inverses

[michel paul]

http://sagecell.sagemath.org/?q=nvjnsk

A student can change the function used as well as the start, stop, and step values of the domain.

Everything else takes care of itself.

What's more, the code reads naturally. You don't have to learn anything extra to follow what's going on in the first 4 lines. That stuff is the core curriculum.

I've been finding the Sage Cell amazingly effective. Send a link to students with code and comments. They respond, creating their own code, and then reply sending their own link.

I have found it a very powerful and easy way to do things.

--

​Michel

​

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"What I cannot create, I do not understand."

- Richard Feynman

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"Computer science is the new mathematics."

- Dr. Christos Papadimitriou
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[Maria Droujkova]

Pretty!

Have you used it with students? What did they make of it?

Cheers,
Dr. Maria Droujkova

moebiusnoodles.com
919-388-1721

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[Donald Cohen]

Michel,

You get wonderful graphs of the inverse, but I don't see its equation. Am I missing something?

Don

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[michel paul]

On Fri, Nov 28, 2014 at 6:47 AM, Maria Droujkova <drou...@gmail.com> wrote:

Pretty!

Have you used it with students? What did they make of it?

​Yes, I've made list comprehension a standard in my math classes. No one can even try to argue that this is 'programming and not math', as it is set builder notation. I'm glad to see that students really can learn to think in these terms. They often find it challenging at first, but it has really paid off to keep encouraging them to get it. It develops a thinking style that is excellent for both programming and mathematics. I have made it a standard in my class that they be able to code transformations similar to what you see in those first 4 lines. 

The more detailed graphics below the #### might be intimidating, so I don't make that an expectation in an ordinary math class, but I do encourage them to mess around with the code there to see what happens, but the lines at the top, sure, I think they are totally within the expectations of the curriculum the students are supposed to be learning anyway, and in learning how to express these things using list comprehension they're learning some non-trivial coding.

​The ease of the Sage Cell makes doing this possible even with the typically reluctant students who weren't planning on having to do programming in math class. There's no overhead - nothing to install, no accounts to set up, no files to transfer. I can post a link, they can interact with it, and they can send me their own link in response. When there are other issues to contend with reluctant students can turn off, but when it's so easy and direct - there's no excuse. It's just right there.

​- Michel​

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[michel paul]

Oh no, not at all! You're actually getting the point (pun initially not intended).  : )

Defining a relation as a set of ordered pairs, we can create the inverse relation simply by swapping the x and y coordinates, and that is what the code does in the second comprehension. 

f(x) is initially used in a list comprehension to create the ordered pairs F which is then used in a further comprehension to create the list of ordered pairs G.

An exercise for the students is to then create the function g(x) that produces the same ordered pairs as we find in list G. 

So it turns out that we can arrive at the same set of points by either

1. altering the original coordinates directly, or
2. creating a list from the inverse function.

Of course, #2 is not always possible, for example, if we change f(x) to x^2. Can we create a single function that will produce the green sideways parabola? This opens up further discussion.

I'm using the same strategy for other transformations. Given some function f(x), let's create a list of ordered pairs, then transform it into a new list by scaling or translation or whatever, and now let's try to describe this new list with a new function.

- Michel

[Maria Droujkova]

Michel,

It can be interesting to have a live event (Math Future) with you where you could talk about this. If you are game, email me off-list so we arrange time (drou...@gmail.com).