Preview all of the video segments and web interactives used in this lesson. Every driver tries to minimize his or her commuting time rather than reduce the average travel time of all drivers, resulting in a paradoxical situation: Closing a road may actually reduce congestion by forcing drivers to take a route many had avoided, speeding up the average commute. How do you know? We even created fictitious trajectories in which the ball magically passes through all of the physical obstacles (hoop, backboard, back plate) except for one, to see where it collides first. mathematical guidance for the best time to take aim, every equation and variable from traffic theory could be reassigned and interpreted as a description of a basketball game, Founder of geometric analysis honored with Abel Prize, This dizzying labyrinth will host next year’s party for math’s ‘Nobel’ prize, Skepticism surrounds renowned mathematician’s attempted proof of 160-year-old hypothesis. Professional basketball players face that quandary multiple times in every game. We went to the coaches and assistant coaches at North Carolina State University, where we are based, and told them we had this uncommon ability to study basketball shots very carefully. As an old basketball player, like many of you, I enjoy watching the game – and, every so often, catching a glimpse of that perfect free throw. Ask students to discuss how they selected the equation and graphs used, and how they calculated the values for time and height using each set of player stats. However, Skinner's equations show that for a scenario in which if team B shoots, on average, every 20 seconds, team A should shoot every 13 seconds rather than every 10.
Skinner, an avid basketball fan, was inspired to analyze the game when he heard a talk at an American Physical Society meeting in 2007 on the flow of traffic.
"When you're a nerd, you can't turn it off.". What are your conclusions and the reasoning behind them? That paradox reminded Skinner of the Patrick Ewing theory in basketball, named after the high-scoring player for the New York Knicks.
He factors in the probability that a given shot will go in, the quality of future shots the team is likely to generate, and the number of seconds left before the players must either shoot or forfeit the ball to the opposing team. "There's more receptiveness" to these ideas among basketball coaches and managers than ever before, says sports statistician Dean Oliver, who became director of production analytics at ESPN in March after working with the Denver Nuggets since 2006.
For the most part, we discovered things that the players and coaches already knew – but every so often, we came across a new insight. (, Explain that today’s lesson features video segments and web interactives from. So there we have it, 50 IB Maths IA topic ideas to give you a head-start for attacking this piece of IB coursework! Does it depend on whether the shooter is short or tall? Ask students to brainstorm how they think mathematics might be used in the sport. At what time does the ball reach the maximum height? We learned how to turn large numbers of trajectories into statistical probabilities. The ball is located at some height above the floor. Ask students to complete the handout as they explore the online challenges. Get colored stickers (optional) and colored markers, for students to mark the points and construct the trajectory, or path, of the basketball in Learning Activity 1 & 2. His equations follow the ball from the inbound pass until it reaches the hoop. Still feeling confused? Print out one copy of the “Math in Basketball: Take the challenge” and the “Math in Basketball: Try other basketball challenges” answer keys. How does the approach used to solve the challenge affect the choice of representations? Breaking the Code Euler’s Totient Theorem Minesweeper… Explain that in the activity, students should use the Fast Break Facts for information about the three key variables and Elton’s stats to figure out the maximum height the ball reaches on its way into the basket when making a free throw shot.
Ask students to think of situations in their daily life where they may need to apply the concept of maximizing. Download the video clips used in the lesson to your classroom computer(s) or prepare to watch them using your classroom’s internet connection. In the past few years, we’ve expanded our work to study where the best bank shots strike the backboard and developed a tool for anyone who wants to perfect it. What are some ways to represent, describe, and analyze patterns that occur in our world?
We figured out how to go from modestly changing motion to rapidly changing motion, such as when the ball bounces on the rim or off the backboard. Using a social bookmarking tool (such as.
We studied the free throw for about five years. As students complete the challenge, encourage them to use the following 6-step mathematical modeling cycle to solve the problem: How can you combine the three key variables: acceleration of gravity, initial vertical velocity, and release height, to determine the maximum height of the basketball? From a mathematical viewpoint, basketball is a game of trajectories. It has a rate at which it is spinning backwards (called backspin), and it has a launch speed and a launch angle.
A small gap is best whether launching at low or high release heights. Analysts had noticed that in games from which Ewing or other big scorers on a team were absent, that team was more likely to win. Because of the money at stake, professional teams are also reluctant to share data that might be compared with the models, Skinner says. Assess deeper understanding: Ask your students to reflect upon and write down their thoughts about the following: How did you determine an effective strategy for solving the challenges in this lesson? Theory is one thing, but putting the models into practice is whole other ballgame. Let your students know that it is now their turn to solve the challenge that Elton Brand posed to the teams in the video. To shoot, or not to shoot, that is the question. Release the ball too slowly and the shot is short; release it too fast and the shot is long. Math Basketball Games. With tournament play underway, I’m reminded of how competitive the game has become, and how it has truly become a game of inches.
Ask students to brainstorm other real-world situations which involve the type of math and problem solving that they used in this lesson. At 52 degrees, the shooter can be off a degree or more either way without a large effect on the shot’s success. The free throw was the first shot that my colleague and I studied in detail. It seems that the shorter players must try harder. Write an article and join a growing community of more than 115,700 academics and researchers from 3,762 institutions. A new article provides mathematical guidelines about when a basketball player should attempt to score and when the team is better off waiting for another chance. Their first question was simple: “What’s the best free throw?” Should the shooter aim towards the front of the hoop or the back? View Math_IA_sample2.pdf from MATH ENG 111 at ESSEC. Whether 'tis nobler to try to score right away or wait for a better chance.
Please note that difference between HL and SL IAs is the level of math expected of students which is reflected in slightly different rubrics and consequently the HL and SL grades differ slightly. To simulate millions of trajectories without the code taking too long to run, we tried any trick we could think of. Review the following terminology with your students: Distribute the “Math in Basketball: Take the challenge” handout. It speeds up the amount of time it takes to see the patterns behind the best shots. It speeds up the amount of time it takes to see the patterns behind the best shots.
", In his own pickup basketball games, Skinner says he's always thinking whether the best player is shooting too much or whether the worst player should shoot more. All rights Reserved. Skinner finds that, as expected, the more seconds left on the clock, the choosier a player ought to be, passing up all but the highest quality shots. Next, assuming the player releases the ball at 7 feet above the ground, a launch angle of about 52 degrees is best. To shoot, or not to shoot, that is the question. "Brian's paper poses an intuitive theoretical model of the way basketball players should select shots in a variety of situations," says Matthew Goldman, a graduate student in economics at the University of California, San Diego, who also models basketball games.
"The most interesting case is the conventional situation of NBA basketball with a ticking shot clock," which analysts often ignore in evaluating the role and contributions of a player, he adds. Professor of Mechanical and Aerospace Engineering, North Carolina State University. At a release height of 7 feet, the gap between the ball and the back of the ring should be less than 2 inches. The Mathematics of Basketball. Where is it now?
Math offers a unique perspective. I took Math SL, and got a 7. Distribute the “Math in Basketball: Try other challenges” handout. You can play these games alone on the computer or in two teams. That’s interesting, because our coaches at N.C. State and others I have talked say that taller players tend to shoot the free throw worse than shorter players do. These trajectories are unique in that the ball’s motion doesn’t change much when it’s flying through the air, but then rapidly changes over milliseconds when the ball collides with the the hoop or the backboard.
After students have completed the activity, ask students to share their solutions and problem-solving strategies with the class through discussion and visual materials, such as chart graph paper, grid whiteboards, overhead transparency grids.