Two Squares

Task 57 ... Years 4 - 8

Summary

Students are given one way to calculate the difference between the area of two squares and are challenged to find another. How they do it will depend on how they see the difference (the uncovered bit) between the two squares. Three ways of seeing it are suggested and each indicates a different way of calculating the difference. The task is based in the mathematician's question Can I check it another way? and precedes an understanding of the classic algebraic formula expressing the difference between two squares, ie:
a2 - b2 = (a - b)(a + b). 		 

Materials

  • Three coloured squares of different sizes:
    squares with 6cm, 5cm & 4cm sides are supplied

Content

  • area concept - counting squares
  • basic arithmetic calculations
  • spatial perception - partition of areas
  • difference between two square numbers
  • generalisation
Two Squares

Iceberg

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

Answers to this task will vary depending on the way students see the dissection of the uncovered section. Therefore, the important thing will be that they can justify their answers. The task encourages more than one way of seeing the uncovered section to reinforce the mathematician's question Can I check it another way?. The task is also self-correcting because students should get the same answer in each case regardless of the calculation strategy.

For example, using red on blue, the following methods for calculating the uncovered (white) part are possible. (There may be more.)


6 x 2 + 2 x 4
= 12 + 8
= 20
4 x 2 + 2 x 2 + 2 x 4
= 8 + 4 + 8
= 20


2(1 x 6) + 2(4 x 1)
= 12 + 8
= 20
4(1 x 1) + 2(1 x 4) + 2(4 x 1)
= 4 + 8 + 8
= 20


2(1 x 6) + 4 x 2
= 12 + 8
= 20
2(1 x 4) + 2(1 x 2) + 4 x 2
= 8 + 4 + 8
= 20

Seeing all these choices, a mathematician might reason that they would use the version that had the fewest calculations. Each geometric piece requires a calculation, so the first partition above is the least because it only has two calculations. In this case the difference between the two squares (62 - 42) is represented by:

and these two pieces can be rearranged as a rectangle:

The width of the rectangle is (6 - 4) and the length is (6 + 4). The area of the rectangle is the product of these two. It is also (62 - 42), so

62 - 42 = (6 - 4)(6 + 4)
which is a particular case of the generalisation:
a2 - b2 = (a - b)(a + b)
This generalisation can be explored further with Task 64, Difference Between Two Squares.

MacMurphy Strategy

Chris MacDonald and Joel Murphy, MacKillop College Swan Hill, were exploring this task in a workshop. They came to a generalisation by collecting data about the calculation involved and looking for patterns or connections within it - fundamental to the way mathematicians work.

First they collected this data:

52 - 42 = 25 - 16 = 9
62 - 42 = 36 - 16 = 20 **
72 - 42 = 49 - 16 = 33
82 - 42 = 64 - 16 = 48
92 - 42 = 81 - 16 = 65

62 - 52 = 36 - 25 = 11
72 - 52 = 49 - 25 = 24 **
82 - 52 = 64 - 25 = 39
92 - 52 = 81 - 25 = 66

72 - 62 = 49 - 36 = 13
82 - 62 = 64 - 36 = 28 **
92 - 62 = 81 - 36 = 45

82 - 72 = 64 - 49 = 15
92 - 72 = 81 - 49 = 32 **

92 - 82 = 81 - 64 = 17

Then they realised that the ones in bold were special. The difference between the square numbers is 1 and the answer is the sum of the square numbers, ie:
If a - b = 1, then a2 - b2 = 1(a + b)
But what about the others? Further study showed that if the difference is 2, the answer is twice the sum (see the ones marked **). Similarly, if the difference is 3, the answer is three times the sum and so on. From this they induced a numeric rather than geometric derivation/justification of the classic rule:
a2 - b2 = (a - b)(a + b)

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. 		   

Students can easily cut squares from graph paper to begin a whole class investigation based around this task. The focus for the lesson would be on highlighting the many ways to carry out a calculation and comparing them to decide the most efficient. Investigation of the MacMurphy Strategy would also play a part.

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

At this stage, 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 Two Squares task is an integral part of:

  • MWA Pattern & Algebra Years 9 & 10

Green Line
Follow this link to Task Centre Home page.