Lesson Plan: Volume of a cone

G u i l l a u m e M i s z c z a k

A s h t o n V e r g e

C h r i s t o p h e r A b b a n d o n a t o

C a t h e r i n e B a r r y

K e l s e y G i l m o r e

Fall 2010

EDUC 5873 Middle/Secondary

Math Methods

Volume of a cone: Unit Lesson Plan

Grade Level: 9

Unit: Measurement

SCO: Estimate, measure, and calculate dimensions, volumes and surface areas of pyramids, cones, and spheres in problem situations.

Prior Knowledge: Students should know how to calculate the circumference and surface area of a circle. They will also know how to calculate the volume of a cylinder (Grade 8).

Setting the stage

o Review of areas and volumes

o Water Tower

o Set the context with memory game and water tower problem

Discovery Phase

o Ben Allerston`s presentation

Reinforcement/Enrichment

o Revisit the water tower worksheet

o Word Problems

o Dan Meyer Theory

o Work Sheet

o Play dough activity

Setting the stage

Objectives:

• Review the area of a circle • Review the volume of a cylinder and volume and area of other shapes as well. • To have students be able to use the formula derived for volume of a cone and apply it in its appropriate context.

Materials:

• Calculator • “Water Tower” Work sheet (Appendix I) • Smart board document image of water tower • Game Cards (Appendix II) • Smart Board file (attached) Activity 1 Advance Preparation: • Print off game card templates and cut out for students. (Appendix II) • Print the instruction sheet (Appendix II) Instruction: Have students play the game "concentration" with pictures of different shapes on some

cards to be matched with formulas for volume or area. This can be played before the

discovery to review the volume of a cylinder. After students discover the formula, the

cards for the volume of a cone can be added to review that as well.

Suggestions:

Allow for discussion, questions and hypothesis when approaching solving for the volume of a cone. Do not give away the formula. (Discovery phase)

Activity 2

Advanced Prep:

Get students to bring in 3 cylindrical objects each that are very different sizes.

Instruction:

Get the students to bring out their cylinders and first guess which object would hold the

most. Then get the students to measure their cylinders and figure out the volume of

each. Have them record this as well as their original guess on a worksheet (attached).

Activity 3

Advance Preparation:

• Photocopy the ``Water Tower`` work sheet

Instruction:

1. Introduce the problem

The town of Wood Lake needs to fill up their new water tower. The dimensions diameter and two different heights are given. We have been contracted to determine the volume of water needed to fill the water tower.

2. Show them the image of the water tower.

3. Hand out a worksheet with the water tower shapes and its dimensions and ask the class how we would go about solving this problem.

(see appendix I)

4. With previous knowledge, the class should be able to solve for the cylinder shape of the water tower (the bottom half). They will not be able to solve for the volume of the cone. From here, move on to Discovery.

Activity 4

Instruction:

Bring up the smart board file. Have students label the parts of the circle and cylinder and tell you what the area and volume formulas are.

Discovery Phase

See the Manipulative Lesson Plan by Ben Allerston at http://people.stu.ca/~pheeney/5873ManipVolumeCone11.pdf

Reinforcement/Enrichment

Objectives:

• To have students be able to apply their knowledge of cones and see examples of cones in realistic settings.

• To have students be able to use the formula derived for volume of a cone and apply it in its appropriate context.

• Reinforce the formula for volume of the cone

• Apply the formula to various problems

Activity 1 (Appendix I)

Materials:

• Calculator • “Water Tower” Work sheet (Appendix I) • Smart board image of water tower

Instruction:

1. Reintroduce the problem concerning the water tower. (Students will need to me reminded of the details)

2. Ask them to take out their “Water Tower” worksheets with the water tower and dimensions.

3. In groups of two or three, ask the class to solve for the volume of the cone in the image.

4. When students look to be either finished with the problem, or stuck on a step, guide and facilitate them through the problem. Allow them to think for themselves and work towards solving it. It is important to remember that this is the first time they will be using this formula and therefore, it must be done at a very slow pace to ensure that each student is able to keep up.

(Cylinder height – 50.0 m, diameter – 20.0 m, cone height- 50.0 m)

Activity 2 (Real Life Example)

Materials:

• Smart board document of images of different cones and truncated cones

• Concrete examples of cones and truncated cones (cottage cheese container, lampshade, cone paper cups, etc.)

Instruction:

1. Ask students in groups to discuss and list examples of cones and truncated cones that they would see every day in the outside world.

2. As a class, ask students to list off some of the examples they came up with and show them some examples on the Smart board. Bring in some examples to show them, such as a cream cheese container, a lampshade, a funnel, a cone paper cup, etc.

Activity 3 (Appendix VI)

Materials:

• Play dough • Rulers

Advanced Prep:

• Make various colors of Play dough and bring it in to class.

Recipe:

• 1 cup salt • 1 cup flour • 2 teaspoon cream of tartar • 1 cup water • 1 tablespoon cooking oil

• Food coloring Add food coloring for desired color of Play dough to the water and oil and mix in the pot. Stir in the remaining ingredients. Heat for 2-3 minutes on medium heat until Play dough begins to stick together. Transfer to wax paper and cool to touch (it will dry out some at this point). Work out the lumps.

Instruction:

Have students make their own cones of different sizes using the Play dough. Have the students measure the dimensions of the cones and calculate volume. Have them fill out the worksheet provided. The students can also make cylinders to practice that formula.

Application Problems (see Appendix V)

Applying Dan Meyer Theory with Application Problems

Watch Video: http://www.ted.com/talks/dan_meyer_math_curriculum_makeover.html Let’s Review What Dan is saying:

5 Signs you are doing math wrong: The students display the following: 1. Lack of Initiative 2. Lack of Perseverance 3. Aversion to Word Problems 4. Eagerness for formula

“Impatience with irresolution” We are giving students a smooth pathway toward the answer

How can we change this?

Create a compelling question that has a compelling answer Present questions that involve problem solving “Math serves the conversation the conversation doesn’t serve the math” 5 Ways of Better Teaching:

1. Use multimedia 2. Encourage student intuition 3. Ask the shortest question you can

4. Let students build the problem 5. Be less helpful!

Problem Some lampshades are truncated cones. When the point of a cone is removed by cutting through the cone along the plane parallel to the base, the resulting figure is called a truncated cone.

a) If the original cone has a diameter of 10 cm and a height of 12 cm, and the top is cut off at a distance of 4.0 cm from the vertex, determine the volume of the truncated cone that remains. b) Determine a formula for the volume of a truncated cone when the diameter of the original cone is d units, the height is h units, and the top is cut off at a distance of x units from the top. Re-model this question to match Meyer’s theory: What’s wrong with this problem?

First of all change context, this is not realistic. Yes truncated cones can be represented as lamp shades, but why would we be using volume with lamp shades? What are we filling it with/why? Lamp shades are more appropriate if we’re talking about SA not Volume! So let’s rephrase! What shapes can we think of when we’re discussing truncated cones and volume?

Where is the word problem? Where is the discussion?

When the point of a cone is removed by cutting through the cone along the plane parallel to the base, the resulting figure is called a truncated cone. A flower pot has the shape of a truncated cone.

Problem: Joey, a five year old t-ball player came to his father one day asking him if they could practice his swing. Father, Jim, thought how great it would be if he had a stand for

the ball so Joey could practice his swing. It just so happened Jim had an old pylon in the garage He thought if he trimmed a bit off the top it would be perfect for the ball to rest on. The only problem was he needed something to help weigh down the pylon so it would be sturdy when Joey hit the ball off it. Jim decided he would pick up some gravel to fill in the truncated cone. The only problem is he wasn’t sure how much to get?

a) Jim measured the height of the pylon before trimming it and found it to be 25inches tall. b) He found the diameter of the base of the pylon to be 10inches. c) Jim trimmed the tip off the cone of the pylon 3inches from the vertex. 1. Help Jim figure out how much gravel he will need to fill the truncated cone. Work in groups to brainstorm how you can solve this problem:

Remember: Start working with what you know first! Practice Problems for Re-Modeling

Problem 1: What is the height of a cone if its diameter is twice the height and the

volume of the cone is 72 cubic inches?

Use the formula for volume of a cone, replacing the V with 72 and the height, h, with r, Why replace the height measure with r? The diameter of a circle is twice the radius, so the diameter is equal to 2r. If the diameter is twice the height, then the diameter, 2r= 2h. The length of the radius is equal to the height. ______________________________________________________________________ Have you ever tried to pile sand? It doesn’t cooperate all that well. When sand is poured from a container it tends to form a coned-shaped pile, it spreads out farther than it is high. Problem 2: Sand is falling off a conveyer belt and forming a conical shape as the falling sand runs down the sides of the pile. If the height of the pile is always one-third the diameter, then by how much does the volume of the pile change when the pile grows

from 10 feet tall to 12 feet tall? Find the volume of a pile of sand that’s 10 feet tall and compare it to the volume of a pile of sand that’s 12 feet tall. The pile of sand that’s 10 feet tall has a diameter of 30feet- which means a radius of 15feet. The pile of sand that’s 12 feet tall has a diameter of 36 feet- or a radius of 18 feet. ______________________________________________________________________

Problem 3: Maria wants to determine how much liquid her new martini glass can hold. She tried pouring a 355ml can of Coke into her glass, but filled the glass before emptying the can, so she knew her glass held less than a full can of pop. She decided to measure the height of the glass and found it to be: 12cm (from vertex to rim) tall and 8cm wide from one rim to the other. But she wasn’t sure what to do from here,

she recalled the formula for a cylinder to be V= r²h but wasn’t sure if she could use that for her cone shaped-martini-glass. a) Can you help Maria find how much volume her new glass can hold? b) How much coke is left over in the can?

Appendices

I. Water Tower Worksheet

II. Memory Game cards

III. Memory Game Instruction

IV. Solutions for Dan Meyer`s theory problems

V. Sample Word Problems

VI. Play dough exercise

VII. Cylinder Worksheet

Wood Lake Water Tower Name:

1. Find the volume water that the cylinder is able to hold using the Water Tower

dimensions provided. 2. Find the volume of water that the cone is able to hold using the Water Tower

dimensions provided.

Diameter: d = 20.0 m

Height of cone: h = 50.0 m

Height of cylinder: h = 50.0m

Concentration Number of players: 2 to 4 Material: even number of cards (20 to 36); the deck is comprised of pairs of matching cards ie 2D and 3D shapes with their corresponding area or volume formulas. The same shape or formula can be inserted into the deck more than once to make the game larger and more challenging. Shuffle the cards and place them face down in a rectangular array. The first player turns over two cards in search of a matching pair so that everyone can see the overturned cards. If the two cards match they belong to this player and are placed in his/her collection pile. This player continues to play until he/she overturns a mismatched pair. If they do not match, they are returned face down to their original location in the array. The next player take his/her turn. The game ends when all matching pairs have been found. The player with the most pairs is the winner. game obtained from Journey’s in Math - Teacher’s Guide (grade 8 &9)

Published by Ginn

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Appendix III

Answer sheet for Problems (Dan Meyer)

Answer: Problem 1

72p = 1/3 pr² (r)

72p = 1/3 pr³

3 * 72p= 3 * 1/3 pr³

216p= pr³

216p/p= pr³/p

216= r³

r= 6

Answer: Problem 2

10-foot pile: V= 1/3 p (15)² * (10) » 2, 355 cubic feet

12-foot pile: V= 1/3 p (18)² * (12) » 4, 069 cubic feet

The pile of sand has grown by over 1, 700 cubic feet!

Answer: Problem 3

a) V= 1/3 pr² (h)

V= 1/3 p(4)² (12)

V= 201. 06 ml » 201mL

b) 355mL - 201mL= 154mL

Answer: Problem 4

a) 300 cm³

b) [pd²/12 (h- x³/h²) units³]

References:

Math Word Problems for Dummies By Mary Jane Sterling

https://nrich.maths.org/

Mathematical Modeling Text: Nelson

http://www.ted.com/talks/dan_meyer_math_curriculum_makeover.html

Appendix V: Worksheet

1 Worksheet 1.1 Understanding Concepts

1. The radius of the cone is 10 cm.

What is the volume of the cone?

2. The radius of the cone is 6 cm.

a) What is the measure of AB to one decimal point? b)What is the measure of AB describing? c) What do you notice about the shape of ABC? d) Find the volume of the cone

3. Find the volume of each of these cones. a) Oblique circular cone

b) Semicircular cone

4. An industrial drill bit was used to drill a hole in the shape of a cone, the side view of which is pictured in the diagram below. It takes 25 mL of glue to fill this hole.

What is the diameter of the hole's opening? (Round to 1 decimal place.)

5. An ice cream cone is 5 inches high and has an opening 3 inches in diameter. If filled with ice cream and given a hemispherical top. How much ice cream is there?

6. Ben and Barry's ice cream manufacturers have come up with a new ice cream cone. A cylindrical rim has been added to the traditional cone-shaped base, as shown in the diagram below.

To price their new cone, they need to know the amount of ice cream that will be used to fill the entire cone as well as the hemispheric scoop atop the cone. How many cm3 of ice cream will be need?

1.2 Project Questions 7. Cut out the circular section shown, and roll it into a right circular cone in which the two 4-inch radial segments are joined. Find the following measurements both by exact computation and by measuring your paper model. a) The radius of the base of the cone b) The height of the cone c) Use parts a) and b) to compute the volume of the cone

8. Cut out a semicircular sector, and roll it up and join the two radial segments to form a cone. Show that the diameter of the cone is equal to the slant height of the cone, both by measuring your paper model and by a calculation. What does this mean in terms of volume?

1.3 Activity Sheets

1.4 Challenge Question 9. An ice cream soda glass is shaped like a cone of height 6 inches, and has a capacity of 16 fluid ounces when filled to the rim. Use the similarly principle to answer the following questions, using the fact that the cone of liquid is similar to the cone of the entire region inside the glass.

a) How high is the soda in the glass when it contains 2 fluid ounces? b) How much soda is in the glass when it is filled to a level 1 inch below the rim?

References:

1. Activities Manual Mathematics for Elementary Teachers 2nd edition, S. Beckman, Pearson Addition Wesley, 2008

2. Mathematical Reasoning for Elementary Teachers 2nd edition, C. Long and Tewple. Addition-Wesley, 2000

3. Math 314 LBPSB resources.

2 Answer Key

1. V = 13π(102)(25) = 2616.66cm3

2. 152 = x2 + 62

x2 = 189

AB = 13.74

3. a) V = 13π(52)(12) = 314cm3

b) V = 13π(62)(14) = 527.52cm3

4. 25 = 13π(r2)(5)

π(r2) = 15

r = 2.19

d = 4.4

5. volume of the hemisphere: V = 23π(1.53) = 7.065in3

volume of the cone: V = 13π(1.52)(5) = 11.775in3

total volume: V = 7.065 + 11.775 = 18.84, V = 19in3

6. volume of the cone: V = 13π(32)(10) = 94.2cm3

volume of the cylinder: V = π(32)(3) = 84.78cm3

volume of the hemisphere: V = 23π(33) = 56.52cm3

total volume: V = 94.2 + 84.78 + 56.52 = 235.50cm3

7. a) 3in, since 216 = 1

8 = (12)3and1

2(6) = 3

9

Name: Date:

A.) Cone 1 (Drawing)

Dimensions & Calculation:

Volume =

B.) Cone 2 (Drawing)

Dimensions & Calculation:

Volume=

C.) Cone 3 (Drawing)

Dimensions & Calculation:

Volume =

D.) Cylinder Drawing

Dimensions & Calculation:

Volume=

Name: Date:

A.) Description of Object, circle which object you predict will have the largest volume

Object 1:

Object 2:

Object 3:

B.) Calculation of Volume

Object 1 dimensions:

Calculation:

Volume:

Object 2 dimensions:

Calculation:

Volume:

Object 3 dimensions:

Calculation:

Volume: