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If it doesn’t come up, highlight the fact that A and B are equivalent expressions, and C and D are equivalent expressions.
7.2 Toward a Closed Form
25 minutes
In this activity, students develop a closed-form de nition of both a geometric and arithmetic sequence to represent a mathematical context in a sca olded way. In order to successfully complete this task, students express regularity in repeated reasoning (MP8) because they understand how the values of speci c terms are calculated before explaining or expressing how the
th term is calculated.
Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
If using, distribute scissors and 2 copies of the blackline master to each group of 2 students.
If students have access to graphing technology, suggest that it might be a helpful tool in this activity.
Student Task Statement
Clare takes a piece of paper with length 8 inches and width 10 inches and cuts it in half. Then she cuts it in half again, and again. . .
1. Explain why the area of Clare’s paper, in square inches, after two cuts is .
2. Write an expression for the area of the paper, in square inches, after 5 cuts.
3. Explain why the area of the paper, in square inches, after cuts is given by the formula .
4. Approximately what is the area of the paper after 10 cuts?
5. Kiran takes a piece of paper with length 8 inches and width 10 inches and cuts away one inch of the width. Then he does it again, and again. . . Complete the table for , the area of Kiran’s paper in square inches after cuts.
6. Explain why after 6 cuts, the area of Kiran’s paper, in square inches, is .
7. Write a formula for that is not recursive.
8. Which is larger, or ?
9. Which is larger, or ? Explain how you know.
Unit 1
Lesson 7: Closed Form De nitions
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