In solving generalization problems that involve figures and diagrams, I have always found working with the figures—constructing and deconstructing them—to generate the formula more interesting than working with the sequence of numbers directly that is, making a table of values and apply some technique to find the formula. Here’s a sample problem involving counting hexagons.
Problem: When making a cable for a suspension bridge, many strands are assembled into a hexagonal formation and then compacted together. The diagram below illustrates a ‘size 4’ cable made up of 37 strands.
How many strands are needed for size 5 cable? How many for size n cable?
I actually found five algebraic expressions for the relationship between the side of the cable and the number of strands. Each of these shows a different way of looking at the hexagons. I leave this problem to you to discover the corresponding visuals to these two algebraic expressions:
You can also find the solution to this problem by finding the formula for the sequence 1, 8, 19, 37, 61, … or creating this function table:
Which of these type of activities—geometric patterns vs number patterns—would you consider pedagogically more powerful? What are the advantages of each?