Over the last couple of years we have organised our Year 7 curriculum so we do some introductory algebra early on. Forming expressions from words and collecting like terms, would be topics that I would put in the introductory bucket. The benefit of this is that it can be interleaved into various other topics to extend thinking and promote generalising. Perimeter is an example of this, when we can give side lengths letters instead of numerical values.
An alternative to presenting a bunch of text-book type questions is to investigate a simple 4 piece Tangram, as described in this task from Mike Ollerton.
(Click on the image to access the full Word document)
The task as presented is primarily an exercise in shape, but I might use a slightly modified version of the Tangram to focus purely on perimeter.
Before asking students to cut the triangles out of the shape, we might agree on some labels for the side lengths. If we focus just the shorter sides, we could call Triangle A’s short lengths a, Triangle B’s short lengths b, and Triangle C’s short lengths c, so we end up with something like this.
Again, before getting to work with the scissors we could discuss how we might describe the hypotenuse lengths in terms of a, b and c. And in fact, if we really need c at all…
Using this notation for the side lengths, we can then cut up the 4 shapes and generate many other shapes and find their perimeters. Using 1 shape, 2 shapes, 3 shapes, all 4 shapes: what is the shape with the longest perimeter? The shortest perimeter? What is the difference between longest and shortest? What shapes are different but have the same perimeter – can we prove this using algebra?
It hopefully presents a need for “collecting like terms” as well as some introductory practice in using the technique.
A second task that links algebra to perimeter uses a 9-pin geoboard. This sounds fancier than it is. You don’t need the actual boards, students can create their own in their books or you can give them some dotty paper. First, we tell students that we are going to make triangles using only these lines a, b, c. This is a key image that we will need to refer to either on the board or on a handout.
The next task is to construct triangles using various combinations of these lengths. Each triangle must fit within the 3×3 array. Depending on the class and your objectives for the lesson, at some point you show them that there are only 8 “different” triangles. An opportunity here for a discussion on what we mean by different and what congruency is.
Once we have these, we can go through the process of calculating the perimeter for each one using the a, b, c notation we introduced earlier.
There are some options from here. Mike’s suggestion is that we order these from smallest to largest. We could do that by just looking at the shapes and having a guess. It’s pretty obvious that C is smallest although some of the in-between ones are harder to see. We could establish that a<b<c in the first picture (again by looking). Then we would also need to decide, for example which is bigger: 2a or c. With an older group, you might even use Pythagoras and express c and b in terms of a using surd form.
By looking at the differences between each shape’s perimeter, we start dealing with negative quantities of a, b, and c. If we then sum up all those differences, we should end up with an expression the same as the difference between the smallest and largest, with the c’s cancelling out. Which is quite satisfying and obvious if you think about the expressions lined up on a number line.
There is a fair degree of flexibility within tasks like these and I believe that as teachers we need to select carefully what routes we expect to go down in a lesson. There is a danger that we try to encompass too many different topics in one go and if all of these topics are new to a class then they (and you!) are likely to lose track of what they are actually meant to be learning in this lesson. However, if you are confident that the learners in your class are secure with certain concepts (in this case collecting like terms with negative coefficients) then it is a good way to consolidate and practice this knowledge whilst pushing into new territory.