Why Stay with Whole Numbers?

Examples, videos, and solutions to help Grade 7 students learn how to use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

New York State Common Core Math Algebra I, Module 3, Lesson 8

Worksheets for Algebra I, Module 3, Lesson 8 (pdf)

Lesson 8 Student Outcomes

• Students use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
  
• Students create functions that represent a geometric situation and relate the domain of a function to its graph and to the relationship it describes.

Lesson 8 Summary

• Formulas that represent a sequence of numbers have a set of inputs; each input number is used to represent the term number. The outputs of the formula listed in order form the sequence.
  
• Formulas that represent different situations such as the area of a square of side x can have a set of inputs consisting of different subsets of the real number system. The set of inputs that makes sense depends on the situation, as does the set of outputs.

Opening Exercise The sequence of perfect squares {𝟏, 𝟒, 𝟗, 𝟏𝟔, 𝟐𝟓, … } earned its name because the ancient Greeks realized these quantities could be arranged to form square shapes. If 𝑺(𝒏) denotes the 𝒏 th square number, what is a formula for 𝑺(𝒏)?

Exercises

1. Prove whether or not 169 is a perfect square.
2. Prove whether or not 200 is a perfect square.
3. If 𝑆(𝑛) = 225, then what is 𝑛?
4. Which term is the number 400 in the sequence of perfect squares? How do you know? Instead of arranging dots into squares, suppose we extend our thinking to consider squares of side length 𝑥 cm.
5. Create a formula for the area 𝐴(𝑥) cm2 of a square of side length 𝑥 cm: 𝐴(𝑥) = ___________.
6. Use the formula to determine the area of squares with side lengths of 3 cm, 10.5 cm, and π cm.
7. What does 𝐴(0) mean?
8. What does 𝐴(−10) and 𝐴(√2) mean? The triangular numbers are the numbers that arise from arranging dots into triangular figures as shown:
9. What is the 100th triangular number?
10. Find a formula for 𝑇(𝑛), the 𝑛th triangular number (starting with 𝑛 = 1).
11. How can you be sure your formula works?
12. Create a graph of the sequence of triangular numbers (𝑛) = 𝑛(𝑛+1)/2, where 𝑛 is a positive integer.
13. Create a graph of the triangle area formula 𝑇(𝑥) = 𝑥(𝑥+1)/2, where 𝑥 is any positive real number.
14. How are your two graphs alike? How are they different?

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Lesson 8 Exit Ticket
Recall that an odd number is a number that is one more than or one less than twice an integer. Consider the sequence formed by the odd numbers {1, 3, 5, 7, …}.

1. Find a formula for O(n), the nth odd number starting with n = 1.
2. Write a convincing argument that 121 is an odd number.
3. What is the meaning of O(17)?

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