number of shapes into equal groups (as shown here).
- Students may use the Number Line app to make equal groups of 3 and 7 to see where they meet at the same point, adding 2 to both (as shown here.)
Push students toward multiplicative thinking and making connections between multiplication and division by asking them to think about numbers that have 2 left over when split into 3 equal groups and those that have 2 left over when split into 7 equal groups. Ask them to think about how that number is related to a number that would have none left over when broken into 3 or 7 equal groups. Ask them why a number might have the same amount left over when split into these two sets of groups.