Section 1: So many important ideas in mathematics have come from trying to understand how things move. People have always tried to comprehend the motion of the planets, and the apparent motion of the sun when viewed from the earth. Nature seems to be in a state of constant transformation and motion. Objects in motion and the changes in their motion require energy and the part of science that studies motion is called dynamics, which comes from the Greek word for power (do you know anyone who is dynamic). Dynamics is one part of physics, which is the overall study of matter and energy in all their forms. Have you ever watched a football game? Have you ever seen a quarterback trying to run away from the defense and avoid a sack? Sometimes the quarterback runs all over the field, but ends up gaining only a couple of yards. Sometimes the defense sacks the quarterback behind the line of scrimmage, and there is a negative play. Believe it or not, this situation in football has important mathematics involved! One of the important ideas that emerges from this type of play is the difference between distance and displacement. Distance refers to the total length of the run the quarterback made, with all of the twists and turns, and the displacement is the difference of where the ball ended up from where the ball started. So it is possible to have a displacement of 0 after running a total distance of 50 yards! Our next game will explore the concepts of distance and displacement. From these discussions the mathematics of dynamics will begin to emerge. What we need to build is a system that has lots of movement in it. We will call this type of system with movement a dynamical system. What s the best type of dynamical system to study? Well, if we are going to study it, it might as well be fun. How about a race? Races have the two main ingredients of a dynamical system: Movement in space that is being timed. Before we get to the rules of our race, let s spend some time thinking more about distance and displacement. 2
Directions: As a team, determine at least three situations where there is a difference between the total distance traveled and the displacement from your starting point. Write those situations below, and report them to the class. As a class, make a list of all the team reports and display them in class. Team Work Directions: Can the total distance you travel ever be less than your displacement from the starting point? (For example: Could the quarterback run a total distance of 10 yards, but have a displacement from the starting point of 15 yards? Explain why or why not? Individual Work 3
Rules for Running the Race v The race should be run in teams of three to five individuals. v The judge decides how many legs the race will have. The judge will also throw a three-color die at the end of the race to determine how a winner will be chosen. (We ll explain this in a moment) v Each team gets: 1. A race track (a big sheet of chart paper works well) 2. Enough cubes to run the determined number of legs 3. A coin 4. A marker (any color is fine) v v v Before the race, each team marks a start line on their race track (somewhere in the middle of the track, as shown below). During a turn, each player will have 5 seconds to stack as many cubes as they can. At the start of the race, each of the participants who are going to run the first leg begin with all of the cubes tops up in front of them with their hands facing up. When the judge says Go the players begin stacking the cubes and stop when the judge says, Stop. Start 4
v Next, each team flips a coin. If the coin lands heads, the leg (the stacked cubes) are placed beginning at the start location in the positive direction, to the right. If it is tails, the leg will go to the left (in a negative direction). v Each leg is run in exactly the same way. After a leg is run, it is placed on the track so that it starts where the previous leg ended (just like in a relay race, when the racer hands the baton off to the next runner) and goes to the right if the coin flip was heads, and left if the coin flip was tails. v After all the legs are run we need to choose a winner. Here are the three possibilities. The winner or winning team is the one who: 1. Runs the largest total distance. (Blue) 2. Finishes the farthest away from the start location. (Red) 3. Finishes the closest to the start location. (Yellow) v We assigned one color to each of the three winning possibilities. The judge throws the three-color die to determine who won the race. Now Let s Run a Race! Good Luck! 5
Individual Work - People Drawing Directions: Now that you have run a race, create a drawing of the race below. Try to have your drawing capture all of the information of the race, so that someone could look at your drawing and tell a story about what happened in the race. 6
Individual Work - People Talk Directions: Write a story that describes the race. Put in the story as much detail as you can about what happened in the race. 7
Team Work Directions: Discuss each of the questions below with your team members and write down your team s answers. Each team should report their answers to the class, and the responses should be collected and displayed in the class. 1. How is this relay race like a regular relay race? How is it different? 2. What are each of the steps used to create a leg of the relay race? 3. What does flipping the coin determine in this relay race? 4. Would flipping the coin before stacking a leg of the race change your strategy in the race. Why or why not? 8
Directions: The questions below concern information about winning the race. Each team should answer the questions carefully and then report out to the class your responses. The class should record each team s responses and display them in the class. Team Work 1. Which outcome (blue, yellow, or red) would have been the best outcome for your team and why? 2. If your team lost, is there one action in the race which you could change that would make your team win? 3. If your team won, is there one action in the race which you could change that would make your team lose? 4. Is it possible that one team could win, no matter what color is rolled on the die? Explain your answer. 9
5. Suppose you knew beforehand that the color on the die would end up blue. What type of outcomes on the legs would give your team the best chance of winning? 6. Suppose you knew beforehand that the color on the die would end up red. What type of race outcomes would give your team the best chance of winning? 7. Suppose you knew beforehand that the color on the die would end up yellow. What type of race outcomes would give your team the best chance of winning? 10