OPEN GUIDED Lesson Opener (slide 2) Minds On (slide 3) Summary (slide 17) Your Turn (slides 18 and 19) Guided Action (slides 7 to 9) Open Action (slides 4 to 6) Consolidation (slides 10 to 16) 1.5 Volumes of Pyramids and Cones Tech Tip Recommendations: 1. Decide which path you will take. 2. Print or copy any attachments you think you will need in class. 3. Delete the slides you will not need (the organizer and either the Guided Action or Open Action), and save the file. 4. Use the saved file in the classroom.

1.5 Volumes of Pyramids and Cones What shapes can you identify in these buildings?

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 Click on the GSP icon to activate the sketch. Drag sides to change the objects. 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

Action a) 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 Click on the link to activate the application. Choose Cone → Tank. Click “New Problem” to see a different situation. Tech Tip

Action c) 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

Action 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. 1.5 Volumes of Pyramids and Cones

Action a) 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 Click on the link to activate the application. Choose Cone → Tank. Click “New Problem” to see a different situation. Tech Tip A When you click “Pour,” how much of the tank is filled? B If you were pouring from the tank to the pyramid or cone, how much of the tank would you empty for each pour? C How is the volume of the pyramid related to the volume of the prism? D How is the volume of the cone related to the volume of the cylinder?

Action c) 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 The formula for volume of a prism is volume = (area of base)(height). F The formula for volume of a cylinder is volume = (area of base)(height). 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. Consolidation Reflecting and Connecting Reveal 1.5 Volumes of Pyramids and Cones

Highlights and Summary Consolidation 1How 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 B Answer 1.5 Volumes of Pyramids and Cones

2How 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 B Answer Highlights and Summary Consolidation 1.5 Volumes of Pyramids and Cones

3This 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 h r r C Highlights and Summary Consolidation Answer 1.5 Volumes of Pyramids and Cones

4This 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 h h

5Which 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

6Which 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) ÷ cm 15.5 cm D Highlights and Summary Consolidation Answer 1.5 Volumes of Pyramids and Cones

Consolidation Summary 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. h r h B Pyramid: volume = 3 Bh 3 πr 2hπr 2h Cone: volume = 1.5 Volumes of Pyramids and Cones 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 Consolidation Back to Lesson 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 2 h V = π(4 cm) 2 (10 cm) V = cm 3 V cone = V cone = cm 3 V cylinder = (167.6 cm 3 )3 V cylinder = cm 3 π(4 cm) 2 (10 cm) 3 3 πr 2hπr 2h 1.5 Volumes of Pyramids and Cones

prism an object with opposite congruent bases; the other sides are parallelograms cylinder an object with opposite congruent circular bases; the rest of the surface is curved. volume the amount of space an object fills surface area the number of square units it takes to cover an object Back to Lesson Back to Lesson Back to Lesson Back to Lesson 1.5 Volumes of Pyramids and Cones

Back to Lesson Back to Lesson Back to Lesson Back to Lesson pyramid an object with a polygon base; the rest of the sides are triangles that meet at a single vertex base the bottom line of a shape or the bottom shape of an object height in 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 cone an object with a circular base with a single vertex opposite; the rest of the surface is curved 1.5 Volumes of Pyramids and Cones

Back to Lesson Back to Lesson Back to Lesson Back to Lesson pyramid an object with a polygon base; the rest of the sides are triangles that meet at a single vertex base the bottom line of a shape or the bottom shape of an object height in 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 cone an object with a circular base with a single vertex opposite; the rest of the surface is curved 1.5 Volumes of Pyramids and Cones