This is just such fun. Art Benjamin’s calculating show is just remarkable. Of course a great question is how he manages to do what he does. There is obviously a lot of skill, brain power and practice but also some very sophisticated techniques that we start to see at the end of the video. Could this be a good Math Exploration? It might be although we would need to know that the topics are “requisite with the programme”
It has been a couple of weeks since I last posted about our Balancing Matters problem. During that time I have been reflecting on the problem. I did nothing new in terms of actual writing but shortly after the last time I posted I had a break through.
The problem of keeping the “liquid area” constant was getting messy but then it became obvious that with some fairly basic trigonometry and the formula for the area of the trapezium I had a fairly elegant solution on my hands. Here it is.
It also became obvious that when writing up the exploration I would need to begin with the conservation of area. You can see the write up of the “conservation of area” problem on this page.
Three important points to take from this stage on the process.
The working that I did at the beginning gave me the fertile soil to think and reflect on the problem mentally. This meant that when I returned to the task I had a clear strategy to follow and a breakthrough to explore. This takes time and having some thinking time with no writing is a good thing. Remember that the unconscious mathematical brain is often smarter than the conscious one.
If you look carefully at the working done on the conservation of area you will notice that there are two key physical restraints that allow us to make progress. The first one is that area is conserved. This is of course the 2 D equivalent of the fact a liquid, like coke, cannot be compressed under normal circumstances and so the volume of liquid in a container is conserved as we change its orientation and shape. The second one is that the “surface” of the liquid is horizontal. In our 2 D problem this means the line that is the equivalent of a 3 D surface is horizontal. Look carefully at the working and see how these two restraints are key to getting the formula for the coordinates of the trapezium.
Finally, note that I am breaking my big problem of finding when the can will tip over into sub problems of finding the centre of gravity of a trapezium and then the conservation of area problem. The “write up” phase of the exploration will be a matter of putting these notes together in a coherent way that will allow me to explain clearly and concisely my solution.
One other point I would like to make is about technology. The Geogebra file that goes with the notes on the conservation of area page is in effect the check that I am making to see whether I am correct. It took me a while to get the formula into the syntax of Geogenra properly, but once I did and I used the sliders to change the parameters in my coke can the model worked as it should “physically” and I was certain that my working was correct.
I hope you enjoy the solution to the problem. It feels nice and the formula come out elegantly. I am going to revisit the Centre of Gravity problem next and try and use vectors to get my C o G for the trapezium. The coordinates that I found in this post will then be turned into column vectors and it will be easy (I hope!!) to combine that with a Geogebra file to get my solution.
We have not posted for some time but I am pleased to announce that the next few weeks should lead to some interesting posts.
We are going to take the ideas we had about balancing the coke can introduced earlier in the year and conduct a “model exploration”. The outcome of this exercise should be an exploration that could be graded against IB criteria for Internal Assessment.
The object of this exercise is NOT to produce a perfect IA exploration, though we hope it will be quite good. However, more important is to observe the approaches and ways of thinking involved.
Finding a Focused Topic
The second post called more balancing matters actually began to look quite deeply into the problem in hand. In this post we managed to create a Geogebra file modeling the situation and start to propose what the problem of finding a balance point for a cylinder filled with liquid might be. The first thing to note is that creating this Geogebra file was no easy matter and by creating the file it became clear that one of teh primary issues was how to find out how the liquid behaved as the can was tipped in order to conserve the volume of the liquid in the can. This seems like a really tough problem.
Perhaps this is too difficult a problem for the exploration. Is there an easier problem we could solve that might be a good exploration in itself or if quite doable lead to a solution method for the actual problem of finding the tipping point of the actual can. The real challenge of the coke can is that it is 3 – dimensional. Perhaps we can solve a 2 – dimensional problem first.
Consider the following image in 2 – dimensions of a rectangle on a slope with a 2 – dimensional “liquid” inside it.
Note that the “liquid” inside the rectangle is in the shape of a trapezium since the “liquid”must keep a horizontal surface. This trapezium needs to preserve its area (the 2 d equivalent of volume) as the slope is changed. We also need to know how to find the “centre of gravity” of the trapezium.
Already we have two quite nice problems to solve. These are
Finding the centre of gravity of a trapezium.
Preserving the area of a trapezium as its base is “tipped”.
Perhaps these questions will make a good exploration on their own. If they do we will then need to construct a topic around this that is of interest as a whole. Of course, if we can make rapid progress with the 2 – d problem we may be able to tackle 3 – dimensions and then our topic would simply be “finding the balance point of a partially filled coke can”.
We will shortly post about finding the centre of gravity of a trapezium.
It does not take long to find some pretty cool ideas for math explorations. Two particularly good websites are Nrich and Plus.Math. Nrich is great because it is easily searchable and has problems that you can try and solve but there are also articles and videos. Plus.Math is also wonderful, has longer articles and really goes into the type of mathematics that Math HL students can get there teeth into. Bit like Goldilocks, the math is hard but not too hard!!
The first one poses quite a nice problem on Complex Number and if there are examples that square to real numbers (and also imaginary numbers). Great little stimulus to be able to explore an aspect of topics that Hl students will know about and enjoy.
The second article I picked on was one on adding series to interesting results. It is a long article with a lot of ideas in it but certainly some wonderful stuff for a Math HL student to get their teeth into to start to explore. This is particularly good for students who will eventually study Calculus as their option topic (e.g. Pamoja Math HL).
Please post some questions about these articles or anything else in Nrich or Plus.Math that takes your fancy. Even better, if you have examples of websites to share please send in a comment.
I was home in bed not feeling very well when my attention was drawn to a five pointed star in a picture above the bed. I started wondering how to construct the star, what were its symmetries and reminding myself that there are pentagons all over the star.
This got me to thinking that there might be a good IA Exploration here somewhere. So I decided to to a quick research of pentagons on the Internet. Nothing fancy, I just googled and used Wikipedia. Not the basis of a good research but a great place to start. During this research my attention was brought to the “Cyclic Pentagon” and thought this might be where I could do an IA Exploration.
Below are two videos (both only 5 minutes in length) that describe how I got the the point of finding Cyclic Pentagons. Hopefully this will encourage you to take one of the stimuli and do some research so that you can come up with some ideas to post in your Week 20 IA Blog post.
Trigonometry seems to be full of these annoying things called identities. Why is it that we can write a trig expression is so many different ways? I would suggest it is a matter of symmetry. Since trigonometry can all be found in a Unit Circle and a circle has lots of symmetry then the consequence is that there are lots of symmetries when we write trig expressions. In this case symmetry means that the expression looks different but actually it is identically the same.
Look at the triangle above. Check using Pythagoras and basic geometry that the triangle is correct (the key is the fact that there is an isosceles triangle in there). The cool thing about this triangle is also that we can get the double angle formula for Tanget out of it just by inspection. Can you see this?
Now here is a problem for you. This comes out of an IB exam paper one (i.e. no calculator allowed). Although the original question used complex numbers it can actually still be done without (it just requires a bit more crunching of the algebra). OK….there are three things to do.
1. Can you prove that ?
2. How about ?
3. Now use these results to prove that ?
Then comes the really cool part. Can you show from the third identity that
Now that is a pretty amazing result. We can then use other exact values for trig ratios to work out other ratios for other angles and so on.
Maybe trig identities are not so boring after all? Is there an exploration here?
I was thinking about the balance problem for a can of coke and it provoked some really interesting questions of their own that might model the kind of process IB Math HL and SL will go through when doing an exploration.
So I started by looking to do an experiment. Since we do not drink sodas in our house I instead went to make a dynamic Geogebra file to model the situation. My first attempt had an issue. Can you spot it?
OK…so if the coke was frozen in the can then this model works fine. It is easy to find a centre of gravity (the intersection of the diagonals) and so we can do our experiment without spilling coca cola all over the kitchen.
I managed to rectify the error and have a liquid in my can. At first my only problem appeared to be working out the centre of gravity but then another issued arose as I played with the angle. Can you see what it is? Play with the angle and see what happens to the liquid that is shaded blue.
So already with a stimuli of the student balancing the coke can, a few pretty dumb errors whilst making a Geogebra model and we come up with two really interesting questions.
How do you find the centre of gravity of a trapezium?
How do we fix the trapezium so that the area is conserved as we tilt it?