2D Shape part 2: Triangles and Quadrilaterals

2D Shape Part 2: Triangles and Quadrilaterals

Hi folks!

This is the second part of my series on shape, find part 1 here.

We looked last time at the names of polygons in general. Today we’re going to delve into the two most frequently seen by our pupils, much of which must be memorised. Now to clarify, this blog is only going into the names and properties of the shapes in question; perimeter and area will be coming later.

We’ll start with triangles:

First point of contention, some people say to their students that there are four types of triangle. I say there are three, and that right-angled is a property rather than its own genre. Now, what can make a nice thinking point in a lesson, is to ask which types can have the right-angled property, and why. Then, how many right-angle triangles can be made with each type?

But first, we want to look at the three types of triangle, and how the language behind them can help our students’ thinking.

Equilateral helps us unlock so many words in later mathematics. The prefix equi- to mean equal comes in so many forms, all of which you can quickly demonstrate to your class. Show them an equation, ask what equivalent means (so they can also attribute -val to value), draw points A and B and bring someone to the board to show a point equidistant from them. Bring another student to show equality and inequality. Then you have -lat for sides. Talk about a bilateral agreement (which they will eventually see in History, Geography, Politics and Citizenship), talk about lateral thinking meaning to think about things from another side. Show them how translating means to cross sides, both with a language and with a shape (setting up transformations). Your learners don’t even have to write these down, just expose them to the power of these word stems to demystify higher order Tier 2 and 3 vocabulary.

Scalene is the opposite, and helpfully enough, its etymology literally means uneven. I’ll be honest, beyond that there aren’t a great many connected words to lean into. Bizarrely, it doesn’t even link to isosceles, despite the consonant cluster. However, introducing it before isosceles can help with remembering the spelling, coincidence or not.

Isosceles I would immediately define the whole word, not just the prefix. Reason being, another word for equal could be confusing with equilateral. If you come out of the gate with ‘equal legs’, you can very quickly get them to see where the Greeks were coming from. I’d draw them at all sorts of rotations, asking if they have equal legs or not. At that point we can dive into iso- (although really, isometric is the only key word to branch off here, albeit in two different contexts, both mathematical and biological). As I say, use scalene as a springboard with the spelling, so they remember that silent c. You’ll hopefully find this also helps narrow down incorrect vowels (I have often seen ‘isossoles’ as an incorrect spelling). As soon as the silent c is in play, many students understand which vowels can follow, and so will only have e and i as options.

Quadrilaterals are also chockful of language.

Granted, much of it isn’t immediately useful to students, as words either never crop up again or relate to properties already known. But for new teachers, or those coming into maths as non-specialists, they can be incredibly useful as aids to memorising the complex world of different quadratics. So, let’s rattle through them:

Square derives from a word deriving four. The same root went on another journey to become squadron, who at that point in history often formed square (those of you who’ve just had flashbacks to Sharpe’s Waterloo, you’re welcome). This is why the four becomes four equal.

Rectangle uses the root rectus, meaning upright. This is where erect stems from, so rectangle literally means ‘right-angled’. Interesting to note here that being bolt upright was later seen as being ‘square’, hence the symbol for a right-angle being a square, and a set square not having to itself be a square.

Parallelogram is one that surprised me a little, from the suffix. Gram actually comes from another word for a line, so parallelogram literally just means ‘parallel lines’. Thinking about it though, this is what separates a pentagram from a pentagon. A pentagram is only concerned with containing five lines, whereas a pentagon insists upon exactly five interior angles, which the pentagram doesn’t always have. This makes even clearer the angles-first approach that should be taken with polygons.

Trapezium is one I absolutely loved when I first heard it. To truly get this, we should first realise that many trapezia have the long side on top, not on bottom as at least British education seems to prefer. Once you have that orientation, you can see why the Greeks named the trapezium after a table! Another lovely piece of ‘say what you see’.

Brief side-note here, I would very much encourage you to show pupils multiple types of trapezia here, to show that they are not always symmetrical, and that distinction is awarded only to the isosceles trapezium (callback to the triangles here which should be pounced upon, as a table with equal legs!

One that we don’t come across much, but is a part of this branch of the quadrilateral tree, is the trapezoid. That may shock many of you, who thought it was merely the American word for a trapezium. However, both systems contain both terms, we just us them the other way round!

The one sure characteristic of any trapezium is having a pair of parallel sides. The trapezoid is merely taking away that one trait. Much like the human is a specific type of humanoid, the trapezium is the ‘nice’ version of the more generalised trapezoid…unless you’re American, then it’s the other way round, go figure.

A rhombus is one I’ll confessed I never looked up until the other day, but again for teachers to rationalise it’s a good one. A rhombus is a spinning top! This explains the equilateral nature, without the need for right-angles.

Finally, this is more for primary colleagues, yes the kite is named after the toy, but it’s useful to know that the toy is named after the bird, due to its hovering nature.

As briefly alluded to, I make two ‘family trees’ of quadrilaterals in students’ books, whereby we have annotated diagrams, and bullet-pointed properties to show how specificity is removed to get from one to the next. If anyone would like me to detail this is a future post, please feel free to ask.

The first branch is square -> rectangle -> parallelogram -> trapezium (optionally then onto trapezoid though I personally don’t).

The second branch is square -> rhombus -> kite

Branch 1 focuses more on the nature of angles, direction and parallel lines. Branch 2 focuses on symmetry and side length.

I hope you all enjoyed this deep-dive, and if you have any questions, alternative methods, or other feedback, please do send it by any means, public or private. Next time, we will be looking at circles, and different parts of shapes, before we move on to area and perimeter. Oh, and heads up, I’ve scheduled it for 10 o’clock as it’s a bank holiday…you’re welcome. Thanks everyone, see you next week!