Transformations on a Pegboard

'Transformations on a Pegboard' printed from https://nrich.maths.org/

Show menu

Transformations on a Pegboard

Someone using an elastic band and a pegboard used four pegs to make the blue square you see below. They challenged another person to double the area by just moving two of the pegs. You can see what they did here.

2 by 2 square made into 2 by 4 rectangle

Have a go at these:

Can you make this into a right-angled triangle by moving just one peg?

traingle of base 6 units, perpendicular height 4

Can you enlarge this to the same shape with all the sides twice the length, moving just two pegs?

4 by 2 rectangle

You could use our interactive geoboard below to try out your ideas.
Choose the size of your pegboard then select the line tool and click on two dots to draw a line between them.

You could set up some similar challenges for your friends, or have a go at More Transformations on a Pegboard.

Why do this problem?

This problem is a good way of consolidating properties of shapes and visualising changes in their properties. The interactive enables learners to satisfy their curiosity and try out ideas, when at first the task might seem very challenging.

Possible approach

You could introduce this problem by giving pegboards and elastic bands to pairs of children. If they have not used pegboards recently a few minutes of free play helps concentration later! Alternatively, learners could use the interactive virtual geoboard to explore the challenges given. If you have an interactive whiteboard, using the virtual geoboard would be a good way to share ideas with the whole class during the lesson.

Children will discover that there is more than one way to do the first part of the problem. How many ways can they find? You could talk about how they know they have got them all - perhaps by looking at each vertex in turn in a systematic way. The problem will encourage children to think hard about what makes a triangle a right-angled one. You could ask them to investigate the other changes that occur when the length of sides of the rectangle are doubled (for example, what about the area?).

Learners could draw their answers on square dotty paper or write instructions in words (which is much harder!).

Key questions

Which pegs have you tried to move?

Can you make the shape by moving any other pegs instead?

Are there any other ways to do it?

Possible extension

Learners could make up similar puzzles for others to do using the virtual geoboard or paper. They may also like to have a go at the challenge More Transformations on a Pegboard which focuses on areas of triangles.

Possible support

Using a real pegboard with elastic bands will make this more accessible for many children. They could use two bands in different colours so that one can be left in the original place all the time.