Fiddling+with+Math+-+page+2

On the course, the subject of how to encourage kids to try stuff and not worry about making mistakes seems to crop up quite frequently. I think our whole attitude to this has to change. It shouldn't be a case of 'if at first you don't succeed, try, try again', it should be more like the attitude in Blaze's Optics Lab in Zap - if you fiddle enough, you'll finally figure it out! Excuse the bad editing - Jing didn't like this software...

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Just what I needed - I woke up with a migraine this morning only to discover a new challenge to play with. I couldn't resist it though, and worked through the construction to see if I could figure it out. I couldn't really think very straight, but it looked to me like line DH crossed Y=1 half way towards where we wanted it to. H was halfway between A and F so I thought maybe if I drew a line from D all the way over to H instead of just half way, maybe it would reach twice as far over all the way up and intersect with the folded diagonal at a point a third of the way up. It did! I thought it might be something to do with 30-60-90 triangles, so I checked the angle, but my intuition was way off this time. By this time Alan was surfacing, so I undid a couple of moves and showed him the challenge. He looked puzzled for a minute and then suggested drawing a line from D to F because DH didn't go far enough over. I got called away at that point and when I got back he'd drawn in the other diagonal and the line CF and marked the relevant intersections. I drew a 'fold line' that joined those two intersections and that seemed to be the final solution.

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Alan clicked the construction back to the beginning and watched it right from the start as it built up, and then looked a little confused and asked,

"What's line DH for?"

"That's for giving you a clue!" I told him. Well done Linda - that clue was just enough so he didn't have to think so long he lost interest!

We haven't done the math on why it works, but GeoGebra made it pretty easy to spot, and then test, the solution. GeoGebra seems to allow you to use your intuition without allowing you to get carried away trusting it too much. Also, for the first time ever as far as I'm aware, Alan spent a few minutes gazing at the construction and became quite absorbed in the various triangles and the way they interacted to form a kite. We still haven't had a proper fiddle together on the rhombus challenge - hopefully soon!

A few weeks ago, Alan had a problem with one of the examples in his GCSE maths book and thought there was an error in their calculations.

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"Mum," he said, "the question here says to show that the x-coordinate of each point of intersection of the line y=2x+2 and the circle x2+y2=8 must satisfy the equation 5x2+8x-4=0 and hence find the coordinates of the points where the line y=2x+2 cuts the circle x2+y2=8, but there seems to be an error in the example in the book. Can you check it for me?" ======

I'm afraid I didn't have a clue what I was doing but we worked through it together and it turned out that Alan had expanded his brackets incorrectly and the book was perfectly correct.

I just experimented putting in the relevant equations into GeoGebra - it would have been so much better if we'd 'checked' it like that at the start!

I think I'm becoming a convert...



This is the sort of widget Alan used to make to test his 'times tables' on Widget Workshop when he was seven years old. I'd forgotten how good that game was! It was an incredibly good 'hands on' bit of software which gets kids thinking like programmers. I think it was in Conrad Wolfram's TED talk about teaching math using computers that I heard that the best way to ensure that kids understand the process of a calculation is to get them to program a computer to do it for them. Building widgets like this is a great way of starting them out on that path!

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I believe that the original release came boxed with an activity book. I had a later, cheaper release but the built-in puzzles seemed to cover all the essentials. I'd still love to get my hands on a copy of the book though. Here's a sample of the puzzles...

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Alan is currently working through a chapter in his GCSE textbook called 'Advanced Mensuration.' I don't know who thinks of the names of these chapters, but Alan is a typical 15 year old and there's no way he can take a word like 'mensuration' seriously. He refers to this chapter as the menopause chapter and is rather of the opinion that it's a bloody joke thought of by some thick clot. Please - whoever names the chapters, remember that there is more going on in a teenager's head than just maths and choose a title they can take seriously! What's wrong with 'Measuring'?

IF YOU'RE HERE FOLLOWING LINDA'S LINK - THE BIT ABOVE IS WHAT YOU'RE LOOKING FOR!!

We managed to find time last night to try to figure out why the 'fold a square into thirds' thing works. It took us ages, and it pretty well stumped us both, especially Alan. We decided in the end that it's something to do with the slope of DF being -2. So if you look at the triangle DFA, you can make similar triangles by sliding F up along towards D. And, because they are similar, if you slide it a third of the way 'up', it will also be a third of the way 'along'. A third of the way 'along' would cross x at 1, as it starts out 1.5, and a third of the way 'up' would cross y at 1, as goes from 0 to 3. This point, (1,1) is also a third of the way up the whole square, and also a point along the diagonal AC. So where they cross is where (1, 1) is, and marks a third of the way both along and up the whole square.

The geometry would work whichever way 'around' you have the square, so repeating the folds 'on the other side' will produce another intersection which marks another third marker. Two third markers (assuming you only turned the square through a quarter turn, not a half turn) will allow you to fold along between them along a third.



I guess if you played with the equations of the lines you'd find they could both be solved by x=1 and y =1, but text books always seem to work from the other direction. First find the equations/slopes, then figure out an apparently useless problem based on them. This way round you start off with the problem and the importance of understanding slopes, similar triangles and equations gradually dawns on you after you've figured it all out, and it all seems a whole load more relevant. If you like folding bits of paper ;-)

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