Let's investigate triangles
Vincent and Tara are making triangles with the class construction set. They have a pile of strips of different lengths. How many different triangles can they make?
Problem
Vincent and Tara are making triangles with strips from the class construction set.
They have strips of three different lengths: 4 holes, 7 holes and 11 holes.
There are plenty of strips of each length.
Vincent makes a triangle with two strips that are length 7 and one strip that is length 4, like this:
Tara makes a triangle with three strips that are each length 11, like this:
How many different triangles can you make using strips that are length 4, length 7 and/or length 11?
You can use the interactivity below to try out your ideas if you do not have a construction set.
Can you find three strips which cannot be made into a triangle when you use them together?
Why is it impossible to make a triangle from those three strips?
Getting Started
Have you made triangles with all the sides the same colour?
Have you made triangles with all the sides different colours?
Have you found three sticks you can't make a triangle with?
This sheet will give you four copies of each strip in the colours in the problem, which you could print off and cut out.
Student Solutions
Thank you to everybody who sent in their solutions to this activity. We received lots of good ideas about which triangles were possible or impossible to make.
Zoe from Halstead Preparatory School in the UK sent us this picture:
Well done for checking every possible combination, Zoe! Zoe says that the longest side of a triangle has to be shorter than the other two sides put together. I wonder why?
We also had this picture sent in from Vivienne and Georgy at Halstead Preparatory School:
Well done for using practical resources to check each triangle! These pictures show really clearly why each triangle is either possible or not possible.
We also had similar solutions sent in from Viren KS and from Ariana, Ayda, Ishita and Rhea from Halstead Preparatory School. Thank you all for sending us your ideas.
Teachers' Resources
Let's Investigate Triangles
Vincent and Tara are making triangles with strips from the class construction set.
They have strips of three different lengths: 4 holes, 7 holes and 11 holes.
There are plenty of strips of each length.
Vincent makes a triangle with two strips that are length 7 and one strip that is length 4, like this:
Tara makes a triangle with three strips that are each length 11, like this:
How many different triangles can you make using strips that are length 4, length 7 and/or length 11?
You can use the interactivity below to try out your ideas if you do not have a construction set.
Can you find three strips which cannot be made into a triangle when you use them together?
Why is it impossible to make a triangle from those three strips?
Why do this problem?
This problem allows children to make important discoveries about triangles through representing the sides of triangles with strips from a construction kit. They will have the opportunity to find out which combinations of lengths cannot be made into triangles, and why this is so.
If learners have access to the physical resources, they may also discover how rigid triangles are, and that three fixed lengths can only be arranged in one way (unlike other polygons such as quadrilaterals where the angles can be changed).
Ideally, this investigation would be carried out using a construction set and therefore requires plenty of equipment and space. However, the interactivity will enable all children to access the mathematics to some extent, even without the hands-on resources.
Possible approach
You may like to show one of the triangle images to the class and ask them what they see, and what they would like to ask. Give learners time to talk in pairs before sharing their thoughts with the whole group. Some of the observations may not be mathematical, but it is important all comments are valued at this stage.
Build on what is said to introduce the problem, encouraging children to work in pairs on making different triangles with lengths 4, 7 and 11. (They will benefit from talking through their ideas with a partner.) If you do not have access to a construction set, learners could use the interactivity in pairs on a tablet or computer. You may want to demonstrate how to change the length of the strips and how to rotate each strip first.
In a mini plenary, you could discuss how they are keeping track of the triangles they are making, and you may like to draw attention to any pairs who are recording in some way. For example, children may sketch the triangles, or record just using the length of the strips used.
You could invite pairs to record a triangle they have made on the whiteboard for everyone to see, and that way you will gradually build up a class solution (this will be made easier if the class agrees on a way of recording as a group first). Emphasise that when another solution is added to the board, it is everyone's responsibility to check it, and to make sure it is not a repeat of one already there.
In the plenary, you can look at the class solutions altogether. It might be useful to record each solution on a separate sheet of paper so they can be moved around and ordered, to help reveal any that are missing. In this way, the group can be sure that all possiblities are found. As the discussion proceeds, make a note of any combinations of three rods that don't make a triangle. To conclude the lesson, invite learners to have a go at explaining why three particular rods cannot make a triangle. They may manipulate strips a bit like this:
This shows that the red and blue rods are not long enough to make a triangle with the green rod. Some children may have a 'feel' for this idea of relative length without needing to use the strips to convince themselves and others.
Key questions
Have you made any triangles using this length of strip?
Have you made any triangles with all the sides the same length?
Have you made any triangles with all the sides different lengths?
Are you sure you/we haven't made one like that already?
Why do you think you can't make a triangle with those three strips?
Possible support
Some learners might find it helpful to use just two lengths to start with. (Make sure that the longest one is shorter than two of the others put together lengthways e.g. strips which are 4 holes and 6 holes long.) You can make four different triangles with these. You can then add a third length.
Possible extension
Children could be encouraged to find a way to show that they have discovered all of the possible triangles using four different lengths. Some learners might like to have a go at the task Sticks and Triangles, which explores the construction of triangles further, culminating in a full generalisation.