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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

### 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

• 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

• 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

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

### Peer and Independent WORK

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

### 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 – 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 – 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

• 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 practice

• Page 13 (1 – 16)

### Peer and Independent Practice

• Pages 13 – 14 (18 – 32)

### Homework

• Page 9 # 54 -55

• Page 15 # 33 - 34

### 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.