PURSUIT WHAT IS IT? ----------- In this model there is one leader turtle and a group of follower turtles. The leader moves along a path according to a preselected formula (such as y = x^2). The followers move toward the leader with each step. The followers update their pursuit path each time the leader moves. Have a little fun- hide the leader and try to guess the path along which it is moving. Clues to the leaderıs path are found by observing the paths of the followers. HOW TO USE IT ------------- General Settings: Use the NUMFOLLOW slider to select how many followers will participate in the pursuit. If exactly 4 followers are selected, they will be placed at the four corners of the screen. If a different number is selected, the followers will be placed at random locations on the screen. Use the STEPSIZE slider to decide how far from its current location a follower will move after each step of the leader. How steps are implemented: - The leader starts at the left edge of the screen. - The leader always moves from left to right by one unit increments along the x-axis.* - The leaderıs y coordinate is based on the selected formula and the current x coordinate. For example, if the current formula is y = x^2 and the leaderıs current xcor value is -3, the ycor value will be set to 9. - The followerıs heading is set using the TOWARDS-NOWRAP function. - Followers move forward in increments of STEPSIZE. * For the trigonometric functions, an integer counter is used for the x coordinate values. This counter is then scaled so that two periods of the function will be visible on the screen. X and Y Axes: The light gray lines that are drawn once SETUP is pressed are the x-axis and the y-axis. Leader Switches: SHOWLEADER makes the leader shown or hidden. This switch must be set before SETUP is pressed. When the switch is set to 1, the leader will be visible. When the switch is set to 0, the leader will be hidden. SHOWLEADERTRAIL, when turned on, causes the leader to stamp at each location determined by its formula. This switch may be turned on and off at any time during the pursuit. When the switch is set to 1, the trail will be visible. When the switch is set to 0, the trail will be hidden. RANDOMFORMULA, when turned on, uses a randomly selected formula for the leader. When the switch is set to 1, the formula will be randomly selected. When the switch is set to 0, the formula will be selected by the user by editing the move-leader procedure. Press SETUP when all of the above selections have been made. This will create the leader and the selected number of followers. Press STEPONCE to make the leader increment its xcor value by one and to make the followers take one step toward the leader. Press GO to make the leader and followers move continuously. To stop them, press the GO button again. RUNNING THE MODEL ----------------- Try starting with 4 followers with a step size of 1. Do not show the leader or the leaderıs trail. Use settings so that the graphics window is square. Press SETUP then press GO. See if you can guess the formula the leader is using by observing the path of each follower. For each of the next questions, consider the follow-up questions Why or why not? How can you tell? Does the speed of the leader seem to change over time? Does the speed of a follower seem to change over time? Do all followers travel at the same speed? What can you tell about the leaderıs formula based on the path of each follower? What traits of each followerıs path give you information about the leaderıs formula? Which of these traits do you find most helpful? Why? To Change the Formula for the Leader: A number of formulas have been stored in the procedures for this model. They are stored using the CONSTANTS feature. To explicitly make a given formula active, set RANDOMFORMULA to 0 and edit the procedure named MOVE-LEADER. Within that procedure is a list of formula names. The formula name that is in square brackets at the end of the conditional statement that begins with "if (choice = 99)" will be used. Note: Since formulas must be seen to be changed, it will be more fun to select the formula randomly. See the EXTENDING THE MODELı section for instructions on how to add your own formulas to the model. THINGS TO NOTICE ---------------- There are several characteristics of each followerıs path and the leaderıs trail that are worth noting. Follower Path Slope: What does it mean if the slope of the path is increasing? What does it mean if the slope of the path is decreasing? What does it mean if one section of the path has a steeper slope than another part? What does it mean if the slope of the path is constant? To think about the slope of a path, consider whether the path appears to be going 'uphill' or 'downhill' and consider whether the 'hill' is steep or flat. Follower Path Concavity: What does it mean if the path has a section that is concave up? What does it mean if the path has a section that is concave down? What does it mean if the path has sections of both of these types? What does it mean if the path has neither concave up nor concave down sections? A path that is concave upı will be shaped like part of an upright coffee cup. A path that is concave downı will be shaped like part of an upside down coffee cup. Relationships Between Paths: Do the paths have any symmetry? Would you expect them to? Why or why not? Distances Between the Leader and a Follower: Once you have determined the formula for the leader, run a simulation with the leaderıs trail turned on. (Note that there are other suggestions to verify your answer in the THINGS TO TRYı section. Make sure you have tried at least some of these before you show the trail. If you show the trail before you are really sure you are right, you might end up spoiling all your fun- thereıs no going back once you have seen the trail of the leader!) Find a path where a follower seems to get close to the leader only to have the leader appear to speed up and escape from the follower. Why does this happen? What kinds of generalizations can you make about the formulas or relationships for which this happens? How do the distances between the leaderıs trail stamps relate to the perceived speedı of the leader? Leader Stamp Proximity: Depending on the path that the leader is following, the leaderıs trail stamps may not be evenly spaced. What does it mean when the steps are evenly spaced? What does it mean when the steps are not evenly spaced? What does it mean when the stamps are closer together? What does it mean when the stamps are further apart? THINGS TO TRY ------------- Try moving followers to specific locations after SETUP has been pressed but before GO has been pressed. Make predictions about how different locations would be helpful. What can you learn if a follower starts in any of the following locations? - along the right edge of the screen - along the left edge of the screen - along the top edge of the screen - along the bottom edge of the screen - along the x-axis - along the y-axis - at the origin What is the most helpful first location for a follower? (The location on top of the leaderı is, of course, out of the question!) What is the most helpful follow-up location for a single follower or for a group of followers? Come up with a strategy for placing followers so that you can determine the path of the leader fairly quickly. Describe your strategy. You may use the command center, the click-and-drag feature, or the turtle window to move a follower. The leader is turtle 0, the followers are all turtles with who > 0. Try increasing the number of followers. (Even if you think you have the formula figured out, try using larger NUMFOLLOW values before you show the leader or the leaderıs trail. ) Why does using a larger NUMFOLLOW value make it easier to guess the leaderıs formula? Try increasing the STEPSIZE of the followers. (Even if you think you have the formula figured out, try using larger STEPSIZE values before you show the leader or the leaderıs trail. ) Why does using a larger STEPSIZE value make it easier to guess the leaderıs formula? The above discussions all involve trying to guess the path of the leader. Alternately, you can know the formula of the leader and try to guess the paths of the followers. If you know a leaderıs formula and are trying to guess the pattern of the followersı paths, make sure to record you guess before you run the simulation. Compare your predicted results with the actual results. - What reasoning led you to correct predictions? - What assumptions that you made need to be revised? EXTENDING THE MODEL ------------------- To add your own formulas, you need to edit the CONSTANTS declarations. Add your formula to the current list. Each addition should be of the form formula-name [command-list] where formula-name is the name you will use to reference the formula and command-list is the set of StarLogo commands you will use to determine the x and y coordinates of the leader. - Note that the command list must begin and end with square brackets. - Within the set of commands you may need to scale the y-axis to keep the leader from wrapping. (See for example, the cubic function.) - You must restrict the domain of the leader if your formula has values for which it would be undefined. (See, for example, the logarithm function.) - Add formula-name to the MOVE-LEADER procedure and make it the active formula. In this simulation, the leader uses only integer x coordinates. For which formulas might the results be different if the leader moved along smaller intervals? What would happen if the STEPSIZE of the followers was always set to equal the distance the leader traveled during its most recent step? Adjust the procedures so that rectangular graphics windows do not cause unexpected wrapping. Do any of these changes impact answers to any of the questions asked above? Create leader functions and pick a follower location to get the following shapes in the followerıs path: - a straight line with positive slope - a straight line with negative slope - a horizontal line - a vertical line - a loop - a circle - a curve with one humpı - a curve with two humpsı - a curve with three humpsı - a curve with n humpsı STARLOGO FEATURES ------------------- Switches are used to show or hide information about the leader. Coloring of patches is used to draw x and y axes on the graphics window. The CONSTANTS declaration is used to make it easier to change the formula of the leader. The TOWARDS-NOWRAP command is used to orient the followers.