More transformations on a pegboard

Use the interactivity to find all the different right-angled triangles you can make by just moving one corner of the starting triangle.

Age

7 to 11

Challenge level

Primary

Geometry

Position, Direction and Movement

Being curious Being resourceful Being resilient Being collaborative

Problem

You might like to try Transformations on a Pegboard before looking at this problem. Here are some ideas to take this activity some stages further.

You may like to use this interactive pegboard to try out your ideas. Read underneath the interactivity for the challenges!

Choose the size of your pegboard then select the line tool and click on two dots to draw a line between them.

One of the challenges was, 'By moving just one peg can you make a right-angled triangle?'. Well, I had a go starting the same way:

A 9-by-7 peg board with a triangle of base length 7 pegs and perpendicular height 4 pegs.

Here are two I found moving one peg each time (from the original):

A 9-by-7 peg board with a right-angled triangle of base length 5 pegs and height 3 pegs.

A 9-by-7 peg board with a right-angled triangle of base length 7 pegs and height 3 pegs.

And so on and so on ...

How many different right-angles triangles can you get if you stick to a grid of $7$ rows of dots and $9$ columns of dots?

The triangle started off having a base of $6$ and was $3$ high.

Look at the other triangles you have found that have this same base and height.

What can you say about the areas of this set of triangles?

(You might like to draw some more triangles with a base of $6$ and a height of $3$ which don't have right angles to see what that adds to your thoughts and ideas.)

Teachers' Resources

More Transformations on a Pegboard

Here are two I found moving one peg each time (from the original):

Why do this problem?

This activity offers a good chance to develop concepts surrounding triangles, including their areas. It is an engaging activity which allows pupils to work in twos and to have fruitful disussions.

The interactivity will help to arouse learners' curiosity and explore what happens when different sizes of grids are chosen or other shapes are explored. So, this activity can lend itself to pupils posing their own questions in the form of "I wonder what would happen if...?". (The further note at the foot of this page offers more support on curiosity in the classroom generally.)

Possible approach

As we all learn in different ways it would be good to allow learners to explore this activity with a number of aids at their disposal, for example:

- Spotted paper in squares

- Nail boards with elastic bands

- The on-screen interactive board

- Squared paper

Encourage them to start with the original shape and move one point at a time to see the results. They will need to do this in a systematic way if they are going to find all the possibilities and it is likely that the group will benefit from discussing some strategies for doing this as a whole. This might also require you to model the first few steps in a particular way of being systematic, for example by moving the "top" corner of the triangle one peg up, then two pegs up etc. You can ask the children what to do next at each stage and talk about what might be "sensible" so that you don't miss any out. They may notice a pattern in the way the possible positions for this point lie so that it might not be necessary to actually test every point on the grid. Depending on the class' experience, you could use a number of ways of checking whether an angle is a right-angle, for example by using a protractor or simply a corner of paper.

When it comes to looking at the areas of sets of triangles (i.e. triangles which have the same base and height), the pupils will have different ways of calculating these areas and these should be acknowledged and shared. Many children will be able to make the full generalisation about areas of triangles with identical base and height through this investigation.

Key questions

How are you exploring where the elastic band can go?

How will you make sure you don't miss any triangles out?

How are you finding the areas of the triangles?

What do you notice about the triangles' areas?

Possible extension

In the original Transformations on a Pegboard problem, one of the challenges was about changing a square into a rectangle so that the area doubles by moving two pegs. You could go on to investigate this further with your class, but rather than find a rectangle, look for any shape which can be made by moving two pegs which has double the area of the square. How many ways can they do this? Sometimes when you've exhausted a lot of ideas you have to change the rules again so here's an idea - keep the same starting square but have a larger grid, so it looks like this:

Can the group find some new shapes now that by moving two pegs will double the original area? (No matter what the final shape is!) What would happen if you could only move one peg to double the area?

Possible support

Using nail/peg boards or the interactivity on screen will help all children access this problem, but those with poorly developed motor skills may need help from an adult or fellow pupil.

Further note

You may be interested in the following talks given by Professor Susan Engels, which focus on encouraging curiosity and are available on YouTube:

The Rise and Fall of Curiosity - the extract from 23.50 to 37.15 on adult encouragement and teacher behaviour is particularly worth viewing

The Hungry Mind: The Origins of Curiosity - the extract from 8.22 to 12.29 on children asking questions is especially useful.

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