*Year 12 Further Maths. FP1 Chapter 3 – Coordinate Systems (Lesson 1)*

*My Narrative for this topic/ chapter*

*The idea of this lesson and the objective*

I am introducing this topic by defining a parabola, and looking at it from the point of view of coordinate geometry (my first two nodes above). My objective is that all students will be able to answer a past exam question involving these ideas by the end of the lesson.

*Activity 1*

Students are required to brainstorm everything they know about Coordinate Geometry. They will have studied this at GCSE and for C1 and possibly C2. The idea here is that it focuses their minds on one of the big themes of the lesson: geometry in the Cartesian plane. I want to have talked about at least everything in my rectangle on the right above by the end of this.

Exposition

I will now introduce the idea of a parabola being a locus of all points equidistant from a focus and directrix. I will describe it in its Cartesian form, and leave the parametric form for a later lesson. The words directrix, focus and vertex will be defined. I will use a geogebra file to assist my explanation:

Activity 2

I want students to be familiar with this parabola, defined by y^{2} = 4ax, for different values of a. I want them to be able to know the (straightforward) connection between the equation, the focus, and the directrix. A simple activity naming these aspects of the graph then:

Activity 3

Now students are familiar with the graphs, I want them to engage with it geometrically, using their knowledge of coordinate geometry. I want them to solve a problem, a good problem from a past exam problem. The problem, which is representative of about a quarter of exam problems on this topic comes down to this:

My judgement is that students will find this problem too difficult to try straight away, so I will use my strategy of ‘a way in to solving problems’. I look at this problem and try to simplify it in a way that keeps the structure of the maths but makes it accessible to all students. Here is my simplified problem:

The judgement call I am making is that my students will still find this challenging enough to engage with it and be eager to find the solution, but they won’t find too challenging so as to being off-putting.

The students will be asked to attempt this problem themselves, using one of the graphs they labelled earlier. Then they will be asked to share their ideas in groups, and then one student will come to the board to explain the solution. Once all students are happy with this, I will give them the original problem.

This time students will be required to stay with the problem themselves and not ask for help. Certainly for at least 6 minutes. Anyone who finishes will be able to attempt other similar past exam questions.

After all students have given the problem a thorough attempt, persisting through their difficulties, I will either have some students having solved and some not, or all students having solved it. If it is the second scenario, then they will all move onto similar past exam questions, and if it’s the first, then I will match up students who can do it to work with students who are struggling.

When all students are happy with the solution, then they can all start working through the past exam problems.

*Plenary*

I will talk about how we have brought together existing ideas about coordinate geometry and a new idea, about a parabola defined geometrically, to solve some maths problems. I will direct them towards my narrative and preview the idea about defining a parabola parametrically.

*Assessment*

I generally use thumbs up/ sideways/ down to gauge whether the students have understood something I’ve said, and rating on their fingers from 1 to 5 for students to self-assess whether they have understood the maths they have been working on. This informs how I will proceed in the lesson. I will circulate and make assessments of how well the students are understanding the maths, and attempt to give process help over product help, i.e. I will attempt to help the students think things through for themselves as opposed to helping them get the answer.

*Differentiation*

The lesson has been planned to give all students a good chance of meeting the objectives. For students who find it easy, there are progressively more difficult exam questions for them to answer and there are opportunities for them to articulate their ideas to other students. For those who are struggling, they will have me to talk to as well as other students who they can articulate their difficulties to.

Hi Rufus

I like the post, and the way you are thinking about the lesson. Here are my thoughts:

First off, I’m not sure about the brainstorming bit; I like to get straight in to the action, and draw out anything that connects or seems important *as we go along*, perhaps recording it somewhere, on a different board or something. I think this is because I like to get straight into the action, solving problems as soon as possible; I’m not sure the brainstorming might achieve what you want it to.

Next: Did you consider the idea of letting students arrive at the realisation that a parabola is the locus of equidistant points somehow? I always feel that ‘telling’ them this is somehow taking away some of the excitement? For example, could you define ‘a locus of points as equidistant…’, and see what curve appears (i.e. in reverse)? [Perhaps you could even get into other conic sections by changing the constraints on this definition?] I would be tempted to get them to do some preparation *before* the lesson, to explore conic sections and their properties; it might give them some context, a visual image of how they are connected and why they are studying them (because they are interesting/aesthetically pleasing/possibly important).

I also think it is interesting how some of the defined terms comes about… Why is the focus called the focus? Why that point exactly? And what is the significance of the directrix? Where does it come from? Why that line exactly?

I like your strategy of breaking down a problem. I wonder if instead of giving a problem to solve, you could give some information and ask them to find out what they can, and then introduce your (exam) problem to be solved? Maybe not.

You say you want them to work for 6 minutes. Why 6 minutes? I generally want students to struggle around for ages, refer to the textbook and each other, exhaust all other avenues before asking me, but this often takes longer than 6 minutes – of course, there is a limit.

I must admit I’m not sure about the assessment method of using thumbs, I would rather get them to solve a similar but different problem at some point, and listen / talk to them as I go round, but have never personally found this kind of self-assessment very helpful. But if it works for you, ignore me.

As for differentiation – who cares? I like the fact you will get students to help each other if they have solved the problem (will they be doing this before they have solved the problem too?) Just getting them to work together on difficult problems is enough for me, and that is what you are doing!

Well, I hope this isn’t all too critical, I just thought I’d offer my views as you asked 🙂

Would like to hear your thoughts on all this, Danny

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Danny, this is exactly the kind of response I wanted. It’s just a lesson plan for an observation I’ve got this week. It should do the job of satisfying the observer but clearly there are all kinds of things I could do better. I will have a think about your points.

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