Y9 GCSE Transformations
Rufus Johnstone June 2016
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?
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 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].
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:
- 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:
- To identify the transformations:
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]
- anti-clockwise/ clockwise
- object/ image/ equivalent points
- equations of lines
- 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
- 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.