Egyptian rope
The ancient Egyptians were said to make right-angled triangles using a rope with twelve equal sections divided by knots. What other triangles could you make if you had a rope like this?
Problem
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?
Getting Started
You could try using twelve sticks of equal length such as headless matches to try out your ideas.
Student Solutions
We had some lovely solutions sent in for this activity, so thank you to everybody who shared their ideas with us.
Jeremy from Thailand sent us this video:
Well done for finding those three equilateral shapes. Jeremy mentions that he found another triangle that wasn't equilateral - I wonder what that one would look like?
Ci Hui Minh Ngoc from Kong Hwa School in Singapore sent in these ideas:
Well done for finding all of the possible triangles and all of the regular shapes! Ci Hui Minh Ngoc has drawn triangles whose sides are 4 x 4 x 4, 2 x 5 x 5 and 3 x 4 x 5 units of length.
Those parallelograms do have equal length sides, but they aren't classed as regular shapes. Have a close look and see if you can work out why.
William sent in this picture of his solutions:
He said:
You can basically do any shape whose sides are a factor of 12, because Egyptian Ropes have 12 knots. So a 3 sided shape like a triangle, a 4 sided shape like a rectangle or square or rhombus or parallelogram, a 6 sided shape like a hexagon, or a 12 sided shape like a dodecagon. And then have these shapes with units that add up to 12.
Good ideas, William! Can you spot which two of your shapes are actually the same?
Teachers' Resources
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
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?
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.
Possible extension
Learners could investigate the possible triangles made with different numbers of sticks as in the problem Sticks and Triangles.