Square of Odd Numbers Minus 1

Many numbers have interesting properties worth investigating. These properties can be used to introduce the notion of proofs. Teachers may try this out in middle school or high school students. Although the proofs might not be as elegant or as formal such as written in books, its notion and the reasoning behind it is more important. Let us try to answer the question below.

Investigate the numbers which are one less than the square of odd numbers.

We can try some examples and see their properties.

$latex 3^2 – 1 = 8$

$latex 5^2 – 1 = 24$

$latex 7^2 – 1 = 48$

$latex 9^2 – 1 = 80$

Looking at the pattern, it is quite obvious that all of them are divisible by $latex 8$. Maybe, you might want to have more examples. To lessen the burden the calculation, you might want to use a spreadsheet to do this.  Continue reading

Representing the Sum and Difference of Two Squares

One of the most beautiful characteristics of mathematics is you can create multiple representations of a single concept. Functions for example can represented in words, equations, graphs, or tables.

In this post, we represent geometrically an algebraic concept namely the sum and difference of two squares. The difference of two squares states you can factor $latex a^2 – b^2$ into $latex (a+b)(a-b)$. You may use a piece of paper to follow the procedure below. 

Geometric Representation

1. Take a square with length $latex a$ (any convenient side length)

 square

2. Cut out a small square with side length $latex b$ from one of the corners of the large square. The area of the remaining figure is $latex a^2 – b^2$. Can you see why? Continue reading