Appendix D: Assessment and Evaluation - Samples
Sample 2: Applications of Mathematics 11

Topic: Probability and Statistics

Prescribed Learning Outcomes:

Problem Solving

It is expected that the student will:

• solve problems that involve mathematics within other disciplines.
• use appropriate technology to assist in problem solving

Patterns and Relations

It is expected that the student will:

• solve nonlinear equations using a graphing tool.
• design and solve linear and nonlinear systems, in two variables, to model problem situations.

Statistics and Probability

It is expected that the student will:

• extract information from given graphs of discrete or continuous data.
• draw and validate inferences, including interpolations and extrapolations, from graphical and tabular data.
• collect experimental data and use best fit exponential and quadratic functions, to make predictions and to solve problems.

Unit Focus

The goal in this unit is to have students appreciate that the world around them can often be modeled using strict mathematical relationships. Although many of these relationships can be complex, much of the interaction that they observe between objects around them can be modeled simply and elegantly using mathematics well in within their grasp. In this unit we will study the motion of projectiles and discover how any object thrown by hand or shot out of a gun obeys simple mathematical laws.

Planning the Unit

To develop the unit, the teacher:

• identified the prescribed learning outcomes for the unit.
• examined the specific prerequisite knowledge and skills needed to achieve these outcomes.
• determined which prerequisites were in place and which should be reviewed
• planned a variety of activities to help students achieve the outcomes
• looked for ways to connect students' learning to other desirable outcomes
• identified criteria to evaluate students' learning
• designed assessment that was an integral part of the learning process

The Unit

Introduction

The teacher began the unit with a discussion about how many of the phenomena we observe in the world around us can be described using mathematics. The teacher introduced the example of projectiles and the many different objects that fit the description of projectiles and discussed with students how the computer and graphing calculators make it possible to use powerful analysis tools to discover the mathematical relationships that describe projectile motion.

Review

• Students reviewed basics of data gathering (tables) and graphing techniques.
• Students reviewed (or were introduced to) how to enter equations and individual data points into a graphing calculator or graphing software.

Demonstrating Understanding of Basic Concepts

• Students solved a short problem set involving simple physics problems that can be solved by graphing equations and analysing the results. Analysis of some of the questions involved regression analysis of individual data points.

The Activities

• Students worked on a problem set involving basic projectile motion problems to give them opportunities to practice manipulating the basic formulas associated with projectile motion.
• Introductory exercises were designed to teach students how to do a quadratic curve fit on projectile data.
• The teacher presented a video lab involving analysis of two projectiles: a ball rolled off a table and a thrown ball.
• Students then worked on a final problem set involving more complex projectile questions.
• Students filled out a questionnaire designed to reflect their competencies and attitudes toward the material covered in this unit.

Lab Activity Curve Fitting Data

1. The following data was collected when a ball was rolled off a high shelf in a classroom.

A. Enter the points (t, ) in either a graphing calculator or some type of graphing software.

B. Use the capabilities of the technology you have chosen to fit a curve to the data on the graph. For this set of data what is the best shape to try fitting?

C. The key descriptors for a line are its slope and y-intercept. What do the slope and the y-intercept stand for in the context of our projectile? What are their values? Explain those values in the context of the question. (Why does the line have the value it does? etc.)

D. Now plot the data (t, ) and do a curve fit on this data. What shape does the data seem to take?

E. For a simple projectile problem like this the equation that relates the distance a projectile travels to time is where . Is the regression constant for your curve ~ 4.9? If so this data follows a classic projectile curve.

t (sec)	(m)	(m)
0.0	0.000	0
0.1	0.072	0.049
0.2	0.144	0.196
0.3	0.216	0.441
0.4	0.288	0.784
0.5	0.360	1.1225
0.6	0.432	1.764
0.7	0.504	2.401

 

2. Spaceman Spiff recently crash landed on the planet Blarg. Do a curve fit on his data, recorded in the following table. Find key information about the planet from your analysis of the data.

A. What was the horizontal component of the initial velocity at which Spiff threw the rock?

B. What was the vertical component of the initial velocity?

C. What is the acceleration due to gravity on the Planet Blarg?

t (sec)	(m)	(m)
0.0	0.000	0
0.5	1.75	0.41
1.0	3.50	1.65
1.5	5.25	3.71
2.0	7.00	6.60
2.5	8.75	10.31
3.0	10.50	14.85
3.5	12.25	20.21
4.0	14.00	26.40

 

A. Find the arrow's maximum height.

B. Find the arrow's range (how far it traveled horizontally).

C. What was the arrow's initial velocity? (Combine the horizontal and vertical components of the initial velocity.)

D. At what angle was it shot? (Use trigonometry.)

t (sec)	(m)	(m)
0.00	0.00	0.00
0.25	4.91	3.14
0.50	9.82	5.66
0.75	14.74	7.57
1.00	19.66	8.87
1.25	24.57	9.55
1.50	29.49	9.62
1.75	34.40	9.08
2.00	39.32	7.93
2.25	44.23	6.17
2.50	49.14	3.78
2.75	54.06	0.80

Lab Activity Projectile Motion

Purpose: To analyze how a projectile behaves when it is:
a) rolled off the end of a table; and,
b) thrown upwards at an angle to the horizontal.

We hypothesize that as gravity is the only thing affecting a projectile's motion once it is no longer being supported/propelled, it should not accelerate at all horizontally and it should accelerate at vertically. In other words

Materials:

• Two videos, one of a ball rolling off a table and one of a ball being thrown. It is necessary to know the real size of some element on the video so that you can figure out the scale of the video.
• A video player with the ability to advance one frame at a time.
• A clear sheet of acetate.

Data:

Procedure:

1. Place acetate over the television screen.
2. Play the video frame by frame and mark the center of the ball in each second frame.
3. Measure the distances both horizontally and vertically from the first mark to each mark on the video.
4. Multiply these distances by the scale factor for the video (the real length of some object on the screen divided by the length in video).
5. Record data in the table below.

Analysis:

1. Convert distances to velocities by finding the change in distance in data points on either side of a given point and dividing it by the time elapsed for that data point. (Example: . Fill completed values in the table below.
2. Plot on one graph and vy vs t on a second graph.
3. What does look like? It should be fairly linear and close to horizontal, so use a linear regression to find the line of best fit and find the slope of the regression line. What is the slope?
4. What does look like? It should also look linear but have an upward slant to it so use a linear regression to find the line of best fit and find the slope of the regression line. What is the slope?
5. In both cases above classical physics tells us that the values we have found should be the horizontal and vertical components of the ball's acceleration. Assuming gravity is the only force acting on the ball then the horizontal acceleration should be zero and the vertical acceleration should be . How close did you come? Do you feel that you have proved you hypothesis?

Conclusion:

Summarize the purpose of the experiment and your findings and state whether you achieved the purpose of the experiment.

Image #	Average Horizontal Velocity (m/s)	Average Vertical Velocity (m/s)
1
2
3
4
5
6
7
8
9
10

Purpose:

To analyze how a projectile behaves when it is:
a) rolled off a table; and,
b) thrown upwards at an angle to the horizontal.

We hypothesize that as gravity is the only thing affecting the projectiles motion once it leaves the table, it should not accelerate at all horizontally and it should accelerate at 9.80 m/s2 vertically.

Materials:

• Video E-1710
• World in Motion software

Procedure:

1. Load the software and video E-1710.

2. Play the video frame by frame and mark the center of the ball in each frame.

3. Go to the analysis menu and choose the , graph.

4. Find the slope of each graph. (Use a mouse to mark the endpoints of the line. Slope will appear in the box below the graph.)

5. Print each graph.

Analysis:

1. The slope of each velocity-time graph is acceleration.

2. Plot on one graph and on a second graph.

3. What does look like? It should be fairly linear and close to horizontal so use a linear regression to find the line of best fit and find the slope of the regression line. What is the slope?

4. What does look like? It should also look linear but have an upward slant to it so use a linear regression to find the line of best fit and find the slope of the regression line. What is the slope?

5. In both cases above classical physics tells us that the values we have found should be the horizontal and vertical components of the ball's acceleration. Assuming gravity is the only force acting on the ball then the horizontal acceleration should be zero and the vertical acceleration should be . How close did you come? Do you feel that you have proved you hypothesis?

Conclusion: Summarize the purpose of the experiment and your findings and state whether you achieved the purpose of the experiment.

'Basic' Projectiles
Problem Set

Key Formulas

1. A ball travelling at 0.65 m/s rolls off a table 1.25 m high. How long does it take to hit the ground? (Assume no friction in all of these problems.)

2. A diver runs straight off the end of a 5.0 m diving tower at 4.5 m/s.

A. How long does it take for her to reach the water below?

B. How far does she travel horizontally in her flight?

3. In a crash scene for an adventure movie a car is wired to speed horizontally off a cliff at 38.0 m/s. If the car lands 75.0 m from the base of the cliff:

A. How long was the car in the air?

B. How high is the cliff?

'Advanced' Projectiles
Problem Set

Key Formulas

1. A bullet is fired from a rifle pointing horizontally, 1.66 m above the ground. The rifle's muzzle velocity is 294 m/s.

A. How long is the bullet?

A. How far does the bullet travel? in the air? (What is its range?)

2. Louie punts the football with an initial velocity of 28.8 m/s at an angle of above the horizontal.

A. How long is the ball in the air?

A. How high does the ball get?

A. How far does the ball travel?

3. Marvelous Mark throws a baseball from the outfield towards home plate with an initial velocity of 14.0 m/s at an angle of 42° above the horizontal. (Ignore Mark's height for this question.)

A. How long is the ball in the air?

A. How high does the ball get?

A. How far does the ball travel?

A. At what angle does the ball hit the ground?

4. A cannon sits on top of a 6.5 m high cliff and is aimed at an angle of 48¡ to the horizontal. A daredevil is then shot out of the cannon at 75.0 m/s toward a target net on the ground below. Where should the net be placed so that she doesn't crash?

5. Racing Rory has decided to attempt to jump the Rushing Rapids Gorge. He builds a ramp at an angle of 30.0° to the horizontal at the edge of the gorge and plans to accelerate up the ramp and over the gorge to the other side. If his supercharged motorcycle will achieve a velocity of 35.0 m/s at the top of the ramp, and the bank on the other side of the gorge is 3.5 m lower than on the ramp side, will he successfully cross the 120 m gorge?

6. If the height of the ramp is 16.0 m above the river, what is the maximum height that the landing side could be in order for Rory to have a chance of making the leap?

Experiment Check List and Grading Form (graphic image)

Exploring Science and Mathematics Using Statistics and Technology

Student Questionnaire

Indicate your understanding of the concepts explored in this unit as well as your comfort level with the statistics and technology used/introduced by circling the appropriate number on the scale for each question.

Connections	Strongly Disagree	Strongly       Agree
1. Mathematics is a necessary component of science.	1               2               3               4               5
2. Mathematics is useful in the following ways:	1               2               3               4               5
a) Designing an experiment	1               2               3               4               5
b) Finding connections and relationships in the data collected	1               2               3               4               5
c) Supporting conclusions drawn from the data	1               2               3               4               5
Technology
3. I can use graphing calculators/graphing software to find relationships in experimental data	1               2               3               4               5
4. I know how to use a graphing calculator/graphing software to perform linear and quadratic regressions on experimental data.	1               2               3               4               5
Problem Solving
5. I can match basic nonlinear equations from physics to specific problem involving projectiles.	1               2               3               4               5
6. I can model problems involving projectiles (nonlinear data and/or nonlinear equations) using a graphing calculator or graphing software.	1               2               3               4               5
Statistics and Probability
7. I can collect data according to the requirements laid out in an experiment.	1               2               3               4               5
8. I know how to do a 'best-fit' curve on experimental data.	1               2               3               4               5
9. I can extract needed information from a graph of nonlinear data.	1               2               3               4               5
10. I can draw conclusions and validate predictions from data presented in tables or in graphs.	1               2               3               4               5

 

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Revised: September 2001

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