Over-Under
How it works
Imagine you're playing the following game: You choose a number between 1 and 5. After choosing your number, you pull a random number from a bag with 5 pieces of paper numbered 1 through 5. If your number is less than or equal to the number from the bag, the number of points you get is the number you chose (not the number from the bag). However, if the number you chose is bigger than the number from the bag, you get 0 points. For example, suppose you choose 4. If you pull a 4 or 5 from the bag, you get 4 points, but if you pull a 1, 2, or 3 from the bag, you get 0 points. What number should you choose to maximize the total number of points you get if you play a bunch of rounds of the game (putting the number you pulled out back in the bag each time)?
In this activity, students start by exploring the game described above. They then explore what happens when you change the numbers in the bag — for example, what if you have the numbers 1 through 6 instead of 1 through 5? 1 through 7? 1 through 10? What happens if you only have one 2 and one 3? Students explore which number they should choose to maximize their score, and also how they can change the distribution of numbers in the bag to change which number is the best one to choose.
Why we like this activity
- It’s fun! Students enjoy playing the game and trying to figure out which number they should choose.
- It helps students to develop probabilistic and statistical reasoning.
- It helps students to develop numerical reasoning.
It requires students to engage in mathematical habits of mind:
Making observations / comparing and contrasting / looking for patterns / making and testing predictions / understanding and explaining when trying to figure out which number to choose for a given distribution of numbers in the bag.
Making observations / comparing and contrasting / looking for patterns / making and testing predictions / finding and using strategies / understanding and explaining when trying to figure out how to change the distribution of the bag so it's best to choose a specific number.
- It has a low floor and a high ceiling: It's easy for students to start trying to find the best number to choose by trial and error, but there's much more to explore and discover.
This activity was developed in collaboration with the Julia Robinson Mathematics Festival.