When we were introduced to the project math without words, our group chose to work with triangle and the idea of showing that every triangle has an angle sum of 180 degrees. Later on we found out that we had to show this project in Holland, and our group got a class of the highest level judging by the Dutch school system (ref: ). Because of that we chose to add an extra dimension to this project, which was the idea that not every triangle has an angle sum of 180 degrees, some can have more than that, which can be proven with spherical geometry.
The conditions were:
- Letting Danish students teach math to Dutch children.
- No communication in words by Danish students/Managing the class without words.
- Teach by use of sign language and sounds (clapping, snapping e.g.).
- Sets of triangles in three species: Right, Obtuse and Acute.
- One of the three triangles for the teacher to use later on.
- Sets ripped angles of 90⁰ (ripped corners of an A4-paper).
- A soccer ball.
- Whiteboard & whiteboard markers (different colors).
- A4-paper to draw a triangle from the three given corners.
The lesson plan that we used in Holland step by step:
Step 1 and 2.
Start the lesson by dividing pupils into X groups by using signs to group up and using the fingers to give the group number and checking if student understand. Each group now gets one of the three different kinds of triangles. The teacher then draws the three different triangles on the white board and checks with the pupils which groups have gotten which triangle. This is done by pointing at triangle A and making hand raising gestures to let the right groups raise their hands. Repeat this with triangle B and C, and at the end write down the number of groups with the same triangle.
After this the teacher takes a random triangle at his disposal and rips off the corners like it is indicated on the drawing on the whiteboard. The pupils should get that they have to do the same with their triangle.
In step 3 the teachers starts by drawing the ripped pieces on the whiteboard and make clear with arrows they have to push them together. While the pupils does this the teachers draws three kinds of possibilities on the whiteboard. One that is less than 180 degrees, one that is 180 degrees, and one that is more than 180 degrees.
The teacher now makes gesture to raise hands. First get the attention of the groups who have triangle A, point at the three different options and see which one they got. The teacher does the same with the groups that has triangle B and C. All three groups should get option 2, the one that makes a straight line of 180 degrees.
In this step we use the knowledge that the pupils have to our benefit, they know that a straight line has 180 degrees, and from that we find that option 2(described in step 3) is 180 degrees.
The previous step makes it possible to conclude that the angle sum of all three triangles is 180 degrees, no matter the shape.
The teacher now gathers all the paper from each group and tosses it out. Then he handles each group a set of 3 corners of an A4 paper sheet, each with a 90 degree angle.
The teacher points at option 3(the one with that’s more than 180 degrees) from step 3, and makes a question mark. The pupils should now get the idea that the teacher is trying to ask if a triangle can have more than 180 degrees in it.
Can it be made like a triangle? The teachers makes a drawing on the chalkboard(whiteboard), that should look like the three corners being put together as a triangle.
The pupils should get the idea that they need to try and put the three corners together, and make a triangle. They probably and hopefully won’t succeed:)
The teacher then gets a plastic soccer ball (from one of his helpers or an observer) and shows that it can be done on a curved surface by putting all the three corners together on it and drawing a triangle with chalk that combines those corners. Important thing is to somehow get all the pupils together in a round circle or something like that, so that they actually can see it happening. Last but not least, when it’s done, the teacher should show on the whiteboard that all the angles on the triangle are 90 degrees, hence we get a triangle with more than 180 degrees.
They should get the idea, and after this probably start a discussion with the teacher.
To support step 7, here is a video we made when the Dutch students were in Denmark. It should help people who read this understand step 7 a bit better. In addition, as you can see in the video, we had a plastic device designed to help us draw the triangle on the soccer ball.
Project made and performed by: Sara, Wesley and Martin S.