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Studying Math Patterns at Baskin Robbins

August 4, 2008

I am one of the majority of people in the U.S. who doesn’t “get” math. Despite my best efforts to the contrary, I did learn some math in school, but it was all poorly understood rote procedures. I certainly never understood what was interesting about this subject. I eventually got good enough at Algebra problems to enjoy them a little — they were kind of like puzzles. But beyond understanding that I was using arithmetic to budget and balance a checkbook, I NEVER saw math as something three-dimensional in the real world.

In high school, we used square and cubic numbers. It was something I understood two-dimensionally (literally) as problems on the page of a textbook. Much later, as a 30-something-year-old home schooling mom, I was playing around with some of our 1-centimeter cubes (you know — math manipulatives — those things we home schoolers have all over our carpets, between the cushions, and under couches. *LOL*) And I finally got WHY they’re called square and cubic numbers. I will concede that my “AHA” moment would be a “duh” moment for most. Everybody else probably already knew this. But when I saw that square numbers, constructed with cubes, actually make squares, and cubic numbers make cubes I thought it was VERY cool. Later I learned about triangular and pyramidal numbers. Cool stuff.

I will never be a math expert, but — through the things I’ve learned as a home schooler — I can see glimmers of what makes it profound and beautiful. And I’m having fun with it. Once in a while, my kids even have a little fun too.

This week, we are continuing our simple unit on probability and statistics. Today we did the “31 Flavors” problem.

I got this problem out of an old edition of About Teaching Mathematics by Marilyn Burns, and I played with it a bit.

If an ice cream parlor only served one flavor of ice cream, how many kinds of double dip ice cream cones could you make? One (2 scoops of vanilla) What if there are 2 flavors? Three (2 scoops of vanilla, 2 scoops of chocolate, and a scoop of chocolate with a scoop of vanilla). Following this pattern, how many possible combinations would there be with 31 flavors?

When I first read the problem, I thought it was confusing. Then I realized it was more or less a series of triangular numbers.

To help Sarah and James solve the problem, I set up a table on the white board, and I showed them how to use centimeter cubes (white cubes for scoops of vanilla, brown cubes for scoops of chocolate, and so forth — cute, huh? :-P) For instance, 2 white cubes were stacked up to make a double dip vanilla cone.

They worked on the problem for a while with a little help from me.

It goes like this —

1 flavor — 1 possible combination

  1. 2 scoops of vanilla

2 flavors — 3 combinations

  1. 2 scoops of vanilla
  2. 2 scoops of chocolate
  3. 1 scoop of vanilla & 1 scoop of chocolate

3 flavors — 6 combinations

  1. 2 scoops of vanilla
  2. 2 scoops of chocolate
  3. 2 scoops of strawberry
  4. 1 scoop of vanilla & 1 scoop of chocolate
  5. 1 scoop of vanilla & 1 scoop of strawberry
  6. 1 scoop of strawberry & 1 scoop of chocolate

4 flavors — 10 combinations

  1. 2 scoops of vanilla
  2. 2 scoops of chocolate
  3. 2 scoops of strawberry
  4. 2 scoops of mint chocolate chip
  5. 1 scoop of vanilla & 1 scoop of chocolate
  6. 1 scoop of vanilla & 1 scoop of strawberry
  7. 1 scoop of vanilla & 1 scoop of mint chocolate chip
  8. 1 scoop of chocolate & 1 scoop of strawberry
  9. 1 scoop of chocolate & 1 scoop of mint chocolate chip
  10. 1 scoop of strawberry & 1 scoop of mint chocolate chip

It’s easiest for me if I do it systematically. For example:

  • 4 flavors means 4 possible cones with 2 of the same flavor.
  • The 1st flavor, vanilla, can be paired with 3 other flavors.
  • The 2nd flavor, chocolate, can be paired with 2 other flavors (besides vanilla, which has already been covered).
  • The 3rd flavor, strawberry, can be paired with 1 other flavor ( besides vanilla & chocolate, which have already been covered)
  • We’ve already exhausted the possibilities with mint chocolate chip

Then the kids figured out the pattern:

1 (+2) … 3 (+3) … 6 (+4) … 10 (+5) … 15 and so forth

When I asked them to work out how many possible combinations there were with 31 flavors, they got stuck. I tried to “scaffold” the task by helping them write a sequence of numbers across the board:

JoVE, Tigger, and Tigger’s friend approached the same problem differently, discovering an algebraic formula instead of writing out the whole sequence.

But I thought my kids would understand it better if they could literally see the logical sequence spread out across the board.

Hmmm … 496 different kinds of double dip ice cream cones to choose from? I guess we’d better get busy.

The kids and I went to Baskin Robbins to get … what else? Double dip ice cream cones. A scoop of mint chocolate chip ice cream is a lot more interesting than a green centimeter cube.

By the way, I just found a variation of this problem online.

On the way, I asked how many different kinds of double dip ice cream cones could we choose from — with 31 flavors — without getting 2 scoops of the same kind. (496 – 31 = 465, since there are 31 options if you get 2 of the same kind).

Sarah, the 14-year-old, gave me the Quotable Kid Moment of the Day — “Did you just get us in the car so you could ask us another math question. ‘Cause you’re torturing me.”

Yup … these kids are really suffering. I got to spend part of my “math period” in the ice cream shop whenI was in school. Didn’t you? 😛

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