http://www.cut-the-knot.org/arithmetic/algebra/QuadraticFormula.shtml

Err, no you can’t. At least not without writing quite parametric relationships.

Your input variable, x, is either horizontal distance or time. Only parametric equations allow you both.

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My objection to these questions is that they take the question the wrong way round, because they assume the model is reality.

Instead:

Jim models the heigth of a baseball using the formula h = -16t^2 + 80t + 3.

1) In his model, at what height is the baseball hit?

2) Jim is trying to model a situation where the ball goes just over 100 ft in the air. Does this model do that?

Then it is clear that the model is exactly that, an attempt to replicate real behaviour mathematically.

Even better, why not give them the path and ask them to figure out the equation that best fits it?

A baseball is struck 3 feet off the ground, and flies to a maximum height of 103 feet after 2.5 seconds.

Write an equation that models the height of the ball as a quadratic.

When is the ball next at a catchable height of 8 feet?

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The worst question, by far, is the third, because of the crappy “Did you know?” bit at the bottom. Now I have half a class discussing baseball and Canadians, not Maths.

I used to have a scenario that the kids had to work through (https://educationrealist.wordpress.com/2013/12/16/the-negative-16-problems-and-educational-romanticism/). They were provided with the initial height, max height, time to ground, and their knowledge of the equations. They could use this to derive the -16 and the time to max height.

But lately, I’ve taken to just tossing a pen up into the air and catching it, and pointing out that the height pretty much has to be a quadratic function of time, or at least, barring any other information, we can model it as such. But how?

So the kids eventually realize that we could film an entire launch sequence, time it, capture the height, and so on. They all pretty quickly realize the vertex form is the obvious one to use in the model.

Thus far, I’ve found it easier to just launch a kid instead of a pen. Some kids hold a tape measure up against the wall, the kid with the best vertical jump is filmed. A chunk of the rest of the kids time the jumper. We practice several times watching the jumper leave the ground and land again, and then collect all the times and average them.

Then we evaluate the film and get the max height–with considerable discussion as to whether to use toes, heels, or the middle.

Then they plug it into the formula and find a. I determine the success of the data collection by how close they come to -16, which they have already realized has to be the pull of gravity (I briefly explain the 32 and then send them to physics teachers–and no, I don’t do metric. What horrible numbers.)

Then they expand their equation to standard form, where the realize that c=0 (or as close as measurement permits).

Then we discuss. What if Tony/Fatima/whoever jumped higher the second time? Would gravity have changed? Would he still start and finish at the ground? What parameter would change in vertex form? How about in standard form?

Eventually they determine that the bigger b is, the higher the person jumps, and that c has to be the initial height.

I’ve done this twice, and it’s worked great, but up to now it’s been a classroom discussion. Kids can opt out and do nothing or just watch, because I’m still working out the details. I want to turn it into an activity by individual groups, but unless I use one filmed jump that I show on the promethean, that’d be hard to do. And part of the fun is having someone in class jump.

Anyway. We do a stomp rocket activity after that, where we calculate velocity and max height (in perfect conditions, obviously) which is lots of fun. But I like the desmos activities to get them thinking about stretch. I’m going to play with that idea.

James Key:

My view: either do the math in a way that makes sense to kids, or do it in a way that makes it obvious that we don’t much care what is happening “behind the scenes” in cases where we want to focus on the modeling components of a task.

Interesting. I don’t see “modeling” as divorced from understanding what’s happening “behind the scenes.” Can you explain more?

For instance – in Version #3, students are asked to determine when the ball returns to a 3-foot height. Are they able to reason that the expression 3+48t-16t^2 has value 3 when the latter terms are zero? Factoring is useful for finding zeros: 3+t(48-16t). So the height is 3 when t = 0 or when t = 3 – thus is takes 3 seconds for the catcher to catch the ball.

For version #1, what do we gain by providing the quadratic formula? Why not just have students use technology to find the zeros? i.e. Use a solver. Now the focus is on the modeling aspects. My view: either do the math in a way that makes sense to kids, or do it in a way that makes it obvious that we don’t much care what is happening “behind the scenes” in cases where we want to focus on the modeling components of a task.

Another thing I do not like about any of the three problems is that the numbers are “nice.” Meaning – they are all integer coefficients and constants, and when calculating the vertex, -b/2a works out to be something easy to work with (3, 2.5, 1.5 respectively). Let’s face it… the real world is messy and therefore “real world” problems should be messy as well. It is not likely that the ball will reach its maximum height after 3, 2.5, or 1.5 seconds. It is more likely to be some longer decimal that needs to be rounded. I wish textbooks would provide more problems that require rounding in multiple steps which would mean teachers would have to teach students how to do so. So many students get used to having answers work out “nicely” for them (integers or decimals that don’t need to be rounded). Then when they get something “unusual” they think they have done it wrong! That’s real life though! It is more unusual for numbers to work out nicely in the real world.

What is it going to take to get textbook publishers to write more problems that make sense to kids?

I think what’s missing from this exercise is a longer introduction about why quadratics are a good model for the height of a ball going straight up (which I don’t have a good self-contained explanation for without delving into physics), why it is purely a function of time (given the coefficients), and why the initial velocity and starting height are the only coefficients that matter (plus the apparently random -16 for gravity goes totally unmentioned), and why they go where they go in the equation.

I don’t know that the problem has to be a “job world” question to be interesting. I think it’s much more engaging if less of the problem is asserted to you.