First, students will learn how to prove polynomial identities.
Polynomial identities by Khan Academy
Next, students will learn how to prove that the differences between the consecutive square numbers, are the consecutive odd numbers.
Describing numerical relationships with polynomial identities by Khan Academy
Conclude by giving your students these challenges:
How many trapeziums are in the figure below:
Here's the solution: Looking at the trapeziums below, from left to right, top to bottom, here are the number of each: 6, 3, 1, 3, 1, 1, 3, 1. Adding these, gives us 19. But the entire equilateral triangle has order 3 rotational symmetry, meaning each trapezium occurs in 3 different ways. Thus, our answer must be \(19 \cdot 3 = 57.\)
Note: This problem comes from NRICH, but NRICH provides no solution.