Every secondary teacher and secondary textbook author knows that parabolas are #realworld because they describe the path of projectiles subject to gravity.

Forgive me.

“Projectiles” are *not* #realworld. “*Baseballs*” are #realworld.

But let’s not relax simply because we’ve drawn a line between the math *inside* the classroom and the student’s world *outside* the classroom. Three different textbooks will treat that application three different ways. Here are three of them, in fact.

Tell us: who wore it best?

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

Click each image for a larger version.

**Version #1**

**Version #2**

**Version #3**

## 5 Comments

## John Rowe

June 22, 2016 - 7:50 pm -Here’s a thought (only a thought, not sure how well it would work):

Play a snippet of a batter hitting a ball into the outfield. Hopefully the bases are loaded.

What happens next? (This could be students writing a commentary of what happened)

Some questions that might come up could be:

Does the ball hit the ground?

Is it a foul ball?

Is it a home run?

Which outfielder is it closest to?

Who will get run out?

Apologies about baseball terminology, I’m an Australian.

## Ismael zamora

June 23, 2016 - 6:46 am -I have a picture of a Home run during an actual game, with launch angle and initial velocity on it. The funny part is the distance is blocked by a logo on the screen. not sure how to post it?

## Ismael zamora

June 23, 2016 - 6:50 am -Just posted it to 101questions.

## Ethan Weker

June 23, 2016 - 8:08 am -If there’s one thing I’ve learned from you over the years, Dan, it’s that you can always add, but never subtract. Why for this problem does each book spoon feed the problem? This turns a potentially fun question into a boring algorithm with very little true problem solving. Of course, there are certain constraints inherent in a print textbook problem, but I’ve done similar problems with lots more questions, and much less starting with answers.

To start…maybe a better question can be set up from showing a video clip of a line drive set to land between the center and right fielders. Stop the video partway, and ask kids what they think could happen. Hopefully, questions about if the outfielder will be able to catch the ball before it lands, and if they don’t, will they get the throw to first in time. (This does leave behind any students who don’t know/care about baseball, but hopefully, we have other problems that interest them later, and in the meantime, maybe they will get caught up in the excitement of the moment.)

From the wonder line of questions, we then determine what information we need – how fast is the ball hit? When will it land? This can also be nicely differentiated to include the angle of the hit, the horizontal velocity and distance traveled, the distance and speed of the outfielder, a system of equations, then everything that goes into the possible throw to first. Add in probabilities of errors, and the richness of this problem seems to never end.

If we, as teachers, can anticipate the questions that students will have, we can have data prepared to let students push through, request that data when they need it, reason through what formulas to use and why quadratics make sense. When the student drives the questioning in a problem, they become much more invested than if we just tell them to use the quadratic formula with the projectile motion model where we solve for 0.

Also, on a side note, still don’t understand why some texts seem to introduce quadratic formula before, and separate from, completing the square. Not to be hyperbolic here, but there’s nothing worse in a math class than introducing something as if it’s magic, which is how many students seem to see the quadratic formula. Something to memorize, because it works. (And maybe expanding on that idea should be my own blog post…note for later.)

## Scott Farrar

June 23, 2016 - 9:01 am -1.

Header gets us into the context ok. But we aren’t in that context for a long time. Immediately we become the most abstract version of this lesson. I always wonder how many kids understand a phrase like “gives the solutions in terms of the coefficients.” I guess this is the first intro to the quadratic formula as well, since we’re learning its derivation in another lesson. I think the strategy from the book is to have students plug numbers in as a kind of “hey see its an equation also and it can be shoehorned onto this context”

2.

Given an equation without a lot of context. If the student accepts its reasonable to use that equation then I actually kind of like this one. This is a “be less helpful” find the vertex exploration. The vertex is contextualized as a time and height of a physical object so that can help. But the context is not exactly fleshed out. Since the book doesn’t tell specific strategies, students are free to try a lot of stuff or the teacher is free to suggest certain strategies as well.

This one needs more from the teacher, but I see that as a positive for the experienced teacher.

3.

First blush: 48 feet per second, I have no idea how fast that is. (google says 32 mph… pretty slow for a batted ball!!). Another small item: funny how the terms are in reverse of the standard form of polynomials. “How long was the ball in the air?” means solving for t, but it can be done with a quick factoring of 16t(3-t). This is a plug in and solve, and we can dispense with the context immediately. Even though there’s a fun little box about Canadian baseball players and their positive impact on the game! A little bit of Canadian West Coast Bias there though! Where’s Joey Votto!? Getting the answer of 3 seconds seems reasonable I suppose but

All three:

Notice each question has a little authoritative jab in it? “Round to the nearest hundredth.” “Show your work.” “Explain.” These are funny because the teacher is either going to override these rules or will already have standing rules in the classroom about this. Why does the book print them? Maybe because teachers don’t read lesson guides that might suggest certain strategies for teaching– books have gotten into the habit of trying to run the lesson for you.

All three of these attempt to use the flight of a batted fly ball to suggest an image of a quadratic from the students experince. I think #2 is the one that requires repeated thought about the context however. We are asked about the vertex there, but have no method to find it except for guess and check? Or: we’ll develop a method using some reasoning about the context. That’s good, something to build off of.

I’d also want to justify the model we are using a little bit before just diving in. Even if its just relating a x^2 graph (which Ss can do from a numerical pattern and point by point) to a strobe photo of a falling ball https://people.rit.edu/andpph/photofile-c/a-stroboscopic-motion/drop-2balls-strobe-7994A.jpg and talk a little about how gravity pulls things down faster and faster. I suppose we could say we only model projectiles with quadratics because they “seem to work” but if I did that path I’d want to show that other models don’t work and that seems rather time consuming and not very intellectually honest.

Couple other resources we might build off of:

MLB statcast can measure some of the things that these problems pull out of thin air. http://i.imgur.com/oHVvPpK.png video: http://m.mlb.com/video/topic/31426364/v606604783/oaknyy-hicks-unleashes-1055mph-throw-home-for-out

Or you can analyze video clues to find some additional facts about baseball paths. Like with Cespedes’ throw a couple years ago: video: http://m.mlb.com/video/topic/6479266/v33614681/oaklaa-cespedes-throws-out-kendrick-at-the-plate Analysis: http://www.baseballprospectus.com/article.php?articleid=23864 some screencaps: http://imgur.com/a/HSUPO

I think its tricky what we are asking to find though. With Cespedes we wanted to know speed of throw because its a feat and we didn’t measure it. With Hicks, we had speed… what do we want to know? We could compare how flat it is compared to others. We might start with, “was Hicks’ throw low enough so that the cut-off man could have caught it?” That’s where we can dig into a vertex finding question like #2 with some actual consequences (HIT THE CUTOFF MAN! — or don’t I guess if you can throw the guy out at home)