I thought I’d go through the process of how I devise questions for students. I was asked recently to cover a lesson about function transformations – a topic I haven’t taught in a while (As HoD in my last post, I typically had the ‘GET THEM A C GRADE OR WE’RE ALL DOOMED’ groups, who didn’t do Higher Tier. So anyway I was in a position where I thought I’d create a new resource from scratch.

Here was my process:

1. Decide what it is I want students to know

In this case, a range of function transformations (e.g f(x+2), and combinations of transformations (e.g -f(x+2) + 3 etc).

2. Decide how I want them to attempt it

I decided a combination of embedding skills, and using Desmos to check answers and patterns would be nice.

The Desmos part takes care of  itself pretty much (although I’ll need access to computers obviously). So onto the embedding skills bit…

I decided to go with an activity that would get students to write on pre-prepared axes with an existing shape on them, and apply a transformation. This will use more paper, but at least the students won’t spend the whole lesson drawing out axes – and it reduces the chances of them making errors copying a shape before they even get to the skill I want them to do : transforming it.

My next decision was to use straight line patterns rather than typical (think exam question) curves. There are a few reasons for this: firstly I think they’re prettier! Secondly I think they’re easier to transform, as they’re simpler to draw and you can see the points perhaps a little clearer. Thirdly, they’re small so I can chose between having the transformations overlap the original, or have it separate. I’m going to keep the transformations separate for the first few questions, as an overlapping transform will confuse some students, and I don’t want confusion right at the beginning! Granted, this kind of stops them being real ‘functions” but the principals remain the same.

So I’m now at a point where I know what my basic questions are going to look like… something like this:

Now I can see that I can fit at least 2 transformations in that picture, so I’m going to allow for a bit of differentiation: A simple transform, and a more complicated one right from the outset. Students can choose which version they attempt:

Now when I first created this resource, I just punched any old transformations into the questions with the only thought process being “the first is simple, the second is difficult”.

After I had completed the questions, I went back, printed them off and actually did them all myself. This is a really important step. Firstly because I realised some transformations went off the grid, which obviously wouldn’t work, secondly because some of them overlapped when I didn’t think they would, and didn’t want them to, and thirdly because I need a flipping answer sheet as these things can get very tricky.

So next I had to amend my questions so that they were smarter and more progressive, without any overlapping until later on. Unfortunately this also caused me an error later on, as I ended up with question sheets with the original transformations on, and answers with the new ones on (I spotted it, thanks to a proof reader, but more on that in a minute!).

I created the graphics in Illustrator, which took a little while, but the massive advantage to this is that I can easily display the question, one answer, both answers, the question alongside both answers etc in a graphic really quickly once it’s all completed. And that comes in handy in a minute…

At this point I had a worksheet that looked like this:

And a handy answer sheet that looked like this:

Finished right? Nope. I was very aware that it was a cumbersome task figuring out all the transformations, and that I’d probably mixed a few old questions with new answers, so I emailed it to someone I knew was mega smart and would find any holes (Martin Noon by the way. He clever). I was expecting him to find at most 2 errors. He found 5! (very politely). Whoops. Even so, the process was essential. I wouldn’t normally get someone to proof read a worksheet, but considering that function transformations can be a bit long winded and cumbersome, combined with it being a bit rusty, I knew I’d need to do it.

So now I have a new worksheet, full of questions, some hard, some easier. Finished? NEVER!!! I MUST GO ON!!!

I wasn’t happy with it at all yet. I liked the look of it, but The questions only really test whether students can follow rules for transformations. Do they really understand them? Can they think abstractly about them? I’m not really testing that yet.

So then I started to dissect some questions and ask them in a different way, just to make them a bit more thinky. Thinky is now a word.

First I thought I could rewrite some of them as sort card activities. This was just copying and pasting thanks to Illustrator and its many, many layers:

I’m not a huge sort card fan, but what I like about doing the questions like this, is that students aren’t just following a formula, they have to really think about what happens to a shape when it’s transformed, looking at the end and working backwards etc. You could easily throw in some dummy cards too, just to test them even more!

Next I thought I’d use the age old tactic of reversing a question or two:

So here students have to now find the original shape using two transformations (they could use one to be fair). It’s a nice (and simple) twist just to test them a bit more from a different angle.

There’s another angle using exactly the same information in the question too:

This time they have to figure out the transformations based on the movement of the shape. I like this version!

I sat for a few minutes and thought how else I could ask a question using exactly the same information I’d painstakingly created over the last few hours. Finally I came up with these:

The question on the left has pieces of the original shape missing, but they crop up again in the transformations. You have to piece it all together.

The final question removes colour so that students can’t tell from a quick glance which shape is the original and which are transformations, so they have to look a bit closer to figure it out.

Now this all took me a good few hours, but I’m not about to throw this resource away. In fact I will most likely use it for a long, long time. All pdfs are below. Worksheet V1 is without the adapted questioning.

function_transformation_sortcards

Transformations_Functions_WORKSHEET

Transformations_Functions_WORKSHEET_V2

## 10 thoughts on “Asking the Right Questions”

1. Deat Gid, that’s stunning and inspiring. Fantastic work.

2. Can you explain the notation -f(x)+4 for a transformation. I am puzzled.
Thanks.

3. Take a function, say f(x) = x + 2, then – f(x) will in essence be a reflection in the x-axis.
Take where x = 5, y = 5 + 2 = 7, but -y = 5 + 2 means y = -7.
Furthermore, – f (x) + 4 would be a reflection in the x axis, followed by a vertical shift upwards of 4.
Does that help?

• There are problems with this approach.
The first one is that -f(x) + 4 and 4 – f(x) do not mean the same thing.
The two operations do not commute.
Send me an email to howard_at_58@yahoo.co.uk if you want details.

4. Congrats for being able to work out how to do this in illustrator..The learning curve of working the program rivals the concepts the you are showing.

• Haha! Definitely a very counter intuitive program to begin with.