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.
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.
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:
I want students to be familiar with this parabola, defined by y2 = 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:
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.
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.
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.
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.