Activities in Desmos allow students to interact with graphs before answering a question, and this facility allows for a more natural way of asking questions that are quite hard to ask just using pen-and-paper tests.
Imagine a point A is at (2, 1) and another point A' is at (–2, 1). What is the transformation?
With draggable points, a student could move a point A around the screen and see the image point A'. After moving the point around, the student could see that the transformation was a reflection in the y-axis.
Now imagine on the next screen, the point A is at (2, 1) and the image A' is again at (–2, 1). But this time the student drags the point around and sees that the transformation was a horizontal translation.
Once a student has engaged with these two screens the original question can be asked, but it seems much less abstract. Imagine a point A is at (2, 1) and another point A' is at (–2, 1). What is the transformation?
This question can also be asked in Desmos, and is basically a duplicate of the previous screen where the point A is no longer draggable.
Thinking of a transformation as a function T: R2 → R2 could seem very abstract and hard to visualise (without an ability to see in four dimensions). By allowing the first point to be draggable we can provide a good intuition of what a transformation is.
Also, it is interesting to me that simply by changing between the dynamic versions (Screen 1 and Screen 2) and the static version (Screen 3) changes the answer.
I'm not sure how easily these concepts could be shown using a pen-and-paper quiz (one could include some points A, B, C and their images A', B', C' but this seems a bit messy and not as helpful if we want students to think of it as a function mapping any point in R2 to another point in R2.)
The activity shown in this post is just to show the idea but if it's helpful, you might like to copy and edit it to see how it is implemented. This is the link.
I'm also really taken by the change from screen 2 to 3. Thoughts, as they come:
ReplyDeleteIn screen 2, I see point A being defined by parameters (A_x, A_y); A' is then defined in relation to those values (A_x-4, A_y).
Screen 3, meanwhile, defines A through constants (2, 1), and so A' must also be defined through constants: (-2, 1).
We can talk about *the* transformation on screen 2 because that relation is baked into the definition of the points -- the transformation and the (algebraic) representation are one and the same. (As an aside, I wondered what an accurate graphical representation for the points would look like. My first thought was to add (A_x, A_y) labels, but that's mixing in the algebraic representation. Maybe a four-piece horizontal arrow heading left?)
We can't talk about a single transformation on screen 3 because those definitions just use constants -- we can impose a relation between them from the outside, but intrinsically they are just sitting there wherever they happened to have been placed.
I'm noticing that I have two different conceptual categories for draggable desmos points. There's the classic (x0, y0) which sits in my brain as a point defined with parameters, and then there's the new (3, 4) with dragging turned on. That one sits in my head as a *particular point* in a way the other doesn't, even though they can both be manipulated in the graph identically.
I suspect that when we get access to the individual coordinates in the expressions list my brain will unify them.
I like to think about the analogous behaviour of functions R -> R.
ReplyDeleteIf I just told you that I have a linear function that maps 2 to 5 and then asked what the function could be, there are lots of answers.
But if I allow you to type other input values then you could work out the exact linear function. For instance, after you see that 4 maps to 7 then you might deduce that f(x) = x+3 was the actual function.
Now, in the case of functions on real numbers, we don't usually do this because we can just graph all the inputs and outputs on a set of axes, so there's not much point just showing one input/output value.
But in the case of R^2 -> R^2, straightforward graphing isn't an option, so the best we can do is basically substitute in different values to work out what the transformation might be.
Not sure if this really addresses your comments but this is how I've been thinking about it!