Walking round a triangle
Problem
This ladybird is taking a walk round a triangle. She starts off near the middle of one side of the triangle. Can you see how much she has turned when she gets back to where she started?
Would it be the same amount of turn if she went around another triangle?
Why?
Getting Started
Draw a big triangle on the playground with chalk. Pretend you are the ladybird.
Start near the middle of one side and go round the triangle.
How much do you turn?
Do you think this would be true for every triangle?
Student Solutions
The questions asked were:
How much was turned?
Is it always the same and (the big question) - why?
There was a lot of confusion about the angles involved and most solutions that were sent in indicated $180$ degrees. This number of degrees is of course involved in what we know of triangles. However, when going around a triangle we do not turn the internal angle but $180$ minus the internal angle.
The solution sent in by a pupil from TNT school in Canada does quite well in showing this.
The triangle on which the ladybird walks has three arbitrary angles x, y, and z whose sum is $180$ degrees. The ladybird starts at S facing point A. Once it reaches A it turns an angle "a" and now faces point B. It does the same at point B and C (turning an angle "b" and face C;
turn an angle "c" and face A respectively). The sums of angles a and x, c and y, b and z are each $180$ degrees. By adding: $(a+x) + (y+c) + (z+b) =$ $540$ degrees and subtracting x,y and $z = 360$ degrees so the ladybird turns $360$ degrees.
This would be the same for any triangle because this is a triangle with unspecified angles.
The other solution that tried to answer all the three questions was from Alexis who goes to Claremont Fan Court School, who goes on to say;
The Ladybird turns $360$ degrees in total. This is because in order to get back to the direction you were facing, you have to turn around $360$ degrees however many facings you go around. Ergo, this is true for any triangle, or any regular polygon for that matter.
Well done the two of you.
Teachers' Resources
Walking Round a Triangle
This ladybird is taking a walk round a triangle. She starts off near the middle of one side of the triangle. Can you see how much she has turned when she gets back to where she started?
Would it be the same amount of turn if she went around another triangle?
Why?
Why do this problem?
This problem offers a geometric context for a generic proof. A generic proof works by using the structure of a particular example to show the result in the general case. In this case walking round any triangle will involve making a complete turn. A complete turn taken in three stages will always create a three stage path or a triangle.
Possible approach
Get the children to draw triangle paths on paper or in chalk on the playground. If they do this outside they can walk along the triangle path themselves. They should start by facing along one of the lines on any of the sides. It is best to start part of the way along one side so that there is no confusion about the amount of turn involved to get to the same point facing the same way. When they get back to the starting point they will have turned through a complete turn. It is important to turn in the same direction (clockwise or anticlockwise) every time.
Key questions
Which way are you facing?
How much have you turned?
How many turns are you making to go round the triangle?
How much have you turned when you get back to where you started?
Possible extension
It would be possible to look at walking round a square, a pentagon, different quadrilaterals, and so on in the same way. The problem also links to Round a Hexagon.
Possible support
More support could be offered through using a robot and programming it to go around a triangle or other polygon. The children may find it helpful to use this sheet