# FP3 Vectors – Developing Deeper Understanding (Part 1)

I love teaching vectors in FP3 because I see such simplicity behind all of the geometric situations. In Edexcel’s module the chapter essentially breaks down into three sections:

• Learning how to calculate a cross product, culminating in the triple product giving the volume of a parallelepiped.
• Learning that the equation of a line (which was introduced in C4) can be written in several forms; and learning about the equation of a plane, again in several forms.
• Finding points of intersection of lines and planes; angles between lines and planes; and shortest distances between points, lines and planes.

This post picks up on a lesson at the second stage of this journey. Part 2 looks at a lesson snapshot from stage three.

## Productive Struggle

My understanding of the phrase productive struggle is to give students a task which they can’t launch straight into solving (with their current knowledge) but that they can chip away at and explore.

Each year, I look forward to this lesson:

Occasionally near the beginning, I might need to clarify precisely what the question means but I very rarely give any indication about how to proceed. And I give my students a lot of time. Originally I thought I give them around 15 minutes but this week, my class had at least 20 before we shared some ideas on the board. I asked 4 particular students to write up what they had tried:

Each of these contributions had something interesting that I wanted to bring to the whole class discussion. The writing in black identified a geometric meaning to the problem. The student writing in blue worked from the principle that the cross product of a vector with itself is zero (ie the zero vector). The student writing in red took an algebraic approach and realised that the three equations do not have a unique solution, but she was able to identify a valid solution. The student in green worked along similar lines but was able to list multiple solutions and identify a pattern behind them. (The annotations in red and blue came later.)

Eventually we could have a full discussion about why $(\mathbf{r}-\mathbf{a})\times \mathbf{d} = \mathbf{0}$ represents a line – from both an algebraic and a geometric perspective. Students are always quite surprised to see the shortest demonstration:

Of course, a similar lesson could be built around $\mathbf{r}\cdot \mathbf{n} = c$ but I’m not usually mean enough to make the students suffer twice in the same week!