Modeling Rugby Conversion Kicks Using NetLogo
Keywords: NetLogo, data analysis, modeling, optimization
Level
Grades 9-12
Algebra 2, PreCalculus, Statistics
Purpose
The purpose of this activity is to engage students in a NetLogo modeling activity. Students will interact with the NetLogo Rugby model in order to collect and analyze data. Additionally, students will develop a function and determine an optimized ouput.
Overview
In rugby, after a try (similar to a “touchdown” in American football) has been scored, the scoring team has the opportunity to gain further points by 'kicking a conversion'. The kick can be taken from anywhere on an imaginary line that is perpendicular to the try line (the goal line) and goes through the location on the try line where the try was scored as shown in Figure 1 below. Where should the kick be taken from to maximize the chance of a score?
Figure 1: Rugby extra point kick is taken from a position on the field in line with the location of the touchdown
In order to answer this question, students will gather data from a NetLogo model of the situation. The data will need to be “scrubbed” and analyzed using a spreadsheet or other data analysis tool. Additionally, students can be asked to create an equation that models the angle of the kick as a function of the distance from the goal line.
Student Outcomes
· Students will use NetLogo to create and analyze data
· Students will develop a function that models the relationship between the kicking angle and the distance from the goal line
Common Core Math Standards
· Modeling Standards
· Reason quantitatively and use units to solve problems.
· Interpret functions that arise in applications in terms of the context.
· Analyze functions using different representations.
· Build a function that models a relationship between two quantities.
· Summarize, represent, and interpret data on a single count or measurement variable
Modeling Standards
· Modeling is best interpreted not as a collection of isolated topics but rather in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (★).
Reason quantitatively and use units to solve problems.
· N-Q.1. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
· N-Q.2. Define appropriate quantities for the purpose of descriptive modeling.
· N-Q.3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
Interpret functions that arise in applications in terms of the context.
· F-IF.4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.★
· F-IF.5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.★
· F-IF.7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★
Build a function that models a relationship between two quantities.
· F-BF.1.Write a function that describes a relationship between two quantities.★ Determine an explicit expression, a recursive process, or steps for calculation from a context.
Summarize, represent, and interpret data on a single count or measurement variable
· S-ID.1.Represent data with plots on the real number line (dot plots, histograms, and box plots).
· S-ID.2.Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.
CT-STEM Skills (from CT-STEM Skills Taxonomy)
Data and Information Skills
· Collecting Data
· Manipulating Data
· Analyzing Data
· Visualizing Data
· Creating Generative Descriptions of Data
· Making Hypotheses and Drawing Conclusions Based on Data
· Identifying the Limitations of Data
Modeling and Simulation Skills
· Using Simulations and Models to Understand a Concept
· Using Simulations and Models to Find and Test Solutions
· Understanding How and Why a Model Works
· Assessing Models and Simulations
· Building New Models and Extending Existing Models
Problem Solving Skills
· Troubleshooting
· Decomposing Problems into Subproblems
· Reframing Problems into Known/Familiar Problems
· Simplifying Complex Problems
· Assessing Different Approaches/Solutions to a Problem
·
Computer Science Skills
· Interpreting Instructions Written for a Computer
· Defining Instructions for a Computer
Time
2-3 class periods
Materials and Tools
Internet access
NetLogo
Spreadsheet software such as Excel or Open Office
CT-Rugby.nlogo [Sent as an attachment to my email to Laura]
Student Handout
Creating the kicking angle function for teachers
Preparation
Ensure internet connectivity
Background
In order to model the kicking angle as a function of distance from the goal line, students will need to have knowledge of right triangle trigonometry. In order to determine a maximum angle, students should be able to graphically determine the maximum of a function.
Teaching Notes
Warm up:
Most students (and teachers) will not be familiar with rugby. One way to introduce the idea of the sport is to watch this video of the 2011 Rugby World Cup which was held in New Zealand in September of 2011.
Use the handout to explain the idea of kicking a rugby conversion after a “try” has been scored. Teacher note: In rugby, after a try has been scored (similar to a “touchdown” in American football), the scoring team has the opportunity to gain further points by 'kicking a conversion'. The kick can be taken from anywhere on an imaginary line that is perpendicular to the try line (the goal line) and goes through the location on the try line where the try was scored. If the kick is taken too close to the goal line, the angle is very small. As the ball is moved away from the goal line, the angle increases to a maximum value, and then begins to decrease again. The question becomes where should the kick be taken from to maximize the chance of a score?
Using NetLogo:
Provide access to the NetLogo file CT-Rugby.nlogo. The “Info” tab of this file provides guidance for the students. That same information is also provided below.
Teacher note: I adapted the original netlogo Rugby setup for this particular lesson plan. I am not an experienced programmer, but netlogo is really user-friendly, and it only took a little while to get familiar with how the code works. I then found it pretty easy to make the few tweaks I needed.
MODEL DESCRIPTION
This is an attempt to model a rugby conversion kick using NetLogo. The model kicks a ball in a random direction from each point along the imaginary kick line (the vertical yellow line). For each conversion scored, the starting point is printed out. Data should be analyzed after several hundred "ticks" or iterations. Does the data suggest an optimal kicking position? Teacher note: students should already have a sense that an optimal kicking position exists for any given kick-line location.
HOW TO USE THE MODEL
Begin by adjusting the three sliders that are used to initialize the model:
- GOAL-SIZE determines the size of the goal.
- GOAL-POS is the x-coordinate of the left goal-post.
- KICK-LINE determines the x-coordinate of the kick-line. The kick line is KICK-LINE units away from the leftmost edge of the world.
Note that the values of KICK-LINE, GOAL-SIZE, and GOAL-POS may need to be adjusted to fit your current world-width.
After adjusting the sliders, click the SETUP button to initialize the model. When you're ready to start kicking, press the GO button. Rounds will be repeated and results accumulated until you press the GO button again. Monitors display how many balls have been kicked and how many resulted in goals.
Data for successful kicks is printed below the model in the "Command Center" area.
Note that the "Command Center" should be cleared between data runs.
Teacher note: For the next part to work well, it’ll be useful to let the simulation run for at least 50,000 kicks.
DATA ANALYSIS
The data that is listed in the "Command Center" looks like (patch 20 34). The two numbers are the coordinates of a successful kick. Why are the first numbers the same for each data point? Teacher note: the first number is the x coordinate of the kicking position. The data to be analyzed is the second number, which is the y-coordinate of the initial kicking position.
The data can be copied and pasted into a data analysis tool. It will be necessary to "scrub" the data to get to the y-coordinates of successful kicks. Analyze the y-coordinate data. Does the data suggest an optimal kicking position?
Teacher note: One of the CT-STEM skills is working with data. That is why students are asked to use “messy” data.
In google spreadsheet:
1) Use the ‘find-replace’ to get rid of the words and parentheses.
2) Make a column with just the y-values
3) To determine mode: use =mode(range) , with y-value column as the range
4) To visualize the mode, create 2 new columns, one showing the unique y-values and the other showing the frequency of each of those unique y-values:
a) use =unique(range) to pick out unique y-values
b) use =countif(range,criteria) to determine frequency of each y-value
c) highlight the unique y-value column and the frequency column. Click on chart icon. Make a scatter plot, which helps visualize that the mode is truly the most frequently recorded y-value.
OPTIMIZING A KICKING ANGLE
Verify the optimal kicking position that is suggested by the data by analytically determining where the kicking angle can be maximized. Teacher note: See creating the kicking angle function handout. Also see the Math Handout for solution help.
Teacher note: Then open the original rugby netlogo model and rerun with same setup to check against the simulation as well.
EXTENSIONS
Generalize the results given by the NetLogo model by determining the relationships between the kick-line, the goal, and the best point to kick from.
Consider the following other variables that might affect the solution to the problem: wind speed, grass height, or the size/weight/skill of the player kicking. Change the NetLogo code to take into account one or more of these extra features.
Use dynamic geometric software such as GeoGebra.org to solve the problem. The ‘Level Curves’ option can provide some insight into a geometric solution.
Implement plotting procedures for some or all of the following: the number of successful kicks compared to the overall kicks, the plot of both types of kicks over time, or the difference in histograms depending on the locations of the kick-line.
LEVEL CURVES OPTION
If you turn on the SHOW-LEVEL-CURVES? slider and press SETUP, you will see the level-curves associated with the given set of slider settings. This allows you to visualize the field of solutions for the analytic case of a uniform field. Patches are colored according to how large the goal looks from that position. Along each connected curve of the same color, the goal appears to be the same size. From straight ahead, the goal appears maximally wide. From a shallow angle, the goal looks smaller at the same distance along the try line.