# Y9 GCSE Transformations

Rufus Johnstone June 2016

## Overview

This is a plan to teach Y9 GCSE Transformations. I want to use it as an exemplar, a discussion point, and a piece of analysis for how I teach in general.

The plan is for next year, the class I will be teaching will be a bottom set. I have taught most of them already, sharing the bottom set in Years 7 and 8 with another teacher. I work in a school that has excellent behaviour systems and pastoral care.

I think the most important aspect of my teaching is exactly what I am starting with this: planning how to teach a lesson sequence[i], teaching it, and then reflecting on how it could be improved.

My plan will include the aspects that I think are important in teaching a good lesson sequence:

- Learning outcomes
- A map of the maths & my mathematical narrative
- Where do I want the students to get to? Model answers
- Explanations and Examples
- How do I want the lesson sequence to proceed? Activities, practise and homework
- Scaffolding and Extension
- Assessment including Multiple Choice Questions
- Planned Questioning, Planned one-on-one chats with the students
- My own practice: time-sensitivity, how will I record and assess how good the sequence has been? Benchmarking, how does it compare with a current GCSE class/ other schools?

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## Learning Outcomes

The learning outcomes from the textbook (our scheme of work/ curriculum):

## A Map of the Maths & my Mathematical Narrative

**The maths involved in this topic is something that both demands and invites thought. I’m sure I will reassess and re-evaluate this map as time goes by.**

**I am still thinking about what it means to transform the plane. I think 99% of students will consider the plane to stay the same and the object to move.**

My students will be studying for the Foundation course, with the possibility of doing Higher. I have mapped out the maths involved like this:

* *

**The mathematical narrative is my decision on how best to teach the sequence. It demands pedagogic knowledge, thought, reflection and a willingness to change one’s mind in light of new information. It also invites discussion about all these things, and I think it’s probable that I will be reflecting on and adjusting these narratives for my whole career. **

**In this sequence, I am still thinking about why the best way to find a centre of rotation for an object and its rotated image is to use two perpendicular bisectors.**

*Congruence*

Congruence is the common thread of translations, reflections and rotations. These three transformations of the plane preserve the distance between any two points on it, therefore any shape will be congruent with its image after the transformation[ii].

*The Transformations*

By transforming the Cartesian plane we can map any shape to any of its congruent images using one of 4 transformations.

The three transformations in the curriculum are included in this, however the following two shapes illustrate that there are some orientations of congruent shapes that require the recruitment of a transformation from outside the curriculum:

The glide reflection, which is a reflection followed by a translation parallel to the line of the reflection, completes a *group [iii]* of transformations that can map a shape in the plane to any of its possible congruent images. For example, the plane that includes the shape on the left above could be reflected along the line HD and then translated by 6 to the right to map onto the plane with the image on the right.

Hence, the recruitment of the glide reflection from outside the school curriculum to complete the mathematical coherence of the study of transformations.[iv] This is important to me because I want to teach the students maths in a way that makes sense mathematically, not just a set of arbitrary skills and procedures[v].

*Prior Knowledge needed by the students*

I want to ensure students understand and master the following concepts to be able to proceed with their work:

- Coordinates
- Clockwise and anticlockwise, 90, 180 and 270 degree turns
- Equations of lines. This means vertical and horizontal lines as well as y = x and y = -x
- Object and Image

*Carry out, identify and describe the transformations*

Using a computer program to do this enables students to carry out, identify and describe many, many more transformations than they would do by hand. It also focuses the task on students getting to grips with the mathematics involved as opposed to using tracing paper in a dextrous way. Subsequent to this, students will use their knowledge and understanding of transformations from the computer program and work through exam-type questions by hand.

- To carry out and describe the different transformations:

https://www.geogebra.org/m/hMNc4eES?doneurl=%2Fsearch%2Fperform%2Fsearch%2Freflections

- To identify the transformations:

Reflections https://www.geogebra.org/m/MXSH6EBn

Rotations https://www.geogebra.org/m/CFyCfXaC

Translations (to be made soon)

*Taking it further*

First of all there is combined transformations. Since including the glide reflection makes the transformation form a mathematical group there is plenty of scope to teach combined transformations in this context.

To find the centre of rotation of an object and its image is a good challenge. It involves two sets of constructing a perpendicular line between equivalent points on the shapes. Where these two lines meet is the centre of rotation.

For any students who master all of the above and are still looking for a challenge, I will look at my mathematical map and see what an appropriate extension would be.

## Where do I want the students to get to? Model answers

There a number of questions that I would like to model solutions to for the students. I have not included the questions on this blog, they are past exam questions and questions from the textbook. Recently, our school had an INSET on literacy, in which an English teacher gave an impressive talk on modelling answers for students and I intend to improve my modelling by using ideas gleaned from that talk. The intention is not for students to mimic everything I do, but redraft answers to improve their work based on my modelling.

## Explanations and Examples

To explain the following concepts I will be using a strategy that gives concrete examples of that concept and non-examples. The non-examples are carefully chosen to clarify potential misconceptions. In my experience there are only positive effects of this strategy: students find it useful in learning the concept, and the class ask lots of questions to help them clarify things:[vi]

- congruence
- anti-clockwise/ clockwise
- coordinates
- object/ image/ equivalent points
- equations of lines
- rotation
- reflection
- translation
- glide reflection
- combined transformations
- finding the centre of rotation by trial and improvement and by constructing perpendicular bisectors

## How do I want the lesson sequence to proceed? Activities, practise and homework

- Pentominoes
- Concept of Congruence
- Look at mapping them on the Cartesian plane. Define: Object and Image.
- Concepts: rotation, reflection, translation
- Concepts: clockwise/anticlockwise, coordinates, equations of lines
- Practise: (part 5 above) Homework: (based on part 5 above set on mymaths, 80% pass threshold). Formative feedback by talking to the students in the lesson.
- ICT to map object onto image: carry out, identify and describe rotations, reflections and translations
- Paper, pencil and tracing paper to practise the above with questions from the textbook.
- Multiple Choice Test based on the above with formative feedback.
- Carry out, identify and describe combined transformations in the context of a mathematical group.
- Find the centre of rotation by trial and improvement and constructing perpendicular bisectors
- Final feedback
- Summative test

## Scaffolding and Extension

This is an area of improvement for me. I think I have covered this in the rest of my plan, but I welcome advice on how to be more explicit about it.

## Assessment including Multiple Choice Questions

There are 5 main assessments:

- Multiple Choice Questions where the emphasis is on formative feedback. I have put together a series of multiple choice questions on diagnosticquestions.com. I will set this work for the students to do in a computer room one lesson. I think this is an exciting area of assessment. Given the responses students make, I can give feedback to everyone who gets each particular questions wrong, saving me lots of time and addressing each part of each student’s mistakes.
- Online Homework with a 80% threshold to pass and formative feedback to students until they reach this threshold
- My flexibility in how I teach the sequence of lessons and how I talk to the students one-on-one and explain improvements they can make in class.
- Online Homework with formative feedback from me.
- End of unit test. Students to correct mistakes in their books subsequent to this.

## Planned Questioning, Planned one-on-one chats with the students

This is an area of improvement for me. I think I should read ‘questions and prompts for mathematical thinking’ and I welcome comments on how I can structure these conversations and questions. At the moment, I just use my experience of what has worked well in the past.

## My own practice: time-sensitivity, how will I record and assess how good the sequence has been? Benchmarking, how does it compare with a current GCSE class/ other schools?

**(These are just some notes on what I want to do in the future)**

- Analysis of current y10 class with current Y10 cohort (Y10 set 2)
- Read Daisy Cristodoulou’s book on assessment.
- Read Cognitive psychology, Daniel Willingham, cognitive overload, cognitive dissonance
- A scheme of pedagogic tasks and adept cuing

- Passing the exam: Practise, practise, practise?
- Conceptual and Procedural Understanding (e.g. Skemp)
- Misconceptions: addressing and avoiding
- Textbook and curriculum (I’m not sure these ‘hooks’ from the textbook scheme of work are useful at all)

[i] It seems cruel that in the not-too-distant past, the emphasis was on lesson planning. Sequence planning makes much more sense both pedagogically and in terms of managing one’s workload.

[ii] Transformation Geometry, Wesslen and Fernandez, Mathematics Teaching 191

[iii] Transformation Geometry, Wesslen and Fernandez, Mathematics Teaching 191

[iv] Transformation Geometry, Wesslen and Fernandez, Mathematics Teaching 191

[v] In this blog, Cristina Milos discusses the problem of ‘highly contextualised tasks’ in maths: https://momentssnippetsspirals.wordpress.com/2016/02/25/transfer-of-learning-is-there-a-solution-2/ . ‘Food maths’ in which pizza is recruited to teach Fractions is an example of this. I can contrast this with my recruitment of maths from outside the curriculum to help teach maths that is inside the curriculum in a context that is helpful for students.

[vi] I believe this is an idea from Engelmann and his theory of Direct Instruction. I read about it in this blog: http://staffrm.io/@emmamccrea/w44eZFLclE

Hi Rufus. I like the inclusion of glide reflections and the links to constructions. In your Venn/thought process I would add the idea of invariance, which is of course linked to congruence and similarity.

I like your note (i): sequence planning is much more important than treating every lesson as a separate entity. How much time have you allowed for this unit? Do you intend to move onto the “Higher” skills or just stick within your centre section of the diagram?

Another question: what do you mean by “In this sequence, I am still thinking about why the best way to find a centre of rotation for an object and its rotated image is to use two perpendicular bisectors.”? Is it that you think there may be a better way or are you contemplating why this process works?

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Thanks Jemma, I really appreciate you reading it and commenting.

First, I am still getting my head around the perpendicular bisectors, I can see that it works, but I want to think about it some more to see how it connects perhaps with circles and coordinate geometry.

I will have about 10 lessons for this sequence, and I will have to see how it goes, I will cover the Foundation work with everyone and I expect a number of students to move onto the Higher.

Invariance is not something that I’ve thought about much, apart from its intrinsic part of congruence. I think I will include it, as it’s important to look at what stays the same and what changes under congruence (and similarity).

How do you approach a lesson sequence?

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More differently recently. I’m rewriting our five-year scheme of work and allowing much longer on topics. For each topic we identify the start and the end (which is never, in reality, the end) and then what path we want to take en route. We’re building in explicit opportunities to revisit past content (distributing practice and interleaving) and trying to incorporate the ideas of liminal variation, examples and non-examples. For me, the first obstacle to get over is this idea of a lesson as a discrete unit of learning time (in which everyone must make “progress”), which has been a problem over the last ten years or so. In planning a sequence and a route I think it’s easier to look past artificial barriers like one-hour lessons (I might write a blog post on this soon actually).

One question that arises from what you’ve written is whether it’s necessary to form a Higher/Foundation distinction. Does this force a reliance on the test? Is it better to think of a path through the topic and get as far along the path as you can in the time you have available?

As for the perpendicular bisectors, I hastily threw together a Geogebra model here: https://www.geogebra.org/o/DhWf5KSD (not as lovely as the ones you’ve made above). If you picture the circular path a vertex takes as it rotates and join a line segment from the vertex on the object to its corresponding vertex on the image (which will also lie on this circle) then you have a chord of the circle. Then it’s a matter of circle theorems: the perpendicular bisector of a chord always goes through the centre of the circle.

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Ah, your planning sounds a bit like my mapping and narrative. I’d really like to see one of your sequence plans if possible! Also, it seems you are picking up on the same ideas that I like, Kris Boulton’s Engelmann explanations, cognitive science etc. I read with interest your blog on the Research Ed, I’ll try to come to the next one.

Yes, the 3 part lesson as the fixation of SLT has been hugely problematic both in terms of workload and planning for learning. In reality, 6 lessons in a row might look very different, but they will all be contributing to the learning of a topic. In terms of Foundation/ Higher, I think the reality is that that GCSE exams are of paramount importance, and in planning, it’s a good idea to know exactly the learning outcomes expected for each.

Thanks for the thoughts on perpendicular bisectors, that’s helped. Honestly, it’s strange to say but I had never thought about the connection between the algebraic equation of a perpendicular bisector and the construction of one with compasses!

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Hi Rufus

Just had a read through. I also like the notion of including glide reflections. I would want to set the mathematical scene for that. Eg when we do a reflection, rotation and translation the object is transformed into an image that is congruent. So what if we do a rotation followed by reflection? My feeling is that you need to investigate/ raise awareness of composition of before closure.

Also I would spend some time in 5 on spatial awareness and language. I would want to assess whether they can a recognise and b describe features of reflection. I d predict most could. I’d want to hear relational phrases rather than keywords eg side ab goes left in the object but right in the image .

Then to assess the same for rotations clockwise and anticlockwise. A lot of students need to physically experience the movement and how eg right rotates to become up. I’d suggest thinking of the progression as first seeing the rotation as a turn or swing round a point that gives an estimate of the position. Then look at180 and 90 rotations on the cartesian grid and the relations between object and image . When we do our masters geometry course it is these visual steps we pay attention to.

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Fantastic! Thanks Cathy

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Hi Rufus,

I read the section with the header, “How do I want the lesson sequence to proceed?” and wondered how each of these activities listed below the header explicitly connected to the Venn diagram of ideas you selected.

Also, there are a number of strategies for carrying out these different transformations. In our unit sequence, we connect the transformations to doing geometric constructions with a compass and straight-edge. How do you see students being able to do the transformations themselves? This seems like an important point to be clear about.

Are students:

– Using patty paper to learn how to map out the location of the image of an object under a reflection, translation, or rotation?

– Using Geogebra to experiment with different transformations and coming up with their own rules?

– Building on what they know about geometric constructions in order to develop strategies (such as measuring across what will become a perpendicular bisector in order to reflect a point) for carrying out transformations?

– Relying on a set of rules that help them map coordinates of vertices of geometric shapes during a geometric transformation?

Finally, I really like the Venn diagram. Are there topics within the Venn diagram that are more or less connected to each other? There’s some evidence that students build better schema when the ideas are explicitly (through questioning, representation, explanations) connected together.

Thanks for sharing this insight into your thinking!

David

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First of all, thank you for taking the time to give such a thoughtful reply, I very much appreciate it.

My thinking is definitely not clear, and one of the purposes of writing this was so that I could try to clarify things. I think it’s important to map out all the maths that is involved in the topic and then to find a narrative that will work for the students from that. Trying to clarify ‘how I want the lessons to proceed’ is still hugely problematic for me.

I hadn’t really thought about doing all the transformations with a compass and straight edge, but I can see the benefit of working with a compass to do the rotations and then extending that to perpendicular bisectors for the centre of rotation.

In terms of what the students will be doing, I want them to first use geogebra to identify, carry out and describe the transformations, this is because I want them to do this free of the dexterity required to use tracing (patty?) paper. The maths, for me, is worked on in a more straightforward way when using geogebra. I want to be quite ‘tight’ with my explanations, with me teaching the students about the transformations, and them using that knowledge to explore them on geogebra. Only once they are confident with them will I require them to work with the tracing paper.

I like the venn diagram idea as well, and it helps me to think about the maths and the way I want to teach it. I have shared them with students before, but I think they’re more about me and my teaching. I think there’s a possibility I could modify them for students’ use and that’s an area I will consider for the future.

Thank you again for your response, best, Rufus

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Hi Rufus, I am interested in Jemma’s response both in terms of micro and macro. The micro is the business of including glide reflections in students’ experiences of transformations and I shall email you an activity which supports student cognition of this. I also notice the business of group theory and whilst this is something studied at undergraduate level I see no reason, and in fact have utilised this with GCSE students when I was an HoMaths. I shall also email you another activity where students engage with finding the centre of rotation between shapes and this involves determining for themselves how this CoR can be found.

At the macro level I am 100% in support of what Jemma wrote: “For me, the first obstacle to get over is this idea of a lesson as a discrete unit of learning time (in which everyone must make “progress”), which has been a problem over the last ten years or so. In planning a sequence and a route I think it’s easier to look past artificial barriers like one-hour lessons”. This is because if we develop a SoW which engages students with the interconnected nature of mathematics then we can run topics or units of work for as long as we like.

When teaching GCSE groups I would frequently do a ‘straw poll’ with a class based upon the question: “How many more lessons do we want to continue to spend on this topic?” (This would usually be after two or three weeks when we had been working on a sequence of tasks/problems all based within a broad topic area). Mapping out all the ‘bits’ of mathematics involved in a sequence/topic/unit became important in order to check we were ‘covering’ the GCSE syllabus. This was supported by, as mentioned earlier, the interconnected nature of mathematics. So for example producing group tables for the rotations through 90, 180, 270 and 360 together with reflections in x=0, y=0, y=x and y=-x and for those students who were ‘ready’ to consider the sub-group of rotations was not seen as ‘undergraduate mathematics’ but more as a way of making sense of how these transformations could be combined to produce an existing transformation.

I found my teaching became about explaining a process to students by contrast to explaining my understanding of specific bits of mathematical content. By “explaining a process” this involves setting up a situation about which the outcome would never be clear but as students worked on tasks the underlying mathematics became clear. For example, asking them to draw an asymmetric quadrilateral on a co-ordinate grid then to perform a sequence of transformations (on this quadrilateral) by ‘doing something’ with the co-ordinates of their chosen shape, was to lead them to an understanding of transformations. Examples of these “doing somethings” would be as follows:

Take the co-ordinates of your quadrilateral and:

a) multiply the x-ordinates by -1 and the y-ordinates by 1. What happens to the original shape?

Return to your original quadrilateral and:

b) multiply the x-ordinates by 1 and the y-ordinates by -1. What happens…?

c) multiply both ordinates by -1. What happens…?

d) swap the x-ordinates and the y-ordinates around. What happens…?

etc – with 2 by 2 transformational matrices not far away for those students who demonstrated a preparedness to go deeper.

Similarly enlargements could also be considered if and when it seemed appropriate.

By having larger lumps of time on a sequence/topic/unit, as mentioned in an earlier response enabled different depths to be plumbed.

Of course I did not have access to Autograph or Geogebra etc so much of what we did involved tracing paper; were I to re-write my schemes of work I would have to rethink what resources I would use now, and how I would use them…

Okay I shall sort out the resources I mentioned in the next couple of days

Regards

Mike

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Mike, I love the implicit use of matrices in the activity you describe. Perhaps you could share these resources on a wider scale? I’d love to see them.

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Hi Jemma, this work on matrices appears on my website as an article from Mathematics Teaching: http://www.mikeollerton.com/pubs/atm_pubs/ATM-MT132%20Sowing%20seeds.pdf

I am about to email Rufus some other materials so if you let me have your email address I shall send them to you as well

Regards

Mike

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Thank you, Mike. I’ve messaged you through the form on your website.

Jemma

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[…] teaching. I think they can be worked on, modified and improved over the course of a 40 year career. I have written about a maths sequence plan, and it probably appears to be very complicated. This is because, for the this topic, my plan is a […]

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