## Meaning making with manipulatives

This year I have an interesting year 11 class. Most of then are targeted 5 or 6, but only achieved 3 or below in their end of year 10 mocks. They had a poor diet of maths through their KS3 due to staffing issues and long term sicknesses they had some.non maths teachers and also extended periods of time with supply. They did have strong teaching in year 10, and some did in year 9, but they have massive gaps in their core skills and knowledge that I’m discovering and trying to close on the way through. For instance, today they did an assessment which included the question “round 7364 to 3 sf” 50% of them answered 736, and a further 20% had different wrong answers. They lack confidence in their own ability and some of them are turned off by the subject due to it. They are all capable of getting to grade 6 at least, but some of them don’t believe it so I’m working to try build that confidence as well as filling the gaps.

Last week we were looking at probability, venn diagrams had gone well and three diagrams were going very well, until we looked at a question that involved conditional probabilities. Cant remember the exact question, but it was along the lines of their being 8 red things and 3 blue things in a bag, someone takes 2 of them at random without replacement, what is the probability they get 2 the same colour.

They could see the first set of branches, but couldn’t get their heads round the second. I tried explaining it a few ways and nothing was working, so I pulled out the box of multi link cubes that lives in my cupboard and passed some round. They had 8 of one colour and we of another and we looked at what was going on.

This simple use of manipulatives really allowed the students to get their heads round the concept. Normally I think I would have resorted to drawings, because there is, at least in the back of my mind, a feeling that manipulatives are only useful for younger students or those with lower attainment levels. But recently I’ve been trying to build more manipulatives into my practice and many I’ve spoken to have told m how successful they can be with older and higher achieving students.

This use of them not only helped their understanding, but it built their confidence, and after trying a couple of times with the cubes they could answer the questions just aswell without them.

I think I will consider other places manipulatives will be helpful with this class as I move throughout the year, as I think it ca help them with meaning making and understanding, but obviously only where it fits and adds to the learning. I don’t want to b using things for the sake of it where it may detract from the learning taking place.

*I’d love to hear any places you use manipulatives and how you use them. If you’d llike to share, please so do in the comments or social media.*

## 100% Chance

*“100% Chance of getting a safety car.”*

That line was repeated numerous times in the build up to today’s signapore grand prix and in the early stages of the race. It was repeated by various pundits and comentators and it has made my blog boil.

**THERE IS NOT A 100% CHANCE OF A SAFETY CAR**

That should be enough, to be honest. It’s not certain that it will happen, there is a chance that it might be avoided. The problem stems from the confusion between relative frequency and probability.

Relative frequency IS a good proxy when it comes to probability but it’s isn’t always exactly the same thing. The relative frequently of a safety car being needed at Singapore was, still is, 100% because There has been one at every race ever held here, which gives us a relative frequency of 100%. But the sample size is tiny (9 races) and this isn’t big enough in this case. Please sky sports, sort your maths out.

## A quirk of probability

Today I was playing a game on my phone. *Marvel puzzle quest* *if you’re interested. It’s based on characters from Marvel, follows story lines and it’s game play is a bejewelled type match game. If you enjoy those games and like Marvel then it’s worth a download. And let’s face it, you’re the sort of person who spends their free time reading maths blogs. So there’s a good chance you will enjoy it!*

Anyway, back to the point, at the end of each level you recieved a reward, most levels have 4 possible rewards and you get one of them chosen at random. You can replay the level and earn all the rewards, each time you get one at random, however the pool does not decrease and if you are randomly assigned one of the ones you have already won you get a non reward, as it were.

This morning I had completed a level. *I’d beaten Venom, for those of you interested.* I then replayed it to try and gain the other rewards, after 6 goes I’d had the first reward then 5 non rewards. What are the chances of that? I thought, then quickly answered 1/4^6 obviously. That’s all well and good, but then I thought “it’s **got** to be a different one this time!” And got very angry at myself.

These are **independent **events. The probability of the next one being a non – reward is clearly still 1/4. Yes, the odd of getting 7 in a row are tiny (1/4^7) but so are the odds of getting 6 in a row then one of the others (3/4^7). This is something I spend a lot of time discussing in lessons, because I know it’s something learners often have trouble with, so I should have known better than to let that thought sneak into my mind.

I was reminded of the Derren Brown show where he threw ten tails in a row on a fair coin. It’s a clip I often use in lessons and ask students to conjecture how he did it, rarely do any get the right answer (I think perhaps only once!) The answer, for those wondering, is that he threw the coin over and over until he got ten in a row, every time he got a head he started the count again. I think he said it took him 16 hours. He could do this because he knew it would probably happen eventually.

In both these instances, this row of events taking place looks highly unlikely. However Derren repeatedly did it until he got the result he was looking for and I’m probably one of millions of people playing that game and so it’s bound to happen to someone.

Probability is a funny thing, and I think a bit of knowledge about it can lead you to fall into the trap I did. I knew the probability of 7 in a row was low, but for a second forgot that we were looking at the probability of 7 in a row given that we already had 6 in a row!

## 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.*

## A probability puzzle

*“Colin and Dave are playing a game. Colin has a probability of 0.2 of hitting the target with any given shot; Dave has a probability of 0.3. Whoever hits the target first, wins. Colin goes first; what is his probability of winning?”*

Yesterday I listened to the latest edition of “Wrong, but useful” (@wrongbutuseful), and the above is Is the puzzle set by the cohost Dave “The king of stats” Gale (@reflectivemaths).

It was pretty late on in the evening, but I decided to have a quick attempt at the puzzle nonetheless.

Here you can see my back of the envelope workings, complete with the word “frustum” on the top of the envelope as I had added an extra r into it in my recent post. You will also see that approaching midnight on a Saturday after a day at the NTEN-RESEARCHED-YORK conference is not the most idea time for solving maths puzzles! My thinking was fairly valid, I think, but my tired brain has made an absolute ton of mistakes! *(If you haven’t spotted them, go and have a look before reading on!)*

When I looked at it this morning the first thing that jumped out at me was “0.14×0.028 does NOT equal 0.0364. Then I thought, “ffs, 10/43 is very definitely not 0.43!” This was closely followed by: “and why have you used 0.14 as r, the 0.2 is the probability he hits, if he hits the game is over, Cav you ploker!”

So I had another go:

I think this is right now. (Although do feel free to correct me if you spot another error!) The probability Colin wins on the first go is 0.2, on his second go is the product of him missing (0.8) Dave missing (0.7) then him hitting (0.2) so 0.8×0.7×0.2. As the sequence goes on you are multiplying by 0.7 and 0.8 each time, so it gives rise to a geometric sequence with first term (a) as 0.2 and common ratio (r) as 0.56.

The total probably of Colin winning is the sum of the probabilities of him winning each time. This is because the probability he wins his the probability he wins on his first go OR his second go, OR his third etc. The game goes on until someone wins, so is potentially infinite, thus we need to sum the series infinitely.

As the series has a common ratio of 0.56, which is less than one, we can sum the series to infinity using s = a/(1-r) which gives 0.2/(1-0.56) = 0.2/0.44 = 5/11. Thus the probability Colin wins is 5/11 or 0.45recurring.

## False Variables and Simpson’s Paradox

Last weekend I attended a day of lectures as part of my MA course. The focus of the day was on barriers for learning and it was quite intensive. Part of the day involved looking at the statistics involved in various things and seeing how they related to the development of children and the lecturer mentioned the idea that a false variable can skew ones ideas, and can make it look like something is having an effect, when in reality it is something else.

This idea of false variables is one that has been “following” me around recently. The first book I read this year was “The Simpsons and their Mathematical Secrets” by Simon Singh. In the book he discusses “Simpson’s Paradox”. The example he uses is in relation to the US government vote on the American civil rights act of 1964. In the north, 94% of democrats voted for the act compared to 85% of republicans. In the south 7% of democrat voted for, and 0% of republicans did. However, overall 80% of republicans voted for the act, compared to 61% of democrats. This example is great for showing Simpson’s Paradox and really emphasises the fact that stats can be deceptive. The worrying thing is that these stats can be manipulated to show that a higher proportion of democrats in the north and in the south supported the bill, or that a higher proportion of republicans supported the bill. Meaning both sides can legitimately lay these claims and hence really confuse the electorate. The fact of the matter is that the real variable that was feelings towards the bill differed largely due to attitudes in the north vs attitudes in the south, rather than a political allegiance.

Simpson’s paradox also appeared at school recently. A teach-firster in our department was planning a lesson on probability and asked me if I knew “that thing where you have a higher probability of picking one colour in each bag of balls, but if you put them all into the same bag you get a higher probability of the other.” This produced a rather interesting discussion, around Simpson’s Paradox, no one else in the department were familiar, and they all found it pretty interesting. We both then included it in our lessons. The question was around bins with coloured counters in them and showed that you had a higher probability of picking black counters from the blue bin in two cases, but if you combined the counters into the same bin, the higher probability came from the red bin.

The example of this false variable situation given in our lecture was that of breast feeding. The stats suggest that breast feeding equates to a better academic achievement for the pupil. But if you drill down into the stats you see that there is a far higher proportion of breast feeding mothers in the “middle class” as opposed to the “working class”, and that academic achievement may be more down to socio-economic status, rather than the breast feeding itself. This could be due to a plethora of reasons which may include: a higher level of education to the parents, enabling them to provide more support to learning at home; a higher income in the house which may enable private tuition if a child is falling behind or even that more working class families are reliant on shift work, longer days and multiple jobs, leaving them less time to spend with their children to aid their development. This is clearly a complex issue, and it highlights the fact when reading anything that includes statistics you have to ask yourself, “does the author have an agenda, and are they twisting the facts to suit it?”

## Probability and Sex Ed

This week we had our third CT day of the year. (CT Days, or citizenship themed days, are collapsed timetable days where pupils do a range of topics linked to a theme.) I was with my coaching group and we had a great day on the topic of “personal wellbeing”.

The new year 11 (we move up year groups at spring bank) had a day on sexual education. Currently in maths they are learning about probability and one of my colleagues and I decided this was a perfect opportunity to merge the two.

We gathered some data on the probabilities if contracting STIs from an unprotected sexual encounter and they looked at the probabilities involved in contracting things after multiple encounters (here).

We also looked at expected values, and given the effectiveness of different types of contraception, (from here) how many pregnancies a year would you expect if a couple who were always safe made love twice a week. The answer shocked the whole class. They were also amazed by the difference when I asked them to complete tree diagrams and work out the expected value if the couple used condoms and the pill.

This was a much easier concept for them to relate to than picking sweets out of a bag as they could see that this was something that would affect everyone at some point in their lives. It also got across some messages that are important, especially as our school is located in an area with quite a lot if young parents.