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Putting The ‘M’ in STEM: Math + Football = The Pythagorean Theorem?

Jan 6, 2018

Putting The ‘M’ in STEM: Math + Football = The Pythagorean Theorem?

Did you know the start of the NFL playoffs is a great way to learn how athletes apply basic geometry in real life?  In football, if defense wins championships, then tackling is the most important part of the game. Stopping a powerful, shifting ball carrier running at lightning fast speed isn’t easy. But almost every play ends with a tackle. So how do defenders do it? They use angles of pursuit. Sound like math to you? It is!

A defender’s angle of pursuit is the exact path they should take to meet the ball carrier as quickly as possible. Well, that’s simple, you say…the player with the ball is right there. Run over and tackle him, job done. Well, no; offensive players don’t tend to stay still with the ball, they either run towards the end zone or take other measures to throw off your pursuit.  Take too shallow an angle and you won’t meet them along their path but end up chasing them. Take too deep an angle and you’ll meet them having given up many more yards than you needed to sacrifice.

So, does every team have a math tutor on the sideline to do some quick calculations and shout out the proper angle? Nope, taking the proper angle of pursuit is a combination of coaching and instinct. To correctly know the right angle, you must:

  • Know your own speed.
  • Correctly judge the ball carrier’s speed.
  • Gauge the distance between you.
  • Factor in how close the player is to the sideline.
  • Factor in obstacles in between.
  • Factor in help from other defenders.
  • Avoid getting tricked by shifty movements.

Impossible, right? “I need math help, send me a Mathematician” you say? No need. With coaching and practice, this becomes an instant process that defenders do in their mind! It’s known as the “Constant Target Heading Angle Strategy” or the “Interception Strategy”. And we can use our 2,500-year old friend, the Pythagorean Theorem, to mentally calculate it.

Let’s say you are a defender, with the ball carrier 15 yards away in the open field. You think you can tackle him after he gains about 20 yards. By the time he runs those 20 yards, how far do you have to run? Visualize a right triangle, with the ball carrier making up one side of 20 yards (A), the distance of 15 yards making the second side (B), and the unknown third side, the hypotenuse, is the distance you must run (C). The Pythagorean Theorem says that the sum of the square of both sides is equal to the sum of the square of the hypotenuse, or A2 + B2 = C2. Let’s remember that to “square” a number is to multiply it by itself once. In this example, 202 + 152 = 400 + 225 = 625. So, C2 = 625. Then we can simplify C2 by taking the square root of both sides, remembering that the “square root” is finding the two equal factors that when multiplied result in that number, the reverse of squaring a number. In this case, 25 x 25 = 625. So, C = 25, meaning you must run 25 yards in the same amount of time the ball carrier runs 20 in order to meet them 20 yards downfield and (hopefully!) make the tackle. If your forty-yard dash time is 4.5 seconds, which is pretty average in the NFL, you can cover 25 yards in 2.8 seconds. So, you better be quick, or at least quicker than the ball carrier. If not, you’ll end up chasing them and possibly giving up a touchdown.

Try it at home! If you watch the playoffs with a DVR, watch for angles of pursuit in action, rewind and try to calculate the Pythagorean Theorem.

Can you think of other places where angles of pursuit are important? There are so many!

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