Regular readers will know that I love a good puzzle. I love all maths problems, but ones which make me think and get me stuck a bit are by far my favourite. The other day Ed Southall (@solvemymaths) shared this little beauty that did just that:
I thought “Circles and a 3 4 5 triangle – what an awesome puzzle”, I reached for a pen an paper and drew out the puzzle.
I was at a bit of a loss to start with. I did some pythag to work some things out:
Eliminated y and did some algebra:
Wrote out what I knew:
And drew a diagram that didn’t help much:
I then added some additional lines to my original diagram:
Which made me see what I needed to do!
I redrew the important bits (using the knowledge that radii meet tangents at 90 degrees and that the line was 3.2 away from c but the center of the large circle was 2.5 away):
Then considered the left bit first:
Used Pythagoras’s theorem:
Then solved for x:
Then briefly git annoyed at myself because I’d already used x for something else.
I did the same with the other side to find the final radius.
I hope you enjoyed this one as much as I did!
Recently Ed Southall shared this problem from 1976:
I’m not entirely sure if it is from an A level or and O level paper. It covers topics that currently sit on the A level, but I think calculus was on the O level at some point. Edit: it’s O level I saw the question and couldn’t help but have a try at it.
First, I drew the diagram – of course:
I have the coordinates of P, and hence N so I needed to work out the coordinates of Q. To do this I differentiated to get the gradient of a tangent and followed to get the gradient of a tangent at P.
Next I found the equation, and hence the X intercept.
And then, because I’m am idiot, I decided to work out the Y coordinate I already knew and had used!
The word in brackets is duh…..
Now I had all three point.
It was a simple division to find the tangent ratio of the angle.
The next 2 parts were trivial:
And then I misread the question and assumed I’d been asked to find the shaded region (actually part d).
Because I decided calculators were probably not widely available in 1976 I did it without one:
I thought it was quite a lot of complicated simplifying, but then I saw part c and the nice answer it gives:
Which makes the simplifying in part d simpler:
I thought this was a lovely question and I found it enjoyable to do. It tests a number of skills together and although it is scaffolded it still requires a little bit of thinking. I hope to see some nice big questions like this on the new specification.
Edit: The front cover of the paper:
Recently I saw this picture from Ed Southall (@solvemymaths) and thought it interesting:
It is an O level question on Venn Diagrams from 1988. I had a go at it.
The Venn itself was easy enough to fill in and the forming and solving part followed nicely.
As did the rest.
Having gotten used to A level statistics this was relatively straightforward, it manages to test use and knowledge of Venns but doesn’t go as far as probabilities.
I like Venn diagrams and I think questions like this are a good start point to build on, students who can do this will find A level Venns much easier. I assume that this style of question may be what we can expect from the new style GCSE, and even if it’s not its certainly something I intend to use with my classes.
Recently I saw this from brilliant.org on Facebook and it struck me as an interesting problem:
the first solution is trivial and obvious:
But the Facebook post said there was two, so I set out in search of the next one. As there were exponents I thought I’d take logs of both sides:
Then realised I could take logs to base X and make things a whole lot simpler….
So x = 9/4
As you can see it reduces to an easily solvable problem, and all that was left was to check the answer:
A lovely little problem that gives a good work out to algebra and log skills.
What an emotional few weeks. This time of year is always emotional, but this year that has been ramped up to a whole new level. There is all the usual emotion of Y11 and Y13 classes finishing the year, and this year that has been compounded by the fact that I am leaving my current school at the end of term.
I’m sad that I won’t work. With some of my colleagues anymore and I’m sad that I won’t get to teach some of my classes next year. On the flip side, I’m excited by the challenge that lays ahead and I’m excited by the fact I’m going to be working with some former colleagues and friends again.
Then I’m devastated by the referendum result. I thinks it’s a disaster for the country for so many reasons. The economy will suffer, the rich diverse culture that we have in Britain will suffer, it will affect touring musicians which may mean many UK based ones will give it up and less overseas stars grace our shores.
Then there’s the rise in hate crime. In the first week after the referendum there were 300 reported hate crimes against non brits. Up from 60 on a normal week. I find both those figures abhorrent, but the larger one particularly so. To me it shows that the racist and xenophobic underbelly of our society now feel they have been legitimised. It was always going to happen they way Nigel Farage and his cronies have spent the last two decades selling the EU debate as “we want our country back”.
This post was written prior to Michael Gove being knocked out of the leadership contest. It was first published here, on Labour Teacher on 8th July 2016.
Way way back in the days of the ConDem coalition, we had an education secretary named Michael Gove – a man who very soon could be our prime minister. Give polarised opinion within tele profession. Many chastised everything he did, and other rushed to defend his ideals. There were some, like me, who took each idea on its merits, chastised some and celebrated others. (You can read some of my thoughts on his tenure here.)
One of the ideas he had that I liked was the idea of a single exam board. We had a situation where it.was considerably easier to gain a C in maths on some boards than other and that, to me, seemed quite ridiculous. This idea was quashed before it started due to “EU monopoly laws”.
Last week after a long campaign I was left heartbroken by the decision taken by the (slim) majority of the country for the UK to leave the EU. I had looked at the pros and cons and am certain that remaining would have been the better option. I tried to find positives, but there were few. People celebrated the fact we would no longer have Cameron (a man who I generally detest) in number 10, but even this was a negative as the names I the frame to replace him make him look a much more reasonable option. Some of the folk in the running make him look positively Marxist.
So I continued to look for positives, and I remembered the idea of a single exam board. Surely this would now be back in the table? Especially if, as I suspect will happen, Gove wins his parties leadership?
This would mean students from around the country were all sitting the same exams and we sold have a situation where you knew exactly what each grade means. I’ve got my fingers crossed.
Today I discovered another. I was looking over Q D1 exam paper and came across this flowchart:
The questions were all fairly reasonable and one of my students was completing the question to see if he had for it right. Afterwards I asked if he knew. What the algorithm was doing, he wasn’t sure at first but when another student explained it was finding the product of x and y he realised.
Then he asked, “but why does it work?”
I looked at the algorithm and initially it didn’t jump out at me. I tried the algorithm with 64 and 8.
I could see it worked through mocking factors of 2 from the left to the right but this time there was no odd numbers, so I picked some other numbers:
And that’s when it all made sense. Basically, what’s happening is you are moving factors of 2 from x to y thus keeping the product equal. When x is odd, you remove “one” of y from your multiplication and put it in column t. Your product is actually xy + t all the way down, it’s just that until you take any out your value for t is 0. T is a running total of all you have taken from your product.
The above becomes:
40 × 20
= 20 × 40
= 10 × 80
= 5 × 160
= 4 × 160 + 160
= 2 × 320 + 160
= 1 × 640 + 160
= 0 × 640 + 160 + 640
= 0 × 640 + 800
I tried it out again to be sure:
This is an nice little multiplication method that works, I’m not sure it’s very practical, buy interesting nontheless.
Have you met this method before? Have you encountered any other strange multiplication techniques?