Shaping up to scale factors
Q The pupils I work with in a lower set in maths have understood that similar shapes are created by enlarging or reducing a shape by a particular scale factor. But they find it difficult to apply this to problems - for example, to find the length of sides in two similar triangles. We have nine pupils and a support teacher so can practical activities easily.
A Some pupils find it difficult to relate to the spatial arrangements of similar shapes. Your group might find the following exercise helpful. Group them round tables with some A3 sheets in the middle with various shapes drawn on them, such as triangles, trapeziums and logos (for examples see www.mathagonyaunt.co.uk). Draw the shapes so that the matched sides are not in the same relative positions. To each sheet attach one of the shapes cut out of coloured card (Fig 1).
Begin by establishing that one shape is indeed an enlargement of the other.
Invite each pupil in turn to arrange the shape on the paper so that the sides are in the same relative positions as its similar shape (not the congruent shape). Ask them why they think the position they have chosen is the correct one. When the majority agree that they have located the correct position, draw round the shape and label the vertices and sides. Repeat this with the other shapes. If further consolidation is needed, prepare a similar on A5 sheets so pupils can work in pairs. The A3 sheet should now look like Fig 2.
Invite pupils to suggest how to use the given information to find the lengths of the other sides. With the more unusual ideas, ask them why they thought that might be useful - sometimes they have a different way of looking at the problem, which help them understand how to solve it (it also encourages them to talk maths). This focus leads to the fact that the side ZX is an enlargement of BA. Mark the matching sides with a coloured circle (Fig 3). They can use the lengths 10cm and 4cm to calculate the scale factor (allow the use of calculators for this). The scale factor of enlargement (the multiplier) is found by dividing the larger number, 10, by the smaller, 4, giving 2.5. Indicate this on the diagram.
Talk about what happens when you multiply a number by 2.5 (the number gets bigger). Multiplying by 2.5 increases the size of the small triangle to that of the large one. Dividing by 2.5 takes you from the large triangle to the small one. This image is important to help understand what is happening in terms of the enlargement - p=5.04cm and q=5.5cm (Fig4).
This information can also be organised in two columns, one for the small triangle and one for the large. Between the columns, the scale factor increases the dimensions in one direction and decreases them in the other.
This use of the scale factor in a more abstract form is helpful when the problem does not include shapes (Fig 5).
Now invite pupils in turn to show how they would solve the other A3 problems. They can then work in pairs to complete these. Prepare more exercises in different contexts. For instance, a recipe for four people that has to be adapted for 10. Or ask them what happens to the area of a shape when the sides are doubled, trebled and so on.
Wendy Fortescue-Hubbard is a teacher and game inventor. She has been awarded a three-year fellowship by the National Endowment for Science, Technology and the Arts (NESTA) to spread maths to the masses. Email your questions to Mathagony Aunt at email@example.comOr write to TES Teacher, Admiral House, 66-68 East Smithfield, London E1W 1BX