Math - 2017-18

G.1 and *G.1 - Logic and Proofs

G.1    The student will 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)  using Venn diagrams to represent set relationships; and

d)  using deductive reasoning.

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. 


Bloom's Level:  Evaluate

Adopted: 2009

BIG IDEAS

  • 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. 
  • 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, and theorems.
  • Logical arguments consist of a set of premises or hypotheses and a conclusion.
  • Euclidean geometry is an axiomatic system based on undefined terms (point, line and plane), postulates, and theorems.
  • When a conditional and its converse are true, the statements can be written as a biconditional, i.e., iff or if and only if.
  • Logical arguments that are valid may not be true. Truth and validity are not synonymous. 

·  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, then
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. 

ESSENTIALS

G.1a1  Identify the converse, inverse, and contrapositive of a condition statement.

·  G.1a1  Identify the converse, inverse, and contrapositive of a conditional statement.

G.1b1 Translate verbal arguments into symbolic form, such as  (p → q) and (~p → ~q).

·  G.1b1  Translate verbal arguments into symbolic form using the symbols of formal logic.

G.1c2  Interpret Venn diagrams.

G.1d1  Determine the validity of a logical argument.

·  G.1c1  Determine the validity of a logical argument using valid forms of deductive reasoning.

G.1b2  Recognize and use the symbols of formal logic, which include

G.1c1  Use Venn diagrams to represent relationships, such as intersection and union.

G.1d2  Use valid forms of deductive reasoning, including the law of syllogism, the law of contrapositive, the law of detachment, and counterexamples.

·  G.1c2  Determine that an argument is false using a counterexample.

G.1d3  Select and use various types of reasoning and methods of proof as appropriate.


KEY VOCABULARY

conditional 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

Updated: Oct 27, 2017