In order for teachers to increase their effectiveness in helping students develop skills in reasoning and proof, it's helpful for them to reflect on their own practices with relation to these two concepts. Many teachers may have had their most memorable interaction with proof when they worked with two-column deductive proofs in geometry class. In this section we invite you to think more broadly about the concepts and how you use them in your own thinking.

In this activity we will reason, make conjectures, and develop proofs about triangular and square numbers. Both of these types of numbers are referred to as figurate numbers, because of their relationship to related geometric patterns.

First, recall the staircase problem from Part A of session three in this course. In that activity, students were problem solving (and using reasoning) to find a rule that would provide the number of squares needed to build a staircase of n steps. As the students found in that activity, this is another way of examining the sum of the natural numbers, 1 + 2 + 3...

Although, the informal representation of stairs was used in that activity, if we think of points within each square, we can see each staircase as a triangular number since each can be arranged to form a successive equilateral triangle. (Note that by convention, 1 is defined as a triangular number.)

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