Trouble with Triangles 1
Questions to consider before reading further.
- What is the perimeter of a triangle with the side lengths of 6.4, 2.9, 3.3 units?
- Make as many triangles as you can with a perimeter of 12 units (integer value sides).
Yes, the first is actually an impossible triangle and the second question doesn’t have as many solutions as many elementary students believe it does. I have asked the second question in multiple grade 3-6 classrooms this year. So far, there was one student who was able to immediately identify that 10-1-1 triangle is impossible. Triangle inequality theorem is not a part of elementary Alberta Program of Studies for mathematics. Why do I believe it is important for students to realize that not all combinations of numbers can describe triangles?
For me, it comes to the goals of our geometry lessons. There are countless worksheets around the internet that ask students to add the values of the sides to find the perimeter. How often do students have an opportunity to visualize these numbers and make sense of them?
When I asked grade ¾ students to work on the second triangle question (finding all possible triangles with the perimeter of 12, whole number values), there were some interesting answers: 6×2=12 and 4+8=12.
“Can you draw this triangle and label the sides?”
“I don’t want to draw, I can do multiplication and it is faster”.
“How many sides does a triangle have?”
We have used a paper strip folded into 12 equal sections. A challenge of paper was that the students tried to push the edges together. Some were unconvinced that 6-3-3 triangle is impossible because “you can pull these a little closer”.
Simon Gregg mentioned matches and created a beautiful Cuisenaire summary of all triangles with the perimeter of 21.
Upsilon fan shared gr3 investigation
My colleague Nicole who teaches grade 1/2 in Calgary, started with asking students to build the triangles with Cuisenaire rods and count the length of their perimeter. I wonder if before posing perimeter questions it is useful to explore triangles freely, build them with different sides and connect the constructions to numbers. Students in her classroom have also built triangles with the length of each side that equals 10 (with the purpose of practicing the “friends of ten”).
Trouble with Triangles 2
How can we find the sum of the angles of different polygons?
I picked this question for the similar reason as the first one: I hoped that it will give students the opportunity to notice patterns in shapes, to use these patterns to figure out what we don’t know based on what we know, and to consider how the numbers that describe the shapes (angles) can be used to make generalizations. Students have already learned that the sum of the angles of a triangle is 180 degrees. Many were aware that the sum of the angles of rectangles is 360 degrees, but they did not extend this rule to the quadrilaterals with non-90 degrees angles.
I started with drawing shapes on the board and breaking them into triangles until someone said they know the rule I use to break the shapes. We had fun breaking the shapes (“you got my rule” – “you didn’t get my rule”), until we settled on some guidelines: straight lines from one vertex to another, no intersections, shapes are broken into as many triangles as possible. I left the students at that being very determined to not spoil the work. Besides a couple of students, everyone was stuck, so I had to suggest to draw a few shapes, break them and see if they can notice any patterns. It was interesting to see how “let’s just try things and see if we can notice some regularities” helped students to get on the path to the solution. By the end of the lesson, most of the students were able to notice the relationship between the number of triangles and the number of the sides of the polygons. Looking for the patterns and conjecturing about them was the most difficult part of the work.
Questions and Thoughts
Geometry in elementary provides many opportunities to get students into the habits of mathematical reasoning: generating data, looking for patterns, making conjectures, testing, proving, refining and communicating them. In my previous years, if I connected measurement to geometric properties, it was usually in a descriptive (the edges are the same length), not a problem-solving context . I wonder how we can make elementary school geometry more about describing relationships than describing only objects. And how do we find questions and problems that can help to explore the relationships between the shapes as visual objects and numbers that describe them? Until now, I have been stumbling upon these questions, but I want to be more intentional about them now.
I also hope to recognize when students ask the questions that have the potential to explore these connections. Here is one from my last year grade 3 student: What is the smallest number of faces a polyhedron can have?