Hertford County High School

Randy Pantinople, Teacher

and

Sarah Mcaree, Student

Hertford County High School

with assistance from

Jenny Jones, Curriculum Developer

Shodor Education Foundation

and

Faiz Warsi, Business Partner

Cummins Engines

Preface

Cylinders are three-dimensional objects bound by a curved surface with two congruent circles in parallel planes. Cylinders are familiar figures that we deal with everyday. Cups, tubes, some containers, and some buildings are designed using cylindrical shapes. In fact, cylinders are used in a variety of different fields. In firearms, cylinders are the rotating devices that contain the firing chambers in revolvers. In locks, a cylinder is the part of the lock that is profiled to accept a specific key. In science, the graduated cylinder is the glassware for measurement of liquids in laboratories.
One of the greatest inventions of man in the field of transportation is the “engine”. The central working part of an engine is a cylinder, where combustion occurs and the piston travels. Cummins Engines, which is the business partner of this module, is a global power leader that designs, manufactures, sells and services diesel engines and related technology around the world.
This module is designed to develop students’ understanding of the concepts and skills surrounding cylinders. The module will also help students to understand the roles of cylindrical shapes in designing engines. By the end of the module, students will be able to design and make a cylinder model similar to Cummins Engines.

Module Overview

Lesson One: Students discover the formula of the surface area of the cylinders and use the formula to solve real-life problems involving the surface area of cylinders.

Lesson Two:Students investigate the relationships between the dimensions (radius and height) of the cylinder and its volume. They will also use the formula of the volume of the cylinders to solve problems.

Lesson three: Students deepen their understanding of similar cylinders and enhance their skills in solving problems that include similar cylinders.

Lesson four: Students apply the skills they learned from the previous lessons using ideas from the business partner.

Business Partner

Cummins is a global power leader that designs, manufactures, sells and services diesel engines and related technology around the world. Cummins serves its customers through its network of 550 company-owned and independent distributor facilities and more than 5,000 dealer locations in over 160 countries and territories.

Table of Contents

Preface

Module Overview

Business Partner

Lesson 1: Surface Area of the Cylinder

Lesson 2: Volume of the Cylinder

Lesson 3: Similar Cylinders

Lesson 4: Project Making

Appendix

Activity 1: Discovering the Surface Area of the Cylinder

Activity 2: Problem Solving with Surface Area......

Activity 3: Problem Solving with Volume......

Worksheet 1: Volume of the Cylinder......

Worksheet 2: Similar Cylinders......

Project Information Sheet......

Rubric for Project Information Sheet......

Answer Key to Worksheet 1......

Answer Key to Worksheet 2......

Lesson 1: Surface Area of the Cylinder

Objectives:

The student will:

Determine the surface area of a cylinder through investigationusing concrete materials.
Generalize the formula for the surface area of the cylinder.
Solve problems involving the surface area of a cylinder.

Standards:

Common Core Standards:

Eighth Grade

Geometry:Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

North Carolina Standard Course of Study:

Geometry

Goal 1: The learner will perform operations with real numbers to solve problems.

1.02 Use length, area, and volume of geometric figures to solve problems. Include arc length, area of sectors of circles; lateral area, surface area, and volume of three-dimensional figures; and perimeter, area, and volume of composite figures.

Materials:

Cylindrical objects from home (soup cans, cans of beans, candles, spice jars, etc.)
Precut shapes of one of the above cylinders, which include the two end circles and a rectangle which forms the curved region.

Vocabulary:

Cylinder
Dimension
Area
Circumference
Height
Surface area
Radius

Procedures:

Post a picture of a truck: a 2009 Dodge Ram 2500.

Link: the students some background information on Cummins engines:Cummins Engines is a global power leader that designs, manufactures, and sells diesel engines around the world. The truck in the picture is 2009 Dodge Ram 2500, which is currently the only passenger vehicle that features a Cummins Diesel Engine.

Brainstorm with students: what makes the truck move?
Answers will vary; some students may answer gas, wheels and engine.

Emphasize to the students that the engine is like the heart of the truck. When it stops pumping gas, the truck stops too. Use the link below to show the model of an engine while the truck is moving.

showing the link, ask students to name some 3-dimensional solids they see.
Cylinder, rectangular solid

Complete Activity 1 with the students. Follow-up with exploration questions such as the following:

Is it easier to use your algorithm to find the surface area of a cylinder?
Would you rather just calculate the area of each section and add them up?
Why?

Complete Activity 2 with the students. Follow-up with exploration questions such as the following:

What words would you use to describe the two cylinders where one was twice the dimensions of the other?
Can you think of language we’ve used before to describe similar relationships?
Do you think there’s an “easy” way to solve it when one cylinder is just a scaled version of the other?

Close out the lesson by telling students that they’ll explore this relationship in a later lesson. First, we’ll be looking at another way to describe cylinders. Can you think of what this might be?

Lesson 2: Volume of the Cylinder

Objectives:

The student will:

Develop understanding of the relationships among the radius, height and volume of a cylinder.
Solve problems involving volume of a cylinder

Standards:

Common Core Standards:

Eighth Grade

Geometry:Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

Geometry

Geometric Measurement and Dimension:Explain volume formulas and use them to solve problems.

North Carolina Standard Course of Study:

Geometry

Goal 1: The learner will perform operations with real numbers to solve problems.

Materials:

8 1/2" by 11" sheets of paper
Tape
Ruler
Graph paper
Fill material (birdseed, Rice Krispies, Cheerios, packing "peanuts", beans etc.).
Worksheet 1

Vocabulary:

Volume: The amount of space occupied by three-dimensional object or region of space, expressed in cubic units.

Procedures:

Ask students to consider the following question: “How does the radius and height of a cylinder affect the volume of the cylinder?”

What do I mean by volume?
In what other contexts have you heard about this?

Split students into groups.

Distribute Worksheet 1. Have students complete the activity. Have students present their predictions and explanations from Question 1 to the class. Follow up with exploration questions such as the following:

Was your prediction correct? Why do you think that is?

Answers will vary.

Do you see any pattern that relates the size of the cylinder and the amounts they hold?

As they get taller and narrower the cylinders hold less, and as they get shorter and stouter, they hold more.

As a class, complete Activity 3. Close out the lesson by telling students that they’ll explore relationships between different cylinders in the next lesson. Can you make any predictions about what we’re going to discover?

Lesson 3: Similar Cylinders

Objectives:

The student will:

Identify similar cylinders
Use scale factor to determine the ratios of surface areas and volumes of similar cylinders.

Standards:

Common Core Standards:

Eighth Grade

Geometry:Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

Geometry

Geometric Measurement and Dimension:Explain volume formulas and use them to solve problems.

North Carolina Standard Course of Study:

Geometry

Goal 1: The learner will perform operations with real numbers to solve problems.

Materials:

Models of cylinders (should be made according to specifications below)
Calculators

Resources:

Larson, Ron et al. Geometry, North Carolina Edition. Illinois: McDougal Little,2004.

Procedures:

On the table, students are given different sets of cylinders with different dimensions.
Cylinder A: diameter of 8cm, height of 14cm
Cylinder B: diameter of 7cm, height of 13cm
Cylinder C: diameter of 12cm, height of 21cm

Which among the cylinders appear to be similar?

Cylinder A and Cylinder C

How did you know that the two cylinders are similar?

12 is 1.5*8

21 is 1.5*14

Discuss what the term “similar” means. Consider the following to guide the discussion:

First, consider a familiar concept: triangles.
What does it mean for two triangles to be similar?
Triangle 1 has sides of lengths 3cm, 4cm, and 5cm. Triangle 2 has sides of lengths 6cm, 8cm, 10cm. Are they similar? Why?
Can we generalize our definition to include shapes beyond just triangles?
What about 3-D solids? How can we include them in our definition?
Consider the following definition: “Two solids with equal ratios of corresponding linear measures, such as heights and radii, are called similar solids. This common ratio is called the scale factor.”

Looking back on the cylinders, cylinder A and cylinder C are similar. The scale factors of their radii and heights are the same.

Scale factor of radii:1.5

Scale factor of height: 1.5

Since the scale factors of radii and heights are both 1.5, therefore Cylinder A and Cylinder C are similar.

Present and discuss the Similar Solids Theorem: If two similar solids have a scale factor of a:b, then corresponding areas have a ratio of a2:b2, and corresponding volumeshavea ratio of a3:b3.
Go over the following examples with the students:

The cylinders are similar with a scale factor of 1:3. What are the ratios of their surface areas and volumes?

Ratio of Surface Areas: 1:9

Ratio of Volumes: 1:27

Sarah is making a model similar to one of the cylinders of a truck engine. She knows that the volume of the actual cylinders of the engine is 2300 in3. What is the volume of her model cylinder if she will reduce the dimensions to the scale factor of 5:2

Volume: 147.2 in3

Two 3-dimensional solids are similar. If the ratio of their surface areas is 9:16, what is the ratio of their volumes?

Ratio of Volumes: 27:64

Have students complete Worksheet 2. Follow-up with a discussion of student findings. Consider the following exploration questions:

What does the scale factor affect surface area and volume the way it does?
Where does that formula come from?
Is it easier to think about the scale factor being squared or cubed? Or would you rather find the surface area and volume some other way?
Why?

Lesson 4: Project Making

Objectives:

The student will:

Design and make a model of a cylinder that would fit a given piston.
Apply knowledge of surface area and volume to real world problems.
Apply knowledge of similar cylinders to real world problems.

Standards:

Common Core Standards:

Eighth Grade

Geometry:Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

Geometry

Geometric Measurement and Dimension:Explain volume formulas and use them to solve problems.

North Carolina Standard Course of Study:

Geometry

Goal 1: The learner will perform operations with real numbers to solve problems.

Materials:

Pre-made piston models

You can create this by creating two paper cylinders where one (with a top) sits inside the other and can slide up and down.
This should have a diameter of 2 in.

Paper
Card stock
Cardboard
Glue
Scissors
Adhesive materials
Rulers

Procedures:

Organize students into groups. Students are in the same groups as the previous class meeting.

Present the problem: How do you model a cylinder given a model piston and the scale factor?

Explain this further: In a truck engine, the piston moves up and down the cylinder (show students this animation: I give you a piston, you know that the cylinder has to fit it exactly.
For our purposes, the “bore/stroke” ratio (that is the ratio of the diameter to the height of the cylinder) will be “square” or 1.
The piston isn’t to scale, though. Each group will have a scale factor for their project.
The model piston has a diameter of 2 in. Use different scale factors for each group, consider the following possibilities:
1:2
1:3
2:5
2:3
1:4
2:7

Give students the assignment: Each team will come up with a procedure for making a cylinder that will fit their piston. They should consider the height and radius of the cylinder when thinking through what materials they will need.

Have students complete the Project Information Sheet with the details of what their team did.

Have students share what they did. Ask questions of each group:

How did you know what dimensions your cylinder would be?
How did you calculate the surface area and volume of your cylinder?

Appendix

Activity 1: Discovering the Surface Area of the Cylinder

Before conducting the activity, make sure that you pre-assigned students to bring one object that is cylindrical in shape like empty can, bottles, etc.

Instruct the students to bring out their objects and hold up. Which part of the object is the surface area?

Student answers may vary.

If students are having trouble getting to the correct answer, ask guiding questions:

What is the surface of the cylinder?
Do you think the “surface area” has anything to do with that?
If you had to guess right now, what would the “surface area” be?
What does “area” mean?
Do you think it makes sense for “surface area” to mean the “area of the surface”?

Ask students to think about what that means. What is included in the “surface area” of a cylinder?

Surface area of a cylinder is the total area from the lateral (body) and top and bottom of the object.

How can we find this entire area? How would you find the area of a shape? What if the shape looked something like this?

Calculate the area for each section and add them up?

What sections should we break the cylinder into?

Student answers may vary.

Bring out your precut shapes (two circles and a tube) and cover the can so that the students can see that the entire can is covered by your shapes. These three shapes cover the can. What does that tell us about calculating surface area?

We can calculate surface area by calculating the area of each of the three shapes and adding them together.

We know how to calculate the area of the circles, but how do we calculate the area of the open cylinder or tube?

Cut the shape down the middle to create a rectangle. Then calculate the area of the rectangle.

Post the shapes on the board. What is the height of the rectangle compared to the cylinder? What is the length of the rectangle compared to the cylinder? What can you conclude about the surface area of the cylinder?

To get the total surface area of the cylinder, we will add the area of the two circles and a rectangle with a height equal to the height of the cylinder and a length equal to the perimeter of the base and top circles.

If students are having trouble understanding the dimensions of the rectangle, wrap the rectangle around the cylinder and ask students to identify the perimeter of the base circle. Then unwrap the rectangle and demonstrate how this is the same as the length of the rectangle.

Can we calculate the surface area of the cylinder all at once? Let’s create an “algorithm” that doesn’t require us to have the dimensions of the cylinder until the very end:

Base Circle: πr2

Top Circle: πr2

Body Rectangle: 2πr*h

What do we know about the two circles? What does that mean for our algorithm?

They’re the same, so we can just multiply one times 2 like this:

Area of base and top: 2πr2

Body Rectangle: 2πr*h

Let’s put it together:

Surface Area of Cylinder: 2πr2+ 2πr*h

Activity 2: Problem Solving with Surface Area

Complete the following examples with the class:

EXAMPLE 1: A cylinder has radius 2 cm and height 15 cm. What is the surface area?

SA = 68 cm2 or 213.628 cm2

EXAMPLE 2: A canned goods company manufactures cylindrical cans with a radius of 1.5 in. and a height of 3 in.

Find the surface area of the can.

SA = 13.5 in.2 or 42.41in.2

Find the surface area of the can whose radius and height are each twice as big (radius: 3 in., height: 6 in.).

SA = 54 in.2 or 169.646 in.2