Course Profile Principles of Mathematics, Grade 10, Academic, Catholic

Unit 2: Analytic Geometry

Time: 24 hours

Activity 1 | Activity 2 | Activity 3 | Activity 4 | Activity 5 | Activity 6 | Activity 7

Unit Description

Students will use analytic geometry to solve problems involving the properties of line segments and to verify geometric properties of triangles and quadrilaterals. Specific investigations will use these line segment properties to develop formulas for the lengths and midpoints of line segments; determine the equation of a circles centred at (0, 0); solve multi-step problems involving properties of line segments; determine the characteristics of triangles and quadrilaterals having fixed co-ordinates; investigate and verify geometric properties of triangles and quadrilaterals having fixed co-ordinates; and develop communication and problem-solving skills.

Strand(s) and Expectations

Ontario Catholic School Graduate Expectations: CGE2b, 2c, 3b, 3c, 3e, 4f, 5a, 5g, 7b.

Strand(s): Analytic Geometry

Overall Expectations: AGV.02, AGV.03.

Specific Expectations: AG2.01D, AG2.02D, AG2.03D, AG2.04D, AG3.01D, AG3.02D, AG3.03D.

Activity Titles (Time + Sequence)

The following table provides a suggested sequence and timing for teaching this unit. The activities are designed to give students an opportunity to investigate the principles of analytic geometry and to apply these principles in a variety of ways. These activities employ a common theme involving the design and development of a piece of land into a new community. Students use this theme to apply various analytic geometry concepts while developing a portfolio for submission and presentation at the end of the unit.

Each activity in the unit will help students to visualise the concepts of analytic geometry; to analyse situations involving these concepts; to develop an appreciation of the world around them; and to promote a respect for God’s creation and an understanding of the need to use resources wisely.

Activity 2.1 Follow-up Skills	Delta Force Use the Lone Wind Place map to determine distances between various elements of the community. When available, The Geometer’s Sketchpad may be used to develop formulas for determining the lengths and midpoints of line segments. Use the Lone Wind Place map to develop a management line between the mid points of the cables connecting the cell towers Practise finding mid-points and lengths of line segments on a xy-plane using pencil and paper and a calculator	150 minutes 75 minutes
Activity 2.2 Follow-up Skills	Circular Thinking Using the Lone Wind Place map, students analyse ambulance costs based on the distance from a fixed centre Practice with equations of circles centred at the origin on an xy-plane using pencil and paper	75 minutes 75 minutes
Activity 2.3 Follow-up Skills	Watered Down Music Determine distance between towns for music reception purposes. Use right bisectors of a chord in the design of a sprinkler system Full period paper and pencil test	150 minutes 75 minutes
Activity 2.4 Follow-up Skills	Freedom on the Beach! Use Lone Wind Place map and analytic geometry to determine the position of an object off the beach. Practise other applications of lengths and midpoints in triangles (e.g., altitudes, centroids, medians, etc.)	110 minutes 115 minutes
Activity 2.5 Follow-up Skills	Cell Power Use communication towers with parallelogram boundaries and transmitting power constraints to prove properties of a parallelogram Practice, consolidation, and application of skills and concepts of quadrilaterals (lengths, slopes and midpoints of sides, equations of diagonals, characteristics of segments joining midpoints, etc.)	75 minutes 150 minutes
Activity 2.6 Follow-up Skills	Oh Deer! Ticked! Explore relationships of line segments joining midpoints of the sides of a triangle Practise a variety of problems that will verify geometric properties of triangles or quadrilaterals with given vertex co-ordinates. (e.g., diagonals on a rectangle bisect each other; diagonals of a square are perpendicular, etc.)	75 minutes 150 minutes
Activity 2.7	Portfolio Presentations Math Fair. Students display and present their final Lone Wind Place plan and reports for public viewing and assessment.	90 minutes
	Summative Assessment Paper and pencil test	75 minutes

Prior Knowledge Required

· All expectations from the Grade 9 Number Sense and Algebra Strand (particularly NAV.03 and NAV.04).

· All expectations from the Grade 9 Measurement and Geometry strand (particularly MGV.03).

· The use of dynamic geometry software (e.g., The Geometer’s Sketchpad™)

Unit Planning Notes

· Provide several copies of the Lone Wind Place Community Template for the various activities. Teachers may wish to enlarge the Template onto ledger size paper. For presentations, students may wish to use transparencies and overlays of their plans.

· The activities in this unit revolve around a central theme in which students design a new community. Components of this community will be cumulative, evolving as students proceed through the activities.

· The unit activities are designed to allow either a student-directed approach, a teacher-directed approach, or a mixture of both. With sufficient computer access, teachers may have students develop the skills and concepts of this unit in self-directed groups of three or four students.

· Teachers may wish to have student groups present interim reports of their progress at the end of activities. A final submission will include a completed map of the community, a planning report, and a class presentation.

· The final presentation of their plan and investigations will occur at the end of the unit with each group’s Culminating Assessment Package.

· Students will address all items of the Analytic Geometry Strand while resolving many of the problems encountered by developers, planners, and/or contractors during the construction of a community (e.g., surveys, utilities, streets, bridges, buildings, churches, school, landscaping, ponds, lakes, rivers, recreation facilities, industry, etc.).

· There will be many opportunities to integrate topics from other disciplines (e.g., Economics, Social Sciences, Sciences, Ethics, etc.) and every attempt should be made to do so.

· There must be a balance of time spent between the investigative aspect of this unit and the paper and pencil approach.

· Timely and appropriate use of dynamic software and technology should be interwoven and integrated throughout.

Teaching/Learning Strategies

This unit provides opportunities for a balance of teacher-directed and student-directed activities while employing a variety of groupings (e.g., whole class, small groups, independent study, etc.) Students are provided with many of the design features of the Lone Wind Place community but teachers may encourage students to add their own features. A final report and presentation of their Lone Wind Place plan will be part of the summative assessment activity for the unit. Where appropriate, teachers may wish to have students maintain journals of their work in this unit. Dynamic software (e.g., The Geometer’s Sketchpad™) is used as an investigative tool, where possible, to develop and reinforce the various analytic approaches which meet the expectations in this unit.

Throughout the activities in this unit, students could use the following model for problem solving:

Hypothesise: Formulate hypotheses related to properties of geometric figures on the xy –plane within the context of the Lone Wind Place Community

Explore/investigate: Through hands-on activities investigate their application in the Lone Wind Place map (e.g., The Geometer’s Sketchpad™, graph paper)

Model/Formulate: Develop algebraic, graphical and/or tabular models

Transform/Manipulate: Develop algebraic and graphical skills as needed in the context of their investigations

Infer/Conclude: Re-evaluate their hypotheses in light of their learning and make inferences to extend their learning

Communicate: Express their findings individually and in groups; orally and in writing.

Assessment and Evaluation

A balance of assessment tools and strategies is recommended. Activities that are used as teaching tools can and should be used as assessment tools. A list of recommended assessment tools are performance tasks, paper and pencil tasks (e.g., quizzes, diagnostic tests, worksheets, small assignments), written reports, oral presentations, observation, and peer assessment. The course overview lists a variety of assessment tools and strategies. Students should submit their drafts and preliminary results of their investigations at the end of selected activities. Interim feedback will give students an opportunity to make modifications for their final presentation.

Resources

The Geometer’s Sketchpad™

Satellite images of localities: http://www.terraserver.microsoft.com

Activity 2.1: Delta Force

Time: 150 minutes

Description

With teacher direction, students will determine formulas for the distance between points, the lengths of line segments and the co-ordinates of the midpoint of any line segment. Where available, they may use The Geometer’s SketchPad™ to establish concepts and to develop formulas. Students will use the Lone Wind Place Community as a template for applying their models. They will begin a survey and layout of this tract of land for public use. On their survey they will need to establish locations for utility centres and to determine the distances between them. Students will include this as part of their Cumulative Assessment Package.

Strand(s) and Expectations

Ontario Catholic School Graduate Expectations

The graduate is expected to be:

· an effective communicator who reads, understands, and uses written materials effectively;

· an effective communicator who presents information and ideas clearly and honestly and with sensitivity to others;

· a reflective and creative thinker who creates, adapts, evaluates new ideas in light of the common good;

· a collaborative contributor who works effectively as an interdependent team member.

Strand(s): Analytic Geometry

Overall Expectations

AGV.02 - solve problems involving the analytic geometry concepts of line segments.

Specific Expectations

AG2.01 - determine formulas for the midpoint and the length of a line segment and use these formulas to solve problems;

AG2.04 - communicate the solutions to multi-step problems in good mathematical form, giving clear reasons for the steps taken to reach the solutions.

Planning Notes

· Students will need a copy of the Lone Wind Place Template.

· Teachers will need a transparency of the Template.
(Note: This Template will be used for subsequent activities.)

· Students need graph paper and a calculator.

· A computer lab may be used for the dynamic software investigation part of the activity.

· If this is the first time the students have used The Geometer’s Sketchpad™, they will need extra time to learn how to use the program.

Prior Knowledge Required

Basic Grade 9 skills with coordinate geometry; use of The Geometer’s Sketchpad™ and the Pythagorean theorem

Teaching/Learning Strategies

Teacher Facilitation: Hand out a copy of the Lone Wind Template. Present students with the project scenario. Establish groups of three or four. Class discusses the problem and intended activity.

Student Activity

Introduction

You have been awarded a huge contract to develop former Crown Land into a new Life Style Community. You need to plan the placement of community services such as fire halls, public utilities, distribution centres, communication towers, residential areas, zoning, etc.

Main Street (east-west) and Bay Street (north-south) are the only roads that currently exist. The administrative centre (O) is located at the intersection of Bay and Main. The administration centre has been nicknamed O, in recognition of Orville Ridgin who formerly lived on the site.

The site has been surveyed in one-kilometre sections. To assist with planning, your map has a grid in which one grid unit represents one kilometre.

Initially, you will need to establish a street plan for future access to the whole region. You need to survey the land for roads and various critical sites necessary for the community

Teacher Facilitation: Allow individual creativity. If the resources are available teachers may wish to preface this activity with a short class discussion on urban planning and the application of good design techniques involving mathematical and engineering principles, urban geography principles (e.g., Geography Profiles, an Urban Planner, etc.) and aesthetic features.

Student Activity 1

Part A

The head surveyor asks that you prepare a report which establishes a method for identifying sites and method for measuring distances between two different locations. Develop an equation model that will allow Community Administrators to find the distance between any two locations on the map.

Teacher Facilitation: Using small groups of two or three students, discuss and hypothesize methods for determining the length of a line segment, given the co-ordinates of two points. Students may develop and test their hypotheses using calculators, spreadsheets and The Geometer’s Sketchpad™.

Use The Geometer’s Sketchpad™ to develop a measurement model that determines the length of a line segment with one end at the origin. Generalize the model so it may be used to measure the length of any line segment.

Action	Computer Keystrokes
1. Display the axes	Graph – Create Axes
2. Create a point C in quadrant 1
3. Draw a circle from the origin A(0, 0) to the point C
4. Draw line segment AC
5. Display co-ordinates of C and length of segment AC	Measure - Co-ordinates Measure - Length
6. Draw line segment CD to intersect the x-axis perpendicularly	Construct – Perpendicular Line
7. Display length of CD
8. Draw line segment AD	Note: AC, CD, and AD are sides of a right-angled triangle, ACD
9. Display length of AD
10. Move point C and note relationship of coordinates and the measured lengths
11. Use the SketchPad calculator to create relationships to investigate and create a hypothesis

Part B

1. Extend your hypothesis by beginning at step 1 again, and having point A not at the origin

2. Establish the formula for the length of a line segment between any two points

3. Test the formula by repeating the above steps. Begin by creating a circle not centred at the origin

4. Create a table and present results to teacher and class

Teacher Facilitation: While circulating, direct students in the following manner:

· Establish a grid reference pattern (e.g., (5E, 6N) which should ultimately be abstracted to
(+5, +6))

· Suggest that groups first consider locations in the NE quadrant (+, +)

· If necessary, hint about right-angled triangle properties that they learned in Grade 9.

· Have groups present their measurement model to the class for further discussion

· Lead the students to a model similar to

Lone Wind Place Template Legend

A, B, and C are Life Guard Stations (Activity 2.4).

P, Q, R, and S are cellular towers (Activity 2.5).

F, J, and H are the corners of the triangular Conservation Area (Activity 2.6).

Part B

The head surveyor asks you to develop a preliminary survey for the placement of three cell phone towers. All need to be strategically placed in order to give maximum coverage between O, and the rest of the site. There is already an existing cell transmitter at the lighthouse (L). Each tower needs an underground backup line that will directly connect them to each other.

For the following use the diagram shown below.

· In your report, justify your suggestions about your placement of the cell towers.

· Assume each tower has a range of 5 km and one at the lighthouse has a range of 7 km.

· Your report needs to determine the minimum amount of cable required to directly link the three towers to each other underground. The planning committee demands a complete report on how you calculated your answer. In addition, include a line from O to the tower closest to it, and another line from the lighthouse (L) to the tower closest to it.

· Once again, your results must be presented using formulas for distance between two points.

Teacher Facilitation: Recommend that students position their towers at intersections of roads which themselves should lie along the grid lines. Recommend that their report consist of a table containing the coordinates of the towers and the distance calculations. If computers are available, their report may include a table developed on a spreadsheet. Have students present their reports and conclusions to the class.

Part C

Ideally, a cell phone will receive its signal from the nearest transmission tower. Each pair of transmission towers must be calibrated to achieve this balance. In order to calibrate the signals a technician will measure the signal strength of each pair of the cell towers at a point midway between them. You need to calculate and show the coordinates of the mid-points between each pair of the cell towers. The surveyor requires you to calculate the exact coordinates of these connection points to the nearest 0.1 kilometres and plot them on your template.

Your report requires a table containing the coordinates of the towers, the coordinates of the midpoints between adjacent towers, and the transmission distances from each cell tower to the mid-point between each pair of towers. Be sure to include the towers at O and L in your calculations. Your results should be accurate to the nearest 0.1 km.

Teacher Facilitation: If dynamic geometry software is not available in the school preparation for the activity can be performed using graph paper and mathematical sets. The number of these constructions must be limited and adjusted according to time restrictions. Students should develop formulas for mid- points between each of the pairs of towers, the tower closest to lighthouse L and the tower closest to the administration centre O. A short lesson on determining the coordinates of the midpoints as an average of two values may be necessary.

is the midpoint between and

Alternate Activity where Technology is Limited

Use an individual computer with a projection unit for illustration purposes. Students should be allowed to do some or all of the demonstrations for their classmates.

Use small groups of two or three students.

Use The Geometer’s Sketchpad™ to determine a method for finding midpoints.

Instructions for The Geometer’s Sketchpad™ (Only initial keystrokes are indicated)

Actions	Computer Keystrokes
1. Show Axes	Graph - Create Axes
2. Draw Line from origin along x-axis to (5, 0)
3. Construct Midpoint	Construct - Point at Midpoint
4. Label it	Display - Show label
5. Select endpoint and label it
6. Display coordinates of endpoint and midpoint
7. Select endpoint and move. Observe values of co-ordinates
8. Repeat using a line from origin to (0, 5)
9. Repeat with line from O to (5, 5)
10. Hypothesise how to obtain midpoint, given O and endpoint co-ordinates
11. Move endpoint into other quadrants and describe what happens.
12. Be sure snap to grid is on
13. Fix one endpoint integer co-ordinates (e.g., 3, 3) 14. Move other endpoint to different co-ordinates and observe midpoint co-ordinates
15. Determine a model for determining midpoint co-ordinates
16. Test your model for all cases
17. Record and present your model for determining the midpoint

Assessment/Evaluation Techniques

The survey report that students have developed in the activities may be presented for preliminary discussion of concepts and ideas. Each group of students will retain their reports until the end of the unit.

Assessment Instruments

· Observe group and assess teamwork and initiative using the Learning Skills Rubric (Appendix A).

· Evaluate Lone Wind Place Planning Report using Written Report Rubric (Appendix D).

· Use paper and pencil tasks (e.g., quiz on finding midpoints and distance between points).

Follow-up Skills: 75 minutes

Refer to the textbook for questions that calculate midpoints and the distance between points. Direct your students to investigate each of the following:

· Determine distances between origin and any point in any quadrant

· Find the distance between two horizontal points and two vertical points

· Determine the distance between two points anywhere on the Cartesian Plane

· Solve problems that determine the distance between any two points on a line

· Relate this to former studies about characteristics of a line and its slope.

Activity 2.2: Circular Thinking

Time: 75 minutes

Description

This activity enables students to develop the equation of a circle centred at the origin. They will develop their equations in small groups and communicate their results to the class. The initial problem will be presented with respect to the Lone Wind Place development project.

Strand(s) and Expectations

Ontario Catholic School Graduate Expectations

The graduate is expected to be:

- an effective communicator who reads understands and uses written materials effectively;

- a collaborative contributor who works effectively as an interdependent team member.

Strand(s): Analytic Geometry

Overall Expectation

AGV.02 - solve problems involving the analytic geometry concepts of line segments.

Specific Expectations

AG2.02 - determine the equation for a circle having centre (0, 0) and radius r, by applying the formula for the length of a line segment; identify the radius of a circle of centre (0, 0), given its equation; and write the equation, given the radius;

AG2.04 - communicate the solution to multi-step problems in good mathematical form, giving clear reasons for the steps taken to reach the solutions.

Planning Notes

· This activity continues using the Lone Wind Place development theme. A problem requiring the development and use of an equation of a circle will be presented to the class.

· The activity may be done using dynamic graphing software. Due to the relative shortness of this activity, the teacher should consider the expertise of students with the software, the availability of computers, and the time allocated for investigation and development of their report.

· Graph paper, a compass, and a calculator are required for each group of four students.

Prior Knowledge Required

Number Sense and Algebra Grade 9 Academic: use algebraic modelling to solve problems, specifically the Pythagorean theorem; communicate solutions using appropriate mathematical forms (e.g., tables, graphs)

Teaching/Learning Strategies

Student Activity

Students briefly review their progress with the Lone Wind Place land design project.

Teacher Facilitation: Present the following communiqué from the Lone Wind Place Administration Office.

Administration wishes to develop cost zones for their ambulance service. Zone A will end at 5 km from O and Zone B will be the area beyond Zone A. They want you to develop an equation that describes the boundary curve between these two zones so that the can correctly program their computer with the information.

They wish you to report how you develop these equations in order to allow for future modifications.

· In large group discussion, spend 5-10 minutes reviewing and brainstorming possible hypotheses for solving this problem.

· Recommend that the problem be abstracted onto an xy-plane for exploration

· Break into groups of four and have students formulate a model

Student Activity

1. On graph paper or The Geometer’s Sketchpad™, draw a circle centred at (0, 0) on the xy-plane and having a radius of 5 units

2. Select several points on the circle in the first quadrant. Use integer coordinates.

3. Create a table containing the coordinates of these points. Hypothesize a relationship between these points and (0, 0).

4. Continue investigations for points and ends of line segments in other quadrants.

5. Develop a report of your investigations.

6. Communicate their results individually and in groups, orally and in writing.

Assessment/Evaluation Techniques

The report that students have developed in the activities may be presented for preliminary discussion of concepts and ideas. Each group of students will retain their reports until the end of the unit.

Assessment Instruments

· Assess teamwork, independence, and initiative using the Learning Skills Rubric (Appendix A).

· Assess individual communication skills using Verbal Presentation Rubric (Appendix C).

· Use paper and pencil quizzes.

Follow-up Skills: 75 minutes

Practise with equations of circles centred at the origin on a xy-plane using pencil and paper

· Given centre and radius determine equation of a circle.

· Find coordinates of points on a circle using its equation.

· Find circle given coordinates of a point on the circumference.

Activity 2.3: Watered Down Music

Time: 150 minutes

Description

Students will consolidate the concepts and skills addressed thus far in the unit (midpoints and lengths of line segments, and circles). These will be used in context with transmitting power of local radio stations (radii) and with the sprinkler system in a circular garden (chords and right bisectors).

Strand(s) and Expectations

Ontario Catholic School Graduate Expectations

The graduate is expected to be:

- an effective communicator who reads, understands, and uses written materials effectively;

- a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problem;

- a collaborative contributor who works effectively as an interdependent team member.

Strand(s): Analytic Geometry

Overall Expectations

AGV.02 - solve problems involving the analytic geometry concepts of line segments.

Specific Expectations

AG2.03 - solve multi-step problems, using the concepts of the slope, the length, and the midpoint of line segments;

AG2.04 - communicate the solution to multi-step problems in good mathematical form, giving clear reasons for the steps taken to reach the solutions.

Planning Notes

· Students should continue working in small groups (two or three students per group).

· Students will use a copy of the Lone Wind Place template.

· Each student will be responsible for their own written report.

· Students will need graph paper, mathematical set and calculators.

Teaching/Learning Strategies

Student Activity 1

1. Find the distance (as the crow flies) from the city centre (0, 0)) to:

· Paradise Island,

· Kloseton,

· Farfromton,

· Happyville,

· Fifthville

2. For what practical reason(s) may you want to know these distances?

3. There are three radio stations that transmit from the city centre:

· the local college station, COLG, that has a power to transmit up to 20 km;

· a Jazz station, CJAZ, that has a power to transmit up to 50 km;

· a rock station, CROK, that has a power to transmit up to 100 km.

For a resident of each town, find what radio stations they will be able to receive.

Teacher Facilitation: Help student locate the five towns if necessary. To help the students with Question 2, the teacher may suggest that this new community has the most up to date medical facilities in the region. Hence emergency patients may have to be flown directly from the local towns to the city centre (0, 0).

Student Activity 2

Surrounding the Administrative Buildings are several beautiful gardens manicured by the city’s parks and recreation department. The centre piece garden, named The Magical Circle, is a circular shaped garden with a radius of 50 m. A unique watering system is currently being used. There is a large soaker hose of length 71 m with each end anchored on the perimeter of the garden. A second soaker hose, of length 85 m, is anchored at the midpoint of the first soaker hose and runs perpendicular from the first soaker hose to the perimeter of the garden.

Miray, a member of the city’s engineering department, has been asked to investigate this watering pattern and to suggest possible alternatives. In order to do this, she constructs a grid so that the centre of the garden is at the point (0, 0). She also locates all the anchor positions of the soaker hose and then heads back to her office. Unfortunately when she arrives, she cannot find all her measurements. All she can find is the anchor positions of the first soaker hose (A(40, 30) and B(-30, 40)). You are a young co-op student placed with Miray at the engineering department. She passes this information on to you to analyse.

Answer the following questions to help you out:

1. Find the equation of the circle.

2. Find the anchor positions of the second soaker hose.

3. Do you see any problems with this watering pattern?

4. What recommendations would you make to Miray?

5. Why do you think that this pattern was put in The Magical Circle gardens in the first place?

6. There is a main watering outlet at the centre of the garden. How close will the soaker hoses be to the watering outlet?

7. The line along which the second soaker hose is lying is the perpendicular bisector of the line along which the first hose is lying. Explain in your own words what the term perpendicular bisector means.

8. Based on your garden diagram, does the perpendicular bisector of a chord of a circle have any special properties? Give reasons for your answer.

9. Construct a circle of your choice. Draw in a chord and construct the perpendicular bisector of this chord. Does this perpendicular bisector have the same property you identified in Question 8?

10. Compare your results with the students around you. Does the property always seem to be true?

11. What conclusion can you make about a perpendicular bisector of any chord of a circle?

Teacher Facilitation: While circulating about the room, ensure that the students have made an accurate diagram. If the students are having difficulty finding the anchor point on the perimeter of the garden for the second soaker hose, you may accept a graphical approximation.

Assessment/Evaluation Techniques

The report that students have developed in the activities may be presented for preliminary discussion of concepts and ideas. Each group of students will retain their reports for their Culminating Assessment Package.

Assessment Instruments

· Observe groups and assess ability to work independently, teamwork, work habits, organization, and initiative using the Learning Skills Rubric (Appendix A).

· Evaluate Report using Written Report Rubric (Appendix D).

· Evaluate oral presentation using Verbal Presentation Rubric (Appendix C).

· Assess appropriate criteria in the Observational Rubric (Appendix B).

Follow-up Skills: 75 minutes

· Full-period Paper and Pencil Test (problems of the multi-step variety using the concepts of slopes, lengths, midpoints of line segments, and the equation of circles)

Activity 2.4: Freedom on the Beach!

Time: 110 minutes

Description

Students will use a coordinate system (vertices) to find lengths, midpoints, slopes, and equations of lines that form triangles. These will be used to classify triangles and to find medians, altitudes, right bisectors (and their equations), along with circumcentre and centroid. The activity focuses on finding the circumcentre of a triangle formed by three lifeguard stations on the beach of our community.

Strand(s) and Expectations

Ontario Catholic Student Graduate Expectations

The graduate is expected to be:

- an effective communicator who reads understands and uses written materials effectively;

- a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems;

- a self-directed, responsible, life long learner who applies effective communication, decision-making, problem-solving, time and resource management skills.

Strand(s): Analytic Geometry

Overall Expectations

AGV.02 - solve problems involving the analytic geometry concepts of line segments;

AGV.03 - verify geometry properties of triangles and quadrilaterals, using analytic geometry.

Specific Expectations

AG2.01 - determine formulas for the midpoint and the length of a line segment and use these formulas to solve problems;

AG2.03 - solve multi-step problems, using the concepts of the slope, its length, and the midpoint of line segments;

AG2.04 - communicate the solutions to multi-step problems in good mathematical form, giving clear reasons for the steps taken to reach the solutions;

AG3.01 - determine characteristics of a triangle whose vertex coordinates are given;

AG3.03 - verify geometric properties of a triangle or quadrilateral whose vertex co-ordinates are given.

Planning Notes

· Students need copy of Lone Wind Place map.

· Students will need graph paper, mathematical set, and calculators.

· Investigate properties of triangular shapes (lengths of sides, altitudes, medians, right bisectors, circumcentres and centroids with and without grids).

· Consolidate basic definitions of all the above and for scalene, isosceles and equilateral triangles.

· Provide sufficient practice examples.

· If The Geometer’s Sketchpad™ is to be used reserve computer lab time.

· Modification of this activity is included where availability of technology is limited.

Prior Knowledge Required

· classification of triangles according to length of line;

· segments; medians, altitudes, right bisectors, circumcentres, centroids; equations of lines;

· use of The Geometer’s Sketchpad™ if dynamic software is to be used

Teaching/Learning Strategies

Teacher Facilitation: Pose the following problem.

There are three lifeguard stations on the beach of the Lone Wind Place map (labelled A, B and C). The city has been given $500 000.00 from a local philanthropist that must be used to improve the appearance of the beach area. The city council, after consulting with a citizen’s group and a well- known local artist, has decided to put a Freedom Fountain out in the water. For aesthetic purposes, the Freedom Fountain is to be put equidistant from each of the three lifeguard stations. How can they find this position? This is of particular interest because the fountain will have special lighting and hence expensive underground cables will be necessary.

Teachers may preface this activity with a short class discussion (with journal entries) on possible relationships between three adjacent items in a city (e.g., city hall, fire station, and police station). A link can be made with the role of a city planner and the discipline of urban geography.

The time spent on the investigative approach should be limited, with the majority of time spent on the analytic approach, for the following student activity.

Student Activity

1. Plot at least six points on graph paper (label these points).

2. Draw at least two triangles using these points.

3. Find the lengths of the sides of the triangles.

4. Identify the various triangles according to lengths of sides (scalene, isosceles, and equilateral and put the corresponding definitions in your notebook).

5. What is an altitude? Draw the three altitudes for each triangle.

6. Find the equations of these altitudes. (Hint: the altitude is perpendicular to the base, so, if you know the slope of the base, you can use this to find the slope of the altitude.)

7. In a triangle, what is a median? Draw the three medians for each triangle.

8. Find the equations of the medians of the triangles.

9. Find the equations of the right bisectors of the sides of the triangles (put the corresponding definitions in your notebook).

10. Locate the circumcentre and centroid of each triangle on your drawing (put the corresponding definitions in your notebook).

11. Verify these coordinates using the equations you found in Questions 8 and 9.

12. Describe any other triangle centres you know about, and explain how you could determine their coordinates.

Teacher Facilitation: Time spent on the above activity should be limited since most of these concepts have been addressed previously. While students are working on the activity the teacher should circulate about the room to aid and prompt students who are experiencing difficulties. Once students have finished the activity, the teachers may lead the class in a discussion to summarise the definitions and the observations in the above student activity (if there is a need). Return to the initial problem and brainstorm with the students how they can use the Lone Wind Place map (with grid) to solve the problem. Students then prepare a written report.

The above Student Activity may be done using The Geometer’s Sketchpad™. Teachers may wish to demonstrate tools and other basic capabilities of Geometer’s SketchPad for the students or let them investigate and learn on their own, under the teacher’s guidance. This choice will depend on the extent that the students used this program in Grade 9 and to the extent that it was used in previous activities of this unit.

Students should be encouraged do the analytic approach along with their investigation below (i.e., midpoints, lengths, equations).

Return to the initial problem and brainstorm with the students how they can use the Lone Wind Place map (with grid) to solve the problem. Students then prepare a written report.

Alternate Activity Where Technology is Limited

· Group students to make maximum use of technology.

· Use computer and projection unit for demonstration purposes. Students should be allowed to do all of the demonstrations for their classmates.

· If dynamic geometry software is not available, the activity can be performed using graph paper and mathematical sets. The number of these constructions must be limited and adjusted according to time restrictions.

Assessment/Evaluation Techniques

The survey reports that students have developed in the activities may be presented for preliminary discussion of concepts and ideas. Each group of students will retain their reports for their Culminating Assessment Package.

Assessment Instruments

· Assess knowledge and understanding by conferencing with students about their findings as they are working through the investigation using Verbal Presentation Rubric (Appendix C).

· Observe group and assess teamwork and initiative using the Learning Skills Rubric (Appendix A).

· Evaluate the report using Written Report Rubric (Appendix D).

Follow-up Skills: 115 minutes

Teachers should supplement this activity with textbook exercises (include a wide variety of paper and pencil type questions):

· equations of altitudes and medians

· lengths of altitudes and medians

· location of centroid

Activity 2.5: Cell Power

Time: 75 minutes

Description

Students will use a coordinate system (vertices) to find lengths and equations of lines that form quadrilaterals. These will be used to classify quadrilaterals and to find diagonals (and their equations) and their intersection points. The activity focusses on proving relationships involving the intersection of diagonals in a parallelogram.

Strand(s) and Expectations

Ontario Catholic Student Expectations

The graduate is expected to be:

- an effective communicator who reads understands and uses written materials effectively;

- a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems;

- a responsible citizen who accepts accountability for one’s own actions.

Strand(s): Analytic Geometry

Overall Expectations

AGV.02 - solve problems involving the analytic geometry concepts of line segments;

AGV.03 - verify geometry properties of triangles and quadrilaterals, using analytic geometry.

Specific Expectations

AG2.03 - solve multi-step problems, using the concepts of the slope, its length, and the midpoint of line segments (e.g., determine the equation of the right bisector of a line segment, the co-ordinates of whose end point are given; determine the distance from a given point to a line whose equation is given; show that the centre of a given circle lies on the right bisector of a given chord);

AG2.04 - communicate the solutions to multi-step problems in good mathematical form, giving clear reasons for the steps taken to reach the solutions;

AG3.02 - determine characteristics of a quadrilateral whose vertex coordinates are given (e.g., the perimeter; the classification by side length; the properties of the diagonals; the classification of a quadrilateral as a square, a rectangle, or a parallelogram);

AG3.03 - verify geometric properties of a triangle or quadrilateral whose vertex co-ordinates are given (e.g., the line joining the midpoints of two sides of a triangle is parallel to the third side; the diagonals of a rectangle bisect each other).

Planning Notes

· Students will need Lone Wind Place map.

· Students will need graph paper, mathematical set, and calculator.

· Reserve computer lab time (if planning to use The Geometer’s Sketchpad to supplement the analytic approach to the activity).

· Modification of this activity is included where availability of technology is limited.

· Investigate properties of quadrilaterals (length and slope of sides, length and slope of diagonals).

· Consolidate basic definitions of all the above for squares, rectangles, and parallelograms.

· Provide sufficient practice examples.

Prior Knowledge Required

· lengths, midpoints, slopes, equations of lines;

· classification of quadrilaterals;

· diagonals of quadrilaterals;

· use of The Geometer’s Sketchpad™ (if planning on using dynamic geometry software)

Teaching/Learning Strategies

Teacher Facilitation: Pose the following scenario.

There are four communication transmission towers (P, Q, R, and S) indicated on the Lone Wind Place map (form a parallelogram). The city’s public works department is responsible for deciding on the strength of the transmitter that is needed for each tower. The choice for transmitters are:

· 1 km maximum distance; cost is $100 000

· 2 km maximum distance; cost is $175 000

· 5 km maximum distance; cost is $400 000

· 10 km maximum distance; cost is $750 000

If the cost to the community is to be kept to a minimum, what choices should be made for each tower? (Justify, with analytic geometry)

Teachers may preface this activity with a class discussion (with journal entries) on possible relationships between four items in a city that would form some type of quadrilateral and possibly have an influence on such things as transportation routes and public utilities (water lines, gas lines, hydro, sewers, etc.)

Teachers should lead the class in some example(s) of finding equations of line segments, lengths and midpoints of line segments for four vertices that can be used to form various quadrilateral shapes. The activity below can be done by the students with graph paper or with The Geometer’s Sketchpad™ or a combination of the two. If this activity is chosen then the time spent on it should be limited, with the majority of time spent on the analytic approach. If The Geometer’s Sketchpad™ investigation is chosen teachers may wish to demonstrate tools and other basic capabilities of The Geometer’s Sketchpad™ for the students or let them investigate and learn on their own, under the teacher’s guidance. This choice will depend on the extent that the students used this program in Grade 9 and in previous activities in this unit.

Student Activity

1. Put a grid (x- and y-axis) on graph paper (or The Geometer’s Sketchpad™ screen)

2. Plot several points that will allow the construction of several quadrilaterals. Try to create as many different types of quadrilateral as you can.

3. Find the lengths of the sides and diagonals of the quadrilaterals.

4. Identify the various quadrilaterals according to the lengths of their sides and measure of interior angles (square, rectangle, parallelogram).

5. Using the grid, find the slopes and equations of the various line segments. Make observations that indicate relationships between the line segments (sides and diagonals with respect to slope, lengths and points of intersection).

6. Compare your results with those found by other students. Make a summary of the properties of the line segments (including the diagonals) in each type of quadrilateral.

Teacher Facilitation: While students are working on the activity above the teacher should circulate about the room to aid and prompt students who are experiencing difficulties. Once students have finished the activity, the teacher should lead the class in a discussion to summarize the definitions and observations in the above student activity. In particular, to help with the initial problem, students will need to have recognized that the diagonals in a parallelogram bisect each other. Return to the initial scenario and brainstorm with the students how they can use the community map (with grid) to solve the problem.

The teacher should encourage (and prompt) students to find the midpoints of the sides of the parallelogram formed by the towers and the intersection of diagonals. Inform students about the “circular” transmitting patterns of the towers. (All points that are 1 km from the tower lie on a circle with radius 1 km. The transmitter will reach all points inside and on this circle.) The teacher may have to prompt students to find the length from the vertices to other vertices, to the midpoints of the sides and to the intersection point of the diagonals. The teacher may have to prompt students to recognize that the lengths found above will influence the choice for the size of the transmitter.

Activity Where Technology is Limited

· Group students to make maximum use of technology.

· Use computer and overhead projection tablet for demonstration purposes. Students should be allowed to do some or all of the demonstrations for their classmates.

· If dynamic geometry software is not available, the activity can be performed using graph paper and mathematical sets. The number of these constructions must be limited and adjusted according to time restrictions.

· Paper folding can also be used to illustrate intersection and parallelism properties

Assessment/Evaluation Techniques

The report that students have developed in the activities may be presented for preliminary discussion of concepts and ideas. Each group of students will retain their reports for their Culminating Assessment Package.

Assessment Instruments

· Assess knowledge and understanding by conferencing with students about their findings as the students are working through the investigation using Verbal Presentation Rubric (Appendix C).

· Observe group and assess teamwork and initiative using the Learning Skills Rubric (Appendix A).

· Evaluate the report using Written Report Rubric (Appendix D).

Follow-up Skills: 150 minutes

Teachers should supplement this activity with textbook exercises that include a wide variety of paper and pencil type questions dealing with quadrilaterals:

· equations of sides of quadrilaterals given the vertices and classification of quadrilaterals

· intersection of equations of lines that will yield vertices of quadrilaterals

· midpoints and lengths of sides of quadrilaterals

· length and equation of diagonals and line segments joining midpoints of sides

· properties relating to the above (e.g., diagonals of a rectangle meet at right angles, line segment joining the midpoint of the sides of a parallelogram is parallel to the other two sides and other multi-step type problems)

Activity 2.6: Oh Deer! Ticked!

Time: 75 minutes

Description

Students will investigate and prove the relationship that various line segments in a triangle have with respect to the sides of the triangle (e.g., a line segment joining the midpoints of any two sides of a triangle is parallel to the other side). They will use the triangular shaped Conservation Area on the Lone Wind Place Template.

Strand(s) and Expectations

Ontario Catholic Student Expectations

The graduate is expected to be:

- an effective communicator who reads understands and uses written materials effectively;

- a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems;

- a collaborative contributor who works effectively as an interdependent team member;

- a collaborative contributor who achieves excellence, originality, and integrity in one’s own work and supports these qualities in the work of others;

- a responsible citizen who accepts accountability for one’s own actions.

Strand(s): Analytic Geometry

Overall Expectations

AGV.02 - solve problems involving the analytic geometry concepts of line segments;

AGV.03 - verify geometric properties of triangles and quadrilaterals, using analytic geometry.

Specific Expectations

AG2.03 - solve multi-step problems, using the concepts of the slope, its length, and the midpoint of line segments;

AG2.04 - communicate the solutions to multi-step problems in good mathematical form, giving clear reasons for the steps taken to reach the solutions;

AG3.01 - determine characteristics of a triangle whose vertex co-ordinates are given;

AG3.03 - verify geometric properties of a triangle or quadrilateral whose vertex co-ordinates are given.

Planning Notes

· Students use a copy of the Lone Wind Place map template with the Conservation Area clearly labelled.

· Students will need graph paper, mathematical set, and calculator.

· Reserve computer lab time (if using The Geometer’s Sketchpad™ as a resource).

· Students should work in pairs or small groups – classroom should be organized accordingly.

· The intent of this activity is to have students realize and prove (among other things) that a line segment joining the midpoints of two sides of a triangle will be parallel to the third side.

Prior Knowledge Required

· lengths and midpoints of line segments

· slopes and equations of lines

· The Geometer’s Sketchpad™ (if planning on using dynamic geometry software)

Teaching/Learning Strategies

Teacher Facilitation: Before putting the students in pairs (or groups), pose the following situation.

The Lone Wind Place Community has set aside a piece of land that has a wealth of natural habitat. It is called the LONE WIND PLACE Conservation Area. The area is determined by the points F, J, and H on the Community Map. There is a main entrance at point F, off Main Street. There is also another entrance at the marina at point J. The conservation area is completely enclosed with fencing joining the vertices F, J and H.

There have been recent cases of Lyme disease in the region. The LONE WIND PLACE conservation area has a thriving deer population increasing the possibility of ticks which carry and transmit this disease. Julia, an ecology graduate student from the local university, has been given a summer job to tally the deer population in the conservation area. Because of the size of the area, Julia has decided to split the conservation area into four congruent (same area) triangular regions. This will allow her to apply sampling principles to estimate the total number of deer by finding the number of deer in any one of the areas. Julia has decided to choose her four congruent triangles by ensuring that each region has two of its boundaries on the fence lines of the conservation area (see accompanying diagram).

Where should the boundaries of these four triangular regions be constructed to ensure that all four regions have the same area? (use analytic geometry to justify)

Student Activity

Part A Instructions
(The Geometer’s Sketchpad™ can be used as a resource with this activity)

1. Find the conservation area (determined by points F, J, and H) on the Lone Wind Place map.

2. Plot these points on The Geometer’s Sketchpad™ screen (or on graph paper).

3. Construct the triangle determined by these points.

4. Choose 3 points (one on each of the sides of the conservation area).

5. Construct 3 line segments using the 3 points chosen in Question 4.

6. Find the area of the 4 triangular regions determined by the 3 line segments.

7. If these areas are not the same, move the positions of some or all of these 3 points until the area of the 4 triangular regions are the same.

8. Find the coordinates of the 3 points.

Teacher Facilitation: While circulating about the room, the teacher may have to prompt the students to choose the midpoints of the sides of the conservation area. Once the students have finished Part A, the teacher should have a short class discussion on the choice of these 3 points and what properties the line segments joining these 3 points have relative to the sides of the conservation area.

Part B Instructions

1. Find the equation of the three sides of the LONE WIND PLACE Conservation Area.

2. Find the midpoints of the three sides of the LONE WIND PLACE Conservation Area.

3. Find the equation of the three line segments that join the pairs of midpoints of the sides of the LONE WIND PLACE Conservation Area.

4. What do you notice about these line segments compared with the sides of the LONE WIND PLACE Conservation Area?

5. Refer to your answer(s) from Question 4. Do you think that this will be true for any triangle?

6. By choosing a triangle of your choice, prove or disprove. Be sure to include a well-labeled diagram with the three vertices clearly labelled.

7. How would Julia use these regions to estimate the number of deer in the LONE WIND PLACE conservation area?

8. Do you think there could have been a better way for Julia to do her job? Explain

Alternate Activity Where Technology is Limited

· Group students to make maximum use of technology.

· Use computers and projection unit for demonstration purposes. Students should be allowed to do some or all of the demonstration for their classmates.

· If dynamic software is not available, the activity can be performed using graph paper and mathematical sets.

Assessment/Evaluation Techniques

The report that students have developed in the activities may be presented for preliminary discussion of concepts and ideas. Each group of students will retain their reports for their Culminating Assessment Package.

Assessment Instruments

· Assess knowledge and understanding by conferencing with students about their findings as the students are working through the investigation using Verbal Presentation Rubric (Appendix C).

· Observe group and assess teamwork and initiative using the Learning Skills Rubric (Appendix A).

· Evaluate the Report using Written Report Rubric (Appendix D).

Follow-up Skills: 150 minutes

Use other resources (e.g., textbook) to find a variety of problems that will verify geometric properties of triangles or quadrilaterals whose vertex co-ordinates are given such as:

· diagonals of a rectangle bisect each other

· diagonals of a square are perpendicular

· line segment joining the midpoints of two sides of a triangle is one half the length of the other side

· medians of a triangle intersect at a common point (centroid)

· finding area of triangles and quadrilaterals from vertices (find appropriate lengths of sides and heights)

Activity 2.7: Portfolio Presentation

Time: 90 minutes

Description

Students present their final unit portfolios in the form of a math fair or class presentations. Each group will display their final proposals and justifications for the Lone Wind Place Community Project as required by the activities of the unit. The final portfolios will consist of displays and presentations involving charts, maps, proposals, and mathematical justifications for the Community Project.

Strand(s) and Expectations

Ontario Catholic Student Expectations

The graduate is expected to be:

- an effective communicator who reads understands and uses written materials effectively;

- an effective communicator who presents information and ideas clearly and honestly and with sensitivity to others;

- a reflective and creative thinker who creates, adapts, evaluates new ideas in light of the common good;

- a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems;

- a reflective and creative thinker who adapts a holistic approach to life by integrating learning from various subject areas and experience;

- a self-directed, responsible, life long learner who applies effective communication, decision-making, problem-solving, time and resource management skills;

- a collaborative contributor who works effectively as an interdependent team member;

- a collaborative contributor who achieves excellence, originality, and integrity in one’s own work and supports these qualities in the work of others;

- a responsible citizen who accepts accountability for one’s own actions.

Overall Expectations

AGV.02 - solve problems involving the analytic geometry concepts of line segments;

AGV.03 - verify geometry properties of triangles and quadrilaterals, using analytic geometry.

Specific Expectation

All specific expectations for AGV.02 and AGV.03.

Planning Notes

· Teachers will need to provide for the use of audio-visual equipment, computer projection equipment, computers, and display areas.

· An alternate presentation area and/or time might be arranged for the Math Fair displays and presentations

· Some groups may wish to present expert speakers as part of their presentation.

· Teachers may wish to acquire the assistance of individuals who have been assisted with science or technology fairs.

· Assessment instruments need to be prepared for the class, to assist with peer assessment.

Teaching/Learning Strategies

Teacher Facilitation: Each activity of the Lone Wind Place Project Survey contains reports of student investigations, analysis and general findings. After all the activities are complete a final presentation will be used to assess the overall capabilities of the group. These presentations may be in the form of individual presentations of the work of each group using a variety of presentation tools or as a math fair scenario in which all groups are responsible for setting up displays and explaining their results to other members of the class and visitors. Presentation skills, along with mathematical understanding of the concepts and skills of this entire unit will be assessed.

Student Activity

Students assemble their reports and investigations from all of the previous activities in this unit and develop a presentation package. The final presentation report containing all the results of each activity (thematic approach) allows students to role-play and present their results as a consulting firm report for committee consideration. It also allows groups to present their reports as individual group presentations using charts, paper reports, software such as PowerPoint, and/or other conventional methods currently being used in business.

An alternative approach allows each group time to create a Math Fair display. Their results would be displayed using typical display and presentation tools for science and/or math fair environments.

Assessment/Evaluation Techniques

Any and all of the Rubrics in the Appendix can be used for student assessment. A sample skeleton Peer/Self-Assessment Checklist is included with this activity, since peer and self-evaluation are essential components of this activity. However, teachers should create their own checklist elaborating on the various criteria listed.

Math Fair Peer/Self-Assessment Checklist

	Needs Improvement	Satisfactory	Good	Excellent
DISPLAY
Organization
Accuracy
Appeal/Creativity
WRITTEN
Organization
Mathematical
Language
Flow
VERBAL
Preparation
Clarity of
Explanations
Mathematical
Language
DEPTH OF ANALYSIS

Place a mark (X) in the appropriate box

Summative Assessment

Time: 75 minutes

Paper and Pencil Test that would include all aspects of this unit and should include all categories (Knowledge and Understanding, Thinking and Inquiry, Problem Solving, Communication) of the Achievement Chart.

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