Nibbling at squares
Nibbling at squares is intended to be an introduction to the formula for the difference of two
squares.
This task uses geometry to bring meaning to the algebraic relationship. A geometric approach to
learning algebraic procedures allows students to visualise and gain meaning for a process. Students
will investigate the area of a square with a smaller square, or nibble, cut from a corner.
It is recommended that students work in pairs.
Mathematical focus:
Developing the formula for the difference of two squares using areas of paper squares.
MA5.3-5NA A student selects and applies appropriate algebraic techniques to operate
with algebraic expressions
Students expand binomial products using a variety of strategies (ACMNA233)
Incidental connected content in context:Squaring positive integers and finding square roots.
Groundwork
Paper squares and scissors are required.
Coloured paper squares of side length 127mm are a suitable size and are commercially
available. The side lengths are actually 5 inches long, despite being manufactured in
Australia.
To make measuring in integer units more straightforward, squares of 10cm side lengths
would be ideal. They could be made by cutting larger standard rectangular paper or
cardboard.
At least two squares are required for each pair of students. Extras will be needed as
replacements, and for students seeking to investigate further.
Timing 90 - 120 minutes
The task
The Task 1
Students are told the side length of the paper square is 4 units. They are required to cut a square of
side length 1 unit, a nibble, off one corner.
To determine where to cut the nibble, students may measure the original square side length and
divide by 4, or fold twice to make rectangular quarters.
If using paper with side lengths of 10cm, have students measure and establish the side length. The
smaller square will have side length of 1cm. Subsequent calculation will need to be altered if using
these lengths.
Once the nibble is cut it can be discarded. Students are asked to describe the shape they have made.
These descriptions are recorded on the board. The aim is to come up with the description of the
shape as the difference of two squares. While it would be unexpected for students to generate the
term difference of two squares, they will often describe it less formally. The level of prompting and
refining that the teacher needs to make to arrive at the term the difference of two squares will vary
considerably between classes. Anticipated student descriptions include:



an irregular hexagon
a big square with a corner missing
a square with a smaller square taken away
Statements such as this last one can be discussed with prompts such as:
What other ways can we say “taken away”? Usually after suggesting “subtract and “minus”
students will eventually suggest “difference”.
The name of the shape, at least for the purposes of this task, is the difference of two squares.
The intention is provide students with a visual representation of the difference of two squares prior
to performing the algebraic manipulations associated with a2- b2= (a-b)(a+b).
Students are required to find the area of their difference of two squares shape. They can label side
lengths of 1 unit, 3 units and 4 units.
Area = 42-12=15units2
We want students to notice that the two smaller rectangles can be repositioned to make a larger
rectangle because of the common side length.
Prompts
What do you notice about the two rectangles?
What do they have in common? The common side length of 3 units allows rearrangement into
one larger rectangle. Have students cut the difference of two squares into the two rectangles and
rearrange the pieces.
What is the area of this new rectangle?
Draw the discussion together and record the area of the large rectangle.
Area = 5 x 3= 15units2
The Task 2
Discussion questions:
Can we always make one large rectangle out of the two smaller rectangles when the smaller
square is 1 unit in side length?
Does it happen because the smaller square side length is one quarter the length of the large
square side length?
Is it just a coincidence that the two rectangles share a side length and can be put together
to make a larger rectangle?
The area of 15units2 has been calculated by two different methods. Will this always work for
any square with any sized square nibble?
The student responses can be summarised on the board.
Regardless of the answers to the discussion questions above, students are asked how they can prove
or disprove this hypothesis:Cutting a smaller square from the corner of a larger square, regardless of side lengths, allows two
rectangles to be made and arranged to make one single larger rectangle.
A second square is used by student pairs to investigate. Students can choose what length to name
the larger square’s side length. Instead of 4 units they may call it 5 units or 10 units. If the squares
were custom made for the task, then a range of different sized squares can be used by different
pairs of students. Ensure they are cut to integer side lengths as this will allow greater recognition of
the relationships between side lengths at the end of their investigation.
Similarly students can choose the side length of the smaller square, but again restrict their choice to
integers.
When students produce a large rectangle, they are to note the side lengths and area of their
rectangle.
Reflection
Discuss with students that regardless of the chosen lengths of the sides of the squares chosen, one
large rectangle can be made. If some pairs have not had success the class could discuss why there
were exceptions. Inaccurately measured or poorly cut squares would account for these exceptions.
This stage of the task lends itself to photographing student work. Photos after the two stages of
cutting, with dimensions labelled, provide a record of “the thinking”.
The dimensions of the squares used by students is recorded on the board with the length and width
of the large rectangle. An example is below.
Side length
larger square
4
5
8
10
Side length
smaller square
1
2
1
3
Large rectangle
length
5
7
9
13
Large rectangle
width
3
3
7
7
The students are asked to study the table, look for patterns and describe their findings.
The relationship is described visually:-
The relationship is described algebraically:-
a2-b2 = (a-b)(a+b)
Practice
To help with retention of the formula, and to develop fluency, students practice expanding and
factorising equations that involve the difference of two squares. It is strongly recommended that
initially students draw pictures as demonstration of working out. These informal visuals could be
drawn in workbook, scrap paper or mini-whiteboards.
Many students will of course find this cumbersome and soon want to complete the questions by
following the pattern and proceed algebraically. By insisting they draw pictures, for at least the initial
few practice questions, it will help students retain the picture they have developed for the difference
of two squares.
Some students may continue to draw their working out well beyond teacher expectation. It is likely
that they are still making sense of the process involved in expanding and factorising with the
difference of two squares.