## 1-9 math walk

Today I want to look at another puzzle I found on math walks (from Traci Jackson @traciteacher):

I love these 1-9 puzzles, and thought I’d have have a crack.

First I considered the 9, with the 1 gone already that means that the 9 must share a line with the 2 and the 3 to make 14.

That means that the 4 shares 1 line with a 2 and one with a 3. That means the 4 is one lines 4,2,8 and 4,3,7.

I considered these lines:

If I put the 7 on the left, I’d need the 6 at the intersection of the green lines. That would also mean that the 2 was above it, but I’d need another 6 on that line which doesn’t work.

So the 8 must be on the left and if we follow through we get:

I enjoyed this puzzle, if you have any cool 1-9 puzzles do send them on.

## Step perimeters and infinite coasts

A few weeks ago I came across this picture on twitter:

I looked at it and thought “13cm, that’s a great question to use for perimeter.” I like questions like this, where you can show that if you invest the sides to the opposite sides of little rectangles the perimeter stays the same. I’ve seen them referred to as “step perimeter” and “cross perimeter” problems. Usually it’s the cross version I use, or a rectangle with a smaller rectangle cut from a corner. I liked this question as it had many different cuts and I feel would be great to open up a discussion on class.

I then thought about the problem a lot more, you could make many cuts. You could make them at irregular intervals like this, or you coukd make them at regular ones. You can look at it many ways. I started thinking about using more and more regular intervals and how it would start to look like a straight line. I then started to consider that. Intuitively it feels like as the intervals get infinitesimally small the limit should be the straight line. But that’s not what happens.

Bizarrely after I’d been thinking on this for a long time I was scrolling through twitter and saw this:

Someone else had obviously been having similar thoughts to me.

I started thinking about it again. I changed it slightly in my head, I changed it to a 5cm by 12 cm rectangle. This was so i could visualise the difference better.

Obviously due to Pythagoras’s Theorem the diagonal of a 5×12 rectangle is 13. But the distance we get by travelling in a series of steps that are parallel to the sides is the semi perimeter- in this case 17. That’s 4cm difference.

What’s going on here? It feels counterintuitive and almost paradoxical. What is going on is that no matter how small the steps are, it’s still longer to take them than it is to cover the distance diagonally. And the sum of all these differences will always be 4 for a 5×12 rectangle.

When you draw them so small it looks diagonal, it’s not actually diagonal. Its still steps. But you cant see them due to the width of the line and the inaccuracies in drawing and human vision. If you zoom in then the steps are their.

It made me think about the coastline paradox. Which states that you can never accurately measure a coastline as it is fractal in nature. The smaller the unit of measurement you use the longer the coastline will be. Meaning that the length of the coastline grows to infinity as your unit of measurement shrinks towards zero, giving coastlines infinite length of you use infinitesimal measurements.

This kinda feels the same as the case if the steps that create the illusion of a straight line. In reality you can’t physically measure an infinitesimal length and you will find a length for the coastline. And in reality you won’t be able to draw the steps that are that small. But it’s a nice interesting result. One I’ve enjoyed thinking about so far, and one I expect I’ll be thinking about some more

## Returning numbers

I’ve recently discovered the website “Math Walks“, by Traci Jackson (@traciteacher). The site is full of pictures taken on walks that have happened during quarantine where Traci has used chalk on a paving stone to create a maths prompt to aid discussion. Sometimes they are puzzles, sometimes they are sequences, and sometimes another maths picture and they always seem to get me thinking about maths. This one is one I’ve been thinking about today:

It’s a great visual, and I really like it on so many levels. Obviously the task if ti find a solution that when you follow the path round you get your original answer. It struck me as interesting that this can be accessed on a number of different levels.

I could give this to my daughter and she would be able to complete trial and error and eventually find solutions to each of these, but theres much more too it than that.

I thought about how I would tackle this problem and decided I would use algebra:

I quite liked this as a forming and solving equations exercise, I think it’s accessible but bit too easy. Many students may struggle with the notation around the forming and many may get confused with the order they need to do things in.

I considered how one might challenge a student who does just guess, and I feel that asking them to prove whether their answer is the only one or not would be a good follow up question in this case, I think it would be unlikely that many using trial and error would get both answers for the one which includes a square.

I then considered if there was a trick to generating these puzzles, presumably you start with your answer then you can make sure you always get back there.

I think these ones are lovely, and I hope to use them at some point when we get back. I’d love to hear your thoughts on them. How would you approach them? How would you generate them?

## Circles and an octagon

Here’s an interesting puzzle that came via Diego Rattaggi (@diegorattaggi) and involves circles and octagon.

I started as always with a diagram:

I labelled some sides up, then changed my mind and changed labels as I was thinking about taking a coordinate geometry approach and didn’t want to have used x and y. As it happened I didnt use that approach anyway. While looking at the sketch I realised that the triangle AOC was a right angled isoceles. Due to an error a few weeks back I wanted to double check I wasn’t making an assumption here so did some work to justify this was the case:

I was using some similar reasoning to this hexagon puzzle, I could justify that to had to be an isoceles, and that extending the lines gave an isoceles, I could just that the vetex was definitely on diameter I’d drawn and was equidistant from both circles in x and y, but felt that wasn’t enough, and if it wasn’t definitely the centre there could be multiple solutions, then I saw a different version in my screenshot:

Once I had this information it was fairly straightforward using Pythagoras’s Theorem:

At this point I realised the ratio if the radii squared was the same as the area so that’s all I needed:

I got to the end and realised I had my fraction upside down so I flipped it over.

This was an interesting puzzle, and I think I will need to think further on the case where the centre being the exact of the right triangle wasn’t specified. I might need to look on geogebra.

## 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.

## Circles on a line

I saw this lovely question from Mr Gordon (@MrGordonMaths) the other day:

I looked at it and even though it said it was GCSE maths only it didn’t look at all obvious how to find an answer. It did look interesting though, I wondered how my y11s would get on with it. I thought I’d give it a try:

As always I drew a sketch:

I was looking at straight line shapes I could draw and realised the trapezium was the better option in this case:

From here it was a bit of Pythagoras’s theorem:

Which gave me all I needed for the final answer, which is 1:4. (Obviously I discounted the trival R=0 as it doesn’t make sense in the context of the question).

A nice little puzzle that I can see could be rather taxing for students despite using only concepts they will have learned in KS3 and 4. It’s the sort of question that can really help with problem solving ability and is one i will definitely try on my year 11s when we are back fully.

I’d love to see how you solved this one, especially if you took a different approach.

## 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**.*

## 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.

## Tilings and areas

My daughter and I had another play around with pattern blocks. Firstly we played around and made some patterns. She made this one that was pretty cool:

We talked about tiling the plane and how shapes tesselate. Looking at which shapes fit together. Then I asked her if she could make a repeating pattern.

She came up with this one. Which wasn’t exactly what I meant but cool non the less.

Then she made this one that was more what I had meant. At this point we discussed which colour had more shapes and which took up more of the area.

We had similar discussions about these two tilings. We discussed how the red and yellow had the same amount if area in the red yellow and green one even though the yellow had twice as many squares. She showed this by making a hexagon put if the trapeziums.

She said the green, blue and purple one looked 3d.

I agree. I mentioned briefly that it was to do with the angles if the lines and that you can get dotted paper to help draw 3d which has dots at these angles. We talked briefly about rotational and reflective symmetry too.

She then made a hexagon:

We talked about how much bigger it was. She said it looked about 4 times bigger. We then discussed what this mean, and looked at the areas. Counting triangles.

I showed her that we could do it without counting triangles. We then looked at the side lengths of the hexagons and discussed how and why this scale factor was different to the area one. I think this photo of the hexagons is an excellent visual to use when looking at similarity in secondary school. Normally I just talk about squares and rectangles but can see an excellent set of visuals using these shapes.

We then started to look at fitting shapes together round points and on a line. And we found that if you put the thin blue rhombuses together on a line you can get some cool patterns:

We didn’t get into angles that much, but I can certainly see this could be a great entry point to those discussions in future. I can also see that as well as similarity there can be further discussions around area and perimeter that build from using these shapes and I hope to explore this more in future sessions.

This is the tenth post in a series looking at the use of manipulatives in maths teaching. You can see the othershere.## Share this via:

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