Reasoning in Geometry
Inductive Reasoning
Inductive Reasoning: reasoning based on patterns
Conjecture: a rule based on inductive reasoning
Patterns:
Add 2
Add 5
Add 2 to each number you add by
+2 +4 +6 +8 +10
Multiply by 2, Subtract 1
x2 -1 x2 -1 x2 -1 x2
Powers:
-1, 1, 5, 11, 19, 29...
+2 +4 +6 +8 +10
+2 +2 +2 +2
1, 4, 9, 16, 25…
+3 +5 +7 +9
+2 +2 +2
11, -4, -14, -16, -7, 16, 56, 116, 199, 308
-15 -10 -2 +9 +23 +40 +60 +83 +109
+5 +8 +11 +14 +17 +20 +23 +26
+3 +3 +3 +3 +3 +3 +3
Finding the General (nth) Term
Take a sequence of numbers:
20, 27, 34, 41, 48
Look at the relationship between the term and value:
Term: 1 2 3 4 5 6 20 n
Value: 20 27 34 41 48 55 153 7n+13
+7 +7 +7 +7 +7 +7
Ask yourself what happens from term to term:
n represents the term number
7(n) + k
7(1) +k = 20
7 + k = 20
-7 = -7
k = 13
Check:
7(2) + 13 = 27
14 + 13 = 27
27 = 27
7(20) + 13 = 153
140 + 13 = 153
153 = 153
Mathematical ModelingParty Handshake Problems
10 people attend a party. Each person shakes everyone else's hand, how many handshakes take place?
1 person can't shake his own hand meaning each person shakes 9 hands, and each handshake accounts for two people. This reasoning gives you this equation:
10 (10-1)
_________
2
10 (9)
_________
2
90
______
2
45 handshakes
The equation for party handshakes is:
n (n-1)
________
2
Venn Diagram Problems
36 people are in a class. 24 of them are taking chemistry, 22 are taking calculus, 16 are taking both. How many students are enrolled in neither chemistry nor calculus.
chemistry both calculus
8 16 6
neither
6
Start from the middle and work your way out, there are 16 students taking both and 24 are taking chemistry meaning that 8 are only taking chemistry. Use the same type of logic to solve the rest.
Matrix Problems
Match the driver to the car given the following clues:
No first name initial is the same as the initial of the car
No person has the same number of letters as the car
Brian's brother was mad when Brian didn't buy a Toyota
Fred drove Kevin's BMW to the concert
Toyota Kia BMW Ford
Tom X X X O
Kevin X X O X
Brian X O X X
Fred O X X X
Use the clues to narrow down you choices until the answer becomes clear
Deductive Reasoning
Deductive Reasoning: the process of showing how certain statements follow from directly agreed upon assumptions and proven facts
Solving problems algebraically is deductive reasoning:
2 (3x – 1) + 4x = 8x + 24 Given
6x – 2 + 4x = 8x + 24 Distributive Property
10x – 2 = 8x + 24 Simplify/Combine Like Terms
2x – 2 = 24 Subtraction Property of Equality
2x = 26 Addition Property of Equality
x = 13 Division Property of Equality
Some mathematical proofs are also deductive reasoning
Angles A, B and C form Triangle ABC
m∠A + m∠C = 90 degrees Given
m∠B + m∠C = 90 degrees Given
m∠A + m∠C = m∠B + m∠C Substitution
m∠A = m∠B Subtraction Property of Equality
∠A ≅ ∠B Definition of Congruence