### Four Triangles Puzzle

Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?

### Transformations on a Pegboard

How would you move the bands on the pegboard to alter these shapes?

### Inside Seven Squares

What is the total area of the four outside triangles which are outlined in red in this arrangement of squares inside each other?

# Cutting Corners

## Cutting Corners

I don't know about you but when I'm walking or cycling I like to cut a corner if it's possible and safe. I do this most when I'm walking along a path and the next path is a long way ahead and going off to the left at right angles. Looking from above this might look like this :-

and so I decide to "cut the corner" as it saves time, it's a shorter distance to go, and you can overtake someone who is walking around the path. So I go along the dotted route.

Hey this makes a rather good triangle, also, I guess, rather special. The two shorter sides are the same length and are at right angles to each other.

You may, at school, have a collection of these kinds of triangles and you might be able put them together in different kinds of designs and patterns.

I thought it would be good to have a very "OPEN" challenge this time since this triangle came from the open-air! Here it is drawn on its own.

I've taken a big one of these triangles and cut it several times in a special way. The first cut was from the bottom right hand corner and cut the big triangle in half. I've then gone on to cut in a "Zig-Zag" fashion each cut halves the previous triangle, each time halving the remaining triangle.

These cuts give us lots of triangles and my challenge to you, by using many of these triangles is to come up with the MOST,
the MOST Extraordinary,
the MOST AMAZING,
the MOST UNUSUAL,
the MOST . . . . . . !? PATTERNS/DESIGNS

You might do it on paper or card. But you might be able to use a draw program on your computer to make these Triangles [if you do that make sure that they are half of the size of the last one].

Finally - - as always -- "I wonder what would happen if we were allowed to . . . . ?"

### Why do this problem?

This is such an open investigation - I really feel that it is good, every now and again, to include such an activity as this in the children's mathematical experience. The pupils' understanding of the properties of these isosceles triangles will be enhanced. The concept of area is often challenged in this investigation as our eyes seem to play tricks on us and we have to verify things in a more certain way. With some pupils you may even start thinking about Pythagoras and square numbers and square roots.

### Possible approach

You'll need to be enthusiastic in a way that will capture the pupils' interest and imagination. You could begin by working on the board and demonstrating how to cut the triangle in half (turning it on its side helps to see the symmetry for some children).  Then offer the children a large ready-made triangle each for them to halve and halve again. If you use different coloured paper for each child then you might encourage them to swap a piece of the same size and shape with a another child, so making their pattern more interesting.

### Key questions

Do you notice anything about the pattern/shape you've made?
Can you make other shapes with the triangles you've chosen to use?

### Possible extension

Is it possible to halve all triangles like this? Why or why not?

### Possible support

It may be necessary to help those who lack confidence in manipulation to draw lines with a ruler and cut the triangles. You might prefer to draw lines ready to cut, and focus the activity instead on the language of shape and size.