Here’s a new applet I created (simply because I haven’t made one in a while and thought it’d be fun). Look at how different angles result in different sunflower configurations by clicking the picture below.
Why is the golden angle 137.5 degrees? If you’re struggling to figure it out based on the standard derivation of the golden ratio then there’s a few videos on youtube that’ll help.
I was looking at some TES resources the other day trying to find new ideas for teaching mean from a frequency table. I found a few good ones so this post is just a collection of ideas for teaching averages, both from the TES (slightly extended or modified) and things I’ve picked up over the years. This was originally just supposed to be private blog post but I decided to make it public just to see if anyone else has more ideas to bring the topic to life or make it relevant.
At University, I did a full mathematics degree which covered around 36 modules – that’s quite a lot of maths! The weird thing about my degree (which is probably the case with most mathematics degrees?), is that most of the stuff we learnt was pretty darn old! To add some perspective, I think the most recent module was General Relativity which Einstein published in 1916. That’s strange! I know very little about maths which has been discovered/invented in the last 100 years!
With this in mind, I decided to look at some more recent stuff (1960’s/70’s) and bought a book on fractal geometry and chaos (James Gleick, Chaos: Making a New Science). It’s a popular Science book which is great as an introduction to the subject in terms of naming the key players and outlining the big ideas. As someone interested in mathematics (I never call myself a mathematician, as to me, this is someone who has or does contribute to the field), I needed to ‘branch out’ for a more in-depth understanding.
In this post, I’ll try to explain some of the basic mathematical principles behind Fractal Geometry from an IB/A-Level perspective. My knowledge only extends across two books, but hopefully this post will enable you to understand the most interesting and important stuff without spending weeks reading around the subject. I’ve enlisted the help of James Tanton and directed you to the Classic Iterated Functions Systems website to help with some of the explanations.
If you had to pick the most frustrating part of your Secondary School Maths education, what would it be? For me, it is hands down the trigonometric ratios. You can probably guess how it was taught. The teacher simple wrote SOH CAH TOA on the board, gave some examples and then set us off with work from the textbook.
I remember exactly how I felt at the time. Hold the phone, what are these things sin, cos and tan all about?
Almost all IB mathematics teachers are curious about how well students will take to the new guidelines for IB coursework. There are many aspects of this change which I could talk about at length, but I’ve decided to discuss one part which I feel needs to be considered to help to ensure that the exploration is, as the IB put it, “stimulating and rewarding” for all students involved.
Today I was skimming over Rob Eastaway’s excellent book, “Why do buses come in threes?” and found an equation for a situation in snooker which wasn’t derived in the book. With the white ball and object ball lined up to the pocket, a formula was given for the vertical distance M from the centre of the pocket.
The formula is based on the player being professional which means that the angle α, between the horizontal and the direction of the white ball, is very small :
M = α(P – B)(B – W)/W
This makes sense on a number of levels:
1) When α = 0, the distance M = 0.
2) When B = W, the white ball will be touching the object ball and so M = 0.
3) When P = B, the object ball is over the pocket so M = 0.
The problem, which would be the same for any mathematically interested person, is that it isn’t good enough for me to see the formula – it has to be derived. I’ve had a stab at it and was hoping that other maths folk could confirm my method. Here it is:
For small α, sin(α)≈α and l ≈ B – W
Using the sine rule on the left triangle: sin(α)/W = sin(β)/l (vertically opposite angles)
Hence: α/W = sin(β)/(B – W)
sin(β) = α(B – W)/W (1)
From the triangle on the right: tan(β) = M/(P – B) (2)
Since tan(β) = sin(β)/cos(β), combining (1) and (2)
M/(P – B) = α(B – W)/cos(β)W
Since for small angles, cos(β)≈1,
M = α(P – B)(B – W)/W
In this post I’m going to give a few thoughts about how I think chess can help students develop mathematically. I’ll also provide an idea for scaffolding students through problems or proofs.
Below is a Beginner level Chess problem from www.chessproblems.com. With black to move first, can you figure out a way for black or white to win within the next 10 moves? (Have a think about this for a minute before moving on)
Anyone who knows anything about Mathematics knows Euler’s Formula connecting the number of faces (F), edges (E) and vertices (V) of polyhedra.
F – E + V = 2
(A cube for example has 6 faces, 12 edges and 8 vertices so that 6 – 12 + 8 = 2).
But what if the shape has a hole in it? (Click the picture to go to the file)
Does the formula change? If so how does it change? And the most important question, why does it change?
Further to this, what if the the shape has 2 holes, 3 holes, …, n holes? How does this affect the formula?
Click on the image to go to the applet.
Teaching Mathematics 