I came across a clever application of the quadratic formula on Youtube. Tom Rocks, an Oxford mathematician and the video’s creator, uses the dimensions of a soccer net (“football net” anywhere in the world outside of US and Canada) and the maximum diving range of a goalkeeper in all directions, to obtain the radius of the target circle for a penalty kick. In other words, it figures out the best two places where the kicker should aim the ball.
The key to solving the problem the “long way”, as a student would often put it, is to draw a right triangle (in blue, above), deduce its dimensions, apply the Pythagorean theorem and then use the quadratic formula to obtain the value of the radius.
If you’ve taught math or recall being in a high school algebra class, you know that at some point, a student will inevitably argue that a reasonable estimation in the context of this problem can be obtained with a scaled drawing. I’ve done that too in the illustration, and it works.
So on the surface, it seems that this strategy of attempting to demonstrate the practicality of the quadratic formula can easily backfire. The extra decimals are totally useless to a real soccer player. Someone who has played the game can also argue that if he can get the goalie to move first, there is a higher likelihood of scoring if he just kicks low and in the opposite direction.
The teacher facing such valid points should never take it as a personal affront but use them as launching points for a brief discussion of mathematics’ role in the real world.
The issue of the bluff-move is an example of how the real world is more complex than a mathematical model. This is what we have here. It’s assuming that both the kicker and goaltender act as robots, when in fact, the minds of the players add additional variables that cannot be quantified.
What do you say to a teen who has come up with the answer of 0.65 meters by simply drawing the situation accurately? For starters it’s a wonderful way to verify that our math is valid. But to truly persuade the human brain that is so often committed to conserving energy, we can to do better as teachers. Whereas basic arithmetic permeates everyday life, high school math on the surface seems to be a twilight zone and applications are often contrived or, as in this case, involve problems that can be solved more easily with simpler math. We have to explain to teens that high school math will soon appear in other high school subjects such as chemistry, physics and economics, Moreover, a lot of it is a foundation for higher mathematics, which is not only far more practical in the real world than basic algebra, but it opens the path for learning pure mathematics, which is pure joy and a worthwhile endeavour in itself.
Once can also argue that although the extra accuracy of the Pythagoras-and-quadratic-formula-route of determining the radius of the target circle is useless in this case, in other applications, a rigorous approach and the extra-decimals obtained can be crucial. The classic example is GPS positioning. For a satellite to pin down location within a 3 to 4 meter accuracy, the math of the Doppler effect, of general relativity and special relativity all have to be used.
And guess what was used to derive the gamma factor involved in obtaining relativistic time, length and mass? Yes, the Pythagorean theorem.