I made these a couple of years ago as a quick way of getting students onto an activity at the beginning of a lesson, or to use up any dead time at the end of a lesson. They include a huge range of “the basics” and various maths puzzles. I used to print them as A5 booklets that each student would own in their packs.
The trisection of an angle using only a compass and straight edge was a famous problem in Ancient Greece. It was proven to be impossible by Pierre Wantzel in 1837, and proven possible using origami later (anyone know by whom?). Origami isn’t allowed though for Euclidean Geometry.
However there are some angles you can trisect using Euclidean Geometry. I demonstrated how to construct an angle of 30 degrees a few weeks ago, and perpendicular lines before that, so trisecting 90 degrees is clearly possible.
Here’s another fun way, which uses the construction of a pentagon. If we join the vertices of a pentagon, we trisect the internal angles.
So here we go:
Step 1: Arbitrary Circle
Step 2: Perpendicular to the diameter (use two identically sized largeish circles, or arcs if you want to be neat and tidy):
Step 3: Draw in a perpendicular bisector of the radius (CO in this pic)
There’s a lot of lines now, here’s a crude “what I did in the pic above”:
Step 4: Draw a circle with diameter CO:
Step 5: Draw a straight line from B going through D, and thus creating point E intersecting the circle, and a second new point at the further edge of the new circle, point F. Then draw in two circles, one with radius DE and one with radius DF
Now we can just join up the points to make up a pentagon as shown (click for enlargement):
Now draw lines to join the vertices, and you’ve just trisected a 108 degree angle:
It’s a big mess, so I did a neater version without most of the construction lines:
When I trained to be a teacher, one of the most surprising things was how badly I seemed to have been taught at school. Finding out about all the different things I was supposed to do and say, and how to act and react made me unsurprisingly reflective upon my own school days, and coming up with very little that looked or sounded like what was supposed to have taken place.
“Learn it for homework” was a phrase that was often thrown around when I went to classes. I’d sit and listen and take notes in Chemistry, and I had to double underline titles and dates, and draw margins. I remember little else, but always at the end of the lesson I’d be told to “learn it for homework”.
In the last ten years or so I’ve used that phrase anecdotally to ridicule the notion that learning wasn’t taking place in the classroom, and that the teacher’s job was supposed to be to help me learn something, but that the emphasis was being put on a solo experience at home rather than the teacher themselves.
But the idea that learning takes place in the classroom is problematic too. Without getting into too much of an epistelogical debate, the notion of learning simply being some kind of enlightenment regarding a concept or passing down of knowledge is flawed.
Time for a quick anecdote.
About five years ago I took it upon myself to learn Japanese. Yes, learn Japanese. How hard could it be? Turns out, hard enough to give up quite quickly. I started by trying to learn the basics. I wanted to speak it, so I listened to a lot of audio, and I wanted to recognise it, so I needed to associate sounds with unfamiliar symbols. Predictably, I ended up with a series of symbols written with phonetic pronounciations and definitions, and I set about trying to memorise them. After about thirty minutes of saying them out loud, writing out the symbols over and over, covering them up etc I had decided I could remember some of them quite well. The next day I remembered only one or two, and so set about re-learning them, or at least, perfecting my memorisation of them. A few days later and again I could barely remember much, but they came back to me quicker when I revisited them. This of course falls exactly in line with what many people have already discovered about short and long term memory. But where is the learning taking place here? Was it the acknowledgement of the meaning of the symbols? Was it the ability to pronounce them in the same way as the audio? Was it the ability to recall these things? Was it all of these things combined?
Now let’s put that idea into the classroom. We teach in a variety of ways, and whatever strategy we use, the whole process can be (arguably) boiled down to this: we want students to understand and apply concepts and ideas to different situations successfully, and without help. The “without help” element may seem disagreeable, but our end product is an examination grade, and there’s no assistance given in exams beyond the text provided.
Now is it realistic to assume that just because a class full of bright eyed students can repeat what you showed them, apply it in different situations, and explain fully what it is all about in the context of that single hour that they have learnt it?
If we revisit it a week later and no-one remembers anything, have they learnt it? At the time it was taught, perhaps they could handle the most difficult questions you could throw at them. The concept started as foreign, but became familiar and comfortable, all within the space of 60 minutes. That in itself is commendable, and is without doubt an indication of a great teacher. But it isn’t the end. Recollection of this newly acquired knowledge is a key element of learning. In fact, it’s arguably the most important. If you can’t remember it, you won’t pass your exam.
We seem acutely aware of this discreperancy with our weakest students, who “never remember anything”, but perhaps we also seem to assume that those who progress quickly with maths are somehow immune to this key stage of learning – memorisation and recall.
Furthermore, where does the responsibility lie with this key element of learning? Is it the teacher’s responsibility to embed memorisation of what has been learnt? Or should it be learnt for homework?
I am working with a small handful of (thankfully grateful) year 10s during registration time at the moment. We are concentrating on complementing topics covered in maths lessons, to try and help them cope better during class time. Currently the topic is proportion.
We began by talking about enlarging shapes, and how when a square is enlarged, how can we recognise the new shape? Will it still be a square? What about when we enlarge circles? This in turn led to a more general idea of proportion, and with it, the difficulties in spotting an enlargement of a typical rectangle, or a triangle. The bigger shapes are still rectangles and triangles respectively, but not necessarily enlargements. The process feels more complicated, because it is. Anyway, I digress.
Over the last few weeks we’ve covered enlargements and ratios. The students found both topics a little challenging at first, but picked up the ideas, and I managed to find some methods that worked well with them to systematically find answers to fairly diverse questions such as “If I have x money, and the total is split into this ratio, how much have you got?” and “x money is divded into this ratio, how much do you have, how much do I have?”, and “x money is split such that I have this much and you have that much, what is the ratio?”.
Each question was being answered confidently eventually, albeit quite slowly. I could live with that. But what bothered me was that all the students kept making silly mistakes. Not with the method, they had a method they understood and could use, but still they’d come up with the wrong answer. Each of them, at different times, kept getting something wrong – but still, the methods were always perfect. So what was happening? They would talk me through their thinking, and the things they were saying they were doing to the numbers were correct. Those descriptions would yield perfect answers. However the students were simply doing the calculations badly.
It became clear that this was not an issue of mathematical understanding, it was an inability to calculate. The more I explored this, the more saddened I became. These students, as with so many I’ve come across in my career, had no recall of any multiplication table at all. Not even the twos. As such they relied entirely on guestimating, which frankly only works even marginally well if you already have a good understanding of your times tables anyway! Explaining that the two times table only ever produces even numbers, and demonstrating it with dots seemed like a revelation to them. Explaining further that *all* times tables of even numbers only contain even numbers, and linking that back to the two times table, again seemed like some kind of dark secret nobody was supposed to know about. We then looked at the 9 times table, and logically deduced that each sum 9 x n must be close to 10 x n (we didn’t use algebra, just to be clear!), and must be less than 10 x n. Therefore it’s quite easy to figure out what the first digit of the number will be. For 9 x 4, it must be 30 something, and 9 x 6 must be fifty something etc.
Again, light bulb moments. One student exclaimed joyfully (after further exploration of the 9 times table I should add) “I can do it! I know my 9 times table for the first time!”. I should have felt happy for her, but still didn’t. Performance, afterall, is not learning. If she remembers next time I see her, which will likely only occur if she practices fairly regularly, then maybe I’ll share her confidence. I hope she does.