PROBLEM SOLVING AND REASONING Flashcards
is drawing a general conclusion from a repeated observation or limited sets of observations of specific examples.
Inductive Reasoning
specific case to general case.
Inductive Reasoning
Conjecture
is drawing conclusion using inductive reasoning.
The conjecture may be true or false depending on the truthfulness of the argument.
True
Is this example of conjecture or not?
1 is an odd number.
11 is an odd number.
21 is an odd number.
Thus, all number ending with 1 are odd numbers.
Conjecture
is drawing general to specific examples or simply from
general case to specific case.
Conjecture
Deductive starts with ?
a general statement (or hypothesis)
Logical reasoning may be valid but nit necessarily true.
True
Srinivasa Ramanujan wrote a letter to Godfrey Harold Hardy on
infinite sums, products, fractions, and roots.
Intuitive can be found in mathematical literature and discovery.
True
Ramanujan’s formulas prove there is
mathematical intuition,
though he didn’t prove them.
made a sound judgment without directly proving the formulas of Ramanujan’s were correct.
Hardy
a reliable mathematical belief without being
formalized and proven directly and serves as an essential part of mathematics.
Mathematical intuition
carries a heavy load of mystery and ambiguity and it is not legitimate substitute for a formal proof.
intuition
is being visual and is absent from the rigorous formal or abstract version.
Intuitive
As a student, you can build and improve your intuition by doing the following :
i. Be observant and see things visually towards with your critical thinking.
ii. Make your own manipulation on the things that you have noticed and observed.
iii. Do the right thinking and make a connections with it before doing the solution.
A __________ demonstrates that a certain statement is always true
in all possible cases.
mathematical proof
is an essential defining attribute of mathematics and
mathematical knowledge.
Mathematical certainty
By definition, a _________ is an inferential argument for a mathematical statement
while __________ are an example of mathematical logical certainty.
proof, proofs
Let us say P and Q are two propositions. In an if-then statement, proposition P would be the ___________ while the proposition Q would be our
___________ denoted by: _____
hypothesis, conclusion, P → Q
TWO WAYS ON HOW TO PRESENT THE PROOF
a. Outline Form
b. Paragraph Form
Proposition: If P then Q.
1. Suppose/Assume P
2. Statement
3. Statement
.
.
. Statement
Therefore Q .
a. Outline Form
Proposition: If P then Q.
Assume/Suppose P. ____________. ___________.
_____________________. ____________ . . . _____________. _______________.
Therefore Q.
b. Paragraph Form
is a mathematical argument that uses rules of inference to derive the conclusion
from the premises.
direct proof
The steps in taking a direct proof would be:
- Assume P is true.
- Conclusion is true.
is a type of proof in which a statement to be proved is assumed false and if the assumption leads to an impossibility, then the statement assumed false has been proved to be true.
Indirect proof or contrapositive proof
In an indirect proof, let us say we need to prove a given theorem in a form of P → Q. The steps or outline in taking an indirect proof would be:
In an indirect proof, let us say we need to prove a given theorem in a form of P → Q. The steps or outline in taking an indirect proof would be:
Assume/Suppose ~Q is true.
.
.
.
Therefore ~P is true.
There is a special
name for an example that disproves a statement
COUNTEREXAMPLE.
This method works by assuming your implication is not true, then deriving a contradiction.
Proving by Contradiction
Proof by contradiction is given by this way;
- Assume p is true.
- Suppose that ~q is also true.
- Try to arrive at a contradiction.
- Therefore q is true
Proof by contradiction is given by this way; (give the four steps)
- Assume p is true.
- Suppose that ~q is also true.
- Try to arrive at a contradiction.
- Therefore q is true
POLYA’S FOUR-STEPS IN PROBLEM SOLVING
Step 1: Understand the problem.
Step 2: Devise a plan.
Step 3: Carry out the plan.
Step 4: Look back.
was a mathematics educator who
strongly believed that the skill of problem solving can be taught.
George Polya (1887-1985)
was a mathematics educator who
strongly believed that the skill of problem solving can be taught.
George Polya (1887-1985)
He developed a framework known as Polya’s Four-Steps in Problem Solving.
George Polya
The process addressed the difficulty of students in problem
solving.
POLYA’S FOUR-STEPS IN PROBLEM SOLVING
Mathematics is useful to predict.
True
Number pattern
leads directly to the concept of functions in mathematics.
is applied to problem-solving whether a pattern is present
and used to generalize a solution to a problem.
Number pattern
can be in the form counting up or down and the missing number is
of the form of completing count up or down.
Pattern
is a type of figurative number represented as dots or pebbles arranged in the shape of a regular polygon.
polygonal number
Examples of polygonal number
Triangular number, Square number
Alphametic
is a type of number puzzle containing sum (or other
arithmetic operation) in which digits (0 to 9) are replaced by
letters of the alphabet.
One of the most famous alphametic puzzles is the one introduced by
Henry Dudeney in 1924.
He was an English author and mathematician who specialized in
logic puzzles and mathematical games.
Henry Dudeney
