## 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.*

## Modelling in class

Recently the idea of modelling is one that appears to be following me around everywhere. I mean that in the sense of modelling as a teaching strategy, not that Calvin Klein is stalking me and urging me to take to a cat walk for him.

The repeat appearance of discussion around modelling has got me thinking about it a lot. At the recent #mathconf19 many of the sessions discussed modelling. Ed Southall (@solvemymaths) did some great modelling on constructions and suggested many ways use it to improve outcomes, Kate Milnes (@katban70) talked through modelling a mathematical thought process and using it to help students achieve their own and Pete Mattock (@MrMattock) looked at visually modelling abstract ideas to make sense of it.

A recent CPD session I attended split the group into two and a different teacher taught each group how to construct an origami crane. One taught using modelling and instructions while the other went out of her way not to and then the different outcomes were discussed.

The trust I work in sees modelling in the classroom as best practice and it is encouraged in all lessons. This is similar to stories I hear from friends in other schools and trusts in the local area.

Then I read this piece by (@mrgmpls) which spoke about the “norm” in lessons being that students are given problems and expected to struggle their way through with minimal input because “without struggle there is no learning”. The blog post was arguing that this is not the best way to teach and pointed to many examples of recent posts about “desirable difficulties” and the such as evidence that this anti-modelling feeling was very prevalent in education today.

This got me thinking on a few levels, firstly it made me think about struggle in the classroom. I’m a firm believer in the idea of modelling. I think that modelling how to do sometime a good solid worked examples should be a staple of any teaching. But I also see the need to struggle in the classroom. If we model processes and have students then complete basically the same question following the model and never get them thinking about it again then we open them up to the possibility of becoming very unstuck in an exam if a topic is examined in a different way.

For me, this means that students need to learn the processes and the conceptual understanding of the topics together. I would also argue that completing exercises of similar questions to embed these is a very good idea. However, there needs to be some point when students need to learn to apply their processes and knowledge outside of their comfort zone. For instance, when teaching trigonometry I would teach non-right triangles and right triangles separately. I would teach sine and cone rules separately, but I would always incorporate some lesson time at the end to a series of problems where they have to deduce which process, or processes, they need to use to be able to answer the question. I might even model my thought process for them, but they will then need to think about why they are doing this and apply it.

I thought that this would be a common theme in all classrooms, so the second question I had from the article was “is there really a feeling of anti modelling at play?”. Having discussed it a bit with the author I discovered that he is based in the US, and I got to thinking that maybe it might be a US vs UK idea, or that perhaps it was even just an idea limited to the state he teaches in, so I tweeted out asking if any on edutwitter were in schools where modelling is discouraged.

I was surprised to discover that actually there are some UK based teachers who are discouraged from modelling in the classroom. This makes me wonder how widespread this is, and what the rationale is for discouraging modelling. If you are in a setting that discourages modelling, or are against modelling, I’d love to hear about it and the reasons behind it. Please get in touch via the comments or on social media.

## When will I use this?

Recently I read a rather interesting article from Daniel Willingham about whether there were people who just cant do maths. It was a very good read and I hope to write my thoughts on it later, when I’ve had more time to digest the material and form some coherent thoughts, but there was one part that set me off on a train of thought that I want to write about here.

The part in question was discussing physical manipulatives and real life examples. Willingham said that there is some use in them but that research suggests this can sometimes be overstated as many abstract concepts have no real life examples. He then spoke about analogies and how they can be very effective in maths of used well.

This got me thinking, earlier on the day a year 12 student had asked me “when am I ever going to use proof in real life?”. This type of question is one I get a lot about various maths topics, and my stock answer tends to be “that depends what career you end up in”. Many students, when asking this, seem to think real life doesn’t mean work. A short discussion about the various roles that would use it and that its possible they never will if they chose different roles but that the reasoning skills it builds are useful is usually enough and certainly was in this case.

It does beg the question though “why do they only ask maths teachers”? Last week when a y10 student asked about “real life” use of algebraic fractions I asked him if he asked his English teachers when he’d need to know hiw to analyse an unseen poem in real life. He said no. I asked if he thought he would. Again no.

So why ask in maths?

The Willingham article got me thinking about this. There has been, throughout my career, a strong steer towards contextualising every maths topics. Observers and trainers pushing “make it relate to them” at every turn. But some topics have no every day relatable context.Circle theorems, for instance, are something that are not going to be encountered outside of school by pretty much any of them. So maybe thats the issue. Maybe we are drilling them with real life contexts too much in earlier years, and this means when they encounter algebraic fractions, circle theorems or proof and don’t have a relatable context the question arises not from somewhere that is naturally in them, but from somewhere that has been built into them through the mathematics education we give them.

Maybe we should spend more time on abstract concepts, ratger than forcing real life contexts. Especially when some of those contexts are ridiculous – who looks at a garden and thinks “that side is x + 4, that side is x – 2, I wonder what the area is?” (See more pseudocontext here and here).

*What do you think? Do you think we should be spending more time lower down om the abstract contexts? Please let me know in comments or via social media.*

## Another Year Over

*So this is summer, and what have you done, another year over and a six week holiday just begun. – *What Lennon may have written had he been a teacher.

I know what you are thinking, “why are you up so early? It is sunday and it is summer!” And you are right to wonder. Usually its my body clock that makes it so, but this year my 6 year old daughter has taken on that responsibility. Argh.

This year has been a good one for me. Tough in places, but enjoyable over all. I work at a school where I like my colleagues, like the vast majority of the students, feel that the department I work in is strong and that the senior leadership know what they are doing and are making decisions that are pushing the school in the right direction. When I moved to my current school, which was in the process of academy conversion following a 4 Ofsted grading, part of the draw was the chance to be part of affecting a positive change and improving the chances of the students. In the 2 years I’ve been here I’ve seen massive improvements and can see the trajectory we are on.

There’s been some tough times, but there has been some good ones too and I look forward to next year and our next steps in the journey.

This year I’ve spent a lot of time improving subject knowledge amongst the department. I feel this is something that needs to continue. It was made necessary this year as we had a number of non specialists and trainees in the department and most of the experienced maths teachers had never taught the new content that is now on the GCSE. This is something that needs to conrinue next year. We have no non specialists next year, but do have NQTS, trainees and staff who still wont have taught the new content. These sessions allow not only for building content knowledge but also for discussing subject specific pedagogy and possible misconceptions.

I’ve also thought a lot about transition from KS2 to KS3, this has been driven in part by a need to improve this area and in part by a fascinating workshop we hosted led by the Bradford Research School. I hope to write more about the workshop and the fascinating findings I’ve had while looking at KS2 sats data, nationally and locally, and the KS2 curriculum. Suffice to say, if you are a secondary teacher who hasn’t looked, your year 7s probably know a considerable amount more than you think they do on arrival.

The KS2 sats provide some great data and there really is no need to retest students on entry. Except maybe the ones who have no data. I’ve always been averse to KS2 SATS but the data they produce is so rich I feel I’m coming round to them. Although I’m not sure I agree with the way they are currently reported and I certainly stand against the idea of school league tables.

I’ve not written as much as I would have liked on here this year, and I hope to change that going forward. I didn’t decide to blog less, it just sort of happened, so hopefully I can turn that around.

Now it’s summer, I’m looking to relax, have fun and to teach my daughter how to enjoy a lie in….

## An excellent puzzle – alternate methods

Yesterday I wrote this post looking at a nice puzzle I’d seen and how I solved it.

The puzzle again:

Lovely, isn’t It?

After I published my previous post I wondered if I may have been better using a vector approach or a coordinate geometry approach. So I gave them a try.

**Coordinate Geometry**

I started by sketching the figure against an axis.

I place the origin at the centre of the circle, worked out the equation if the circle and the right leg of the triangle and solved simultaneously for x. Giving x =1 and x=1/3. These x values correspond to half the base of each triangle, which shows the scale factor from the large triangle to the small one is 1/3. As the area of the large one is rt2 this gives the area of the small as rt2/9.

I like this method, probably a little better than the one prior to it.

**Vectors**

First I sketched it out and reasoned I could work it out easy enough with 4 vectors.

I saw that I could write AC as a sum of two others:

I knew that the length of AC was 1 so I used Pythagoras’s Theorem to calculate mu. It left me with the exact same quadratic to solve. This time mu was the fraction of DB needed so was automatically the length scale factor. The rest falls out as it did before.

As well as this, Colin Beveridge (@icecolbeveridge), maths god and general legend, tweeted a couple of 1 tweet solutions. First he used trig identities:

**Trig Identities **

I assumed this was right, but checked it through to ensure I knew why was going on:

We can see beta is 2 x alpha and as such the tan value is correct. The cos value (although it is missing a negative sign that I’m sure Colin missed to test me) follows from Pythagoras’s Theorem:

This is again the scale factor as it is half the base of the small triangle and the base if large triangle is 2.

**Complex Numbers**

Then Colin tweeted this:

At first I wasn’t totally sure I followed so I asked for further clarification:

And then saw where Colin was going. I tried to work it through, by way of explaining here in a better manner.

I sketched it out and reasoned the direction of lines:

Then I normalised that and equated imaginary parts to get the same scale factor:

I am happy that is is valid, and that it shows Colin is right, but I’m not entirely sure this as the exact method Colin was meaning. He has promised a blog on the subject so I will add a link when it comes.

I like all these methods. I dontvthink I would have though of Colin’s methods myself though. I’d love to hear another methods you see.

## 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 fin 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 cords 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) excellentmathsbotwebsite. 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.## Share this via:

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