#geometryhelp

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

doing geometry all. friggin. day. And I still don't understand it. Is there anyone here that's good at it and can teach me? PLEEEEEEEEASEEEEEEEE I beg of you.