Visual representations of the difference of two squares

By | October 26, 2011

Students’ understanding of mathematics is a function of the quality and quantity of the connections of a concept with other concepts. As I always say in this blog, ‘To understand is to make connections’.

There are many ways  of helping students make connections. One of these is through activities involving multiple representations. Here is a lesson you can use for teaching the difference of two squares, x^2-y^2.

Activity: Ask the class to cut off a square from the corner of a square piece of paper. If this is given in the elementary grades, you can use papers with grid. If you give it to Grade 7 or 8 students you can use x for the side of the big square and y for the side of the smaller square. Challenge the class to find different ways of calculating the area of the remaining piece. Below are two possible solutions

Solution 1 – Dissect into two rectangles


Solution 2 – Dissect into two congruent trapezoids to form a rectangle


Extend the problem by giving them a square paper with a square hole in the middle and ask them to represent the area of the remaining piece, in symbols and geometrically.

Solution 1 – Dissect into four congruent trapezoids to form a parallelogram


Solution 2 – Dissect into 4 congruent rectangles to form a bigger rectangle

These two problems about the difference of two squares will not only help students connect algebra and geometry concepts. It also develop their visualization skills.

This is a problem solving activity. It’s important to give your students time to think. Simply using this to illustrate the factors of the difference of two squares will be depriving students to engage in thinking. They may find it a little difficult to represent the dimensions of the shapes but I’m sure they can dissect the shapes. Trust me.

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One thought on “Visual representations of the difference of two squares

  1. Roger Binschus

    I like them. I may use them.

    As a teacher, I am often torn on how to handle something like this. It is pretty fascinating to me. However the time spent tackling a project like this in class… Is it worth it?

    I imagine that this would take about 10-15 minutes to execute in class. My best students would do it in 2, and the rest would constantly ask “what next?” — How can one effectively do this in a time considerate fashion?

    Can you do the same with difference of cubes? Maybe with Lego’s?


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