Unit 4: Geometry

Did you ever play with shape blocks when you were younger? Possibly you put a triangle next to a rectangle and then added a square on a different side. Do you know how to find the area of this new shape? In this unit, you will use your prior knowledge about area, as well as some new information that we will cover, to help you decompose a complicated shape into smaller polygons in order to successfully find the area of the total shape. During this unit, you will also solve problems with three-dimensional shapes and work with a variety of real-life situations.

Completing this unit should take you approximately 14 hours.

☐ Subunit 4.1: 30 minutes

☐ Subunit 4.2: 6 hours and 45 minutes

☐ Subunit 4.3: 2 hours and 15 minutes

☐ Subunit 4.4: 3 hours

☐ Subunit 4.5: 1 hour and 30 minutes

Unit4 Learning Outcomes
Upon successful completion of this unit, you will be able to:

• Find the area of triangles and quadrilaterals.

• Find the area of shapes composed of triangles and rectangles.

• Apply the standard area formula to real-world situations.

• Find the volume of rectangular prisms, including prisms with fractional side lengths.

• Define the basic formula for volume of a rectangle prism. Explain why it works with unit cubes.

• Represent a three-dimensional figure with a two-dimensional net.

• Solve surface-area problems using two-dimensional nets.

• Apply techniques for finding area, volume, and surface area to real-world situations.

4.1 One-, Two-, and Three-Dimensional Measurements

Our world is three-dimensional. You can pick up objects up and hold them in your hand. When you look at a three-dimensional object, you can see that it has length, width, and height/depth. However, we also use one- and two- dimensions in our world, as well. One-dimensional objects are just lines, or lengths; for example, neighborhoods separate family property by a one-dimensional boundary. People often measure the length of their property line and build a fence. The fence itself is three-dimensional; however, the length of the fence is one-dimensional.

Two-dimensional refers to something that has a length and a width. When you look at a photograph, go to a movie, or watch television, you are viewing objects as two-dimensional (assuming you don’t have the special 3-D glasses). The walls in your bedroom might need painting. You could measure the length and width of the walls to calculate how much paint to buy. A wall is two-dimensional because it’s flat.

When you walk around in your everyday life, think about how people use one-, two-, and three-dimensional measurements.

• Activity: The Saylor Foundation’s “Understanding One-, Two-, and Three-Dimensions”

Link: The Saylor Foundation’s “Understanding One-, Two-, and Three-Dimensions” (PDF)

Instructions: Using this resource as a guide, you will create a table to help you learn about dimensional measurements. In your notebook, separate a page into three columns. Your paper should look similar to the one you see in the attached link. Title each column “One-Dimensional,” “Two-Dimensional,” and “Three-Dimensional.” Use the 4.1 subunit introduction, your prior knowledge, and the resources in this unit to add ideas about working with each dimension. It will be helpful to go back and refer to this table as you work through geometry units.

Setting up your notebook and beginning to fill in this chart should take you approximately 15 minutes.

• Explanation: Khan Academy’s “Perimeter and Area Basics”

Instructions: The video starts off showing you how to find the perimeter of a rectangle. This is probably a review of your prior knowledge. While you watch, draw the shapes and find the perimeter of the rectangles in the video. Remember, perimeter is a one-dimensional measurement because it’s a single line going around an object.

The second part of the video discusses area. Remember, area is a two-dimensional measurement because it has a length and width. When we find area, we use the label square units or units2 because we can actually cover our shape with squares and count them to find the area, as demonstrated in the video. Take notes while the instructor finds the areas of the rectangles.

Watching this video and taking notes should take you approximately 15 minutes.

4.2 Area

This subunit focuses specifically on finding the area of a variety of shapes. Remember, area is measured in square units or units2. You will use different formulas to find area based on the shape you are working with. Try not to get too caught up in memorizing formulas. It is actually much easier if you stop and think about the shape and a strategy that would make sense for finding its area. The resources in this unit will help you with these strategies. This subunit will give you a chance to add notes to your “Two-Dimensional” column in your notebook.

4.2.1 Triangles

As you work through the resources in this subunit, really focus on how you can understand and not just memorize the formula for the area of a triangle. Even though there are different types of triangles, you will discover how similar the strategies are for finding the area.

• Explanation: Khan Academy’s “Triangle Area Proof”

Instructions: Draw a right triangle in your notebook. Follow along with the strategy to make the triangle into a rectangle. Notice how your triangle is exactly half of the rectangle you drew around it. Copy the formula that fits with this strategy used in the video.

Continue watching the “proof” for how to find the area of the other types of triangles shown in the video. Draw each triangle with the instructor. Consider how you can use the same strategy of drawing rectangles around triangles to find the area of any triangle.

It is OK to refer to the formula A = bh (area = x base x height) when dealing with triangles. However, you should be able to explain why that formula works.

Watching this video and taking notes should take you approximately 30 minutes.

• Did I Get This? Activity: Illustrative Mathematics: “Base and Height”

Link: Illustrative Mathematics: “Base and Height” (PDF)

Instruction: Read about Mrs. Lito’s students and look at the way they labeled the triangles. Answer questions a and b. Scroll down to read the “Commentary” and “Solution” sections on pages 2 and 3.

Completing this activity should take you approximately 15 minutes.

• Did I Get This? Activity: Khan Academy’s “Area of Triangles”

Instructions: This page provides a series of practice problems that you can answer and check online. Each question has a solution worked out step-by-step if you need hints along the way. Practice solving for the area of triangles until you feel confident that you understand how to find the area of a triangle (recommended a minimum of 10 minutes of practice).

Completing these practice problems should take you approximately 15 minutes.

A quadrilateral is a four-sided shape. You probably have some experience with quadrilaterals like rectangles and squares. This subunit will review the area of some quadrilaterals.

• Explanation: Khan Academy’s “Area and Perimeter”

Instructions: While watching this video, press pause when the instructor shows you a new shape. Draw the shape in your notebook and find the area and perimeter of the shape. Watch and listen as he explains the process. Don’t just skip ahead to the answer because the instructor gives you some important reminders.

Watching this video and completing the sample problems should take you approximately 30 minutes.

• Did I Get This? Activity: Khan Academy’s “Area 1”

Instructions: This page provides a series of practice problems that you can answer and check online. Each question has a solution worked out step-by-step if you need hints along the way. Practice solving for area until you feel confident that you understand how to find the area (recommended a minimum of 10 minutes of practice).

Working these practice problems should take you approximately 15 minutes.

• Did I Get This? Activity: Khan Academy’s “Area of Squares and Rectangles”

Instructions: This page provides a series of practice problems that you can answer and check online. Each question has a solution worked out step-by-step if you need hints along the way. Practice solving for area until you feel confident that you understand how to find the area (recommended a minimum of 10 minutes of practice).

Working these practice problems should take you approximately 15 minutes.

• Explanation: Bettina Richmond and Tom Richmond’s “Area Formulas for Basic Shapes”

Link: Bettina Richmond and Tom Richmond’s “Area Formulas for Basic Shapes” (HTML)

Instructions: This page provides you with animated justification for how to find the area of shapes. Watch the animation for “Area of a Rectangle” and “Area of a Parallelogram.” In your notebook, take notes of the formula and the text given to you. Also, sketch the animation the best you can. Note: there is no sound.

Completing this task should take you approximately 15 minutes.

• Explanation: CK-12: “Area of a Parallelogram”

Link: CK-12: “Area of a Parallelogram” (HTML)

Instructions: The previous resource showed you an animation of how to find the area of a parallelogram. This resource will also explain through words and pictures how to find the area of a parallelogram and why it works. Read through the introduction; take notes as you read through the “Guidance” section. Do the example problems and write down the vocabulary words. Complete the “Guided Practice” and watch the video. You will do some practice problems below.

Taking notes, working examples, and watching the video should take you approximately 30 minutes.

• Did I Get This? Activity: Activity: Khan Academy’s “Area of Parallelograms”

Instructions: This page provides a series of practice problems that you can answer and check online. Each question has a solution worked out step-by-step if you need hints along the way. Practice solving for area until you feel confident that you understand how to find the area (recommended a minimum of 10 minutes of practice).

Completing this activity should take you approximately 15 minutes.

• Did I Get This? Activity: YouTube: Mathispower4u: James Sousa’s “Example: Determine the Area of a Rectangle Using Mixed Numbers”

Instructions: As you watch the video, draw the rectangle and label the side lengths. Pause the video while you solve for the area of the rectangle. (This is a good review for dealing with fractions.) Don’t forget to simplify your answer back to a mixed number. Watch the remainder of the video. Compare your answer with the instructor’s.

Completing this problem should take you approximately 15 minutes.

4.2.3 Composite Figures

Picture a square and a triangle pushed up together to share a side length. This is called a composite figure. A composite figure is a figure that is made of a variety of shapes. When you are asked to find the perimeter of a composite figure, keep in mind that perimeter is the distance around an object. It is sometimes helpful to take your pencil and trace the outside of the object; you will need to add each of those side lengths together to find the perimeter. When finding the area of a composite figure, think about how you can break the object into smaller, separate shapes. Once you’ve done that, you will find the area of the smaller shapes. Then, you will add the individual areas together. Sometimes, side lengths won’t be given, but clues within the problem or the shape can help you figure out the missing side length.

• Explanation: CK-12: “Area of Composite Shapes”

Link: CK-12: “Area of Composite Shapes” (HTML)

Instructions: This lesson will provide you with a few videos to explain how to find the area of composite shapes. Begin watching the first video. Pause the video so you can take the time to draw the shape. Solve for the area of the composite shape with the instructor. Note: at the 1:40 mark in the video, the instructor figures that the height of the triangle is 3. Do not worry about the process used to find the height. You will learn about that later in your math career.

Scroll down to the second video and begin watching. Pause the video to draw the shape. Try to find the area of the composite figure before the instructor. Watch the instructor’s process. Complete the next composite shape with the same process.

Complete practice problems 1 - 15 at the end of the lesson.

Watching these videos and completing the practice problems should take you approximately 45 minutes.

• Explanation: CK-12: “Area of Composite Shapes Involving Triangles”

Link: CK-12: “Area of Composite Shapes Involving Triangles” (HTML)

Instructions: Read about home plate. Think of other everyday objects that are made of a variety of shapes - the “key” on a basketball court, a hopscotch pattern, a stained glass window, and the front image of a house. Read through the “Guidance” section, draw the composite figures, and label the side lengths. Work the example problems. For each one, draw the shape and calculate the area. Check your work with the answer provided. Continue working down the page, completing the “Guided Practice” and “Practice” problems.

Taking notes and completing the practice problems should take you approximately 1 hour.

• Did I Get This? Activity: YouTube: Mathispower4u: James Sousa’s “Ex: Find the Area of an L-Shaped Polygon Involving Whole Numbers”

Link: YouTube: Mathispower4u: James Sousa’s “Ex: Find the Area of an L-Shaped Polygon Involving Whole Numbers” (YouTube)

Instructions: Look at the composite figure. Pause the video. Draw the shape and find the area - remember to split the composite figure into smaller rectangles. To find the length of the missing sides, use the other side lengths given to you. Do not estimate, measure, or guess side lengths! Watch the remainder of the video to compare your strategy to the instructor’s. Could you have split your composite figure into rectangles in a different way than the instructor did, while still finding the area to be the same?

Watching this video and taking notes should take you approximately 15 minutes.

• Did I Get This? Activity: Khan Academy’s “Interesting Perimeter and Area Problems”

Instructions: Watch the video and pause it at the 0:40 mark. Draw the star shape on your paper. Try to find the perimeter based on the information given to you. Watch the instructor as he explains how to find the star perimeter. Compare your strategy to his.

Pause the video at 2:40. Think about how you can find the area of this shape. Break it into two smaller shapes. Draw the composite figure and find the area. If you are struggling, watch another 1:00 to see a good first step in solving for this area. Once you have found the area, watch the instructor as he continues explaining how to find the area of the composite figure. Compare your strategy to his.

Pause the video at 5:20. Draw the shape. Label the given side lengths. Think about how you find the other side lengths (without guessing, estimating, or measuring). If you need a hint, watch the video for another 1:10 and see what the instructor’s strategy is with the side lengths. If you feel confident, pause the video again at 6:30 and find the perimeter. Watch the remainder of the video to see how the instructor finds the perimeter.

Watching this video and working through the example problems should take you approximately 30 minutes.

• Did I Get This? Activity: YouTube: Mathispower4u: James Sousa’s “Ex: Determine the Area of an L-Shaped Polygon Using Decimals”

Instructions: Look at the composite figure. Pause the video. Draw the shape and find the area - remember to split the composite figure into smaller rectangles. To find the length of the missing sides, use the other side lengths given to you. Do not estimate, measure, or guess side lengths! Watch the remainder of the video to compare your strategy to the instructor’s. Could you have split your composite figure into rectangles in a different way than the instructor did, while still finding the area to be the same?

Completing this problem should take you approximately 15 minutes.

• Did I Get This? Activity: Illustrative Mathematics: “Finding Areas of Polygons, Variation 1”

Link: Illustrative Mathematics: “Finding Areas of Polygons, Variation 1” (HTML)

Instructions: This activity is an opportunity for you to explore different strategies for finding area. Keep in mind that when you have a composite figure, there is not a simple formula for finding area. You will need to problem solve and think outside the box as you work to find two strategies for finding the area of each polygon. Once you have answers, scroll down and read the “Commentary” and “Solution” sections on pages 2 and 3.

Completing this activity should take you approximately 30 minutes.

• Did I Get This? Activity: YouTube: Mathispower4u: James Sousa’s “Area Application - Area of an Inner Room with an Outer Footing”

Link: YouTube: Mathispower4u: James Sousa’s “Area Application - Area of an Inner Room with an Outer Footing” (YouTube)

Instructions: Read the problem with the instructor. Pause the video at the 0:25 mark. Draw the room and the outer footing. Solve for the inside dimensions and area of the inner room. Watch the remainder of the video to compare your strategy with the instructor’s.

Watching this video and completing the activity should take you approximately 15 minutes.

4.3 Surface Area

Have you ever wrapped a gift for someone? You have to make sure you cut off enough wrapping paper so that it covers the entire gift. The box that gift is in is a three-dimensional box; in math it would be called a rectangular prism (although you might just think of it as a shoebox). The wrapping paper covers the gift - each side, or face, of the box is a two-dimensional surface. To figure out how much wrapping paper is needed, mathematicians find the surface area. This subunit focuses on how to find surface area and introduces you important vocabulary you need to know when dealing with two- and three-dimensional objects. This subunit will give you a chance to add notes to your “Two-Dimensional” and “Three-Dimensional” columns in your notebook.

4.3.1 Nets   - Explanation: Explanation: CK-12: “Surface Area of Prisms”

``````Link: CK-12: [“Surface Area of
Prisms”](http://www.ck12.org/geometry/Surface-Area-of-Prisms/lesson/Surface-Area-of-Prisms/) (HTML)

Instructions: The purpose of this lesson is to help you learn some
vocabulary concerning two- and three-dimensional measurements. While
you read through the lesson, do not focus on the mathematical
computations; you will work on the math a little later.

Read the introduction. In the “Guidance” section, take notes on
words such as **three-dimensional, prism, surface area,** and
**net.** After the third “sticky note” with a net on it, skip down
to examples A, B, and C. Go through each true/false question.

Skip down to the “Vocabulary” section. Make sure you have each of
the vocabulary words clearly defined in your notebook.

If you can, cut out a net from a piece of your notebook paper (or
graph paper if you have it). Fold the net to form a
three-dimensional rectangular prism or cube. Tape your rectangular
prism together and use it as an example as you continue through this
unit.

Taking notes and building a net should take you approximately 45
minutes.

-   [CCSS.Math.Content.6.G.A.4](http://www.corestandards.org/Math/Content/6/G/A/4)

attributed to CK-12.
``````

4.3.2 Parts of a Three-Dimensional Object   - Explanation: CK-12: “Faces, Edges, and Vertices of Solids”

``````Link: CK-12: [“Faces, Edges, and Vertices of
Solids”](http://www.ck12.org/geometry/Faces-Edges-and-Vertices-of-Solids/lesson/Faces-Edges-and-Vertices-of-Solids/) (HTML)

Instructions: This lesson will give you some more vocabulary as you
work with three-dimensional objects. As you read the “Guidance”
section, take notes on the vocabulary words: **faces, edges,** and
**vertices**.

Draw a rectangular prism, or use an actual object that you can point
to and label - a shoebox or even a computer are examples of a
rectangular prism. Objects are often described by their number of
faces, edges, or vertices. The word face is used when finding
surface area (see subunit 4.3.3), so make sure you understand these
vocabulary words.

Complete the examples, read through and take notes on the
“Vocabulary” section (you will not be expected to know about a
cylinder, pyramid, cone, or sphere as a sixth-grade student), and
complete the “Guided Practice” questions.

Watch the video for another view of how to label faces, edges, and
vertices. Complete problems 1 - 7 in the “Practice” section.

Taking notes, completing the practice problems, and watching the
video should take you approximately 30 minutes.

-   [CCSS.Math.Content.6.G.A.4](http://www.corestandards.org/Math/Content/6/G/A/4)

attributed to CK-12.
``````

4.3.3 Solving for Surface Area   - Explanation: CK-12: “Surface Area of Rectangular Prism”

``````Link: CK-12: [“Surface Area of Rectangular
Prisms”](http://www.ck12.org/geometry/Surface-Area-of-Rectangular-Prisms/lesson/Surface-Area-of-Rectangular-Prisms/) (HTML)

house. Take notes as you read through the “Guidance” section. Solve
the example problems given to you. Draw both a rectangular prism and
a net for each problem. This will help you recognize how a
two-dimensional net is related to a three-dimensional rectangular
prism. The formula for the surface area of a rectangular prism is
long and complex. However, write it down and think about how each
part related to the actual shape.

Write down the vocabulary words, and then continue on to the “Guided
Practice” section. Watch the video provided. Notice there is not a
lid on the box. This takes away an entire face.

Answer questions 1 - 10 in the “Practice” section.

Taking notes, watching the video, and completing the practice
problems should take you approximately 1 hour.

-   [CCSS.Math.Content.6.G.A.4](http://www.corestandards.org/Math/Content/6/G/A/4)

attributed to CK-12.
``````

4.4 Volume

Remember the gift you were wrapping in the previous subunit? What’s inside? How do you know if you have a box that is big enough for the object you are wrapping? Whenever you are thinking about the space inside of something, you are dealing with volume. Volume is a three-dimensional measurement because it has a length, width and a height (recall the one-dimensional measurement has only a length; two-dimensional measurements have a length and a width). Volume adds a third dimension - height, or sometimes called depth. Have you ever been to a swimming pool? The amount of water in the pool is the volume. Have you ever crammed your suitcase full of clothes before you went on a trip? The amount of stuff inside your luggage is volume. This subunit focuses on how to find volume and will give you important vocabulary for dealing with volume. This subunit will give you a chance to add notes to your “Three-Dimensional” column in your notebook.

4.4.1 Using Cubic Units   - Explanation: CK-12: “Volume of Prisms Using Unit Cubes”

``````Link: CK-12: [“Volume of Prisms Using Unit
Cubes”](http://www.ck12.org/geometry/Volume-of-Prisms-Using-Unit-Cubes/lesson/Volume-of-Prisms-Using-Unit-Cubes/) (HTML)

Instruction: Read through and take notes on the “Guidance” section.
Notice the label for volume is in cubic units or units3 because
volume involves counting the cubes to fill an object. Complete the
problems in the “Examples” section in your notebook, take notes on
the “Vocabulary” (as a sixth-grade student you do not need to know
triangular prisms), and complete the “Guided Practice” sections.
Stop there for now. You will do some practice with unit cubes in the
next resource.

Reading this lesson and taking notes should take you approximately
30 minutes.

-   [CCSS.Math.Content.6.G.A.2](http://www.corestandards.org/Math/Content/6/G/A/2)

attributed to CK-12.
``````
• Did I Get This? Activity: Illustrative Mathematics: “Computing Volume Progression 1”

Link: Illustrative Mathematics: “Computing Volume Progression 1” (HTML)

Instructions: Think about filling a box in the shape of a cube (remember a cube has all equal sides). Answer questions a and b. Check your answers on the second page.

Completing this activity should take you approximately 15 minutes.

4.4.2 Understanding the Equation V = lwh and V = bh   - Explanation: CK-12: “Volume of Rectangular Prism”

``````Link: CK-12: [“Volume of Rectangular
Prism”](http://www.ck12.org/geometry/Volume-of-Rectangular-Prisms/lesson/Volume-of-Rectangular-Prisms/) (HTML)

Instructions: As you read through the introduction, notice that
Candice and Trevor are “filling” their boxes with the packing
peanuts. When you see the word “fill” you can often know that the
problem involves volume. Continue reading and take notes on the
“Guidance” section. Draw some rectangular prisms (do your best with
three-dimensional drawing), label each side “length,” “width,” and
“height.” As you read you will notice there are two formulas. Make
sure you not only know how to use each formula, but that you also
know the difference between them and can explain how they work.
Continue with the example problems, vocabulary, and guided practice.
In the “Practice” section, complete problems 1 - 10. Draw and label
a rectangular prism for at least half of the practice problems to
get an image of how the prism looks.

Reading this lesson, taking notes, and completing the practice
problems should take you approximately 45 minutes.

-   [CCSS.Math.Content.6.G.A.2](http://www.corestandards.org/Math/Content/6/G/A/2)

attributed to CK-12.
``````
• Activity: Howard County Public School System’s “Student Homework”

Link: Howard County Public School System’s “Student Homework” (HTML)

Instructions: Scroll down to the third section, 6.G.2. Download the document “6.G.2 Student Homework.docx.” Answer the three questions in this activity. Notice that the cubes are inch on each side. The boxes have fractional side lengths as well. You will have to use both your volume skills and your fraction skills to solve these problems.

Completing these practice problems should take you approximately 30 minutes.

• Did I Get This? Activity: Illustrative Mathematics: “Computing Volume Progression 2”

Link: Illustrative Mathematics: “Computing Volume Progression 2” (HTML)

Instructions: Answer questions a and b. It might be helpful to draw a model of what the water level will look like in question b. Scroll down and read the commentary and the solution on the second page.

Completing this activity and checking your work should take you approximately 30 minutes.

• Did I Get This? Activity: Illustrative Mathematics: “Computing Volume Progression 3"

Link: Illustrative Mathematics: “Computing Volume Progression 3” (HTML)

Instructions: As you read this problem, keep in mind that you will need to do some converting of units before you work to find the height of the tank. A good strategy to consider is that although the problem is talking about water, volume is measured in cubed units. Think about the amount of space the first layer (height of 1 cm) takes up. Consider that you will need to see how many 50 cm by 60 cm layers should be stacked to fill the tank full. Scroll down and read the commentary and the solution on the second page.

Completing this activity and checking your work should take you approximately 30 minutes.

4.5 Real-Life Situations and Word Problems Involving Geometric Shapes

Painting your bedroom walls, filling a swimming pool with water, planting grass seed in the yard, and wrapping a gift are just a few examples of real-life situations when you will use geometry. Understanding how to find perimeter, area, surface area, and volume are all very important. However, the initial task is to know what the question is asking and to understand what type of formula will be used. As you go through this subunit, stop and think about each situation. Consider key words that might trigger ideas to know if you are dealing with one-, two-, or three-dimensional measurements. This subunit will give you a chance to use the notes from your notebook. You can also add additional real-life examples to each of the columns in your notebook.

Click here to see an example of how the “Understanding One-, Two-, and Three-Dimensions” might look at this point in the unit.

• Did I Get This? Activity: Illustrative Mathematics: “Banana Bread”

Instructions: Read the problem on the first page about making banana bread. Solve the problem and describe why you think your answer makes sense. Scroll down to second page and read the “Commentary” and “Solution” sections. Compare your strategy with the one provided.

Completing this activity and checking your work should take you approximately 15 minutes.

• Did I Get This? Activity: Illustrative Mathematics: “Painting a Barn”

Link: Illustrative Mathematics: “Painting a Barn” (HTML)

Instructions: Read and solve the problem on the first page. Explain your work. Refer to the notes you made throughout this unit to help you consider what math operation to use when painting walls. Scroll down to the second page and read the “Commentary” and “Solution” sections. Compare your strategy with the one provided.

Completing this activity and checking your work should take you approximately 30 minutes.

• Checkpoint: Illustrative Mathematics: “Computing Volume Progression 4”

Link: Illustrative Mathematics: “Computing Volume Progression 4” (HTML)

Instructions: Read the problem on the first page about taking the stone out of the rectangular tank. Solve the problem and describe why you think your answer makes sense. Scroll down to the second page and read the “Commentary” and “Solution” sections. Compare your strategy with the one provided.

Completing this task and checking your work should take you approximately 15 minutes.

• Checkpoint: Illustrative Mathematics: “Christo’s Building”

Link: Illustrative Mathematics: “Christo’s Building” (HTML)

Instructions: Read through the problem and answer all questions. Take your time thinking about what each question is asking and how to solve it. Draw models and show your work. Scroll down and read through the “Commentary” and “Solution” sections on pages 2 and 3.

Completing this activity and checking your work should take you approximately 30 minutes.

`````` Instructions: Test your knowledge by completing this checkpoint.