2. Quadrilaterals Teacher notes
Work through each description in turn and ask pupils to suggest everyday objects exemplifying these shapes.
Parallelogram: Draw pupils’ attention to the convention of using double dashes to distinguish between the two pairs of equal sides and the use of double arrow heads to distinguish between two pairs of parallel sides. State that when two lines bisect each other, they cut each other into two equal parts. Ask pupils for other derived properties such as the fact that the opposite angles are equal and adjacent angles add up to 180º. Stress, however that a parallelogram has no lines of symmetry. Ask pupils if they know the name of a parallelogram that has four right angles (a rectangle), a parallelogram that has four right angles and four equal sides (a square) and a parallelogram with four equal sides (a rhombus). A parallelogram can be thought of as a slanted rectangle.
Rhombus: Ask pupils for other derived properties such as the fact that the opposite angles are equal. Ask pupils if they know the name of a rhombus that has four right angles (a square). A rhombus can be thought of as a slanted square.
Rectangle: Ask pupils for other derived properties such as the fact that the diagonals are of equal length and bisect each other. Ask pupils to explain why it is possible to describe a rectangle as a special type of parallelogram.
Arrowhead: Ask pupils for other derived properties such as the fact that one pair of angles is equal. Ask pupils if an arrowhead can ever have parallel sides. The answer is no.
Square: Ask pupils for other derived properties such as the fact that the diagonals are of equal length and bisect each other at right angles. Ask pupils to explain why it is possible to describe a square as a special type of parallelogram, a special type of rhombus or a special type of rectangle.
Trapezium: A trapezium has one line of symmetry when the pair of non-parallel opposite sides are of equal length. It can never have rotational symmetry.
Isosceles trapezium: Ask pupils for other derived properties such as the fact that there are two pairs of equal adjacent angles.
Kite: Ask pupils for other derived properties such as the fact that there is one pair of opposite angles that are equal. Ask pupils if a kite can ever have parallel sides. Conclude that this could only happen if the four sides were of equal length, in which case it would no longer be a kite, but a rhombus.

3. Area formulae

4. Area calculations

5. The area of a parallelogram
Teacher notes
Modify the parallelogram by dragging the points to change its height and its width.
If required use the pen tool to show why the area of the parallelogram is equal to the area of a rectangle with the same base length and height.
The square grid can be turned off once the derivation of the formula has been established.
As an extension activity discuss how we could use Pythagoras’ Theorem to calculate the perimeter of the parallelogram. Investigate how the perimeter changes while the area remains constant.

6. The area of a triangle Teacher notes
Modify the triangle by dragging points B and C to change its height and its width.
If required use the pen tool to draw a rectangle around the triangle to justify its formula.
The square grid can be turned off once the derivation of the formula has been established.
As an extension activity discuss how we could use Pythagoras’ Theorem to calculate the perimeter of the triangle.

7. The area of a trapezium Teacher notes
Modify the trapezium to change its height and the length of its parallel sides.
Use the pen tool to show how we could find the area of the trapezium by dissecting it into two triangles and a rectangle.
The square grid can be turned off once the derivation of the formula has been established.
As an extension activity discuss how we could use Pythagoras’ Theorem to calculate the perimeter of the trapezium.

8. Finding the area of irregular polygons
Teacher notes
Drag the vertices to modify the shape. Ask a volunteer to divide the shape into rectangles and triangles using the pen tool, set to draw straight lines. Use this to work out the area.
Alternatively, a rectangle could be drawn around the outside of the shape and the area found by subtraction.
As an extension ask pupils how we could find the perimeter of the given shape (using trigonometry).

9. Rectangle perimeters

10. Hectare forests
Do each of the forests below cover an area of one hectare? What are the perimeters?
75 m
25 m
40 m
50 m
250 m
80 m
25m
20 m
60 m
100 m
50 m
Teacher notes
First forest: 40 × 250 = m².
Second forest: (20 × 200) × (200 – 60) = = m².
Third forest: (80 × 75) + (80 × 25) + (50 × 80) = = m².
Ask pupils to create other shapes that would create an area of a hectare.
Perimeters
Forest 1: 2( ) = 580 m.
Forest 2: (200 – 60) + ( ) = 640 m.
Forest 3: = 660 m.
200 m
50 m
1 hectare = m2 10

11. Shapes made from rectangles
Teacher notes
Drag and drop the points to change the shape.
Modify the compound shape on the board and discuss the various ways to find its area by splitting it into two rectangles or by subtracting a rectangle.
Turn off the square grid and hide the lengths of some of the sides by clicking on them. Discuss how the lengths of these missing sides can be found and use these lengths to find the perimeter and the area.

12. Recognizing formulae

13. Tangram
This Chinese puzzle is made from 7 shape pieces that fit together to make a square with sides of 16 cm.
The intersections are all evenly spaced.
Teacher notes
Discuss the measurements of the shapes before pupils begin to work out the areas. The measurements of the square are not whole numbers and so the subtraction method can be used to work it out.
Large triangle = 0.5 × 16 × 8 = 64 cm²
Medium triangle = 0.5 × 8 × 8 = 32 cm²
Small triangle = 0.5 × 8 × 4 = 16 cm²
Parallelogram = 8 × 4 = 32 cm²
Small square = 256 – ( ) = 32 cm²
What are the areas of the individual shapes? 13