May 252014
 

Posted from Diigo. The rest of my favorite links are here.

 Posted by at 1:31 am

  8 Responses to “Diigo Links (weekly)”

  1. Glad you liked my gravity piece.

    >…teachers add it to the curriculum …

    Usually it’s not teachers deciding the curriculum.

  2. I agree, but teachers SHOULD be adding relevant modeling problems into the curriculum. This modeling idea of using gravity as a quadratic fits with the CCSS in several places so is perfect to be used as a modeling exercise, and modeling is one GIANT standard in the CCSS.

    After, isn’t every quadratic just the product of two lines (either in the real or complex plane) with either convenient or inconvenient axes locations? Why not use something like this to hammer home that idea and model the real world while simultaneously hitting all the standards that need to be taught?

  3. My two separate comments look like one. You said teachers were adding the order of operations to the curriculum, and I was disagreeing with that.
    .
    >isn’t every quadratic just the product of two lines…?

    Hmm, I guess, but I don’t see it in any deep way.

  4. Sue, I disagree. I think teachers add to the curriculum all the time. “FOIL” is a great example of something some teachers will spend days on, and them claim that they have not enough time to teach multiplication of polynomials instead of binomials. I think we, as teachers, do this often.

    Nixthetricks.com is full of examples where teachers are spending time adding things to the curriculum, and I have been guilty of it as well. It takes a conscious effort to focus on the mathematics (distribution) instead of the trick (foil).

    The “quadratic as product of two lines” becomes interesting when you extend it. For example, If I ask you to write the cubic that goes through three co-linear points, you can just write the three lines that have the same zero g(x)=(x-h)(x-k)(x-l) and then shift that cubic up by the original function that created the co-linear points. f(x) = g(x) + ax+b.

    Honestly, I think this gets to a fundamental idea of every polynomial function. A polynomial function is any function that can be decomposed into a set of linear functions through adding, subtracting or multiplying.
    https://www.desmos.com/calculator/uqpkqvvega

    I have been toying with this for a while. I think it creates some interesting approaches to polynomial functions and definitely shines a light on why we start with lines and need to master linear functions.

  5. Glenn, I hope you’re right, on teachers’ freedom to choose these topics.
    .
    I am fascinated by your cubic example, but I don’t quite get it. Could you explain: Given the collinear points (1,2), (2,5), and (3,8), how do I create a cubic through them?

  6. Now I get it! (The desmos page said it just a bit differently.) I use the x-coordinates to build a cubic that has those as x-intercepts. Then I add the equation of the line through the points. Nice!

  7. Yes. It is a very interesting and completely different way of thinking about translations and intercepts. It also means that EVERY polynomial is nothing more than a product and sum of lines.

    What is completely amazing is that if you think of translations this way, there is is no reason why the “k” must be a constant. now you can “translate” a sine function or or cubic by any other function as well. I have an example with a sine function in the desmos file.

    What do you think, does this example deserve it’s own writeup?

  8. Absolutely! And I’m not yet seeing what you’re saying about sine. I’m pretty sure you don’t mean the usual translations.

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