## 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.*

## Nice area puzzle

Yesterday evening I came across this lovely area puzzle on twitter:

The puzzle is from Gerry McNally (@mcnally_gerry) he says its his first, and I hope that’s “first of many”.

I reached for the nearest pen and paper and had a quick go:

As you can see, I misread the puzzle originally and thought the lower quadrilateral was a square. The large triangle is isosceles as given in the question. This allowed me to use the properties of similar triangles and the base lengths given to work out the areas of the square, both right angled triangles and the whole triangle. This then allowed me to calculate the area of the shaded quadrilateral and hence that area as a fraction of the whole.

Then I went to tweet my solution to Gerry and realised that nowhere does it say that the bottom quadrilateral is a square. I had added an assumption. This made me ponder the question some more. Instincts told me that it didn’t have to be a square, but that the solution would be the sane whether it was a square or not. But I didn’t want to leave it at that, I wanted to be sure, so I had another go.

I sketched out the triangle again:

This time I called the height of the rectangle x.

This made it trivial to find the area’s of the rectangle and the triangle GCD. Triangle HAB was easy enough to find using similar triangle properties.

and then I found the area of the whole shape again using similarity to discover the height.

This allowed me to find the shaded area:

Then when I put it as a fraction the xs cancelled and it of course reduced to the same answer.

I really like this puzzle, and would be interested to see how you approached it, please let me know in the comments or on social media.

## AS Levels

We are now fully into “Exam season”, Year 11 have their GCSE exams, and Year 13 have their A Levels. Then Year 12 have AS Levels.

AS levels are a weird thing. They are no longer a component part of the A Level, they are very early in the exam session and it seems to me an unnecessary added pressure.

Last year we took a decision as an academy not to enter pur maths students for the AS exams. We did this to maximise pur teaching time and avoid unnecessary stress. This year the decision was taken at trust level to enter them in all subjects.

I can now see two sides of the argument. Last year our students focussed heavily on their other subjects and not maths as they had external exams for those subjects. This meant we lost teaching time and their homework suffered during exam season. This year we have not finished all the content early enough to really focus the revision. I really dont know whats best. I do think, however, that it is important to have a decision made for all subjects.

*Are you entering your students for AS Levels? I’d love to know if you are or not and why you made that decision. You can answer in thw comments or on 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.

## 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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