Download Proving Conjectures using Deductive Reasoning

Proving Conjectures using Deductive Reasoning & Number Tricks
Deductive Reasoning:
- drawing a specific conclusion through logical reasoning by starting with general
assumptions that are known to be valid
Proof: (often uses deductive reasoning)
- a mathematical argument showing that a statement is always true
Generalization:
- a principle, statement or idea that has general application
Transitive Property:
- if A = B and B = C then A = C
Examples Set 1: Make a deduction in each of the following cases.
a) Ms Mackey lives in Moose Jaw. Moose Jaw is in Saskatchewan
Ms Mackey lives in Saskatchewan
b) Every animal has a heart. All dogs are animals.
All dogs have a heart.
c) The sum of any two consecutive whole number is an odd number. The whole
numbers 11 and 12 are consecutive.
The sum of 11 & 12 is an odd number.
d) The diagonals of a parallelogram bisect each other. PQRS is a parallelogram.
The diagonals of PQRS bisect each other.
Example 3: The sum of the angles in any triangle is 180◦. In Δ ABC angle A = 90◦,
therefore, angle B + angle C = ?
90◦
Example 4:
Conjecture: The square of an even number is always even. Prove this using
deductive reasoning.
2n = even number
(2n)2 = 2n x 2n
= 4n2 (any number multipled by an even number will always be even)
Example 5: Conjecture: The sum of two odd numbers will always be even. Prove
this using deductive reasoning.
2n + 1 = odd number
(2n + 1) + (2n + 1) = 2n + 1 + 2n + 1
= 2n + 2n + 1 + 1
= 4n + 2
this is an
even #
adding 2 to an even # will result in another even #
Example 6: Choose a number. Double it. Add 5. Add the original number. Add 7.
Divide by 3. Subtract the original number.
a) Set up three trials with a different original number, then make a conjecture.
TRIAL 1
TRIAL 2
TRIAL 3
Original Number
4
-11
20
Double
8
-22
40
Add 5
13
-17
45
Add original #
17
-28
65
Add 7
24
-21
72
Divide by 3
8
-7
24
Subtract original #
4
4
4
When doing this number trick, you will always end up with an answer of 4.
b) Prove the conjecture using deductive reasoning. Let the original number equal n.
Double it. Add 5. Add the original number. Add 7. Divide by 3. Subtract the
original number.
step 1: n
step 2: 2n
step 3: 2n + 5
step 4: (2n + 5) + n = 3n + 5
step 5: (3n + 5) + 7 = 3n + 12
step 6: 3n + 12 = n + 4
3
step 7: (n + 4) – n = 4
Example 7: Prove the difference between the square of any two odd numbers is
divisible by 4.
Inductive: Use specific examples
(3)2 – (-5)2
(11)2 – (7)2
= 9 – 25
= 121 - 49
= - 16 (divisible by 4)
= -72 (divisible by 4)
Deductive: Use x or n to help define the numbers
(2n + 1)2 – (2n + 3)2
(2n + 1)(2n + 1) – (2n + 3)(2n + 3)
4n2 + 2n + 2n + 1 – (4n2+ 6n + 6n + 9)
4n2 + 4n + 1 – 4n2 - 12n - 9
8n – 8 <--- divisible by 4
Example 8: Prove using a two column proof that vertically opposite angles are
always equal.
Statement
Reason
1) <A + <B = 180˚
1) Supplementary Angles
A
2) <B + <C = 180˚
2) Supplementary Angles
B
D
3) <A = 180˚- <B
3) Subtraction Property
4) <C = 180˚- <B
4) Subtraction Property
C
5) <A = <C
5) Transitive Property