Patterns and inductive reasoning
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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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Patterns and Inductive Reasoning

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Patterns and inductive reasoning

Patterns and Inductive Reasoning


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)


Drawings nets and other models

Drawings, Nets, and Other Models


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?


Patterns and inductive reasoning

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 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.


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