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Course Profile
Principles of Mathematics, Grade 9 academic, 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.
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Acknowledgments
Catholic District School Board Writing Team – Mathematics - Academic
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 Academic Mathematics profile incorporate the following expectations:
Overall Expectations: NAV.01, NAV.02, NAV.03, NAV.04.
Specific Expectations: NA1.01, NA2.05, NA2.06, NA3.01, NA3.02, NA3.03.
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: NA1.01.
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: NA1.01, NA2.06, NA3.01, NA3.02, NA3.03.
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.
Unit 3: Measurement and Geometry
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 three sub-units.
|
Unit 3A |
Solving Problems Involving Measurement |
20 hours |
|
Unit 3B |
Optimization of Measurement |
10 hours |
|
Unit 3C |
Exploring Geometric Properties of Plane Figures |
10 hours |
In this unit, skills such as mental mathematics, estimation, approximating, and solving multi-step problems are consolidated. Students extend their skills with manipulating polynomial expressions to solve first-degree equations and then apply these to measurement problems. Students determine the optimal values of various measurements, solve problems involving surface area and volume of three-dimensional objects, and use dynamic geometry software to make generalizations about geometric relationships.
Unit 3A: Solving Problems Involving Measurement
Activity 1 | Activity 2
| Activity 3 | Activity 4 | Activity
5 | Activity 6 | Activity 7 | Activity 8 | Activity 9
Time: 20 hours
Unit Description
Students develop formulas for the surface area and volume of prisms, pyramids, cones, cylinders, and spheres. They apply the formulas to solve multi-step 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, NA1.01, NA1.02, NA1.03, NA1.04, NA1.05, NA1.06, NA2.01, NA2.02, NA3.04, NA3.05, NA3.06.
Activity Titles (Time and Sequence)
|
Activity 1 |
Measurement and Geometry: Get Ready! |
75 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 pyramid and cone |
75 minutes |
|
Activity 6 |
Surface area of pyramid and cone |
150 minutes |
|
Activity 7 |
Surface area and volume of sphere |
150 minutes |
|
Activity 8 |
Review, problems assignment, paper/pencil test |
150 minutes |
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Activity 9 |
Solving First Degree Equations: A Balancing Act |
300 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 for students 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 exact versus approximate values, effective use of scientific calculators, and 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 8 takes the form of a story and consists of a set of multi-step problems that may require estimation as part of the solution. The teacher should hand out the assignment at the beginning of the unit and 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 multi-step arithmetic problems, including the importance of communication
· 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
· 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 development of formulas; teaching of 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
Resources
The main resource is the core program mathematics textbook.
Accommodations
This unit primarily involves the solution of multi-step problems and related calculations. Communication, meaning the form of solutions, is an important consideration.
The following accommodations might be considered:
· Have supportive materials available for students who demonstrate a lack of the required prior knowledge during the first activity.
· Allow students to work at some problems in pairs, to assist in developing initial understandings. Students must be able to solve multi-step problems independently to meet the curriculum expectations.
Activity 1: Measurement and Geometry: Get Ready!
Time: 75 minutes
Description
Students consolidate their mental mathematics and estimation skills, and judge the reasonableness of answers within the context of the perimeter and area of composite figures. Students consolidate their skills with using a scientific calculator effectively in working with formulas that involve exponents, and rational and irrational 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.
Planning Notes
This introductory lesson is an opportunity to recap students’ prior knowledge, to judge experience with multi-step problem solving, and to open the discussion of the expectations involving exact value, use of a scientific calculator, and judging reasonableness of answers.
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
Teaching/Learning Strategies
· In a whole group discussion, quickly 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 (½ x 36), with the result multiplied by the π-button 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 punch 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)
· its nature as an irrational number (Rational numbers have been discussed in Units 1 and 2. This would be a good opportunity to explain irrational numbers by contrast to rational.)
· the use of the π symbol in substitution
· the approximate nature of π when used in calculation, and the advantage of the π -button 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., B 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 students to an example of a word problem that involves an application of area or perimeter. Focus on expected form in the solution, the importance of drawing a diagram, and variation in rounding rules in relation to a context (e.g., dollars and cents). Discuss and model judging the reasonableness of an answer in relation to the context of the problem.
Example:
A circular garden has a radius of 3.2 m. It is surrounded by a pathway of width 0.8 m.
a) To fertilize the garden, 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 covers 10 m2 of garden. How much of the powdered fertilizer will be needed for one application on the garden?
b) The pathway is made up of interlocking bricks. Each brick covers an area of 300 cm2 and costs $1.25. Determine the value of the bricks on the pathway.
c) What percent does the area of the pathway represent of the total area of garden and pathway combined?
· Select a homework assignment from the student text that involves determining the perimeter and/or area of composite figures, and solving problems involving applications of perimeter and area. Supplement as necessary with problems that integrate ratio, rate, and percent. Include problems in which the perimeter or area is known and a dimension must be found. Use these for the purpose of discussing rearrangement of formulas by substitution. (This continues throughout the unit.)
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 their environment as models.
Strand(s) and Expectations
Ontario Catholic School Graduate Expectations: CGE 3c.
Overall Expectations: MGV .02, NAV.01.
Specific Expectations: MG2.01, MG2.02, MG2.03, MG2.04.
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, the “big box” approach in the design of shopping malls, 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 x 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 x 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. 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/or volume of prisms. Supplement as necessary with problems that integrate ratio, rate, and percent.
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, NAV.01.
Specific Expectations: MG2.02, MG2.03, MG2.04, MG2.05.
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
Prior Knowledge Required
· surface area and volume of rectangular and triangular prisms and the origin of their formulas (Measurement, Grade 8)
Teaching/Learning Strategies
· 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 x 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 one or two 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. Supplement as necessary with problems that integrate ratio, rate, and percent. Include problems that involve compositions of cylinders and prisms.
· 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 is the length of the rectangle? (the circumference of the base). Additional models might include the labels on soup or fruit cans, which are easily removed.
The rectangle has width h and length 2πr. What is its area?
Putting the pieces together, the formula for total surface area is:
TSA of a cylinder = 2πrh + 2πr2
The formula for lateral surface area will be: LSA of a cylinder = 2πrh
· Do one or two sample problems involving calculation of the surface area of a cylinder. Include a composite figure and a word problem.
· Select a homework assignment from the student textbook that involves determining the surface area of cylinders. Supplement as necessary with problems that integrate ratio, rate, and percent. Include problems in which the volume or lateral surface area of a cylinder 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/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, MG2.03, MG2.04, NA1.01, NA1.02, NA1.03, NA1.04, NA1.05, NA1.06, NA2.01, NA2.02, NA3.04, NA3.05, NA3.06.
Planning Notes
· Create a pencil/paper quiz on perimeter and area of plane figures; surface area and volume of prisms and cylinders.
Teaching/Learning Strategies
· Have students complete the pencil-paper quiz and then begin 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 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 Pyramids and Cones
Time: 75 minutes
Description
Students develop and apply the formulas for the volume of a square-based pyramid and 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.
Prior Knowledge Required
· formulas for the volume of a rectangular prism and 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 cardstock that will hold its shape. You need a rectangular prism and a pyramid, having the same base and height; 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 rectangular prism and 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 and a pyramid.
· Using the models of the rectangular prism and the square-based pyramid, demonstrate that these prisms have the same base and height. (With the volume set, the pyramid fits inside the rectangular prism.)
· 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 rectangular prism is somewhere between two and four times the volume of the pyramid.
Test the relationship by filling the pyramid 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 square-based pyramid is one-third the volume of a rectangular prism having the same base and height. A formula might be written as:
V = ⅓b2h, where b is the side length of the base and h is the interior height of the pyramid.
· Repeat the activity, this time using the cone and the cylinder that have the same base and height. A similar conclusion is reached: the volume of a cone is one-third the volume of a cylinder having the same base and height. Then, a formula might be written as:
V = ⅓πr2h, where r is the radius of the base and h is the interior height of the cone.
· Do sample problems involving the volume of cones and square-based pyramids. Include composite figures that involve, not only pyramids and cones, but also rectangular prisms and cylinders. Also include application problems.
· Select a homework assignment from the student textbook that are problems involving volume of pyramids and cones. Include problems in which the volume is known, and one dimension must be solved for.
Activity 6: Surface Area of a Pyramid and a Cone
Time: 150 minutes
Description
Students develop and apply the formulas for the volume of a square-based pyramid and 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: MG2.01, MG2.02, MG2.03, MG2.04.
Prior Knowledge Required
· formulas for the circumference of a circle and the area of a rectangle, a triangle, and a circle
· solution of problems involving proportion
· exponent rule for division
Planning Notes
Have on hand blank paper, tape, scissors, rulers, compasses, and protractors for students to use in building models of square-based pyramids and prisms.
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 Square-based Pyramid
· Have students work in pairs. Each pair is to build a model of a square-based pyramid, having a base length of 6 cm and a slant height of 5 cm. They determine the interior height of the pyramid, both by measurement and by calculation.
· Ask students to report the interior heights of their pyramids. Discuss the actual calculation of the interior height, using the Pythagorean theorem.
Use a diagram of a pyramid showing right triangle relationship among semi-base, height, and slant height.
From the diagram:
s2 = h2 + (
)2
· Ask students to describe the way in which they would determine the total surface area and the lateral surface area of a square-based pyramid:
Total surface area = area of square based + 4(area of one triangular face)
Lateral surface area = 4(area of one triangular face)
· Do sample problems involving the total and lateral surface area of square-based pyramids. Include a composite figure that may involve, not only a square-based pyramid, but also rectangular prisms or cylinders. Also include an application problem. Give the pyramid height in one problem and the slant height in the other.
· Select a homework assignment from the student textbook. Include problems involving composite figures and problems involving applications. Also include problems in which the lateral surface area is known, and one dimension must be solved for.
The Cone
· The development of the formula for the surface area of a cone is similar to that of the square-based pyramid, but is complicated by the continuous lateral surface of the cone. Begin by comparing a cone and a square-based pyramid having the same height, and for which the diameter of the cone is equal to the base length of the pyramid. Identify the fact that a similar Pythagorean relationship will exist in the cone to that of the pyramid.
