About the counting problem last Christmas

Last Christmas, we gave out a free worksheet based on the familiar carol “The Twelve Days of Christmas”, where students are asked to count the number of gifts they will receive on each day and cumulatively the total number of gifts received up to the n-th day. While working through the exercises, we (and our students) gained some interesting insights into the problem of repeated reasoning. Here, we’re sharing what we observed.

The US Common Core for Mathematical Practice outlined eight mathematical practices that should be integrated into daily lessons, to develop long term empowering “habits” in students. The 8th of these standards is titled “Look for and Express Regularity in Repeated Reasoning”. It focuses on recognizing patterns and structure, so as to be able to make generalizations, and apply it to future situation.

For example, in the “Counting the Twelve Days of Christmas” worksheet, the pattern students observed is a series of triangular numbers.

pyramidtriangles

Having understood the basic structure, students are encouraged to look more deeply into the pattern. What if we turn the triangular structure around and lay them out in a rectangular fashion?

Triangular Numbers in Rectangles 1

What if we double the dots now?

Triangular Numbers in Rectangles 2

Students can now more easily see how the sequence can be generalized into the number of dots in the n-th step, since they are now dealing with a more tractable rectangular shape. The total number of dots in the original triangular for the n-th step is simply half of the size of the rectangle, which is n times (n+1), i.e. n(n+1)/2.

In this way, students have found a generalization for the n-th triangular number, which they can apply to obtain any number in the sequence, for example in the worksheet, the number of gifts on the 12th day would be 12(12+1)/2 = 12×13 / 2 = 78.

Using this as a model to help kids look for pattern can help them develop their own strategies when dealing with similar situations. While working through the Christmas problems in class, we used a lot of “what if” questions to encourage creativity and curiosity, and help the kids explore what happens when we try different strategies, and had great fun doing it!

One of our student's work
One of our student’s work on the 12 Days of Christmas worksheet

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