Cut It Out
3 out of 3
Can you dissect an equilateral triangle into 6 smaller ones? What number of smaller equilateral triangles is it NOT possible to dissect a larger equilateral triangle into?
Problem
An equilateral triangle can be dissected into four (equal) smaller equilateral triangles.
Can you dissect an equilateral triangle into six smaller ones, not necessarily all the same size?
Is it possible to dissect a larger equilateral triangle into any number of smaller equilateral triangles?
Are there any numbers that are impossible?
Getting Started
What is the smallest number of triangles you can break one triangle in to?
You could print out this sheet of triangles or use isometric dotty paper.
Student Solutions
The diagram shows an equilateral triangle dissected into 4 smaller equilateral triangles. Which numbers of smaller triangles are possible and which are not, and why? This remains a Tough Nut as nobody has yet explained why it is impossible to dissect an equilateral triangle into certain numbers of smaller equilateral triangles.
Robert and Andrew, St James Middle School, Bury St Edmund's sent this solution with six equilateral triangles. They say they think this is the only solution with 6 smaller triangles other than rotating the shape.
Talei and Stephanie of Poltair Community School and Sports College, St Austell, Cornwall said they think that dissection into 2 or 3 smaller triangles is impossible and that 4 is the smallest number of triangles possible. They gave this as an example of dissection into 15 smaller triangles.
Talei's also had the idea of allowing overlapping triangles and realised that this would make it possible to have any number of smaller triangles, even infinitely many, but that would be a different question.
Lucas from St. Joseph's in Reading wrote in this wonderful explanation.
It is possible to make any number of smaller triangles except 2, 3 and 5 in the large equilateral triangle. Any even number can be made by drawing an odd number of small triangles in a row along the bottom going up, down, up, therefore creating a larger triangle on the top. For odd numbers ( This works for even numbers above 9), simply subtract 3 from that number - for 7 we would get 4 - Have you made this triangle yet? If not, subtract 3 again and so on. In the number 4 triangle (Or whichever you are using), find a smaller triangle, pointing up or down, and inside of it draw a triangle pointing in the opposite direction with the corners touching the sides. Count up the triangles and you should find that there are the number you are looking for. This is because a triangle has 3 sides, so as well as the triangle in the middle, it creates 3 around the edge.
Teachers' Resources
Using NRICH Tasks Richly describes ways in which teachers and learners can work with NRICH tasks in the classroom.
Why do this problem?
At the most basic level, this problem provides an opportunity to reinforce what an equilateral triangle is, and whilst that will be sufficient for some, this question requires much more sophisticated thinking than that. For the high flyers in your class, the fact that it is a 'tough nut' (i.e. no children's solution has so far been submitted) might provide the extra incentive to persevere when the going gets tough.
Possible approach
This problem requires some time to do properly. If you can introduce it whilst doing 2D shape as a main topic, you might want to offer it as a 'simmering' activity for a small able group who can return to it over several days, perhaps when they have completed their allotted tasks.
The question is in three parts and these are best introduced one at a time. Give out sheets of equilateral triangles and allow some time for the children to explore the first question either individually or in pairs. Some children might prefer to draw their triangles on isometric paper. Listen for statements which are accompanied by some sort of justification, for example "Four is the smallest because after one big triangle the next size up has two triangles along each side." Encourage the children to work systematically.
If you think it appropriate, make available a section of display space where they can place their diagrams, cut out and sorted into 'families'. This way the individual findings can be a contribution to a greater whole.
Key questions
What is the smallest number of triangles we can break one large one into?
How do you know?
Is there a largest number?
Possible extension
Some children will want to find a general rule. This is a sophisticated piece of generalising.
Possible support
You may want to give out triangular grid or dotty paper and ask the children to draw equilateral triangles of different sizes and count how many smaller triangles are in each. This will help them to see that 1, 4, 9, 16 are useful numbers to use.