Today I was working on some Vectors exam questions with my Y13 mechanics class and I came across this question:
I have recreated is incorrect working.
Obviously he had found out when the ship was at the lighthouse, instead of 10km away. I explained this to him and started to explain how he should have tackled this when a sudden realisation angered me.
Now for those if you that didn’t work through the question, here is the actual answer:
This is an impossible answer! If the lighthouse is on the trajectory of the ship and it will hit said lighthouse at t=3 then that would stop the ship! At the very least it would slow it down!!!! In reality it would have to avoid the lighthouse and change trajectory. Meaning the second answer, T=5, would not happen under any circumstances!
My initial thought was: “are they expecting students to spot this and discount the second answer? That’s a bit harsh.”
So I checked the markscheme:
What do you guys think? Is this infuriating or am I just getting get up over nothing? I’d love to hear your views in the comments or via social media.
I find most memory aids a little silly. Why learn a rhyme about horses when you can just learn the trig ratios? Why learn a rhyme about the duke of York when you can just remember the order the colours come in?
However, I find that music is a good way of remembering things. For some reason music is good for us to remember words. I can, for instance, remember the words to a great deal of 90s pop songs even though I didn’t like them and never chose to listen to them because I heard them out places and on TV so often that they got lodged in my brain forever.
This is something I have seen work well in learning maths facts. Year on year I hear pupils sing “mean is average, mean is average…” etc in lessons to remember the averages. And I also hear a great many variations on the circle song.
Last year when I was teaching kinematics one of the students said “Sir,play the SUVAT song.” I’d not heard of the SUVAT song and he found it on you tube and we listened to it. It’s simple and it’s catchy and it really helped him and his class remember those equations. So on Tuesday I played it to one of my mechanics classes. By the end of the leson I’d heard three people sing it and it has been stuck in my head all week.
What do you think about mnemonics? Do they have a place? Have you any songs or rhymes that you use to remember things or that you encourage students to use? And do they help?
Today I was going through an M1 question with a year 13 student and was surprised to see the method he had used. The question involved finding an angle in a right angled triangle given the opposite and adjacent sides. The learner had used Pythagoras’s Theorem to find the hypotenuse then used the sine ratio to find the angle.
Puzzled I questioned further, thinking he may have instinctively found the hypotenuse without fully reading the question then having all 3 sides so going with the first. This turned out not to be the case:
“I know sine equals opposite over adjacent innit sir, I have trouble remembering the other ones so I just always use sine.”
This was extra interesting as earlier I had come across a markscheme which suggested the way to resolve a force at an angle of 30 degrees was to use Fsin30 for the vertical and Fsin60 for the horizontal! Further checking showed this learner did that too.
I wasn’t too sure what to make of it. It’s mathematically correct, so there’s no issue there. The learner has a grasp of the other ratios but is more confident with sine so I can see why he would default to that position, although I hope the extra time it takes isn’t an issue tomorrow. I can’t fathom, however, why the markscheme would show it this way in the first instance. (Not the only time a markscheme has confused me recently!)
What do you think? Have you got any quirky methods like this? Have any if your students? Do you have an idea why a markscheme would default to this position? I’d love to hear your response.
My daughter is now a little over two, and as such I spend quite a lot of time at children’s play parks and the amount of maths you can find there is incredible.
I’ve written along these lines before. From silly things, like the fact our local park was the only place I’ve seen with a simple abacus that’s set up for a base ten, rather than base 11, world. To the more intricate mechanics behind swings and roundabouts.
Today, however, it was this that got me thinking:
“A slide?” I hear you thinking. “Surely there’s nothing too mathematical about that? Gravity means you go down it.” Well in some respects you would be exactly right. However, we don’t live in a model world, and there is the question of friction.
Today the park was rather damp, it had been raining over night, so we took a towel to dry the slide. Once the slide was dry my daughter went down it, but at an incredibly slow speed and stopped about halfway down. I have always noted that the speed of the slide varies incredibly, and had assumed it was due to the coefficient of friction varying between different materials. Her cotton trousers provide a much faster slide than her jeans, for instance. But today she had denim on and it doesn’t normally stop halfway down. I then thought about another recent trip where the slide had seemed slow, and that day it had been particularly warm.
It got me thinking, does the temperature affect the coefficient of friction between two materials? Or perhaps it’s humidity that has an affect? A little bit of research leads me to believe that both can be contributing factors. Definitely more ammo for investigation in a mechanics school trip!
After the slide, she wanted a go on this:
As you can see, the local see saw is fairly basic and works best if two people of a similar mass are on each end (obviously we aren’t). It’s still easy enough to work, but it made me think about another local-ish park that has a much more complex see saw where there are 3 seating positions on each end. That gives more variability to the user’s and would mean two children of different mass could select positions,that give similar, but opposing, moments and hence work the see saw as well as if they had equal mass. What a fantastic idea, and a fantastic use of maths!
I love the amount of maths that is present at local playparks, and one day I do hope to take a mechanics trip (perhaps a cross curricular one with physics) to investigate all these things. It would be awesome to make a TV show around it. Perhaps I’ll make my own Numberphile – esque video on it one day!
This weekend my daughter received this:
It was a present from her grandma who had just come home from a holiday.
For those who haven’t seen one, it’s what is commonly known as a “Snow Globe”. There us a glass (possibly plastic) sphere set on a base which has a scene. In this case it’s a lighthouse. The sphere is filled with water (a clear liquid at least) and there are little specs of stuff in the water. Often these are white (hence the name “Snow Globe”) but these ones are strange and mesmerising, flickering with colours. Sort of semi transparent but change colour in the light.
I hadn’t seen one for years, and I started spinning it and watching the glitter spin round. I guess I must have had the recent M3 exams on my mind because as I was watching them spin I started wondering what the centripetal acceleration was. As I was thinking about this it occurred to me that this system I was holding wouldn’t obey classical, Newtonian, laws, but would be governed by the laws of fluid mechanics.
I did study fluid dynamics at university, I recall the navier-stokes equations. I recall looking at currents and river flow. I don’t, however, recall discovering how fluids would act when they were in a closed spherical system.
Watching the Snow Globe spin is fascinating. As you spun it, some glitter collects at the bottom, if you alter the direction of spin the rise up into the middle. If you stop spinning it, the contents carries on spinning, if you reverse the spin it still carries on, and you can see the outside slow and change direction first. Due to the friction with the sphere. This then moves the rest of the liquid, but you can get an amazing sight of the outside liquid going one way, but the inside going the other.
When you spin it, or shake it, then leave it to rest the glitter jumps and jives in a brilliant way. It seems like it is a sort if “Brownian Motion“, but I would need to investigate it further.
All in all, I’m fascinated by this Snow Globe, and I am going to read up on fluid dynamics, their effects in a closed spherical system and brownian motion as soon as I get chance.
If you have any suggestions of books, papers, articles, blogs etc on the subject, I’d love to hear them!
Since becoming a parent I have seen maths in many places I wouldn’t necessarily have thought I would have. The playground (or park) is one source that just keeps giving.
A week or two ago we were at a new playground and I found this abacus, and of course the maths links to an abacus are obvious, but what else is there?
Last year I was pushing my daughter on a swing and I couldn’t help but see the swing as a pendulum and wonder whether we could take a physics and maths cross curricular trip to the playground and investigate the simple (or more likely damped) harmonic motion on display from the children’s swings. I think the mechanics involved would be interesting and different swings could be tested to see which ones are more efficient. Given that some swings are extremely noisy, I would assume that these are the least efficient and would love to test this hypothesis.
Today I had another thought about playground maths. My daughter was playing on the roundabout, and after a while she got off and decided to start picking up the bark chips that cover the floor and plonking them onto the roundabout. We told her to stop and to spin the roundabout as that should make the chips fall off. She did this but the chips didn’t move. This was due to the fact her spin didn’t have enough speed to generate a centripetal force big enough to cause a reaction (or whispers centrifugal force shhh) larger than the maximum frictional force acting in the bark. This made me wonder how large the coefficient of friction would be between the painted metal roundabout and the bark chips and what the minimum speed required would be to move them. Again, I figured a cross curricular trip would be great to investigate this too. I didn’t have my phone handy, so couldn’t photograph the chips on the roundabout, or film the sight of them flying off quickly when I gave the roundabout a spin!
There you have it, children’s playgrounds, the perfect school trip for A Level maths and physics!
Last night a colleague and I attended a “Teachers Evening” at Manchester University. We weren’t exactly sure what to expect, but jumped at the opportunity to spend an evening immersed in maths.
Having studied at Manchester University, and knowing that the tower that used to house the mathematics department has been levelled, I was intrigued to see what the place looked like. The Alan Turing building is brilliant, and extremely aptly named. The vibe of the place was great and the fact that it was still buzzing with undergraduates and postgraduates at half five was great. Some were even playing backgammon.
The evening itself was set up as two lectures, with an interval. The first lecture was from Professor Oliver Jensen, who’s seat is named after Sir Horace Lamb. His lecture, “bringing mechanics to life”, combined his own specific interest in biofluiddynamics with Lamb’s more general interest in fluid dynamics. It was a great lecture and left me eager to know more, I have a list of topics mentioned which I would like to investigate further. I felt a love of fluid dynamics reawaken, it had been my favourite topic in the second semester of my second year. It also gave me a good number of lesson ideas for future mechanics classes, and a whole host more answers to the question “what use is maths in the real world?”
The second lecture was entitled “shuffling around, why you should play cards with mathematicians” and was from Dr Charles Walkden. The talk was fantastic. It combined theories on shuffling, modulo arithmetic and mathematical card tricks. It was really interesting and gave me some great lesson ideas, including using standard from to compare the number of ways to order a deck of cards with the number of drops of water in the Pacific. The modulo arithmetic ideas were very timely, as this forms the basis of a “taught round” I’m an up coming year ten competition!
All in all, a great evening, and I can’t wait for the next!