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Course Profile (for a locally developed course)

Essential Mathematics, Grade 9

Unit 3

Course Profiles are professional development materials designed to help teachers implement the new Grade 9 secondary school curriculum. These materials were created by writing partnerships of school boards and subject associations. The development of these resources was funded by the Ontario Ministry of Education. This document reflects the views of the developers and not necessarily those of the Ministry. Permission is given to reproduce these materials for any purpose except profit. Teachers are also encouraged to amend, revise, edit, cut, paste, and otherwise adapt this material for educational purposes.

Any references in this document to particular commercial resources, learning materials, equipment, or technology reflect only the opinions of the writers of this sample Course Profile, and do not reflect any official endorsement by the Ministry of Education or by the Partnership of School Boards that supported the production of the document.

ã Queen’s Printer for Ontario

Acknowledgments

Public and Catholic School Board Writing Team – Essential Mathematics

John Dallan, Lead Writer, Upper Grand District School Board

Bernie McGarry, Halton District School Board

Tina Noel, Renfrew County Catholic District School Board

Rob Samson, Simcoe Muskoka Catholic District School Board

Shirley Scott, District School Board of Niagara

Emilia Veltri, Lakehead Public District School Board

Jim Vincent, Peel District School Board

Lead Board

Halton District Secondary School Board

Kit Rankin

Susan Orchard

Larry Zavitz

Kelly Terry

With assistance from:

The writing team for the Applied and Academic Grade 9 Public Course Profile

Unit 3: Investigating Two-Dimensional Figures

Time: 14 hours

Unit Description

In this unit, students are engaged in a variety of activities dealing with two-dimensional geometry that allows them to solve measurement problems in real-life contexts. Through the use of concrete materials students develop and apply formulas. They select appropriate tools which allow them to measure to the degree of accuracy required in a particular situation. Concrete materials, drawings, and technology are used to investigate the effect that varying one dimension has on perimeter and area. Opportunities are also given to explore geometric properties and optimal values of various measurements of two-dimensional figures. Students communicate their findings and apply them to identify and solve problems that are familiar to them. The Pythagorean theorem is developed through the use of concrete materials and used to solve simple problems. Students continue to develop their skills for estimation and judging the reasonableness of an answer.

Strand(s) and Expectations

Number Sense Strand Specific Expectations: NS1.01, .13, .15.

Relationships Strand Specific Expectations: RE1.01, .03, .04, .05.

Measurement and Geometry Strand Specific Expectations: MG1.01, .04; MG2.01, .02, .03, .05, .06, .07, .08.

Activity Titles

What follows is a suggested sequence, with timing, for teaching Unit 3. These activities are designed to have students make sense of mathematics by working through concrete experiences to develop students= understanding of various mathematical concepts. Many skills are developed within the activities themselves. However, the need for remediation and further development of skills will arise from the activities.

Thus far in the Grade 9 program students have yet to deal with measurement to any great extent. In this unit perimeter and area are explored in detail for the first time in this profile.

Activity 1	Linear Measurement	75 minutes
Activity 2	Investigating Area and Perimeter	150 minutes
Activity 3	Area and Perimeter Relationships	75 minutes
Activity 4	Video Arcade	75 minutes
Activity 5	Fencing the Yard	90 minutes
Activity 6	The Long and the Short of it	150 minutes
Activity 7	An Around About Problem	75 minutes
Activity 8	Summative Evaluation: Planning a Backyard	150 minutes

Unit Planning Notes

· This unit incorporates numerous concrete materials that must be organized prior to the activity.

· There are opportunities to modify Activity 5 to use a spreadsheet.

Teaching/Learning Strategies

· This unit requires flexibility of timing while at the same time requires structure so that students are engaged in meaningful tasks. Teachers are to be working diagnostically with students to determine what type of support each student requires. Time has been built into activities to allow for these opportunities and to further develop skills within context.

· Encourage students to estimate their answers prior to using a calculator and check their answers for reasonableness.

Resources

Coxford, Arthur Jr. Geometry from Multiple Perspectives. National Council for Teachers of Mathematics, 1991.

Ebos, F., D. W. McKillop, E. Milne, B. J. Morrison, B. Robinson, and K. Whelan. Math In Context 7. Nelson Canada, 1992.

Ebos, F., D. W. McKillop, E. Milne, B. J. Morrison, B. Robinson, and K. Whelan. Math In Context 8. Nelson Canada, 1992.

Flewelling, G., J. Routledge, J. Clark, and T. Brown. Making Mathematics 8. Gage, 1991.

Elchuck, L., J. Hope, B. Scully, J. Scully, M. Small, and S. Tossell. Interactions 9. Prentice Hall Ginn, 1996.

Kenney, M.J., S. J. Bezuszka, and J. D. Martin. Informal Geometry Explorations. Dale Seymour, 1992.

Lunney, J., P. Rae-Dion, B. Tuck, and B. Walters. Math Sense Book 1. Nelson Canada, 1991.

Reak, C., K. Stewart, and K. Walker. 20 Thinking Questions for GeoBoards. Creative Publications, 1995.

Woodward, E. and T. Hamel. Visualized Geometry - A Van Hiele Level Approach. J. Weston Walch, 1990.

Activity 1: Linear Measurement

Time: 75 minutes

Description

In this activity, students review their metric measurement skills through the use of the Metric System Steps diagram. They use these skills to determine the appropriate unit of measure, estimated length, and actual length of a variety of objects.

Strand(s) and Expectations

Strand(s): Number Sense, Measurement and Geometry

Specific Expectations: NS1.01, .13, .15, .16; MG2.01, .04, .05.

Planning Notes

· Students require: rulers, calculators, metre sticks and/or measuring tapes and a copy of the Metric System Steps included in sample worksheet 1.

· Have available a variety of objects for students to use to determine the appropriate unit of measurement required.

· A similar topic is contained within the Grade 9 Science curriculum. Teachers may wish to work with the science teachers to ensure a consistent approach is taken.

Teaching/Learning Strategies

Student Activity

· Students label their Metric System Steps appropriately during the teacher-directed lesson.

· Students complete a measurement chart (see worksheet 1) by measuring objects using appropriate units.

· They practice converting units using the Metric System Steps (see sample worksheet 2).

Teacher Facilitation

· The teacher leads a discussion on the appropriate use of metric measure and conversion between units. Students should be able to give examples where different units are used in their daily lives.

· Explain how to develop the Metric System Steps and how it is used. Particular attention should be given to the relationship between the location of the decimal place and multiplication and division of 10, 100, 1000, etc. It should be noted that the use of the Metric System Steps is a visual aid and it is hoped that students come to rely on it less and less as they are involved in estimation and actual measurement of objects throughout the course.

· As the teacher introduces new objects to be measured, the class first determines the most appropriate unit of measurement, then they estimate its length and finally they measure the object. A variety of different sized objects are required.

· Facilitate the appropriate use of the Metric System Steps to ensure successful completion of the charts.

· The Guinness Book of World Records can provide some interesting examples of extreme measurements that can then be compared with familiar measurements (e.g., The Great Wall of China and a local monument).

Sample Worksheet 1

METRIC SYSTEM STEPS

For every step up the ladder,

move the decimal one place

to the left.

(Divide by 10)

For every step down the ladder,

move the decimal one place

to the right.

(Multiply by 10)

MEASURING OF OBJECTS

Object	Appropriate Unit of Measurement	Estimated Measurement	Actual Measurement
length of text book
length of bulletin board
diameter of a dime
width of a desk
length of classroom

You may wish to add other objects to this list.

Question:

Which unit would you use to measure the following objects? (mm, cm, m, or km)

1. the height of the average man

2. the width of a skating rink

3. the length of a small insect (e.g., ant, praying mantis)

4. the circumference of the earth at the equator

Sample Worksheet 2

METRIC CONVERSIONS

Object	Actual Measurement	mm	m	km
length of textbook	28 cm	280mm	0.28 m	0.00028km
length of bulletin board
diameter of a dime
width of desk
length of classroom

You may want to add other objects to this list.

This is an opportunity for students to practise their measurement skills outside the context of the classroom.

Assessment/Evaluation

Assess the worksheets for completeness and accuracy. Through the use of a quiz, students can be shown several objects and asked to estimate their lengths and determine the most appropriate unit of measure. Similarly, students may be asked to list several objects that they would measure using units such as mm, cm, m, and km.

Students may also be asked to construct measurement posters. These posters may have objects taped to them or pictures pasted to them from magazines that might all be measured using a particular unit of measure. For example, one poster can depict objects such as buildings, racetracks, cars, and other similar objects that might be measured using metres.

Activity 2: Investigating Area and Perimeter

Time: 75 minutes

Description

In this activity students use concrete materials to solve problems involving the area and perimeter of squares, rectangles, and triangles.

Strand(s) and Expectations

Strand(s): Number Sense, Measurement and Geometry

Specific Expectations: NS1.01, .13, .15, .16; MG2.01, .02, .05, .06.

Planning Notes

· This activity requires index cards to help students determine height of a triangle and centimetre squared graph paper or square tiles (multi-link cubes may be substituted if students are instructed only to consider one face).

· Students require the worksheets in this activity to complete the exercises.

· Prepare an overhead that shows a parallelogram on grid paper.

· The teacher may choose to supplement the worksheets with further exercises.

Teaching/Learning Strategies

PART 1: Perimeter and Area of a Rectangle

Student Activity

· Students use centimetre square graph paper or square tiles to construct at least four different rectangular shapes using cubes. They draw them on grid paper and by counting the squares they determine the length, width, and perimeter and area of each shape.

· Students also determine the area of a variety of rectangles with the help of visual aides and ultimately recognize the formula for the area of a rectangle.

Teacher Facilitation

· Provide students with centimetre square graph paper or square tiles.

· Build several rectangles with the students using the tiles as a guide and then draw the rectangles on graph paper.

· Observe students as they build/draw the shapes and record their data.

· Discuss the results of the chart with the students.

· Discuss with the students a different way of finding the area using the grid paper drawings. Have the students calculate the area using the formula.

· Provide sample problems for students to practise their skills.

AREA OF A RECTANGLE

Sample Chart

Diagram

Length of Rectangle

(cm)

Width of Rectangle

(cm)

Perimeter

(distance around) (cm)

Area (squares counted in rectangle)

(cm²)

Perimeter

(2b + 2h)

(cm)

Area

b x h

(cm²)

Questions:

1. What do you notice about the area of the counted squares and the area calculated by the formula?

2. Determine the perimeter of a square with sides 5.5 cm. Draw a diagram if it helps you.

3. Determine the area of a rectangle that has a length of 8 cm and a width of 4 cm.

PART 2: Areas of Parallelograms and Triangles

Student Activity

· Students are led through the development of the formula of a parallelogram to help them determine the formula for the area of a triangle.

· With the help of grid paper, and perhaps some teacher facilitation, students develop the formula for a triangle.

· After drawing any triangle on grid paper, students determine the height (using an index card for 90^O to the base), base, and area of a triangle.

Teacher Facilitation

· Provide the students with various parallelograms drawn on grid paper.

· Demonstrate the process for finding the area of a parallelogram, by making a parallelogram into a rectangle. This is done by cutting a triangle off the parallelogram and repositioning it to form a rectangle. This should be teacher-directed and an overhead would be beneficial to demonstrate the process.

· Show students that the formula for the area of a parallelogram is the same as the formula for the area of a rectangle (A = b x h). Give particular attention to the idea of perpendicular height in this case.

· Emphasis on proper terminology such as base, perpendicular height, and altitude is important.

· Students should repeat these steps for various parallelograms and then determine the area of each by first converting them to rectangles.

· Demonstrate the drawing of a diagonal in a parallelogram to divide it into 2 triangles.

· Discuss this process, emphasizing the fact that the parallelogram is now cut into two triangles, therefore the area of one triangle is 2 the area of the parallelogram. Develop the formula for the area of the triangle.

· Each student should be instructed to calculate the area of each parallelogram that they have drawn and complete the chart.

AREA OF A TRIANGLE USING A PARALLELOGRAM

Sample chart

Diagram

Area of Parallelogram

(base x height)

(cm²)

Area Triangle

Area of Parallelogram ) 2

(cm²)

· Provide students with a worksheet of triangles drawn on a grid. Demonstrate how to draw corresponding parallelograms. Find the area of each triangle by first finding the area of each parallelogram. A sample is shown below.

· Provide a worksheet of triangles on a grid and have them determine the base, perpendicular height (altitude), and area of the triangles. Demonstrate this using an index card to show perpendicular height.

This may be an opportune time for students to practise solving perimeter and area problems involving shapes with missing dimensions that first must be determined before the problem can be completed, compound shapes and decimal measurements.

Assessment/Evaluation

Collect student work and assess for completeness and accuracy of answers. A quiz can be used to assess students’ ability to calculate the area of various shapes.

Activity 3: Video Arcade

Time: 75 minutes

Description

In this activity students apply their skills in determining perimeter and area. Students redecorate a video arcade. They use the formulas to determine how much carpet, base boards, and paint is required to redecorate the Arcade.

Strand(s) and Expectations

Strand(s): Number Sense, Measurement and Geometry

Specific Expectations: NS1.01, .13, .15, .16; MG2.01, .02, .03, .05, .06.

Planning Notes

· Provide a floor plan for the video arcade.

· Have displays of the formulas for students to see and use for their activities.

· This would be a good opportunity for students to work in pairs or small groups for peer support. The composition of the groups may be determined prior to the activity.

Teaching/Learning Strategies

Student Activity

· Students redecorate a video arcade. They are given a floor plan with various arcade machines placed throughout. They use formulas to determine how much material is needed to paint the walls, carpet the floor, and place baseboards around the floor of the store. The games have been bolted into the floor and students have to account for this when determining the amount of baseboard needed as well as carpeting.

· They record their work in charts provided.

Teacher Facilitation

· Review the formulas for area/perimeter of triangles, squares, and rectangles.

· Provide a floor plan of the video arcade.

· Discuss with the students that the games are bolted into the floor and carpet is not needed for these areas. Games placed against the walls will not need carpet or baseboards.

· When calculating the number of cans required to paint the walls, students should be reminded that they cannot purchase partial cans of paint.

VIDEO ARCADE REDECORATION

You have been hired to redecorate a video arcade. Using the floor plan provided you determine the amount of carpet, base boards, and paint needed to redecorate the video arcade. When you are making your calculations you must take into account that the games are bolted into the floor and you must work around them. For games that are against the wall, no carpet, or baseboards are needed. The walls of the store are 2.6 m high and there is one door measuring 2.1 m by 0.8 m. There are no windows.

The redecoration project must be done in the following order:

1. Paint all of the walls.

2. Bring in the machines and cashier counter to be installed.

3. Install the carpet.

4. Install the baseboards.

VIDEO ARCADE

Object	Room perimeter taken up by counter, machine, or door	Floor Area taken up by counter or machine
Cashier's Counter
Pinball Machine
Sega machine
Virtual cycle
Virtual Reality
Door Opening		xxx
Total

AMOUNT OF MATERIALS REQUIRED

Material	Measurements	Calculations
Paint	Area of walls (long sides)