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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

Activity 1 | Activity 2 | Activity 3 | Activity 4 | Activity 5 | Activity 6 | Activity 7 | Activity 8

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) (cm^2)	Perimeter (2b + 2h) (cm)	Area b x h (cm^2)
1
2

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) (cm^2)	Area Triangle Area of Parallelogram ) 2 (cm^2)

· 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)
	Area of walls (short sides)
	Area of door
	Total area to be painted
Carpeting	Total area of arcade
	Total area of games and counter
	Amount of carpet required
Base boards	Perimeter of arcade
	Perimeter of games touching the wall
	Width of door
	Length of baseboards required

MATERIAL COSTS

MATERIAL	AMOUNT REQUIRED	COST PER UNIT	COST
Paint		1 litre cost $11.25 and cover 15 m^2
		# of cans required?
Carpet		$18.50/ m^2
Baseboard		$8.75 per linear metre
		Total Cost

Assessment/Evaluation

Collect the worksheets and assess them for accuracy and completion. This activity can be extended into a more complex exercise where students place their own objects in the room, construct scale drawings, and then make their calculations. This activity could be assessed with the aid of the rubric at the end of the unit.

Activity 4: Area and Perimeter Relationships

Time: 75 minutes

Description

In this activity, students investigate the effects that varying the length of a side has on its perimeter and area. This is done through the use of concrete materials. Students investigate problems which involve fixing the perimeter of a shape and then determining the different areas possible. Similarly, students fix the area of a shape and then determine the different perimeters possible.

Strand(s) and Expectations

Strand(s): Number Sense, Measurement and Geometry

Specific Expectations: NS1.01; MG1.01, .04, MG2.01.

Planning Notes

· This activity requires the use of centimetre square graph paper or square tiles.

· Students require the worksheets in the package.

Teaching/Learning Strategies

PART 1: DETERMINING A PATTERN BETWEEN PERIMETER AND AREA OF SQUARES

Student Activity

· Students use centimetre square graph paper or square tiles to observe patterns in the perimeter and area of rectangles when the length is increased by 1. The students record their results in the chart and answer the questions at the end of the activity.

Teacher Facilitation

· Have students build rectangular shapes.

· Teachers draw diagrams on the blackboard as students work through examples.

· Have students complete the rest of the chart.

· Circulate to give the students assistance in completing the task.

Using centimetre graph paper or square tiles create the shapes with the dimensions listed in the table and complete the chart.

Sample Chart

Number of units in Width (cm)	Number of units in Length (cm)	Perimeter (cm)	Area (cm^2)
5	6
5	7
5	8
5	9
5	10

Questions:

1. What happens to the area when the length is increased by one?

2. What happens to the perimeter when the length is increased by one?

3. Would the same pattern appear if we had changed the width instead of the length? Explain.

Part 2: CONSTANT AREA: CHANGING PERIMETER

Student Activity

Exercise 2a

· Students construct 5 different rectangles with an area of 36 cm². They determine the perimeter of each rectangle and record it in the chart.

Exercise 2b

· Students are given a certain number of squares to work with (centimetre square graph paper or square tiles can be used). Students draw at least 4 different non-rectangular shapes using all of the given number of squares. They then determine the perimeter of each shape and record them in their chart. For example;

Part 3: CONSTANT PERIMETER: CHANGING AREA

Exercise 3a

· Students determine the area and perimeter of different irregularly shaped figures.

Exercise 3b

· Students construct different sized rectangles that have a fixed perimeter of 24 cm. They construct at least 4 different rectangles, determine their area and record their results in the chart provided.

Teacher Facilitation

· It is important that the teacher discuss that a square is a type of rectangle.

· Provide centimetre square graph paper or square tiles.

· Complete an example for the students on the board.

· Circulate to ensure understanding and completion.

· Draw attention to the important result that the more regular the shape, the closer you are to minimizing the perimeter for a fixed area and maximizing the area for a fixed perimeter. To emphasize this result, you may need to have a class discussion to ensure all possible shapes have been explored.

CONSTANT AREA: CHANGING PERIMETER

Exercise 2a

Using 36 squares construct five different rectangles and calculate their perimeter. Fill in the chart and answer the questions at the end of the activity.

Sample Chart

Shape	Number of Squares (Area)	Diagram	Perimeter
1	36
2	36
3	36
4	36
5	36

Questions:

1. Which shape had the largest perimeter?

2. Which shape had the smallest perimeter?

3. If you were trying to build a rectangular fence and you had very little money, which shape would you use to build your fence? Explain your answer.

Exercise 2b

Construct 4 irregular shapes using 20 squares. Calculate the perimeter of each shape. Then rank each by perimeter, smallest to largest. Display your work in the chart provided.

Sample Chart

Shape	Area	Perimeter	Rank
example:	20	28
	20
	20
	20

Question:

1. What type of shape gives the largest perimeter?

Exercise 3a

Using the diagram below determine the perimeter and area of each shape.

Sample Chart

Diagram	Perimeter	Area
1
2
3
4
5

Question:

1. What do you observe in your chart about your values of perimeter and area?

· The teacher can highlight for students that despite the different areas, the same perimeter was used.

Exercise 3b

Working with centimetre square paper, construct 4 different rectangles shapes with a perimeter of 24 cm, calculate the area of each shape. Fill in the chart and answer the questions at the end of the activity.

Sample Chart

Shape	Perimeter	Diagram	Number of Squares (Area)
1	24
2	24
3	24
4	24

Questions:

1. Which rectangle gives you the largest area?

2. Which rectangle gives you the smallest area?

After each investigation, a class discussion could ensue concerning where the different characteristics (measurements) of objects could be used in different situations (e.g., constructing a fenced pen, heat loss of an object).

This is an opportunity for students to practise their skills in determining the area of rectangular, triangular, and irregular shaped objects.

Assessment/Evaluation

Collect student work and assess for completeness and accuracy of answers. Ask students to write in their journals or notebooks a summary of what they determined in the investigations and list situations where this information might be useful. Ask questions such as: List any new insights you have learned. or What surprised you in these activity?

Activity 5: Fencing the Yard

Time: 75 minutes

Description

In this activity, students solve problems by determining the optimal perimeter to maximize the area and determine the optimal area to minimize the perimeter.

Strand(s) and Expectations

Strand(s): Number Sense, Measurement and Geometry

Specific Expectations: NS01.01, .13, .15, .16; MG1.04, MG2.01, .02, .03, .05, .06.

Planning Notes

· Make up a problem similar to the example given.

· Provide worksheets for the two problems.

Teaching Learning/Strategies

Student Activity

· Students solve the problems dealing with area and perimeter by filling in appropriate charts and completing drawings of the pens to determine optimal area and perimeter.

Teacher Facilitation

· Assign students to work in pairs.

· Hand out the first problem and discuss to ensure understanding.

· Do two or three examples in the chart to get the students started.

· Upon completion, discuss what students have accomplished.

· Introduce the second problem.

· Circulate through class to resolve any problems before they occur and to provide support and encouragement.

· Extension Activity: Teachers could graph the Area vs. Length and demonstrate the maximum and/or minimum area/perimeter needed.

Assessment/Evaluation

Collect the charts and evaluate for completeness and accuracy. Use student observation rubric Appendix 1 to record learning skills.

Sample Worksheet

FARMER BROWN=S PEN

Farmer Brown wants to build a rectangular pen for his ostriches. He has 40 metres of fencing available. Determine the maximum area (using the data provided) he can enclose by filling in the following chart. Draw each of the pens on graph paper using the following scale: 1 side of a square represents 1 m.

Vertical Length (m)	Total Vertical Length (m)	Total amount of width remaining (m)	Width = total 2 (m)	Area (m^2)
1 m	2 m	38 m	19 m	19 m^2
· · ·
19 m

Questions:

1. What patterns were in the table?

2. Why did the first column stop at 19 m?

3. What is the maximum area? What are the dimensions of this rectangle?

4. What is the minimum area? What are the dimensions of this rectangle?

5. Which one should Farmer Brown choose? Why?

Sample Worksheet 2

MRS. JONES’ DOG RUN

Mrs. Jones would like to have her daughter build a dog run for her Labrador retriever. The pen must have an area of 48 m² to provide the necessary space for the dog. Mrs. Jones wants to keep her cost of fencing as low as possible. Complete the chart and determine the minimum perimeter she could use. Draw each of the pens on graph paper using the following scale: 1 side of a square represents 1 m.

Vertical Length (m)	Width (m)	Total Vertical Length (m)	Total Width (m)	Perimeter (m)
2
4
6
8
12
16

Questions:

1. What is special about the numbers in the first column and the number 48?

2. What are the dimensions of the rectangle with the largest perimeter? What is the perimeter of that rectangle?

3. What are the dimensions of the rectangle with the smallest perimeter? What is the perimeter of that rectangle?

4. Which one should Mrs. Brown choose? Why?

5. If the fencing material cost $12.50/m, what is the cost to fence the dog run?

Teachers may choose to review square root notation with their students and have them practise using their calculators to solve problems such as:

What is the side length of a square that has an area of 50 cm². Encourage them to estimate first.

Assessment/Evaluation

Collect the charts and evaluate for completeness and accuracy. Use student observation rubric (Appendix 1) to record learning skills. Students could independently solve a similar problem to those above such as: A farmer wishes to build a pen that separates the male rabbits from the female rabbits as in the diagram below. If she has a total of 60 m of fencing, what are the possible dimensions?

Students use a table to complete this example.

Activity 6: The Long and the Short of it

Time: 75 minutes

Description

In this activity, students examine the Pythagorean theorem and learn how to use it in real life contexts. They also solve simple problems involving the formula.

Strand(s) and Expectations

Strand(s): Number Sense, Measurement and Geometry

Specific Expectations: NS1.01, .13, .15, .16; MG2.01, .02, .07, .08.

Planning Notes

· Students measure the lengths, widths, and diagonal of rectangular objects. Metre sticks, rulers, scissors, and measuring tapes, along with rectangular objects, are required.

· Students are required to use the concept of square roots in the latter part of this activity. A brief review at that time may be appropriate.

Teaching/Learning Strategies

Student Activity

· Students measure various rectangles to determine that the diagonal is always longer than the sides.

· Students complete a worksheet to determine patterns in right angle triangles and discover the relationship that exists (see worksheet 1).

· The relationship of the figures on the sides of a right angled triangle are reinforced by students cutting out and reorganizing shapes (see worksheet 2).

Teacher Facilitation

· Bring in various rectangular objects for students to measure.

· They record these measurements in a chart.

· Discuss the results. The diagonal is always longer than the sides.

· Discuss the different situations where people use the idea of a carpenter's square. For example, brick layers, carpet and pool installers.

· Hand out activity Patterns in Right Angle Triangles (worksheet 1). Prepare an overhead of one of the figures and do it together with the class. For figure 6 have the students estimate the area of the square.

· The teacher can discuss how the size of a television has been traditionally determined by the length of the diagonal. Thus, a 20-inch TV could have several different dimensions. This can be extended to a discussion of how the new high definition televisions will all have the same ratio of 16:9 and how the measurement of length will now be consistent for all new televisions.

· Hand out the Pythagorean puzzles with scissors. It may be easier for students if the puzzle worksheets are copied onto thick paper.

· The teacher may wish to do a few problems with the students. It is important that students do not simply use a formula such as c² = a² + b² but that they refer to the diagrams to assist them in recalling the patterns that they have determined. Students can sketch the squares on the sides of their diagrams to help them recall the relationship.

WHICH IS THE LONGEST?

Using the objects provided fill in the chart below and answer the question at the end of the activity.

Sample Chart

Object	Width	Length	Diagonal
Example: student desk	60 cm	40 cm	72 cm

Questions:

1. What do you notice about the diagonal of the objects?

2. Do the sides add to give the diagonal? Explain why?

Worksheet 1

PATTERNS IN RIGHT ANGLED TRIANGLES

The longest side of a right-angled triangle has a special name.

Similar figures have been drawn on three sides of right triangles. Determine the area of each similar figure and record it in the chart.

Figure 1 (Count each square as 1 unit)

To measure the sides of the large rectangle use a ruler and the grid.

The area of each of these triangles is one unit. Join the dots to the make triangles similar to the given ones.

Use this information to help you determine the area of each similar figure. The first diagram has been started for you.

Figure 2	Figure 3	Figure 4

Figure 5	Figure 6

Figure	Area of figure on one side	Area of figure on other side	Area of figure on hypotenuse
1
2
3
4
5
6

What patterns do you notice?

Students can practise solving simple problems involving the Pythagorean theorem such as:

1. The size of a television screen is given by the length of the diagonal. A television screen is 60 cm wide and 80 cm high. Calculate the length of the diagonal.

2. The diagram shows the old and new route that Chris takes to get to the barn. How much shorter is the new route?

This is also an opportunity for students to practise their skills involving problems that use square roots.

Worksheet 2

PYTHAGOREAN PUZZLES

Directions: Cut carefully along the dotted lines and fit the resulting pieces into the uncut square.

Assessment/Evaluation

Collect the sheets and assess for accuracy and completeness. Ask students to write their own problem that uses the Pythagorean theorem, which can be assessed for communication. A quiz can also be used to assess knowledge.

Activity 7: An Around About Problem

Time: 75 minutes

Description

In this activity the students work in groups to determine the value of π by measuring different objects. With the teacher's assistance, students develop the formula for the circumference and area of a circle. Students apply these formulas to solve real-life problems.

Strand(s) and Expectations

Strand(s): Number Sense, Measurement and Geometry

Specific Expectations: NS1.01, .06, .07, .13, .15, .16; MG2.01, .02, .03, .05, .06.

Planning Notes

· Students require measuring tapes and several circular objects (e.g., pie plates, cups, lids).

Teaching/Learning Strategies

Student Activity

· Students work in groups to measure the circumference and diameter of various objects and record their results in a prepared chart. In completing the chart they determine the value of p. With the assistance of the teacher, students then determine the formula for the circumference of a circle.

· Students are led through a demonstration by the teacher of the development of the area of a circle.

Teacher Facilitation

· This activity could be modified to use a spreadsheet.

· Discuss diameter, radius, and the relationship between them (d = 2r).

· Hand out activity sheets, measuring tapes, and circular objects. Have the students circulate the various objects around the room.

· Have students calculate C/d and discuss results (mention p = C/d and that it can be approximated to 3.14).

· Have students fill in C= pd column for each object.

· Hand out graph paper so students can estimate the area of each object.

· Once they have estimated the number of squares and calculated the area of the circle, discuss with the students about a different way of calculating the area: by using the formula.

· Circulate to assist students in completing the activity.

· The teacher demonstrates the development of the formula for the area of a circle by taking a circle and cutting it into several sectors and rearranging it to form a parallelogram. Students can be asked to comment on its shape and predict what would happen if the sectors were cut smaller. The teacher then demonstrates. By showing students that the width of the rectangle is equivalent to the radius of the circle and its length is the same as one half of the circumference (pr), the formula for area is:

A	=	length x width
	=	p x r x r
	=	pr^2

Sample Worksheet

AROUND ABOUT PROBLEM

You will be given various circular objects to measure. Complete the chart below and answer the questions at the end of the activity using the formula for circumference of a circle. Leave the last column until you have finished measuring all the objects.

Object	Diameter (cm)	Circumference (cm)	Circumference Diameter	Circumference C= pd
pop can
coffee can lid

Questions:

1. What do you notice in the last two columns? Can you explain why this is so?

2. If the diameter of a swimming hole is 6 m, how much fence would be needed to encircle the swimming hole?

3. If a pizza has a circumference of 50 cm, what is the diameter of the pizza?

4. A patio umbrella has a radius of 1 m, how much fringe would be needed for the edge of the umbrella?

5. You just purchased a circular pool with a radius of 1.5 m. You need to buy a pool cover. How much material do you need to cover the pool?

Once students have completed the development of the formulas they can apply these concepts to solve problems involving the area and circumference of circles.

Assessment/Evaluation

Collect charts and questions to assess for accuracy and completion.

Use the Student Observation Rubric (Appendix 1).

Activity 8: Summative Assessment: Planning a Backyard

Time: 150 minutes

Description

In this activity students are given the opportunity to design a landscaping layout for a backyard. They include items such as a circular pool, a diagonal walkway, a triangular flowerbed with a retaining wall, a rectangular rock garden and another item of their choice. They draw a scale diagram using centimetre square graph paper.

Strand(s) and Expectations

Strand(s): Number Sense, Measurement and Geometry

Specific Expectations: NS1.01, .13, .15, .16; MG1.04, MG2.01, .02, .03, .05, .06, .08.

Planning Notes

· Provide worksheets.

· Provide rulers and centimetre square graph paper, compasses or circular objects.

· Provide formulas where needed.

Teaching/Learning Strategies

Student Activity

· Students are given the worksheets and graph paper.

· Students are given the assessment rubric before beginning the work.

· Students answer questions that result in the creation of a scale drawing to be used for a backyard landscaping project.

Teacher Facilitation

· Begin with a discussion about planning and organizing a landscaping project for a backyard.

· This discussion should include why planners use blueprints and the value of organizing yourself before you begin a project.

· Distribute the assessment rubric located at the end of this activity and explain it to the students.

· Distribute the worksheets.

· Distribute centimetre square graph paper and have compasses or circular objects available.

· Explain the instructions as you read them over with the students to ensure understanding.

Sample Project Worksheet

LANDSCAPING DESIGN

Ms. Schwartz has asked a number of companies to create a sample landscaping plan for her backyard. You want to win the contract.

The requirements for her layout:

1. Ms. Schwartz's yard is 18 m x 24 m in size.

2. She has a circular swimming pool with a radius of 3 m.

3. Ms. Schwartz wishes to have triangular flowerbed in one corner of the yard with the two sides that are along the fence measuring 6 m and 8 m.

4. Ms. Schwartz wishes to recycle her favourite wrought iron fence. She wishes to have a rectangular rock garden with a perimeter of 54 m. One of the sides of the rectangle must be at least 6 m but no more than 18 m. To reduce your work, the side length must be an even number. She wants the garden to have the largest area possible. Maximize the area (HINT: Use a chart).

5. A diagonal walkway (1 m wide), that goes from corner to corner in her yard.

6. She has allowed each company to choose another item to locate in her backyard (e.g., bird bath, fruit tree, gardening shed). This choice may help you win the contract.

Good Luck! Remember, she wants you to include everything specified above and she wants it to look good too.

You are encouraged to make a rough draft first and then redraw it as a good copy to submit for assessment. Use your notebook, summary sheet, formula page, rubric, etc. when you need help.

1. On the graph paper, draw the outline of the yard using the scale one centimetre represents 1 metre.

2. A circular pool with a radius of 3 m.

a) Where will you locate the pool in the yard? Draw it to scale.

b) What will the area, in m², be of the pool cover?

3. A triangular flowerbed in one corner of the yard with the two sides that are along the fence measuring 6 m and 8 m.

a) Where will you locate the flowerbed in the yard? Draw it to scale.

b) You will fill it with bushes that each need an area of 0.5 m² to grow. How many bushes must be purchased?

c) If bushes cost $8 each, how much would the total cost be?

4. A rectangular rock garden with a perimeter of 54 m. One of the sides of the rectangle must be at least 6 m but no more than 18 m. The length must be an even number. She wants the garden to have the largest area possible. (Maximize the area). (HINT: Use a chart.)

a) Where will you locate the rock garden in the yard? After doing the calculations below, draw it to scale.

Rock Garden Chart

Length of one side of garden (m)	Total vertical length (m)	Total perimeter remaining for width (m)	Width = total remaining ) 2 (m)	Area (m^2)
6
8
10
12
14
16
18

b) Which dimensions give the maximum area (largest garden)?

5. A diagonal walkway (1 m wide), that goes from corner to corner in her yard.

a) Where will you locate the walkway in her yard? Draw it to scale.

b) Calculate the length of the walkway. (HINT: It is a diagonal.)

c) Paving stones are 0.5 m x 0.5 m. How many paving stones would you need to pave the walkway if the width of the walkway was 1 m and the length was determined in the answer to 5b).

6. One other item of your choice:

a) What other item will you choose? Why have you chosen this?

b) What are its dimensions?

c) Calculate the area of your item.

d) Calculate the perimeter of your item.

e) Where will you locate the item in her yard? Draw it to scale.

Sample student solution of layout

Assessment/Evaluation

Assess the project for accuracy and completion and also using the criteria from the Planning a Backyard Rubric. The observation rubric from Appendix 1 could be used to assess students.

Rubric for Planning a Backyard

	Level 1	Level 2	Level 3	Level 4
The Plan of the Yard	- applies scale conversions with limited consistency - for choice of extra item, cannot justify choice or choice is inappropriate	- applies scale conversions with moderate consistency - choice of extra item is appropriate but may not clearly justify choice	- applies scale conversions with considerable consistency - choice of extra item is appropriate and justifies choice	- applies scale conversions with a high degree of consistency - choice of extra item is appropriate and justifies choice and location
Communication	- placing of the yard items can be described with assistance - has difficulty following steps; incomplete solution	- placing of the yard items can be described with minimal assistance - lacks description of solution but most mathematical forms are present	- correctly describes most of the placements of the yard items - combines some description of solution with mathematical forms (diagrams, formulas); not all connections evident	- correctly describes all the placements of the yard items - combines description of solution with appropriate mathematical forms; logical flow is evident
Solves Problems	- needs coaching for each step - is able to calculate area and perimeter with assistance; weak understanding of concepts	- solves the problem but requires regular reassurance from the teacher - relies solely on simple methods such as counting to calculate area and perimeter	- can solve the problem after some teaching or discussion; occasionally checks with teacher for reassurance - calculates area and perimeter correctly	- is able to independently decide on necessary procedure to accurately solve the problem - calculates area and perimeter using efficient and clear methods

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