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Inductive Reasoning
Lesson 1.7 p. 35
Vocabulary
• Inductive Reasoning
Making conclusions based on patterns you observe
• Conjecture
A conclusion you reach by inductive reasoning
• Counterexample (p. 37)
An example that proves a statement false
Reasoning Inductively
Observation:
The triangle is rotating clockwise
around the square.
Conjecture:
The triangle will be in the lower right
corner of the last square.
The shape inside the circle has one more side
Observation: than the previous shape.
Conjecture: The next shape will be a hexagon.
Writing Rules for Patterns
Number Patterns: The conjecture can be a rule.
30, 25, 20, 15 . . . . .
Start with 30 and subtract 5 repeatedly
2, -2, 2, -2 . . .
Alternate 2 and its opposite.
1, 3, 4, 12, 13, . . .
Start with 1. Alternate by multiplying by 3 and
adding 1.
4, 9, 14, 19 . . .
Start with 4 and add 5 repeatedly
3, 9, 27, 81. . .
Start with 3 and multiply by 3 repeatedly
1, 1, 2, 3, 5, 8 . . .
Start with 1, 1. Then each number is the
sum of the previous two numbers.
Extending a Pattern
Write a rule and find the next two numbers in
the pattern.
40
20
640, 320, 160, 80, ______, ________
Start with 640. Divide by 2 to find the
next number.
Discuss with your neighbor……what are the next
5 numbers in the list?
10, 5, 2 ½ , 1 ¼ , 5/8
Write a rule and find the next 4 numbers
in this pattern:
1, 3, 5, 7, ……..
Start with one and add 2 repeatedly.
The next four numbers are 9, 11, 13, and 15.
Predictions and Counterexamples
Is the conjecture that average hourly earnings in the year
2005 will be about $15.75 reasonable?
The wage rate increases between $1.50 - $2.50 every 5 years.
This conjecture is reasonable.
$13.75 + 2 = $ 15.75
You toss a coin four times, and it comes
up heads each time.
Is it reasonable to make
the conjecture that it
will come up heads
every time?
NO! It could show “tails”
on any toss. It has the same chance
of landing on heads as it does tails.
This is a counterexample.
Analyzing Conjectures
Every four-sided figure is a rectangle…….
This conjecture is not correct…..
The absolute value of any integer is positive.
False…..0 is the absolute value of 0.
0 is neither + nor -.
The next figure in this pattern has 15 *
*
*
* *
*
*
* *
* * *
* *
* * *
* * * *
*
* *
* * *
* * * *
* * * * *
This is correct. The last picture has 15 *.
Check Understanding p. 37
The last digit of the product of 5 and a whole
number is 0 or 5.
This is correct.
A number and its absolute value are always
opposites. This is NOT correct. |8| = 8
The next figure in the pattern will have 25 dots.
This is correct.
What have we done?
• We have used new terms to identify ways of
thinking mathematically…….
• Inductive Reasoning………
• Conjecture …………
• Counterexample ………….