Pattern Block Trains

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Patterns are a fundamental part of mathematics and science study. By reasoning inductively about the visual relationships between shapes in a pattern, we begin to lay the foundation for the skills of conjecture and generalization, which allow students to make predictions and develop ideas about the concept of variable.

These are the basic pattern block shapes we use to build patterns:

Here's a simple pattern to begin with:

The first step is to use observations about the first three figures ('trains') to predict what the 4th and 5th trains in the pattern will look like. What do you notice about them? How can you use that information to predict?

If we add one trapezoid to each figure to produce the next, then the 4th and 5th would be

Now we can ask some questions...

• What will the tenth figure look like?
• How about the 15th?
• How many individual trapezoids will be needed to build the 13th train?
• How many hexagons will be formed in the 22nd train?
• Will the 19th train end with a trapezoid or a hexagon shape?

There are lots of other questions we could explore about this pattern. The last step is to try to generalize about the pattern. For example:

The number of individual trapezoids in any train would be equal to the train number, ('n'):

• Trapezoids = n

If we know the train number is even, then the train will end with a hexagon. If it is odd, then there will be a single trapezoid on the end. If the train number is even, then the train number 'n' divided by two will tell us how many hexagons are formed:

• Hexagons_even = ⁿ/₂

If the train number 'n' is odd, then the number of hexagons would be equal to the train number 'n' minus one, divided by two:

• Hexagons_odd = ^(n-1)/₂

This simple example illustrated some of the possibilities. Here are a couple of other patterns you might find interesting to explore...

First make some observations about the pattern you see, and then try to build or sketch a few larger ones.

Here are some questions to help you explore:

• How many rhombuses will be in any given train (like the 13th train, or the 24th train)?
• How many triangles?
• Will a certain train end with a rhombus or a triangle?
• If a train ends with a rhombus, is it 'leaning' to the left, or to the right?
• It a train ends with a triangle, is it 'pointing' up or down?
• What kinds of generalizations, verbal or using formulas, can you make about the pattern?

Here's one last pattern to explore. Have fun!