Course Profile Mathematics, Locally Developed, Grade 10, Public

Unit 1: Sports and Leisure

Time: 20 hours

Activity 1 | Activity 2 | Activity 3 | Activity 4 | Activity 5 |

Activity 6 | Activity 7 | Activity 8 | Activity 9

Unit Description

Students examine and apply geometric properties and relationships by investigating sports and leisure activities. The sporting world is rich with applications of, and opportunities to use angle properties, parallel lines, the Pythagorean theorem and measurement formulas. Opportunities for the consolidation of a variety of numeracy skills abound. Students practise estimation skills and judge the reasonableness of answers throughout the unit.

Strand(s) and Expectations

Strand(s): Patterns and Relationships, Measurement, Geometry and Proportionality

Overall Expectations: PRV.01, .02, MEV.01, .02, .04, GPV.01, .02.

Specific Expectations: PR1.01, 1.02, 2.01, 2.02, 2.03, 2.04, 2.05, 2.07 ME1.01, 1.02, 1.03, 2.05, 4.01, 4.02, 4.03, 4.04 GP1.02, 1.03, 1.04, 1.05, 2.01, 2.03, 2.04.

Activity Titles (Time + Sequence)

The following outline is a suggested sequence, with timing, for teaching Unit 1. Components of this unit could be used to judge the student’s skill base. Since this course acts as a bridge between the Grade 9 Essential Mathematics course and the Grade 11 Workplace Mathematics course, it is necessary to spend time determining each student’s strengths, weaknesses, and interests in order to plan the necessary steps for success. The skills are applied through a variety of real-life activities. When the need for more skill development arises, an additional 225 minutes outside of the activities has been allotted.

Activity 1	What’s Your Angle? (measuring angles and line segments; parallel lines)	150 minutes
Activity 2	What’s the Score? (numeracy; integers; graphing)	75 minutes
Activity 3	Do You Get the Point? (numeracy; measurement formulas)	75 minutes
Activity 4	Take Me Out to the Ball Game! (Pythagorean theorem; similar triangles)	150 minutes
Activity 5	They Shoot, They Score!!! (numeracy; graphing)	75 minutes
Activity 6	mathlesson.com (Internet investigation; data analysis and interpretation; graphing)	75 minutes
Activity 7	I Love This Game! (numeracy; graphing; data analysis)	150 minutes
Activity 8	But I Can’t Live on Nine Million Dollars a Year! (numeracy; graphing; data analysis)	75 minutes
Activity 9	Summative Assessment	150 minutes

*225 minutes: Time for activity completion and consolidation of skills.

Prior Knowledge Required

Students arrive in the classroom from varied backgrounds with a wide range of experiences, and for many, limited success in mathematics. They have all completed the Grade 9 Essential Mathematics course with some degree of success. It would be helpful to speak with the Grade 9 teacher(s) who could provide insight into the academic content that was covered in depth and that which requires more development.

Unit Planning Notes

In order to maximise the relevance of this unit, it is imperative that current sports-related information be used. Students can be involved in decision-making processes by being given opportunities to research and collect the most recent data.

Sufficient attention should be given to promoting Thinking and Inquiry through data analysis. Emphasis should also be placed on Communication of findings and justification of reasoning.

Teaching/Learning Strategies

The unit provides teachers with ample opportunity to observe and to identify students’ varied skill levels. This unit aims to embed opportunities within each lesson to develop or to remediate students’ skills as required. Time has been allotted to allow flexibility to accomplish this.

Assessment and Evaluation

A variety of assessment tools and strategies are suggested for this unit. Since this is recommended as the first unit in the course, it is suggested that the teacher give the students sufficient formative feedback and support before a formal assessment, to encourage a positive effort.

Resources

Briggs, S., et al. Higher Link. Oxford: Oxford University Press. 1999.

Farmer, L.S. Go Figure!: Mathematics Through Sports. Englewood, CO: Teacher Ideas Press. 1999.

Fraser, Don. Baseball Activity Book. Toronto: Ginn Publishing Inc. 1991.

Hirsch, C.R. (Ed.). Data Analysis and Statistics: Curriculum and Evaluation Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics. 1992.

Kelly, B. Data Measurement and Probability. Toronto: Queen’s Printer for Ontario. 1999.

Kelly, B. Measurement. Toronto: Queen’s Printer for Ontario. 1999.

Merseth, K., et al. Sports Shorts: A Problem-Solving Unit from the Regional Math Network.

Palo Alto, CA: Dale Seymore Publications. 1987.

Saunders, H. When Are We Ever Gonna Have to Use This? Palo Alto, CA: Dale Seymore Publications. 1988.

Worth, S. (Ed.). Rules of the Game. New York: St. Martin’s Press. 1990.

Activity 1: What’s Your Angle?

Time: 150 minutes

Description

In this activity, students estimate and measure angles and lengths of line segments and calculate distances using a given scale. They determine the path of a shot on a mini-putt course.

Strand(s) and Expectations

Strand(s): Measurement, Geometry and Proportionality

Overall Expectations: MEV.02, .04, GPV.01, .02.

Specific Expectations: ME2.05, 4.01, GP1.02, 1.03, 1.04, 1.05, 2.01.

Planning Notes

· Prepare overhead transparencies and worksheets. (Each diagram should fill a page to make the scale reasonable and to allow students to measure lines and angles more easily. Templates are available in Appendix D at the end of the Profile.)

· Students require protractors, rulers, and calculators.

· This lesson provides an opportunity for the teacher to observe the student’s facility with a ruler, protractor, and calculator, as well as their work habits and ability to follow directions.

Teaching/Learning Strategies

Student activity

Students:

· complete a series of worksheets, with the assistance of the teacher;

· estimate angle measures in degrees and lengths in centimetres;

· measure angles with a protractor and lengths with a ruler;

· construct and measure angles;

· calculate actual measures given a diagram and its corresponding scale;

· construct parallel lines to get a hole-in-one.

Teacher facilitation

· A class discussion of favourite sports could take place, leading to a discussion of the sports the students watched and/or participated in during the previous few months. Finally, students discuss their experiences playing mini-putt.

· Using an overhead projector, the teacher shows a diagram of a mini-putt hole (sample worksheet 1) showing a hole-in-one. All of the diagrams on the worksheets were produced using The Geometer’s Sketchpad™ software. They can be created and saved as a Sketchpad file or imported to a word processing software package and edited and/or saved there.

· The teacher uses a ruler to measure the total distance travelled by the shot and a protractor to measure the angle of approach (incidence) and the angle of reflection each time the ball hits a rail. Before measuring, always estimate the angles or lengths. After measuring, draw the conclusion that the angle of incidence and the angle of reflection are equal, e.g., the angle where the ball hits the rail and the angle where it reflects off of the rail are equal.

· Calculate the real distance travelled by the golf ball using a given scale (e.g., 1 cm = 0.5 m).

· Give the students a diagram of another golf hole (sample worksheet 2) and have them make the measurements and do the calculations.

· Distribute sample worksheet 3. It shows a shot that has been started. Students complete it using appropriate angle measures. Students calculate the real length of the shot.

· Sample worksheet 4 shows a shot that missed the hole. Discover that it is possible to putt the ball parallel to the original shot and have it go into the hole (the side rail of the mini-putt course is a transversal and the angle measures are once again equal). (Some students will discover that it is best to work backward from the hole using the same angles as used in the first shot to determine the starting position for the ball.)

· Sample worksheet 5 shows a completed shot that missed the hole. Students start at the hole and work backward drawing lines parallel to the original shot to determine the starting position of the golf ball. Students measure all of the angles to the rails to self-assess the accuracy of their work.

· Distribute a drawing of one golf hole (sample worksheet 6) so students estimate where the first shot should be played in order to get a hole-in-one. Students then complete the drawing to see if their prediction is correct. Measure and use the scale to determine the real distance travelled by the shot.

· As an extension activity, students can design their own golf hole and complete the shot using The Geometer’s Sketchpad™. After each student has designed their own hole, they could print them and the collection of designs could be used to design a class mini-putt course. Each student could complete the course. It would be necessary to set a limit on the number of rebounds allowed for each shot. The class could also go on a field trip to a mini-putt course and apply their findings as they play a round of mini-putt.

Assessment/Evaluation Techniques

Assessment of the students’ use of measurement tools and Knowledge/Understanding and Application of proportional reasoning need only be formative at this early point in the course. Assessment should focus on the observation of Learning Skills, specifically, Work Habits, Initiative and Independence. A checklist could be used to record the observations.

Resources

Jacobs, H.R. Mathematics: A Human Endeavor, W. H. Freeman and Co., 1970.

Sample Worksheets

The following six diagrams could be used to produce worksheets for the student activities. Each diagram should be enlarged to fill a page. The scale used for measuring could be 1 cm = 0.5 m.

Sample Worksheet 1

1. Measure and compare the indicated angles.

2. Measure the legs of the shot in centimetres, and determine the total length of the path that the golf ball travelled.

3. Given a scale of 1 cm = 0.5 m, determine the total length of the path that the golf ball travelled.

Sample Worksheet 2

Prepare a worksheet similar to sample worksheet 1 for students to complete on their own.

Sample Worksheet 3

1. Here is the first leg of a shot. Measure the angle and draw the rest of the path.

2. In order to accurately draw the rest of the path of the ball you have to measure each of the angles at the point of contact of the ball with the rails. Label each angle with its measurement.

3. Measure the legs of the shot in cm, and determine the total length of the path of the ball?

4. Given a scale of 1 cm = 0.5 m, determine the total length of the path that the golf ball travelled.

Sample Worksheet 4

1. Here is the 4^th hole. The shot missed the hole. A shot parallel to the original one could be made so a hole-in-one is possible. Construct the new path.

2. Measure the angles of contact with the rails for both shots. What do you notice?

3. Given a scale of 1 cm = 0.5 m, determine the total length of the path that the golf ball travelled.

Sample Worksheet 5

1. Here is the 5^th hole. This shot missed too. Use what you know about parallel lines to draw the shot that would be a hole-in-one.

2. Explain the steps you would follow to get a hole-in-one.

3. Given a scale of 1 cm = 0.5 m, determine the total length of the path that the golf ball travelled.

Sample Worksheet 6

1. Here is a drawing of the 6th hole. Use what you have learned about angles to complete the shot. Did you get a hole-in-one?

2. If you did not get a hole-in-one, keep adjusting the path until you do get a hole-in-one.

3. Given a scale of 1 cm = 0.5 m, determine the total length of the path that the golf ball travelled.

Activity 2: What’s the Score?

Time: 75 minutes

Description

In this activity, students use a set of data to complete a chart and a graph. Students perform calculations and then extrapolate in order to predict future outcomes.

Strand(s) and Expectations

Strand(s): Patterns and Relationships, Measurement, Geometry and Proportionality

Overall Expectations: PRV.01, GPV.02, MEV.04.

Specific Expectations: PR1.01, GP2.01, 2.04, ME4.02.

Planning Notes

· Provide a partially completed table of golf scores and a grid.

· Calculators are required.

· As often as possible, provide opportunities for students to practise estimation skills and to judge the reasonableness of their answers.

Teaching/Learning Strategies

Student activity

Students:

· are provided with a copy of the data/worksheet;

· calculate running totals and compare scores to par;

· plot points on a given grid;

· answer questions about the data, calculate averages and predict possible future outcomes;

· could use a die to “play” and “score” a round of golf.

Teacher facilitation

· A class discussion about golf jargon as it relates to integers may lead into this activity. (e.g., scoring: par = 0, bogey = +1, double bogey = +2, birdie = -1, eagle = -2)

· Using the sample worksheet, work through columns 3, 4 and 5 given the column 1 data for the first 9 holes. An overhead transparency of the datasheet is recommended for use. Students and teacher complete the table and graph for the first 9 holes, using the grid to plot values from column 3 (score per hole relative to par). Assume that the course is par 3 for every hole.

· The teacher uncovers the column 1 scores for holes 10-18 and the students complete the table and finish plotting the scores.

· Students could be asked a series of questions about the data, (e.g., What is the average number of strokes per hole for the first 9 holes of golf, the second 9, the whole game? What is the most commonly occurring score per hole? How many strokes would the player likely make in a 4 day – 72 hole tournament?) Have the students estimate the answers first and then verify the answers using a calculator.

· Students could roll a die to score a game of golf and analyse their own data as above (e.g., a roll of 1, eagle = -2; a roll of 2, birdie = -1; a roll of 3, par = 0; a roll of 4, bogey = +1; a roll of 5, double bogey = +2; a roll of 6, roll again).

· OR roll 2 die and count a total of 2 or 3 as –2; a total of 4 or 5 as –1; a total of 6, 7, or 8 as par; a total of 9 or 10 as +1; and a total of 11 or 12 as +2.

· Circulate around the room to assist students as required.

Assessment/Evaluation Techniques

Completion of the chart and graphs provides an opportunity to assess each student’s Knowledge/Understanding of the information and the student’s ability to use graphs to Communicate this understanding. Additional questions or another set of golf scores could be presented and marked using an objective marking scheme to more formally assess this activity. If students work in pairs, teamwork skills could be observed.

Sample Worksheet

1. Complete the chart for the round of golf.

COLUMN 1	COLUMN 2	COLUMN 3	COLUMN 4	COLUMN 5
Hole	Score (assume every hole is par 3)	Cumulative Score	Score per Hole Relative to Par	Cumulative Score Relative to Par
1	3	3	0	0
2	4	7	+1	+1
3	4	11	+1	+2
4	2
5	3
6	1
7	4
8	5
9	3
10
….
18

2. Use this grid to chart the player’s score for each hole with respect to par.

A PLAYER’S SCORE COMPARED TO PAR

Score per hole

relative to par

3. Use this grid to graph the player’s cumulative score with respect to the number of holes played.

CUMULATIVE SCORE VS. THE NUMBER OF HOLES PLAYED

Activity 3: Do You Get the Point?

Time: 75 minutes

Description

In this activity, students measure the radii of circles and determine the area of various rings as a percentage of the total area of a circle. Students also complete numerical calculations when scoring a game of darts.

Strand(s) and Expectations

Strand(s): Measurement, Geometry and Proportionality

Overall Expectations: MEV.02, .04, GPV.02.

Specific Expectations: ME2.05, 4.01, 4.04, GP2.04.

Planning Notes

· Prepare overhead transparencies and worksheets.

· Students require rulers and calculators.

· Enlarge the template of the dart board to fill a page to facilitate the measuring. Students should measure accurately to the nearest 0.1 cm. A blank template of a dart board is in Appendix D at the end of the course profile.

· Teachers may choose to use a simpler diagram for the dart board or, if possible, have real dart boards available for students to measure.

· Provide a set of general rules for playing darts and rules for “301”. (Described later).

· As often as possible, provide opportunities for students to practise estimation skills and to judge the reasonableness of their answers.

Teaching/Learning Strategies

Student activity

Students:

· measure the required radii to the nearest 0.1 cm;

· calculate the areas of circles;

· fill in a chart;

· calculate the percentage of the dart board area covered by the double ring, triple ring, 25-point ring and bull’s eye;

· score a dart game.

Teacher facilitation

· Help students recall the method for calculating the area of a circle.

· Assist students as they fill in a chart and do the calculations to determine the required areas.

· Work with the students to begin calculating the percentage of the area of the dart board covered by each ring.

· After the area of one ring has been calculated, students estimate the areas of the next rings.

· As students complete the calculations, circulate around the room to assist as required.

· Discuss the method for scoring a game of “301”.

· Score one player’s game together, then students score a game for the second player.

· Extension: Students could fill out more score sheets or answer questions about the strategy of the game, e.g., Should a player be left with an even number or an odd number in order to set up the winning “double” attempt?

· Alternate activity: In groups of two or three, students could use a “mock” dart board and drop a pencil onto it to determine their scores. They could make their own chart and “play” a game of darts.

Assessment/Evaluation Techniques

Provide a quiz to assess the students’ Knowledge/Understanding of measuring radii, calculating areas and determining percentages. Problem Solving skills can be observed as students work through the activity. If the mock dart board is used and students collect their own scores, Knowledge could be assessed based on the accuracy of the students’ results. Problem Solving can be assessed when students construct their own dart board and assign values for scoring based on the areas. Communication can be assessed using the explanation of how to calculate the area of the 25-point ring. Teamwork could be assessed as the teacher observes the groups or pairs during the collection of the scores.

Sample Worksheet 1

Use a ruler and the diagram of the dartboard to measure the radii listed in the chart. Use this information to calculate the required areas. Explain how you can calculate the area of the 25-point ring. Determine the area of the board covered by the double ring, triple ring, 25-point ring and the bull’s eye. What percentage of the dartboard does each of these areas cover? (The measurements are easier to complete if the dartboard is enlarged to fill a page. A blank template is available in Appendix D at the end of the Profile.)

BOUNDARY	RADIUS (cm)	AREA OF CIRCLE (cm^2)
Centre to circumference of bull’s eye	1.2 cm	A = πr^2 A = 3.14 x 1.2 x 1.2 A =
Centre to inner circumference of 25-point ring
Centre to outer circumference of 25-point ring
Centre to inner circumference of triple ring
Centre to outer circumference of triple ring
Centre to inner circumference of double ring
Centre to outer circumference of double ring
Centre to outer edge of dart board

Use the above results to complete the following table.

After calculating the area of the bull’s eye section of the dart board, estimate each of the other areas before completing the remainder of the calculations.

Section of the Dart board	Area of the Section of the Dart Board	Area of the Entire Dart Board	Estimated Percentage of the Dart Board Covered by This Section	Percentage of the Dart Board Covered by This Section
Bull’s eye
25-Point Ring
Triple Ring
Double Ring

1. Determine a relationship between the assigned points and the areas of the rings.

2. Design your own dart board and assign values to the sections based on the areas.

An example follows.

Sample Worksheet 2

Scoring a Game of Darts

Provide the following information on a worksheet and include a diagram of the dartboard.

Many different dart games can be played, but one popular version is “301”.

General rules

· Each player has a turn consisting of throwing three darts.

· Darts that miss the board, fall from, or bounce off the board receive no score.

Rules for the “301” Game

· Before any score is recorded, the player must begin by hitting a “double” (Any double will do). Once a “double” is thrown, that dart and all darts to follow will count in the scoring.

· The score is determined by subtracting the score of each dart from 301.

· The winner must reduce his or her score to EXACTLY ZERO, with the last dart being a “double”.

· If a player scores more than the exact score needed, that turn at darts does not count.

· If a turn brings a player’s score down to 1, that turn will not count as you must end with a “double”…You cannot get 1 as a “double”!!

Score this game of 301 by writing in the score after each of the 3 darts has been thrown. (The teacher could complete this chart together with the class for the first player and have the students complete the chart for the second player.)

Player 1 shots	Player 1 score	Player 2 shots	Player 2 score
Single 10 Bull’s eye Triple 5	301	Single 3 Single 9 Double 10
Miss Single 5 Single 3	301	Single 12 Bull’s eye Miss
Double 20 Triple 5 25-point ring	301 – 40 – 15 - 25 = 221	25 – point ring Double 3 Triple 5

In groups of two or three, students could use a “mock” dart board and drop a pencil onto it to determine their scores. They could make their own chart and “play” a game of darts. The teacher can make a video of an actual dart game and have the students keep score.

Activity 4: Take Me Out to the Ball Game!

Time: 150 minutes

Description

In this activity, students investigate the Pythagorean theorem and similar triangles as they relate to baseball.

Strand(s) and Expectations

Strand(s): Patterns and Relationships, Measurement, Geometry and Proportionality

Overall Expectations: PRV.01, MEV.01, .04, GPV.02.

Specific Expectations: PR1.01, 1.02, ME1.01, 1.02, 1.03, 4.01, 4.04, GP2.01, 2.03, 2.04.

Planning Notes

· Prepare overhead transparency and worksheets.

· Students require rulers and calculators.

· Teacher and students may wish to use coloured markers to highlight each event.

· As often as possible, provide opportunities for students to practise estimation skills and to judge the reasonableness of their answers.

Teaching/Learning Strategies

Student activity

Students:

· complete a series of calculations involving the Pythagorean theorem;

· construct similar triangles in order to make a series of proportional calculations;

· round off answers where appropriate (1 decimal place suggested).

Teacher facilitation

· The first part of this activity is teacher-directed.

· A discussion of baseball strategy can lead into this activity, e.g., What skills would a player need for each of the infield positions?

· Students can assign names of current baseball players that they would select for their team for each position.

· The teacher can use an overhead transparency showing a baseball infield. The students have a corresponding sheet (Sample Worksheet 1)

· Since most infield plays involve a throw to first base, estimate, then calculate:

· the length of the throw made by the person playing second base, (who plays half way between first and second base);

· the length of the throw made by the shortstop (who plays half way between second and third base);

· the length of the throw made by the person playing third base (standing at third base).

· Since the sides of the diamond are each 90 feet in length, it is necessary to use the Pythagorean theorem to calculate the length of two of the throws.

· Ask how long the catcher’s throw is to second base (to catch a runner stealing second). Identify which students can recognise that they do not have to calculate this distance as it is the same as the distance from 3rd base to 1st base, which was already calculated.

· Next, the students receive another copy of the infield and the teacher changes the length of the throws to be calculated, e.g., the second base player has to move to his or her right along the baseline 20 feet to pick up the ball, the shortstop has to move to his or her right along the baseline 15 feet to pick up the ball, the ball is bunted by the batter and the third base player runs 50 feet up the third baseline toward home plate to pick up the ball and throw it to first base.

· The teacher circulates around the room as the students work to assist as required.

· The second half of the activity involves an investigation of similar triangles.

· “What’s the difference between a regular ball diamond and one designed for 6 year olds?” This question should elicit the response that the dimensions of the diamond are smaller.

· The teacher, at the overhead projector, and the students (on paper) will work through sample worksheets 2, 3, and 4.

· The students can self-assess all or part of their work in this activity. The students and teacher could design a rubric to use for this purpose.

· A closing discussion may revolve around similar triangles and proportionality with respect to the findings of the last three worksheets.

Assessment/Evaluation Techniques

This activity provides opportunities to assess Knowledge/Understanding, Problem Solving, and Application. This is an opportunity for Self-Assessment. The students and teacher could design a rubric to use for this purpose. A partial rubric is provided below. An additional set of questions could be designed using a player throwing from a different location, using ball diamonds of different sizes or using passes between players on a football field.

Achievement Chart category	Level 3 (70 – 79%)	Level 4 (80 – 100%)
Knowledge/ Understanding	· demonstrates considerable understanding of the Pythagorean theorem and the proportionality of similar figures with minimal teacher support. · accurately solves problems.	· demonstrates thorough understanding of the Pythagorean theorem and the proportionality of similar figures independently. · efficiently chooses and uses a method to solve problems.
Problem Solving	· selects an appropriate strategy for each part of the question with considerable effectiveness.	· selects an appropriate strategy for each part of the question with a high degree of effectiveness.
Application	· applies the Pythagorean theorem and the proportionality of similar triangles to solve a problem under a similar setting. If the football setting is used, more direction may be required.	· applies the Pythagorean theorem and the proportionality of similar triangles to solve a problem in a similar setting or unfamiliar setting.

Resources

Kelly, B. Geometry and Spatial Sense. Toronto: Queen’s Printer for Ontario. 1999.

Fraser, Don. Baseball Activity Book. Toronto: Ginn Publishing Inc. 1991.

Sample Worksheet 1

Sample Worksheet 2

1. Using a pencil and a ruler, draw the path of a ground ball hit to the 2^nd base player (half way between 1^st and 2^nd base) and the path of the 2^nd base player’s throw to 1^st base on each of the three diamonds.

2. Highlight the triangle from home plate to the 2^nd base player to 1^st base on the:

90 foot diamond;

45 foot diamond;

30 foot diamond.

3. Determine the length of each throw.

4. How do the triangles look relative to one another?

5. Does a pattern emerge? Explain.

Sample Worksheet 3

1. Recall the length of the throw from the shortstop to 1^st base on the 90-foot diamond__________.

2. Using a pencil and a ruler, draw the path of a ground ball hit to the shortstop (halfway between 2nd and 3^rd base) and the path of the shortstop’s throw to 1^st base on each of the three diamonds. Finish the triangles by drawing along the first base line from 1^st base to home plate.

3. Examine the three triangles that you have just drawn. List any similarities and any differences you notice. Estimate the length of the throw from the shortstop to 1^st base on each ball diamond.

4. Using the Pythagorean theorem, calculate the length of the throw from the shortstop to 1^st base on the:

45 foot diamond;

30 foot diamond.

5. Does a pattern emerge? If so, describe the pattern.

Sample Worksheet 4

1. Using a pencil and a ruler, draw the path of a ground ball hit to the player at 3^rd base and the path of that player’s throw to 1^st base on each of the three diamonds. Finish the triangles by drawing along the first base line from 1^st base to home plate.

2. Recall the length of the throw from 3^rd base to 1^st base on the 90-foot diamond.________

3. How do the triangles look relative to each other?

4. WITHOUT USING the Pythagorean theorem, using similar triangles and a calculator, determine the length of the throw from 3^rd base to 1^st base on the:

45 foot diamond;

30 foot diamond.

Activity 5: They Shoot, They Score!!!

Time: 75 minutes

Description

Students practise numeracy skills by calculating hockey statistics for teams and for individual players. Students interpret the results of the statistics and communicate this understanding. Students display results by constructing a bar graph.

Strand(s) and Expectations

Strand(s): Patterns and Relationships, Measurement, Geometry and Proportionality

Overall Expectations: PRV.02, MEV.04, GPV.02.

Specific Expectations: PR2.01, 2.04, 2.07, ME4.03, GP2.01, 2.04.

Planning Notes

· Bring copies of sports sections of newspapers to class.

· Prepare overhead transparencies and worksheets.

· Students require graph paper, rulers and calculators.

· As often as possible, provide opportunities for students to practise estimation skills and to judge the reasonableness of their answers.

Teaching/Learning Strategies

Student activity

Students:

· calculate hockey-related statistics and record the results in a chart;

· analyse and interpret the meaning of the results of the calculations and provide a written conclusion;

· create a bar graph to display the results.

Teacher facilitation

· Provide newspapers with hockey standings and discuss the purposes of hockey statistics.

· Team Statistics: Using a photocopy of current results for one hockey division on an overhead transparency, the teacher explains how to calculate a team’s ranking in the division. The teacher provides the following information (which may need to be summarised at the top of the student worksheets): A hockey team is awarded two points for a win (W) and 0 points for a loss (L). If the scores are tied at the end of regulation time, overtime play begins. If the score remains tied at the end of overtime play each team gets one point (T). If overtime allows one team to win, the losing team gets one point (RT). (GP) signifies games played.

· Teams are ranked such that the team with the most points is first.

· The students and teacher calculate the points and overall rank of two selected teams from the displayed list and verify answers with the newspaper. Discuss results.

· A bar graph is created comparing team performances by graphing overall points (PT) of each team. The teacher should work with the students to begin a portion of the graph and students complete it.

· Students can compare their completed table to their graph in order to judge the reasonableness of their results.

· Extension: Calculations could be done using the scoring system for soccer (3 points for a win, 2 points for an overtime win and 1 point for a tie). Ask questions such as: How many ways is it possible to get 20 points?

Assessment/Evaluation Techniques

Application could be assessed from the results of question 5 on the worksheet if it is completed individually or working in pairs. It could be assessed using an objective marking scheme. Assess the written conclusions for Communication. As the students work, the teacher could observe Teamwork or Independence and Organisational Skills.

Sample Worksheet

Sample chart

Northeast Division

Team	GP	W	L	T	RT	PT (Wx2 + Lx0 + Tx1 + RTx1) or (2W + T + RT )
Toronto	82	45	30	7	3	2 x 45 + 7 + 3 = 100 points
Ottawa	82	41	30	11	2

A bar graph could be drawn displaying the team name on the horizontal axis and the points earned on the vertical axis.

The following questions could be asked:

1. What percentage of games played has Toronto won?

2. If a team played 82 games that season, what combination of (W), (L), (T) and (RT) could earn 96 points? There is more than one possible answer. Defend your choice of answer by explaining why you believe it is reasonable.

3. What percent of the games have ended in ties for each team?

4. What percent of the games for the entire league have ended in ties?

5. The NHL is considering replacing overtime with a penalty shot-style shootout at the end of regulation time to determine a winner for each game. Redesign the current “points chart” to reflect this new rule.

Activity 6: mathlesson.com

Time: 75 minutes

Description

In this activity, students use the Internet to gather statistics to be graphed. Students interpret and analyse the data and predict possible future outcomes. Students hypothesise possible reasons for the observed trends and for their predictions of future results.

Strand(s) and Expectations

Strand(s): Patterns and Relationships

Overall Expectations: PRV.01, .02.

Specific Expectations: PR1.01, 1.02, 2.03, 2.04.

Planning Notes

· This activity is designed using the information from www.hickoksports.com/history/olwtandf.shtml (Olympic results for women’s track and field), and www.hickoksports.com/history/olmtandf.shtml (Olympic results for men’s track and field). Should a different web site be used, appropriate changes must be made to the activity.

· Computer availability with Internet access is required.

· Prepare worksheet for students to record the data.

· Students require graph paper and rulers.

· Teachers may wish to explore TI-Interactive software as a way to extract data easily and quickly.

Teaching/Learning Strategies

Student activity

Students:

· select a track and field event to research;

· go to www.hickoksports.com/history/olwtandf.shtml for Olympic results for women’s track and field or www.hickoksports.com/history/olmtandf.shtml for Olympic results for men’s track and field;

· collect data for the chosen event;

· construct a line graph from the data displaying the year on the horizontal axis and either time or distance on the vertical axis;

· interpret and analyse their graph by observing trends in the data, predicting possible future outcomes and communicating possible reasons for the observed trends and for their predictions of future results.

Teacher facilitation

· Assist students in locating web sites and extracting data;

· Assist students in setting up the graph with respect to scale, labels, titles and plotting of points as required.

· Coach the students, as needed, in data analysis.

Assessment/Evaluation Techniques

There are opportunities to assess Communication, specifically the degree and clarity in the explanations and reasoning. Inquiry/Problem Solving can be assessed by observing the student’s ability to collect and accurately display the required data. Students could use a checklist to self-assess the graph before submission. They could also answer the question “What have I learned about searching for data using the Internet?” This is also a method for the students to self-assess. Independence could be observed as the students work on the Internet.

Resources (in order of recommendation)

www.hickoksports.com/history/olwtandf.shtml – Olympic results for women’s track and field

www.hickoksports.com/history/olmtandf.shtml – Olympic results for men’s track and field

www.mikap.iki.fi/sport/eng – various track and field results

www.eexi.gr/athletix – various track and field results

Activity 7: I Love This Game!

Time: 150 minutes

Description

In this activity, students use an algebraic model to calculate basketball scoring. Students also calculate percent and create a scatter plot using given data to determine whether trends exist between two variables.

Strand(s) and Expectations

Strand(s): Patterns and Relationships, Measurement, Geometry and Proportionality

Overall Expectations: PRV.01, MEV.04, GPV.02.

Specific Expectations: PR1.01, 1.02, 2.02, 2.04, 2.05, ME4.03, GP2.01, 2.04.

Planning Notes

· Prepare overhead transparencies and worksheets.

· Students require calculators, graph paper, and rulers.

· As often as possible, provide opportunities for students to practise estimation skills and to judge the reasonableness of their answers.

Teaching/Learning Strategies

Student activity

Students:

· with teacher assistance, develop a model for determining basketball scoring;

· use substitution to calculate an individual’s scoring within a game;

· calculate a player’s shooting percentage and plot shooting percentage as a function of distance from the basket using a scatter plot;

· assess the results to determine trends, which may lead to a line or curve of best fit.

Teacher facilitation

· Teacher and students discuss basketball, e.g., participation in, favourite pro-team, playing on school team which leads to a discussion of scoring and to the point value for different shots.

1 point for a ‘free throw’,

2 points for a ‘field goal’,

3 points for a field goal beyond the “3-point line” (The 3-point line is measured from the basket and is 22 feet from the basket in the NBA and 19 feet 9 inches from the basket in the WNBA; the teacher may wish to measure the distance in their own gymnasium).

· The following formula will be derived: TS = 3a + 2b + c, where TS is the player’s total score, a is the number of 3 point shots completed, b is the number of 2 point shots completed and c is the number of 1 point free throws completed.

· Using an overhead projector, the teacher could use different colours to locate the shots made by a player during a game (sample worksheet 1).

· The teacher directs the students to substitute the number of shots made into the formula to determine a player’s score. (e.g., two – 3 point shots, six – 2 point shots, five – 1 point shots TS = 6 + 12 + 5 = 23 points)

· The teacher should use discretion as to how many examples need to be done to ensure an appropriate level of student understanding is achieved.

· The teacher may then give the students the number of each type of shot made and have each student determine the placement of the dots. The student can then calculate the player’s total score.

· The students may then place a number of dots (representing shots made) at various locations on a blank basketball court template (sample worksheet 2, with an additional template available in Appendix D at the end of the Profile) and have a partner calculate the total score. Partners could verify each other’s answer.

· The teacher may pose a series of questions to the students such as:

· If a player scores 26 points in a game and made:

a) four 3-point shots and six 2-point shots, how many free throws were made?

b) three 3-point shots and five free throws, how many 2-point shots were made?

c) eleven 2-point shots and four free throws, how many 3-point shots were made?

Students can solve this problem with or without the aid of a diagram.

· If a player scores 37 points in a game and made seven 3-point shots, how many 2-point shots and free throws might that person have made?

· If a player scores 33 points and made four free throws, how many 2- and 3-point field goals might have been made?

· In order to solve these last two questions, students may work alone or in pairs listing all of the possible combinations of outcomes. While there will usually be more than 1 possible answer, some answers are more realistic than others.

· To close this part of the lesson, a discussion may include the notion that keeping track of these statistics is somebody’s job, i.e., Someone attends the game and keeps track of every player’s scoring and point calculations so that commentators can have instant access to the game statistics.

· The next part of this activity may be introduced by discussing whether the students believe there is a relationship between being close to the basket and the likelihood of scoring. Students are asked to hypothesize how proximity to the basket affects the likelihood of scoring.

· The teacher distributes a data sheet where the students calculate players’ shooting percentages (shots made/shots attempted x 100%) for various distances from the basket (sample worksheet 3). Player #3 makes one out of one shot from 16 feet from the basket. This gives a value of 100% and allows for discussion surrounding outlying points in data when the scatter plot is drawn.

· The students make a scatter plot of the shooting percentages (on the vertical axis) against distance from the basket (on the horizontal axis).

· Does a trend emerge? Does it look like a line or curve of best fit may represent the trend? Does the observed trend match the earlier hypothesis?

· A further discussion about the “line or curve” of a professional player’s shooting percentage may follow. This could lead into Activity 8, which deals with player’s salaries.

· Circulate throughout the room to assist students as required.

· A short section of a basketball game could be shown on video and students could keep track of the shots made on a template and determine the total points earned by a team during that segment of the game. This could also be used as a “hook” to the lesson.

· Possible extension: taking the class to the gym to collect first hand data that could be analysed in the above manner can complete this activity. The analysis could include a scatter plot of individual and whole class results.

· Another extension activity could be to introduce the use of, or consolidate skills using, a graphing calculator by graphing the above results and comparing it to the hand drawn graph.

Assessment/Evaluation Techniques

Assess Knowledge/Understanding using a paper and pencil quiz which is marked using an objective marking scheme. The problems solved during the first part of the lesson can be assessed for clarity of reasoning, interpretation of the problem, and representation of the solution mathematically. Use the written responses to assess Communication. Problem Solving can be assessed at Level 3 if students recognise that more outcomes are possible and accurately analyse one possible outcome. Problem Solving can be assessed at Level 4 if students list all or almost all of the possible outcomes and clearly analyse one of the more realistic possibilities. Through observation, the teacher can assess Work Habits, Initiative and Organisational Skills.

Resources

Kenney, M.J. et al. Informal Geometry Explorations. Palo Alto, CA.: Dale Seymore Publications. 1992.

Sample Worksheet 1

Sample Worksheet 2

Sample Worksheet 3

This chart compares the shots made with the shots attempted (e.g., 3/5 means 3 shots were made in 5 attempts). Calculate the shooting percentages for each of the following three players for the games they played last week.

Distance from basket (feet)	2	4	6	8	10	12	14	16	18	20
Player #1	3/3	3/4	6/10	5/11		2/5	1/3	2/9		0/1
Player #1 %	***
Player #2	8/10	8/9	4/6	1/2		3/8	2/8		0/2
Player #2 %
Player #3	5/6	4/5	3/4	3/7	2/7		0/1	1/1		1/5
Player #3 %

*** 3/3 x 100 % = 100% or the teacher may wish to teach students to use the “%” key on the calculator.

Graph your results on a scatter plot where the distance from the basket (in feet) is on the horizontal axis and the percentages of shots made is on the vertical axis.

1. Does the distance from the basket appear to affect the likelihood of scoring?

2. Does it look like a line or curve of best fit could be drawn to represent the data?

Activity 8: But I Can’t Live on Nine Million Dollars a Year!

Time: 75 minutes

Description

Students practise numeracy skills using salaries of basketball players. Students use these salaries to determine differences, percentages, rank, and rates. Students graph the percentage of the team’s salary budget each player earns and make observations about the data.

Strand(s) and Expectations

Strand(s): Patterns and Relationships, Measurement, Geometry and Proportionality

Overall Expectations: PRV.02, MEV.04, GPV.02.

Specific Expectations: PR2.04, 2.05, 2.07, ME4.01, 4.04, GP2.01, 2.04.

Planning Notes

· Prepare overhead transparencies of salary chart and student worksheets.

· Students require calculators, rulers and graph paper.

· To add relevance to the lesson, use actual, current salaries whenever available.

Teaching/Learning Strategies

Student Activity

Students:

· rank team players by salaries;

· calculate differences between salaries of team players;

· calculate how much each player earns per game and per point;

· calculate the percentage of the overall team salary budget that each player earns;

· create a bar graph comparing the players (on the horizontal axis) to the percentage of the team’s salary budget earned by each player (on the vertical axis);

· answer questions about trends observed in the graph and information drawn from the calculations in the chart.

Teacher Facilitation

· Provide students with a salary chart of sports figures (see sample chart) or have students collect current data, e.g., from newspapers, or the Internet.

· Students require a sheet of graph paper.

· Together, the teacher and the students complete the chart for the first two players; this may be largely teacher-directed.

· Together, the class and the teacher rank the players.

· The teacher helps the students set up graphs to create a scatter plot of the percentage of the team budget each player earns. On the horizontal axis, the players are ordered according to their rank. The highest paid player is first; the lowest paid player is last. Plot the team salary percentages for the first two players on the graph as a class. Students complete the chart, the graph and the questions.

Assessment/Evaluation Techniques

The chart, graph, and questions can be used to assess Communication and Application by assessing the graphing, the application of percentage calculations and the drawing of conclusions. Work Habits and Independence could be assessed as the students work.

Sample Worksheet

Team Salary Budget: 45 million dollars (total payroll for the team for one season)

Player	Salary in Millions ($)	Rank	Percentage of Team Salary Budget Earned (%)	Number of Games Played Last Season	Average Amount Earned per Game ($)	Average Number of Points per Game	Average Amount Earned per Point ($)
Player #1	12		12/45 x 100% =
Player #2	3

PLAYER EARNINGS

Player

1. What trends are evident in the graph? (e.g., comment on the distribution of the salaries)

2. What is the difference, in dollars, between the highest and lowest paid players?

3. Is a player’s average number of points per game related to his/her salary? Explain.

4. Draw conclusions about the players and the team from the chart and the graph you have produced. Write about your findings below.

Activity 9: Summative Assessment

Time: 150 minutes

Description

A cumulative assessment review and a performance-based evaluation are completed over 150 minutes. Students consolidate the skills covered in this unit.

Strand(s) and Expectations

Strand(s): Patterns and Relationships, Measurement, Geometry and Proportionality

Overall Expectations: PRV.01, .02, MEV.01, .02, .04, GPV.01, .02.

Specific Expectations: PR1.02, 2.04, 2.05, ME1.03, 2.05, 4.01, 4.02, GP1.02, 1.03, 1.05, 2.01, 2.04.

Planning Notes

· Often, the first few weeks of a course involves a number of students joining or leaving the class. It is also a time of significant adjustment for students as they get used to new teachers, classmates, settings and expectations. To allow for these adjustments, the first two components of the summative activity closely mirror the activities from earlier in the unit. The remaining two components have been altered from the earlier activities to allow for assessment and evaluation of the application of concepts in related, yet different settings.

· The teacher prepares the unit review.

· A sample summative evaluation is included (which provides opportunities for students to demonstrate Level 4 performance).

· Students require rulers, protractors and calculators.

· Determine, in advance, the degree to which students will be allowed to use supporting materials for the summative evaluation (e.g., notes, study sheets, portfolio).

· Remind students to estimate their answers and to check the reasonableness of their answers.

Teaching/Learning Strategies

Teacher facilitation

· The teacher assists the students as required in order to complete the review.

· The teacher circulates around the classroom during the summative evaluation and provide encouragement and support to students as needed.

Assessment/Evaluation Techniques

This summative activity completes the first unit in the course. The activity can be assessed for Knowledge/Understanding, Communication and Application. Assess questions 1 and 2 for Knowledge/Understanding of the concepts using an objective marking scheme. Questions 3 and 4 allow Application of the concepts and the procedures in a different setting. Communication is assessed by the students’ use of mathematical language and symbols in their solutions.

Achievement Chart category	Level 3 (70 – 79%)	Level 4 (80 – 100%)
Knowledge/ Understanding	· solves problems accurately	· selects the most efficient method to use to solve problems accurately.
Communication	· includes correct use of mathematical language and symbols most of the time.	· includes correct use of mathematical language and symbols almost all of the time.
Application	· includes accurate application of the concepts and procedures in questions 1 and 2 with some support required for application to the unfamiliar settings in questions 3 and 4.	· includes accurate application of the concepts and procedures in questions 1 and 2 independently with minimal or no support required for application to the unfamiliar settings in questions 3 and 4.

The teacher could ask for oral explanations for some of the work. The teacher could continue to record observations of Work Habits such as perseverance or independence.

Sample Summative Evaluation Tool

It is suggested that the diagrams are drawn larger and that the questions are spread out on a page to allow space for calculations and so that students are able to show all of their work.

1. Consider the mini-putt shot in the diagram above.

a) Using the scale of 1 cm = 0.5 m, determine the total length of the path that the golf ball traveled.

b) Measure angle #1 and angle #2. angle #1 = _________, angle #2 = __________.

c) Using these angles to construct parallel lines, draw a hole-in-one shot.

2. The above diagram represents the 2-point and 3-point field goals made by Vince Carter in a basketball game.

a) How many points did Carter score from 3-point shots?

b) How many points did Carter score from 2-point shots?

c) If he scored a total of 27 points during this game, how many free throws did he make?

d) Using a scale of 1 in. = ___ft. (the teacher determines the scale depending on the size of the diagram) determine Carter’s distances from the basket when he sank shot A and shot B.

e) If 27 points per game is Vince Carter’s average, how many points would he score in an 82 game season?

f) If Carter made $6,250,000 last year, what was he paid per regular season game? (Round off your answer to the nearest dollar)

3. In the local T-ball league for 7- and 8-year olds, the distance between the bases is 12m. Use the above template of a ball diamond to help you answer the following questions.

a) Determine the length of the throw from 2^nd base to 1^st base, assuming the player is exactly half way between 1^st and 2^nd base.

b) What is the length of the throw from shortstop to 1st base, assuming the shortstop plays exactly half way between 2^nd and 3^rd base?

c) Determine the length of the throw from 3^rd base to 1st base.

d) If the pitcher made 22 pitches in the 1^st inning, estimate the number of pitches made in a 9-inning game.

4. A Grade 10 phys-ed class is learning archery. Make the following calculations rounded off to one decimal place.

a) Determine the area of the outermost ring.

b) What is the area of the outermost ring as a percentage of the entire target?

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