Quadratic Rectangles
This activity uses the area of squares and rectangles to help
expand and factorise simple quadratic expressions.
You will need a selection of large squares, small squares and
rectangles.
Large squares measure x by x
Small squares measure 1 by 1
Rectangles measure x by 1
Areas
What is the area of each of the basic shapes?
x2
x
x x
1 1
1
x
1
Filling a Rectangle
How could you fill the rectangle below using the shapes you
have?
X+2
X+1
Filling a Rectangle
Does it help to break it down?
x+1+1
x+1
Filling a Rectangle
It should look something like this. If yours looks different, have
you used the same pieces?
X+2
X+1
Filling a Rectangle
So what is the total area?
x
x
x+2
1
1
Area = x2+x+x+x+1+1
x2
x+1
x
Area = x2+3x+2
Filling a Rectangle
Fill these rectangles and find the total area of each.
x+2
x+1
x+3
x+4
Filling a Rectangle
x+2
x
x
1 11
1 11
x2
x x x
x+1
x
11 1 1
x2
xxx x
x+3
x+4
Area = x2+5x+6
Area = x2+5x+4
Filling a Rectangle
A rectangle measuring x+4 by x+2 would be written (x+4)(x+2)
Using your shapes, find the areas of the following rectangles:
(x+4)(x+2) = x2+6x+8
(x+3)(x+1) = x2+4x+3
(x+2)(x+2) = x2+4x+4
(x)(x+5) = x2+5x
Areas
What is the area of each of the basic shapes?
x2
x
x x
1 1
1
x
1
Making Rectangles
Rectangles can be made using the shapes.
Start with one x squared, four x by 1 rectangles and three
single squares.
1
x2
What area do you have?
x
x
x
x
Area = x2+4x+3
1
1
Making Rectangles
Make a rectangle using all the pieces – fitting them edge to
edge.
x+1
x+3
What are the lengths of the sides of the rectangle (in terms of x
and numbers)?
Making Rectangles
x+1
x+3
Area = x2+4x+3 = (x+3)(x+1)
or
Area = x2+4x+3 = (x+1)(x+3)
Does it matter which way round they are?
Making Rectangles
Use your shapes to make rectangles with the following areas
and record their side lengths (in terms of x and numbers):
x2+5x+4
= (x+4)(x+1)
x2+5x+6
= (x+3)(x+2)
x2+7x+12 = (x+4)(x+3)
x2+6x+9
=(x+3)(x+3)
Areas
What is the area of each of the basic shapes?
x2
x
x x
1 1
1
x
1
Making Squares
Squares can be made using the shapes – but they may be
incomplete or have some single squares left over.
Start with one x squared, four x by 1 rectangles and six
single squares.
x2
What area do you have?
x
x
x
x
1
1
1
1
1
1
Area = x2+4x+6
Making Squares
Make a square using all the pieces – fitting them edge to edge.
You must use the large square and all of the rectangles
+2
x+2
x+2
What size is the square?
How many pieces do you have left over?
Making Squares
Area = x2+4x+6 = (x+2)(x+2)+2
x2+4x+6= (x+2)2+2
Making Squares
Squares can be made using the shapes – but they may be
incomplete or have some single squares left over.
Start with one x squared, six x by 1 rectangles and six
single squares.
x2
x
x
What area do you have?
x
x
x
x
1
1
1
1
1
1
Area = x2+6x+6
Making Squares
Make a square using all the pieces – fitting them edge to edge.
You must use the large square and all of the rectangles
x+3
x+3
What size is the square?
How many pieces do you have missing? 3
Making Squares
x+3
x+3
Area = x2+6x+6 = (x+3)(x+3)-3
x2+6x+6= (x+3)2-3
Making Squares
Use your shapes to make squares with extras or missing
pieces using the following areas and record their side lengths
(in terms of x and numbers) and extras or missing squares:
x2+4x+1
= (x+2)2-3
x2+6x+11 = (x+3)2+2
x2+6x+2
= (x+3)2-7
x2+2x+8
=(x+1)2+6
Teacher notes: Quadratic Rectangles
This activity was (loosely) inspired by Kevin Lord’s conference session
in which uses business cards to explore algebraic expressions. This
one uses the areas of squares and rectangles to explore expanding
and factorising quadratic expressions.
‘Filling Rectangles’ is aimed at pupils who are beginning to learn to
expand brackets or pupils who are struggling to understand what to do.
The second part of the activity, Making Rectangles, is designed as an
introduction to factorising quadratic expressions. It may help to briefly
run through ‘Filling Rectangles’ first.
Finally, the materials can be used as an introduction to ‘Completing the
Square’
Teacher notes: Quadratic Rectangles
Pupils will need a selection of the shapes provided on the separate
sheet to cut out and use. Finished shapes could be stuck into their
exercise books or the activities could be carried out in groups and
larger sheets of paper used to make posters.
The shapes are designed so that x is not a multiple of 1, this means
that physically an ‘x squared’ can’t be replaced by a number of
rectangles.
The materials work most easily for expressions of the form (x+a)(x+b)
where a and b ≥ 0. They could be used for any values of a or b, but the
context can become unwieldy. It is suggested that these are simply
used as introductory activities to help develop an understanding of what
is required.
Teacher notes
Extension activity: Completing the square
These materials can also be used as an introduction to ‘completing the
square’
They help pupils to see that it is possible to make a square with some
pieces missing or pieces left over.
Again, this is simply an introductory activity to help develop a basic
understanding; working with negative values is possible, but becomes
more complicated.