## Fun with Cusineire

*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.*

## Symmetry and reflection

This week I’ve been working on symmetry and reflection with my year 9s. As part of this we looked at creating some Rangoli style patterns that had symmetries. After discussing how to reflect shapes in mirror lines and drawing some reflections I shower them some pictures of Rangoli patterns and gave out some squares of squared paper with various mirror lines on and some coloured pencils and gave them some time to get to work.

I was interested to see some of the different approaches. Most just used the squares to create patterns in the first instance. Of these there were two basic approaches, A – complete a pattern in one quarter and reflect into the other quadrants. B – each time they coloured a square they reflected it. I would have opted for option B I think, but watching the class work it seems those who used method A made less mistakes.

There were a couple of students who didn’t just colour squares, they created some triangles, trapeziums and other shapes in their patterns. They all took approach A.

*I’d be interested to hear how you would approach this.*

Once we had some patterns we looked at them as a class on the visualiser, discussing their approaches, what we liked about each one finding mistakes, looking at the reflective symmetries and also discussing rotational symmetry too.

Here are some of the patterns:

This student did a quadrant at a time then shaded what was left in blue when I told them time was nearly up.

This student was doing 4 squares at a time. I like the fact each quadrant is symmetrical too. (I know technically it’s not a symmetrical pattern, but it would.have been had we had a minute or 2 more!)

This student did a bit at a time and reflected each bit in the diagonal line.

I liked all three of these, and they gave rise to a good discussion as they had all gone for more than 2 lines of symmetry and all had rotational symmetry too. More than half of the students had done this, and I wonder if that’s because the Rangoli patterns I had shown them had also done this.

Unfortunately a lot of the others opted to take their work home and I didn’t think to take photos first as there were some awesome patterns.

## 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.

## Constructions

One topics I have never been a fan of teaching is constructions. I think that this is due to a few factors. Firstly, there is the practical nature of the lesson, you are making sure all students in the class have, essentially, a sharp tool that could be used to stab someone. I remember when I was at school a pair of compasses being used to stab a friend of mines leg and this is something I’m always wary of.

Secondly, the skill of constructing is one that I struggled to master myself. I was terrible at art, to the point where an art teacher kept me back after class in year 8 to ask why I was spoken about in the staffroom as the top of everyone else’s class but was firmly at the bottom of his. I explained that I just couldn’t do it, although it was something I really wished I could do. He was a lovely man and a good teacher and he offered to allow me to stay back every Monday after our lesson and have some one to one sessions. I was keen and did it, this lasted all through year 8 and although my art work never improved my homework grades did, as he now knew I was genuinely trying to get better. I have always assumed the reason I am poor at art is some unknown issue with my hand to eye coordination, and I have always blamed this same unknown reason for struggling sometimes with the technical skills involved in constructions. Since coming into teaching I have worked hard to improve at these skills, and I am certainly a lot lot better than I used to be, but I still feel I have a way to go to improve.

For these reasons I chose to go to Ed Southall’s (@solvemymaths) session “Yes, but constructions” at the recent #mathsconf19. Ed had some good advice about preparation and planning, but most of that was what I would already do:

*Ensure you have plenty of paper, enough equipment that is in good working order, a visualiser etc.*

*Plan plenty of time for students to become fluent with using a pair of compasses before moving on.*

He then moved on to showing us some geometric patterns he gets students to construct while becoming familiar with using the equipment. Some of these were ones I’d not considered and he showed us good talking points to pick out and some interesting polygons that arise. The one I liked best looked like this:

*This is my attempt at it, I used different coloured bic pens in order to outline some of the shapes under the visualiser.*

The lesson was successful, the class can now all use a pair of compasses and we managed to have some great discussions about how we knew that the shapes we had made were regular and other facts about them.

Next week we need to move on to looking at angle bisectors, perpendicular bisectors, equilateral triangles, and the such. I hope to get them constructing circumcircles of triangles, in circles of triangles and circles inscribed by squares etc.

Here are some more of my attempts at construction:

“Constructing an incircle” – I actually did this one in Ed’s session!

“A circumcircle” – I drew the triangle too big and the circle goes off the page. Interesting to note the centre is outside the triangle for this one.

“A circle inscribed within a square” – this is difficult. Constructing a square is difficult and that is only half way there if that. This is the closest I have got so far and two sides are not quite tangent.

“A flower” – nice practice using a pair of compasses and this flower took some bisectors too.

*If you have any ideas for cool things I can construct, and that I can get my students to construct, please let me know in the comments or on social media.*

## Manipulatives – the start of the journey

At #mathsconf19 I opted to go to a session by Pete Mattock (@MrMattock) that looked at using manipulatives to link abstract to concrete. I opted for this as manipulatives are something I’ve never been into and have always been quite cynical about, but I figured that as lots of people seem to speak highly of their use it couldn’t hurt to try and add them to the repertoire.

During Pete’s session he gave us some time to explore their use on tables. I was with some very nice people I didn’t know and they all were very experienced at using them and they managed to give me plenty of ideas for their use on top of the many great ones Pete had given me.

Roll forward a couple of weeks and I got my timetable for next year, for the first time in I can’t remember how long the number 7 appeared on it. And for the first time in even longer than that the number 7 was there with the equivalent code for a bottom set. I thought to myself “what an excellent place to start trying out manipulatives”. Then, as part of the schools Y7 catch up project the SLT lead for maths bought the department a number of manipulatives, including base 10, Cuisenaire rods, double sided counters and number sticks. I thought that the stars were aligning for this journey.

At the same time, I started wondering about the maths my daughter would be doing next year, when she goes into year 3. I downloaded the national curriculum and I happened across this book on the ATM website:

I bought it and had a read. There was some nice activities I thought I’d try and some used Cuisenaire rods. A few minutes in Amazon and 12.61 later I had ordered these:

I thought my daughter may have used them at school, but she had only used base 10, so we will be learning about them together. So far we have only used them to work on addition and subtraction, but she is very excited by them and I’m seeing some potential in them.

*I know many of you will be more experienced in the use of manipulatives, so I’d love to hear how you use them. Please let me know on social media or in the comments. Also, if you know of any blog posts or academic articles on their use I would be grateful if you could signpost me to them. I’ll post again about manipulatives when I have done more with them.*

## Area 48

Today I was looking at some of Ed Southall’s (@solvemymaths) puzzles on his website. I saw this one that I had not seen before:

I thought I’d give it a crack. You should too….. go on…. Did you get an answer? Well here is how I approached it:

First I did a little sketch, as I always tell my students to do:

I labelled the points with letters as this is normally quite a good way of keeping track of things.

I then decided to let AB = 1 (I chose that bit to be 1 as I knew a unit square would lead to lots of fractions, in hindsight this also let to fractions and AB = 2 would have been better.)

This gave me a few lengths straight off the bat, and I could find BD by Pythagoras’s Theorem and hence had the area of the larger square – which I need to answer the question.

I also noticed I had a RAT (ABD) and I knew the perpendicular sides, and therefore could work out the area.

I then looked at the triangle BCH. This looked like it would be similar to ABD but I took a couple of moments to justify it to myself before moving on, just in case….

If angle ABD is x then as DAB and BCH are both 90 and the angle sums of a triangle and on a straight line are both 180 then CBH and BDA must both equal 90 – x and CHB must equal x, hence they are similar.

They are similar and the scale factor is 2 (as BC is half of AD and they are corresponding sides).

Hence the Area scale factor is 4 and the area pf BCH is a quarter of the area ABD. As Area ABD = 1 then Area BCH = ¼.

From here I took the area of the two triangles away from the area of the square ACED to get the shaded area and put it over the area of the larger square. (Well, after momentarily putting it over the area of the smaller square like a fool!).

So here I had an answer, 11/20. I clicked on the comments on Ed’s website and saw some answers that were not what I had. This had me second guessing myself, so I thought about a different approach.

I went for a coordinate geometry approach (coordinate geometry seems to have taken over from trig as my brains go to method).

I chose the origin as the common corner of the two squares and called the point where the vertex meets the horizontal point B. This mean B’s coordinates were (1,2). I called this line l1 and could spot its equation was 2x. Part of the shaded area is the area under this curve between x = 0 and x = 1 so I calculated that area to be 1.

The perpendicular through B is the other line that bounds the top of our shaded region. I know the perpendicular gradiens multiply to -1 and I know it goes through point (1,2) so I could work out the equation of this line easy enough:

Then calculate the area below it between the values x = 1 and x = 2. This gave an area of 7/4.

So I had a total shaded area of 11/4 and could divide this by the area of the large square to get 11/20 again.

I felt happier now that I had the same answer though two different methods, and I stress to my students that this is what they should be doing with any extra time in exams. Doing different methods and seeing if they get the same answer!

*I hope you tried Ed’s puzzle, and if you did, please let me know how you approached it.*

## Interesting angle puzzle

Here is a nice little puzzle that has appeared from Ed Southall (@solvemymaths) I think it is a good little puzzle to get your brain going and one that should be usable in a secondary classroom as the maths is not particularly advanced. It would probably be a good one to get some students problem solving and I may give it to some students this week.

**My approach:**

I looked at this and assumed it is all regular. I labelled the three important points ABC and my first instinct was to draw a line from A to C to make a triangle. I decided not to do this, however, when I noticed that If I drew it to the point I have labelled B then I could get a nice isosceles trapezium:

From here it was just a case of using my knowledge of angles in quadrilaterals, other polygons, round a point etc to find the reflex angle required.

First I used knowledge of regular pentagons to see that angle AEF must be 162.

Then I used my knowledge of isosceles trapeziums and the knowledge that AEFB is an isosceles trapezium to work out that BAE and ABF are both 18.

Then I considered ABCD, again I know its an isosceles trapezium. I also know that ADGE is a square therefore i can work out that DCB and ABC both equal 72.

This means the reflex angle reuired must be 288.

*I’ve been looking at it further, and I’m not sure I can see any other ways that would work. But if you spot a different way then I would love to hear it.*

## 7 years on

Seven years ago today I started this blog. I had only been teaching a few years and had discovered Twitter and many teacher and math teacher blogs. I was enjoying reading the many blogs I’d found and had noticed some of them spoke of how they felt it had made them more reflective practitioners. In particular I remember Dave Gale’s (@reflectivemaths) blog Reflective Maths Teacher (I think he used a different host then though) had given me many ideas for the classroom and Dave had said his blog had helped him.

So there I was, lessons had finished o what was my daughters due date. There had been no signs of contractions and I had 30 minutes before a twilight CPD session, so I thought “why not?”. I opened this wordpress account and posted this short post reflecting on am extension task I had made up on the spot as two of my year 7 students had completed all the work I had planned for them before the lesson was up.

I didn’t write another post for about 5 months (I had a newborn baby and a full timetable), and not many people read any posts in that first year.

Since then my blogging has been up and down in its frequency. I’ve shared things I’ve done, shared thoughts on pedagogy, on education policy and maths in general. I have found blogging to be a good way to frame my thoughts on many things and start conversations on maths and on teaching with people I wouldn’t otherwise have communicated with. I feel I have learned a lot. It has made me a better teacher and a better mathematician.

I’m currently in a period of time where my blogging is more frequent again, and I hope to continue with that. I just thought I’d mark this little anniversary with a short post and say thanks to everyone who has read, commented, shared and discussed the thoughts I’ve posted here.

Onwards we go.

## The circumcircle and the law of sines

The other day I posted my solution to a lovely puzzle I’d seen that Ed Southall (@solvemymaths) had come up with, I assumed there would be better more concise solutions and ask for people to send me their solutions. I got a number of them, including this one from Xxxygorzao (@toodrunkforzero) :

At first one of the steps wasn’t obvious to me. It was the step that uses a / Sin A = 2r. I questioned Xxxygorzao who told me that it was a property of all triangles that the ratio of side / opposite angles is equal to the diameter of the circumcircle. I thought this was a rule I had forgotten, but after thinking about it I’m not entirely sure if it is or if it’s one I have never encountered before. Xxxygorzao offered to link me to the proof of this theorem, but I thought it would be nice to try and find one myself as it didn’t jump out as self-evident that it was true.

I started with a circle, and drew a triangle with 3 points on the circumference (n.b. I used a pair of compasses to draw the circle, but this was just because I’m rubbish at sketching circles, it is just for sketch purposes.):

I labelled the points A, B and C, and intend to use the notation A, B and C for the angles at each point and a, b and c for their opposite sides as is conventional in the law of sines. I then sketched chord bisectors and labelled the midpoints D, E and F. Then I labelled their intersection point, and hence the centre of the circle O.

I considered the triangle A, D, O. I know that angle ADO is a right angle and that AD is half of b, as it’s a perpendicular bisector. I also know that the hypotenuse is a radius.

I looked back at my circle and realised that by inscribed angle theorem AOC is equal to 2 x ABC, or 2 x B in this case.

AOC is an isosceles triangle made from 2 radii, and thus the perpendicular bisector of the base is also an angle bisector, giving angle AOD to be equal to B.

The rest then followed from right-angled trigonometry.

It’s a nice fact to know, and I found justifying it to myself this way was both fun and allowed be to gain an insight and an understanding into why this fact is. Obviously, I’ve only shown it for the ratio with B and b here, but the others can be shown in a similar way, and they don’t need to be as they follow from the sine rule.

## And rest

Here we are, we made it. Another year done and another long summer holiday just begun. This time of year I find it hard not to reflect on what had been, what’s to co.e and what might have been.

It’s been a tough year in our department. We had a couple of people on mat leave and a number more who had to have lengthy spells off due to a variety of illnesses and injuries. This left me leading a skeleton department for a large chunk of the year when we were in the thick of it with year 11. It was tough and there were plenty of challenges to overcome, but we made it through and came out the other end. I was lucky enough to have great support from SLT and from the maths lead of our trust, who spent a lot of time in our school. He helped a lot with the additional leadership responsibilities I had to take on and he got really stuck in in the classroom. I feel that the experience has certainly had a positive impact on my own leadership skills and I think it had improved the team, who all went above and beyond to ensure we made it through. Whatever happens in August I know everyone gave there all and we couldn’t have done any more.

By the end of the year, everyone was back. We had a much more settled department and have a great foundation to move onwards into next year. I think we are in a great place to push on and improve further. We have a solid team in place, a great curriculum and have made some plans I feel will give the students a great chance of success.

This year I completed my NPQML. I feel that too has been a benefit to my development and I’m currently looking at options for my next learning venture.

I made it to a #mathsconf for the first time in a while and hope to get to more next year. I feel this has given me further things to think about in my own practice and I have already begun to implement some ideas and strategies in my lessons. Next year I plan to follow these through further and am particularly interested in trialling more use of manipulatives in my classroom.

This year I have been able to have great discussions with some former pupils who are now maths teachers,, and recieved news that another is looking to embark on a career in maths research and lecturing. This has been a source of much pride and enjoyment for me and I know they will all be brilliant in their chosen careers. I’m glad they chose mathematics.

I’ve also managed to start writing more on this blog, which I have neglected a bit over the last few years. I feel that this is something that really helps me frame my thoughts, improve my maths and reflect on my teaching, so I hope to keep the momentum up with it.

Now is the time to rest, to spend time with my family and recouperate ready for September, when it all starts again. Thanks to all who read this blog and those who interact with me on here and on social media. I hope you all have a great summer and continue to interact, as I feel you all help me to continue to improve.

## Share this via:

## Like this: