Ch 1.4 Notes
September 06, 2016
1.4 Proving Conjectures: Deductive Reasoning
Outcomes:
1.3 Compare, using examples, inductive and deductive reasoning.
1.4 Prove algebraic and number relationships such as divisibility rules, number properties, mental mathematics strategies or algebraic number tricks.
1.5 Prove a conjecture, using deductive reasoning (not limited to two column proofs).
3 + 7 = 10
5 + 9 = 14
17 + 13 = 30
conjecture: the sum of two odd numbers is always odd.
Can you find a counterexample to this conjecture?
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Ch 1.4 Notes
September 06, 2016
Proof: when a conjecture is thought to be always true a mathematical proof is needed to make sure of its validity.
A proof is a series of mathematical statements together with reasons why a conjecture is true. Many proofs in mathematics are two column proofs.
statements
1.
2.
3.
reasons
1. reason for statement 1
2. reason for statement 2
3. reason for statement 3
Giving a proof is called logical deduction.
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Ch 1.4 Notes
September 06, 2016
Conjecture: In the triangle shown below the length of side x is 5 cm.
C
3 cm
A
x
4 cm
statements
B
reasons
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Ch 1.4 Notes
September 06, 2016
John, who is in Mr. Smith's math class, has only $12 in his bank account. All students in Mr. Smith's math class need to buy the workbook called "Seeing Through Mathematics". The book
costs $20.
Conjecture: John needs to get $8 from his parents.
statements
statements
reasons
reasons
1.
1. 2. x + 3 = 9 2. 3.
3. subtract 3
4.
4.
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Ch 1.4 Notes
September 06, 2016
n is number like 1 , 2 , 3 , 4 , 5 , ...
a formula for all even numbers : 2n
a formula for all odd numbers: 2n + 1
Write down any two consecutive numbers.
Write down any two consecutive odd numbers.
Write down the sum of an even number and an odd number.
Write down the difference of the square of a number and the number itself.
Write down the sum of an even number with an odd number.
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Ch 1.4 Notes
September 06, 2016
a) The sum of three consecutive numbers
b) the sum of two consecutive even numbers
c) the sum of the squares of an even number and an odd number
d) the product of an even number with an odd number
e) the product of two odd numbers
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Ch 1.4 Notes
September 06, 2016
Prove: The sum of two different odd numbers is even.
First: try to find a counterexample.
statements
reasons
Prove: The product of an even number and an odd number is even.
First: try to find a counterexample.
statements
reasons
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Ch 1.4 Notes
September 06, 2016
12 + 24 = 36
4 + 6 = 10
42 + 24 = 66
16 + 22 = 38
make a conjecture:
try to find a counterexample:
give a proof:
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