standard_description: Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.
problem_activity: Activity
This activity begins a sequence which looks at figures that are not polygons. From the point of view of congruence, polygons are special shapes because they are completely determined by the set of vertices. For curved shapes, we usually cannot check that they are congruent by examining a few privileged points, like the vertices of polygons. We
can
ascertain that they are not congruent by identifying a feature of one shape not shared by the other (for example, this oval is 3 units wide while this one is only
\(2 \frac{1}{2}\)
units wide). But to show that two curved shapes
are
congruent, we need to apply the definition of congruence and try to move one shape so that it matches up exactly with the other after some translations, rotations, and reflections.
In this activity, students begin to explore the subtleties of congruence for curved shapes. Make sure that students provide a solid mathematical argument for the shapes which are congruent, beyond saying that they look the same. Providing a viable argument, MP3, requires careful thinking about the meaning of congruence and the structure of the shapes. Monitor for groups who use precise language of transformations as they attempt to move one traced oval to match up perfectly with another. Invite them to share their reasoning during the discussion. Also monitor for arguments based on measurement for why neither of the upper ovals can be congruent to either of the lower ones.
Launch
Arrange students in groups of 2. Provide access to geometry toolkits. Give students 3 minutes of quiet work time, then invite them to share their reasoning with a partner, followed by a whole-class discussion.
Engagement: Develop Effort and Persistence
. Connect a new concept to one with which students have experienced success. For example, review the criteria used to determine congruence for polygons so that students can transfer these strategies in determining congruence for curved shapes.
Supports accessibility for: Social-emotional skills; Conceptual processing
Speaking: Math Language Routine 5 Co-Craft Questions.
This is the first time Math Language Routine 5 is suggested as a support in this course. In this routine, students are given a context or situation, often in the form of a problem stem with or without numerical values. Students develop mathematical questions that can be asked about the situation. A typical prompt is: “What mathematical questions could you ask about this situation?” The purpose of this routine is to allow students to make sense of a context before feeling pressure to produce answers, and to develop students’ awareness of the language used in mathematics problems.
Design Principle(s): Cultivating Conversations; Maximize meta-awareness
How It Happens:
Display the four images of the ovals without the directions.
Ask students, “What mathematical questions could you ask about this situation?”
Give students 1 minute of individual time to jot some notes, and then 3 minutes to share ideas with a partner.
As pairs discuss, support students in using conversation skills to generate and refine their questions collaboratively by seeking clarity, referring to students’ written notes, and revoicing oral responses, as necessary. Listen for how students use language about transformations and/or refer to measurements of the figures when talking about the curved polygons.
Ask each pair of students to contribute one written question to a poster, the whiteboard, or digital projection. Call on 2–3 pairs of students to present their question to the whole class, and invite the class to make comparisons among the questions shared and their own questions.
Listen for questions comparing different features of the ovals to determine congruence, especially those that use distances between corresponding points. Revoice student ideas with an emphasis on measurements wherever it serves to clarify a question.
Reveal the question, “Are any of the ovals congruent to one another?” and give students a couple of minutes to compare it to their own question and those of their classmates. Identify similarities and differences.
Consider providing these prompts: “Which of your questions is most similar to/different from the one provided? Why?”, “Is there a main mathematical concept that is present in both your questions and the one provided? If so, describe it.”, and “How do your questions relate to one of the lesson goals of measuring corresponding points to determine congruence?”
Invite students to choose one question to answer (from the class or from the curriculum), and then have students move on to the Activity Synthesis.
Student Facing
Are any of the ovals congruent to one another? Explain how you know.
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Student Response
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Student Facing
Are you ready for more?
You can use 12 toothpicks to create a polygon with an area of five square toothpicks, like this:
Can you use exactly 12 toothpicks to create a polygon with an area of four square toothpicks?
[IMG]
Student Response
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Activity Synthesis
Invite groups to explain how they determined that the upper ovals are not congruent to the lower ones, with at least one explanation focusing on differing measurable attributes (for example length and width). Also invite previously selected groups to show how they demonstrated that the two upper (and two lower) ovals are congruent, focusing on the precise language of transformations.
Emphasize that showing that two oval shapes are congruent
requires
using the definition of congruence: is it possible to move one shape so that it matches up perfectly with the other using only rigid transformations? Experimentation with transformations is essential when showing that two of the ovals match up because, unlike polygons, these shapes are not determined by a finite list of vertices and side lengths.
Students have seen that rectangles that have the same side lengths are congruent and will later find criteria for determining when two triangles are congruent. For more complex curved shapes, the definition of congruence is required.


aligns: no