Thread by @blatherwick_sam

Either side of half term I've been teaching vectors to Yr 10 higher.

I wanted to put a thread together on how I scaffold the leap to vector algebra, and how vector algebra is a lovely vehicle for interleaving.

#mathschat

I did loads of vector thinking a couple of years ago, especially prompted by @danicquinn's #mathscpdchat in Nov 2019

We then taught vectors in y11, around the same time as translations.

Later, I moved it to y10, straight after ratio and algebra work

https://twitter.com/danicquinn/status/1196865342578069504

Before half term I did a load of work on column vectors, drawing, adding and subtracting, multiplying by scalars, seeing these visually, and also filling blanks in for the sums.

Also important for students to see the reverse vector is the negative

It's important to connect up that summing vectors create routes of multiple jumps, but I'm not making it explicit, happy to just leave it hanging at this stage and if it's realised then great, if not... well we're getting to it.

By the end we're introducing naming vectors and then linking it to multiplying and reversing them.

I had two weeks between lessons here but actually that was quite nice because it gave me the opportunity to bridge again.

This is about scaffolding the skills we need for tough vector algebra problems - seeing vector sums as "routes", scaling vectors, parallel vectors etc.

I give this grid with the alphabet on.

I model with my initials and give the vector SB and ask them to give the vector for their initials.

This gives the hook for exploration!

1) some students have the same initials - in this class they were both CC! I make sure I highlight this

2) some students have the same vectors. I call them vector siblings which is obviously hilarious to teenagers. I ask why they've got the same answer.

3) some students have opposite vectors. In this class it was AB and GF - "why is this interesting?"

every time I'm drawing on the vectors and linking it to the visual

4) I pick out scalar multiples and draw them on and ask students to notice why it is interesting

Next I want to demonstrate sums as routes. I don't tell students that I'm doing this at all because it's going to damage the "oh that's cool" moment, but undoubtedly some will see it before I do the big reveal.

I ask students to spell out their name, so I model with me:
SA = ...
AM = ...

I ask them to write out all the vectors on their MWB.

I also tell them if I were doing it I'd be doing my name as SAMUEL to make it a bit more interesting

Then when they've done it I ask them to add all the vectors together.

The point of interest here comes from students who get a zero vector as a result and are baffled as to why that would be

So I write all their names on the board and say "what may they have in common?"

When they spot this, I say "well what's happened with your name?" to the other ones - so they see that the sum of all the vectors becomes the same as just the jump from the first to the last.

Now I take away the grid and move to general vectors and introduce p and j

Here we go heavy on questioning. OP = p, what is PQ? What is NO? Give me another vector for p. What is OQ? What is PO? Then same for js (maybe I shouldn't use j?)

There's only short thinking time here

The first deeper thinking task I do is ask them what OK is and I ask them to discuss in pairs. Some suggest pj which is thought provoking. We get p + j but I draw that in as a route AND show how the vectors are commutative so you can see that j + p would give us the same route

Next I ask what ME is.

One student gave an answer which was *so awesome* here - she wrote 2 (p + j)

This was so brilliant as it gave the opportunity to show that it was twice what OK was

But thinking back I wonder IF she was building it from the OK that we had seen previously, rather than building from 2p + 2j, and I wish I'd asked.

But maybe I've just forgotten what else she wrote on her board.

After a bit more questioning (including looking at WZ = z, so what is YZ etc. to see fractions as scalars) I give them this task.

There's some challenging stuff here!

So at this point we're primed

1) we can see adding together vectors leads to routes, so when we don't know a route we can find another way

2) we can work with scalars of vectors, including scalars of sums of vectors

3) we know what parallel vectors look like

Vectors are a great vessel for ratio interleaving and the hardest questions have ratio in.

So we work on example problem pairs leading up to ratio.

And then I give them this. I see this as initial problem solving and ask them what they come up with. Yeah I could just tell them but I'm not going to yet.

Whilst some of the class struggled we had lots of answers we could build on and I was really pleased when students started breaking the line up into parts - so I used that to model the answers before asking them to do similar.

Here are a few extensions to this:

This is about where I'm up to right now. I've got another lesson or so on it where I introduce the second dimension with ratio and then finally we hit super hard exam Q ready

Here are some of those 2D questions

https://twitter.com/blatherwick_sam/status/1196878729294815238

https://twitter.com/blatherwick_sam/status/1536082639752511488?t=3BaO906hD5WtvMfO2O14nQ&s=19

Actually something I missed here - I asked them to do their initials again and demonstrated how the "vector siblings" from before were still the same siblings, as it was the same lattice - hinting at how this was a generalisation

Also here, by the end I've virtually drawn the "grid" in, like a street map with the routes being revealed