How many friends are sitting at each table?
There are 8 friends sitting at 2 lunch tables. No one is sitting alone. What are some different combinations of friends at tables?
- How can you split 8 friends into 2 groups?
- How can you represent the friends in this problem?
- How many possible combinations are there? Did you find them all? How do you know?
- What if the 8 friends were sitting at 3 lunch tables instead of 2? How many friends could be at each table?
- Imagine 3 more friends come to sit at the 2 tables. How many friends are there now? What combinations of friends can sit at 2 tables?
- Can you write a problem like this of your own?
In this task, students consider combinations of 8, while dealing with the constraint that one of the quantities must be more than 1. Students may engage in a range of mathematical skills as they investigate this task, including counting, adding, and subtracting.
There are multiple correct combinations. Students might come up with just one combination or all the possible combinations of 8 that would satisfy the parameters of this problem. They might start with a known combination of 8 (such as 4 and 4) and then add and subtract to determine other combinations that would work. Students may also work systematically, starting with 2 and 6 and adding and subtracting to find all the other combinations that would work.
Various apps can be used to represent the context of this problem.
- In the Number Rack app, students might use the rows to represent tables and the beads to represent the friends. They could start by pulling over 2 beads for each row (as shown here) and adding beads to one or both rows to make a combination of 8. They could also start with 8 beads in the top row (as shown here). Then they could add beads to the bottom row, removing the same number from the top row, to show the decomposition of 8 in various ways.
- Students could also use the Number Frames app to represent the problem. They could use two number frames to represent the tables and counters to represent the friends (as shown here). They could fill up one frame first with all 8 counters and then move some over to the other table to satisfy the problem or count out 8 counters and place them into the two frames (as shown here).
Students could solve this problem in one way by finding a single combination of 8. Encourage them to consider other combinations of 8. Did they find all the combinations? How can they be sure?
To push students’ thinking further, invite them to consider: Are there combinations of 8 that wouldn’t work for this problem? Using the context, they may determine that the combination 1 and 7 and the combination 0 and 8 would not work as solutions to this problem.