Proving Conjectures
Math 11 Notes
Date:
Warm-Up: Making Conjectures
Example
Given the picture below, create a possible explanation for the
events that occurred.
Example
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Proving Conjectures
Math 11 Notes
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Understanding Conjectures
In the above examples, you used the evidence available (the
picture) to create a possible ___________________________. This is the
process of making a _____________________________.
Conjecture
A conjecture is a ___________________________________________________. A
conjecture is based on __________________________________________ but
is not yet proven.
Example
If the first three colors in a sequence are red, orange and yellow,
make a conjecture about what the next colors might be.
Example
Make a conjecture about the next numbers in the pattern:
2, 6, 12, 30, 42.
Practice
Based on the information below, make a conjecture about the
sum of any two odd numbers:
3 + 7 = 10
11 +5 = 16
9 + 13 = 22
7 + 11 = 18
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Proving Conjectures
Math 11 Notes
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Inductive Reasoning
Inductive
Reasoning
Inductive reasoning is reasoning that results in a conclusion,
generalization or educated guess that is ___________________________
______________________________________________________. In other words,
inductive reasoning uses the information available to make a
statement that is likely to be true, but is not necessarily _________.
Think
How did you use inductive reasoning to draw conclusions in the
warm-up questions?
Example
The following excerpt from the movie “Young Sherlock Holmes”
illustrates inductive reasoning. While you watch the clip, think
about why this is an example of inductive reasoning.
0:48 2:32
Example
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Proving Conjectures
Math 11 Notes
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Practice
Practice
--
Practice
a) Use this information to make a conjecture.
b) Show two more examples that demonstrate your conjecture.
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Proving Conjectures
Math 11 Notes
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Exploring the Validity of Conjectures
Warm-up
Make a conjecture about the horizontal lines in the diagram.
Think
How could you test your conjecture? Try it. Is the conjecture
true or false?
Practice
Make a conjecture about the vertical lines below. Test your
conjecture. Is it true or false?
As you can see, conjectures _______________________________________
_______________________. Therefore, basing a conclusion on
inductive reasoning alone is not a valid method of _____________.
However, it can guide us in making a valid proof by helping us
examine the evidence and make an _____________________________.
Counterexample
A counterexample is an example which shows that a conjecture
is __________________________. You need only _______ counterexample
to show a conjecture is false. Once you have done this, there is
not point in further pursuing a proof.
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Proving Conjectures
Math 11 Notes
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Example
a) Katy came to the following conclusion:
“When an even number is squared and one is added, the result
is a prime number”.
Katy’s conclusion is a __________________________ based on
_________________________ ___________________________.
b) Perry investigated Katy’s conjecture further by using even
numbers not shown. He correctly stated that Katy’s conjecture
is false. Determine a counterexample to show Katy’s conjecture
is false.
Example
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Proving Conjectures
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Math 11 Notes
When a conjecture is shown to be false, it may be possible to
revise the conjecture:
Example
Revise the conjecture above (the sum of two prime numbers is
an even number), so it holds true for prime numbers.
Example
Make a conjecture based on the number pattern if the condition
“prime number” was removed.
Practice
Consider the pattern of multiplication shown below.
a) Make a conjecture based on the pattern.
b) Provide a counterexample to prove your conjecture is false.
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Proving Conjectures
Math 11 Notes
Summary
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Inductive reasoning can be used to make ____________________, but
cannot be used to determine if a conjecture is ___________. We can
never be certain a conjecture is true for _____________possible
case. We know it to be true only for the cases we have
_______________________________.
To prove a conjecture is true, we need to use logical or
___________________________ reasoning, which we will discuss later.
To prove that a conjecture is false, we need only _______________
___________________________________
Note
Example
Being unable to find a counterexample does not prove that a
conjecture is true.
The mathematician Christian Goldbach is famous for
discovering a conjecture that has not yet been proven to be true
or false. Goldbach conjectures that every even number greater
than two can be written as the sum of two prime numbers. For
example, 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5. No one has ever
proved this conjecture so it remains just that.
Prime
Numbers
Prime numbers are numbers whose only ________________________
_________________________________________________________________.
Practice
List the prime numbers up to 100. Use these to test Christian
Goldbach’s conjecture.
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Proving Conjectures
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Math 11 Notes
Background Information: Expressing Numbers Algebraically
When we wish to prove a conjecture related to a number
property, it is necessary to express the number as a _____________,
often as ____ or ______.
Example
Find and explain a notation that can be used to represent a set
of consecutive numbers.
Practice
Find and explain a notation that can be used to represent any
even number.
Practice
Find a notation to represent any odd number.
Deductive Reasoning: Proving a Conjecture
Recall
A conjecture based on inductive reasoning is not guaranteed to
be ___________, even if we cannot find a _____________________________.
Remember Goldbach’s conjecture: we are unable to say whether
it is true or false since there is neither a known counterexample
nor a proof.
Inductive reasoning can play a part in the discovery of
mathematical proofs. However, another form of reasoning is
required to make a proof.
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Proving Conjectures
Math 11 Notes
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Proof
A proof is a sequence of facts and theorems known to be ________
that are used to establish the truth of a __________________________.
It is a mathematical argument that shows a statement is true in
all cases, or that no counterexample exists.
Think
Mahmoud goes to “One Hour Photo” to have his pictures
printed. The store advertises that pictures will be ready in one
hour or its free. Mahmoud submits his pictures at 3:00 pm and
they are not ready until 4:35. What can Mahmoud deduce from
this?
In the above example, there are two statements that we know to
be true . We use these facts to get a conclusion that is also true.
Deductive
Reasoning
Deductive reasoning (also known as ____________________________
_________________________) is the process of developing a conclusion
based on information that is already known to be true. The facts
that can be used to prove a conclusion deductively may come
from __________________________________________________________________.
The truth of the premises (the facts a conclusion is based on)
guarantees the truth of the _____________________________________.
In the example on the previous page, we used deductive
reasoning to conlude the pictures would be printed for free.
Theorem
A theorem is a statement which can be ______________ using
logical or ___________________________ reasoning.
Note
Inductive reasoning leads to a conjecture. A conjecture becomes
a theorem when deductive reasoning is applied to show that it
is definitely true.
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Proving Conjectures
Math 11 Notes
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A simple introduction to proof is to apply the transitive
property.
Transitive
Property
The transitive property states that if two quantities are equal to
the same quantity, then they are equal to ________________________.
If a = b and b = c, then _____________________.
Note
Be careful when applying this property. It is only true for
statements of equality. An example of an incorrect application
of this property would be the following:
“If Team A beats Team B. Team B beats Team C. Therefore,
Team A will beat Team C.”
The statements are not statements of equality, so the transitive
property cannot be applied in this case. Although there is a good
chance that Team A will beat Team C, anyone who has watched
the World Cup knows that this is not always true.
Example
All natural numbers are whole numbers. Three is a natural
number. What can be deduced about the number 3?
Practice
Write a conclusion that can be deduced from the statements
below:
“Water freezes below 0°C”.
“The temperature outside is -15°C”
“A bucket of water is placed outside”
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Proving Conjectures
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Math 11 Notes
Using a Venn Diagram to Show the Transitive Property
Example
Represent the information in a Venn Diagram then write a
conclusion that can be deduced from the pair of statements
below:
“Every prime number, except 2, is odd”
“17 is a prime number”
Practice
Write a conclusion from the following statements.
Youssef is taller than Lara. Lara is taller than Sarah.
Daniel lives in Abu Dhabi. Abu Dhabi is in the UAE.
Using Deductive Reasoning to Generalize a Conjecture
Practice
Use inductive reasoning to make a conjecture about what
number the sum of any three consective integers is divisible by
(ie. is the sum divisible by 2, 3, 4, etc?).
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Proving Conjectures
Math 11 Notes
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Example
Using deductive reasoning, prove the sum is divisible by ________.
Practice
Use deductive reasoning to prove the following conjecture:
“The difference between consecutive perfect squares is always
an odd number”.
Perfect
Square
A perfect square has a whole number square root. In other
words, any integer multiplied by itself is a perfect square
Practice
Complete the chart and make a conjecture using inductive
reasoning:
Instructions
Choose a number less
than 10
Add 7
Test Case 1
Test Case 2
Test Case 3
Multiply by 2
Subtract the original
number
Subtract 2
Subtract the original
number
Conjecture:
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Proving Conjectures
Math 11 Notes
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Example
Prove the conjecture using inductive reasoning. Let x represent
the original number chosen.
Note:
Instructions
Choose a number less than 10 (x)
Add 7
Multiply by 2
Subtract the original number
Subtract 2
Subtract the original number
2 (x+7)
=2x+14
General Case
Conclusion: We have proved the final answer is always ________.
Think
Example
Does the original number have to be less than 10?
“When two odd numbers are added, their sums are even”
Use inductive reasoning (show three examples) to suggest this
statement is true.
Use deductive reasoning to prove that the statement is true. Use
the numbers 2n – 1 and 2m – 1.
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Proving Conjectures
Math 11 Notes
Example
Date:
Complete the first three columns only.
Make a conjecture based on your answers:
Example
Use deductive reasoning to complete the general case column to
show that no matter which number you choose, the conjecture
is always true.
Comparing Inductive and Deductive Reasoning
Summary
Inductive Reasoning
Deductive Reasoning
Begins with ___________________
Begins with ______________
or a number of
__________________________ that
_____________________.
are considered __________.
An _________________ is made
The result is a
that the pattern or trend will
____________________ reached
continue. The result is a
from previously known
____________________.
____________.
Conjectures may or may not Conclusion must be _________
be ___________. One ___________ if all previous _________________
__________________ proves the
are true.
conjecture _________.
Used to make ________________ Used to draw ________________
guesses based on
that logically flow from the
_________________ and patterns.
hypothesis.
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Proving Conjectures
Math 11 Notes
Example
Date:
Inductive Reasoning
Deductive Reasoning
1+2=3
4+5=9
-7 + (-6) = -13
128 + 129 = 257
We can represent two consecutive
integers as x and x+1.
From the pattern shown, we think
the sum of two consecutive integers
is always an odd number. But we
can’t be sure the conjecture is valid
in every case based on only this
evidence.
Practice
The sum can be represented as
x + (x + 1) = 2x +1
Since 2x is an even number,
2x +1 is an odd number.
The conjecture has been proven for
the general case, so it is valid for
any two consecutive integers.
Saleem had breakfast at Brioche for three Saturdays in a row.
He saw Ahmed there each time. Saleem told his friends that
Ahmed always eats breakfast at Brioche on Saturdays.
a) What type of reasoning did Saleem use?
b) Is his conclusion valid? Explain.
Think
Textbook
Chapter
Review:
P. 34 - 35
Challenge
Provide examples in your everyday life in which you use
inductive reasoning and in which you use deductive reasoning.
Ten women meet for a bowling tournament and each shakes the
hand of every other woman. Determine the number of
handshakes that occurred.
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Proving Conjectures
Math 11 Notes
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Two Column Proofs
2 Column
Proof
Supplementary
Angles
Complimentary
Angles
A two column proof is a method of presenting __________________
to form a logical argument involving _____________________________.
The statements of the argument are written in one column and
the __________________________ for the statements are written in the
other column.
Angles are supplementary when their measures add to _________.
Some examples of supplentary angles include:
a) The ______________ angles of a ____________________.
b) Opposite angles of a _____________________________________________.
c) Angles that form a ____________________________.
Angles are complementary when their measures add to ________.
Example
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