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Patterns of Square Numbers

Patterns of Square Numbers. Module 1. A Question For You…. You are helping your niece with her homework and she says, “I notice that every time I square an integer that ends in a 5, the resulting product ends in 25 .” You wonder if this is always true. Think on your own… is this always true?.

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Patterns of Square Numbers

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  1. Patterns of Square Numbers Module 1

  2. A Question For You… You are helping your niece with her homework and she says, “I notice that every time I square an integer that ends in a 5, the resulting product ends in 25.” You wonder if this is always true. Think on your own… is this always true?

  3. A Question For You… You are helping your niece with her homework and she says, “I notice that every time I square an integer that ends in a 5, the resulting product ends in 25. You wonder if this is always true. Discuss the validity of this statement with your small group.

  4. Write Your Proof: This question requires you to show your work and explain your reasoning. You may use drawings, words, and numbers in your explanation. Your answer should be clear enough so that another person could read it and understand your thinking. It is important that you show all your work.

  5. And Your Response? • Do you think this is always true? • How do you know? • How could you justify your claim for any integer value? • How many examples would be enough?

  6. NAEP item: Patterns of Square Numbers 152 = 225 252= 625 352 = 1225 The examples above suggest the following statement: When a positive integer that ends in the digit 5 is squared, the resulting integer ends in 25. Explain why this statement is always true. (Hint: (10n + 5)2 = ?)

  7. Rewrite Your Proof Take a few minutes to make any changes to your proof that you would like to make. Is the hint helpful and can it be used within your justification?

  8. Grading Rubric • A grading rubric is used to show students exactly what the expectations are for a task that is being assessed. • Rubrics help provide some credit for correct thinking even if the answer to the task is not exactly right. • Typically, a rubric scores with points ranging from 0 to 4 and includes a description of the characteristics of a response that ranks at each levels.

  9. Create Your Own Guidelines

  10. Share Your Rubric • Share your characteristics for each level beginning with level 0. • Record each groups’ ideas in one list. • Decide upon a rubric for the whole class.

  11. How Did They Do? • Assess the following student responses using our class rubric. • You may want to make notes about how a level 2 response differs from a level 3 response with this particular problem. • Try to be consistent.

  12. NAEP Scoring Guide • Extended: A complete and correct solution • “When 10n+5 is squared, 2 of the terms will be multiples of 100 and the other will be 25” • Satisfactory: Explanation is correct but incomplete or not clear • “(10n+5)2=100n2+100n+25” (without explanation) • Partial: Explanation is not correct • “(10n+5)2=100n2+25” • Minimal: Student provides numerical examples only • Incorrect/Off Task or Repeats given information

  13. NAEP Scoring Guide • Compare the rubric your class created to the scoring guide from NAEP. • Would you change our rubric? • Compare your scoring of student work to the score NAEP would assign.

  14. How Did You Do? • As you assess other student work, think about your own proof. • What would your score be? • What would your NAEP score be? • What revisions would you like to make now?

  15. Follow-Up • What did you learn about your thinking on this problem? • Did looking at others’ responses help you think about your own response? Why or why not? • Was the hint useful or misleading? Why? • Only 2% of students answered this question sufficiently. Why might that be?

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