2.1 Inductive Reasoning

Inductive reasoning is the process of observing data, recognizing patterns, and making generalizations based on those patterns.

Conjecture is a generalization based on inductive reasoning

An example of inductive reasoningSuppose your history teacher likes to give

“surprise” quizzes. You notice that, for the first four chapters of

the book, she gave a quiz the day after she covered the third lesson.

Based on the pattern in your observations, you might generalize that you will have a quiz after the third lesson of every chapter.

Inductive reasoning can be used to makea conjecture about a number sequenceConsider the sequence 10, 7, 9, 6, 8, 5, 7, . . .Make a conjecture about the rule for

generating the sequence. Then find the next three terms.

SolutionLook at how the numbers change from term

to termThe 1st term in the sequence is 10. You

subtract 3 to get the 2nd term. Then you add 2 to get the 3rd term. You continue alternating between subtracting

3 and adding 2 to generate the remaining terms.

The next three terms are 4, 6, and 3.

Describing Number and Geometric PatternsObjectives:• Use inductive reasoning in continuing patterns

Inductive reasoning: • make conclusions based on patterns you observe

Conjecture: • conclusion reached by inductive reasoning based on evidence

Geometric Pattern:• arrangement of geometric figures that repeat

• Arrangement of geometric figures that repeat• Use inductive reasoning and make conjecture as to the next figure in a pattern

Geometric Patterns

Use inductive reasoning to find the next two figures in the pattern.

Use inductive reasoning to find the next two figures in the pattern.

Describe the figure that goes in the missing boxes.

Geometric Patterns

Describe the next three figures in the pattern below.

Numerical Sequences and Patterns

Arithmetic Sequence

Add a fixed number to the previous termFind the common difference between the previous & next term

Find the next 3 terms in the arithmetic sequence.

2, 5, 8, 11, ___, ___, ___

+3 +3 +3 +3

14

+3

17

+3

21

What is the common difference between the first and second term?

Does the same difference hold for the next two terms?

Investigation: Shape ShiftersPage 98In the investigation you look at a pattern in a

sequence of shapes.Complete each step of the investigation. Below

are hints for each step if you need them.

Step 1: Are the shapes the same or different?How does the shaded portion of the shape

change from one odd-numbered shape to the next?

Step 2: First, focus on the polygon shape. Does the polygon change from one even-

numbered shape to the next? If so, how does it change? Second, focus on the small circles inside the

shape. How do these circles change from one even

numbered shape to the next?

Step 3: The next shape is the 7th shape.Because it is an odd-numbered shape, use the

patterns you described in Step 1 to figure out what it will look like.

The 8th shape is an even-numbered shape, so it should follow the patterns you described in Step 2.

Step 4: Notice that the odd-numbered shapes go through a cycle that repeats every eight terms.

So, for example, the 1st, 9th, and 17th shapes look the same;

the 3rd, 11th, and 19th shapes look the same; and so on.

Use this idea to figure out what the 25th shape looks like.

Step 5: How many sides does the 2nd shape have?

The 4th shape? The 6th shape? The nth shape? How many sides will the 30th shape have?How will the dots be arranged on the 30th

shape?

Inductive Reasoning process of observing data,recognizing patterns and making generalizations about those patterns.

Drawing conclusions based on experienceEx:

Cause/effectTurning on waterRiding a bike

"Geometry“ means measure of earthCan you name Professionals who use

inductive reasoning?

ConjectureA generalization made with inductive reasoning

(Drawing conclusions)Example: Bell rings M, T, W, TH at 7:40 am

Conjecture? The bell will ring at 7:40 am on Friday

Example: Chemist puts NaCl on flame stick and puts into flame and sees an orange-yellow flame. Repeats for 5other substances that also contain NaClConjecture?

All substances containing NaCl will produce an orange-yellow flame

Mathematicians use Inductive Reasoning to find patterns which will then allow them to conjecture.

We will be doing this ALOT this year!!

Finding PatternsEx: 2, 4, 7, 11, ...

Rule? Add the next consecutive integer

Next 3 terms? 16, 22, 29

Ex: 1, 1, 2, 3, 5, 8, 13, ...Rule?

Add previous two terms (Fibonacci Sequence)Next 3 terms?

21, 34, 55Ex: 1, 4, 9, 16, 25, 36, ...

Rule? Add consecutive odd numbers OR the perfect squares

Next 3 terms? 49, 64, 81