Lesson Plan: Introduction to Nets and Basic 3D Shapes

Lesson Summary

In this lesson, students will explore how 2D nets transform into 3D figures through hands-on activities, animations, and guided instruction. They will construct physical 3D models from printable nets, analyze their faces, edges, and vertices, and determine how different nets can create the same shape.

A key part of this lesson is understanding surface area, as students will learn to calculate the surface area of cubes, rectangular prisms, and triangular prisms using their nets.

This lesson includes multimedia resources from Media4Math.com, such as animated nets, real-world applications, and digital explorations. The lesson concludes with a 10-question quiz and answer key to assess student understanding.

Lesson Objectives

• Identify and analyze nets of common 3D figures
• Predict how a net folds into a 3D shape
• Construct 3D figures from printable nets
• Count and analyze the number of vertices, edges, and faces
• Calculate the surface area of 3D shapes using their nets

Common Core Standards

• CCSS.MATH.CONTENT.6.G.A.4 – Represent three-dimensional figures using nets made up of rectangles and triangles and use the nets to find the surface area.
• CCSS.MATH.CONTENT.7.G.B.6 – Solve real-world and mathematical problems involving area, volume, and surface area.

Prerequisite Skills

Before starting this lesson, students should be familiar with:

• 2D shapes and their properties (squares, rectangles, triangles)
• Basic area formulas for rectangles and triangles
• Geometric terms (faces, edges, vertices)

Key Vocabulary

• Net – A two-dimensional pattern that can be folded to form a three-dimensional shape.
  • Multimedia Resource: https://www.media4math.com/library/43030/asset-preview
• Face – A flat surface on a three-dimensional figure.
  • Multimedia Resource: https://www.media4math.com/library/22039/asset-preview
• Edge – A line segment where two faces meet on a 3D shape.
  • Multimedia Resource: https://www.media4math.com/library/22031/asset-preview
• Vertex – A point where three or more edges meet.
  • Multimedia Resource: https://www.media4math.com/library/22207/asset-preview
• Surface Area – The total area of all faces of a three-dimensional figure.
• Cross Section – The shape created when a 3D figure is sliced by a plane.
• Cube – A three-dimensional shape with six equal square faces, twelve edges, and eight vertices.
• Rectangular Prism – A three-dimensional shape with six rectangular faces, twelve edges, and eight vertices.
• Triangular Prism – A three-dimensional shape with two triangular bases and three rectangular faces, nine edges, and six vertices.

Multimedia Resources

• A collection of definition cards on the topic of 3D Geometry: https://www.media4math.com/Definitions--3DGeometry
• A collection of video definitions on the topic of 3D Geometry: https://www.media4math.com/MathVideoCollection--3DGeoVocabulary

Warm-Up Activities

Choose from one or more of these activities.

Activity 1: What’s Inside?

• Display images of common 3D objects (e.g., cereal boxes, gift boxes).
• Ask students: “If we unfold this object into a flat shape, what would it look like?”
• Have students sketch a possible net for one of the objects.
• Discuss how different nets can create the same shape.

Activity 2: Cross-Section Challenge

Top	Side	Front

• Show students 2D cross-sections of mystery 3D shapes (e.g., the top view, front view, and side view of a cube or prism).
• Ask: “What shape do you think this is?”
• Discuss how different cross-sections reveal information about a 3D figure’s structure.
• Relate this to nets—slicing a figure and unfolding it both reveal its true shape.
• For the table of images above the results are: cube, rectangular prism, and triangular prism.

Activity 3: Digital Exploration

• Show an animated GIF from Media4Math.com that demonstrates a net folding into a 3D figure.
• Ask: “What happens if we rearrange the faces?”
• Have students predict which nets will work and which will not before playing the animation.

Teach

Introduction

In this section, you will explore how 2D nets fold into 3D shapes. You will watch animations, construct models from printable foldable nets, and analyze the geometric properties of each shape. By the end of this section, you should understand how faces, edges, and vertices relate to nets and 3D figures.

Multimedia Resources

• A slide show for the net for a cube: https://www.media4math.com/library/slideshow/net-cube
• A slide show for the net for a rectangular prism: https://www.media4math.com/library/slideshow/net-rectangular-prism
• A slide show for the net for a triangular prism: https://www.media4math.com/library/slideshow/net-triangular-prism

Example 1: Cube

Let's start by exploring the cube! Follow these steps:

1. Watch the animation of the cube net folding into a cube.
2. Look at the printable net of a cube. Cut it out and fold it to create a physical model. Here's a link to a downloadable, printable net: Click here. 
3. As you fold, count the number of faces, edges, and vertices.

Here’s what you should find:

• Faces: A cube has 6 square faces.
• Edges: A cube has 12 edges.
• Vertices: A cube has 8 vertices.

Think About It: What happens if one face is missing? Can the net still form a cube?

Example 2: Rectangular Prism

Now, let's examine a rectangular prism!

1. Watch the animation of the rectangular prism’s net folding.
2. Cut out the printable net and fold it into a rectangular prism. Here's a link to a downloadable, printable net: Click here. 
3. Count the number of faces, edges, and vertices as you fold.

Your rectangular prism should have:

• Faces: 6 rectangular faces (opposite faces are congruent).
• Edges: 12 edges.
• Vertices: 8 vertices.

Think About It: How does a rectangular prism’s net differ from a cube’s net?

Example 3: Triangular Prism

Finally, let's investigate the triangular prism!

1. Watch the animation of a triangular prism’s net folding.
2. Cut out the printable net and fold it into a triangular prism. Here's a link to a downloadable, printable net: Click here. 
3. Count the number of faces, edges, and vertices.

Your triangular prism should have:

• Faces: 5 faces (2 triangular bases and 3 rectangular lateral faces).
• Edges: 9 edges.
• Vertices: 6 vertices.

Think About It: How does the net of a triangular prism compare to that of a rectangular prism?

Example 4: Surface Area of a Cube

Now that we’ve built a cube, let's calculate its surface area!

A cube has 6 equal square faces. To find the total surface area, we use the formula:

\[ SA = 6s^2 \]

Where:

• \( s \) is the length of one side of the cube.

Let's Try It!

Suppose the side length of the cube is 4 cm. Plug this into the formula:

\[ SA = 6(4^2) \] \[ SA = 6(16) \] \[ SA = 96 \text{ cm}^2 \]

Final Answer: The surface area of this cube is 96 cm².

Think About It: How would the surface area change if the side length doubled?

Example 5: Surface Area of a Rectangular Prism

Now, let's calculate the surface area of a rectangular prism.

A rectangular prism has 6 faces (opposite faces are congruent). The surface area formula is:

\[ SA = 2lw + 2lh + 2wh \]

Where:

• \( l \) is the length
• \( w \) is the width
• \( h \) is the height

Let's Try It!

Suppose a rectangular prism has these dimensions:

• \( l = 3 \) cm
• \( w = 5 \) cm
• \( h = 7 \) cm

Now, substitute these values into the formula:

\[ SA = 2(3 \times 5) + 2(3 \times 7) + 2(5 \times 7) \] \[ SA = 2(15) + 2(21) + 2(35) \] \[ SA = 30 + 42 + 70 \] \[ SA = 142 \text{ cm}^2 \]

Final Answer: The surface area of this rectangular prism is 142 cm².

Think About It: How does the surface area formula change if the prism is a cube?

Review

Lesson Recap

Let’s summarize what we’ve learned in this lesson:

• Nets are 2D representations of 3D shapes that can be folded to form solid figures.
• Each 3D shape has a specific number of faces, edges, and vertices.
• Surface area is found by calculating the total area of all faces of a 3D shape.
• Different nets can be used to construct the same 3D figure.
• Nets are used in real-world applications, such as packaging design, architecture, and manufacturing.

Example 1: Packaging Design

Think about the boxes you see in stores, like cereal boxes or shoeboxes. These boxes start as flat nets before being folded into their final shape.

Let’s try it!

1. Sketch a possible net for a box that measures 10 cm × 6 cm × 4 cm.
2. Label all the faces with their dimensions.
3. Calculate the surface area of the box using the formula:

\[ SA = 2lw + 2lh + 2wh \] \[ SA = 2(10 \times 6) + 2(10 \times 4) + 2(6 \times 4) \] \[ SA = 2(60) + 2(40) + 2(24) \] \[ SA = 120 + 80 + 48 \] \[ SA = 248 \text{ cm}^2 \]

Final Answer: The surface area of the box is 248 cm².

Think About It: Why do packaging designers use nets to create boxes?

Example 2: Architecture and Building Design

Architects use nets to design buildings before construction. Many famous buildings are based on geometric shapes, such as the pyramids in Egypt or modern skyscrapers.

Let’s explore!

• Find a famous building that resembles a cube, rectangular prism, or triangular prism.
• Sketch a possible net for the building.
• Estimate the surface area of the structure using an appropriate formula.

Think About It: How do architects use nets to create efficient building designs?

Quiz

Directions: Answer the following questions. Show your work where necessary.

1. What is a net in geometry?
2. How many faces does a cube have?
3. How many edges does a rectangular prism have?
4. How many faces, edges, and vertices does a triangular prism have?
5. Calculate the surface area of a cube with a side length of 4 cm.
6. Calculate the surface area of a rectangular prism with dimensions 3 cm × 5 cm × 7 cm.
7. Why do packaging designers use nets when creating boxes?
8. How do engineers use nets in construction?
9. What happens if a net is missing a face? Can it still form a closed 3D shape?
10. How can you determine if a net will fold into a correct 3D shape?

Answer Key

1. A net is a 2D pattern that folds into a 3D shape.
2. 6 faces.
3. 12 edges.
4. 5 faces, 9 edges, 6 vertices.
5. 96 cm².
6. 142 cm².
7. To minimize material waste and ensure boxes fold correctly.
8. To plan and visualize structures before construction.
9. The net cannot form a fully closed 3D shape.
10. The net must have the correct number and arrangement of faces.