Interleaving Algebra and Perimeter

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.

9-pin Geoboard

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.

Concentric Equilateral Triangles

The red equilateral triangle side length 4cm sits inside the larger pink equilateral triangle such that the “border” is 1cm wide.

What is the ratio of the height of the red triangle to the height of the pink triangle?

Can you solve using trigonometry or only using Pythagoras?

The border is now 2cm, whilst the side length of the red triangle remains 4cm.  What is the ratio of heights now?

Explore what happens for other border widths.  Can you generalise for any border width w?

Geogebra file here.

Spoiler here.

I got a few solutions posted on Twitter for this, but the most elegant (so far…) has to be this from @mathforge. As he said: no Trig, no Pythagoras, just ratio.

Reflections on Pythagoras

I was with a Year 10 class doing Pythagoras’ Theorem recently.  This was a low attaining group who had encountered it before but were shaky.  They had recently been doing rounding to d.p. and s.f.  They had made progress with that but were getting a bit bored so the teacher decided to go over Pythagoras which I thought was a nice way of interleaving topics.  Calculator answers needed to be rounded (for those triangles which were not pythagorean triples).

It got me thinking about the knowledge and understanding required to apply Pythagoras’ Theorem and also about planning topics. In this lesson, they stuck to finding the hypotenuse and didn’t do any problems where they had to find the shorter side. This decision was made by the teacher because she knew the class and the context. By the end all students had practised it a few times and had built self-belief that they could do it. In other contexts, another teacher might decide to introduce both cases side-by-side.

Is there a “best” approach?  Has there been educational research looking into such a finer point of teaching this one topic? If there has, please point me to it because I wouldn’t have the time to look for it.  Nor the belief that if I did eventually find something, reading it would actually improve the outcome of my learners.  There is no single perfect way of teaching any topic. As teachers, we need to keep our eyes and minds open to approaches that we hadn’t thought of or used before. But we must not agonise over trying to find the holy grail, the golden nugget that will suddenly enlighten our students.

As I wasn’t actually teaching the lesson, just helping out, I had time to draw a quick mindmap in the lesson.  I was trying to put myself in one of the students shoes.

`What are the things I need to know, understand and be able to do to be successful in this topic?`

This, fundamentally is what they care about.

I’ve included a picture of my mindmap here, not because it is a stunningly useful breakdown of this topic, but just to show the complexity of what our students need to learn.  And this is without much page space taken up on the “And then…” topics of proofs, pythagorean triples, etc., some may say the “interesting bit”.

I am going to try to do this mindmapping exercise more often in my lesson planning. I found it quite easy to do when I was sitting in this lesson, but I find it harder when faced with a blank sheet and maybe some resources and ideas that I might have used before.  The point is, I think, not to agonise over the perfect mindmap showing the perfect sequencing of “learning nuggets”. It needs to be done whilst thinking about the class and the context. And if there is ever time for such luxuries, doing it with a colleague surely makes the process more satisfying and enjoyable.

The paper napkin trigonometry trick with a smattering of Pythagorean triples.

Take a piece of paper and do the following:

1. Make it into a square (interesting discussion on best way to do this).
2. Fold in half then unfold so you have created crease along a vertical line of symmetry
3. Then take any corner and fold to the midpoint of the opposite edge. Press down to make a crease along the fold line
4. Unfold and now investigate all the triangles you have created, i.e. can you work out their lengths?

Here are some pictures, which also give some hints, although not a complete solution.  The result is very satisfying although I would love to find a way to show this that doesn’t require reams of algebra.  Any takers??

Proving Pythagoras

Although I’m sure I’ve taught Pythagoras lots of times, I have never really looked at the proofs before either for my own subject knowledge or with students. This may be because I was always happy when students had the understanding of how to apply the theorem and were able to find the missing side and so I left it at that.

Looking at proofs is a good way to deepen understanding of a topic, but generally shouldn’t be attempted the first time the topic is introduced, one of the points made in this comprehensive review of literature on how students approach proof in mathematics written by Danny Brown.

There are something like 140 different proofs of Pythagoras, cut-the-knot.org lists 118 geometric proofs here.

I decided to work through three:

Proof 1

On squared paper, students draw two adjoining squares of side length a and b as follows:

Next they draw diagonal lines.  The first thing that needs proving is that these two lines are perpendicular which can be done by finding the gradient of each of them.

We are now starting to get closer to a square of side c. A bit of cutting and rearranging and hopefully they establish that the area that they started with, a²+b² can be re-arranged to form c².

Here is a lovely Geogebra showing how these squares could tessellate for form Pythagorean Tiles.

Proof 2

This one is worth drawing although the scissors won’t help much here.  This is a Geogebra drawing of it (click on it to adjust the lengths):

A few of my students went down a blind alley with this one assuming that a is double b.  That is why it is useful to have the dynamic drawing to show that this is not the case. The crux to this one is seeing that the red square in the middle has side length (a-b) and then multiplying out (a-b)² to get the area of that square.

Proof 3

The third one I chose is fairly simple if you can remember the formula for the area of a trapezium! And really, once you’ve played this video over 10 times, nobody will ever forget that!

Pythagoras and Trigonometry Revision

I used these cards for revision of Pythag and Trig recently.

They are really nice and there are lots of them so you can decide how far you want to go. I just have the file and I’m not sure who created them, so if you know, let me know so I can credit them / link directly.

This sort of activity works really well with a class that will have productive and supportive conversations about the maths and enjoy challenging each other.  It gets them out of their seats and they start to get a sense of which questions are straightforward and which will present more of a challenge. I’m on the look out for similar things for other topics.

I’ll be setting this for homework as it has explanations as well as examples all in one place:

Hippocrates First Theorem

Another one from the fabulous Don Steward:

You could of course just go straight for the algebraic proof but it does require a level of confidence with surds.  So you might want to scaffold this task. Maybe start by putting some numbers in for the radius of the smaller semi-circle, maybe 2.  You could then do it again with 4 and ask students if they are convinced by that.  (Here are some examples to warn against the dangers of extrapolating what appears to be a pattern).  If you do take the numbers approach it’s good calculator practice.  Can you type the whole expression for the area of the curved shape into the calculator to get an exact answer?

And here is a little GeoGebra drawing to go with it.

A simple circle problem

My high attaining Year 9 class didn’t quite get this on their own yesterday but they enjoyed the challenge and were able to follow the explanation.

The crux of the problem is getting a right angle triangle with sides 1, (1+x) and (2-x).  It is the (2-x) side which is hardest to spot.  There were groans when I finally showed them.

Then it requires some algebra – namely expanding (1+x)² and (2-x)² which Year 9 hadn’t had much practice in, so it was good to show them why (1+x)²≠1²+x²

I gave them the problem printed out, here they are 2 to a page.

Here is link to the Geogebra file that I created this on.

I had a few Twitter responses to this including @ProfSmudge who kindly set us an extension question:

It’s an example of an Apollonian gasket, apparently (thanks to @mathforge for pointing that out!).  That gets properly hard, involving Cosine Rule.  Certainly not something I’d give to my students, but I’ve got a few teachers working on it!