## 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.*

## Late tiering decisions

Last week year 11 sat their mocks. Some did really well, others did really poorly. It’s the latter group that has me purplexed. Students sitting the higher tier paper but only scoring single digits per paper, or even earlt teens per paper. What to do with them?

Some of them asked if they could move to foundation, I think its best for them. 1 student got 32 marks over 3 higher papers, did the 3 foundation and was well over 100. 1 student got 40 marks over 3 higher papers spent 30 mins in a foundation paper and got 60 marks. The grade 5s they want seem more achievable on foundation.

My issue lies with a few students desperate to do higher and try for 6s. Scoring around 50 marks over 3 higher papers it seems a risk. But having taught them both i feel that it’s within their capabilities. But from November to march they have made only tiny gains in marks. On the ine hand, foundation means they cant get a 6 and for at least one of them means rethinking post 16 choices, but on the other hand sitting higher means they might end up with only a 4 or less and thst would mean rethinking post 16 again. It’s tricky, any thoughts are welcomed.

## Cereal Percentages

This week my Y11s are sitting mock exams. One of the questions that came up on paper 1 stumped a lot them.

They came out if the exam on monday, and said the paper was very difficult. One of them asked me one of the questions:

*“Sir, if you have a box of cereal and increase it by 25% but keep the price the same, what percentage would you need to decrease the price of the original box by to get the same value?”*

I immediately said “20%”, an answer which flummoxed the student and the others stood around. They couldn’t work out how I had got that answer, never mind so quickly.

I tried to explain it to them, but in that moment, on the corridor, I didn’t do a very good job. For me, it was intuitive. A 25% increase and a 20% decrease would yield the same value as in one you are changing the top of a fraction and the other the bottom of a fraction so you need to use the reciprocal, 4/5 is the reciprocal of 5/4 and 4/5 is 80% hence it needs to be a 20% decrease. Cue blank looks and pained expressions. I was seeing the students again later in an intervention session so I promised to go through it in more detail then.

I talked about the idea of value, how you could consider mass/price and get grams per penny – how many grams for each penny you spend – or you could consider price/mass and get penny per grams – how much you pay per gram. I said either of these would give an idea of value and you can use either in a best value problem.

I showed them the idea of the fraction, said you could call the price x and the size y.

The starting scenario is:

*y/x*

The posed scenario is:

*1.25y/x*

but we know 1.25 is 5/4 so that becomes:

*(5/4)y / x*

which in turn is:

*5y/4x*

I then showed that the second scenario meant getting to the same value but altering x. To do this you would need to mutiply x by 4/5:

*y/(x(4/5))*

*(y/x)÷(4/5)*

*(y/x) × (5/4)*

*5y/4x*

This managed to show some of them what was going on, but others still massively struggled. I tried showing them with numbers. 100 grams for £1. This again had an effect for some but still left others blank.

*I’m now racking my brains for another way to explain it. If you have a better explanation, please let me know in the comments of via social media!*

## Simultaneous Equations

It’s been a while since i last wrote anything here. Which says more about how busy I’ve been than my desire to write, but I hope to start writing more regularly.

This week I was teaching simultaneous equations and a student asked a question that made me think about things so I thought i would share.

I was teaching elimination method and I had done some examples with the coefficients of y having different signs and I put one on the board with the same signs and asked the class to think how we may go about solving. One of the students in the class put uo his hand after a while and said he thought he had solved it.

5x + 4y = 13

2x + 2y = 6

I asked hime to talk us through his thinking and he said “first I multipled the bottom equation by -2”

5x + 4y = 13

-4x – 4y = -12

“then I added the equations as before”

x = 1

“Then I subbed in and solved.”

2 + 2y = 6

2y = 4

y = 2

“so the point of intersection is (1,2)”.

This wasn’t what I was expecting. I was expecting him to have spotted we could subtract instead, but this method was clearly just as correct. It wasn’t something I had considered as a method before this, but I actually really liked it as a method and it led to a good discussion with the class after another student interjected with her solution which was what I expected, to multiply by 2 and subtract.

It was a great start point to a discussion where the students were looking at the two methods, and understanding why they both worked, the link between addition of a negative and subtracting a positive and many more.

*I was wondering, does anyone teach this as a method? Have you had similar discussions in your lessons? What do you think of it?*

## 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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