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Course Profile Foundations of Mathematics, Grade 9 applied, Catholic

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.

© Queen's Printer for Ontario

Acknowledgments

Catholic District School Board Writing Team – Mathematics - Applied

Lead Board

Ottawa-Carleton Catholic School Board

Sean Kelly, Manager

Course Developers

Arlene Corrigan, Renfrew County Catholic District School Board

Dominique Levac, Catholic District School Board of Eastern Ontario

Carolyn Boyer, Ottawa Carleton Catholic School Board

Len St.Clair, Catholic District School Board of Eastern Ontario

Brian McGudden, Toronto Catholic District School Board

Margaret Sinclair, Toronto Catholic District School Board

Paul Costa, Toronto Catholic District School Board

Lori Goodfriend, Catholic District School Board of Eastern Ontario

Catherine Rea, Ottawa-Carleton Catholic School Board

Anne Delahunt, Ottawa-Carleton Catholic School Board

Eastern Ontario Catholic Curriculum Cooperative

Institute for Catholic Education

Preface

Special Note

In Units 1 and 2 additions have been made to the activities to reflect the curriculum expectations for the Number Sense and Algebra Strand.

Please note that Appendix B and C of Units 1 and 2 are referenced in Unit 3.

These insertions to Units 1 and 2 of the Grade 9 Applied Mathematics profile incorporate the following expectations:

Overall Expectations: NAV.01, NAV.02, NAV.03, NAV.04.

Specific Expectations: NA1.02, NA1.03, NA2.05, NA2.06, NA3.01, NA3.02.

Unit 1, Activity 2 “Mathematical Marathon”

Overall Expectations: NAV.02.

Specific Expectations: NA2.05, NA2.06.

Teaching /Learning Strategies

· Students now consolidate and enhance their understanding of the three basic exponent rules by completing assignments from the textbook. Include questions with the exponent rule for the power of a power.

· This would also be a good time to enter and interpret exponential notation on a scientific calculator, since some distances will be quite large. Again, use textbook assignments to involve applications with very small numbers.

Unit 1, Activity 3 “Exploring Motion”

Specific Expectations: NA1.02.

Teaching/Learning Strategies

· Students can design their walk to create a graph that is a) a straight line with a positive slope; b) a straight line with a negative slope; c) several lines with a combination of positive and negative slopes.

· Have students walk at different speeds and in different directions so that they not only investigate positive and negative slopes, but different ratios as well (refer to “Explorations, Modelling Motions: High School Activities with the CBR™”).

Discuss with students the meaning of positive and negative integers in this context.

Unit 2, Activity 1 “Walking the Line”

Specific Expectations: NA1.03.

Teaching/Learning Strategies

On p. Unit 2-4, add to the box in #3:

This may be an opportunity to consolidate students’ skills in performing operations with rational numbers.

Unit 2, Activity 2 “The Help Line”

Overall Expectations: NAV.01, NAV.02, NAV.03, NAV.04.

Specific Expectations: NA1.02, NA2.06, NA3.01, NA3.02.

Part 4: Possible Extension

Now would be a good time for the teacher to diagnose students’ ability to work with integers and provide remediation as necessary. This could then be extended to lessons on manipulating polynomial expressions, supported by textbook resources. When multiplying and dividing monomials, highlight the exponent rules covered in Activity 1. Include the exponent rule for the power of a power.

Coding of Expectations

Remove the coded expectation indicated below, which is not a part of the Applied course:

NA3.06

- rearrange formulas involving variables in the first degree, with and without substitution, as they arise in topics throughout the course (e.g., analytic geometry, measurement)

Unit 3: Measurement and Geometry

Unit 3A | Unit 3B | Unit 3C

Time: 40 hours

Unit Developer(s): Carolyn Boyer, Arlene Corrigan, Paul Costa, Anne Delahunt, Lori Goodfriend, Dominique Levac, Brian McCudden, Catherine Rea, Len St. Clair, Margaret Sinclair

Development Date: July - September 1999

Unit Description

The unit is divided into 3 sub units.

Unit 3A	Solving Problems Involving Measurement	23.75 hours
Unit 3B	Optimization of Measurement	6.75 hours
Unit 3C	Exploring Geometric Properties of Plane Figures	10 hours

In this unit, skills such as mental mathematics, estimation, approximating, and solving problems are consolidated. Students will solve problems involving the perimeter and area of composite plane figures and the surface area and volume of three-dimensional objects; they will determine the optimal values of various measurements and use dynamic geometry software to make generalizations about geometric relationships. Students will extend their skills with manipulating polynomial expressions to solve first-degree equations.

3A: Solving Problems Involving Measurement

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

Time: 23.75 hours

Unit Description

Students solve problems involving the perimeter and area of composite plane figures and develop formulas for the surface area of prisms and cylinders and for the volume of prisms, cylinders, cones, and spheres. They apply the formulas to solve problems. Within the context of measurement, students solve linear equations, rearrange formulas, and evaluate numerical expressions involving exponents. They consolidate skills of mental mathematics and estimation, demonstrate the effective use of a scientific calculator, and judge the reasonableness of answers to problems.

Strand(s) and Expectations

Ontario Catholic School Graduate Expectations: CGE 2b, CGE 4f, CGE 5a, CGE 5b, CGE 5g, CGE 7j.

Strand(s): Measurement and Geometry, Number Sense and Algebra

Overall Expectations: MGV.02, NAV.01, NAV.03.

Specific Expectations: MG2.01, MG2.02, MG 2.03, MG2.04, MG2.05, NA1.01, NA1.02, NA1.03, NA1.04, NA1.05, NA1.06, NA2.01, NA2.02, NA3.03, NA3.04, NA3.05.

Activity Titles (Time and Sequence)

Activity 1	Perimeter and Area of Composite Plane Figures	225 minutes
Activity 2	Surface Area and Volume of a Prism: The Prisms Around Us	75 minutes
Activity 3	Surface Area and Volume of a Cylinder	150 minutes
Activity 4	Assessment Activity	75 minutes
Activity 5	Volume of a Cone	75 minutes
Activity 6	Volume of Sphere	75 minutes
Activity 7	Review and Problems Assignment; Test	300 minutes
Activity 8	Solving First Degree Equations: A Balancing Act	450 minutes

Unit Planning Notes

The ability to solve multi-step problems becomes more important as students advance in their study of mathematics. This unit provides an opportunity to focus on that skill, within the context of problems involving surface area and volume.

Many expectations of Number Sense and Algebra are also an essential part of the learning. This unit suggests that teachers take advantage of every opportunity to assist students in consolidating their understanding of the effective use of scientific calculators and the use of estimation in judging reasonableness of answers. The integration of percent, ratio, and rate within the problems to be solved provides opportunities for students to consolidate those important numeric skills. When planning lessons in this unit, it is important to keep in mind the mosaic of expectations to be achieved.

The problem-solving assignment included in Activity 7 takes the form of a story and consists of a set of multi-step problems that may require estimation as part of the solution. Hand out the assignment at the beginning of the unit. Encourage students to complete questions as they acquire the knowledge while working through the unit.

In solving problems involving measurement, it is frequently necessary to rearrange formulas and solve equations; students’ skills in solving equations are consolidated and extended in this unit.

Prior Knowledge Required

· perimeter and area of rectangles, triangles, parallelograms, trapezoids, and circles

· experience with solving arithmetic problems, including the importance of communication in problem solving

· understanding of the concepts of percent, ratio, and rate; skills in applying percent, ratio, and rate

· skills and strategies in mental mathematics and estimation

· solution of simple equations (to the level of ax + b = c)

Teaching/Learning Strategies

· Teachers use whole group instruction for things, such as: reviews of formulas from Grade 8; concepts and units of perimeter, area, surface area, volume, capacity; demonstrating the development of formulas; teaching of the skills in using scientific calculators, skills in solving equations; fostering development of good habits in problem solving.

· Students work individually and in pairs to solve problems.

· Students work in groups of three or four in developing some formulas.

Assessment/Evaluation

· periodic small quizzes to check understanding and progress (one opportunity is embedded in the activities; other, smaller quizzes may be inserted, as necessary)

· problem-solving assignments, including problem posing

· pencil and paper test

Accommodations

· Allow students to work at some problems in pairs, to assist in developing initial understandings. Bear in mind that the ability to solve problems as an individual is what is required to meet expectations.

Resources

The main resource is the core program mathematics textbook.

Activity 1: Perimeter and Area of Composite Plane Figures

Time: 225 minutes

Description

Within the context of the perimeter and area of composite figures, students consolidate their mental mathematics and estimation skills, and judge the reasonableness of answers. Students consolidate their skills with using a scientific calculator effectively in working with formulas that involve exponents and rational numbers.

Strand(s) and Expectations

Ontario Catholic School Graduate Expectations: CGE 4f, 5a.

Overall Expectations: NAV.01.

Specific Expectations: NA1.01, NA1.02, NA1.03, NA1.04, NA1.05, NA1.06, NA2.04, NA3.05.

Prior Knowledge Required

· concept of perimeter; units in which perimeter is commonly measured

· concept of area; units in which area is commonly measured

· calculation of perimeter of a figure bounded by straight lines; formula for the circumference of a circle

· formulas for the area of a rectangle, square, triangle, parallelogram, trapezoid, and circle

· determining the length of the side of a right triangle, using the Pythagorean theorem

Planning Notes

This introductory lesson is an opportunity to review students’ prior knowledge, to judge experience with problem solving, and to open the discussion of the expectations involving the effective use of a scientific calculator and judging the reasonableness of answers.

Teaching/Learning Strategies

· In a whole group discussion, review the elements listed under Prior Knowledge Required. Carry out an example involving a composite figure, such as:

Determine the perimeter and area of the figure below:

· Emphasize the importance of communication in problem solving – writing a solution so that someone else can understand the thinking process involved. Remind students about the syntax involved in substituting into a formula. From the above example:

Area of half-circle

= ½πr²

= (½)(π)(6) ²

= 56.5 cm²

· Discuss how to handle fractions in a formula when using a calculator. In the example, most of the calculation can be done mentally left (½ ´ 36), with the result multiplied by the π-key on the calculator. With fractions that would lead to repeating decimals, students should take advantage of the full decimal accuracy available on the calculator by punching the fraction in (e.g., if the fraction ⅔ were involved, students would key in 2 ÷ 3).

· Discuss how to handle exponents on a scientific calculator.

· Discuss order of operations on a scientific calculator.

· Discuss the value π:

· where it comes from (ratio of circumference to diameter for any circle)

· the use of the π symbol in substitution

· the approximate nature of π when used in calculation, and the advantage of the π-key on the calculator over the value 3.14.

· Discuss the rounding of answers to measurement problems – when to round and what type of rounding to use.

· Model the use of estimation to judge the reasonableness of the answer produced by a calculator – estimate the answer before doing the calculation. Discuss estimation by rounding to compatible numbers (e.g., π is about 3), by operating with compatible numbers

(e.g., in estimating the calculation (½)(π)(6) ², it makes sense to square 6 and multiply by ½, then multiply by 3 as the estimation of π), multiplying and dividing numbers ending in zero. Encourage students to make estimation a regular part of their calculation procedure. Model estimation frequently. Encourage students to share the different methods by which they carry out a particular estimation.

· Discuss when and when not to use a calculator. Some calculations can be done much more quickly mentally, with the results incorporated into a larger calculation (e.g., as in (½)(π)(6) ² in the example above).

· Introduce the assignment found on Student Worksheet: Application of Area and Perimeter within a Scale Diagram (two pages). Review the calculations involved in interpreting a scale diagram. Emphasize that dimensions should be converted before any area or perimeter calculations are made, not after. (Why? In area, if the calculation is done using the measurement directly from the scale diagram, then the resulting area must be multiplied by the square of the scale factor. Most students have difficulty in understanding this.)

· Pose and solve a problem based on the scale diagram. Model the form of solution expected.

· Monitor student progress closely, while they are working on the activity. Periodically, check their measurements from the scale diagram and their conversions to actual dimensions. Check student solutions to word problems for form and correct calculations. Sit with each student and “pseudo-mark” one solution using the five-criterion marking scheme given on page two of the worksheet.

Assessment/Evaluation

· Assess the word problems in the assignment, using the five-criterion marking scheme given on the worksheet (page 2)

Accommodations

Extension: Have students create a scale diagram of an area of their choice and pose and solve three perimeter/area problems based on it.

Resources

Use the core student textbook for additional practice in calculating perimeter and area, as necessary.

Student Worksheet: Application of Area and Perimeter Within a Scale Diagram
(page 1 of 2)

The diagram on the following page is drawn to a scale of 1 cm represents 1.5 m. The diagram represents the landscaping around a house. In the diagram, anything that is not shaded in is grass, except the area around the pool shaped like the diagram below. This area is cement. Use the diagram and the scale to answer the questions below.

As you are doing questions 1-7, use the scale to determine the actual dimensions of the objects. Use the actual dimensions in your calculations.

1. What percent of the lot is covered by the house?

2. a) Determine the length of the fence that surrounds the backyard and the pool area.

b) The owner plans to replace the fence this year. The cost will be $15 per fence post and $3 per m of fencing needed. Assume that there will be a fence post at every corner and that the posts are placed approximately every 2 m. Determine the cost of the fence.

3. a) Determine the total area of all the gardens.

b) To fertilize the gardens, the owner mixes a powdered fertilizer with water and then sprays it on. The directions require that 25 mL of the fertilizer be mixed with 4 L of water. This will then cover 10 m² of garden. How much of the powdered fertilizer will be needed for one application on all the gardens?

4. The owner plans to put a decorative fence around the circular garden at the side of the house. Determine the length of the fence.

5. Determine the area of the cement surrounding the pool.

6. The walkway at the front of the house is made up of interlocking bricks. Each brick covers an area of 300 cm² and costs $1.25. Determine the value of the brick on the walkway.

7. The owner estimates that it takes 90 minutes to cut all the grass on the lot. Determine the rate of grass cutting in m² per minute.

ASSESSMENT

Each question is assessed on a five-point basis:

· A genuine effort has been made to answer the question.

· The actual dimensions used are accurate.

· The method used to solve the problem is correct.

· The calculations are correct.

· Good form is used in the solution.

[35] TOTAL MARKS

Student Worksheet: Application of Area and Perimeter Within a Scale Diagram
(page 2 of 2)

Activity 2: Surface Area and Volume of a Prism: The Prisms Around Us

Time: 75 minutes

Description

In this activity, students generalize their knowledge of the surface area and volume of rectangular and triangular prisms to include the surface area and volume of any prism. They discuss prisms in the environment as models.

Strand(s) and Expectations

Ontario Catholic School Graduate Expectations: CGE 3c.

Overall Expectations: MGV .02, NAV.01.

Specific Expectations: MG2.02, MG2.03, MG2.04, 2.05.

Prior Knowledge Required

· characteristics of rectangular and triangular prisms and the similarity between them

· concepts and units of measurement for surface area and volume

· surface area and volume of rectangular and triangular prisms

Planning Notes

Have the following available for demonstration purposes:

· objects in the shape of a rectangular prism, a triangular prism, some other prism

· a series of congruent rectangles constructed from interlocking blocks, to demonstrate the formula Volume = Area of base x height

During this lesson, continue to model and emphasize the following embedded learnings:

· using a scientific calculator effectively, including:

· knowing when and when not to use it

· how to handle fractions and exponents

· considerations of order of operations

· using rounding appropriately in solutions to problems

· using mental mathematics and estimation to judge the reasonableness of answers produced by a calculator

· judging reasonableness of answers in the context of a problem

· observing correct form in communicating the solution to a problem

Teaching/Learning Strategies

· In a whole group presentation, use a model to elicit from students the characteristics of rectangular and triangular prisms. Discuss what is in common in their characteristics to identify a general definition for a prism (faces are rectangles, top and bottom are congruent, parallel polygons). Describe other possible prisms and where students may have seen them in the environment around them. Have models available for demonstration (cereal boxes, candy bar boxes, any other unusual packaging in the shape of a prism)

· Review the meaning of surface area and the units in which it is typically measured. Ask students to suggest examples of areas for which each unit would be used. Elicit from students a method for calculating the surface area of any prism (sum of the areas of all its faces).

· Introduce and explain the term lateral surface area (e.g., the sum of the areas of all the side faces of a prism) and ask students to describe situations in which the lateral surface area would be needed instead of the total surface area.

Do a sample problem involving the calculation of the surface area of a prism.

· Review the meaning of the volume and the units in which it is measured. Ask students to suggest examples of objects for which each unit would be used to describe the volume. Elicit from students the formula for calculating the volume of a rectangular prism (V=lwh) and a triangular prism (V = Area of base ´ height) and an explanation of their origin. Be prepared to model using interlocking blocks, if necessary. (Have several rectangles built, each having the same area. Stack them one on top of another. Since the layers are identical, the volume is the Area of the base ´ Height.)

· Discuss the relationship between capacity and volume and identify units of capacity. Ask students to identify quantities that are measured in units of capacity instead of units of volume.

· Identify the relationship between units of volume and units of capacity, (e.g., 1 mL of water occupies 1 cm³ of space.) Extrapolate this relationship to determine the number of litres in 1 m³ of space (1 kL = 1000 L). Ask students to identify something in their surroundings, at school, or at home that would hold 1 kL of water.

· Do a sample problem involving the volume of a prism that is neither rectangular nor triangular. Include a reference to capacity. For example:

A water trough is in the shape of a trapezoidal prism. Its base has internal side lengths of 85 cm and 60 cm and an internal height of 50 cm. The total internal length of the trough is 1.2 m.

a) What is the capacity of the trough?

b) If the trough is filled to 45% of its capacity, how many litres of water does it contain?

· Select a homework assignment from the student textbook that involves determining the surface area and volume of prisms. Include a problem that integrates ratio, rate, or percent.

Accommodations

· Present multi-step problems in parts, as necessary to build the problem-solving skills of some students.

Activity 3: Surface Area and Volume of a Cylinder

Time: 150 minutes

Description

Students apply their knowledge of the surface area and volume of rectangular prisms to develop formulas for the surface area and volume of cylinders.

Strand(s) and Expectations

Ontario Catholic School Graduate Expectations: CGE 3c.

Overall Expectations: MGV.02.

Specific Expectations: MG2.02, MG2.03, MG2.04, MG2.05.

Prior Knowledge Required

· surface area and volume of rectangular and triangular prisms and the origin of their formulas

Planning Notes

During this lesson, continue to model and emphasize the following embedded learnings:

· using a scientific calculator effectively, including:

· knowing when and when not to use it

· how to handle fractions and exponents

· considerations of order of operations

· using rounding appropriately in solutions to problems

· using mental mathematics and estimation to judge the reasonableness of answers produced by a calculator

· judging reasonableness of answers in the context of a problem

· observing correct form in communicating the solution to a problem

Teaching/Learning Strategies

Volume of a Cylinder

· Compare the structure of a cylinder to that of a rectangular prism, noting similarities (e.g., in both, the top and bottom are congruent, parallel faces; in both the “sides” are perpendicular to the base). Note differences (e.g., the “sides” of a rectangular prism are rectangles; a cylinder has only one continuous side).

· Use the similarity between rectangular prisms and cylinders to suggest a method for determining the volume of a cylinder: Volume = Area of Base ´ Height. As a model, use a cylindrical package of cookies to illustrate further. Complete the process by substituting the formula for the area of the base, which is a circle.

So, Volume of a cylinder = πr²h.

· Do sample problems involving calculation of the volume of a cylinder. Include a composite figure and a word problem.

· Select a homework assignment from the student textbook that involves determining the volume of cylinders. Include:

· a problem that integrates ratio, rate, or percent;

· problems that involve compositions of cylinders and prisms;

· a problem in which the volume is known and one dimension must be found.

Surface Area of a Cylinder

· To develop the formula for the surface area of a cylinder, ask each student to roll a piece of paper into a tube. Then identify the shapes that make up the tube. The circle for top and bottom are obvious – but what shape is the side? It came from the piece of paper, so it must be a rectangle. Ask students to determine the height of the rectangle (same as the height of the tube). What about the width of the rectangle? (Ask students to draw a line around the circumference of the tube. Then open the tube. The circumference line has become the width of the rectangle.) Additional models might include the labels on soup or fruit cans, which are easily removed.

So the rectangle has width h and length 2πr. What is its area?

Area of rectangle = lw Lateral surface area of cylinder = 2πrh

and total surface area of cylinder = 2πrh + 2πr²

· Do sample problems involving calculation of the surface area of a cylinder, including a composite object and a word problem.

· Select a homework assignment from the student textbook that involves determining the surface area of cylinders. Include:

· a problem that integrates ratio, rate, or percent;

· problems that involve compositions of cylinders and prisms;

· a problem in which the lateral surface area is known and one dimension must be found

Activity 4: Assessment Activity

Time: 75 minutes

Description

The following assessment is designed in two parts, a pencil and paper assessment and a problem-posing assignment.

Strand(s) and Expectations

Ontario Catholic School Graduate Expectations: CGE 2b, CGE4f, CGE5G.

Overall Expectations: MGV.02, NAV.01.

Specific Expectations: MG2.01, MG2.02, MG 2.03, MG2.04, MG2.05, NA1.01, NA1.03, NA1.04, NA1.05, NA1.06, NA2.01, NA2.02.

Teaching/Learning Strategies

· Create a pencil and paper quiz on perimeter and area of plane figures; surface area and volume of prisms and cylinders.

· Upon completion of the quiz, instruct students to work on the following assignment, finishing it for homework:

Pose and solve two problems, one involving surface area and one involving volume/capacity. Involve percent, rate, or ratio in at least one of them.

· Each problem is to be marked out of 7, based on the following set of criteria:

· The problem requires a multi-step solution.

· The problem involves an interesting application.

· The problem involves realistic measurements.

· The problem is worded clearly.

· The final answer is correct.

· The solution is presented in correct form, including use of English, proper formulas, and correct units.

· Rounding is used correctly in the solution.

Activity 5: Volume of a Cone

Time: 75 minutes

Description

Students develop and apply the formula for the volume of a cone.

Strand(s) and Expectations

Ontario Catholic School Graduate Expectations: CGE 3c.

Strand(s): Measurement and Geometry

Overall Expectations: MGV.02.

Specific Expectations: MG2.02, MG2.03, MG2.04, MG2.05.

Prior Knowledge Required

· formula for the volume of a cylinder

Planning Notes

Obtain a volume set. This is a commercially available resource that contains a plastic model of a rectangular prism, a pyramid, a cylinder, a cone, and a sphere. The models have compatible dimensions (i.e., the same base and height); they are hollow, so that they can fit within one another. If a volume set is not available, make models from Bristol board or stiff cardstock that holds its shape. You need a cone and a cylinder, having the same base and height.

Make available material with which to fill the models (e.g., rice, sand, small plastic pellets), in a quantity sufficient to fill the cylinder.

During this lesson, continue to model and emphasize the following embedded learnings:

· using a scientific calculator effectively, including:

· knowing when and when not to use it

· how to handle fractions and exponents

· considerations of order of operations

· using rounding appropriately in solutions to problems

· using mental mathematics and estimation to judge the reasonableness of answers produced by a calculator

· judging reasonableness of answers in the context of a problem

· observing correct form in communicating the solution to a problem

Teaching/Learning Strategies

· In a whole group presentation using the models and diagrams, illustrate the features and dimensions of a cone.

· Using the models of the cylinder and cone, demonstrate that these prisms have the same base and height. (With the volume set, the cone fits exactly inside the cylinder.)

Ask students how they think the volumes would compare (i.e., Would there be a relationship between the volumes?) Students will likely guess that the volume of the cylinder is somewhere between two and four times the volume of the cone.

Test the relationship by filling the cone with the material chosen and pouring into the prism. Count the number of times that this can be done (3). You might have a student do the demonstration.

The conclusion reached is that the volume of a cone is one-third the volume of a cylinder having the same base and height. The formula for the volume of cone, then, is:

V = ⅓πr²h, where r is the radius of the base and h is the interior height of the cone.

· Do sample problems involving the volumes of cones. Include:

· composite figures that involve, not only cones, but also rectangular prisms and cylinders;

· application problems;

· Select a homework assignment from the student textbook that involves volumes of cones.

· a problem that integrates ratio, rate, or percent;

· problems that involve compositions of cylinders and prisms;

· a problem in which the volume is known and one dimension must be found

Resources

Volume set (available from commercial sources of mathematics resources and materials)

Activity 6: Volume of a Sphere

Time: 75 minutes

Description

Students develop and apply the formulas for the volume of a sphere.

Strand(s) and Expectations

Ontario Catholic School Graduate Expectations: CGE 3c.

Strand(s): Measurement and Geometry

Overall Expectations: MGV.02.

Specific Expectations: MG2.02, MG2.03, MG2.04, MG2.05.

Prior Knowledge Required

· formula for the area of a circle

· Archimedes Principle (When an object is submerged in water, it displaces an amount of water equal to its own volume.)

Planning Notes

The development of the formula for volume of a sphere involves a demonstration of water displacement. A submersible sphere, such as a baseball or a billiard ball, is needed, along with a clear, graduated container and sufficient coloured water for submersion of the spherical object.

During this lesson, continue to model and emphasize the following embedded learnings:

· using a scientific calculator effectively, including:

· knowing when and when not to use it

· how to handle fractions and exponents

· considerations of order of operations

· using rounding appropriately in solutions to problems

· using mental mathematics and estimation to judge the reasonableness of answers produced by a calculator

· judging reasonableness of answers in the context of a problem

· observing correct form in communicating the solution to a problem

Teaching/Learning Strategies

The Volume of a Sphere

· Begin with a discussion of what students think the formula for the volume of a sphere should look like. Since a sphere is circular, π is likely to be involved. Since the only dimension on a sphere is its radius, r must be involved also. Since volume is what we want, thinking in three dimensions must be involved. A likely guess at the formula is that it would involve πr³. The displacement activity that follows is intended to determine by what factor πr³ would be multiplied.

· The activity can be done as a teacher demo or as a student group activity. The advantage of having students carrying out the activity is that data is gathered for a variety of spherical objects; in the teacher demonstration, a result is drawn from the data for only one object. The teacher demonstration, however, is more time-efficient.

· The activity:

· Estimate as accurately as possible the radius of a spherical object that will not float, such as a baseball or a billiard ball. Calculate the value πr³.

· Fill a graduated beaker with coloured water to a level that would allow the complete submersion of the spherical object. Remind students of Archimedes principle, that is, that the amount of water displaced by the sphere will be equal to the volume of the sphere. Record the starting level of the water. Submerge the ball and record the resulting level. Subtract the two water level figures. The result is the volume of the sphere.

· Now, compare the volume estimate by displacement to the calculated value of πr³, by ratio. The result should be around 1.3. The formula, in fact, is

V = πr³.

· Do sample problems involving the volume of spheres. Include a composite figure that may combine spheres or hemispheres with other objects. Also include an application problem.

· Select a homework assignment from the student textbook. Include problems involving composite figures and problems involving applications.

Activity 7: Review and Problems Assignment; Test

Time: 300 minutes

Description

Students prepare for a pencil and paper test of applications of the formulas for the perimeter and area of composite plane figures, the surface area of prisms and cylinders, and the volume of prisms, cylinders, cones, and spheres.

Strand(s) and Expectations

Ontario Catholic School Graduate Expectations: CGE 2b, CGE 4f, CGE 5a, CGE 5b, CGE 5g, CGE 7j.

Strand(s): Measurement and Geometry, Number Sense and Algebra

Overall Expectations: MGV.02, NAV.01.

Specific Expectations: MG2.01, MG2.02, MG 2.03, MG2.04, MG2.05, NA1.01, NA1.02, NA1.03, NA1.04, NA1.05, NA1.06, NA2.01, NA2.02, NA3.04, NA3.05.

Teaching/Learning Strategies

· Create a review lesson as appropriate to the students in the classroom. Assign appropriate questions from the textbook as review.

· A problems assignment is included as a student worksheet under the headings Student Activity – The Question Game and Student Activity – Answering the Questions! Hand this assignment out at the beginning of the unit and encourage students to work at it during the unit. The assignment involves some questions that require students to estimate key information, along the line of Fermi problems.

· Create a pencil and paper test reflecting the work of the unit.

Accommodations

· Assist students as necessary in breaking complex questions into steps.

Student Activity – The Question Game

In the story below, a number of questions are posed by two boys as they shop in a grocery store. Your job is to answer the questions. The numbers throughout the story refer to those on the Answering the Questions! worksheet, where you will find more detailed statements of the problems.

John is a 17-year-old high school student who volunteered to care for Al, the 7-year-old son of a neighbour, every Saturday and to do the family shopping. On one particular Saturday, the two boys set off for the local mall, shopping list in hand.

Al was particularly inquisitive and seemed able to find mathematics at every turn. As he and John entered the grocery store, they walked past a row of shopping carts. John pulled one out of a row. “Hey, John!” said Al, “Have you ever wondered what total volume of groceries is wheeled out of here in shopping carts on a typical Saturday?” John answered: “That thought hasn’t crossed my mind, but it’s certainly an intriguing one.” [1]

The boys walked on in silence, and eventually passed the deli counter. Al spied a counter full of cheese cut into triangular wedges and wrapped in foil. “Hey, John! I’ve got another question for you,” he said. “How many rolls of tin foil do you think it would take to wrap all the cheese wedges in that display?” John pondered momentarily, then answered, “Hmmmm, I’ll have to think about that one!” [2]

Continuing their shopping, Al and John entered the aisle where the soup was kept – chicken noodle was on the list. “I feel another question coming on,” chirped Al. “Oh, great,” responded John, “Let’s hear it.” Al proceeded, “I was just wondering – if you cut off the labels from all the cans of soup in this counter and laid them out on the floor, would they cover the entire aisle?” [3(a)] John responded, “That’s a great question. And I have one too: If you emptied the soup from all those cans, how many bathtubs would it fill?” [3(b)] Al laughed happily, “Now you’re getting into the game!”

And so the rest of the shopping trip went – Al and John taking turns posing grocery store problems. In the ice cream aisle, John reached for a package of waffle cones and Al asked, “I wonder if a 4-litre carton of ice cream would fill all these cones?” [4(a) (b)]

In the fruit department, John stood looking at a display of oranges. They were all quite round, and about the same size. The oranges were stacked in a pile with a 5x5 square on the bottom row, 4x4 square on top of that, then a 3x3 square, a 2x2 square, and, at the top of the display, there was one orange. Al exclaimed, “I’ve got a great question! Suppose you had to build a cardboard box to hold all those oranges so that they would just fit into the box. What would its dimensions be?” [5(a)] John responded, “Great! – and how much empty space would be inside the box?” [5(b)]

The boys went through the check-out, paid for the groceries, and started homeward. “I kind of like this question game,” said Al. “Me, too,” answered John. “But you know, we didn’t answer any of those questions, we only asked them. I wonder what we need to know in order to find the answers?”

Student Activity – Answering the Questions!

Communication is important in your solution to these problems. Be sure to use good form and precise language to identify the steps you are taking, and justify all estimates used.

The solution to each question will be marked out of 5 according to the criteria below:

· Correct procedures and formulas are used in calculating measurements.

· Estimates are reasonable and are justified or explained.

· The overall method used for solving the problem would lead to a correct solution.

· The final answer is correct, based on the estimates used.

· Good form is used in communicating the solution, including correct use of language, proper substitution, and correct units.

1. The buggy of a shopping cart is a trapezoidal prism. The trapezoids on the ends have a height of 50 cm and base lengths of 110 cm and 80 cm. The buggies are 60 cm wide. A grocery store has 200 carts and they are used repeatedly on a Saturday. Estimate the total volume of groceries wheeled out of the grocery store on a typical Saturday.