Sheri's work on the Tower Puzzle & Nanako's work on the Peg Game or Shuttle Puzzle

9/20/00  Many sessions before this day, Sheri had worked on the Tower Puzzle (see ch. 6). The object is to move the tower of discs from one peg to another. The rules are: you can't put a bigger disc on top of a smaller one, you can only move one disc at a time, and you need to move the discs in the minimum # of moves- that comes later!. Don starts young people off with maybe only 3 discs. Then lets the student start with more.

On this day, Sheri figured out the rule for the Tower Puzzle, her work shown below:

In the process she found patterns in the table of values- the differences go up 2, 4 8, 16, 32.. Don reminded her about writing numbers like these as powers of 2- see her work on the right above. The y-numbers she noticed were 1 less than these powers of 2.  Don wrote 7 as 2³  - 1, then Sheri wrote the others this way and wrote the rule as 2^x - 1= y. Don asked Sheri to graph the rule and graph negative values of x. In the process Sheri got into negative exponents and statements like 1/4 - 1 = ^- 3/4, which they talked about. See her work above. Her graph (an exponential function) is shown below:

On 9/21, Don asked Sheri to write about what she did the day before with the Tower Puzzle. As can often be the case, students put away their work and forget about it. Don asked her this, explaining that the reason he asked was to make sure she understood and could explain what she did. She started to do this and when Don saw her slowing down, he changed the subject. He suggested she finish at home. (Sheri hasn't finished this yet, as of 11/2/00.. it's difficult to get young people to write down things like this. Don will persist!)

One day Don, a student and her father, brought  to a welder the 2D graph she did of The Tower Puzzle. He made the 3D rotation of that curve 2^x - 1=y with stiff wire, below. Don hangs it in his Math room; in 25 years it has only had to be repaired once!

Don had Sheri as a 4-6th grader, then she went to University High School. She came back as almost a 12th grader, to prepare for Calculus. Don worked with her for about 3 hrs. individually over the last month. Today, a week into her calculus class, she told me "What we did the last 3 weeks (derivatives), the teacher did with my class in one day, and I was like the only one in the class that understood what she was talking about! It really helped for me to talk with you about the problems as I worked on them". 

It's been a pleasure working with you Sheri, as well as your sister, your mother, 2 aunts, and 2 cousins over the years- what a wonderful family!