## Accumulator maths

Earlier today I was discussing and thinking about football accumulators, and accumulators in general. In case you don’t know what one is here is a quick overview. You basically pick a set number of bets and put an initial stake on, then if your first bet wins the winnings and stake roll over to the next bet etc.

The idea behind them is quite interesting. The more bets within your accumulator, the more you can win and the growth can be exponential.

For instance, if you backed a number of football teams all at 2-1. If you put a quid on, and bet on the one match, you’d finish with 3 quid (£1 stake returned and £2 winnings). If it was 2 matches then that 3 quid would roll to the 2nd match and if they won to you’d end up with 9 quid. A third match and you’d have 27, a 4th and you’d have 81n a 5th 243, a 6th 729, 7th 2187 8 matches 6561 etc. Its a geometric series.

I thought about this and thought it might prove an interesting real live discussion on exponential growth and geometric series. You could see how quickly these things would grow. As most accumulators aren’t all the same odds you could discus how these models change with different amounts and this would lead to nice discussions around commutativity and the like.

I then wondered if this was something that should be discussed in a lessons. Gambling can become an addiction and it can ruin lives. It might have already affected the lives of students in our care, and discussion on it in lessons might be seen as promoting it.

I then started thinking about some of the topics we do teach, and the origins of it. Vast swathes of the maths we teach stems from mathematicians trying to get an advantage in some game of chance or other. And although we might not talk about the gambling we still teach the maths that came about from it.

The maths that comes from accumulators is very interesting, as is a lot of maths with roots in gambling. I would love to discuss it, but think it’s a topic to be wary of. I’d love to hear your views. Do you discuss this sort of thing in your lessons? Do you manage to do it in a way that doesn’t promote gambling? Do you think we should leave it out of lessons? Please let me know in the comments or via social media.

## Exploring the link between addition and multiplication

Today’s Cuisenaire rod session was quite interesting. After aying and looking at some stuff that was similar to previous posts my daughter came up with this sequence:

(Again, please forgive the ordering the table is quite small).

She decided that she wanted to add how much each sequence was worth:

She started with tallys as she had used before, then asked if there was a quicker way. I got her to think about what was going on and she decided she could use multiplying :

After she did the one with 3s we had a discussion about the = symbol and what it meant and why it was wrong to use it the way she had initially.

When she did the 5 one she said “is that the wrong way round”, which led to a nice discussion on the commutative law.

After she had done a few she realised she could miss out a step:

When she did the one with the 8 I said she didn’t need to +1 on a different line and explained why, but she said she wanted to keep doing it to show it was separate. We then discussed the order of operations.

I think this task is an excellent way of seeing why multiplication would take precedence over addition when we come to looking at the order of operations.

I think this task and discussion were a good way to embed the link between repeated addition and multiplication, and to lay foundations for algebraic reasoning when it comes to collecting like terms. I can see that for older students it would also be a great way to show and think about the position to term relationship in a sequence.

This is the 9th post in a series about the use of manipulatives in teaching mathematics. The others can be viewed from here.

## Reflections of a locked down teacher

Back in March I was driving home when Boris Johnson announced that schools would be closed for the foreseeable future. It was something we had all thought was inevitable given the way the pandemic was going, but it was still somehow a shock. It had certainly never happened in my lifetime and really cemented to me that we were living through a very bizarre time. As we come towards the end of the lockdown period I thought I’d write some reflections on what I’ve been up to.

**What we did:**

We had little notice, 2 days, so we set about planning work to go home with the students. As it happened, as a department we had seen the inevitability and had the week before send lots of work through for most year-groups. I had finished the course with year 12 and 13 so ha sent through a mass of revision materials/past paper questions that i would have used in lessons if we had stayed in school. We had printed similar for year 11 and we printed the revision packs year ten would have been given later in the year before their year 10 exam. So when the announcement came and there was a mad rush for printers we only needed to sort some stuff for years 7-9 which made things easier. The year 11 and 13 work didn’t prove that necessary with the announcement that exams wouldn’t be happening this year, although those who are continuing with maths or maths based studies next year have been working hard on them. As well as the paper based work we started setting daily hegartymaths tasks for all students. The situation was less than ideal for the learning of our students but they were at least able to access learning.

Lots happened that first week. We spent a lot of time sorting out the data to be submitted to the exam board and planning what future online learning would look like. Live lessons for year 10 started on the monday of the second week and have continued since, only taking a break during may half term. Live lessons for year 9 and 12 started after easter (was that week 4?) and have continued. At this point we started to provide recorded lessons for all year groups. From the 15th Year 10 will be in in bubbles and Y12 by appointment. I will be in on a wednesday teaching some year 10s, whilst also continuing with live lessons for my classes.

**Live lessons:**

Teaching an online live lesson is not something I had done before, and it was a steep learning curve. I usually do a lot of live modelling on the interactive whiteboard but I didn’t have the equipment at home to be able to do this so i needed to rethink my plannig and delivery. This meant that planning has been taking longer. Where I would normally include examples to work through I now had to also include the working in my planning. This meant that it was more difficult to model the thought process live as I had already done the working, so I had to be mindful when discussing the examples to explain why I was doing each step as well as what each step was. I think i’ve improved at this as the weeks have gone on and I have certainly gotten better at discussing what other option we could have taken. I think the puzzle blogs I write have helped me with the communication skills that are required in this. We have used MS Teams for live teaching and I think it has been a good tool. It allows you to share the presentation and allows students to engage, also the record function has been good for those who miss the lessons or just need to watch it again. I have found that students are more likely to engage in text chat than to speak on the microphone, which has been interesting.

**CPD:**

I’ve managed to do quite a lot of things for CPD during lockdown. We use Edexcel for both GCSE and A Level and they have put on a number of great webinars that I have been able to access, we were involved in some work with the local maths hub this year and when schools closed we have moved this to online CPD sessions, which have been good. I’ve managed to read a couple of maths/teaching books, I’ve been able to work with my daughter at maths and this has allowed me to trial the use of manipulatives for meaning making and I’ve been able to engage with many things on twitter such as the sessions curated by Atul Rana. These have all allowed me to develop my practice.

**Planning ahead:**

The missed time in school has meant we needed to look at the curriculum plan for next year and rearrange some things to include missed learning from this year. This has been a large job but i think we are almost there now. Although it may all change if we aren’t back to normal by September.

**Time:**

While I’ve been working similar hours to normal, the fact that I have been mainly based at home has meant that the time taken commuting (depends on traffic but normally up to 2 hours a day) has not been there. This has allowed me to spend more time with my family, which has been good. It’s also allowed me more time to engage with recreational maths and to write on this blog. I’ve been doing a puzle post a day mon-fri for the last 8 weeks ish and I’ve enjoyed doing it and the conversations tat have arisen for it. As we move forward I may not have the time to do as many, but i certainly hope to keep up at least one per week going forward. I’ve also written a number of posts discussing the maths my daughter and I have been doing, again I think the frequency will drop, but i hope to keep up with these too.

**Going forward:**

No-one knows whats going to happen, hopefully we will continue to progress towards schools reopening fully, but I can also envision a scenario where a second wave hits imminently and we have to close schools again faster than we reopened them. We’ve been planning for a full start in september, but this might not be what happens, we might be limited to a year group per day. We may still have to do some combination of online and in person. We all need to be flexible and do our best to keep our students, our families and ourselves safe.

## Visualising the link between square and triangle numbers

I wanted to write today about some things my daughter and I were working on with the rods the other day, and some of the maths it inspired me to look into afterwards.

We started out just playing as usual and she made this house:

Then she started making patterns. She came up with these:

Well, specifically the top 2. I asked her if she could make any smaller ones and she came up with the wrw one, and I asked if there were any smaller. She said no, so I put the single white one there and asked her if it fit the sequence. I could see an argument either way but wondered what she would decide. She decided it did fit.

I then asked her if she could continue her sequence:

(You’ll have to forgive the ordering, she was working on quite a small coffee table.)

She then decided she was going to look at what they were worth. She started adding them up and got 1 , 4 , 9 …. as she was giving me the answers I very quickly realised they were the square numbers. Perhaps this shouldn’t have been much of a surprise, but it was. I wasn’t expecting it. I asked around 25 if she knew the sequence, she didnt, but she did notice the sequence was “going up by two more each time” so I got her to predict what the next one would be then work ot out. Her notes are here:

She did all this on her own, I was impressed by the thought she had to use a tally chart for adding the long number strings. She was very excited when she was getting the ones that were longer than a line right.

While she was doing this I was looking at th pattern and thinking about square numbers. I know that 2 consecutive triangle numbers sum to a square number, and while I was looking at the sequence I realised that the shape was basically 2 consecutive triangle number shapes back to back. I then started thinking about the algebra that goes with this.

I thought I remembered the the nth term.of triangle numbers but I checked anyway:

And then I summed them:

Which I thought was nice. This could be something that leads to further work on algebraic proof.

Bizarrely, given the thoughts I was having about triangle numbers the next thing she wanted to do was create another sequence and she came up with this:

We talked about the similarities and differences between this and the last sequence. She thought these would add up to half their equivalents from the first pattern but was surprised she was wrong. We looked back at the pictures and she realised why it wasn’t half. At this point it was pretty late so we packed away.

*This is the 8th post in a series about using manipulatives in the teaching of maths. You can find all the posts in the series **here**.*

## Making numbers and quadratic sequences

So we had another session playing with Cusinaire Rods. It’s quickly becoming one of my daughter favourite things to do. Which is fine by me. I had just read about an activity in the ATM book (Ollerton et al., 2017) so I wanted to try it with her to see what happened.

The activity is that you get 3 rods, the blue the green and the white, and you try to make all the numbers up to 13. Initially she was trying to just use addition but I explained how we could do subtraction with the rods and she managed to work through quite well:

These are what she came up with. As we went along we discussed how we could write these down, using letters for colours:

This was an interesting way if introducing some algebraic notation into proceedings. I asked her if she wanted to try with any other rods, and she decided she wanted to use 2 oranges, 2 yellows and 2 reds. I asked her what the biggest number we could make with those were and she identified 34 so I asked her to try to find all the numbers smaller (not knowing if it would be possible or not).

We started easy and moved on:

As we went along we noted down what we had done and she suggested we use 2r for 2 reds.

There were only 2 numbers she was unable to make, 31 and 33.

She then decided she wanted to make patterns like we had the other day. I asked her to try and come up with some and she came up with these:

She made them up to the one where there was an orange border.

We then talked about their value. Looking at the total value, the value of the border and the value if the middle bit. We looked at how much each bit increased and how they were linked.

At this point we finished. But this final sequence had got me thinking. It was a square quadratic sequence, made up from another square quadratic and a linear. I thought this was interesting:

It’s a nice sequence and you can pick put each bit from the pictures. The 1st term has a 1×1 square surrounded by a frame which is 4 lots of the 2 rod. You can see the whole shape is a 3×3 square. The next has a 2×2 square surrounded by 4 3 rods and you can see it’s a 5×5 square. Which is easy to write as 4n+4 for the frame (as each of the 4 rods gets bigger by 1 each time) and n^2 for the small square. As the full shape is always a square 2 bigger than the inside then it will always be (n+2)^2. I think this would be a great starting point for discussions on quadratic sequences with KS3/4 classes.

*This is the 6th post in a series about looking at the use of manipulatives in teaching maths. You can read the others here:*

Manipulatives – the start of a journey

Meaning making with manipulatives

Patterns, Sequences and Fractions

__Reference:__

Ollerton, M. Gregg, S. And Williams, H. 2017. *Cusineire- from early years to adult. *Derby: Association of Teachers of Maths.

## A hexagon and a square

Last week I came across this lovely puzzle from Catriona Shearer (@cshearer41):

I didn’t have much time when I was it so I saved it for later and have just come back to it. Have a go yourself before reading on.

I drew a sketch:

Initially I didn’t have the labels or the centre marked, I put them in as I needed them. I first labelled the vertices of the hexagon that the square goes through as A and B. I reasoned that as they lie on the circumcircle of the hexagon, and the angle ACB is a right angle that C also likes on the circumcircle and AB is a diameter:

At this point I realised D was also on the circumcircle. We are looking for angle DCB, which we cant spot straight away but we do know is half angle DOB, and DOB is 120 as we are dealing with a regular hexagon.

This means that DCB is 60.

I loved this puzzle. It’s not the first one I’ve done recently that I’ve solved using circle theorems despite there not being any circles given in the question. This type if puzzle has made me wonder if I should be looking at more things like this when I teach circle geometry. I am going to think further around this and build these sorts of things into that from now on.

How did you solve this one? Please let me know in the comments or via email/social media.

## Angle puzzle- Congruent Circles

Today’s puzzle is another from Ed Southall (@edsouthall) and another that jumped out at me as I couldn’t see an obvious solution. It also has that word “congruent” in it again!:

As I said, I didn’t immediately spot a route to the solution, so I started drawing stuff:

At this point I saw this one:

And thought it was a rombus. I drew this on the full diagram:

Briefly took leave of my senses and decided I needed to justify that it was actually a rhombus, despite there already being more than enough justification:

The I worked out the angles:

Considered circle theorems, but nothing came of them. Considered cosine rule:

Decided that was probably unnecessary:

Drew it out again, reeling I had angle BCD and that triangle BCD was, in fact, isoceles:

Then it followed as a simple subtraction to get the final answer.

This was hardly a neat or concise solution the way I achieved it, although if I’d only done the required steps it may have been nice. How did you solve it?

## Power Puzzle and building resilience

Today’s puzzle comes from “Britain’s Brainiest Dad” Chris Smith (@aap03102). Its a puzzle that was in his awesome newsletter, but that I spotted first on Twitter:

It caught my eye for a couple of reasons. Firstly it was a puzzle that wasn’t a geometry puzzle, and I’ve been a bit geometry heavy recently. Secondly when I saw it I couldn’t see an obvious path to a solution, perhaps I should have spotted the path a bit easier, but when I forst looked I was flummoxed, and that makes me want to do a puzzle.

I wrote out the problem:

Then I thought, “I can split that root 6”:

Then I realised that there were no roots on the left hand side so powers would be better. And noticed that the 3 and the 4 could also be expressed as powers of 3 or 2:

Typing this up now, I realise I should have definitely seen the answer from here, but I didn’t.what I did was simplify it:

Then spot the route to the solution;

A nice little puzzle with a neat solution. While I feel I should have spotted the solution quicker, I think that this process is good to demonstrate how to go about solving problems of this kind when you don’t see how straight away. I took the problem and applied bits of maths I know until the solution came to me. I feel that often this is something that students struggle with. I see it in my class and I have been seeing it with my daughter during the lockdown. Often they cannot spot a way forward so think they cannot solve the problem. I’ve been working on it lots over the last few years, and most of my classes are getting better at it. But each year brings us new classes and the same issues. I find modelling processes like this is a helpful way to help build that resilience into them.

*I’d be very interested to hear how you approached this problem, and whether you have any different solutions. I’d also be interested to hear what strategies you use to build resilience in your students. Please let me know either in the comments or via social media.*

## Circles, Triangles and Hippocrates

Here’s a lovely puzzle from Le Bécachel Sébastien (@le_becachel):

It looked a nice problem so I set off solving it. I rushed straight into this one without thinking, so I’ll write what I did first, then what I wish I had!

First I drew the problem:

I assigned x as the radius if the quarter circle and d as the diameter of the full circle. This allowed me to express the diameter of the larger one in terms of x too, using the knowledge that as the triangle ABC is right angles and all 3 vertices are on the circumference then the hypotenuse is in fact a diameter.

From here I could work out the area of our quarter circle, the area of the full circle and the area of the triangle.

Now I could see that the given yellow area is equal to the area of a semi circle on the hypotenuse BC subtract the area given when we subtract the triangle ABC from the sector ABC. (I* called this the hatched area in my working, its the bit hatched in green. I called this hatched as I had referred to the yellow area as shaded earlier*).

From here is was easy to finish the question:

This is a nice answer, and I quite liked the solution. However, as I was working through it I had a realisation.

The yellow area we are given is a lune! I know some things about lunes, and one is Hippocrates First Theorem, which is:

(Simmons 1993)

What this means for is here is that the given area, 2, of the yellow area of our original problem is equal to the area of the triangle ABC. As this is an isoceles right angled triangle, this means the side lengths are also 2.

This means the diameter of the green circle is 2rt2, so the radius is rt2 and the area is 2pi.

This is a much more concise solution that I prefer very much and I wish I had thought about the question more before rushing in!

*How did you solve it? Please let me know on the comments or via social media.*

**Reference:**

Simmons, M. 1993.* The effective teaching of mathematics, *Longman: Harlow.

## Thinking about circles

A number of things over the last few weeks have got me thinking about circle theorems. I’ve been using them quite regularly to solve a number of the puzzles recently, and most of them work both ways. When I did this puzzle, I initially did it wrong:

So what I did was see that angle CBD was double angle CAD, using exterior angles theorem. And then at that point I thought, “well that makes B the centre of the circumcircle” then I followed the angles to get alpha as 30. A friend had sent me the puzzle and when I sent him my answer he said he had a different one, so I relooked at mine and realized that if 30 was the answer it wouldn’t work. Triangle DEC would have 2 right angles in it, and that’s impossible. I tried again and got the same answer as my friend (36).

But it got me thinking, and talking, about circle theorems. I didn’t know whether this one worked both ways or not, but assumed it did as all the other ones do. When I was discussing it I had a realisation though:

If you have a chord, and the centre of a circle you can always make a triangle (they can’t be collinear as that would make a diameter not a chord). If you have a triangle can always draw a circumcircle and the arc of that circle which falls within the original circle would always make the same angle from the ends of the chord as the centre.

It’s quite obvious when you think about it, but it wasn’t something I’d thought much on.

Then a few days later I was thinking about this puzzle and in particular the bit in the circle:

I was looking at it and thinking how its interesting that when you draw 2 tangents from a point the angle they make at the point is always 180 – the angle made at the centre when you draw radii from the points the tangents meet the circle. This is always true, as tangents always meet radii at 90 so the other 2 angles in the quadrilateral add to 180.

While I was thinking about how nice and interesting this was, it occurred to me that this means that the quadrilateral mounted by 2 tangents from a point and the radii they meet is always a cyclic quadrilateral (as that circle theorem does work both ways).

When I was thinking about this I thought “that’s weird, that’s the exact same circle I was thinking about the other day when considering the angle at the centre theorem”. So the circumcentre of the triangle OAB will always generate this circle.

It then occurred to me that as the radii meet the tangents at 90 the line OC is a diameter, so its midpoint, D, is the centre of the circle. So the circumcentre of OAB will be the midpoint of the line between the centre of the circle and the point where tangents from A and B meet.

It also struck me that alternate segment theorem falls out nicely from this:

I think these are cool properties of circles. It’s nice to just sit and ponder on maths sometimes, and investigate stuff you’ve not really thought about.

If you’ve been pondering anything recently I’d love to hear about it. Also, if you’ve got any cool circle or circle theorem properties o might not know I’d love to hear them too.

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