Section 9.1 : Patterns and Inductive Reasoning

Learning Targets: G.CO.9, G.CO.10, G.CO.11

Important Terms and Definitions

Conjecture: An unproven statement that is based on a pattern or observation

Inductive Reasoning: Using powers of observation to find patterns and making conjectures based on that observation

Counterexample: An example that shows a conjecture is false.

Describing Visual Patterns

(ex 1) Shade the appropriate squares for the next figure in the pattern. How does the figure change? Is it rotated?

(ex 2) Sketch the next figure in the pattern.

Describing a Number Pattern

Describe the pattern in each sequence of numbers. Then predict the next two numbers.

(ex 3)

(ex 4)

product of consecutive numbers is even

Using Inductive Reasoning

(ex) What conjecture can you make about the 21st term of the given sequence?

P, K, M, P, K, M, …

21st term: M

Explanation: The pattern is a repetition of the same three letters. Every third letter is M

(ex 5) What is the shape of the thirtieth figure? The fortieth figure?

Making and Testing Conjectures

(ex) The sum of the first n even positive integers is even.

first even integer: sum of first two even integers: sum of first three even integers: sum of first four even integers: sum of first n even integers:

Conjecture is verified – the sum of the first n even positive integers is even.

(ex 6) What conjecture can you make about the product of two odd numbers?

Number of Chirps in 14 sec

Temperature

5 45

10 55

15 65

(ex 7) The speed at which a cricket chirps is affected by the temperature. If you hear 20 cricket chirps in 14 seconds, what is the temperature?

What is a counterexample for each conjecture.

(ex 8) If a flower is yellow, it is a sunflower.

(ex 9) If the difference of two numbers is odd, then the greater of the two numbers must also be odd.

Number of Points on Circle 2 3 4 5 6 Number of Regions 2 4 ? ? ?

Homework – Section 9.1 : Patterns and Inductive Reasoning

Sketch the next figure in each pattern.

1. 2.

Describe the pattern in each sequence of numbers. Then predict the next two numbers.

3. 4. 5. 6.

7. Using the figure from problem #2, how many dots can you conjecture would be in the

bottom row of the 6th figure? the 40th figure?

8. Using the pattern below, what is the shape of the 16th figure? The 41st figure?

9. What conjecture can you make about the sum of any three consecutive integers? Test your conjecture.

10. What conjecture can you make about the sum of any two odd numbers? Test your conjecture.

Find a counterexample for each conjecture.

11. If a number is divisible by 2, then it is divisible by 4. 12. If , then x is positive. 13. If the product of two numbers is even, then the two numbers must be even.

14. Using the figures from problem #1, complete the following table.

a) Make a conjecture about the number of points on the circle and the number of regions. b) Test your conjecture for the case of 6 points. What do you notice?