Algebra 1 – Chapter 2 project

1. Algebra 1 – Chapter 2 project Questions about Perfect Squares

2. Asking questions In order to understand the underlying structure of numbers (which is more complicated than you would ever think), mathematicians ask questions about them. In this project, you will answer questions about the sums of perfect squares. Then, you will come up with your own questions that you think will generate interesting answers.

3. Perfect Squares • Integers that are the squares of integers • 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, … • Note: integer: whole numbers, zero, and negative whole numbers

4. 1. Sums of perfect squares • Can you write EVERY number as a sum of two nonzero perfect squares? • If so: tell us how to do so for any number. • If not: find an integer that you cannot write as a sum of two nonzero perfect squares.

5. 2. Specific Sums • Find two perfect squares that are the sums of two nonzero perfect squares.

6. 3. Multiple Representations • Are there any integers you can write as the sum of two nonzero squares in two different ways? • *Note: changing the order of the numbers that you are adding doesn’t count.

7. Asking good questions • Replace the italicized words with a new word. • Is your new question a good question? • Why or why not? • Are any integers the sum of two nonzero perfect SQUARES? • Are any integers the SUM of two nonzero perfect squares? • Are any integers the sum of THREE nonzero perfect squares?

8. 7. Extending an idea You can ask a good question by extending an idea. We only asked about positive integers. Is there a good question we could ask about negative integers? Explain.

9. 8. Using answers to ask more questions • Look at your responses to question 2. Use your answer to ask a new question.