## 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.

## A great 1-9 puzzle

This number puzzle was one I really enjoyed and it came from @1to9puzzle :

When I looked at it I did think about setting up 8 equations with 8 unknowns and solving them as one big system. But then I figures there was probably a better way.

I looked at the sums and decided that the one summing to 10 would be a good place to start. That means I need 2 numbers that sum to 6. Which gives 1 and 5 (as 4 is already taken and we can only have 1 of each number). I knew the 2 on this diagonal needed to be 1 and 5 but wasnt sure which way round they were yet.

Then I wrote some number bonds to different numbers down. I considered the middle horizontal row. It needed to sum to 12. 9 and 3, 8 and 4, 7 and 5, 6 and 6. It couldn’t be 6 and 6 as I was only allowed 1 if each. I knew the 4 and 5 were already taken on the diagonal and in the centre so this line had to be 9 and 3.

The 5 and 3 couldn’t be on the right side together as if they were on the right that would leave me needing 10 to make 18. The 5 and 9 on this side would mean I needed 4 to make 18, so I needed the 1 in the bottom right. If I then had the 3 above it I’d need to add 14 but only had 1 more square so that meant I needed to put the 9 in the middle right. From there it was a case of simple addition and subtraction to fill in the rest:

I really enjoyed this one. Would love to hear how you did it, and do send me any others like this you find.

## 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.

## Circles in a semi circle

Came across this puzzle from Le Bécachel Sébastien @le_becachel:

I loved the look of it so I thought I’d give it a try.

First I drew a sketch, then redraw it withoutthe 3 circles:

Considered a triangle:

Went down a rabbit hole of trigonometry identities:

Figured that this was probably not the best route to go on so sketches it again:

Realised that due to the nature of circles the tangents cut it into 3 congruent sectors so the angle must be 60 degrees.

Used the fact that tangents meet radii at 90 and the symmetry of each sector to create a right angled triangle that allowed me to see that the radius if the small circle was a third the radius if the large circle. Thus the area of the small circle was one ninth the radius of the whole large circle.

We had 3 of the small circles so that’s a third of the large circle or 2/3 of the semi circle.

This was a lovely problem and I think this solution is really nice, just wish I’d not fallen down the rabbit hole on my way to finding it!

How did you solve it?

## 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**.*

## A hexagon and some interior lines

Today’s puzzle comes from Eylem Gercek Boss (@_eylem_99) and it’s a nice quick one that I loved, and includes a hexagon, which an awful.lot of puzzles I find at the moment seem to do!:

Initially there wasn’t an obvious solution to me so I sketched it put and labelled a load of things.

Then started writing what I knew:

I had 3 parallel lines equally spaced, so I had 2 similar triangles. I knew the diagonal was double the side length. I had enough to form an eqution:

2x = (1/2)x + 12

3x = 24

x = 8

From here I could easily work out the area:

A nice little problem that got me thinking about problem solving. I didn’t see a solution immediately, although perhaps I should have, so I just started jotting down what I knew until I saw a way forward. This is a key still that students need, just being able to consider what they do know an look at what that means in the context of the question. I think this question would be really good to use with students to model that thought process.

Do you have a different solution? I’d love to hear it.

## Playing with pattern blocks

I’ve been having a lot of fun at home playing with Cusinaire rods with my 7yo daughter. It’s been great, she’s been learning a lot of maths through playing, we’ve been having a lot of fun and I’ve been learning a lot about the rods and how I can use them in lessons. I’ve not only learned how they can be used in the activities we’ve been doing but I’ve seen during the sessions other places they can go which lead to higher level maths that would be more suited to much older students, there will be links to the posts on these sessions at the end of this one if you have missed them.

I’m keen to explore other manipulatives, and when I finished reading the Cusinaire book (Ollerton et al., 2017) I bought another ATM book on Pattern Blocks (Gregg, 2020). I read the first few pages and thought they sounded fun so I purchased a set if block for home (although the colours were wrong again! Turns out typing the name of a manipulative into the search functionof the biggest online retailer doesn’tget youbthe right colours, which knew?).

When they arrived I still hadn’t read too much if the book, but my daughter was interested in the blocks so we got them out. They came with some cards and she wanted to make the shapes on the cards. Here are some of the pictures:

Then we talked about the shapes. She knew what some if them were but not others. She called the rhombuses diamonds. She asked questions about the blocks and I showed her that the side lengths were all the same apart from one of the sides on the trapezium which was double.

She asked if I’d read anything in my book we could do so I told her one of the tasks it suggests was to try and make the shape of the red hexagon out of the other shapes so she tried this:

She made the top 3 very quickly, but then didn’t think she could do anymore. I said she didnt have to use just one colour but she still struggled. I told her to look at the ones she’d made and look for similarities and differences. This was enough of a hint to make her see how to make the rest.

I then asked her if she could make any other hexagons. When she was making one I jumped the gun and said well done when she had made this:

But she said “no I’ve not finished” and added a piece to get this:

I found this quite interesting. She didn’t seem to think that a shape could be finished if it was convex like the first one. We talked about what makes a hexagon and how both versions were.

I then removed 2 purple squares and 2 blue rhombuses (rhombii?) from the second shape and asked if this was a hexagon. She agreed it was and we discussed the similarities and differences in shape between this and the red regular hexagon. Both have 6 equal sides, but they aren’t the same. I did tell her what a regular hexagon was at this point and it wasn’t a term she’d heard before, we didn’t speak long on it though so I don’t know if she will remember.

She also came up with this one:

And this other regular hexagon:

I like this one, and we talked about the similarities and differences between this and a lone red hexagon. I didn’t think she would be ready for a discussion on length and area scale factors yet, but this strikes me as an excellent visual representation of this and it’s certainly one I could see using in a KS3/4 class.

At this point we started talking about how the shapes fit together. I did mention the terms tesselation and tiling but didn’t dwell on them. I asked her if she could find single shape patterns that did and she came up with this:

She said it looked like a honeycomb, and we discussed that bees build them in this shape and talked about why. I also showed her pictures of Giant’s Causeway and the hexagon stones there and discussed how they occur in lots if places in nature.

She then wanted to make “honeycombs” of other colours:

She really liked this one as she said there was an extra 4th hexagon hidden in it. Which she in fact noticed before I did.

Her original green shape was:

Which doesnt look like a honeycomb but which she liked because it looked like it was “on fire”.

She didn’t say much about the yellow, but I thought it looked like a set of screws.

That was about it for the session. She played a bit more and I really liked these shapes made:

Then we packed away. It was really fun for both of us to make these shapes, it’s the first time I’ve really played with Pattern Blocks and I can see they will be great for building my daughters maths. I have also started to see where I might be able to use them in my lessons, so a win win all round.

*This is the 7th post in a series about the use of manipulatives in maths teaching. You can read the others **here**:*

Manipulatives – the start of a journey

Meaning making with manipulatives

Patterns, sequences and fractions

Making numbers and quadratic sequences

__Reference:__

Gregg, S. 2020. *Pattern Blocks. *Derby: Association of Teachers of Maths.

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

## A hexagon in a quarter circle

This puzzle comes from Catriona Shearer (@cshearer41) and it is one that had me thinking quite a bit:

First I drew the sketch:

To work out the area I needed r, I tried a few things out:

But wasn’t getting far, when I drew on AC I wrote:

I knew it was and equilateral when I extended the sides of the hexagon, but then I wasn’t sure if A would fall at the centre of the circle. So I thought I’d better do some more investigating:

I played with the sketch a whle then realised that as AE is a perpendicular bisector of the chord GF that the line AE must go through the origin of the circle. And that as A shares a y value with the origin of the circle point A must be the origin:

This meant it was an easy solve from here as I had a 2 sides of a triangle, the angle between them and knew the third side was r:

Thus I could calculate the area of the quarter circle.

I liked this problem and the fact it made me question my assumptions and really think about what was going on. I suspect there are nicer solutions though, if you have one, please let me know.

## 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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