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Introductory Scenario

Thinking Out Loud

Let’s take a look at a dialogue between Sam and his math teacher, Ms. Polly, as Sam thinks his way through a problem out loud. 

Ms. 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. 

Ms. 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. 

Ms. 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. 

Ms. Polly: Are you sure? 

Sam: Yes! 

Ms. 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! 

The truth is that Sam had found a pattern. He even correctly stated the pattern. The only problem was that he added a day at the end instead of subtracting a day. This wasn’t an isolated incident.

Sam was always in such a rush to get to the answer that he often made simple mistakes like this one. He made the mistakes on homework, on tests, and on projects. His teachers repeatedly told him to check his work, but he just found it too boring and time-consuming to go back through each step one by one on every single problem.

Take Notes

(to be answered independently or in a group) 

  1. Knowing that Sam is particularly resistant to simply rechecking each step one by one, how might Ms. Polly have gotten Sam to check his answer rather than just telling him the correct answer? 
  2. Think of a (non-mathematical) problem that you had recently. What were the steps you took in trying to solve this problem? 

Module: Helping Students Do Math

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