Reasoning and proof :
"Reasoning and proof are not special activities reserved for special times or special topics in the curriculum but should be a natural, ongoing part of classroom discussions, no matter what topic is being studied. In mathematically productive classroom environments, students should expect to explain and justify their conclusions.
When questions such as, What are you doing? or Why does that make sense? are the norm in a mathematics classroom, students are able to clarify their thinking, to learn new ways to look at and think about situations, and to develop standards for high-quality mathematical reasoning."
Conditional Statements :
A conditional statement has two parts, a hypothesis and a conclusion. If the statement is written in if-then form, the "if" part contains the hypothesis and the "then" part contains the conclusion.
Here is an example :
Definitions and Biconditional Statements :
All definitions can be interpreted "forward" and "backward". For instance, the definition of perpendicular lines means
(i) If two lines are perpendicular, then they intersect to form a right angle.
(ii) If two lines intersect to form a right angle, then they are perpendicular.
Conditional statements are not always written in if-then form. Another common form of a conditional statement is only-if-form.
Here is an example.
We can rewrite this conditional statement in if-then form as follows :
If it is Sunday, then I am in park.
A biconditional statement is a statement that contains the phrase "if and only if". Writing biconditional statement is equivalent to writing a conditional statement and its converse.
A biconditional statement can be either true or false. To be true,both the conditional statement and its converse must be true. This means that a true biconditional statement is true both “forward” and “backward.” All definitions can be written as true biconditional statements.
Inductive Reasoning :
Looking for patterns and making conjectures is part of a process is called inductive reasoning.
It consists of three stages.
(i) Look for a pattern.
Look several examples. Use diagrams and tables to help to discover a pattern.
(ii) Make a conjecture.
Use the examples to make a general conjecture. A conjecture is an unproven statement that is based on observations. Discuss the conjecture with others. Modify the conjecture, if necessary.
(iii) Verify the conjecture.
Use logical reasoning to verify that the conjecture is true in all cases.
Deductive Reasoning :
Deductive reasoning, unlike inductive reasoning, is a valid form of proof. It is, in fact, the way in which geometric proofs are written. Deductive reasoning is the process by which a person makes conclusions based on previously known facts.
Deductive reasoning uses facts, definitions, and accepted properties in a logical order to write a logical statement.
Reasoning with Properties from Algebra :
Many properties from algebra concern the equality of real numbers.
Several of these are summarized in the following list.
Let a, b and c be real numbers.
Addition Property :
If a = b, then a + c = b + c
Subtraction Property :
If a = b, then a - c = b - c
Multiplication Property :
If a = b, then a⋅c = b⋅c
Division Property :
If a = b and c ≠ 0, then a ÷ c = b ÷ c
Reflexive Property :
For any real number a, a = a
Symmetric Property :
If a = b, then b = a
Transitive Property :
If a = b and b = c, then a = c
Substitution Property :
If a = b, then a can be substituted for b in any equation or expression.
Proving Statements about Segments :
A true statement that follows as a result of other statements is called a theorem. All theorems must be proved. We can prove a theorem using a two-column proof. A two-column proof has numbered statements and reasons that show the logical order of an argument.
For any segment AB, AB ≅ AB
If AB ≅ CD, then CD ≅ AB
If AB ≅ CD, and CD ≅ EF, then AB ≅ EF
Proving Statements about Angles :
For any angle A, ∠A ≅ ∠A
If ∠A ≅ ∠B, then ∠B ≅ ∠A
If ∠A ≅ ∠B, ∠B ≅ ∠C, then ∠A ≅ ∠C
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