2.1 Patterns and Inductive Reasoning

GeometryChapter 2: Reasoning and Proof

How do you know?

1. Take a piece of paper.2. Fold it in half then unfold it…How many

rectangles do you have?3. Refold the paper and fold it again. Unfold all

the way…How many do you have now?4. If you folded the paper in half 8 times…How

many rectangles would you have?

Inductive Reasoning

• Reasoning based on patterns you OBSERVE

• Used in science - Conclusions made by using the scientific method

Finding and Using Patterns

What are the next two terms in each sequence and what is the pattern description?• 3, 9, 27, 81, …

• 45, 40, 35, 30,…

What are the next 2 terms?

Finding and Using Patterns

Use Your Brain

• In order to figure out the nth-term in a sequence, we make a logical conclusion.

• Conjecture is a conclusion reached using inductive reasoning.

Using Inductive Reasoning

What conjecture can you make about the number of regions after 20 diameters are drawn?

Make a Conjecture!

• R, W, B, R, W, B, R, W, B, …

• What will the 21st term be?

Collect and Decide!

• What conjecture can you make about the sum of the first 30 even numbers?

• # of Terms Sum1 2 = 2 = 1*22 2 + 4 = 6 = 2*33 2 + 4 + 6 = 12 =3*44 2 + 4 + 6 + 8 = 20 = 4*5

What about the first 30?30*31 = 930

You try it!

• What conjecture can you make about the sum of the first 30 odd numbers?

Use the DataSales of backpacks at a nationwide company decreased over a period of six consecutive months. What conjecture can you make about the number of backpacks the company will sell in May?

You are wrong!

• To prove a conjecture WRONG, all you need is ONE counterexample.

• Counterexample is an example that shows a conjecture is incorrect

Prove me Wrong!

• If the name of a month starts with the letter J, it is a summer month.

• You can connect any three points to form a triangle.

Practice Problems

Page 85-87#7-37 (odd)

#38-55

At the Rainbow Apartments, the trim is painted a different color on each floor. Use the clues

below to find out what color the trim is painted on each floor.

CLUES:– The blue trim is on a higher floor than the green

trim, but on a lower floor than the one painted yellow.

– The pink trim is on the floor between the one with yellow trim and the one painted blue.

– The green trim is on the lower floor than the one with lavender trim.

Decide who is twins:1. All sets of twins consist of a boy and girl.2. Anna and her twin arrived on Monday3. Al isn’t Anna’s twin4. Cal and his twin arrived on Monday.5. Sam arrived on Friday, but not with Jill6. Nick and Ellen are twins. They arrived the day

after Dana and her twin.

Al Bob Cal Nick Sam Mon Tues Wed Thurs Fri

Anna

Beth

Dana

Ellen

Jill

Mon

Tues

Wed

Thurs

Fri

Ticket out!

What are the next two terms in each sequence?1) 7, 13, 19, 25,…

2)

3) What is a counterexample for the following conjecture?

All four-sided figures are squares.