**Variables and
Formulas****
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**Math
Models Home Page**

**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?**

**Here's some ways students have
approached this:**

**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.**

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

**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...**

**"Multiply the number of squares by three,
for the top, bottom and middle groups of toothpicks. Then add one
for the extra toothpick in the middle."**

**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!**