## [WCYDWT] Will It Hit The Hoop?

Is he going to make it? Can you draw me the path of a shot that will make it? That will miss it?

How about now? Can you draw me the path of a shot that will make it? That will miss it?

How about now? Can you draw me the path of a shot that will make it? That will miss it?

A little more obvious, isn’t it? And like that, we’ve derived illustrated the fact that, while one point is enough to define a point, and while two points are enough to define a line, you need three points to define a parabola.

Here are seven versions of the same problem. Each one contains:

1. the half video, for asking the question,
2. the half photo, for giving the students something to work with,
3. the geogebra file, one use for the half photo, featuring a dynamic parabola in vertex form.
4. the full video, for showing the answer,

Attachments

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. More here.

1. #### keninwa

November 15, 2010 - 6:32 am -

What kind of software do you recommend for a teacher on a budget that would like to make similar videos?

2. #### Como

November 15, 2010 - 7:12 am -

Hi Dan, my HS son suddenly is interested in the length of various segments of a parabola (and other curves). He is currently studying precalculus and introductory physics.
I don’t want to blow this opportunity to have him ease himself into the calculus, so I was hoping that you have a choice link or two that can help me point the way: I’m struggling not to get bogged down in the algebra.

3. #### Alex

November 15, 2010 - 8:06 am -

In general the lengths of curves are very tricky things. It is often easier to work out something slightly different – for example, it’s much easier to figure out the time taken to travel along a parabola than its actual length.

Of course, physicists come up against things that are tough to calculate all the time. Their usual approach isn’t to make their head explode trying to learn new maths – it’s to approximate. We can do the same here.

I’ve spent three or four minutes putting together a spreadsheet which approximates a particular bit of a parabola. A particularly persistent high school student (with his dad’s help and a good knowledge of spreadsheets) might be able to see how it works and extend it to other curves.

On the other hand if you thing it will intimidate him rather than challenge him – feel free to ignore.

4. #### Jason

November 15, 2010 - 8:58 am -

I quite like this example. It reminds me of a similar problem I encountered in a high school physics class, “What is the maximum range of a field goal kicker?”, which is a variational problem which requires a bit more machinery.

One additional point to consider:

Assuming that the shooter is using a set shot, a student could realize that the shot won’t hit rim after the second photo. Namely, the possible release points gives a family of parabolas which they can examine, none of which hit the rim.

And if they use the rim as the ‘fuzzy’ point, they could guess how high the shooter has to jump to make the shot.

5. #### Surani Joshua

November 15, 2010 - 9:10 am -

Wow, that’s great! This is the sort of thing I can use in my classroom at little “cost”

I don’t have to worry about losing a whole day, or kids shutting down. Its a three-minute activity that really hits and reinforces a single math idea.

THANKS. Will be using it in just a couple of months.

6. #### Julia

November 15, 2010 - 9:58 am -

Nice – with this and your other stuff on quadratics you pretty much have my entire topic covered. :)

But I wonder over your use of the word “derive”. I’ll probably conclude with a more rigorous and algebraic proof.

7. #### Sean

November 15, 2010 - 2:10 pm -

1. Very cool idea accompanied by a simple, elegant delivery. An awesome exposition into one idea and review of a couple others.

2. Just out of curiosity- why did you use a video where your shot was wildly short? A Jedi-mind trick to engage them more? Did it have less complicated production than a shot that went in? The aesthetics are so crisp, but kids who love hoops are going to wonder why the six foot seven dude doesn’t reach the rim.

3. Also, and unrelated: you tweeted about a Jo Boaler visit to one of your lectures. Goosebumps, man. Did she illuminate any of the fogginess of pseudocontext? I think what’s especially interesting of late is the how you’ve delineated pseudocontext vs. actual-context-but-just-bad-problems. Which is worse?

8. #### Dan Meyer

November 15, 2010 - 9:01 pm -

keninwa: What kind of software do you recommend for a teacher on a budget that would like to make similar videos?

QuickTime Pro. Adobe Photoshop. Adobe AfterEffects. Wait. You said “on a budget,” didn’t you. Crud.

jon: Looks like a cool problem. What kind of camera did you use to do the strobe photos?

A garden-variety Flip cam.

Julia: But I wonder over your use of the word “derive”. I’ll probably conclude with a more rigorous and algebraic proof.

Yeah, fair. Maybe call it “illustrated.”

Sean: Just out of curiosity- why did you use a video where your shot was wildly short? A Jedi-mind trick to engage them more? Did it have less complicated production than a shot that went in? The aesthetics are so crisp, but kids who love hoops are going to wonder why the six foot seven dude doesn’t reach the rim.

Maybe it helps to see all the half-videos side-by-side-by-side-etc. If you ask students to guess “which ones make it?” you want a range of responses so they can pick up the deeper structure of the parabola. I also like this one for the instructional sequence it allows above. I can really see the shot going in on the second shot. The third shot changes everything.

Sean: Also, and unrelated: you tweeted about a Jo Boaler visit to one of your lectures. Goosebumps, man. Did she illuminate any of the fogginess of pseudocontext?

I guess it wasn’t on her agenda for the day. As this pseudocontext thing winds down, I really need to see if I get a few minutes on her schedule for an interview.

9. #### Chirs Sears

November 15, 2010 - 9:07 pm -

I’m with Sean on wondering why you would feature the video with a sad, sad air ball. I’m sure there is a good pedagogical reason why you would invite good-natured heckling.

10. #### Bowen

November 15, 2010 - 10:44 pm -

First, AIR BALL AIR BALL. You were clearly distracted by the length of the hypotenuse from the foul line to the hoop.

Second, I believe that after the second video (with two data points) you can tell the shot won’t be good, because there is actually a third semi-known data point: the release point of the shot.

Ah — on a more careful watching, it looks like the first data point given is intended to be the shot’s release point, so never mind.

Here’s something that could finish off the lesson well: shoot and record another shot that goes in, and determine the two points of intersection of that parabola with the one above (or some other missed shot — perhaps taken from a different release point). Then, it would be exceptionally clear that two points aren’t enough to make the decision of whether or not the shot goes in.

Cool video and technique!

11. #### jon

November 17, 2010 - 5:43 am -

Bowen — Good idea. By layering, students could see that more than one solution exists as well since varying shot paths can have the same result!

12. #### Dan L

November 18, 2010 - 3:15 am -

Clarification (maybe):

If you snap the picture with the shooter’s hand in the air you have a launch point (point 1), the ball (point 2), and the hoop (point 3) and you’ll have a uniquely defined parabola. Maybe leaving this information in the background will lead to another “a-ha” moment for the student?

Then of course there are the so-called “brick” shots in basketball where the ball bounces in, then out.

Dan L.

13. #### Jim Ellis

November 26, 2010 - 6:48 am -

Keninwa: I recommend gimp (X11) because it’s free and makes some incredible pictures. Furthermore, there are great youtube tutorials that will help you get started.

14. #### Ted

February 10, 2011 - 3:11 pm -

This is great, although a bit of a pseudo-context. When did you see spectators or basketball-players actually calculate whether the basketball would hit the hoop or not?

15. #### Dan Meyer

February 10, 2011 - 5:51 pm -

I appreciate you keeping me on my toes here, Ted, but this can’t be pseudocontext. There’s no question. It’s just a video of some guy shooting a basketball that cuts out halfway into the shot.

Maybe now the student wants to know the answer to a question that interests her (“will it hit the hoop?” for instance) and you have a natural tool that can help her figure that out. That’s context, not pseudocontext.

16. #### Jeff

February 26, 2011 - 12:44 pm -

Great lesson! My students loved it. I took your half pics and set them as a background on the TI-Navigator and then had students find symbolic rules that would match the pattern. Students had some great ideas as to where to place the vertex. Lots of terrific conversation took place. Thanks for your hard work and for sharing!

17. #### Dan Meyer

February 27, 2011 - 4:11 pm -

Thanks for the report, Jeff. I’m glad you had some fun with the kids. Did you assess them at all? Do you have any indication of what they learned?

18. #### Timon Piccini

May 7, 2011 - 10:00 pm -

Dan,

I have been mulling through your stuff as usual, playing with Geogebra, and trying to learn all this stuff that you are giving the world, and I thought of a fun (yet maybe impossible) extension to this one.

Remember the cheesy teen comedies of the 90’s (Saved by the Bell being the best example), and whenever they played basketball things just always seemed to work. I was thinking of testing some of these out if they work, if they make true parabolas or if they are fake. I don’t know how it would work, but in my mind it would be a hoot (maybe out of context for the students these days, but I am sure there is a cheesy comedy out there with basketball shots).

19. #### Mary

April 27, 2012 - 9:54 am -

Hello,

On your Youtube video of parabolas using a basketball shot as one of your examples, there was a tool you used to manipulate your parabola through stretching and compressing. This tool actually found the equation of your line. Do you happen to know where i could get that tool ??

20. #### Benjamin Etgen

August 21, 2012 - 8:32 pm -

Dear Dan,

Thanks so much for posting these excellent resources. Your approach is clear and engaging.

I teach math in Community College and will use these in my algebra classes.

Thanks!

–Ben

21. #### Rachel S

September 20, 2012 - 1:38 pm -

I did this with my algebra 2/ trig class today (I know I’m a little late on this) and it went really well. By the end they were betting each other actual money that certain shots would or wouldn’t go in. This blog is great!

22. #### Chris McCaffrey

March 16, 2013 - 4:34 pm -

I was helping 2 algebra teachers try to get this to work on netbooks and we needed to find out how to do this the cheap way…

GOM Player, http://player.gomlab.com/eng/ – homepage

Gimp, http://www.gimp.org/

I didn’t get to the video part, but that’s not necessary for the kids to work in GeoGebra and make the parabola. I am thinking WeVideo might be able to handle it, but I ran out of steam.

Here’s my screencasts for using GOM and GIMP to get to the image with all the balls in flight.

– Taking pictures, frames, from a video using GOM Player,