Back to: Module: Helping Students Do Math

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## Procedures

- Reread the scenario involving Sam from the Introduction
**Polly:**Okay, Sam, so here’s the problem. You are the winner of an interesting lottery. The payout is as follows. On the first day, you receive $1. On the second day, you receive $2. On each day after that, you receive twice what you received on the day before. On which day will your total winnings be more than $100?**Sam:**That’s easy.**Polly:**How could you go about solving it?**Sam:**Well, I could draw a picture with a dot per dollar, but that would be a lot of dots! I could use a spreadsheet. I could try to write a formula. I could look for a pattern. Or I could just calculate it all by hand.**Polly:**Okay, so go ahead and solve it.**Sam:**Well, let me look for a pattern. The first few days, I received $1, $2, $4, $8, $16, $32, … and so on. That means that the total money I received by the end of each day is $1, $3, $7, $15, $31, … and so on. Cool! I see a pattern. On any day, the total I’ve received is just one dollar less than the amount I would receive on the next day. So, now I just need to figure out on the first day that I get over $100. If I continue my first list, I get $64, and then $128. So, on the 8th day, I receive over $100. That means that on the 9th day, my total will finally be over $100. So, my answer is the 9th day.**Polly:**Are you sure?**Sam:**Yes!**Polly:**I’m sorry, Sam. That’s the wrong answer. The answer is the 7th day.**Sam:**Awww, man! I thought for sure I had found a pattern!

- Then answer the three questions below, either alone or with a group.

## Questions

- Of the four steps in Polya’s problem-solving method (or UPSCheck), which steps did Sam do naturally in his think aloud with Ms. Polly? Which step(s) did he not do?
- After going through this unit, how might you get Sam to do the fourth step (Check)?
- Knowing about UPSCheck—and knowing how to use it—allows you to have it as a tool in your educational toolkit. As Sam’s example shows, however, you still may need to be creative in choosing when and how to use this tool. Keeping in mind the educational situations that you work in—or the ones in which you intend to work in the future—what do you think would be the most effective way of using UPSCheck to support students’ learning of math?