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Exploring Three-Dimensional Figures

Surface Area and Volume

EASTERN MICHIGAN UNIVERSITY

Winter 2015

Amanda Cavanaugh

1

Exploring Three-Dimensional Figures

Surface Area and Volume

7th Grade Mathematics

Amanda Cavanaugh

Expected Length of Unit: 9 Class Periods

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Table of Contents

Context Analysis…………………………………………………………………………………..3

Rationale…………………………………………………………………………………………..4

Content Analysis…………………………………………………………………………………..5

Content Outline……………………………………………………………………………………6

Unit Objectives/Outcomes………………………………………………………………………...8

Pre-Assessment Process and Results……………………………………………………………...9

Day-By-Day Unit Framework…………………………………………………………………...16

Lesson Plans and Materials………………………………………………………………………17

Culminating Activity…………………………………………………………………………….55

Student Learning Analysis……………………………………………………………………….56

Reflection and Self Evaluation…………………………………………………………………..64

Teaching Materials……………………………………………………………………………….66

References………………………………………………………………………………………..67

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Context Analysis

School District

East Arbor Charter Academy is located in Ypsilanti, Michigan. This school is a free public

charter school that is run by National Heritage Academies, Inc. East Arbor educates children

from pre-kindergarten through eighth grade. Ypsilanti is a city located near Ann Arbor,

Michigan. The population of Ypsilanti as of the 2010 U.S. census was 19,435 people. Many

adults in the area work in professional or technical fields. The children attending East Arbor are

primarily Caucasian (about 50.52%). Other ethnicities represented are African American

(31.87%), Multiracial (9.97%), Hispanic (4.92%), and Asian (2.72%). The approximate number

of students attending is 770. About 32% of the students attending are participating in a

free/reduced lunch program.

Classroom Factors

Each student has his/her own desk. Desks are placed in six rows, with five rows containing five

desks and one row containing only four desks, making for 29 total desks. The six rows allows for

even groups of two and one group of three desks to be easily put together for partner activities.

The wall at the front of the classroom contains the behavior chart and the moral focus topic of

the month on one side of a white board. On the other side of the white board is a math

vocabulary board. The white board in the middle displays what standards/objectives will be

covered that day and the assignments. There is a projector that overhangs this board that projects

directly onto the board. Connected to this projector is also a document camera. Another wall

contains a game called Homework-opoly; a game that the students can earn prizes from for

turning in all homework assignments every week. On the other side of the room is a wall with

windows and posters of topics that have been covered so far. There is a special place for the skill

of the week poster. On the back wall is Khan Academy skill mastery achievement posters.

Students put up stickers and can earn prizes based on how many skills they can master. There are

also cabinets that contain all supplies necessary for instruction. There is one computer for teacher

use only, and a cart of laptops that we share with the other seventh grade classes that the students

use to go on Khan Academy when they have finished their work.

Student Characteristics

For this unit, I will be working with the accelerated class. Just before I began my unit, there were

changes made in scheduling, so four students got moved into the class that have missed some of

the chapters that the accelerated class has gone over that they had not gone over in their previous

class.

Implications for Instruction

In order to help the students who were moved up into the accelerated class be successful, I will

be working with them in their own small group and individually on topics that they may have

4

missed that would be important for them to know for the unit I will be working on. I will also tie

in previously learned topics into my lessons to teach them and remind the other students of

things they may have forgotten.

Rationale

Students

Have you ever wondered how you could know exactly how much wrapping paper you would

need to wrap a present? Or have you ever wondered how to figure out how much water will fit in

your aquarium? What type of shapes do you get if you slice a three-dimensional figure? In this

unit, you will learn all about surface area and volume of different three-dimensional figures, as

well as cross sections.

Teachers

This unit covers surface area of prisms, pyramids and cylinders along with volume of prisms and

pyramids. There will also be a small section on identifying cross sections. Within the unit are

objectives that are derived from the Common Core State Standards for Mathematics. This unit

will focus on group activities, work done individually, and partner activities.

5

Content Analysis

Common Core State Standards

7.G.3 Describe the two-dimensional figures that result from slicing three-dimensional

figures, as in plane sections of right rectangular prisms and right rectangular pyramids.

7.G.4 Know the formulas for the area and circumference of a circle and use them to solve

problems; give an informal derivation of the relationship between the circumference and area

of a circle

7.G.6 Solve real-world and mathematical problems involving area, volume and surface area

of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes,

and right prisms.

Generalizations

Surface area is the measure of the area of the outermost layer of a solid.

o Focus question: When would it be important to know the surface area of an

object?

Lateral surface area is the total area of the sides of a solid, not including the base(s).

o Focus question: What are some objects in your household that are only covered

by the lateral faces, no bases?

Volume is the measure of the amount of space an object occupies or that is enclosed in a

solid.

o Focus question: What are some examples of everyday items that have volume?

Concepts

Nets

Cross sections

Surface area of prisms, pyramids, and cylinders

Lateral surface area of cylinders

Volume of prisms and pyramids

Facts

Surface area can be calculated by finding the area of the base(s) of a figure and adding it

to the area of the lateral face(s).

The formula for the surface area of a rectangular prism is: 𝑆𝐴 = 2(𝑙𝑤 + 𝑙ℎ + 𝑤ℎ)

The formula for the surface area of a cylinder is: 𝑆𝐴 = 2𝜋𝑟2 + 2𝜋𝑟ℎ

If you just want the lateral surface area of a cylinder, just calculate 2𝜋𝑟ℎ.

http://www.corestandards.org/Math/Content/7/G/A/3/

http://www.corestandards.org/Math/Content/7/G/B/4/

http://www.corestandards.org/Math/Content/7/G/B/6/

6

The volume of a prism can be calculated by finding the area of the base and multiplying

it by the height of the prism.

The formula for the volume of a rectangular prism is: 𝑉 = 𝑙𝑤ℎ.

The formula for the volume of a pyramid is: 𝑉 =1

3𝐵ℎ, where B is the area of the base

and h is the height of the pyramid.

Content Outline

I. Net

A. Two-dimensional representation of a three-dimensional figure

B. Helps us see bases and lateral faces

i. Use to discover formulas

II. Surface Area

A. Measured in squared units

B. Area of base(s) + Area of lateral face(s)

i. Prism

ii. Pyramid

1. Slant height

iii. Cylinder

III. Lateral Surface Area

A. Area of lateral face(s)

i. Lateral Face: all sides of a three-dimensional figure that are not bases

ii. Cylinder

IV. Volume

A. Measured in cubic units

B. Prism

i. 𝑉 = 𝐵 ∗ ℎ

1. B = area of the base

2. h = height

C. Pyramid

i. 𝑉 =1

3∗ 𝐵 ∗ ℎ

1. B = area of the base

2. H = height

V. Cross Section

A. Two-dimensional shape created by the intersection of a plane and a three-dimensional

figure

8

Unit Objectives/Outcomes

Common Core State Standards Unit Outcomes: SWBAT: 7.G.3; Describe the two-dimensional figures

that result from slicing three-dimensional

figures, as in plane sections of right rectangular

prisms and right rectangular pyramids.

Identify cross sections of a plane and a three-dimensional solid.

7.G.4; Know the formulas for the area and

circumference of a circle and use them to solve

problems; give an informal derivation of the

relationship between the circumference and

area of a circle

Construct a formula for surface area of a cylinder using prior knowledge of

area and circumference of a circle.

Explain how circumference of a circle is related to finding the area of the

lateral face of a cylinder.

Calculate the surface area and lateral

surface area of a cylinder.

Give real-life examples of when knowing the surface area and lateral

surface area of a cylinder would be

important.

7.G.6; Solve real-world and mathematical

problems involving area, volume and surface

area of two- and three-dimensional objects

composed of triangles, quadrilaterals,

polygons, cubes, and right prisms.

Create nets to discover formulas for

finding surface area of different three-

dimensional figures.

Understand and use area formulas of different polygons.

Calculate the surface area of right prisms, cubes and pyramids.

Calculate the volume of a prism and a pyramid.

Describe how changing the dimensions

of a prism change its volume.

Give real-life examples of when knowing surface area and volume

would be important.

Identify real-life examples of prisms and pyramids.

Additional Social Goal Realize the importance of asking for

help when it is needed.

Value the importance of collaboration.

9

Pre-Assessment Process and Results

Students were given a paper and pencil pre-assessment in class that assessed them on three main

objectives. They were given a rating of N, L, or S for each objective. N = no evidence of

understanding, L = limited understanding, and S = substantial understanding.

Objective 1 N = 0-3 correct

L = 4-8 correct

S = 9-12 correct

Objective 2 N = 0 correct

L = 1-2 correct

S = 3-4 correct


L = 1 correct

S = 2 correct

Student Summary of Pre-Assessment Information

Student Number

Objective 1:

Calculate surface

area/volume of

prisms, pyramids and

cubes.

Objective 2:

Calculate the surface

area and lateral

surface area of a

cylinder.

Objective 3:

Identify cross sections

of a plane and a three-

dimensional solid.

1 L N S

2 N N N

3 N N N

4 N N N

5 N N N

6 L N N

7 N N N

8 N N N

9 N N N

10 N N N

11 N N N

12 N N N

13 N N N

14 N N N

15 L N N

16 N N N

17 N N N

18 N N N

19 N N N

20 N N N

21 N N N

22 L N N

23 N N N

24 N N N

25 N N N

26 N N N

27 N N N

10

Pre-Assessment Planning for Instruction

Objective/Outcome Pre-Assessment

Strategy

Summary of Results Implications for

Instruction

Objective 1:

Calculate surface

area/volume of


cubes.

Students were given a

pencil and paper

assessment which

asked them a total of

11 questions

involving this

objective.

In the pre-assessment,

only 4 students had

limited understanding

of this concept, while

the other 23 students

had no understanding.

While the pre-

assessment shows

most students

showing no

understanding of the

concept, about half of

the students were able

to give me the correct

volume of a

rectangular prism. I

will plan to spend

more time on surface

area of prisms and

pyramids and volume

of pyramids.


Strategy


Instruction

Objective 2:


area and lateral

surface area of a cylinder.


pencil and paper

assessment which

asked them a total of 4 questions involving

this objective.

In this pre-

assessment, I had

none of the 27

students showing any signs of understanding

of this concept.

Since none of the

students had

understanding of this

topic, I will make sure to spend more time on

this concept and bring

in physical examples

to help them visualize

how to find surface

area of cylinders.


Strategy


Instruction

Objective 3:



dimensional solid.


pencil and paper

assessment which

asked them a total of

2 questions involving

this objective.

In his pre-assessment,

only 1 student showed

substantial

understanding of this

concept, while the

other 26 students

showed no

understanding.

Since only one

student was able to

show understanding

of this concept, I will

be sure explain what a

cross section is and

show lots of

examples.

11

Pre-Assessment Data Entry and Graph

Objective 1 Objective 2 Objective 3

N 23 27 26

L 4 0 0

S 0 0 1

0

5

10

15

20

25

30

1 2 3

Nu

mb

er o

f St

ud

ents

Objective

Unit Pre-Assessment Results

N

L

S

12

Students of Special Concern

For this unit, I have five students for whom I have special concerns. Four of these students are

the students who transferred into this class. As I have mentioned before, I am doing this unit with

my accelerated class, and just before starting my unit, class schedules got switched around and

four students got moved into the accelerated class. I am concerned for them because they missed

out on part of the unit about circles (circumference and area), which will be important for them

to know when looking at surface area of cylinders. I may need to recap these concepts for them.

The fifth student is one who is very bright, but he has problems focusing and staying on task. He

asks questions that get the whole class off task. For this student, I will need to make sure that he

stays on task and asks off-topic questions to me after class.

Student for Learning Analysis

The student I picked for the learning analysis is a very bright student. She is always participating

and trying her hardest. However, this student is usually a little slower to pick up on different

topics. She gets easily frustrated when she does not pick up on things as quickly as her peers.

This frustration usually makes her rush through problems, leaving her with simple calculation

errors. For the pre-assessment, she only got one question right on the entire test. I will need to

make sure that if she starts to get frustrated, I slow her down and walk her through things a little

slower to make sure she understands concepts.

13

Chapter

9 Pre-Test

Answers

1.

2.

3.

4.

5.

6.

7.

Name ________________________________________________________ Date _________

Find the surface area and volume of the prism.

1. 2.

Find the surface area of the regular pyramid. Round your answer to

the nearest tenth.

3. 4.

Find the surface area of the cylinder. Round your answer to the nearest tenth.

5. 6.

7. Find the lateral surface area of the

paint can. Round your answer to the

nearest hundredth.

14

Chapter

9 Pre-Test (continued)

Answers

8.

9.

10.

11.

12.

13.

14.

15.

Name ________________________________________________________ Date _________

Find the volume of the regular pyramid.

8. 9.

10. Find the surface area of the 11. Find the volume of the

composite solid below. composite solid below.

12. The volume of a pyramid is 84 cubic feet. The area of the base is

21 square feet. Find the height of the pyramid.

13. What happens to the volume of a rectangular prism when the length

and width are doubled and the height is tripled?

Describe the intersection of the plane and the solid.

14. 15.

15

Pre-Assessment Key

1. SA = 66.6 m2, V = 28.8 m

3 (2 points)

2. SA = 300 in2, V = 240 in

3 (2 points)

3. 9.7 in2 (1 point)

4. 283.3 m2 (1 point)

5. 1695.6 ft2 (1 point)

6. 12.6 cm2 (1 point)

7. 156.02 in2 (1 point)

8. 252 cm3 (1 point)

9. 300 yd3 (1 point)

10. 816.4 ft2 (1 point)

11. 192 in3 (1 point)

12. 12 ft (1 point)

13. Volume is increased/multiplied by 12 (1 point for increase or 2 points for multiplied by 12)

14. Hexagon (1 point)

15. Triangle (1 point)

Total: 18 points

16

Day-By-Day Unit Framework

Unit Overview

Monday Tuesday Wednesday Thursday Friday

Inductive Lesson

Activity with nets

Direct Lesson

Surface Area of

Prisms (9.1)

Direct Lesson

Surface Area of

Pyramids (9.2)

Inductive/Direct

Lesson

Surface Area of

Cylinders (9.3)

Cooperative

Lesson

Review and Quiz

9.1-9.3

Cooperative/Direct

Lesson

Volume of Prisms

(9.4)

No School

No School

Direct Lesson

Volume of

Pyramids (9.5)

Review Game

Review for Test

Explain Activity

Chapter Test

Activity Due

17

Lesson Plans and Materials

Chapter 9 Introduction Activity

Topic: Nets of Prisms and Pyramids

Duration: 1 class period

Materials: Nets, scissors, glue/tape, pencil, paper

Standards: 7.G.6 Solve real-world and mathematical problems involving area, volume and

surface area of two- and three-dimensional objects composed of triangles, quadrilaterals,


Objectives: Create formulas for finding surface area of prisms and pyramids by using nets.

Purpose: The purpose of this lesson is to introduce students to some of the shapes that we will

be learning about in the chapter. These shapes include prisms and pyramids. Students will be

putting nets together and trying to figure out how they could find the surface area of the three-

dimensional figure the nets create. This is an important activity to get them thinking about

finding surface area of different figures.

Anticipatory Set: Students will be given a warmup exercise that has them finding the area of

squares, rectangles and triangles. This is a good refresher for the activity that will follow this

warmup.

Activity: For this activity, students will be put into partners. Each group will be given six

different nets: cube, rectangular prism, triangular prism, square pyramid, rectangular pyramid

and triangular pyramid. The groups will also need scissors and some tape, as well as some paper.

Students will be instructed to cut out the nets and put them together. They will be told to label

different things on the three-dimensional shapes they make, such as length, width and height.

After putting the nets together and labeling these things, they will be instructed to take the nets

apart and lay them flat again. From what they have labeled, they must discover how to find the

area of each figure. They will record the formulas they come up with on a separate sheet of

paper.

Check for Understanding: To check for understanding, after all of the groups are finished or

almost finished, I will ask different groups for different formulas that they came up with for each

figure. We will compare formulas as a class and decide which formulas fit best for each figure.

Assessment: After going over the different formulas as a class, students will turn in the work

that they did on the separate sheet of paper. This will allow me to see more closely where

different students may have gotten different formulas. For homework following this lesson,

18

students must go home and list a real-life example of each figure that we worked with that they

might find in their house and try to calculate the surface area of at least one object.

Closure: To close this lesson, I will tell students that what they did today will be the base for the

first two sections in the chapter. The work they did today will help them in finding the surface

area of different three-dimensional figures, such as prisms and pyramids. I will ask if there are

any questions.

Adaptations/Differentiation: If I see certain students struggling, then I may switch up the

partners. I would make sure to put struggling students with students who seem to be grasping the

information. If struggling continues, I will sit down with individual students and work through

the activity with them.

19

Chapter

9 Fair Game Review

Name ________________________________________________________ Date _________

Find the area of the square or rectangle.

1. 2.

3. 4.

5. 6.

7. An artist buys a square canvas with a side length of 2.5 feet. What is the

area of the canvas?

20

Chapter

9 Fair Game Review (continued)

Name ________________________________________________________ Date _________

Find the area of the triangle.

8. 9.

10. 11.

12. 13.

14. A spirit banner for a pep rally has the shape of a triangle. The base of the

banner is 8 feet and the height is 6 feet. Find the area of the banner.

27

Chapter 9 Section 1 Lesson

Topic: Surface Area of Prisms


Materials: pencil, paper, notebooks, whiteboards, textbook




Objectives: Calculate the surface area of a prism.

Purpose: The purpose of this lesson is to apply the formulas that the students came up with the

day before for prisms. This lesson will teach them how to find the surface area of a prism, which

is important for many things, such as making enough cardboard in the right dimensions to make

a box, or making sure you have enough wrapping paper to wrap a gift.

Anticipatory Set: Students will be given a warmup exercise that has them adding up three sets

of three numbers multiplied together. This is good practice for finding surface area of a

rectangular prism.

Input/Modeling: Students will take the following notes with me in their notebooks.

Surface Area = Area of Bases + Area of Lateral Faces

Surface Area is measured in squared units (ex: cm2)

Surface Area of a Rectangular Prism

Ex: A rectangular prism with a height of 6 inches, a base length of 3 inches and a base

width of 5 inches. What is the surface area?


𝑆𝐴 = 2𝑙𝑤 + 2𝑙ℎ + 2𝑤ℎ

𝑆𝐴 = 2(3)(5) + 2(3)(6) + 2(5)(6)

𝑆𝐴 = 126𝑖𝑛2

Surface Area of a Triangular Prism

Ex: A right triangular prism with a height of 6 meters, a base length of 4 meters and a

base height of 3 meters with the remaining side of the base being 5 meters.


Area of Bases: 1

2∗ 3 ∗ 4 = 6 ∗ 2 = 12

Area of Lateral Faces: 1) 3 ∗ 6 = 18 2) 5 ∗ 6 = 30 3) 4 ∗ 6 = 24

𝑆𝐴 = 12 + 18 + 30 + 24

𝑆𝐴 = 84𝑚2

28

Surface Area of a Cube

Ex: A cube with side lengths of 12 centimeters.


𝑆𝐴 = 6𝑠2 (s = side)

𝑆𝐴 = 6 ∗ 122

𝑆𝐴 = 864𝑐𝑚2

Lateral Surface Area: sum of the area of only the lateral faces

Check for Understanding: To check for understanding, students will be asked to keep out their

notebooks for reference and get out their whiteboards. I will ask them to answer some related

questions on their whiteboards. After students have finished the problem, I will ask them to raise

their hand and I will check their answers or I will have them all hold up their boards for me to

see at the same time. We will go over a question once everyone has attempted at least once if

there are several difficulties.

Guided Practice: Students will be assigned practice problems from their textbook to work on.

When finished, they will be asked to bring their work back to be checked so I can see if and

where different students are struggling. If several students are struggling, we will come back

together as a class for some re-teaching and clarifications.

Closure: To close this lesson, I will ask students what the generic formula for finding surface

area is. I will also ask if there are any questions.

Independent Practice/Assessment: Students will be given Practice 9.1 A to work on for

homework that night.

Adaptations/Differentiation: During independent work time, I will work one-on-one with the

students who I noticed were struggling from the formal assessment on the white boards. Some

students also get pulled out or get extra help from a math interventionist during independent

work time. Students with IEPs have notes scanned for them to take home and shortened

homework assignments.

29

Activity

9.1 Start Thinking! For use before Activity 9.1

Activity

9.1 Warm Up For use before Activity 9.1

How can you determine the amount of

cardboard used to make a cereal box? List at

least two different methods.

Evaluate the expression.

1. 2 2 5 2 2 3 2 5 3

2. 2 1 4 2 1 2 2 4 2

3. 2 6 3 2 6 1 2 3 1

4. 2 3 7 2 3 5 2 7 5

5. 2 2 2 2 2 4 2 2 4

6. 2 4 8 2 4 10 2 8 10

30

9.1 Practice A

Name ________________________________________________________ Date _________

Find the surface area of the prism.

1. 2.

3. 4.

5. 6.

7. The inside of a baking pan is to be lined with tinfoil. The pan is 12 inches long,

9 inches wide, and 1.5 inches tall. How many square inches of tinfoil are needed?

8. Draw and label a rectangular prism that has a surface area of 96 square meters.

31


Topic: Surface Area of Pyramids






Objectives: Make use of previous knowledge of surface area of prisms to develop a formula to

use to find the surface area of pyramids.

Purpose: The purpose of this lesson is to teach students how to use their previous definition of

surface area of prisms to find the surface area of pyramids. Surface area, again, is important for

them to know because it is helpful when you are creating things. In this case, it could be helpful

if students are trying to find out how much material they would need to build a pyramid.

Anticipatory Set: Students will have a warm-up exercise that has them calculating the area of

triangles and the area of nets of pyramids.


Regular Pyramid: a pyramid whose base is a regular polygon

Slant Height: the height of the triangles of a pyramid

Surface Area = Area of Base + Area of Lateral Faces

Square Pyramid:

-Area of Base: 𝐴 = 𝑠2

-Area of Lateral Faces: 4 ∗ 𝐴 =1

2∗ 𝑏 ∗ ℎ (have to multiply by four because there are four

lateral faces)

Ex: A square pyramid has a slant height of 8 inches and a base length of 5 inches.


Surface Area = 52 + 4 ∗ (1

2∗ 5 ∗ 8)

= 25 + 80

= 105𝑖𝑛2

Triangular Pyramid:

-Area of Base: 𝐴 =1

2∗ 𝑏 ∗ ℎ

-Area of Lateral Faces: 3 ∗ 𝐴 =1

2∗ 𝑏 ∗ ℎ (have to multiply by three because there are

three lateral faces)

32

Ex: A triangular pyramid has a slant height of 14 meters, base length of 10 meters and a

base height of 8.7 meters.


Surface Area = 1

2∗ 10 ∗ 8.7 + 3 ∗ (

1

2∗ 10 ∗ 14)

= 43.5 + 210

= 253.5𝑚2







Guided Practice: Students will be assigned practice problems from the book to work on. When

finished, they will be asked to bring their work back to be checked so I can see if and where

different students are struggling. If several students are struggling, we will come back together as

a class for some re-teaching and clarifications.

Closure: To close this lesson, I will ask a few students to summarize how to find the surface

area of a pyramid and ask for any questions that anyone may still have.








33

Activity


Activity


Are the sides of a pyramid always triangles?

Explain.

Is the base of a pyramid always a triangle?

Explain.

Find the area.

1. 2.

3. 4.

34

Lesson

9.2 Warm Up For use before Lesson 9.2

Lesson

9.2 Start Thinking! For use before Lesson 9.2

Your neighbor needs to put a new roof on his

gazebo. The roof is an octagonal pyramid. Why

would knowing the surface area of the roof be

useful information?

Use the net to find the surface area of the

regular pyramid.

1. 2.

35

9.2 Practice A

Name ________________________________________________________ Date _________

Use the net to find the surface area of the regular pyramid.

1. 2.

Find the surface area of the regular pyramid.

3. 4.

5. Your friend is purchasing an umbrella with a slant height of 4 feet. There are a variety of

such umbrellas to choose from.

a. A red umbrella is shaped like a regular pentagonal pyramid with a side length of 3 feet.

Find the lateral surface area of the red umbrella.

b. A yellow umbrella is shaped like a regular hexagonal pyramid with

a side length of 2.5 feet. Find the lateral surface area of the yellow umbrella.

c. A blue umbrella is shaped like a regular octagonal pyramid with a side length of 1.9

feet. Find the lateral surface area of the blue umbrella.

d. Based on lateral surface areas, would you suggest that your friend pick the umbrella

that is her favorite color? Explain.

36


Topic: Surface Area of a Cylinder


Materials: pencil, paper, notebooks, whiteboards, textbook, soup can, toilet paper roll

Standards: 7.G.4 Know the formulas for the area and circumference of a circle and use them to

solve problems: give an informal derivation of the relationship between the circumference and

area of a circle.

Objectives: Calculate the surface area and lateral surface area of a cylinder.

Purpose: The purpose of this lesson is to help students understand how to find the surface area

of a cylinder. This is important for many reasons. There are lots of every-day cylinders that we

see. Knowing how to find the surface area of these cylinders would be important to know when

making any of these cylindrical items, such as car parts, cans, labels, etc.

Anticipatory Set: Students will have a warm-up exercise that has them calculate the area of a

circle, which will be useful when finding the surface area of cylinders.

Input/Modeling: To start off the lesson, I will ask students for different examples of cylinders

that they know of. Once I get them thinking, I will hold up a can of soup that I will bring in. This

shows them an example of a cylinder. We will discuss how the bases of a cylinder are circles. I

will review with the students how to find the area of the bases. To discuss how to find the area of

the lateral face of a cylinder, I will bring in an empty toilet paper roll that is cut along the height

of the roll. I will ask students what type of shape they think the lateral surface makes. After a few

answers, I will unroll the tube and show them it is a rectangle. I will ask them how to find the

area of a rectangle (length x width). They know how to find the width (height), but I will ask

how I might find the length. I will roll the tube back up, motion to the circular form, and then

unroll it again. Students should conclude that you use the circumference of the base for the

length of the lateral face of a cylinder.

Following this introduction, students will take the following notes with me in their

notebooks:

Surface Area of a Cylinder

Surface Area = Area of Bases + Area of Lateral Surface

Surface Area = 2𝜋𝑟2 + 2𝜋𝑟ℎ

h

r h

37

Ex: A cylinder has a radius of 4 mm and a height of 3 mm. Find the surface area.

Surface Area = 2𝜋𝑟2 + 2𝜋𝑟ℎ

= 2 ∗ 𝜋 ∗ 42 + 2 ∗ 𝜋 ∗ 4 ∗ 3

= 32𝜋 + 24𝜋

= 56𝜋

= 175.8𝑚𝑚2

Lateral Surface Area of a Cylinder

Ex: A can of peas has a radius of 1 inch and a height of 2 inches. How much paper is

used for the label of the can?

Lateral Surface Area = 2𝜋𝑟ℎ

= 2 ∗ 𝜋 ∗ 1 ∗ 2

= 4𝜋

= 12.56𝑖𝑛2











Closure: To finish the lesson, I will ask students to summarize what we learned about finding

the surface area of a cylinder and ask for any questions that they may still have.








38

Activity


Activity


Give a real-life example of when it would be

useful to know the surface area of a cylinder.

Find the area. Use 3.14 for .

1. 2.

3. 4.

39

9.3 Practice A

Name ________________________________________________________ Date _________

Make a net for the cylinder. Then find the surface area of the cylinder.

Round your answer to the nearest tenth.

1. 2.

Find the surface area of the cylinder. Round your answer to the

nearest tenth.

3. 4.

Find the lateral surface area of the cylinder. Round your answer to the

nearest tenth.

5. 6.

7. A deep dish pizza has a radius of 6 inches and a height of 1 inch. Find the

surface area of the pizza. Round your answer to the nearest tenth.

40

9.1-9.3 Quiz Review

Topic: Review of Surface Area

Duration: 45 minutes

Materials: pencil, paper, textbook, partner worksheet

Standards:


problems: give an informal derivation of the relationship between the circumference and

area of a circle.

7.G.6 Solve real-world and mathematical problems involving area, volume and surface

area of two- and three-dimensional objects composed of triangles, quadrilaterals,


Objectives: Review finding surface area of prisms, pyramids and cylinders.

Social Objective: Students should work together with their partners to answer problems on the

practice quiz and come up with questions to ask before taking the quiz.

Purpose: The purpose of this lesson is for students to review together before taking their quiz on

the first three sections of the chapter. These sections are about the surface area of prisms,

pyramids and cylinders.

Anticipatory Set: Students will be given a warmup activity that has them finding the surface

area of cylinders.

Input/Modeling: Students will be told to open up to page 375 in their textbooks. On this page is

a practice quiz. They will be working together with a partner to practice these problems before

taking their actual quiz. I will tell students that they will have 25 minutes to work with their

partners on these problems. If they finish early, they are encouraged to go back and check their

work and to look over their notes. They will be told to raise their hand if they have any questions

while working through the problems. Each student is responsible for doing their own work, so

each student should be working out the problems on their own sheet of paper. After the 25

minutes are up, whether everyone is done or not, we will go over the answers to each problem

and answer any questions that the students may have. Once I have finished going over the

directions, students will be told which partner they are working with and they should get started

on the problems.

Check for Understanding: After giving the students 25 minutes to work through the problems

with their partners, we will go over the answers as a class. I will go through each problem and

41

ask for different groups to give me their answers. If no groups have the correct answer for a

problem, we will work through that problem together.

Guided Practice: Students will be working on the practice quiz with a partner. This practice will

help them discover any questions that they may have before they take the quiz.

Closure: To close this lesson, I will ask for any remaining questions we may not have gotten to.

Students will be instructed to go back to their seats and clear off their desks of everything except

for a pencil.

Independent Practice/Assessment: Students will be given the 9.1-9.3 Quiz following this

lesson.




the problems with them. Students who do not finish the quiz will be given extra time the

following class period to finish their quizzes.

42

Lesson

9.3 Warm Up For use before Lesson 9.3

Lesson

9.3 Start Thinking! For use before Lesson 9.3

Explain which cylinder has a greater surface

area:

Radius: 4 cm; Height: 10 cm

Radius: 10 cm; Height: 4 cm

Make a net for the cylinder. Then find the

surface area of the cylinder. Round your

answer to the nearest tenth.

1. 2.

3. 4.

43

Chapter

9 Quiz For use after Section 9.3

Name ________________________________________________________ Date _________

Find the surface area of the prism.

1. 2.

Find the surface area of the regular pyramid.

3. 4.


5. 6.

Find the lateral surface area of the cylinder. Round your answer to the nearest tenth.

7. 8.

9. The surface area of a square pyramid is 136 square inches. The base length is 4 inches.

What is the slant height?

10. You buy two rolls of wrapping paper. Each roll has the same lateral surface area. What is

the diameter of Roll B?

44

Chapter 9 Section 4 Activity/Lesson

Topic: Volume of a Prism


Materials: pencil, record and practice journals (RPJs), partner worksheet, notebook, paper,

textbook




Objectives: Find the volume of prisms using prior knowledge of calculating area of different

polygons.

Social Objective: Students must work together with their partners to figure out their own

formula for the volume of a prism.

Purpose: The purpose of this lesson is to create a formula that can be used to find the volume of

a prism. Knowing the volume of a prism is important for things like knowing how much water

will fit in an aquarium, knowing if a certain product will fit in a box, or knowing if all of your

belongings will fit into a certain size room.

Anticipatory Set: Students will have a warm-up exercise that has them multiplying three

numbers together, which is good to practice, since finding volume of prisms requires multiplying

length, width and height.

Input/Modeling: Students will be asked to open their RPJs to page 197. The activity they will

be working with will have them looking at a treasure chest with dimensions 120cm x 60cm x

60cm. They will have to try to find the number of pearls in the chest two different ways; one way

where they put even square pearls of 1 cubic centimeter in even layers, and one where they could

weigh the pearls. They will be asked to come up with a formula for the first method, and they

should obtain 𝑉 = 𝑙𝑤ℎ. After directions are given, they will be told which partner they will be

working with off of their partner time sheet.

After that activity, students will be asked to go back to their assigned seat and get out their

notebooks and take the following notes.

Volume: measure of the amount of space a three-dimensional object occupies; measured

in units cubed.

Volume = Area of Base * Height

Ex: A rectangular prism has a height of 15 yards, a length of 8 yards and a width of 6

yards. Find the volume.

-Area of Base: 𝐴 = 𝑙 ∗ 𝑤

45

𝑉 = 𝑏 ∗ ℎ

𝑉 = 6 ∗ 8 ∗ 15

= 720𝑦𝑑3

*Note: a rectangular prism has a base area of length x width. Therefore, an alternate

formula for a rectangular prism could be 𝑉 = 𝑙𝑤ℎ.

Ex: A triangular prism has a base height of 2 inches, a base of 5.5 inches, and a height of

4 inches. Find the volume.

-Area of Base: 𝐴 =1

2∗ 𝑏 ∗ ℎ

𝑉 = 𝑏 ∗ ℎ

𝑉 =1

2∗ 5.5 ∗ 2 ∗ 4

= 22𝑖𝑛3

After notes, we will go over a real-life example of popcorn bags at movie theaters, looking at

two rectangular prisms that have the same area but different surface area and they have to

determine which bag would have the least amount of waste.







Guided Practice: Students will be given the practice problems on the next page to practice the

skills they have just worked with individually. This way, I can check that each student actually

understands the topics rather than just copying off of their partner.

Closure: To finish off this lesson, we will discuss different benefits of knowing how to find the

area of a prism. I will also ask for any more questions that the students may have.






the problems with them.

46

Activity


Activity


Do two-dimensional figures have volume?

Explain.

Do three-dimensional figures have volume?

Explain.

Multiply.

1. 7 5 8 2. 12 7 8

3. 13 10 7 4. 11 15 3

5. 14 20 4 6. 12 16 21

47

9.4 Volumes of Prisms For use with Activity 9.4

Name ________________________________________________________ Date _________

Essential Question How can you find the volume of a prism?

Work with a partner. A treasure chest is filled with

valuable pearls. Each pearl is about 1 centimeter in

diameter and is worth about $80.

Use the diagrams below to describe two ways that

you can estimate the number of pearls in the

treasure chest.

a.

b.

c. Use the method in part (a) to estimate the value of the pearls in the chest.

1 ACTIVITY: Pearls in a Treasure Chest

48

9.4 Practice A

Name ________________________________________________________ Date _________

Find the volume of the prism.

1. 2.

3. 4.

5. 6.

7. A cell phone is in the shape of a rectangular prism, with a length of 4 inches,

a width of 2 inches, and a height of 1 inch. What is the volume of the cell phone?

8. A recycle bin is in the shape of a trapezoidal prism. The area of the base

is 220 square inches and the height is 24 inches. What is the volume of

the recycle bin?

9. A water jug is in the shape of a prism. The area of the base is 100 square inches

and the height is 20 inches. How many gallons of water will the water jug hold?

31 gal 231 in. Round your answer to the nearest tenth.

49


Topic: Volume of Pyramids/Cross-sections



Standards:






Objectives:

Find the volume of a pyramid.

Identify and describe cross-sections of planes and three-dimensional figures.

Purpose: The purpose of this lesson is to find the volume of a pyramid. This could be useful if

you were to have a pyramid bowl, candle holder, cup, etc. You would be able to figure out how

much of a substance it could hold. People who built pyramids throughout history have also had

to know this so they knew how big to build their pyramid to fit what they needed to.

Anticipatory Set: Students will have a warm up that deals with multiplying fractions with whole

numbers to get them ready to multiply by 1

3 when using the formula for finding the volume of a

pyramid (𝑉 =1

3𝐵ℎ).


Volume of a pyramid: 𝑉 =1

3𝐵ℎ, where B is the area of the base, and h is the height.

o Volume is measured in cubic units, or units cubed

Ex: A triangular pyramid has a height of 10m, a base height of 6m, and a base length of

17.5m. What is the volume of the pyramid?

𝑉 =1

3𝐵ℎ

Ask: “How do I find the area of the base (B)?” 𝐵 =1

2𝑏ℎ since it is a triangular base.

𝑉 =1

3(

1

2) (6)(17.5)(10)

𝑉 = 175𝑚3

50

Ex: A rectangular pyramid has a height of 7ft, a base length of 3ft, and a base width of

4ft. What is the volume of the pyramid?

𝑉 =1

3𝐵ℎ

Ask: “How do I find the area of the base (B)?” 𝐵 = 𝑙𝑤 since it is a rectangular base.

𝑉 =1

3(3)(4)(7)

𝑉 = 28𝑓𝑡3

Ex: A pyramid with a hexagonal base (𝐵 = 48𝑚𝑚3) has a height of 9mm. What is the

volume of the pyramid?

𝑉 =1

3𝐵ℎ

𝑉 =1

3(48)(7)

𝑉 = 144𝑚𝑚3

Cross-section: a two-dimensional shape created by the intersection of a plane and a three-

dimensional solid.











Closure: To close this lesson, I will have students turn in the guided practice that they were

working on and ask if there are any questions about finding the volume of pyramids.






51

Activity


Activity




Explain the difference between the slant height

of a pyramid and the height of a pyramid.

Which do you use for volume? Which do you

use for surface area?

Multiply.

1. 2

153 2.

38

4 3.

76

10

4. 1

183 5.

530

9 6.

472

13

52

9.5 Practice A

Name ________________________________________________________ Date _________

Find the volume of the pyramid.

1. 2.

3. 4.

5. A tent is in the shape of a pyramid. The base is a rectangle with a length

of 12 feet and a width of 10 feet. The height of the tent is 8 feet. Find the

volume of the tent.

6. A sign made of solid wood is in the shape of a pyramid. The base is a

triangle with a base of 6 feet and a height of 4 feet. The height of the sign

is 7 feet. The wood costs $3 per cubic foot. What is the cost of the sign?

7. Two pyramids with square bases have the same volume. One pyramid has

a height of 6 centimeters and the area of the base is 36 square

centimeters.

a. What is the volume of the pyramids?

b. The base of the other pyramid has a side length of 3 centimeters. What

is the height of this pyramid?

8. How does the volume of a pyramid change when the height is halved?

53

Chapter 9 Review Game (Grudge Ball)

Topic: Surface Area and Volume


Materials: whiteboards, dry erase markers, erasers, ball, hoop

Standards:




problems; give an informal derivation of the relationship between the circumference and

area of a circle.




Objectives: Have students review and recall everything they have learned about surface area and

volume including: surface area of prisms, pyramids and cylinders and volume of prisms and

pyramids.

Social Objectives: Students should be able to work together with their groups to solve the

problems. They should also be able to agree on who can have what jobs in the groups.

Purpose: The purpose of this activity is to get students to review materials in a fun way before

taking their test the next day.

Anticipatory Set: The students will be told that they will be playing grudge ball today and that

they will need to get into groups of 4 or 5 (depending on how many students are present that

day).

Input/Modeling: Students already know the rules of the game, but just in case, the rules are:

1. Students will get into groups and assign one person as the answerer and one person as

the erasinator. The answerer will be the only person who will hold up answers for the

group, and the erasinator is the person who erases the X’s from other teams.

2. Once the erasinator is decided, they must go up to the board and put 10 X’s under

their team’s number.

3. Once the X’s are drawn, a problem will be displayed on the board. Students will have

anywhere from 10 seconds to 1 minute to answer the problem, depending on the

difficulty and type of problem.

54

4. After time is up, the answerer must hold up the answer for the whole group. If a

group gets the correct answer, they have the opportunity to shoot a small basketball

into a small hoop.

a. If the student makes the basket from the two point line, they may erase 4 of

another team’s X’s.

b. If the student makes the basket from the three point line, they may erase 5 of

another team’s X’s.

c. If the student does not make the basket, they may erase 2 of another team’s

X’s.

5. This process continues. If a team does not answer a question correctly, they do not get

to erase X’s.

6. Once a team runs out of X’s, they may earn back X’s by answering a problem

correctly and making a basket. They may not erase X’s if they have no points, but

they must still participate and answer questions.

7. The last team who has X’s is the winning team. If there is a tie, there will be a shoot

out.

Check for Understanding/Guided Practice: By doing these problems that I choose, I can see

which groups are struggling with which topics. It also gives them good practice with the help of

their group members. If everyone gets a problem wrong, this is an opportunity for me to re-teach

that type of problem.

Closure: To close the lesson, students will have to get the desks back in rows and they will have

an opportunity to ask any remaining questions they may have.

Independent Practice/Assessment: Students will be given Chapter 9 Test A to work on for


55

Culminating Activity

At the end of the unit, students will be using the objectives they have learned to complete an

activity that has them applying their new knowledge. For this activity, they will go home and

find a household item that is a cylinder and a household item that is either a prism or pyramid.

They must use their knowledge gained in this unit to find the surface area and lateral surface area

of the cylinder as well as the surface area and volume of the prism or pyramid they chose.

Pictures of the items should be drawn or taken with a camera with the dimensions shown on the

picture. After recording their measurements and calculating these things, they must write a short

explanation of the processes they went through in order to do these calculations. Finally,

choosing only one of their objects, they must write a short diary entry as if they were a person

who was creating the item and why it is important for them to know how to calculate the surface

area, lateral surface area, or volume of this item.

Rubric for Culminating Activity

The point of this activity is for students to see that the concepts they have just learned are things

that people have to know in order to make the items that we take for granted every day. Students

will be graded on this project based on the accuracy of their results on a pass/fail basis. If the

calculations are correct with the given data, the student will pass the assignment. If the

calculations are incorrect with the given data, the student will fail the assignment.

56

Student Learning Analysis

Students were given a paper and pencil post-assessment in class that assessed them on three main

objectives. They were given a rating of N, L, or S for each objective. N = no evidence of

understanding, L = limited understanding, and S = substantial understanding.

Objective 1 N = 0-3 correct

L = 4-8 correct

S = 9-12 correct


L = 1-2 correct

S = 3-4 correct


L = 1 correct

S = 2 correct

Student Summary of Post-Assessment Information

Student Number

Objective 1:

Calculate surface

area/volume of


cubes.

Objective 2:


area and lateral

surface area of a

cylinder.

Objective 3:



dimensional solid.

1 S S S

2 L L L

3 L L S

4 N N S

5 N L S

6 L L S

7 L N S

8 L L S

9 L L S

10 L S S

11 L L S

12 L N S

13 L L S

14 S S S

15 L S S

16 L L S

17 N N S

18 L L S

19 L L S

20 S L S

21 L L S

22 S S S

23 S L S

24 L N S

25 L N S

26 L L L

27 L L S

57

Group Analysis of Student Learning

Objective/Outcome Assessment Strategy Summary of Results

Objective 1:

Calculate surface area/volume

of prisms, pyramids and

cubes.

Students were given a pencil

and paper assessment which

asked them a total of 11

questions involving this

objective.

For this objective, 5 students

received an S, 3 students

received an N, and the other

19 students received an L.


Objective 2:

Calculate the surface area and

lateral surface area of a

cylinder.





objective.

For this objective, 5 students

received an S, 6 students

received an N, and the other

16 students received an L.


Objective 3:

Identify cross sections of a

plane and a three-dimensional

solid.





objective.

For this objective, there were

only two students who

received an L. The other 25

students received an S.

58

Post-Assessment Data Entry and Graph


N 3 6 0

L 19 16 2

S 5 5 25

0

5

10

15

20

25

30


Nu

mb

er o

f ST

ud

ents

Objective

Unit Post-Assessment Results

N

L

S

59

Summary Statement for Student Learning

Based on the data collected in the post-assessment, there was evidence of learning in all three of

the objectives being tested. The first objective was the biggest group of the questions that were

found on the assessment. Before the unit, there were 23 students who had no understanding of

how to find surface area and volume of prisms and pyramids. After the unit, there were only

three students who had no understanding of these topics. Since this section was so big, 19

students fell into the “limited understanding” category. Looking over their work, I noticed that it

was mostly calculation errors that kept these students from getting enough questions correct to

bump them up to the “substantial understanding” category.

The second objective was testing the students on their ability to find surface area and lateral

surface area of cylinders. Before the unit, all 27 students showed no understanding of this

objective. After going through the unit, only six students fell into the “no understanding”

category. Just like with the first objective, the 16 students who fell into the “limited

understanding” category were just making simple calculation errors. Most of them you could see

the correct formulas on the paper, but they made silly mistakes along the way. The other five

students showed substantial understanding of this objective.

The third objective was the one that had the most growth. In the pre-assessment, there was only

one student who showed substantial understanding of the concept of cross sections. The other 26

students showed no understanding. After the unit, 25 of the students were able to show

substantial understanding and the other two students were able to show limited understanding of

cross sections.

While teaching this unit, I can tell that most students were learning the concepts. Even though

the post-assessment data does not show what looks like a significant increase in learning, I feel

that if students were allowed to use calculators, the simple calculation errors that they made

would have been fixed and almost everyone would be moved up a level on the N, L, S scale.

Also, the assessment only covers three objective of the unit, so there was learning in other areas

that were not assessed by this post-assessment.

60

Individual Analysis

The student I chose for my learning assessment showed substantial growth from the pre-

assessment to the post-assessment. In the pre-assessment, she only got one question on the entire

assessment correct. In the post-assessment, she showed substantial understanding of two

objectives and limited understanding of the other objective.

As I had mentioned in my pre-assessment analysis, this student is a very smart young girl.

However, she gets easily frustrated when she doesn’t get things as quickly as the others in the

class. This frustration usually leads to her making simple mistakes, such as remembering an

incorrect formula or making a simple calculation error. To accommodate this student for the unit,

I made sure to pay close attention to her moods. If she was having an off day, I would be sure

that I would go over and talk to her and help her slow down on problems during independent

work time. If I noticed that she was doing really well with a certain topic, I would have her help

someone else who may have been struggling to give her a little confidence boost for herself.

Finally, another thing I did for this student was I made sure to pair her with other people in the

class who generally take about the same amount of time to pick up on things as she does. By

doing this, I was hoping to reduce her frustrations when she would not pick up on something

right away so she could take more time to fully understand the material.

After the accommodations I made for this student, I feel like I saw great progress in her learning

of this unit. While her results may say that she only has substantial learning in two of the three

objectives, I feel like she has made substantial growth in all of the categories.

61

Chapter

9 Post-Test

Answers

1.

2.

3.

4.

5.

6.

7.

Name ________________________________________________________ Date _________

Find the surface area and volume of the prism.

1. 2.

Find the surface area of the regular pyramid. Round your answer to

the nearest tenth.

3. 4.


5. 6.

7. Find the lateral surface area of the

paint can. Round your answer to the

nearest hundredth.

62

Chapter

9 Post-Test (continued)

Answers

8.

9.

10.

11.

12.

13.

14.

15.

Name ________________________________________________________ Date _________

Find the volume of the regular pyramid.

8. 9.

10. Find the surface area of the 11. Find the volume of the

composite solid below. composite solid below.

12. The volume of a pyramid is 84 cubic feet. The area of the base is

21 square feet. Find the height of the pyramid.

13. What happens to the volume of a rectangular prism when the length

and width are doubled and the height is tripled?

Describe the intersection of the plane and the solid.

14. 15.

63

Post-Assessment Key

16. SA = 66.6 m2, V = 28.8 m

3 (2 points)

17. SA = 300 in2, V = 240 in

3 (2 points)

18. 9.7 in2 (1 point)

19. 283.3 m2 (1 point)

20. 1695.6 ft2 (1 point)

21. 12.6 cm2 (1 point)

22. 156.02 in2 (1 point)

23. 252 cm3 (1 point)

24. 300 yd3 (1 point)

25. 816.4 ft2 (1 point)

26. 192 in3 (1 point)

27. 12 ft (1 point)

28. Volume is increased/multiplied by 12 (1 point for increase or 2 points for multiplied by 12)

29. Hexagon (1 point)

30. Triangle (1 point)

Total: 18 points

64

Reflection and Self Evaluation

The objective that the class showed the most success in was the third objective. In the pre-test,

only one person showed substantial understanding of this objective. After the post-test, two

students showed limited understanding of this objective while the other 25 students in the class

showed substantial understanding. While this was the section that we spent the least amount of

time on, it was probably the easiest one to grasp. It was also the last thing we covered before the

test, so it was fresh in their minds. I think the only reason more students did not show

understanding on the pre-test is because they did not know what a cross section was. Once they

knew what it was, the two problems that measured the third outcome became very easy.

The objective that the class showed the least success in was the second objective. In the pre-test,

all 27 students showed no understanding of the objective. After the post-test, there were still six

students who were showing no understanding of the objective. While there were five students

who showed significant understanding of this objective, this still left me with 16 students who

showed limited understanding. I think the biggest reason for this is the fact that this objective

dealt with circles, which meant that there was a lot of multiplying and decimals involved. With

lots of multiplying, quite a few students made simple calculation errors. Looking through their

work, most students would have gotten more problems correct if it weren’t for some type of

calculation error. For these assessments, I think I should have given them calculators. Another

thing that I think lead to most of the students only showing limited understanding was the fact

that this section was heavy on memorization of formulas. A lot of students were able to

remember the formulas and had them written on their papers somewhere, but there were a few

who were missing key parts of formulas. I felt like I explained the formulas well, and I had even

brought in visuals to help them derive the formulas. I think it would have been more beneficial if

I had given all of them their own individual cardboard rolls and had each individual person try to

figure it out, rather than going over it as a class. Even having the students work collaboratively in

small groups would have been more beneficial than a whole class discovery.

The first objective was very similar to the second one in the results. There were 23 students with

no understanding on the pre-test and four students with limited understanding. After the post-

test, only three students showed no understanding, 19 showed limited understanding and the

other five showed substantial understanding. This was a bigger objective, so a lot more was

covered in it. I feel like calculation errors and not remembering correct formulas or methods

were the big factors in student performance. While the data shows that three students showed no

understanding and 19 showed limited, I can tell that the students learned something because they

were getting some questions right, but it wasn’t always enough to bump them up to the next level

of understanding. If it weren’t for calculation errors, I think all of the students would have made

it up to at least limited understanding, and quite a few students who made it to limited

understanding would have been bumped up to substantial understanding.

65

In my instruction, I tried really hard to reinforce processes when going over problems. I think it

would have been more beneficial for students to get more individual practice with processes

rather than practicing with the whole group. Another thing that I would do differently would be

to allow the students to use calculators on the assessments. As I mentioned before, if it weren’t

for silly calculation errors, most students would have done much better on the post-assessment. I

saw the same results when the students took a quiz half way through the chapter. The test that the

students take for the school allowed them to use calculators. If I would have known that, then

they would have been able to use calculators on my assessments as well.

I feel like my unit went well overall. I am still learning different ways to deliver materials and

help students understand. As I mentioned before, I think I would have students do more

individual or collaborative work for practice rather than practicing so much as a whole group. I

know that it is beneficial for students to work collaboratively, and I thought that my whole class

teaching would be good enough for that. I feel like my misunderstanding may have been part of

the reason that some students were still showing no understanding of the first two objectives.

However, most students were able to write correct formulas and processes on their papers, so I

don’t think my teaching methods were too destructive. In the future, I would like to try more

collaboration between the students in smaller groups rather than as a whole.

In doing this unit, I have begun to see just how much the little things matter in teaching. The way

you teach has an effect on student learning. Each student learns differently, so I have to be sure

to teach things in many different ways. I know I am guilty of finding something I like and

sticking to it, but I know that I need to change it up a little bit for future teaching. This way, I can

hopefully reach more diverse learners and keep the students interested. I think that another thing

I would change in the future is allowing students to use things like calculators more often. While

it is good that they practice doing everything by hand, the students’ assessment grades should not

suffer because of calculation errors. Assessments should be more about whether or not the

student learned the processes that I am teaching, not about whether or not they can produce the

correct calculations every time. Students are humans, and humans make mistakes. I think that, in

the future, to eliminate the emphasis on correct answers, I would allow my students to use

calculators for assessments so I can see who really learned what I was trying to teach them.

66

Teaching Materials

Pencils

Scissors

Tape

Record and Practice Journal (RPJ)

Textbook

Notebook

Document Camera

Projector

Individual Whiteboards

Whiteboard Markers

Whiteboard Erasers

Toilet Paper Tube

Can of Soup

All handouts are included in lesson plans

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References

Big Ideas Learning, LLC. (2014). Record and practice journal. Erie, PA: Larson Texts,

Inc.

Big Ideas Learning, LLC. (2014). Teacher resource dashboard. Retrieved from

https://www.bigideasmath.com/teachers/

Cathcart, W. G., Pothier, Y. M., Vance, J. H., & Bezuk, N. S. (2001). Learning mathematics

in elementary and middle schools (2nd

ed.). Upper Saddle River, NJ: Prentice-Hall, Inc.

Common Core. (2015). Grade 7geometry standards. Common Core State Standards

Initiative. Retrieved from http://www.corestandards.org/Math/

Fitzgerald, T. R. (2014). Math dictionary for kids (4th

ed.). Waco, TX: Prufrock Press, Inc.

Gijs Korthals Altes. (2015). Paper models of polyhedral. Retrieved from

http://www.korthalsaltes.com

Larson, R., & Boswell, L. (2014). Big ideas math: A common core curriculum. Erie, PA:

Larson Texts, Inc.

Miller, R. (2000). Geometry for the clueless. New York, NY: The McGraw-Hill Companies,

Inc.

Muschla, J. A., & Muschla, G. A. (2001). Geometry teacher’s activities kit: Ready-to-use

lessons and worksheets for grades 6-12. San Francisco, CA: Jossey-Bass.

Poskitt, K. (2010). Everyday math tricks for grown-ups: Shortcuts and simple solutions for

the not-so-math minded. White Plains, NY: The Reader’s Digest Association, Inc.

Sally, J. D., & Sally, P. J., Jr. (2011). Geometry: A guide for teachers. Providence, RI:

Mathematical Science Research Institute.

Tucker, B. F., Singleton, A. H., & Weaver, T. L. (2013). Teaching mathematics in diverse

classrooms for grades 5-8 (Vol. 2). Upper Saddle River, NJ: Pearson Education, Inc.

Van de Walle, J. A. (2001). Elementary and middle school mathematics: Teaching

developmentally (4th

ed.). Boston, MA: Addison Wesley Longman, Inc.

https://www.bigideasmath.com/teachers/

http://www.corestandards.org/Math/

http://www.korthalsaltes.com/

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