Source: NRICH Maths – Elevenses
Why did we choose this problem?
Good Play With Your Math problems:
- involve some sort of “play” before choosing a specific strategy.
- have a low floor (accessibility and entry point)
- have a high ceiling (need for more complex mathematics)
- have a succinct, accessible, intuitive wording and visualization
This problem meets all four criteria:
- You have to experiment, observe patterns, struggle, and adapt before finding a more elegant strategy. For us, this is what mathematical play looks like.
- This problem is extremely accessible, requiring just integer addition and division in order to test each pair.
- The need to organize the results and systematically test all potential options is no easy task, creating a need for certainty and the desire for a more efficient and elegant strategy. While this problem did not prove all that difficult for me or Ms. Yu, we conjectured that the solution strategy would prove particularly counterintuitive to our students.
- This problem was already succinct and even presented as a poster. We knew it would be possible to present this problem visually, we just needed to decide how much to change.
Modification #1: 13 instead of 11.
We opted for the Thirteens version of the problem for a few reasons: because it was time for Play 13, because we released it on Friday the 13th, and in order to make the problem more challenging. We included the Elevens version on our handout as a scaffold.
Modification #2: Ditch the table.
There is no apparent reason why the numbers should be organized in a square grid. Is this intended to help someone solve the problem? Or might it distract people from the essential features of the question? Instead, we opted to organize the numbers haphazardly.
- One of my students took one look at the problem and cleverly asked if he could use the 13 at the top. Then we looked to see if it would make a difference.
- I had a Problem Set work day in one of my classes on Friday, and a few students worked on this problem for the entire period.
- Several students kept track of their answers by connecting the circles with a line; however, this became difficult after a while. Perhaps we could have made this easier by arranging the numbers in a circle … or maybe this is the perfect result, creating a need for systematic counting and organized results.
- On a similar note: Ms. Harding solved the problem and organized her results in a much different way than I did.
- One student found all but one of the pairs. When I asked him if he thought he had them all, he said he thought he did, but he wasn’t sure. Exactly!