Unit 1, Lesson 1
Solids
Integrated Math 3
1.1
Warm-up
A World of Shapes
Instructional Routines
Materials
None
Activity Narrative
This activity introduces three-dimensional solids with images meant to engage students with objects they may have previously seen. The activity invites students to name and describe similarities and differences between the objects in the images.
Welcome any informal language students begin with to describe their thinking. Encourage students to refine their language (MP6) by using terms such as “base,” ”edge,” “vertex,” “face,” “curved,” and “pointed,” as well as the name of each shape, to describe each object they have seen in previous grades.
Launch
Arrange students in groups of 2–4. Display the images for all to see. Give students 1 minute of quiet think time, and ask them to indicate when they have chosen their images and can explain their similarities and differences. Next, tell students to share their response with their group and then work together to find as many similarities and differences as they can.
Activity
NoneChoose at least two images. How are the objects in them similar? How are they different?
Activity Synthesis
The goal of this discussion is for students to compare and contrast three-dimensional shapes and refine their ideas, using mathematical language.
Invite each group to share the set of images they chose as well as their similarities and differences. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of what is similar and different, attend to students’ explanations and ensure the reasons given are correct. If the idea of cross-sections does not come up during the conversation, ask students to discuss this idea. Students will focus on cross-sections in future lessons, so there is no need to go into depth at this time.
During the discussion, prompt students to explain the meaning of any geometric terminology they use, such as “face,” “base,” ”edge,” “vertex,“ “pyramid,” “cylinder,” “sphere,” “prism,” and “cone,” and to clarify their reasoning as needed.
To involve more students in the conversation, consider asking:
- “How do you know
?” - “What do you mean by
?” - “Can you say that in another way?”
Lesson Synthesis
In this lesson students explored the features and properties of three-dimensional solids.
Here are some questions for discussion:
- “Where do we see two-dimensional shapes in three-dimensional solids?” (Two-dimensional figures make up the faces, bases, and cross-sections of three-dimensional solids.)
- “How can we describe the features of a three-dimensional solid?” (We can describe a three-dimensional solid in terms of the shapes of its faces and cross-sections, its edges, which can be curved or straight, and its vertices, which are pointed.)
Display an image of a cube and a square (or the corresponding cards from the Card Sort) for all to see.
Invite students to share what they know about the two shapes. Students may describe the faces, cross-sections, edges, and vertices. If area and volume are not mentioned by a student, ask “What measurements do we use to describe these shapes?” (area and volume) Tell students we will continue to explore the properties of three-dimensional solids in upcoming lessons.
Student Lesson Summary
Two-dimensional figures and three-dimensional solids share some features, but they also have differences. The similarities and differences between two-dimensional figures and three-dimensional solids can be used to describe three-dimensional solids in terms of the shapes of their faces and cross-sections, their edges, which can be curved or straight, and their vertices, which are pointed.
For example, both a square and a cube have a square component, as the base and faces of a cube are squares. However, the square is a flat, two-dimensional figure measured by area, while a cube is a three-dimensional solid that has surface area and volume. In a square, edges are the sides, and vertices are where the two sides meet. In a cube, edges are where two faces meet, and vertices are where three faces meet. A square has 4 straight edges and 4 vertices, while a cube has 12 straight edges and 8 vertices. Furthermore, a cube can be sliced to reveal two-dimensional cross-sections, which could take the shape of a square or several other shapes.
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