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Why: This is a teaching routine useful in many contexts whose purpose is to give all students enough time to think about a prompt and form a response before they are expected to try to verbalize their thinking. First they have an opportunity to share their thinking in a low-stakes way with one partner, so that when they share with the class they can feel calm and con dent, as well as say something meaningful that might advance everyone’s understanding. Additionally, the teacher has an opportunity to eavesdrop on the partner conversations so that she can purposefully select students to share with the class.
Launch
Arrange students in groups of 2. After a few minutes of quiet think time, ask students to share their responses with a partner before whole-class discussion.
Student Task Statement
Let .
1. Which is larger, or ? 2. Which is larger, or ? 3. Which is larger, or
Student Response
? Explain how you know.
1.
2.
3. , because it will always be 3 larger than .
Activity Synthesis
Select di erent students to share their response and reasoning for the rst two questions. The values of the functions in the rst two questions can be computed directly, but for the last question, students need to reason about some arbitrary input and 1 less than it, .
For the last question, ask several students to explain their reasoning. After the rst, ask, “Did anyone think about it a di erent way?” Record a few di erent ways of reasoning about the last question. The goal is to be able to accurately interpret the meaning of as it relates to with language like “ is the value of the function for some input, and is the value of the function when the input is 1 less.”
5.2 Bowling for Triangles
20 minutes
In this activity, students investigate and By expressing regularity in repeated reasoning (MP8), they interpret . Students brie y consider why it would be di cult to use this rule to compute In the discussion that follows the activity, it is suggested that an example of a recursive de nition for a familiar geometric sequence be presented.
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Teacher Guide Algebra