Algebra: Sums & Differences of Squares

Australian curriculum number (ACMNA233)

Students build their skills in using the special algebraic results relating to perfect squares and differences of squares. They use algebraic reasoning to justify arithmetic results and make connections to visualisations that involve areas.

This sequence is designed to deepen and consolidate students’ understanding of the algebraic identities a2b2 = (a + b)(ab) and (a + b)2 = a2 + 2ab + b2. It is assumed that students have already encountered these identities in their initial learning of the expansion and factorisation of binomial products. The lessons may be done in any order.

 

Lesson 1: Quarter Squares

A historical calculation method is used to show students an alternative method for multiplying two-digit numbers. After exploring and becoming familiar with the method, students use algebraic skills, in particular the binomial expansion of perfect squares, to justify why the method always works. Several sustaining activities, including a visual method for showing the identity used in the multiplication, are included.

Lesson 2: Filling Corners

Students use a visual method for transforming a rectangle into a square by dissecting and rearranging the rectangle, then filling in the missing corner with a square. This introduces a reasonably efficient method for finding pairs of factors and, hence, testing for primality, and the method is trialled using a spreadsheet and Python code. Students use the algebra of the difference of two squares to show why the method works. Some related activities sustain the learning.

Lesson 3: Algebraic Allsorts

Students engage in a range of activities, including visualisation, methods for rapid calculation, and solving word problems that rely on the difference of two squares or the binomial expansion of perfect squares. Several of these activities rely on atypical problem-solving wherein students look at problems holistically (e.g. using unknown values, but without solving for the unknown values).

 

Last updated June 23 2020.