Patterns and Inductive Reasoning

Lesson Plan - Get It!

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Inductive reasoning is used often in geometry.

It allows you to make conclusions and conjectures based on patterns of examples or past events. Inductive reasoning is an extremely important part of the world of mathematics, since many mathematicians have made mathematical discoveries through the use of inductive reasoning.

Below is a simple example of the use of inductive reasoning and making a conjecture:

Assume that you are given the sequence 2, 4, 6, 8, ….

• What would be the next term in the sequence?
• What would be the 10^th term in the sequence?

Even though this is a simple example, it is helpful in illustrating the steps you need to go through when using inductive reasoning and making your conjecture.

Read the steps below:

1. Describe the pattern you see in the sequence

You add to the previous term to get the next term.

2. Understand what the question(s) are asking you to do.
   • What is the next term in the sequence?

You would have to add 2 to the last term in the sequence to get the next one. 8 + 2 = 10, so the next term in the sequence would be 10.

2. • What would be the 10^th term in the sequence?

You could continue to add 2 to each term until you reach the 10^th term, or you could multiply 10 and 2 to get the 10^th term.

Notice that in the placement of each term in the sequence above, when the term is multiplied by 2, the answer is in the place of the term. Term 1 is 2 (1 x 2 = 2). Term 2 is 4 (2 x 2 = 4).

If you multiply Term 10 by 2 you will get the term (10 x 2 = 20). Therefore, the 10^th term is 20.

Below is another sample problem. See if you can make the conjecture:

A local bike shop noticed that the sales of its children's bicycles was decreasing. Using the graph below and inductive reasoning, can you predict how many bikes the shop will sell in June?

1. Your first step is to identify the pattern in the information.

Monthly sales:

January - 57 bikes
February - 54 bikes
March - 51 bikes
April - 48 bikes
May - 45 bikes

2. Your next step is to make your conjecture.

Here is a note of caution when using inductive reasoning:

• There will be times when a conjecture may be incorrect, so it's important to verify your answer.
• This is the same as "checking your answer" in algebra, when you substitute it back into the original problem to make sure that it is correct.
• Do the same thing when using inductive reasoning in geometry.
• For example, what answer did you arrive at for the "Bike Sales" example above?
• Did you check to make sure that you arrived at the correct answer?

Good for you!

You will use inductive reasoning and making conjectures quite a bit in geometry, with many more challenging problems ahead.

In the Got It? section, you will have opportunities to practice your new skills to assure your understanding of these new concepts.