Systems of Equations via Playing Football

(International note: Before my visitors from across either pond skip ahead in their reader, note this lesson is quite doable with any sort of thrown ball.)

This falls into my grab bag of “physical challenge” lessons, where the students are tasked some act to perform which could be done normally but mathematics makes much easier.

CHALLENGE: Throw a football such that a receiver catches it without slowing down.

The exact conditions depend on if you’re doing the beginner or advanced variant. Here’s beginner:

(International note #2: Tweak for metric. Change “football” and “quarterback” to the ball and name-for-person-who-throws of your choice.)

After setting up the challenge (and nominating quarterbacks and receivers) toss ideas around until the students realize they’ll need the speed of each quarterback’s throw and each receiver’s run.

Outside trip #1: Take everyone outside and lay out a distance, say 20 yards. Have the quarterbacks take turns throwing and get the rest of the class to time the balls from release to catch. (To keep everyone busy, I have students in pairs where one student is timing and the other is writing stuff down. I have them use their cell phones for the time, but if your school policy does not allow this I recommend finding the track coach and borrowing some stopwatches.) Also have each receiver take turns running the same distance (with those times tracked as well).

If you’re feeling punchy, have your students attempt the stunt a couple times. If your students are like mine they can do it but only if the receiver is allowed to slow down. Emphasize the fact they just got tackled.

Back to inside: Students should have enough to work out the necessary speeds. Then leading by discovery or whatever path you desire, they need to realize the formula distance = speed * time (d=st) is applicable here, specifically two equations:

distance of catch = speed of quarterback * air time of ball
distance of catch = speed of receiver * air time of ball + distance receiver is away when ball is thrown

that is

d =  s_1 \cdot t
d =  s_2 \cdot t + 10

where d and t ought to be the same for the receiver to make the perfect catch, and s_1 and s_2 vary depending on the quarterback / receiver match. (I have them work out the calculation for every possible pairing, which requires a moment of combinatorics on their part.)

Solving this system will lead to various answers for d and t; make sure the students write down all of them.

Outside trip #2: Students use what they have learned to attempt the stunt. Bring a cone so the students can mark where the quarterback should be throwing on a given trial. Attempt each pairing multiple times, having the non-athletes timing again (to check against the times they solved for).

Back to inside #2: Since this was more or less a science experiment, students should write up conclusions as well as answer questions like “how could our setup be improved?” or “what other sports situations might mathematics help us with?”

Advanced challenge:

This was the original. This time the students solve for the angle the quarterback needs to throw at. I created this lesson to teach parametric equations. It’s good for trigonometry or pre-calculus.

Anticipating your questions:

Does it work? Yes.

Really? What about variability of throws and running? You need to tell your quarterbacks and receivers to be as consistent as possible. They’re usually a little off, but it’s amazing how close what they do matches what the math says they should do.

What about wind? It’s a bummer (unless you are going advanced all the way to vectors). I use a walkway that is outdoors but shielded from wind. Even without shielding as long as the wind is relatively light you should be ok.

Could this be used to help actual football players? Yes.

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8 Responses

  1. Not sure if this is related, but this article reminded me of this situation:
    My friend and I are at the beach, my friend is at the water’s edge and I am on the boardwalk some distance away. The boardwalk is parallel to the water’s edge, and I can walk faster on the boardwalk than on the sand. At what point do I leave the boardwalk to reach my friend in the shortest amount of time? In other words, at what point do I cut the corner? or what if my friend is additionally walking along the water’s edge?

  2. I’m intrigued! Now is this throw a straight line, or would it follow a parabolic curve (or something similar)? How does that factor into the system of equations?

    • The parabola affects the speed of the ball, but in my runs of this lesson the students have managed to throw it consistently enough that the effect can safely be ignored.

      If a student lobs it really high or really low at random, they aren’t a good choice for quarterback anyway (if nobody is a good choice, use an easier-to-handle ball).

  3. 1)I wonder if a shuttlecock like that used in Badminton would be more consistent in speed? Doesn’t the design exponentially slow down the birdie if hit at too high a speed?

    2)I would never use the first diagram in a classroom. My first reaction was not to see that as a diagram of throwing a football, and my mind isn’t even that far in the gutter. The students would surely catch on even quicker.

  4. I’ve been looking for a way to incorporate projectile motion with linear motion and your post inspired this
    quarterback applet.

  5. [...] Algebra 3 System of Equations Problem Idea May 30, 2010 I just found this. It might be neat to try, especially if I end up with some football players in my [...]

  6. [...] – What an interesting mix of riddles, observations and sharing!  Watch the football-based equations, Q*Bert binomial theorem videos, and the math notation examples for inspiration.  The blog [...]

  7. [...] what hurdles still need to be leaped before we get to the Stuff My Freshmen Actually Care About (the football lesson is still one of my most popular activities [...]

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