This is a post describing some ideas for introducing 2D and 3D vectors that I used with my students this week. Each of the tasks contain an element of

*physical*

*construction*. It is my conjecture that the time constraints at A-level and Higher lead to teachers feeling as though they do not have enough time to spend working with vectors in a physical way. In my experience, this leads to learners having difficulties visualising vectors problems, particularly those in three dimensions.

We started working in 2D. Here are some screenshots of a geogebra task designed to help students gain familiarity with (combinations of) vectors as a path between two points. They made paths between a set of points by dragging copies of two vectors, blue and red (and their negatives). This led to a discussion about parallel vectors being multiples (see final image, click to enlarge):

The students then worked on a variant of this task, but with the two vectors

**i**and**j**, to gain familiarity with this notation whilst exploring collinearity. Here is a recording of the screen as they worked on this task.We worked on a number of other interactive tasks in 2D before extending to 3D. I wanted the students to be able to see and feel what was happening in three dimensions, so I (and they) posed a number of problems about a 2 x 2 x 2 cube, as can be seen in this video. Here is a screenshot:

There are some lovely moments in this video, particularly when the students start posing their own problems, and explaining what they are doing and why.

After the video, the students were working on some problems (see below), for which the cube proved useful. During this, I decided to ask one of the students whether he could find two points on the cube with a length sqrt(7). He replied:

*"For that you would need 1 ,2 and 4 [as the squared components of the vector], but that's not possible because of the 2."*

The students worked on a set of exam questions. For each of the 3D problems, I asked them to construct the image in geogebra before solving. One of the problems was as follows:

Here is a video of the students constructing the image. This felt like a very useful exercise, as it provided a visualisation, and provided a means of testing conjectures about the position of the coordinates.