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: Warm-up
This is the first math talk activity in the course. See the launch for extended instructions for facilitating this activity successfully.
The purpose of this Math Talk is to expand students’ strategies for finding a mean beyond following an algorithm to reasoning that the mean of the values in a symmetric data set is the middle value. The third item is designed to illustrate that this technique only works for symmetric data sets. These understandings help students develop fluency and will be helpful later in this lesson when students will need to use symmetry to match a mean to the distribution.
This Math Talk provides an opportunity for students to notice and make use of the symmetric structure (MP7) of the values to determine the mean. While participating in these activities, students need to be precise in their word choice and use of language (MP6).
Monitor for students who:
use the standard algorithm for finding mean (sum and divide)
use the symmetry of the data set
Launch
This is the first time students do the
math talk
instructional routine, so it is important to explain how it works before starting.
Explain the math talk routine: one problem is displayed at a time. For each problem, students are given a few minutes to quietly think and give a signal when they have an answer and a strategy. The teacher selects students to share different strategies for each problem, and might ask questions like “Who thought about it a different way?” The teacher records students' explanations for all to see. Students might be asked to provide more details about why they decided to approach a problem a certain way. It may not be possible to share every possible strategy for the given limited time; the teacher may only gather two or three distinctive strategies per problem.
Consider establishing a small, discreet hand signal that students can display to indicate that they have an answer they can support with reasoning. This signal could be a thumbs-up, a certain number of fingers that tells the number of responses they have, or another subtle signal. This is a quick way to see if the students have had enough time to think about the problem. It also keeps students from being distracted or rushed by hands being raised around the class.
Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.
Representation: Internalize Comprehension.
To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization
Student Facing
Evaluate the mean of each data set mentally.
27, 30, 33
61, 71, 81, 91, 101
0, 100, 100, 100, 100
0, 5, 6, 7, 12
Student Response
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Anticipated Misconceptions
If students struggle to use symmetry as a method for finding the mean, consider asking them to find the mean for the values: 1, 2, 3, 4, 5.
Activity Synthesis
Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:
“Who can restate ___’s reasoning in a different way?”
“Did anyone have the same strategy but would explain it differently?”
“Did anyone solve the problem in a different way?”
“Does anyone want to add on to __’s strategy?”
“Do you agree or disagree? Why?”
Although all correct methods for solving for the mean are valid, highlight the use of symmetry in the data. In previous lessons, students learned that symmetric distributions have a mean in the center of the data. When symmetry is present, it can be used to quickly discover the mean.
Speaking: MLR8 Discussion Supports.
Display sentence frames to support students when they explain their strategy. For example, "First, I _____ because . . ." or "I noticed _____ so I . . ."  Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Design Principle(s): Optimize output (for explanation)


aligns: no