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Patterns and Inductive Reasoning

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Patterns and Inductive Reasoning

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

- 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?

- 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.

- Pages 6 – 7 (selected problems 1 – 30)

- Pages 7 – 8 (selected problems 31 – 46)

- 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 – 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 – 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?

- 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?

- Page 13 (1 – 16)

- Pages 13 – 14 (18 – 32)

- Page 9 # 54 -55
- Page 15 # 33 - 34

- 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 rightn + 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.