Unit Title: Volume and Surface Area

Unit Lesson #1: Volume of Prisms and Pyramids

This is the first lesson in a five-lesson unit that addresses CCRSM for Geometry to prepare students functioning at NRS Levels 5 and 6 for the GED assessment.

Author: Ashley Gootee, Muhlenberg County

Lesson Plan

Supporting materials are included following the lesson plan in this document:

Activating Activity

Formula Discussion

Formula Challenge

Guided Practice Handout

Independent Practice Handout

Lesson Self-Reflection

Muhlenberg

County:

Lesson Title: Volume of Prisms and Pyramids (Lesson 1 of 5) Unit Title (Optional): Volume and Surface Area

NRS Level(s): 5 Content Area(s): Mathematics Length of Lesson (e.g., hours, days): 2 hours

CCR Standard(s):

Overarching Standard(s):

E-G: Explain volume formulas and use them to solve problems.

D-G: Solve real-life and mathematical problems involving angle, measure, area, surface area, and volume.

C-EE: Apply and extend previous understandings of arithmetic to algebraic expressions.

CCR Reading Anchor 7: Integrate and evaluate content presented in diverse media and formats, including visually and quantitatively, as well as in words.

Level Specific Standard(s):

E-G: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. (G.GMD.3)

E-G: Apply concepts of density based on area and volume in modeling situations. (G.MG.2)

D-G: 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. (7.G.6)

C-EE: Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole number exponents, in the conventional order when there are no parentheses to specify a particular order (order of operations). For example, use the formulas V=s3 and A=6s2 to find the volume and surface area of a cube with sides of length s=1/2 (6.EE.2c)

Level D-Reading : Integrate information presented in different media or formats (e.g., in charts, graphs, photographs, videos, or maps as well as in words to develop a coherent understanding of a topic or issue. (RI.6.7)

Key Shift(s): Focus x Coherence x Rigor

KYAE Employability Standard(s):

E.1: Effectively contribute to a team through cooperation, leadership, and giving and accepting critical feedback to work toward a common goal.

E.6: Identify and effectively use skills and resources needed for a particular task.

E.7: Accurately analyze information and respond appropriately.

E.8: Interact with others in a professional manner.

Standards for Mathematical Practice

X_Makes sense of problems and perseveres in solving them

X_Reason abstractly and quantitatively

X_Construct viable arguments and critique the reasoning of others

__Model with mathematics

__Use appropriate tools strategically

X_Attend to precision

__Look for and express regularity in repeated reasoning

__Look for and make use of structure

Materials

Paper

Pencils

Personal Whiteboards

Markers

Erasers

Whiteboard

Tape

Rulers

Calculators: TI-30XS

GED Formula Sheet

Activating / Connecting / Discussion Sheets (provided with lesson)

Formula Discussion Sheet (provided with lesson)

Instructional Delivery Sheet (provided with lesson)

Formula Challenge Sheet (Provided with lesson)

Guided Practice Sheet (provided with lesson)

Independent Practice Sheet (provided with lesson)

Connecting / Extending Sheet (provided with lesson)

Net diagrams for rectangular prism, right prism, and pyramid

Real World Folding Geometric Shape (manipulatives)

Computers / iPads

Smartboard / Projector / Instructor Computer

Key vocabulary

Area

Volume

Cube

Prism

Rectangular Prism

Right Prism

Pyramid

Unit

Cubic Units

Parallel

Equal, Congruent

Similar

Base

Face

Edge

Vertex

Use of Technology

Students will need internet access to locate some information for some of the problems asked.

The classroom will need to be able to accommodate a video played for the class from learnzillion.com or each student could watch the video on individual iPads or computers if the classroom does not have a projector or smartboard.

Lesson Purpose

The purpose of this lesson is to expand the knowledge of using the volume formulas. Students will use volume formulas to calculate volume of prisms and pyramids as well as calculate missing dimensions. To be able to use and explain formulas in order to solve problems is a major work of level E Geometry.

Students will use many previously learned skills in this lesson in order to calculate volume and calculate the missing dimensions when given volume. Students will need to have an understanding of substitution and solving equations for a variable. Students will also need to be familiar with converting units of measure.

Lesson Objective(s)

Student Target

At the end of this lesson, students will be able to:

use formulas to solve for problems involving volume of rectangular prisms.

use formulas to solve for problems involving volume of right prisms.

use formulas to solve for problems involving volume of pyramids.

use formulas to solve for missing dimensions involving volume problems for rectangular prisms, right prisms, and pyramids.

apply concepts of density in the context of area and volume.

I can use formulas to calculate the volume of a rectangular prism.

I can use formulas to calculate the volume of right prism.

I can use formulas to calculate the volume of a pyramid.

I can use formulas to calculate for the missing dimension when given the volume.

I can apply the concept of density when given the area or volume.

Assessing Mastery of the Objective(s)

By the end of this lesson, the students will be able to

calculate the volume of a rectangular prism as evidenced by using the volume formula to complete real-world problems with 80% accuracy.

calculate the volume of a right prism as evidenced by using the appropriate formulas to complete real-world problems with 80% accuracy.

calculate the volume of a pyramid as evidenced by using the appropriate formulas to complete real-world problems with 80% accuracy.

calculate the missing dimensions of a rectangular prism, right prism, and a pyramid as evidenced by using appropriate formulas and correct units of measure to complete real-world problems with 80% accuracy.

apply concepts of density in the context of area and volume as evidenced by using the appropriate units of measure to calculate units per square foot or units per cubic foot with 80% accuracy.

Pre-teaching

Introduction and Explanation

Activating/Connecting:

(Note to instructor: This lesson is intended for NRS Levels 5 and 6. Students will need to have prior understanding of perimeter and area as well as substitution and solving for unknown variables in algebraic expressions.)

I need to fill my swimming pool for the summer. I will need to call the local pool supply store to schedule a water delivery truck. The company requires me to tell them the shape and the dimensions of the pool when I call. Why would they need to know the shape and dimensions?

Use the discussion questions provided in order to discuss why the dimensions and the shape of the pool would be required in order to fill the pool.

Discussion Questions:

While using the discussion sheet, discuss each item in detail to ensure understanding. After discussion, have the students open Socrative on the ipads and answer the following questions.

Quick assess:

1) Which formula would I use to calculate the amount of tile need for my bathroom floor? (area, volume, surface area)

2) Which formula would I use to calculate the amount of liquid could fill a container? (area, volume, surface area)

Formula Discussions:

Use the formula discussion sheet provided to discuss the formulas and what each piece of the formula means in detail. Also, students should realize that all formulas for volume consist of the area of the base and the height.

During this time, discuss that cubic units are used for volume and why, discuss that the bases of prisms are parallel and equal.

Draw and label the objects and their nets on the board, overhead, or Smartboard.

(Note: the use of nets for this activity can help students determine the bases and height of the prisms. The net of the pyramid does not allow for the height to be seen, but does allow the students to see the base and sides of the solid. Also, the basic use of nets, in this lesson, will help to prepare students for the lessons on surface area in this unit.)

After discussion is completed, have the students set these sheets aside for later use.

Teaching

Instructional Delivery

Understanding Formulas:

I will use the instruction sheet provided to discuss the reasons why students need to pay close attention to detail and choose the appropriate tools when solving for volume and/or solving for a missing dimension. While working with this handout, students will see why paying close attention to the units used to measure the dimensions are important. We will compare objects of different heights and take a closer look at the formulas for a prism and a pyramid to see why the pyramid is 1/3 the volume of the prism.

After working through the formulas, try some more difficult problems that do not necessarily give you the exact dimensions for the volume formulas. Explore using other formulas in order to solve for the volume or missing dimensions. (Use the handout provided for students to take notes and work through the problems.)

1) Jacob is looking to buy a fish tank for his sons room. He knows he wants a tank that will hold 30 cubic feet of water. The base can be no more than 3 feet long and 2 feet wide. What would the height of the tank be?

2) Candi is building planters for her patio. She wants each one to hold 8 cubic feet of dirt. The height of each planter will be 2 feet. If the base of the planter is a square, what is the length of each side of the base?

3) Sarah is working on a project for her Social Studies class. She knows that the volume of the large pyramid in Egypt is 2,592,100 m3. She also knows that the base is 52,900 m2. She needs to find the height of the smaller pyramid and all the information she has is that it is 1/3 the height of the larger pyramid. What is the height of the smaller pyramid?

4) LearnZillion Problem** (See below)

A square house has a pyramid-shaped attic built under the roof, which you are converting into a spare bedroom. You are installing an air conditioner to cool the room. The perimeter of the base of the room is 190ft, and the attic height is 6ft. How many cubic feet of air are in the empty attic?

(*Highlight the key details in the problem before working the problem.)

**Problem 4: Volume of a pyramid using LearnZillion.com video.

https://learnzillion.com/lesson_plans/8529-solve-real-world-problems-involving-pyramids

Pause at 1:10 to discuss key details in the problem.

Pause at 1:40 to discuss to try and work through the problem. Once students have looked at the problem and tried to work through with your help, play the rest of the video.

Guided Practice

Guided Practice #1:

I will give the students the guided practice sheet. This sheet goes back to the pool problem from the beginning of class. Students will be asked to determine the volume of the pool and to determine how much it would cost to fill the pool. The students will also have to think about and discuss what is needed to determine how many gallons of water will be needed. The students will be given some information that is needed, but will also need internet access to find other information. I will allow students to work in groups of two (three if necessary) to work through the problems.

Questions for this activity:

How much water is needed to fill the pool?

How much water can one delivery truck carry?

How many trucks are needed?

How much will the delivery cost?

Guided Practice #2:

The second question on this sheet asks students to determine the size and cost of a party tent. The students will need to understand area, volume, and capacity to complete each piece of the problem.

Questions for this activity:

How many people are invited?

How many square feet per person is needed?

What is the capacity of a tent this size?

How many cooling fans would be necessary?

How much would this tent cost?

After the students have an opportunity to discuss the problems, I will work them on the board allowing students to describe the processes used to complete each problem and ask any questions they may have.

Independent Practice

Assessment Activity:

These questions can be handed out as a worksheet or used on socrative or kahoot.

An extra practice sheet is provided for students to work on finding the volume of prisms and pyramids. The sheet also includes finding missing dimensions and some problems use different units of measure. Remind students to pay close attention to detail.

I will move around the room in order to ask questions and provide help as needed.

Post-teaching

Reflection, Closure, & Connection

Applying/Extending:

The students will prove the volume of a pyramid is 1/3 the volume of a rectangular prism. The students are given the challenge of proving this is true to a new classmate.

Students are expected to come up with their own ideas for creating this explanation and visual. A discussion of different options, allowing students to share ideas, is necessary. The main goal is for them to be able to explain and prove the formulas do work.

Extra practice:

Steck-Vaughn Mathematical Reasoning: Unit 4, Lesson 7 and 8

Student Book Pages 106-109; Workbook Pages 150-157

McGraw Hill Common Core Basics: Lesson 12.6 (Pages 360-365)

McGraw Hill Common Core Achieve: Lesson 7.3

Student Book (Pages 234-241)

Exercise Book (Pages 79-81)

New Readers Press Kaplan: Lesson 6 (Pages 388-391)

Preview of next lesson:

In the next lesson, students will continue work with volume formulas for cylinders, cones, and spheres. Have students look at the formulas for volume of a cylinder and a cone. Ask students to be thinking how these two formulas are similar to the formulas for rectangular prism and a pyramid.

Remind students to bring the handouts from this lesson to the next class. Some of the sheets will be used for the next lesson.

County

Muhlenberg

Email

Ashley.gootee@kctcs.edu

Instructor

Ashley Gootee

Director

Cris Crowley

Reviewed and approved by program director

Signature

2014/15 KYAE Lesson Plan Template for Mathematics

Activating Activity:

I need to fill my swimming pool for the summer. I will need to call the local pool supply store to schedule a water delivery truck. The company requires me to tell them the shape and the dimensions of the pool when I call. Why would they need to know the shape and dimensions?

Volume: Introduction and Explanation

Discussion Questions

What is volume?

When is volume used? What do you think of when you hear the word volume?

What is a two-dimensional object? Draw a two-dimensional object. What is a two-dimensional object in your daily life?

What is a three-dimensional object? Draw a three-dimensional object. What is a three-dimensional object in your daily life?

Formula Discussion:

Use the formulas on the GED formula sheet.

a) Write the formulas for each figure.

b) Label the variables for each formula.

c) Describe B, using the help of a net, for each formula. Describe h in the rectangular and triangular prism formulas, using the help of a net.

Fill in the chart:

Figure

Volume Formula

Name each variable

A net of the figure may be helpful in describing some of the pieces of the formula

Rectangular Prism

V=

V=

B=

h=

Right Prism

V=

V=

B=

h=

Pyramid

V=

V=

B=

h=

How are area and volume similar? How are they different?

Why is the height necessary when calculating the volume of an object?

Instructional Delivery:

When solving mathematical problems, you first have to be sure you know what the problem is asking and what information is given. You will also need to be sure to use the appropriate sources needed to solve the problems. When asked questions about the volume of an object, you will need to know the shape of the object and which formula to use. You will need to determine if you are asked to find the volume or maybe a missing side length. You may even have to convert units of measure depending on the information given in the problem.

Volume of Rectangular Prisms

Pay close attention to the following rectangular prisms.

Can the volume of each of the rectangular prisms be calculated? Will the volume be the same or different?

What information do you need to calculate the volume? What do you need to pay close attention to?

6 ft

2 ft

6 ft

2 ft

2 ft

2 ft

Compare the following two rectangular prisms. Can their volumes be calculated? Will the volumes be the same or different? What information do you need to calculate the volume? What do you need to pay close attention to?

48 in

24 in

2 ft

4 ft

24 in

2 ft

Volume of right prisms

Take a look at the following right prisms.

How are they different from the rectangular prisms?

How will calculating the volume for a right prism be different than calculating the volume of a rectangular prism?

What is the formula for the volume of a right prism?

What is the base of the prism? What is the height?

Use the dimensions from the picture above. Find the volume of the right prism.

Take a look at these prisms. Locate the base and height. What information do you need to solve for the volume? Find the volume.

6 ft

2 ft

3 ft

Volume of pyramid

Take a look at the following pyramids.

How are they different from the rectangular prisms?

How will calculating the volume for a pyramid be different than calculating the volume of a rectangular prism?

6 in

3 in

What is the formula for the volume of a pyramid?

What is the base of the pyramid? What is the height?

Use the dimensions from the picture above. Find the volume of the pyramid.

Look at this pyramid. What information do you need to solve for the volume? Locate the base and height. Calculate the volume.

The pyramid has a base edge with a length of 22 cm and a height of 48 cm. What is the volume?

Formula Challenge!

In order to successfully use formulas for the Mathematics sections of the GED test, you will need to pay close attention to the details of the questions that are asked. Be sure to pay close attention to the units being used and what dimensions are actually given. You will not always be given the dimensions that the formulas ask for. You may have to find a dimension or even use another formula to find the dimension. Take a look at the next few problems and try to answer the questions being asked.

1) Jacob is looking to buy a fish tank for his sons room. He knows he wants a tank that will hold 30 cubic feet of water. The base can be no more than 3 feet long and 2 feet wide. What would the height of the tank be?

2) Candi is building planters for her patio. She wants each one to hold 8 cubic feet of dirt. The height of each planter will be 2 feet. If the base of the planter is a square, what is the length of each side of the base?

3) Sarah is working on a project for her Social Studies class. She knows that the volume of the large pyramid in Egypt is 2,592,100 m3. She also knows that the base is 52,900 m2. She needs to find the height of the smaller pyramid and all the information she has is that it is 1/3 the height of the larger pyramid. What is the height of the smaller pyramid?

4) LearnZillion Problem

A square house has a pyramid-shaped attic built under the roof, which you are converting into a spare bedroom. You are installing an air conditioner to cool the room. The perimeter of the base of the room is 190ft, and the attic height is 6ft. How many cubic feet of air are in the empty attic?

(*Highlight the key details in the problem before working the problem.)

Guided Practice: Rectangular Prisms

I need to fill my swimming pool for the summer. I will need to call the local pool supply store to schedule a water delivery truck. The company requires me to tell them the shape and the dimensions of the pool when I call. Why would they need to know the shape and dimensions?

My pool is a rectangular shaped pool. It has a width of 10 feet and a length of 18 feet. The whole pool has a depth of 5.5 feet.

The delivery trucks can carry 3,000 gallons of water. They charge$0.04 per gallon

and an extra delivery fee of $15 per truck load.

How much water is needed to fill the pool?

(Hint: Use the internet to determine how many gallons of water are in each cubic foot.)

How much water can one delivery truck carry?

How many trucks are needed?

How much will the delivery cost?

Guided Practice: Pyramids

You are renting a pyramid shaped tent for your outdoor party. The tent company has many different options to choose from. The company uses calculations of how many people you plan to have attending and the appropriate area of the floor space needed to accommodate the guests. The company suggests that each person have a minimum of 16 square feet of space. All of the tents are 15 feet tall.

If you are expecting 75 guests to attend the party, how many square feet of space will be needed?

What would be the capacity of a tent of this size?

The company also suggests fans for the summer months. Each fan will maintain a comfortable temperature for every 1000 cubic feet of space. How many fans will you need?

The tent company charges $5 per 100 cubic feet of space plus $25 per fan. How much is your total?

Independent Practice:

Try similar problems on your own. Pay close attention to the details in each of the problems. You will need to decide which resources are needed for problem. You may use a calculator to complete the problems.

1) A concrete pad is being poured with the dimensions of 50 feet long, 16 feet wide, and 12 inches deep.

a. How many cubic feet of concrete is being poured?

b. How many cubic inches of concrete is being poured?

c. How much will the concrete cost if the company charges $50 per cubic yard?

2) A container in the shape of a rectangular prism has a volume of 864 mm3. It has a width of 4 mm and a height of 12 mm. What is the length of the container?

3) A right triangular prism has sides that are 8 inches, 15 inches, and 17 inches long. It has a height of 20 inches. What is the volume of the prism in cubic inches?

4) A tent in the shape of a triangular prism has a length of 4 feet. The front and rear tent flaps are shaped like triangles, each with a base of 3 feet, a height of 2 feet, and two side lengths of 2.5 feet. What is the volume of the tent?

5) A square pyramid has a base with side lengths of 12 centimeters and a height of 8 centimeters.

a. What is the volume of the pyramid?

b. What is the volume of a rectangular prims with the same dimensions?

6) A pyramid has a volume of 128 in3. The height is 6 inches. What is the length of each side of the base?

Connections / Extensions:

You have used the formulas for the volume of a rectangular prism and pyramid in various ways during this lesson. We also discussed that the volume of a pyramid is 1/3 of the volume of a rectangular prism. For this task, create a way to prove that the pyramid volume is 1/3 the rectangular prism volume. You have the challenge of proving to another classmate that the volume of a pyramid really is the same as 1/3 the volume of a rectangular prism. You will need to be a descriptive as possible with your wording and create a visual of your method as well. You can use anything you want in order to prove the formulas are true.