Last Friday, I was invited to speak about the Problems of the Week to the Northeast region Noyce Scholars (early career and future STEM educators, and a really fun group to do math with!).

Since I’d been thinking a lot about the AlgPoW that just wrapped up, Cherokee Creek Checkers Club, we looked at that problem and some students’ work on the problem.

We began with this scenario:

Cherokee Creek Middle School has a checkers club with an enthusiastic group of players. Unfortunately, they started this semester in a bit of a slump, winning only 40 of their first 90 games in various inter-school competitions. That gave them a winning percentage of only about 44%.

Then they found a new coach, who inspired them and significantly improved their overall level of play. This month they went on a hot streak, winning 3 out of every 5 games they played.

We used a scenario to loosen our thinking up and try to focus on three big preparation themes: prerequisite knowledge, big math ideas, and multiple approaches.

Looking at the scenario, the Noyce Scholars noticed and wondered about cool math like:

• Quantities are represented as percentages, fractions, and ratios in this problem.
• How could you determine which coach was better? How do you know for sure?
• What’s the overall winning percentage, based on the number of games played, before and after the new coach.
• If they won 40 out of 90 games at first, how many games might they play over a year? A week?
• How come winning 3 out of 5 is an improvement over winning 40 out of 90?
• They increased their winning percentage by 16 percent

All of these noticings and wonderings gave us insight into two of our three foci.

1. What do students need to know/understand to work on a problem about this story?
• Fractions, ratios, and percents and how they are related
• How winning percentages are calculated
• That two coaches are being compared and one might be better than the other; the team improved over time
2. What big math ideas are at work here?
• Comparing two quantities using fractions, ratios, and percentages
• Change in one quantity as a function of another e.g. overall winning percentage as a function of games played with the new coach

Our third focus is multiple approaches, and for that we needed a problem to solve. I revealed that after playing some more games, the team’s overall winning percentage climbed to exactly 52% and we wondered, “how many games did they play? How many did they win?” Recalling that they won 3 out of every 5 games, we set to work.

The multiple methods that we used were Make a Table, Guess and Check, and Make a Mathematical Model (an algebraic model, in this case).

Then we took a look at the work of an Algebra I student, M.:

They played 85 games that month

Explanation:

For this problem I did guess and check. First I tried 40/90 and 30/50 because it says they won 3/5 games so 70/140 was 50% which was too little so I tried 33+40/55+90 and got 50.3% so I realized I needed to go up a lot more. Then I tried 60/100 + 40/90 and 100/190 and got 52.6% so I went down a bit and did 94/180 and got 52.22222% so I went down more and did 91/175 and 52% evenly. so they had played 90 games and now they played a total of 175, so they played 85 games that month

Check- 40/90 + 51/90 is 91/175 which is 52%

Here are some of the initial things we noticed and wondered about M’s work:

• She checks her work at the bottom
• In the check, she adds 40/90 + 51/90 = 91/175 and we wondered how she was using that calculation to check her work
• We also wondered in the check if she meant 51/85 instead of 51/90 because 51/85 was used earlier in the problem and would represent winning 3 out of every 5 games (17 sets of 5 games)
• We wondered how she thought of her first guess, 30/50
• We noticed that sometimes she used “and” and sometimes she used the “+” sign
• We noticed she used fraction notation and wondered if she was using it informally, using “+” to mean “and” and using “/” to mean “out of”
• We noticed that she got the correct answer, and it didn’t take her very many guesses at all!

And finally, we thought of some questions we could ask M. to help her revise her work and keep reflecting on the problem such as:

• This time, it took you 5 guesses to get the answer. What are some ways you could find an answer with fewer guesses next time you solve a similar problem?
• How did you decide what guess to start at? What did you notice in the problem that led to your first guess?
• How would you explain your steps and calculations to another student who is stuck and doesn’t understand how to make a guess and check it?
• Each time you did a calculation you wrote it a different way. If you pick one of the ways (e.g. 33+40/55+90) and translate all the other calculations into that format, what patterns do you notice in the calculations?

The final step, which we ran out of time for, is to get feedback on the questions we generated. Which of them are engaging? Which inspire reflection and meta-cognition? What do you think of the tone of each question?

By the way, if you’re interested in the process of looking at student work and finding the right question to ask, we are focusing a new set of Professional Development courses on it this year. We’d love to have you join us!

If you teach with the PoWs, or work with pre-service teachers, you might also be interested in our free mentoring, which pairs volunteer pre-service teachers with students working on the PoWs. The mentors learn to ask good questions and the students get feedback on their mathematical thinking.

Some “Cherokee Creek Checkers Club” links in case you are interested: