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Patterns and Inductive Reasoning

Patterns and Inductive Reasoning. Inductive reasoning. Reasoning that is based on patterns you observe. Example: 2,5,8,11,… What do you see? How do you know? Justify. Conjecture. A conclusion you reach using inductive reasoning. Example: 1 = 1 = 1x1 1 + 3 = 4 = 2x2

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Patterns and Inductive Reasoning

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  1. Patterns and Inductive Reasoning

  2. Inductive reasoning • Reasoning that is based on patterns you observe. • Example: 2,5,8,11,… • What do you see? • How do you know? • Justify

  3. Conjecture • A conclusion you reach using inductive reasoning. • Example: • 1 = 1 = 1x1 • 1 + 3 = 4 = 2x2 • 1 + 3 + 5 = 9 = 3x3 • 1 + 3 + 5 + 7 = 16 = 4x4 • Make a conjecture about the sum of the first 30 odd numbers. • How do you know?

  4. Counterexample • An example for which the conjecture is incorrect. • Example: Find a counterexample for this conjecture. • The square of any number is greater than the original number. • Can you think of a number that makes this false? • How do you know that number makes the conjecture false? • Justify your thinking.

  5. Guided Practice • Pages 6 – 7 (selected problems 1 – 30)

  6. Peer and Independent WORK • Pages 7 – 8 (selected problems 31 – 46)

  7. Drawings, Nets, and Other Models

  8. Isometric drawing • Isometric drawing – a drawing on isometric dot paper to show three sides of a figure from a corner view. • Example – p. 10 • In Greek, isosmeans “equal” and metron means “measure.” • In an isometric drawing, all 3-D measurements are scaled equally. • What do you know about 3-D drawings? • Is it important? Why or why not?

  9. Orthographic drawing • Orthographic drawing – another way to show a three-dimensional figure. It shows the top view, front view, and right-side view. • Example : p. 11 #2 • Have you ever seen an orthographic drawing in real-life? • When should they be used? • Why?

  10. Foundation drawing • Foundation drawing – shows the base of a structure and the height of each part. • Example: p. 11 #3 • When would someone use this type of drawing? • Why would that be a better choice?

  11. Net • Net – a two-dimensional pattern that you can fold to form a three dimensional figure. • Have you seen a net? • When are they used? • Do you think they are helpful? • Why or why not? • What is different about it from the other drawings? • Which do you prefer? • Do you think we should use all 4? • Why or why not?

  12. Guided practice • Page 13 (1 – 16)

  13. Peer and Independent Practice • Pages 13 – 14 (18 – 32)

  14. Homework • Page 9 # 54 -55 • Page 15 # 33 - 34

  15. Homework for 8-12-13 • 54.When he was in the third grade, German mathematician Karl Guass (1777-1855) took ten seconds to sum the integers from 1 to 100. Now it’s your turn. Find a fast way to sum the integers from 1 to 100; from 1 to n. ( Hint: Use Patterns) • 55. a. Write the first six terms of the sequence that starts with 1, and for which the difference between consecutive terms is first 2, and then 3, 4, 5, and 6. • b. Evaluate (n2 + n)/ 2 for n =1, 2, 3, 4, 5, and 6. Compare the sequence you get with your answer for part (a). • c. Examine the diagram at the right n + 1 • and explain how it illustrates a value of # # # * • (n2 + n)/ 2. n # # * * • d. Draw a similar diagram to represent # * * * • (n2 + n)/ 2 for n=5.

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