Introduction to Mechanics: Lesson Plan and a Speed Riddle

First, the riddle:

You are driving along a 2-mile long bridge. You drive the first mile at an average speed of 30mph. You want your average speed for the whole bridge to be 60mph. What speed do you need to drive at for the second mile?

This is how I began my mechanics taster session with next year's AS students. Have you worked it out yet?

The first  answer I got from the class was 90mph. Is that what you think the answer is? If you do, you're wrong. Sorry.

(30+90)/2 = 60, but this is not the average. You would spend longer driving at 30mph than at 90mph, and this average does not take that into account.

Hint Number One:

What is the definition of "average speed"? Total distance covered divided by total time taken.

At this point the calculators started to come out. Pens and paper had yet to make an appearance. I then started to get some bizarre answers like 2mph. This is an example of why a calculator without pen and paper is a dangerous thing.

Hint Number Two:

To average 60mph, how long should the entire journey take?

Speed = distance/time. So 60 = 2/t. So t = 2/60 hours. This would be 2 minutes. Have you worked out the answer yet?

Hint Number Three:

How long have they been travelling so far?

Speed = distance/time. So 30 = 1/t. So t = 1/30 hours which is... 2 minutes.

So it is impossible.

Surprised? I was. It still don't really see how it can be impossible. Surely if you go fast enough you can catch up? It seems wrong somehow.

Now, onto the rest of the lesson.

I borrowed from the science prep room a mechanical weighing scale. The kind you have in your bathroom. Side note: what is it about science prep room technicians that makes them so formidable? I have never returned a borrowed item so promptly!

Anyway, I put the scales on the floor and stood on them. I asked the class to look at the number displayed. I then pointed out I was wearing heavy clothes and block heels and a big watch and I'd just eaten lunch. Then I asked the class what would happen to the number if I put my hands on the back of the chair in front whilst standing on them. They correctly told me the scales would say I've lost weight. Then I asked if there was a way for me to make the scales think I've gained weight. That's a more interesting question. I'll leave you to think about that.

Then I asked them to consider a person weighing themself whilst in a lift. What would happen to your weight when the lift is going up? The class was split almost exactly in half on this. Some thought you would gain weight because the lift is pushing the scales into your feet. Some said you would lose weight because the lift is pushing your feet off the scales. They were also divided over what happens when the lift is going down.

Mechanics is a very sciency bit of maths, and when there's a debate in science there's only one way to settle it: an experiment! So I simply declared, "To the lift!" The students were surprised and I think a little bit excited. We all went to the lift and took it in turns to go in groups of four up to the third floor then down to the ground floor, then back.

We returned to the classroom and discussed our findings and tried to explain them.I won't tell you the result of our experiment, but if you ever get the opportunity to try it out, please do!

Next I held up a tennis ball and a basket ball (borrowed from the PE department, who are a lot less scary than the lab techs) and told them we were going to do another experiment. I asked them which ball they thought would hit the floor first if we dropped them at the same time from the same height.

Again, the class was divided. We discussed mass, surface area, rigidness, air resistance... It was a good discussion. And then we gleefully left the classroom to do our experiment. Half of us went to the top floor (the third floor, or the fourth floor if you're American), and the rest went down to the bottom. Our school is kind of open so that from the top floor you can see all the way to the bottom if you lean over the balcony on the inside. This made it ideal. (The building won an award recently for its awesome architecture).

There were a few students in the corridor working on the computers or printing stuff so we drew a bit of an audience. And when the two balls hit the ground (at the same time? Well, that would be telling...) we definitely drew some attention to ourselves! It was loud.

So we went back to the classroom and discussed our findings. We talked about how the experiment was kind of rubbish because there were too many variables. So I told them we were going to watch a better experiment where these variables were controlled. That's when I showed them this clip from Brainiac. Sorry about the Arabic subtitles.

That concluded the lesson. I think it was a great way to introduce the mechanics module and give them a bit of a taster. Hopefully it has made them excited to start their A level maths in September!

Emma x x x

Happy Palindrome Day to Me

Today I am 8888 days old.

Enough said.




Emma x x x x x x x x

A New Way to Teach Dividing Fractions

How do you divide a fraction by another fraction?

For example, how would you do something like this:

My guess is you would flip the second fraction upside down and then multiply like so:

If you are a maths teacher, is this how you teach students?

Do you think your students understand why this method works? And, be honest, do you understand why it works?

Well today in the maths office at my academy one of my colleagues showed us a new method he'd thought of.

It works like this:

I think this is a little bit more intuitive.

My colleague got the idea from one of his year sevens who had answered this question without showing any working out:

The answer is quite obviously three. How many quarters are there in three quarters? Three, duh. But I am quite certain most of my A* students would perform the technique of flipping and timesing without even thinking.  My colleague was impressed that this student had used some common sense. He wondered whether the same idea could be applied to fractions with different denominators. It is a little bit less obvious that 21/28 divided by 20/28 is 21/20, but it's not entirely unbelievable. Whereas the "trick" of flipping and timesing can look a little bit like magic to some students.

I haven't tried teaching this method so I can't comment on its effectiveness yet. But as a mathematician it appeals to me. It's quite neat. And in case you were wondering, yes this works with algebraic fractions too.


If you're going to be teaching fractions soon, why not try this out? If you do, please let me know how it goes.

Do you think this is a good method?

Emma x x x

A Mathematician's Lament

Sorry it has been so long since I last wrote. Exam season is a very busy and stressful time for teachers as well as students! (Perhaps even more so). Now that the wonder of "release time" is upon us, I should be able to write more.

I stumbled upon a piece of writing called A Mathematician's Lament. I think it is one of the most beautiful pieces of writing I have ever read. As I was reading it, I found myself screaming (inside my head, I don't want to make a scene - I'm British remember) "that's exactly what I think!". It's like Paul Lockhart looked inside my head, stole my thoughts, and wrote them down but in a nicer order and with better punctuation.

I urge you to read it if you can spare the time. At least read page one. For those of you who really can't be bothered, here is a TL;DR summary for you:

*Mathematics is an art.

*Mathematics should be taught the same way that music and art are taught.

*Maths is not about acquiring skills that you might apply "one day".

*Much like music lovers have certain types of music they like and certain types they dislike, mathematicians can dislike some types of maths. It is a matter of taste.

*Maths is about imagination, you do not deal with real things because real things are inaccurate and messy; maths is simple and beautiful.

My favourite quote:

"there is nothing as dreamy and poetic, nothing as radical,
subversive, and psychedelic, as mathematics."

I have shared this with my department and I am considering sharing it with my students as well. I think it is especially important to share with year 11s who are looking to take A Level maths. Those students who have thought they were good at maths all these years because they got all the answers right might not realise that rote learning won't get them very far in A level.


Read the whole thing and then tell me what you think in the comments below.

Emma x x x