Unit 1 Grade 9 Applied

Measurement: 2-D and 3-D

BIG PICTURE Students will: describe relationships between measured quantities; solve problems involving the areas and perimeters of composite 2-D shapes; develop volume formulas for pyramids cones and spheres; apply knowledge and understanding of 3-D formulas to solve simple problems in

context.

Math Learning Goals Expectations

Develop through investigation the formulas for volume of a pyramid, and a cone based on the volume of the corresponding prism or cylinder of the same radius and height.

Develop through investigation the formula for volume of a sphere based on the volume of a cylinder/cone.

Consolidate volumes of prisms, pyramids, cylinders, cones and spheres.

Solve problems involving combinations of the figures using metric and imperial measure.

Activate students’ prior knowledge of terminology related to identifying geometry shapes.

Determine students’ readiness to identify geometric figures in composition, and use appropriate calculations for perimeter and area.

Find area and perimeter of composite shapes. Solve composite area problems, e.g., logos,

business signs, irregularly shaped gardens. Connect the geometric representation of the

Pythagorean theorem to the algebraic representation a2 + b2 = c2.

Relate their understanding of inverse operations to squares and square roots.

Solve 2-D and 3-D problems using the Pythagorean theorem.

Substitute into and evaluate algebraic expressions involving exponents.

MG2.01 MG2.02 MG2.03 MG2.04 MG2.05 NA2.01 NA2.02 NA2.03 NA2.04 NA2.08

1.1: Investigation – Comparing Volumes

Purpose Compare volumes of shapes that have the same base and height.

Hypothesis I think that...

1.

× _________ =

2. × _________ =

3. × _________ =

Investigate How many times will the volume of the shape on the left fill the shape on the right? 1. Vcone × ___ = Vcylinder 2. Vsquare pyramid × _____ = Vsquare prism 3. Vtriangular pyramid × _____ = Vtriangular prism

Conclusion

1.2: Pair Share – Volume Partner A answers question A, partner B coaches.

Partner B answers question B, partner A coaches.

A – Prisms Circle the shapes that are prisms.

Volume of a prism = area of __________ __________.

B – Pyramids Circle the shapes that are pyramids.

Volume of a pyramid = volume of a prism __________.

8 mm 15 mm

1.3: Pair Share – Volume Calculate the volume of the following figures. Show your work. A

B

Calculate the volume of the following figures. Show your work.

A B

5 cm

12 cm

1.4: Volume of Pyramids and Cones - Worksheet

1. Find the volume of the following: a) b) 2. Find the volume given the following:

a) a cone with a diameter of b) a pyramid with a base area 4 cm and height of 15 cm. of 72 cm2 and height 10 cm. 3. BAAAAH petting zoo allows visitors to feed animals grain from paper

cones. a) How much grain can fit into a cone with a diameter of 7 cm and

height of 8.5 cm? b) Smoots the goat usually eats around 600 cm3 of grain for his

afternoon snack. How many times must you fill the cone in order to give Smoots his snack?

4 m

3.5 mm

10 cm

8 cm

9 cm

4 cm

4.5 cm

4. Fred’s Fries come in a cylindrical container for $4.00, while Chuck’s Chips come in a cone-shaped container for $2.00.

Who offers the best deal and what assumptions must you make? Fred’s Chuck’s 5. a) Calculate the capacity of the given coffee scoop. b) Which do you think would provide a greater increase in capacity:

doubling the radius, or doubling the height? Draw each of these new scoops and label dimensions.

c) Calculate the capacity of each new scoop in part b), and compare

to your predictions in part b).

1.5: Frayer Model of a Sphere

Definition

Formula

Examples

Non-Examples

Sphere

Frayer Model of a Sphere (solution)

Definition

Sphere – A 3D shape where every point on the surface is the same distance from the center.

Formula

Notes V represents the Volume r represents the radius

Examples

- Baseball - Basketball - Planets (Earth) - Dodgeball - Orange - Eyeball - Bubble

Non-Example

- Circle - Pizza - Egg - Drum - Frisbee - Stop sign - Football

Sphere

Developing the Formula for the Volume of a Sphere

VolumeSphere.ppt. (Presentation software file)

1 2

3 4

5 6

7

1.6: Volume of a Sphere

Volume of a Sphere Formula:

V r4

3

3

Examples: Find the volume of:

a)

r = 2 cm

b) a sphere with a diameter of 15 cm.

1.7: Volume of a Sphere - Worksheet

1. Find the volume of the following spheres: a) b) 2. A Whopper is a spherical shaped chocolate malt candy with a

diameter of 1.75 cm. a) Determine the volume of one Whopper. b) How much malt candy will you consume if you eat a package of

30 Whoppers? 3. a) A sphere has a radius of 4 cm. Calculate its volume. b) A second sphere has a radius double the radius of the first

sphere. Predict how the volumes of the two spheres are related. c) Calculate the volume of the second sphere. Compare this value

to your prediction, and explain any differences.

r = 8 m d = 12.2 cm

4. The diameter of the Earth is approximately 6400 km. Determine the volume of the Earth, in cubic kilometers.

5. The perfect ice cream treat involves a completely filled cone with two

perfectly spherical ice cream scoops loaded up on top of the cone. (see sketch)

a) Find the volume of ice cream in the perfect ice cream treat.

b) If the ice cream is being scooped from a cylindrical container with a height of 14 cm and radius of 5.3 cm, find out how many perfect ice cream treats can be served from this container.

6 cm

12 cm

15 cm

3 cm

8 cm

1.8: Applications of Volume Assignment

Name:________________ 1. Calculate the volume of the perfume bottle. 2. How much soup can this container hold? 3. Calculate the volume of the rectangular prism. 4. Which popcorn container will hold more? Prove your answer by calculating the volume of

each figure

10 cm

7 cm

6 in. 4 in.

5 in.

15 cm

16 cm

15 cm

8 cm

5. How many times bigger is the second sphere? 6. How much more expensive should the large aquarium be than the small aquarium if the cost

is based on the volume? 7. Determine the volume of the cabin. Show your work.

Hint: Change both measurements to the same units!

30 in 32 in.

16 in.

24 in. 8 in.

12 in.

16 in.

5 mm 2 cm

3 m

2 m

6 m

6 m

1.9: Design Details Name the geometric shapes contained in the following two objects. Use arrows to connect names of shapes to the diagram.

1.10: Frayer Model

Composite Figures

Definition Facts/Characteristics

Examples Non-Examples

1.11: Scale Drawing Details 1. Consider the two composite figures. (a) Identify the geometric shapes in each. Write the names on the diagrams.

(b) Area

For determining the area of the shaded regions, describe the features and calculations that are:

i) the same ii) different (c) Which of the two figures is larger? By how much? Justify your answer using pictures, symbols, and words. (d) Perimeter

Use a coloured pencil to outline the perimeters of the two figures. How do these two perimeters compare?

2. Use the diagram below to calculate the area and perimeter of the triangle.

a) What dimensions are needed to determine the perimeter? b) What dimensions are needed to determine the area? c) Calculate:

Area Perimeter 3. Provide an example in daily life of a figure that involves more than one geometric shape.

1.12: Area Challenge

On the diagram, draw line segments to subdivide the shape into simple shapes, and calculate the area. On the placemat: In your space, list the steps required to determine the area of the figure. As a group, discuss the steps, then provide one complete solution to the problem in the centre of the placemat.

1.13: Composite Figures – Worksheet #1

Name: Date: 1. Find the area of a province.

2. What to charge?

3. Wall Painting...

4. The Big Bridge.

a. Find the area of the cross-section of this bridge.

b. Find the perimeter of the cross-section of this bridge.

12 m

8 m

4 m

6 cm

4 cm

1.14: Composite Figures – Worksheet #2 1. For each composite figure shown, solve for the missing lengths and calculate the

perimeter. a) b) 2. For each composite figure below, calculate the total area by breaking the figure

up into familiar shapes. a) b)

12 m

15 m

b

20 m

34 m

a

8 cm

b

a a

20 cm

a

1.15: What Is This? What shapes are hiding in the drawing below?

1.16: Composite Figures

Composite Figures (Presentation software file) CompositeFigures.ppt.

2

Bill is painting his basement floor.

The dimensions are:

3.2 m

1.8 m

2.3 m

2.5 m

If one can of paint will

cover 3.5 m2, and each

can of paint costs

$15.25, what is his total

cost including 15% GST

and PST?

3

a) Determine the total area.

A = 2.3 x 2.5 + 3.2 x 1.8

A = 5.75 + 5.76

A = 11.5 m2

Atotal = A1 + A2

3.2 m

1.8 m

2.3 m

2.5 m

A2

A1

4

b) If one can of paint covers

3.5 m2, how many cans of

paint will he need?

He will need four cans of paint.

3.2 m

1.8 m

2.3 m

2.5 m11.5 m2

5

c) If one can of paint

costs $15.25, what is his

total cost including 15%

GST and PST?

Paint cost: 4 x $15.25 = $61.00

Taxes: = 0.15 x $61.00 = $ 9.15

Total cost: $70.15

3.2 m

1.8 m

2.3 m

2.5 m11.5 m2

6

Shelly wants to make curtains to

cover her decorative window.

In order to hem the curtains

she will add 10 cm to each

edge.

If 1 m2 of curtain material

cost $2.20, how much will it

cost to make the curtains

including 15% GST and

PST?

2 m

2 m

7

The radius for material is

1.10 metres.

a) Determine area of semi-circle.

A = πr2 ÷ 2

A = π(1.10)2 ÷ 2

A ≈ 3.80 ÷ 2

A ≈ 1.90 m2

10 cm

2 m

2 m

2 m

8

The length of the sides of

material is 2.20 metres.

lwA

)20.2)(20.2(A 2m84.4A

b) Determine area of the square.

Total Area = Semi Circle + Square

= 1.90 + 4.84

= 6.74 m2 or 7m2

2 m

2 m

10 cm2 m

9

Material: $2.20 x 7 = $15.40

Taxes: $15.40 x 0.15 = $ 2.31

Total Cost: $17.71

c) If 1 m2 of material costs $2.20,

determine the total cost of the curtains

including taxes?

2 m

2 m

1.17: Exploring Composite Shapes

Shape Divisions Area Calculations

1.18: Use What You Know (use formula sheet)

Knowledge and Skills

Calculate the area of the given circle. Show your work.

r = 2.5 cm

Hint:

A r2

Reasoning and Proving

Westview School has a track.

You want to run 2 km every day. Determine how many times you have to go around the track. Show your work.

Communicating

Genna wants to tile her bathroom counter with mini tiles. She needs to determine the area of her counter space. Explain with words, diagrams, and symbols how she should determine the area.

Connecting

This figure has a radius of r units.

r

Which of the following formulas could be used to determine the perimeter?

21a) 2 b) 0.75

4

3 1c) (2 ) d) 2

4 4

r r r r

r r r r

Give reasons for your answer.

1.19: Pythagorean Theorem

1 2

3 4

5

The Pythagorean Theorem (Presentation software file) PythagoreanTheorem.ppt.

1 2

3 4

5

1.20: Coach and Be Coached

A coaches B B coaches A

m

12 m 8 m p

3 m

8 m

z

9 m 12 m

y

24 cm

10 cm 5 mm

x

12 mm

x

2 cm

2 cm

A hydro pole casts a shadow that is 10 m long. A technician measures the wire that runs from the top of the pole to the end of the shadow and finds it to be 26 m. How tall is the pole?

Don is building a loft in his garage. The ladder he is using extends to 10 metres. The loft is 8 m from the floor. How far away from the wall should he anchor the ladder?

w

130 cm 55 cm

h 44 cm

60 cm

1.21: Pythagorean Worksheet 1. If the legs of an isosceles right triangle are 6 units long, find the length of the

hypotenuse.

2. Eva wants to put an underground sprinkler system in her back yard. A drawing of the system is shown below. About how many feet of water pipe will Eva need?

3. Jackson is 54 miles east of Lazy R Resort. Ontario is 31 miles south of Jackson. A land developer proposes building a shortcut road to directly connect Ontario and Lazy R. Draw a picture and find the length of this new road. 4. A television screen measures approximately 15.5 inches high and 19.5 inches wide. A television is advertised by giving the approximate length of the diagonal of its screen. How should this television be advertised? 5. A 6-ft ladder is placed against a wall with its base 2 ft from the wall. How high above the ground is the top of the ladder?

1.22: The Pythagorean Theorem – Hidden Application The Geobellies Company wishes to make a new type of container for their product. The designer has created two containers: one the shape of a square-based pyramid, and the other a cone. Your job is to determine which container holds more.

Pyramid

slant height 10.5 cm

Cone

slant height 10.5 cm

base length 5.6 cm diameter is 5.6 cm 1) Use the Pythagorean Theorem to find the height of each shape

Pyramid Cone

2) How much can each container hold? Show your work.

Pyramid Cone

3) Identify the shape with the greater volume. 4) If the geobellies cost $0.005/cm3, how much will it cost to fill each container?