## 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.

## 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**.*

## 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.

## 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.

## Patterns, sequences and fractions

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

Meaning making with manipulatives

**Reference **

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

## Playing with Cusineire

Last year after a maths conference session from Pete Mattock (@MrMattock) and a number of other things happening I decided that I wanted to work on developing my own use of manipulatives in my teaching. I bought a box of rods for home and have spent some time playing with them along side my daughter. Last year I wrote this post on our early work with them.Since them we have messed around with them a lot, and have had some nice chats around them. Today I watched a nice twitter live session with Atul Rana (@AtulRana) and Simon Gregg (@Simon_Gregg) about their use (you can watch it back here) and I’ve also been reading a lot about their rods during lockdown. Here I want to share some of the things we’ve done with them over the last few days.Yesterday we were playing with the rods and she asked what she should do. I told her to make a pattern. One of the first things she came up was a staircase:I asked her if it was a staircase why it went up and down and up and not just up and she said, “that would make more sense”, and changed it.We discussed the order and I asked if we could do anything to it to make other shapes and she came up with this:Initially she just put another inverted staircase on top of the one she had done first, then she put the additional rods to make a frame. Before she had put the frame we discussed what each row was showing, and she said it was number bonds to ten. After we had discussed that I asked her what shape she had made, she said it was a ‘rectangle or a square’. I asked her which it was and she said she thought it was a square. I asked if there was a way we could know for sure, and she said there was, we needed to check if the side lengths were the same so she did that and decided ot was intact a rectangle.Today we played with them again, she made a face:Then she made this:Initially she said it was a “roof” but later decided it was actually a basket. We talked about the shape and I asked her if she knew what it was called. She said she didn’t, that she knew it had 4 sides but wasn’t a rectangle or a square, so I told her it was a trapezium and we discussed the properties of them.She then decided to do number bonds to 10:Then we looked at ways we could make 4:This was an activity that I’d seen Atul and Simon discuss and led to some quite interesting discussion. She decided to group the “ways” into distinct ones and repeat ones. I told her that the number of distinct ways to do this for a number was called the partition number. We also discussed the repeats and how she knew they were the same. She was good at describing the commutativity of addition but hadn’t heard the word commutative before. She then wanted to try with some other numbers:For 6 she decided not to do repeats, and initially only found these:Then added another:I find partition numbers really interesting, but after a brief discussion we moved on as I was about to go off on a tangent on Ramanujan’s Congruences and Modulo arithmetic which would have been over hear head just now.We picked a number to play with and she tried to find as many ways as possible to make it using inky red white and green rods:This led to some interesting discussions as to why we could do it all white or all green but not all red.I asked her what numbers we could make with all red and she said all the even numbers, so we looked at this and discussed why that is.We then looked at what other colours we could make those numbers with:She said all numbers could be made with just white so didn’t bother putting those on we talked about why the colours paired up, ie there are 5 reds for a ten as a red is 2 that means we need 2 yellows as yellows are 5. We discussed that this means multiplication was also commutative. I asked what numbers would be made by each rod and she said those in the timestable. We discussed this and the terms multiple and factor.She asked if there was always just 4 factors so I asked her to investigate 12:Obviously there is no rod for 12 so we made one out of an orange and a red and imagined it was a single rod. She was surprised how many factors it had.At this point we had been working a while and she was tired so we left it there. I can see that the discussion could easily have moved onto primes and their lack of factors, and squares and how they have an odd number of factors while all other number have an even number. And this would link back to our earlier discussions around commutativity.*This was an enjoyable way to spend time this evening and I hope we can do some more exploring with the rods in the days and weeks to come. If you have any great activities to do with the rods, or have written/read anything on this, then please let me know.**This has been the 4th post in what is a bit of a series around manipulatives, the others are:*Manipulatives, the start of a journeyFun With CusineireMeaning making with manipulatives

## Meaning making with manipulatives

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

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

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

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

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

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

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

## 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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