I’m a big fan of triangles. Not something I openly share on my CV or twitter profile, but there you have it.
It makes a cool display, and, geekily, can be arranged into a triangle.
My favourite on the list is Calabi’s Triangle (below). There are only two triangles that can fit the largest possible inscribed square in three positions. The equilateral triangle, and Calabi’s triangle.
This may seem like a huge tide turn from the rather complicated Sine, Cosine and Tangent posts previously, but it’s an area that is relatively simple, yet often taught incorrectly. The devil is in the detail.
#1 The Parallelogram
This seems to cause all sorts of errors, particularly spelling (check out the picture above!). I’m speculating it’s partially due to the ‘well it’s so straight forward I don’t need to check’ frame of mind. Perhaps.
Anyway, here are the most common errors I have encountered.
1. All four sides can’t be equal, it has two ‘sets’ of equal sides.
Noop. The second part is right, but that doesn’t imply that all four sides can’t be equal.
2. Squares are squares, not parallelograms.
Noop. Squares fall into all kinds of categories. They definitely ARE parallelograms.
3. Rectangles aren’t parallelograms, they’re rectangles.
Noop. Half truth. Rectangles are indeed rectangles, the rest is wrong.
4. Diamonds aren’t parallelograms.
Whoah whoah whoah whoah sweet child o’mine. Diamonds aren’t a maths thing, they’re a girl’s best friend. Us men, we get dogs. Unlucky. Anyway, stop that right now. If you mean this thing…
a… RHOMBUS. Then no, you’re wrong. It’s still a parallelogram. :p
So what makes a parallelogram a parallelogram? (*did I just use Epanalepsis there? I’m not sure)
Well, here’s a weird thing. Wikipedia (rightly) states that squares, rectangles and rhombi are parallelograms, yet look at their list of properties:
Pretty sure a square has rotational symmetry of order 4, not 2.
Anyhoo, Wolfram is a bit more reliable and says this instead:
That’s better methinks.
As a random aside, did you know that if you connect the midpoints of ANY quadrilateral, regular or irregular, you form a parallelogram? Weird.
#2 The Square
As I hinted at earlier, the square falls under many different classifications.
Technically speaking it is a…
square, parallelogram, rhombus, rectangle and a quadrilateral,
Maybe this will help:
There’s a curious debate as to whether parallelograms are in fact degenerate trapeziums. Some definitions say a trapezium has only one set of parallel lines, others say a set of parallel lines. I’d have to say that if you’re going to accept squares as rectangles and equilateral isosceles triangles then… welcome to the party!
*edit: there’s also an odd confusion surrounding trapezoid definitions across the waters. It is nicely summarised here.
You might be wondering about the rectangle thing. Well, a rectangle’s strict definition is
Stop teaching the area of a square as base x height too. It’s (side)2. Then maybe students will understand why we call it squared, and stop worrying when asked to find the perimeter or area of a square when given only one side. Also, the diagonal of a square is not the same length as the sides. Use Pythagoras if you don’t believe me.
#3 The trusty triangle
“There are four types of triangle”
No there aren’t. There are dozens! To be fair I’m just being pedantic. I think we should teach students about all the COOL triangles, not just the simplified classification system.
“All triangles have internal angles that sum to 180 degrees”
…except this one. God bless non-Euclidean Geometry.
Also, I’ve seen teachers incorrectly teach the scalene triangle as “a triangle with none of the properties of the other triangles”.
Prepare to have your mind blown. This is a scalene right-angled triangle:
While we’re at it… This is an isosceles right-angled triangle:
Each type of triangle is not necessarily mutually exclusive. An equilateral triangle is technically a ‘special case’ isosceles triangle. It has two equal sides. It just happens to have another equal side too. The only property needed to fulfill ‘scalene’ is that all sides are different lengths. Angles don’t even get a mention. Although three different angles is implied by three different lengths.
#4 Heptagons and Nonagons
These two shapes get a really rough deal. They’re almost NEVER used at school. I blame the internal angle of a heptagon (a nasty decimal) and, well, the nonagon has no excuse. WordPress is even underlining nonagon as an incorrectly spelt word. Do these shapes a favour and talk about them more! Also, try my heptagon puzzle while you’re here.
#5 Area annoyances
I’ll close with this: Think about how confusing it is for our poor students that we frequently interchange the following when describing volume and area:
base, height, width, length and depth. These are all subjective terms. Be mindful.
Also, wouldn’t it be easier if we just taught students how to find the area of a triangle, then applied that to all shapes. All 2-D shapes after all, are made up of triangles. Even circles, kind of.
Part 2 in my series of ‘complementary maths’ – extra bits that complement the GCSE curriculum to hopefully aid student understanding : Tan.
A couple of weeks ago I explained the origins and meaning of Sine and Cosine, and how they relate to the unit circle, chords and right angled triangles.
Tan is a fairly obvious follow-up.
“Tan” is an abbreviation of “tangent”. Tangent stems from ‘Tangens’ (Latin) meaning “to touch”. With regards to our unit circle (yes, it’s back), we’re referring to a straight line touching the edge of the circle. Not just any straight line though. This one starts at the base of our half-chord right-angled triangle (with hypotenuse = radius).
Well, almost at the base. The base of said triangle is *inside* the circle, so we must extend the line. Time for a picture.
Ignore Sec, we’re sticking with GCSE here. Although now seems like a nice time to mention that Tan, Sec, Cosec and CoTan are all derivatives of Sine and Cosine. Sine and Cosine are the main ones, and you can do any trigonometric operation using only those two. In practice that’s annoying hence we use the others in A-Level and beyond.
Anyway, I digress. From the diagram above, you can see the tangential line that we use for ‘Tan’. It starts from an extension of our Cosine line (adjacent to the angle – CE in the second picture below, point E being the intersection with the tangent), and finishes where it meets the extension of our radial hypotenuse (BD below is the extension, D is the point of intersection).
So, in the unit circle, TanѲ means the length of DE (in the diagram above) at angle DAE (which is Ѳ). As with Sine and Cosine, these values change whenever we change the size of the angle.
So you’ll notice we now have two similar right-angled triangles.
We also know that using the smaller triangle ABC, the tangent of Ѳ is BC / AC, which in this unit circle is simply SinѲ / CosѲ
Hence TanѲ = SinѲ / CosѲ
Furthermore, if we use the larger triangle ADE, the tangent of Ѳ is DE / AE.
But we know AE is the radius, which in the unit circle = 1.
Hence TanѲ = DE,
Now, just like Sine and Cosine, if we plotted each value of Tan on a unit circle, you’d get the familiar corresponding graph. In the case of Tan, it looks like this:
So … why are there asymptotes?
Well, think back to how Sine and Cosine values are created using the unit circle. We need right-angled triangles. However, at 90 degrees, there is no triangle, there’s just a single line. Visualise that here. Or just look at the pictures below:
Now here’s where it gets a little weird. Technically, there’s no reason why Tan (90) can’t equal zero (*ducks bottle*), because, by the same reasoning(ish), Cosine (90) = 0, but that’s *allowed*. However, Tan 90 is considered ‘undefined’ because:
TanѲ = SinѲ/CosѲ and dividing by zero is world-endingly bad. So bad that we say it’s undefined.