Developing Some Triangle Properties in Grade 9

Triangle properties are now a part of the grade 9 de-streamed math curriculum and after doing some circle properties in a couple of classes, I tried a couple of activities intended to help students develop two important triangle properties. As usual, I wanted to use a concrete-diagrammatic-symbolic approach to this.

 I began by showing students this daigram and asking students what type of triangle it was:

Without exception, all students said it was an isosceles triangle...

I then gave each pair of students five segments of pre-cut stirrers and asked the students to measure them:

The lengths of these segments were 3 cm, 4 cm, 4 cm, 5 cm and 8 cm. I then asked the students to choose any 3 segments and see if they made a triangle with the condition that the segments must all join at a vertex. 

After about 5 minutes, I asked students to give me examples recorded the results as shown below:

I told the students that I would be able to say if there examples were correct by using a special property that I knew of. So when one group of students said that the 8 cm , 4 cm, 4 cm segments made a triangle I replied 'Not according to my special property'! When I looked at what they had done, I was able to point out that the segments were carefully joined at each vertex.

I then challenged students to figure out what was the special property that I was using. It didn't take long for them to articulate that the two shorter sides need add to give a number bigger than the longest side. My sense is that this realisation was helped by using the concrete segments. 

I then showed them the triangle again and asked what type of triangle it was:

Now they saw that it was an impossible triangle! I told them a story about this. A few years ago, I met with a group of developers from a well-known educational software firm who wanted us to see their product. At one point, they showed us some of the questions that their software automatically generated, including the diagram above with the question 'What is the perimeter of this triangle?' Tongue firmly in cheek, I said 'I will give $20 to anyone who can accurately construct that triangle.' I do remember getting a few bemused looks from the developers before I had to remind them of the triangle inequality. The students enjoyed this anecdote!

For the next property that I wanted to them to learn about, I led them by drawing these two triangles and asking them to identify the largest angle, smallest angle and the 'middle' angle.

Whilst most were able to identify the hypotenuse and explain that they knew it was the longest side, they were not all aware that it was opposite the right angle. It made me realise that how I use the word 'opposite' in this context isn't always intuitive for students, so it was necessary to draw on the appropriate arrows as I said 'opposite':

When I had shown which sides were opposite which angles, I then challenged them to find the second property. It gave me the opportunity to move around the room and listen to the conjectures (of which there were many) and, if necessary, challenge these conjectures. it wasn't long before we were able to summarise our findings: in any triangle, the longest side is opposite the largest angle and the smallest side is opposite the smallest angle.

In retrospect, I would should have pushed this thinking further by drawing not just right-angled triangles but acute and obtuse ones too. I also would include isosceles and equilateral triangles.

Although some might think that both of these properties as quite trivial, I see it as being very important that we make sure our students understand why these work. Firstly, by approaching these properties with activities such as these, we are also meeting expectations from Strand A: Mathematical Thinking. Secondly, if students know these properties, they can be used as a good check for glaring mistakes when they are finding missing lengths and angles in triangles.

We ended with this challenge:

It was interesting to see that the only triangle that was not identified correctly was the top right one: the right triangle with sides 4cm , 7cm and 8cm. Their reasoning was that it satisfied the triangle inequality so I then had to remind them that if it is a right angled triangle, then it also has to satisfy the Pythagorean Rule.

Many thanks to Mrs. T. Maecker and her two wonderful classes.