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Monday, August 26, 2013

Building Functions

How do you motivate vertex form for a quadratic? Do you just drop it on them? That's pretty much what I used to do in Algebra 1. Hey, kids, you want to model this u-shaped path, like you get when you toss a basketball. I'm just going to tell you to start with y = a(x - h)2 + k. Start messing with a, h, and k, and see what happens. Let's see what we can say about how they each affect the graph.

There's an awful lot of "building functions" in the common core, and an awful lot of modeling, and I think it's great. The whole F-BF header should be a playground. I'm just not clear on how you take a class there.

You can look at sequences of patterns easily enough that result in y = ax2, and I suppose patterns that result in y = (x - h)2 and y = x2 + k. Do those arise naturally anywhere? Or do you choose carefully something from Visual Patterns?

What else? "Fit a quadratic function to a photograph" seems to be a favorite of presenters at conferences who want to browbeat teachers for not making class real-world enough. But how do students develop those functions in the first place, in an authentic way? I feel a little awkward for asking, because I feel like I should already know this. But I also suspect that not that many people have a great answer.


  1. I suppose one way is to connect it to linear equations through that great handout of yours I used last week (page two had lots of y-c1 = m(x-c2) on it, which we identified as the point-slope form they'd seen before). Glenn's list of y = a*f(x-h) + k looks like a good way to get at all this. I'd like to try to do something with it this week.

    But that might be better for pre-calc, where all of this is really sort of review. Would it help any with a first contact?

  2. I feel like it's more natural (for me) to begin with x-intercept form, that is, y = (x-x1)(x-x2).

    Then you can look at how you can increase the max height if you introduce y = A(x-x1)(x-x2).

    You can then shift the x-axis up and down by introducing the k term in y = A(x-x1)(x-x2)+k.

    I don't feel like that addresses your issue at all, but whenever I've tried to build quadratics it's been from roots.

  3. We are starting with vertex form this year...a new idea for us. We are working with the framework of using h and k as vertical and horizontal transformations to get the parabola to match up with a picture. Using the tech and the sliders and whatnot. Can't tell you how it goes, as it hasn't gone yet. But I can get back to you!

    Of course, this led into the opposite question of how do we motivate standard form if we've started with vertex form--rather than the other way around. :)

  4. I got to vertex form by taking y=ax^2 and treating it with translations and dilations. Then we used the root form motivated by being able to model from the roots, and showed the two were equivilant. The other benefit of working with translations was how well that translated to every other function from then on.

  5. I usually introduce quadratics in vertex form right after our unit on absolute value functions. Students start to see how different numbers affect the graphs and equations using the jelly bean guessing contest. This leads us to our general vertex form equation. Not sure if this is really a good way to motivate vertex form for abs value either, but it has gone pretty well the past 2 years.

    When we move onto quadratic scenarios, it's a smooth-ish transition from y=ax^2 to vertex form since they're already familiar with it.

  6. Kate, it sounds like you're asking this question: "I want students to understand the important family of functions f(x) = (x-h)^2 + k. How can I get them to discover this for themselves?" (Correct me if I'm wrong here.)

    Another angle would be this: I'm going to *present* functions in this family, and then have the students discover what is going on. This is the kind of sequence I have in mind:

    1. graph y = x^2. Let the students pick 5 points and plot them.

    2. graph y = x^2 + 5. Plot some points. No help from the teacher here!

    3. graph y = (x-3)^2. plot.


    Once the student has had the opportunity to graph several such examples, you can get into the nature of what is going on. Where is the vertex? Okay, so you said the vertex is at (___,___), can you tell me why? Try to get the students to /understand/ how the formulas are related to the graphs. And this is much deeper than simply saying, "I notice that when you add 5 to x^2, the graph moves up 5." Fine -- that is certainly true, but /why/ does that happen?

    This is the kind of discussion that I think will help kids understand transformations. Once they have built a deep understanding there, then you can be like, "See how it's oh-so-easy to graph a quadratic function when it's in this neat little form? Now let's try something else: y = x^2 - 10x + 24. Where is the vertex? Anybody? Anybody? No?" And now you teach them how to "complete the square" to /make/ the thing into the vertex form.

    Would love to know what you think about this approach!

  7. This feels a lot like the inductive v. deductive debate I hosted on my blog awhile ago. The inductive motivates the need for the deductive, is how I settled back then, and I think it works here.

    Basically, use induction to find a model that works. Any model. Then try to explain it deductively.

    Like the teacher trying to model (2,5), (3, 8), (3, 11) yesterday. She suggested x^2+1. Rather than immediately getting deductively into why that didn't fit what we were looking at, we just tried her rule out on (3, 8) and watched it break. After we settled on the linear rule 3x + 2 inductively, we talked about where we saw the 3 and the 2 in the real-world pattern.

  8. You could use desmos or another graphing app and have students graph two linear equations (i'm assuming you've already gone through linear stuff) like 5x+1 and 2x-1. Then, have them multiply the two together, either having them distribute, or just typing in (5x+1)(2x-1). Now they can see a quadratic function, and some interesting info about the zeroes. After that, you can have them see what they can do to the quadratic to move it around (left, right, up, down, skinny, fat), and see if they can start to spy out the vertex form that way.

  9. If I read the opening question as why would students want to learn about vertex form, I think any context that needs the vertex to solve the problem (eg: max or min) can motivate that. I also think the vertex is key to making sense of quadratic equations, but that's another story.

    I like what you said about exploring through patterning, which others have also commented on as inductive reasoning leading to deductive. I'm sure you could write a better list of tips than what I've got below, but what's worked for me when doing this is to:

    - Let the students do the thinking; if I do the deductive part for them, that'll cut off their thinking and many will just end up with fragile procedural knowledge; it also undermines the purpose for doing the exploration to begin with.

    - Ask a lot of 'what if' questions both during and after.

    - I used to separate the parameters into different lessons; now I would still give some separate exploration, but all as part of the same lesson/task/activity.

    - I used to say, "Today we're going to explore the graph of y = (x-h)^2". "Why the minus sign? You'll find out." Ugh - it 'spoiled' it. Now I leave the generalized form until AFTER they have explored it numerically.

    - A caution (probably already familiar to most teachers) about y = ax^2. When exploring, students sometimes use 'horizontal' language (eg: fatter or skinnier) instead of 'vertical' language (eg: taller or shorter). That has implications for the deductive thinking part.

  10. Caution- I'm not a high school teacher, so the following ideas may not work in a real classroom. But maybe you can modify them appropriately.

    If the question is "why quadratics at all", one answer is "because that's how the physical world works". With modern video cameras, you could literally throw a basketball up into the air, watch it come down, then go through the video frame by frame, measuring (time, height) coordinates. Plot them and they should look a lot like a parabola, although air resistance and other imperfections may distort that - having not actually tried this I'm not sure how accurate the results would be.

    Then you could show the students how to plot the points in a program that can do curve fits (e.g. Microsoft Excel, or whatever graphing program they use), then fit a quadratic to find the coefficients. It will report them in ax^2+bx+c form, at which point you could do the algebra to get to vertex form and note that ah-ha, the vertex form coefficients are easier to interpret in terms of physical concepts like where the arc peaks.

    And if the question is "why vertex form", the previous sentence answers it: the reason we are interested in algebraically-equivalent alternative ways of parametrize functions in general is that depending on the context, different parameter sets are easier to interpret. But that is very context specific. As an applied mathematician, I use math in the context of real world problems, so there is always a context surrounding the problem. A big part of the effort is to find parameters, or ways of visualizing them, that fit the intuition of the subject matter experts. In the basketball case, the context is physics and the experts are looking at characteristics like maximum height of the ball, speed when it hits the ground, and so forth. In another context, the value at time zero might be more important, so vertex form would not be the best for all situations. To my mind, students need to learn not just the mechanics of the algebra, but some intuition around what the different forms buy them - why they might prefer one to another in different situations.

    Hope that is helpful.

  11. I have never done a good job with this topic. Maybe start the unit by giving them the graph and asking for an equation. If you haven’t seen the equation before, vertex or intercept from is much more natural than standard form.

    If they focus on the vertex, see if you can get an equation that goes through the vertex: y - 3 = (x – 5) using Sue’s suggestions. Then discuss how to change the equation to get the symmetry of the graph. Do we know of anything that treats (3 – 5) & (7 – 5) the same way?

    If they focus on the x-intercepts, as Dave mentions, can we get any equation that goes through one of the x-intercepts – even a line? How can we use this to create one equation that goes through both x-intercepts? Maybe look at a quadratic with a double intercept if you want to move to vertex form.

    I would guess that these approaches would require a lot of hints and nudges - maybe so much that it kind of falls apart. But, they get at the reason behind the forms of the equation and the symmetry involved.

    Observing the affect of the parameters on the graph (using sliders and a picture if you like) is very manageable, but a really superficial analysis. It could be followed up with the “why” questions. But, once they see how the parameters change the graph, it might be tough to get interest in the “why".

  12. I read your question in a similar way to Marc G. and as a result, I completely agree with him, in that finding maximums and minimums completely motivate vertex form. Having said that, I don't think students need to stumble upon the form themselves, though they do need to be able to argue its importance for themselves. As teachers of algebra, we are always asking students to manipulate or write things in various ways. Why not start with a quadratic (either in context or not) and see what they can do? See how many different ways students can represent the quadratic and then introduce a couple of your own. Afterward, you could ask which form of the quadratic most easily enables you to find (name a characteristic of your choosing here). This would lend itself to a lot of quality argumentation and explanation from students as well as sense making. It would also allow students an opportunity to directly address CCSS.Math.Content.HSA-SSE.B.3

  13. How do you motivate vertex form for a quadratic? Do you just drop it on them?

    As long as we're dropping things, I'd motivate the quadratic model for falling motion the same way Galileo did in his Dialogue Concerning Two New Sciences ("Naturally Accelerated Motion," pp. 160 -- 179 in Crew & de Salvio's 1914 translation). Galileo's argument is beautiful, but long-winded, so here's a summary.

    It seems natural to guess that a falling object might pick up speed at a constant rate a. If the object starts with speed 0, it will have speed at at time t. Graphing the object's speed on a time interval [0, T], you can see that every moment the object spends with its speed a certain amount above aT/2 is balanced by a moment at the speed the same amount below aT/2. Thus,* in the time interval [0, T], the object should move the same distance as an object moving with constant speed aT/2. Now we have something we know how to calculate: an object moving at speed aT/2 for time T goes distance aT^2/2. And we've arrived!

    * This argument is kind of flimsy, but it held up for the 50 years between Galileo's publication of the Dialogue and Newton's introduction of calculus, so it should be good enough for most high-schoolers...


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