BLM 1-5a

OPEN

GUIDED

Lesson

Opener

(slide 2)

MindsOn

(slide 3)

Summary

(slide 17)

Your Turn(slides 18 and 19)

GuidedAction

(slides 7 to 9)

OpenAction

(slides 4 to 6)

Consolidation

(slides 10to 16)

1.5 Volumes of Pyramids and Cones

Teaching Notes

Teaching NotesBLM 1-5b

Tech

Tip

1.5 Volumes of Pyramids and Cones

What shapes can you identify in these buildings?

GSP

a) Build a prism and a cylinder that have about the same volume.

b) How could you determine the volumes of the objects you built?

c) How could you determine the surface areas of your objects?

Minds On

Tech

Tip

1.5 Volumes of Pyramids and Cones

A candy company makes solid chocolate candies in the forms of prisms and cylinders.

Its new pyramid candies will have the same base dimensions and height as its prism candies, because they have to fit in the same packaging.

Its new cone candies will have the same base dimensions and height as its cylinder candies.

Action

1.5 Volumes of Pyramids and Cones

Actiona) How will the volume of chocolate in a new pyramid candy compare with the volume of chocolate in a prism candy?

b) How will the volume of chocolate in a new cone candy compare with the volume of chocolate in a cylinder candy?

1.5 Volumes of Pyramids and Cones

Tech

Tip

Actionc) What formula could you use to determine the volume of chocolate in a pyramid candy?

What formula could you use to determine the volume of chocolate in a cone candy?

1.5 Volumes of Pyramids and Cones

ActionA candy company makes solid chocolate candies in the forms of prisms and cylinders.

Its new pyramid candies will have the same base dimensions and height as its prism candies, because they have to fit in the same packaging.

Its new cone candies will have the same base dimensions and height as its cylinder candies.

1.5 Volumes of Pyramids and Cones

Actiona) How will the volume of chocolate in a new pyramid candy compare with the volume of chocolate in a prism candy?

b) How will the volume of chocolate in a new cone candy compare with the volume of chocolate in a cylinder candy?

1.5 Volumes of Pyramids and Cones

Tech

Tip

A

B

C

D

Actionc) What formula could you use to determine the volume of chocolate in a pyramid candy?

What formula could you use to determine the volume of chocolate in a cone candy?

E

F

1.5 Volumes of Pyramids and Cones

1. What is the relationship between the volume of a prism and the volume of a pyramid?

2. What is the relationship between the volume of a cylinder and the volume of a cone?

3. What must be true about the dimensions of the objects for these relationships to hold?

4. Show how the formula for the volume of a pyramid can be developed from the formula for the volume of a prism.

5. Show how the formula for the volume of a cone can be developed from the volume formula for the volume of a cylinder.

ConsolidationReflecting and Connecting

Reveal

Reveal

Reveal

Reveal

Reveal

1.5 Volumes of Pyramids and Cones

Highlights and Summary

Consolidation

1 How can you calculate the volume of a prism if you know its dimensions?

A Multiply all the dimensions of the prism together.

B Multiply the area of the base by the height.

C Add the areas of all the faces of the prism.

D Add all the dimensions of the prism.

10 cm

6 cm

4 cm

7.5 cm

10 cm

6 cm

BAnswer

1.5 Volumes of Pyramids and Cones

2 How can you calculate the volume of a cylinder if you know its dimensions?

A Multiply all the dimensions of the cylinder together.

B Multiply the area of the base by the height.

C Add the areas of all the faces of the cylinder.

D Add all the dimensions of the cylinder.

4.0 cm

12.0 cm

6.5 m

8 m

BAnswer

Highlights and Summary

Consolidation

1.5 Volumes of Pyramids and Cones

3 This cylinder and this cone have the same base dimensions and the same height. How are their volumes related?

A Their volumes are the same.

B The volume of the cone is half the volume of the cylinder.C The volume of the cone is one-third the volume of the cylinder.D The volume of the cone is one-quarter the volume of the cylinder.

h hr r

C

Highlights and Summary

Consolidation

Answer

1.5 Volumes of Pyramids and Cones

4 This prism and this pyramid have the same base dimensions and the same height. How are their volumes related?

A Their volumes are the same.

B The volume of the pyramid is half the volume of the prism.

C The volume of the pyramid is one-third the volume of the prism.

D The volume of the pyramid is one-quarter the volume of the prism.

C

Highlights and Summary

Consolidation

Answer

1.5 Volumes of Pyramids and Cones

hh

5 Which calculation represents the volume of this pyramid?

A V = (3 cm)(6 cm)(15 cm)B V = (3 cm)(6 cm)(15 cm) ÷ 3

C V = (6 cm)(6 cm)(15 cm) ÷ 3

D V = [(6 cm)(6 cm)(15 cm)]3

15 cm

3 cm

6 cm

C

Highlights and Summary

Consolidation

Answer

1.5 Volumes of Pyramids and Cones

6 Which calculation represents the volume of this cone?

A V = π(11.2 cm)2(15.5 cm)

B V = 2π(15.5 cm)(11.2 cm)

C V = 3[π(15.5 cm)2(11.2 cm)]

D V = π(15.5 cm)2(11.2 cm) ÷ 3

11.2 cm

15.5 cm

D

Highlights and Summary

Consolidation

Answer

1.5 Volumes of Pyramids and Cones

ConsolidationSummary

• The volume of a prism or a cylinder can be determined by the following formula: volume = (area of base)(height)

• The volume of a pyramid is one-third the volume of a prism with the same base and height.

• The volume of a cone is one-third the volume of a cylinder with the same base and height.

hr

h

B

Pyramid:

volume =

3Bh

3

πr 2h

Cone:

volume =

1.5 Volumes of Pyramids and Cones

Reveal

Reveal

Reveal

Your Turn

Consolidation

A conical paper drinking cup has the inner dimensions shown.

Describe the strategy you would use to determine the capacity of a cylindrical cup with the same base radius and vertical height as this conical cup.

Solution

4 cm

10 cm

1.5 Volumes of Pyramids and Cones

Your Turn Solution

ConsolidationBack toLesson

Calculate the area of the circular base, and then multiply this area by the height of the cone.

Alternative Solution:

Determine the volume of the cone, and then triple this volume.

V = πr 2h

V = π(4 cm)2(10 cm)V = 502.7 cm3

Vcone =

Vcone =

Vcone = 167.6 cm3

Vcylinder = (167.6 cm3)3Vcylinder = 502.7 cm3

π(4 cm)2(10 cm)3

3πr

2h

1.5 Volumes of Pyramids and Cones

prisman object with opposite congruent bases; the other sides are parallelograms

cylinderan object with opposite congruent circularbases; the rest of the surface is curved.

volumethe amount of space an object fills

surface areathe number of square units it takes to cover an object

Back toLesson

Back toLesson

Back toLesson

Back toLesson

1.5 Volumes of Pyramids and Cones

Back toLesson

Back toLesson

Back toLesson

Back toLesson

pyramidan object with a polygon base;the rest of the sides are triangles thatmeet at a single vertex

basethe bottom line of a shape or the bottom shape of an object

heightin a shape, the vertical distance from a vertex to the opposite side; in a prism or cylinder, the vertical distance between bases; in a pyramid or cone, the vertical distance from the vertex to the base

conean object with a circular base witha single vertex opposite; the rest ofthe surface is curved

1.5 Volumes of Pyramids and Cones

Back toLesson

Back toLesson

Back toLesson

Back toLesson

pyramidan object with a polygon base;the rest of the sides are triangles thatmeet at a single vertex

basethe bottom line of a shape or the bottom shape of an object

heightin a shape, the vertical distance from a vertex to the opposite side; in a prism or cylinder, the vertical distance between bases; in a pyramid or cone, the vertical distance from the vertex to the base

conean object with a circular base witha single vertex opposite; the rest ofthe surface is curved

1.5 Volumes of Pyramids and Cones