Every week this summer I’m posting three versions of the same real-world task. Please tell me: who wore it best?

- In what ways are they different?
- What do their differences say about their authorsâ€™ beliefs about students, learning, and math?
- Would you make changes? Which and why?

Every secondary teacher and secondary textbook author knows that parabolas are #realworld because they describe the path of projectiles subject to gravity. Forgive me. â€œProjectilesâ€ are not #realworld. â€œBaseballsâ€ are #realworld.

But letâ€™s not relax simply because weâ€™ve drawn a line between the math inside the classroom and the studentâ€™s world outside the classroom. Three different textbooks will treat that application three different ways.

Click each image for a larger version.

**Version #1**

**Version #2**

**Version #3**

Chris Hunter claims, “The similarities here overwhelm any differences.” That’s probably true. So let’s talk about some of those similarities and what we can do about them.

**My Least Favorite Phrase in Any Math Textbook**

They each include the phrase “is modeled by,” which is perhaps my least favorite phrase in any math textbook. Whenever you see that phrase, you know it is *preceded* by some kind of real world phenomenon and *proceeded* by some kind of algebraic representation of that phenomenon, a representation that’s often incomprehensible and likely a lie. eg. The quartic equation that models snowboarding participation. No.

Chris Hunter notes that the equations “come from nowhere” and seem like “magic.” True.

@dmcimato and John Rowe point out that what *normal people* wonder about baseball and what *these curriculum authors* wonder about baseball are not at all the same thing.

That isn’t *necessarily* a problem. Maybe we think we should ask the authors’ questions anyway. As John Mason wrote in a comment on this very blog on the day that I now refer to around the house as John Mason Wrote a Comment on My Blog Day:

Schools as institutions are responsible for bringing students into contact with ideas, ways of thinking, perceiving etc. that they might not encounter if left to their own devices.

But these questions are *really* strange and feel exploitative. If we’re going to *use*, rather than *exploit*, baseball as a context for parabolic motion, let’s ask a question like: “Will the ball go over the fence?”

And let’s acknowledge that *during the game* no baseball player will perform *any* of those calculations. This is not *job-world math*. So the pitch I’d like to make to students (heh) is that, yes, your intuition will serve you pretty well when it comes to answering both of those questions above, but *calculations* will serve you even better.

Ethan Weker suggests using a video, or some other visual. I think this is wise, not because “kids like YouTubes,” but because it’s easier to access our intuition when we see a ball sailing through the air than when we see an equation describing the same motion.

Here’s what I mean. Guess which of these baseballs clears the fence:

Now guess which of *these* baseballs clears the fence:

They’re different representations of the *same* baseballs â€“ equations and visuals â€“ but your intuition is more accessible with the visuals.

We can ask students to solve by graphing or, if we’d like them to use the equations, we can crop out the fence. If we’d like students to work with time instead of position, we can add an outfielder and ask, “Will the outfielder catch the ball before it hits the ground?”

This has turned into more of a Makeover Monday than a Who Wore It Best Wednesday and I shall try in the future to select examples of problems that differ in more significant ways than these. Regardless, I love how our existing curricula offer us so many interesting insights into mathematics, learning, and curriculum design.

**Featured Comment**

Iâ€™ll throw ours into the ring: In which MLB park is it hardest to hit a home run?

## 17 Comments

## pam rawson

June 29, 2016 - 1:48 pm -A close read reveals that the textbook problems are only talking about the height of the ball as a function of time. One even goes so far as to say, “popped straight up.” I think that this approach, while common, leads to misconceptions about the algebraic models being used, especially when they are not well developed. I mean, fly balls look like they travel along a parabolic path. So why wouldn’t that equation give me the path of the ball? Perpetuating or feeding misconceptions like this just makes our jobs as teachers and colleagues more difficult. I can’t tell you how often I’ve heard the words, “If only the math teachers would …” from science colleagues. I don’t want to contribute to bad science.

The functions that you put out there model the path of the ball, height as a function of distance from home plate (I’m guessing), and that’s clear from looking at the coefficients. But students in Algebra 1 (I’m guessing at the level of the text) probably don’t have the physics background to know the difference. And that just adds more mystery and confusion to what those numbers represent.

The two sets of equations are modeling different things about the flight of the baseball and we need to be clear about what we want our students to learn. And for what it’s worth, I would always start with a visual.

## Howard Phillips

June 29, 2016 - 2:18 pm -Pure math : f(x)

Applied math : f(t)

## Paul Carson

June 29, 2016 - 2:55 pm -Always good to use intuition. On a separate note, 1, good to see it’s described as the quadratic formula, not equation, and 2, I’d like to see it become

x = -b/2a +- sqrt(b^2-4ac)/2a

One day!

## Karim

June 29, 2016 - 4:02 pm -I’ll throw ours into the ring: In which MLB park is it hardest to hit a home run?

http://www.mathalicious.com/lessons/out-of-left-field (*)

In terms of Dan’s “will the outfielder catch the ball?” question, the outfield screen at Nationals Park now displays the speed of the ball off the bat. A typical MLB home run will hit 100+ mph and clear the wall in a few seconds. In college it’s faster, since they use metal bats. In either case, I’d be careful with this question; IMO, this is where the task goes from being *about baseball* to *about parabolas*, a distinction which may be helpful in coming up with a clean definition of “real world.”

* We’re still pondering which makes for a better final question: Why isn’t the actual flight path of a baseball parabolic (current); or Was the 1998 home run race between Sammy Sosa (Cubs) and Mark McGuire (Cardinals) fair?## l hodge

June 29, 2016 - 5:27 pm -Nice choice of ball flights (allowing for some non-computational approaches).

As Pam pointed out, the problem has changed from height and time to the path of the ball. Why the change? What is gained or lost?

Some information involving time is needed for the question about whether the fielder will catch the ball. What additional information would you anticipate the students requesting (or would you provide)?

## Dan Meyer

June 29, 2016 - 7:46 pm -Nice corrections all the way through here, team, none of which I’ll honor to the point of re-editing animated GIFs, etc, strictly on account of summer. I suppose I’m wondering which corrections are fatal to this particular makeover project and which can be saved by parametrizing the makeover with more care than I did.

## Laura

June 30, 2016 - 2:32 am -Pam Rawson-But students in Algebra 1 (Iâ€™m guessing at the level of the text)

In Virginia this is Algebra2.

## Miles

June 30, 2016 - 6:54 am -A major objection to all three seems to be that it’s totally unclear where the coefficients come from and why a quadratic is a good model for the height. As Pam very correctly points out, these are all modelling the _height_ of the ball but don’t all acknowledge it clearly. I like the way the third sets up the problem as a plausible scenario: the ball is hit straight up in the air, and the 48 f/s is reasonable (a bit slow, but for a pop-up, not out of the question, depending on how old Ted is), and we can know this initial speed with speed guns. Similarly, we can know (or estimate closely) at what height off the ground the catcher grabs the ball, as well as how far off the ground it is when Ted hits it. In my opinion, the third problem more clearly shows where the values of the coefficients come from (though it could be confusing whether the 3 in the constant term is supposed to be the 3 ft at which the catcher gets the ball). I also like that all the setups use h and t for the variables. The third is missing units, though.

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.

## Karen

June 30, 2016 - 9:31 am -This is all just so incredibly awesome. My students could care less about how high in the air a baseball goes, they just care about whether the batter can hit it!

They, rightfully so, question me all the time about where did the quadratic equation that is “modeling” the problem even come from? Who figured it out? And…why would they even want to?

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

## Lindsey

June 30, 2016 - 10:47 am -I agree with Miles that it is not clear to students where the coefficients come from for the models, which is a common theme in math textbooks. I dislike the phrase “can be modeled by” as well and I prefer to select problems that ask the students to create their own model as opposed to those that provide the model and essential require students to “plug and chug.” It would be great to make some tie-in to the physics aspect of the situation for students to begin thinking about the connections between disciplines.

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.

## Reggie

July 1, 2016 - 7:37 am -I was thinking about the coefficients also. They can be(mostly)meaningless when just handed over to students. You could start with an activity where the students throw a ball in the air and time it. They measure their starting height and compute initial velocity to generate a quadratic model. When we do this in class the goal is for them to figure out how high the ball went on their throw. This activity involves computing the vertex but it could also be a nice gateway to the quadratic formula questions above. I find that it is usually a good thing to remind students of the graphical significance of the quadratic formula.

I really like the tweaking to the initial questions to make them more accessible, interesting and likely to lead to a discussion.

## James Key

July 1, 2016 - 6:04 pm -I’m interested in seeing whether students can reason their way to the answer – *without* using a formula that is likely to be a black box for many students. Can they make sense of the problem on their own?

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.

## Christy Poole

July 2, 2016 - 3:52 am -I would love a tech tutorial on how to make that gif!

## Dan Meyer

July 2, 2016 - 2:54 pm -@

Christy, happy to oblige. Are you looking for an animated GIF tutorial or a tutorial about how to make this Desmos graph.James Key:Interesting. I don’t see “modeling” as divorced from understanding what’s happening “behind the scenes.” Can you explain more?

## education realist

July 4, 2016 - 9:20 pm -The coefficients are my primary concern; I, too, don’t want them to be meaningless.

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.

## Chester Draws

July 4, 2016 - 9:36 pm -If weâ€™d like students to work with time instead of position, we can add an outfielder and ask, â€œWill the outfielder catch the ball before it hits the ground?â€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.

================================

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

Instead:

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?

=============================

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.

## Chris Stephens

July 6, 2016 - 9:13 am -A bit off topic, but you might find this interesting: many ways to derive the quadratic formula:

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