## A nice area puzzle

At some point over the last few days Danny Brown (@dannytybrown) tweeted this lovely puzzle out:

While waiting in the car this afternoon I saw the screenshot I’d taken of it and had a think about it. Initially I’d made a daft mistake with the base of a triangle and got the wrong answer (5:6 if you were wondering), but after a rethink and some in head workings I got to an answer I’m now happy with. I have since written the solution down and want to write about it here.

Firstly, when attacking problems like this I like to sketch them out. This step is one many students are reluctant to take, and I try to drill it into them that sketching is always helpful in understanding problems. If I’d had a pen and paper handy when I’d attacked this problem originally I’d have sketched it and not made my daft error.

From the sketch it is easy to see the area of shape A, this was easy to visualise.

In my working I then sketched shape B:

When I visualised this I made my daft error, which was to assume the triangle had base x. From the sketch it’s easy to see that it’s not, and it will need calculating, for this I used Pythagoras’s Theorem:

To calculate the area of the triangle I’d need an angle, so I thought about what I knew, the angle at E is made from a fold along EF so the angle must be equal to DEF.

Using the tan ration of the right triangle I got tan (theta) to be rt3.

Then it was a case of calculating the height and then the area:

The working written out properly looks a lot more thorough than my phone jottings:

*This is a nice puzzle that should be accessible to higher GCSE students and definitely A Level Students, but I worry that most would give up. We need to be giving our students the tools to unlock this sort of problem. I’m not sure how we can do that explicitly. I set these tasks, give them hints and then walk them through my thinking to model how I would attack them, and this has a great effect for many. But I wonder if this is enough.*

## Concentric Circles Puzzle

This morning I happened across this tweet from David Marain (@dmarain)

As you may know, I have a penchant for puzzles, and find it hard to leave them unsolved, so I thought about it and came up with an answer. I thought I would jot down my thought process here.

For those who can’t see the picture, the puzzle is:

*A point is chosen at random inside the larger of two concentric circles. The probability it lies outside the smaller one is 0.84. What is the ratio of the larger radius to the smaller radius?*

It’s a lovely little puzzle that combines a bit of geometric thinking with probability theory, so do have a little go first.

You done? Good.

My thinking started as: “that 0.84 must be equal to the area of the big circle – the small circle all over the area of the big circle.”

I used a as the area of the big circle, b as the area of the small circle and formed the following equation:

*(a-b)/a = 0.84*

With a little rearranging I got:

*0.16a = b*

So a ratio of areas that is

*1:0.16 (a:b)*

Or

*6.25:1*

Which is equivalent to

*625:100*

Which simplifies to

*25:4*

As we are looking for the ratio of radii, we need to square root each side, which gives a ratio of

*5:2*

A nice little solution to a lovely puzzle. Thanks for sharing David.

*Nb: no photo of envelope workings as I did it mentally.*

## Different approaches to ratio

Today I have taught two lessons on Ratio, both were very different, but both went well. The first lesson was to a very able year 7 class. It was a long lesson (75 minutes) and we got a lot done. At the start of the lesson not everyone knew what a ratio was, but by the end they could all write ratios, work out ratios from pictures, split amounts into two part ratios and the majority of the class could split them into three part ratios.

The lesson is P2 and comes after assembly. Our academy is split into 3 schools “Foundation School”, “Middle School” and “High School”, this doesn’t mean that they are in separate buildings or anything, but it means each school is over seen by a teaching Head of School who co-ordinates everything, with each year group having a non-teaching year manager who focuses mainly on behaviour. Our assemblies are run in school groupings, and due to a raising intake of pupils, the foundation school assembly has had to be split to accommodate the whole group in the hall. This means that P2 on a Monday my Yr7 class tend to arrive in two blocks, the first half come on time, they have been in the first assembly and as such have been let out of their tutor bases at the correct time, the second group invariably arrive a few minutes late and in dribs and drabs as assembly finishes and the whole hall filters out. This can lead to a messy start to lessons, so i like to have a bell activity. Today’s was a word search based around ratio and pupils were asked to work out meanings for the words too. This was a good settler and helped them learn meanings and spellings of key tasks.

As the lesson went on we had a short task on simplifying fractions, a MWB activity on writing and simplifying ratio, a card sort activity on writing and simplifying ratio, an extension sheet here asking pupils to set things out in ratios, a discussion around splitting amounts into ratios, a task on writing instructions for this, a task involving questions on the board where they split amounts into ratios and a further extension task involving some past paper GCSE questions. The chunking of this lesson helps with the pace and to keep the pupils on task. They were all engaged with each task, and made great progress. As a plenary we reviewed the objectives and pupils graded their own learning, and I asked them to write some questions based on today’s lesson which we will use in the starter activity tomorrow. I did leave the class wondering if i had spent a little too long on the earlier tasks. I had two people in observing the lesson, not an official observation. One was a PGCE student who is due to take on the class next week, and a colleague of mine from the PE department with whom I am doing a project based on observing each other so we can gain ideas on teaching and a better grasp on the lesson grading process. (Read more here: https://cavmaths.wordpress.com/2013/02/01/observing-others/ ) My PE colleague said he could tell the lesson went well as the time flew, I think this is the key to the longer lessons, and I think in this lesson it was achieved by the amount and variety of tasks.

The second lesson I taught today on ratio was to a low ability year 8 class. They are starting at a lower level and the lesson was much shorter (50 mins), so I hadn’t planned nearly as much. At the start of the lesson they did not know what a ratio was, and by the end of the lesson they could write multiple part ratios and simplify them. They come from Tech, and all seem to be in different tech classes and arrive at different times. I find a bell activity works to settle this class too and so I used the word search, and as a class we discussed the meaning of the words.

We then went on to a discussion of ratio and what it meant and a show me activity where the pupils had to tell be the ratios of red dots to green dots etc, this was a good task and it moved onto simplifying ratios as it went through, all pupils were engaged and they all quickly got the hang of it. We then moved into drawing pictures of ratios and then onto ratio biscuits. This was an idea I had got from Paul Collins blog and I have been itching to try it, but it seems to have taken forever for ratio to come up on the SOW (http://mrcollinsmaths.blogspot.co.uk/2012/09/icing-biscuits-all-in-order-to-teach.html). The class were given two biscuits each and some Smarties. The TA in the class spread frosting on their biscuits and they decorated them with said Smarties, they then had to write the ratio of colours of Smarties out on the kitchen roll (in its simplest form). If they were right, they got to eat their decorated biscuits, if they were wrong; Miss and I were going to. Unfortunately for us, they were all right. It was easy to differentiate this task by selecting the colours of Smarties the pupils received!

The plenary for this lesson was exit tickets and the pupils showed they had met the objectives and enjoyed the lesson.

Here are some pics of the biscuits!