## 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.

## 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.

## 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?

## 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.

## 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.

## 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

## Coloured squares and dots

Here is a lovely puzzle that comes from mathsbot creator Jonathan Hall (@studymaths) via Chris Smith (@aap03102) and his newsletter:

When I looked at it, my gut instinct was it had to be blue or green, if I was pushed I’d go blue.

So then I worked it out:

So it was green, then blue and black. I consider this for a moment and I didn’t feel like this was a very satisfying conclusion. I did notice that it went up and back down the same numbers, so I wondered what function we were looking at. If we discount the case of 1 square then it’s this:

It’s a quadratic function with roots at n = 6 and 0. These roots are when there would be 6 or 0 dots, but they aren’t involved here as we have discounted the 6 dots case and there is no 0 dots case, it would be trivial anyway.

I had the roots, so could see by symmetry that we have a maxima at (3,72):

I could, of course, have completed the square:

*Careful with negatives!*

Or I could even have differentiated:

A nice little puzzle, that can be generalised for more dots too.

The case if the single square is an interesting one. When you go out a level you only add 7 squares, but in each other case you add 8. The 8 come as if you move the top row up, the bottom row down then the left and right sides are each 4 short (the corner and the one next to them), but the special case of 1 is that you only need 8 to enclose it, so that’s only 7 more than you had.

## 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.*

## 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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