standard_description: Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.
problem_activity: Activity
In the last activity, students started with solids and identified various cross sections. In this activity, students view three-dimensional slabs of a solid between parallel cross sections and try to determine what the original solid was. Being able to visualize the relationship between a solid and its cross sections is important to later work on Cavalieri’s Principle.
Launch
Ask students, “What solid would a stack of all the same coins create?” Display a stack of quarters and note that it creates the shape of a cylinder. Then display, in order, a quarter, a nickel, a penny, and a dime. Ask, “What solid would a stack of coins decreasing in size create?” Make a stack with a few of each type of coin to make a solid that resembles a cone.
Student Facing
Each question shows several parallel cross-sectional slabs of the same three-dimensional solid. Name each solid.
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Student Response
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Student Facing
Are you ready for more?
3D-printers stack layers of material to make a three-dimensional shape. Computer software slices a digital model of an object into layers, and the printer stacks those layers one on top of another to replicate the digital model in the real world.
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Draw 3 different horizontal cross sections from the object in the image.
The layers can be printed in different thicknesses. How would the thickness of the layers affect the final appearance of the object?
Suppose we printed a rectangular prism. How would the thickness of the layers affect the final appearance of the prism?
Student Response
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Activity Synthesis
Ask students to share their predictions for what solids are formed. Then display these images for all to see.
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Now focus students’ attention on cross sections that are taken parallel to a solid’s base (for those solids that have bases). Ask students how cross sections can be used to differentiate between prisms and pyramids. (The cross sections of prisms taken parallel to the base are congruent to each other. The cross sections of pyramids taken parallel to the base are similar to each other.)
Speaking: MLR8 Discussion Supports.
To help students respond to the questions for discussion, provide sentence frames such as: “The cross sections of _____ taken parallel to the base are _____ because.…” As students share their responses, press for details by asking students how they know that the cross sections of prisms taken parallel to the base are congruent, while the cross sections of pyramids taken parallel to the base are similar. Ask students to distinguish the meanings of geometric congruence and similarity.
Design Principle(s): Support sense-making; Optimize output (for justification)
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner. Display sentence frames to support student conversation such as: “I noticed _____ so I….”, “How do you know…?”, “That could/couldn’t be true because….”, and “I agree/disagree because….”
Supports accessibility for: Language; Social-emotional skills


aligns: no