Think Through
My Algebra students truly enjoyed this activity, and as it was our first "Think Through" of the school year, I spent more time with this activity, and conducted an interview with my students and other math teachers who I gave this problem to. The important aspect of a "Think Through" are that there are many different opportunities to arrive at an answer, and to find more than one path is a struggle. Your students should feel challenged! Here are the various conceptual paths taken by various students and even different math teachers who wanted to try this math challenge for themselves:
Conceptual Paths
| Conceptual Path #1 |
| The pattern of blocks can be represented as a recursively defined sequence, where a(1)=1, a(2)=3 and a(n)=a(n-1)+n. In this formula, n represents the figure number and a(n) would represent the number of blocks in the nth figure. |
| Conceptual Path #2 |
| To find how many blocks the nth figure has, you can simply add all the figure numbers up until you reach your desired figure number “n.” For instance, the 10th figure has 55 blocks, since
1+2+3+4+5+6+7+8+9+10 =55 |
| Conceptual Path #3 |
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| Conceptual Path #4 |
Turn the staircase into a rectangle by doubling the amount of squares as the original picture. The pattern will always be that the width of each figure is n, and the length is always n+1. By finding the area of the new rectangle: (n)(n+1) and dividing your answer by 2, you will have the number of squares in the original staircase. Below is a picture of the process: |
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