Quadratic RectanglesThis 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 xSmall squares measure 1 by 1Rectangles measure x by 1

AreasWhat is the area of each of the basic shapes?

x

x x

1

1

11xx2

Filling a RectangleHow could you fill the rectangle below using the shapes you have?

X+1

X+2

Filling a RectangleDoes it help to break it down?

x+1

x+1+1

Filling a RectangleIt should look something like this. If yours looks different, have you used the same pieces?

X+1

X+2

Filling a RectangleSo what is the total area?

x+1

x+2 x2 x

xx 1

1

Area = x2+x+x+x+1+1Area = x2+3x+2

Filling a RectangleFill these rectangles and find the total area of each.

x+3

x+2

x+4

x+1

Filling a Rectangle

x2 x2

x

xxx

xx

xxx x

111

1 111 111

x+2

x+3

x+1

x+4

Area = x2+5x+6 Area = x2+5x+4

Filling a RectangleA 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)

(x+3)(x+1)

(x+2)(x+2)

(x)(x+5)

= x2+6x+8

= x2+4x+3

= x2+4x+4

= x2+5x

AreasWhat is the area of each of the basic shapes?

x

x x

1

1

11xx2

Making RectanglesRectangles can be made using the shapes.

Start with one x squared, four x by 1 rectangles and three single squares.

What area do you have?

x2 xx xx

1 1 1

Area = x2+4x+3

Making RectanglesMake a rectangle using all the pieces – fitting them edge to edge.

What are the lengths of the sides of the rectangle (in terms of x and numbers)?

x+3

x+1

Making Rectangles

x+3

x+1

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 RectanglesUse your shapes to make rectangles with the following areas and record their side lengths (in terms of x and numbers):

x2+5x+4

x2+5x+6

x2+7x+12

x2+6x+9

= (x+4)(x+1)

= (x+3)(x+2)

= (x+4)(x+3)

=(x+3)(x+3)

AreasWhat is the area of each of the basic shapes?

x

x x

1

1

11xx2

Making SquaresSquares 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.

What area do you have?

x2 xx xx

1 1 1

Area = x2+4x+6

1 1 1

Making SquaresMake a square using all the pieces – fitting them edge to edge.You must use the large square and all of the rectangles

What size is the square?How many pieces do you have left over?

x+2

x+2

+2

Making Squares

Area = x2+4x+6 = (x+2)(x+2)+2

x2+4x+6= (x+2)2+2

Making SquaresSquares 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.

What area do you have?

x2 xx xx

1 1 1

Area = x2+6x+6

1 1 1xx

Making SquaresMake a square using all the pieces – fitting them edge to edge.You must use the large square and all of the rectangles

What size is the square?How many pieces do you have missing?

x+3

x+3

3

Making Squares

Area = x2+6x+6 = (x+3)(x+3)-3

x2+6x+6= (x+3)2-3

x+3

x+3

Making SquaresUse 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

x2+6x+11

x2+6x+2

x2+2x+8

= (x+2)2-3

= (x+3)2+2

= (x+3)2-7

=(x+1)2+6

Teacher notes: Quadratic RectanglesThis 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 RectanglesPupils 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 notesExtension activity: Completing the

squareThese 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.