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PRE-ALGEBRA
Lesson 1-7 Warm-Up
PRE-ALGEBRA
Inductive Reasoning (1-7)
What is “inductive
reasoning”?
inductive reasoning: making judgements or drawing conclusions based on
patterns you observe.
What is a
“conjecture”?
conjecture: The conclusion you reach by inductive reasoning.
Example:
Observation: The shaded triangle is rotating clockwise around the square.
Conjecture: The next figure will have a shaded triangle in the bottom right
corner.
How do you write a To write a conjecture: 1. find the pattern by determining what was done to a
conjecture and use it number to get the next number, and 2. write the rule of the pattern. Note:
to continue a pattern. Begin the rule with the number you start with.
Example: Write the rule for 640, 320, 160, 80, … and use your rule to find the
next two numbers in the pattern.
The rule is, “Start with 640 and divide by 2”. The next two number, then, are
40 and 20.
PRE-ALGEBRA
Inductive Reasoning
LESSON 1-7
Additional Examples
Use inductive reasoning. Make a conjecture about
the next figure in the pattern. Then draw the figure.
Observation: The circles are rotating counterclockwise
within the square.
Conjecture: The next figure will have a shaded circle at
the top right.
PRE-ALGEBRA
Inductive Reasoning
LESSON 1-7
Additional Examples
Write a rule for each number pattern.
a. 0, – 4, – 8, –12, . . .
Start with 0 and subtract 4 repeatedly.
b. 4, – 4, 4, – 4, . . .
Start with 4. Then alternate 4 and its opposite.
c. 1, 2, 4, 8, 10, . . .
Start with 1. Alternate multiplying by 2
and adding 2.
PRE-ALGEBRA
Inductive Reasoning
LESSON 1-7
Additional Examples
Write a rule for the number pattern 110, 100, 90, 80, . .
Find the next two numbers in the pattern.
110, 100, 90,
– 10 – 10 – 10
80
The first number is 110.
The next numbers are found by
subtracting 10.
The rule is Start with 110 and subtract 10 repeatedly.
The next two numbers in the pattern are 80 – 10 = 70 and
70 – 10 = 60.
PRE-ALGEBRA
Inductive Reasoning (1-7)
How can you tell
You can tell whether a conjecture is reasonable by making a prediction.
whether a conjecture Example: Is the conjecture that average hourly earnings in 2013 will be abou
is reasonable?
$18.25 reasonable?
The conjecture is reasonable, since average hourly earning seem to be
increasing about $2.00 every five years.
What is a
“counterexample”?
counterexample: a statement that proves a conjecture false (Note: you only
need one counterexample to prove that a conjecture is wrong.)
How can you use a
counterexample to
prove a conjecture is
wrong?
Example: Is the conjecture that every four-sided figure is a rectangle
reasonable.
Counterexample:
The conjecture is incorrect. The figure above is a four-sided figure but not a
rectangle.
PRE-ALGEBRA
Inductive Reasoning
LESSON 1-7
Additional Examples
A child grows an inch a year for three years in a row. Is
it a reasonable conjecture that this child will grow
an inch in the year 2015?
No; children grow at an uneven rate, and eventually they
stop growing.
PRE-ALGEBRA
Inductive Reasoning
LESSON 1-7
Additional Examples
Is each conjecture correct or incorrect? If it is
incorrect, give a counterexample.
a. Every triangle has three sides of equal length.
The conjecture is incorrect. The figure below is a
triangle, but it does not have three equal sides.
b. The opposite of a number is negative.
The conjecture is incorrect. The opposite of –2 is 2.
PRE-ALGEBRA
Inductive Reasoning
LESSON 1-7
Additional Examples
(continued)
c. The next figure in the pattern below has 16 dots.
The conjecture is correct. The diagram below
shows the next figure in the pattern.
PRE-ALGEBRA
Inductive Reasoning
LESSON 1-7
Lesson Quiz
Find the next three numbers in each pattern.
1. 1, –1, 2, –2, 3, . . .
–3, 4, –4
2. 1, 3, 7, 15, 31, . . .
63, 127, 255
3. –11, –8, –5, –2, . . .
1, 4, 7
PRE-ALGEBRA