Pattern Block Perimeters

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In this lesson, we build on the concepts introduced in Pattern Block Trains to further develop ideas about patterns, functions and generalizing. If we choose the length of the edge of an orange square to be our unit of length, we can determine the perimeter of each block or combination of blocks.

What would be the perimeter of the combination of pattern blocks below? Be sure not to count the edges of the blocks that are touching each other.

Right. Ten units is the perimeter of the group of pattern blocks. Now, take a look at the following pattern:

What do you think the 4th and 5th figures in the pattern would look like?

We can make some further observations about this pattern and the trains' perimeters...

• The number of squares in the 'train' is equal to the figure number.
• The perimeter of train #1 is 4 units, the perimeter of train #2 is 6 units, and the perimeter of train #3 is 8 units.
• The perimeter of each train is two more than the one before it.

What do you think the perimeter of the 10th train in the pattern is? Students have come up with various ways of determining the value of the 10th train's perimeter:

• Count by twos until the 20th train is reached - 4, 6, 8, 10, 12, ...
• See 20 units along the top of the train, and along the bottom, plus one unit on each end...
• Twenty squares are used, having a total perimeter of 80 units. There are 19 places where squares are joined together, and two units of perimeter are lost at each 'joining'...

How could you determine the perimeter of any train in the sequence, given the figure number? Some students may be ready to generalize with formulas. An 'nth' sketch might be helpful:

Here's another way to look at it...

How could we determine the number of the train that has a perimeter of 76 units? Here's one method students use...

Some students might be ready to use a symbolic approach that describes the method above.

"The train number equals the perimeter minus two, divided by two"

 n = ^{(76 - 2)}/₂

In the generalized form... n = ^{(P - 2)}/₂

Here are two more patterns on which to try your hand. Below the pictures are some questions to get you started. Perhaps you can think of some interesting questions of your own to explore.

• What are some observations you can make about the first five trains in each pattern?
• How much do the perimeters increase each time we add a piece?
• What are the perimeters of the first five trains in each pattern?
• What is the perimeter of the 14th train? the 24th? the 100th?
• Which triangle train has a perimeter of 72 units?
• Which trapezoid train has a perimeter of 83 units?
• Can you make some generalizations about the trains' perimeters?