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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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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
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?
counterexample
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.
guided practice
Guided Practice
  • Pages 6 – 7 (selected problems 1 – 30)
peer and independent work
Peer and Independent WORK
  • Pages 7 – 8 (selected problems 31 – 46)
isometric drawing
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?
orthographic drawing
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?
foundation drawing
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?
slide11
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?
guided practice1
Guided practice
  • Page 13 (1 – 16)
peer and independent practice
Peer and Independent Practice
  • Pages 13 – 14 (18 – 32)
homework
Homework
  • Page 9 # 54 -55
  • Page 15 # 33 - 34
homework for 8 12 13
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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