Teaching Resources for AP Probability and Statistics

Movement with Functions: Lesson 3 - In this lesson, students use remote-controlled cars to create a system of equations. The solution of the system corresponds to the cars crashing. Multiple representations are woven together throughout the lesson, using graphs, scatter plots, equations, tables, and technological tools. Students calculate the time and place of the crash mathematically, and then test the results by crashing the cars into each other.

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Title Movement with Functions: Lesson 3

URL http://illuminations.nctm.org/LessonDetail.aspx?ID=L770

Materials needed Stop watches
Remote-controlled cars (strongly suggested, but alternatives are described below)
Rulers
Colored Masking Tape
Collision Activity Sheet (optional pre-activity)
Road Rage Activity Sheet
What If? Activity Sheet (optional)
Road Rage Answer Key

Learning Objectives
Students will:

Collect data and graph a scatter plot to determine the speed of a remote-controlled car

Create a line of best fit using estimation and technology

Use tables, graphs, and algebraic calculation to determine when their cars will crash with another group's car

Validate their calculations by crashing the cars into each other

Analyze why their time and location estimates for the crash may not be the same as a real-life trial

Grade Levels Grade 7, Algebra I, Algebra II, Probability & Statistics, AP Probability & Statistics

CA 97 Standards Grade 7: AF 3.3 Graph linear functions, noting that the vertical change (change in y-value) per unit of horizontal change (change in x-value) is always the same and know that the ratio (“rise over run”) is called the slope of a graph.

Grade 7: SDAP 1.2 Represent two numerical variables on a scatterplot and informally describe how the data points are distributed and any apparent relationship that exists between the two variables.

Algebra 1: 9.0 Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically. Students are able to solve a system of two linear inequalities in two variables and to sketch the solution sets.

Algebra II: 2.0 Students solve systems of linear equations and inequalities (in two or three variables) by substitution, with graphs, or with matrices.

Probability and Statistics: 8.0 Students organize and describe distributions of data by using a number of different methods, including frequency tables, histograms, standard line and bar graphs, stem-and-leaf displays, scatterplots, and box-and-whisker plots.

AP Probability and Statistics: 12.0 Students find the line of best fit to a given distribution of data by using least squares regression.

AP Probability and Statistics 14.0 Students organize and describe distributions of data by using a number of different methods, including frequency tables, histograms, standard line graphs and bar graphs, stem-and-leaf displays, scatterplots, and box-and-whisker plots.

CA Common Core State Standards
Standards for Mathematical Practice:

1. Make sense of problems and persevere in solving them
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics
5. Use appropriate tools strategically
6. Attend to precision
7. Look for and make use of structure
8. Look for and express regularity in repeated reasoning

Grade 6.SP.4; Display numerical data in plots on a number line, including dot plots, histograms, and box plots.

Grade 8.EE.5: Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.

Grade 8.EE.6: Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

Grade 8-EE.8: Analyze and solve pairs of simultaneous linear equations.
a. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because
points of intersection satisfy both equations simultaneously.
b. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.
c. Solve real-world and mathematical problems leading to two linear
equations in two variables.

Algebra-CED.3; Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.

Algebra-REI.7; Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.

Algebra-REI.10; Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

Algebra-REI.12; Graph the solutions to a linear inequality in two variables as a half plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. plots on a number line, including dot plots, histograms, and box plots.

Statistics & Probability-ID.1; Represent data with plots on the real number line (dot plots, histograms, and box plots).

Statistics & Probability-ID.5; Summarize categorical data for two categories in two-way frequency tables. Recognize possible associations and trends in the data.

Statistics & Probability-ID.6; Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential model
b. Informally assess the fit of a function by plotting and analyzing residuals.
c. Fit a linear function for a scatter plot that suggests a linear association.

Statistics & Probability-ID.7; Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

Statistics & Probability-ID.8; Compute (using technology) and interpret the correlation coefficient of a linear fit.

How could you use this resource? This activity can be used by students to create a systems of equations using manipulatives and data. Teachers can use this activity to create real life connections to systems of equations.

EL and Special Needs This is a hands-on activity using manipulatives and group work to help students make connections for concepts and real life applications.

Lesson Plans
Share your lesson plan on MERLOT!

See other teachers' lesson plans on MERLOT.

Teacher Comments
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See other teacher comments on MERLOT.

Cost Free

Copyright © 2000-2010 NCTM

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Title	Movement with Functions: Lesson 3
URL	http://illuminations.nctm.org/LessonDetail.aspx?ID=L770
Materials needed	Stop watches Remote-controlled cars (strongly suggested, but alternatives are described below) Rulers Colored Masking Tape Collision Activity Sheet (optional pre-activity) Road Rage Activity Sheet What If? Activity Sheet (optional) Road Rage Answer Key
Learning Objectives	Students will: Collect data and graph a scatter plot to determine the speed of a remote-controlled car Create a line of best fit using estimation and technology Use tables, graphs, and algebraic calculation to determine when their cars will crash with another group's car Validate their calculations by crashing the cars into each other Analyze why their time and location estimates for the crash may not be the same as a real-life trial
Grade Levels	Grade 7, Algebra I, Algebra II, Probability & Statistics, AP Probability & Statistics
CA 97 Standards	Grade 7: AF 3.3 Graph linear functions, noting that the vertical change (change in y-value) per unit of horizontal change (change in x-value) is always the same and know that the ratio (“rise over run”) is called the slope of a graph.
	Grade 7: SDAP 1.2 Represent two numerical variables on a scatterplot and informally describe how the data points are distributed and any apparent relationship that exists between the two variables.
	Algebra 1: 9.0 Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically. Students are able to solve a system of two linear inequalities in two variables and to sketch the solution sets.
	Algebra II: 2.0 Students solve systems of linear equations and inequalities (in two or three variables) by substitution, with graphs, or with matrices.
	Probability and Statistics: 8.0 Students organize and describe distributions of data by using a number of different methods, including frequency tables, histograms, standard line and bar graphs, stem-and-leaf displays, scatterplots, and box-and-whisker plots.
	AP Probability and Statistics: 12.0 Students find the line of best fit to a given distribution of data by using least squares regression.
	AP Probability and Statistics 14.0 Students organize and describe distributions of data by using a number of different methods, including frequency tables, histograms, standard line graphs and bar graphs, stem-and-leaf displays, scatterplots, and box-and-whisker plots.
CA Common Core State Standards	Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them 2. Reason abstractly and quantitatively 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically 6. Attend to precision 7. Look for and make use of structure 8. Look for and express regularity in repeated reasoning
	Grade 6.SP.4; Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
	Grade 8.EE.5: Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.
	Grade 8.EE.6: Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.
	Grade 8-EE.8: Analyze and solve pairs of simultaneous linear equations. a. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. b. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. c. Solve real-world and mathematical problems leading to two linear equations in two variables.
	Algebra-CED.3; Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context.
	Algebra-REI.7; Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.
	Algebra-REI.10; Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
	Algebra-REI.12; Graph the solutions to a linear inequality in two variables as a half plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. plots on a number line, including dot plots, histograms, and box plots.
	Statistics & Probability-ID.1; Represent data with plots on the real number line (dot plots, histograms, and box plots).
	Statistics & Probability-ID.5; Summarize categorical data for two categories in two-way frequency tables. Recognize possible associations and trends in the data.
	Statistics & Probability-ID.6; Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential model b. Informally assess the fit of a function by plotting and analyzing residuals. c. Fit a linear function for a scatter plot that suggests a linear association.
	Statistics & Probability-ID.7; Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
	Statistics & Probability-ID.8; Compute (using technology) and interpret the correlation coefficient of a linear fit.
How could you use this resource?	This activity can be used by students to create a systems of equations using manipulatives and data. Teachers can use this activity to create real life connections to systems of equations.
EL and Special Needs	This is a hands-on activity using manipulatives and group work to help students make connections for concepts and real life applications.
Lesson Plans	Share your lesson plan on MERLOT! See other teachers' lesson plans on MERLOT.
Teacher Comments	Share your comments on MERLOT! See other teacher comments on MERLOT.
Cost	Free
Copyright	© 2000-2010 NCTM