Math - 2018-19
G.1 - Logic and Proofs
G.1 The student will use deductive reasoning to construct and judge the validity of a logical argument consisting of a set of premises and a conclusion. This will include
a) identifying the converse, inverse, and contrapositive of a conditional statement;
b) translating a short verbal argument into symbolic form;
c) determining the validity of a logical argument.
- I can analyze the logic behind advertising, decide the official weight
of a Hershey bar by weighing a fraction of the candy bars coming off a
production line, and present a winning a court case.
- I will develop
logical critical thinking by drawing conclusions and making inferences from
known or assumed facts.
UNDERSTANDING THE STANDARD
· Inductive reasoning, deductive reasoning, and proof are critical in establishing general claims.
· Deductive reasoning is the method that uses logic to draw conclusions based on definitions, postulates, and theorems.
· Valid forms of deductive reasoning include the law of syllogism, the law of contrapositive, the law of detachment, and the identification of a counterexample.
· Symbolic notation is used to represent logical arguments, including the use of , , , , , and .
· The law of syllogism states that
if p q is true and q r is true, then
p r is true.
· The law of contrapositive states that if p q is true and q is true, then p is true.
· The law of detachment states that
if p q is true and p is true,
q is true.
· A counterexample is used to show an argument is false.
· Inductive reasoning is the method of drawing conclusions from a limited set of observations.
· Proof is a justification that is logically valid and based on initial assumptions, definitions, postulates, theorems, and/or properties.
· Logical arguments consist of a set of premises or hypotheses and a conclusion.
· When a conditional (p q) and its converse (q p) are true, the statements can be written as a biconditional, p iff q; or p if and only if q;or p q.
· Logical arguments that are valid may not be true. Truth and validity are not synonymous.
· Exploration of the representation of conditional statements using Venn diagrams may assist in deepening student understanding.· Formal proofs utilize symbols of formal logic to determine validity of a logical argument.
The student will use problem solving, mathematical communication, mathematical reasoning, connections, and representations to
· G.1a1 Identify
the converse, inverse, and contrapositive of a conditional statement.
· G.1b1 Translate
verbal arguments into symbolic form using the symbols of formal logic.
· G.1c1 Determine
the validity of a logical argument using valid forms of deductive reasoning.
· G.1c2 Determine
that an argument is false using a counterexample.
statement, converse, inverse, contrapositive, verbal argument,
hypothesis/premises, conclusion, symbolic form, deductive reasoning, inductive reasoning, law of syllogism, law of
contrapositive, law of detachment, counterexample, proof, Venn diagram ,
intersections, unions, negation, therefore,
compound, conjunction, disjunction, conjecture, truth value, validity,
truth table, biconditional, postulates/axioms, theorems, Euclidean geometry,
axiomatic system, undefined terms, point, line, plane, algebraic properties