Proving polynomial identities

First, students will learn how to prove polynomial identities.

Watch these Khan Academy videos:

Polynomial identities introduction
Analyzing polynomial identities

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:

2016 Math Kangaroo Levels 7-8 Problem #30 by STEM4all
2009 AMC 8, Problem 4
2000 AMC 8, Problem 3
2011 AMC 8, Problem 16

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.

Additional lessons and practice problems