Variables and Formulas
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Grouping for Counting

Pattern Block Trains

In Course I, students learned about patterns and predicting. In this lesson, we build on that concept to see how using patterns and generalizations can help us determine more information about a situation, or keep track of complex information. Grouping elements of a geometric pattern is one way to begin to accomplish this. Given a layer of cubes like the following, we can group the individual cubes to count them rather than counting one-by-one. How would you group the cubes for counting?

cube structure, front view

Here's some ways students have approached this:

grouping methods

Now let's use grouping strategies to explore a simple 'toothpick' pattern. Here is a row of five squares - how many toothpicks were needed to build it? We could just count one-by-one, of course, but the idea is to come up with several grouping strategies to make the job easier, especially when the pattern grows large.

five toothpick squares

Again, here are some ideas students have come up with:

toothpick squares groupings

Once we have some grouping methods, we can begin to make predictions about larger groups of squares and generalize about the pattern. For example, can you determine how many toothpicks would be needed to build a row of 12 squares? How about 43 squares, or even 200?

Students write 'directions' for using the various grouping methods to determine the number of toothpicks for any given row of toothpick squares. A typical set of instructions for one of the methods demonstrated above might go something like this...

How would you describe the other methods demonstrated above?

From here, it is a short step to writing formulas in which variables are used to represent the total number of toothpicks and the number of squares. If we let 'T' be the total number of toothpicks, and 's' be the number of squares, the formula for the instructions written above would look something like

Here's a thought-provoking question for you. How many squares could I build if I used 160 toothpicks? Also, we can explore other toothpick patterns like the following for more fun with patterns. Maybe you can create some other interesting patterns on the kitchen table with your child to help them hone their generalizing and formula-writing skills!