Maths Teachers: 5 Reinvigoration Tips

Well, it’s Easter break so I thought I’d share some simple ideas to reinvigorate over the next two weeks (or week if you broke up for the holidays earlier).

  1. Get some rest

I don’t think many people will take issue with that one! Sleep in, recharge the batteries, get well if you need to.

2. Go outside, go for a walk

It’s so easy to whittle away a break by staying at home, watching tv and going online, but there’s nothing more invigorating (for me at least) than going on a long walk outside. Making time to think is bizarrely a bit of a precious commodity these days.

This is a great article about how useful a walk can be, and several mathematicians have attributed thinking time during walks to their major successes including Andrew Wiles and Yitang Zhang.

3. Turn off your phone for the day and stay off the Internet

Remember there’s more to life than twitter notifications and facebook likes. There’s a real world right outside the door and you have the time to see it! More importantly, I think we (at least, I) need a bit of a disconnect from time to time just for a sanity check.

4. Remember what you work for

What is the purpose of work to you? Whatever that purpose is, this is probably the time you should be fulfilling that purpose. Spending time with the family? Going out with friends? Spending some money? Whatever “the things” are, now is the time to do the things.

5. Reignite your love for your subject

I fear a lot of maths teachers don’t even like maths anymore, and probably don’t even realise it. Teaching the same curriculum day in day out for years and years will no doubt diminish genuine interest and curiosity, and replace them with routine and boredom. The best way to get your maths mojo back is to actually do some maths! No, not a GCSE question that you can figure out in 2 seconds because you’ve done them to death with classes for the last 10+ years, but a genuinely thought provoking question, or learn a new element of maths, like Pick’s Theorem or read this amazing book by William Dunham

 

 

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Second Proof: Area of a Trapezium (Trapezoid US)

I saw an interesting animation on twitter recently and decided to have a go at a formal proof (formal ish).

In essence, we’re going to chop a bit off the top of a trapezium and turn the whole thing into a triangle with the same height, and a base of (a+b). I stupidly called b ‘x’ in my scribblings, but you get the idea:

IMG_0983

Now extend AD by length ‘x’ and call the new point ‘E’, and create line BE:

IMG_0987

Angles CFB and DFE are clearly equal. Now box in BE*:

 

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We now have a rectangle, with diagonal BE. Hence we have two (green) triangles that are congruent, and hence angles EBH and BEG are equal. If two angles of these slim (red) triangles are equal, then the third must also be equal, and seeing as they both have an equal length side, they must be congruent.

IMG_0990

And if they’re congruent, then the area of the trapezium ABCD must be equal to the area of the triangle ABE.

The base of triangle ABE is ‘a’ + ‘x’ (imagine x is b… doh!), and the height of the triangle is the height of the trapezium, hence the area of the trapezium has the formula ((a+b)/2)*h

The Perimeter of L Shapes

When teaching perimeter, there are many, many resources at your disposal. It’s in most schemes of work somewhere for all ages, and as such many people have made worksheets and the like.

I have a bug bear with perimeter worksheets though. There is almost always a complete excess of information provided, often making questions more difficult than they need to be. This is particularly true when students need to find the perimeter of L-shapes (because “important skill”).

Now, when teaching area of a triangle, an excess of information is really useful, as it helps pick out the students who don’t know what they’re doing. Similarly, removing a lot of the information with L-Shape perimeter questions, the opposite is true.

Look at the examples below:

ma08014

perimeter 1

For each of the above, you can solve them with about half of the information given. Furthermore, by removing the excess information, you’re showing students something more important – that deduction is possible, and is useful.

Obviously this does not apply to every shape. Take this example:

areapic4

This wouldn’t work quite in the same way, as the 6cm cannot be deduced if it is removed. However, both the “3cm” can happily disappear.

This question popped up in my lesson last week:

l shape

Now, to me, this question is actually (pointlessly) more difficult with the 3cm height included. It infers that you may need to either dissect the shape, or start with the expression

h+(h-3) +3 +20+3+(20-3) = 72

rather than just using

2h + 40 = 72

Perhaps the question is being clever and expecting students to *see* the route with fewer calculations by ignoring some measurements, but the amount of space provided for an answer (not shown above) suggests not.

Silly perimeter questions.

 

Knowing the Answers Isn’t Enough

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I put this question on the board the other day as I found it interesting. The idea is students find the area of the triangle. It looks like a right angled triangle, and sure enough a lot of students assumed that it was.

One or two tested it using Pythag, and found it wasn’t. Those who tested it, applied the cosine rule to find an angle, and found the area using 1/2abSinC.

Those who were less meticulous, jumped straight into 1/2 base x height.

Interestingly, the answer sheet lists it as this:

answers arent everything

However, this is taken from a “Using Sine to find Area” worksheet, so students would likely have used the context to not use 1/2 base x height.

If I were less experienced however, I might be inclined to let those assuming this was a right angled triangle off the hook, as that gives an answer of 85.

The other way gives a decimal that rounds to 85. Don’t just rely on an answer sheet!