Difference Between Two Squares

Task 64 ... Years 6 - 10

Summary

Wooden pieces model what happens when a smaller square is placed on top of a larger one and in one of its corners. An L shape is formed which can be cut into two rectangles. This L shape is the difference in area between the two squares and because dimensions of the two original squares are known, the dimensions of these rectangles (and hence their area) can be calculated. The result is a powerful algebraic rule that often simplifies calculations.

Materials

4 specially cut wooden pieces

Content

square numbers
area of squares and rectangles
multiplication as area (continuous array model)
algebraic generalisation
mental arithmetic
factors and factorisation

Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.

The spatial challenge in the puzzle is to create this:

and turn the difference (shown in grey) into this:

(The second 'b' square is provided as a reference piece.)

This construction shows the difference in two ways.

In the first diagram it is the area remaining when the smaller rectangle area is subtracted from the larger.
In the second diagram the 'difference pieces' have maintained the same area, but have been rearranged into a new rectangle, the area of which can be calculated from its length and width.

The numeric consequence is that a² - b² = (a - b)(a + b).

This generalisation means that calculations such as those in Question 6 are a little easier, because the numbers don't get so big. For example, instead of:

27² - 18² = 729 - 324 = 405

we can calculate (with a little mental arithmetic):

27² - 18² = 9 x 45 = 405

This can be particularly helpful if we recognise two subtracted numbers as being perfect squares, eg: 900 - 169.
Difference Between Two Squares is a partner to Task 57, Two Squares, which explores alternative ways of 'seeing' the difference. Also, one person has added an explanation to Task 11, Lining Up, which is a special case of Difference Between Two Squares. It's quite insightful visual algebra and deserves a look.

Difference between two squares can also be explored using the Free Tour Picture Puzzle, Square Numbers.

But what happens if we place one cube inside another to create the difference between two cubes?

Whole Class Investigation
Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.
Using square paper, the largest square that can be drawn is Size 16. Read out the class list and assign a number from 1 - 16 to each person (it doesn't matter if you have to repeat). This way partners will not always have the same difference between their two squares. The assigned number is the size of square they have to cut out. (You will need one pair of scissors per pair).
Arrange for each pair to compare their two squares, as above, and discuss the representation of the difference. Ask each pair to cut out this L shape then divide it into two rectangles. The first challenge is to make these two rectangles into one rectangle, and the second is to discover how the dimensions of this single rectangle relate to the dimensions of the original squares.
As students discover that the same type of relationship is true for all their squares, the generalisation above can be developed and applied.
At this stage, Difference Between Two Squares does not have a matching lesson on Maths300.

Is it in Maths With Attitude?
Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.
The Difference Between Two Squares task is an integral part of:

MWA Pattern & Algebra Years 9 & 10

Follow this link to Task Centre Home page.