### Archive

Archive for the ‘Pedagogy’ Category

## 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 others here.

## 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.

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

June 3, 2020 1 comment

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

Fun with Cusineire

Meaning making with manipulatives

Playing with Cusineire

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.

## Patterns, sequences and fractions

May 31, 2020 3 comments

When my daughter and I play with the Cusineire rods we always seem to run out of reds. So I ordered a second packet the other day, and when it arrived it looked like this:

I can’t get over how weird it looks with the colours being different, but it’s here now so we are going to have a play with it anyway.

Yesterday we got then out and she started making shapes:

These looked like some things I’d seen in the ATM book (Ollerton et al. 2017) on using rods. In the book they called them boats and this made sense to me as far as their shape went. I asked her which the odd one out was and she identified the one with a purple base. I asked why and she said that it was because all the others got one smaller each time as we went up. So I asked her to try and change it so it fit the pattern.

First she did this. Which was interesting. She misinterpreted what I was asking (I was trying to get her to alter the boat that didn’t fit) and made a whole new boat that would in fact have been the next in the pattern. I was quite impressed and we discussed this for a while and then I asked what other ones we could make:

Again she’d found many boats that fit the pattern, but the pesky purple base still didn’t, so we discussed it and corrected it.

We than talked about how many each boat would be worth if the grey cubes were worth one.

She calculated a few and noticed they were going up in 3s, so we predicted some bigger ones and counted to check.

I then asked her why they went up in 3s, and she didnt know so I got her to try and add some rods to one to make it be the same shape as a bigger one:

She could then see we were adding one to each level and there were 3 levels.

We then looked at other patterns we could make:

We started with these L shapes and I asked her if she could find them all:

Again we talked about their value if the small cubes were one. I asked her to predict what the values would go up by. She initially said one. I asked her why and aa she was explaining she changed her mind to do. We worked some out to check.

Then she decided to make squares:

This led to similar discussions a d investigations into how much they were increasing by.

Around this point I asked what would happen if the white cube was 2 or 3 and we discussed that. I then asked what would happen if a larger number was 1. She didn’t understand at first so we made some partitions:

I said, if the brown one is “one”, how much is the red one worth. She reasoned that it must be worth a half as there were 2 that fit a brown. We then discussed the grey and blue values.

And then had a similar discussion around this:

Which was interesting and gave her a better understanding of fractions, she had never met 1/6 before.

She then started making trains the same lengths:

We talked about why these were the same, ie the commutative property although we didn’t use the word today. I then asked her to see which other rods she could make the same length with:

We discussed why these worked and others didn’t. Linked it to times tables, discussing the terms multiple and factor. We then wrote the number sentences that they would represent if the grey cube was 1:

After this she started to see what other shapes she could make. She made some triangles:

We talked about these triangle and what they were called. She hadn’t heard the terms equilateral or isoceles before. She knew what a right angle was but hadn’t heard the term “right angled triangle”. I asked her if she could make another right angled triangle. She noticed the lengths were 3,4 and 5 so she tried 6,7 and 8:

We discussed why it wasn’t a right angled and she then suggested doubling each side:

We discussed this a little and finished for the day.

A fun and enjoyable time for both of us, and the playing we are doing is both helping her mathematically, and helping me understand the rods and see where they might be useful in my teaching going forward. So it’s a win win.

This has been the 5th post in a series on use of manipulatives in maths, you can find the others here:

Manipulatives – the start of a journey

Fun with Cusineire

Meaning making with manipulatives

Playing with Cusineire

Reference

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

## Playing with Cusineire

May 28, 2020 5 comments

## Part whole division – why?

I’ve been thinking a lot about division recently. I wrote this here a short while ago about dividing by fractions, then I was sent a document by Andrew Harris (2001) entitled “Multiplication and Division”, which I was asked to read as part of a series if CPD sessions from the local maths hub, then a number of different people have asked me questions about division recently too. I think probably for most this is due to helping their own kids with maths and meeting methods and structures that they aren’t familiar with, as they weren’t taught when they were at school themselves.

The main thing from friends that keeps popping up is using part whole models for division. And funnily enough it is one of the structures I was considering after reading the Harris document and looking at the distributive law and what higher level topics this underpins in later maths.

So what is it?

Using the part whole method for division is where you split a number into 2 or more parts before dividing then add your answers back at the end. For instance, if you want to divide 486 by 6 you can split it into 480 and 6. The benefit of choosing these numbers is that 48 is in the six times table. So you can see that 48÷6=8 so 480÷6=80, then you have 6÷6=1, add them together and you get 486÷6=81.

This structure, or method, is a very common mental strategy used by lots of people when dividing numbers in their head. Lots of those people will never have heard the term “part whole model” and will not have seen it laid out in a pictorial manner as students today will, but they will use that structure nonetheless. I myself was using it as a mental strategy a long time before I’d heard anyone refer to a part whole model or seen the visual representations.

What we are doing when we do this is using the distributive law of multiplication and division to break our problem into chunks that are easier to manage.

One of the questions I was asked was “is there a rule to how you split it up?” The person who asked me was wondering if you always split it up into hundreds, tens, ones etc or if you could do any. I explained that it didn’t matter, and that actually the divisor would normally be important in deciding. For instance if you were dividing 423 by 3 it wouldn’t make much sense to use 400 as this isnt divisible by 3. It would be more sensible to choose 300 (÷3=100), 120 (÷3=40) and 3(÷3=1).

But why not use short or long division?

This is a question I’ve seen a lot of times from a lot of people. They see the part whole method as a long and clumsy way to solve problems that they can solve easily using one of the two standard algorithms. I can see the point in asking, the algorithms are far more efficient as written methods. But that’s not why this model is taught. No one expects students to get to their GCSE and start drawing part whole models to solve division problems. The visual representations are their to help build an understanding of what is going on, an understanding of the relationship between numbers and mathematical operations. In this case it’s to build an understanding of how the distributive law works and to give a good mental strategy for division. It even helps understand how the long and short division algorithms work, as they are both based on splitting the dividend up into parts. There must come a point when these structures and representations are removed and students move to the abstract, but that doesn’t devalue their importance to that learning journey.

What else is the model used for?

The idea of a part whole diagram is introduced way earlier than this. Students get used to partitioning numbers into part whole models while working on addition and subtraction. It helps then see at that level that they have a relationship, that they are the inverse of each other. So when students come to meet this model for division it’s a small step on what they were already doing.

These are similar to some of the earliest part whole models my daughter did when she started school. They were being used to show place value, and also to show how addition and subtraction work and interact. For both these tasks this model is an excellent visual representation to help students understand the concepts.

Part whole models can also be represented as bar models. Here the one on the left can again be used to show either place value or addition/subtraction. The one on the right is an early algebraic model, and if we are told that x+2=9 we can use this representation to show why x must equal 7. This representation is more effective if students are familiar with it from their earlier mathematics.

Building on this we can show the distributive law when it comes to multiplication:

And show how that links to division:

As we go further into maths this idea of part whole division comes up again and again. One place that springs to mind is when calculus is first introduced at A level. One of the first things that we teach is how do differentiate and integrate polynomials with different powers of x. And a favourite style of question from examiners is this:

Or its derivative equivalent.

The easiest way to do this, when it comes to integration or differentiation, is to rewrite the fraction as separate terms:

What we have done here is used the part whole model to divide the expression on the numerator by x^2. We could draw that in our part whole model:

I wouldn’t advise that, its unnecessary, but having a secure knowledge of that model and how it works due to the distributive law is key to understanding how and why we can simplify this fraction in that way.

I’ve thought a lot about division recently, and I’m sure I will continue to do so, so if you agree,disagree or have anything else to add please get in touch either in the comments or via social media as I’d love to hear your views.

## Fun with Cusineire

July 18, 2019 5 comments

This is the second post in what I hope will become a long series about using manipulatives in lessons.

Last week I posted about how I was going to try and I corporate more manipulatives into my lessons, and that I’ve bought a set of Cusineire Rods for home to play with with my daughter. I’ve not manages to really do much in lessons since, the week has been disrupted by a couple of drop down days and sports day, and the lessons I’ve taught have mainly been around construction and loci, and symmetry and reflection.

I did, however, manage to have a play with some at home. My daughter was interested by the rods, and wanted me to show her some of their uses. First we looked at how they can be used to find number bonds to all different numbers, then we used this to look at adding and subtracting.

She uses Dienes base 10 blocks at school for similar so she started with just the 10 rods and the 1 cubes and showed be how she would use these at school. I then talked to her about how we could use our knowledge of number bonds to do the same thing but using all the rods. This was a nice discussion and allowed be to see some potential benefits to building number fluency with rods over dienes blocks.

She then showed me how she can use manipulatives to divide and to work out a fraction of something. The only fractions she really knew about were 1/2, 1/3, 1/3, 2/4 and 3/4. This led us to a discussion about the nature of fractions and their link to division. She knew that finding a quarter was the same as dividing by 4 and finding a half was dividing by 2 so I asked about finding other unit fractions showing her the notation and she made the link easily.

We then used rods to look at two of the fractions she knew, 1/2 and 2/4. She was surprised to see they always came out the same, and we used rods to investigate this and discussed the nature of equivalent fractions.

She then asked whether you could use the rods to multiply, I thought about it and came up with using them to create arrays:

This was 2 fives. Initially she was counting all the white blocks to get an answer, but after a bit when one of the numbers was one she could count in she started counting in those.

We looked at some where we were multiplying the same number together and I asked her if she noticed anything similar between these shapes and different to the ones we had done before and she picked out that these were squares and the others rectangles. This led to a good discussion as to why this was, linking to the basic properties of squares/rectangles and introducing the terminology square numbers and what that means.

I then looked at these two:

We had done 3 x 4 first then I said to do 4 x 3, she said “it will be the same because it doesn’t matter which way round they are”, so we did it anyway to check and talked about why that was. I tried to incorporate the words congruent and commutative into the discussion, but I think they went over her head.

At this point her role changed to teacher and we had to teach all these things to her dolls…..

It was fun to play with Cusineire rods like this, and the mathematical discussion they provoked flowed very freely, so I can certainly see that thIs could be very helpful in lessons.

In other manipulative news: I had 20 minutes or so free earlier and spend it looking at Jonny Hall’s (@studymaths) excellent mathsbot website. In particular his virtual manipulatives section. I found what I think to be some good ideas for algebra tiles and double sided counters and think that virtual manipulatives may be a very good way of getting these things into lessons.

## Primed

Recently I’ve seen a couple of things on twitter that I’ve thought quite interesting because they have got me thinking about the way I tackled them.the first was this:

It was shared by John Rowe (@MrJohnRowe) and posed a nice question looking at two cuboids that had the same volume asking for the side lengths. It said to use digits from 0-9 without repetition. My immediate thought was to consider prime factors:

2, 3, 2^2, 5, 2×3, 7, 2^3, 3^2

I only had one each of 5 and 7 so discounted them. And I had an odd number of 2s. This meant I’d need to discount either the 2 or the 8. Discounting the 2 left me with factors of 2^6 and 3^4 thus I needed 2^3 x 3^2. I had a 1 I could use too so I had 1x8x9 = 3x4x6. This seemed a nice solution. I also considered the case for discounting the 8. I’d be left with 2^4 and 3^4 so would need products of 2^2×3^2. 1x4x9 = 2×3×6. Also a nice solution.

Afterwards I wondered if John had meant I needed to use all the digits, I started thinking about how this could be done and realised there would be a vast amount of possibilities. I intend to give this version of the the problem more thought later.

The next thing I saw was this:

From La Salle Education (@LaSalleEd). Again it got me thinking. What would I need to get the product 84? The prime factors I’d need would be 2^2x3x7. This gave me only a few options to consider. 1 cant be a number as the sum will always be greater than 14 if 1 is a number. 7 has to be one of the numbers, as if its multipled by any of the other prime factors we have already got to or surpassed 14. So it could be 2,6,7 (sum 15) or 4,3,7 (sum 14) so this is our winner.

These problems both got me thinking about how useful prime factors can be, and they both have given me additional thoughts as to what else I could and should be including in my teaching of prime factors to give a deeper and richer experience.

Prime factors can crop up so many places, and I feel sometimes people let them get forgotten or taught in isolation with no links elsewhere. I always use them i lessons o surds and when factorising quadratics but know lots don’t do this.

I will write more on prime factors later, but for now I’m tired and need sleep. If you have any thoughts on prime factors or any additional uses not mentioned here I would love to here them. Please let me k ow i the comments or on social media.

N.b. The La Salle tweet includes this link which takes you to a page where they are offering some great free resources.