## Egyptian Rope

The ancient Egyptians were said to make right-angled triangles using a rope which was knotted to make twelve equal sections.

If you have a rope knotted like this, what other triangles can you make? (You must have a knot at each corner.)

What regular shapes can you make - that is, shapes with equal length sides and equal angles?

**Why do this problem?**

This problem is one that combines knowledge of properties of shapes with addition, subtraction, multiplication and division of small numbers. It also provides an opportunity for learners to consider the effectiveness of alternative strategies.

### Possible approach

You could use this problem during work on either number or shape.

It could be introduced by sharing the picture of the triangle made from rope and asking children what they see. Invite learners to share their thoughts with the whole group and facilitate a discussion about the image.

If it does not come up naturally, draw the class' attention to the fact there are twelve sections in the rope and ask learners to investigate other possible triangles. Have to hand various resources which they could use as they work on the problem in pairs. This might include, for example, headless matches, lolly sticks, cut-up drinking straws, paper, Cuisenaire rods...

It would be useful to discuss how learners will know that they have found all the possible triangles. Listen out for those who work systematically, in other words they are looking for solutions in a particular order so they know they won't miss any out.

Learners could then go on to the second part of the problem to find regular shapes that can be made using the same piece of string (or all twelve sticks/matches/straws...). Some may continue to work practically, some may prefer to draw sketches and others may consider the problem numerically.

The final plenary could focus on which regular shapes have been possible but in particular about why it is impossible to create, for example, a pentagon.

### Key questions

Why do you think it isn't possible to make a triangle with these two sides?

How do you know you have found all the possible triangles? Can you tell me why no other triangles are possible with this piece of string?

How are you trying to find regular shapes that you can make with the string?

What numbers are factors of 12? How can this help you to make some regular shapes?

### Possible support

Having twelve sticks of equal length (such as headless matches, or even pencils) to build the shapes makes this problem accessible to all children.

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### Possible extension

Learners could investigate the possible triangles made with different numbers of sticks as in the problem Sticks and Triangles.