© Scarborough (adapted with thanks from Lynnette Cardenas), 2005 1 Math 365 Study Guide Section 1-1: Explorations with Patterns • Use inductive reasoning to determine a pattern. • Use inductive reasoning to lead to a conjecture (hypothesis). • Use a counterexample to disprove a conjecture. • Define a sequence. • Identify an arithmetic sequence and its difference. • Determine the nth term of an arithmetic sequence. • Identify a geometric sequence and its ratio. • Find the nth term of a geometric sequence. • Given some terms of an arithmetic or geometric sequence, determine how many there are. • Determine the pattern of a given sequence (not necessarily arithmetic or geometric) and determine its nth term. Section 1-2: Mathematics and Problem Solving • Name each step in the four-step problem-solving process. • Give examples of different strategies to use in order to devise a plan. • Use the four-step problem-solving process to solve problems. Section 1-3: Algebraic Thinking • Identify variables in a word problem. • Write statements in algebraic form. • Apply the addition, multiplication, and cancellation properties of equality to solve equations. • Use substitution to solve equations. • Write mathematical models to solve application problems. Section 1-4: Logic • Identify statements. • Write the negation of a statement. • Identify quantifiers. • Write the negation of a statement with a quantifier. • Write statements and negations in symbolic form.© Scarborough (adapted with thanks from Lynnette Cardenas), 2005 2 • Identify conjunctions and disjunctions and their symbols. • Write truth tables for conjunctions and disjunctions. • Determine whether statements are logically equivalent. • Recognize the symbol for logically equivalent. • Identify conditional statements (implications), including the hypothesis and conclusion, and its symbolic notation. • Write a truth table for a conditional statement. • Be familiar with the 7 different ways conditional statements can be formed (see pg 49). • Identify and write the 3 statements related to the conditional (converse, inverse, and contrapositive), including their symbolic notation. • Understand that the conditional and the contrapositive are logically equivalent, as are the converse and the inverse. • Identify a biconditional statement and its symbolic notation. • Write a truth table for a biconditional statement. • Understand what valid reasoning is. • Use Euler diagrams to help determine whether valid reasoning is being used in an argument. • Given a hypothesis, use direct reasoning, indirect reasoning, and the chain rule to determine the conclusion. Section 2-1: Describing Sets • Define a set and its elements (members). • Determine if a set is well-defined. • Identify the set of natural numbers, including its symbol N. • Write sets both by listing the elements and by using set-builder notation. • Determine whether sets are equal. • Understand and use the definition of one-to-one correspondence. • Understand and use the Fundamental Counting Principle. • Determine whether sets are equivalent. • Determine the cardinal number of a set. • Determine whether a set is finite or infinite. • Identify the empty set and the universal set and their symbols. • Define the complement of a set, a subset, and a proper subset. • Identify the symbols used to represent the complement of a set, a subset, and a proper subset.© Scarborough (adapted with thanks from Lynnette Cardenas), 2005 3 • Draw Venn diagrams. • Determine the complement of a set. • Determine the subsets and proper subsets of a set. • Use the cardinal number of sets to show the ideas of less than and greater than. • Determine the number of subsets and proper subsets of a set. Section 2-2: Other Set Operations and Their Properties • Determine the intersection of sets and identify the symbol for intersection. • Determine the union of sets and identify the symbol for union. • Determine set difference (the complement of A relative to B) and identify its symbolic notation. • Understand the commutative properties of set union and of set intersection. • Understand the distributive property of set intersection over set union. • Use a Venn diagram as a problem-solving tool. • Count and list the elements in a Cartesian product. Section 2-3: Addition and Subtraction of Whole Numbers • Identify the set of whole numbers, including its symbol W. • Use a set model to illustrate addition of whole numbers. • State the definition of addition of whole numbers using set notation. • Identify the addends and sum in an addition problem. • Use a number-line model to illustrate addition of whole numbers. • Understand the concepts of greater-than, less-than, greater-than-or-equal-to, and less-than-or-equal-to. • State and use the definition of less-than. • Identify and use the properties of whole-number addition (closure, commutative, associative, identity). • Identify the additive identity. • Identify basic addition facts. • State and use the three strategies of learning basic addition facts (counting on, doubles, making 10). • Identify the difference in a subtraction problem. • Use a take-away model to illustrate subtraction of whole numbers. • Use a missing-addend model to illustrate subtraction of whole numbers.© Scarborough (adapted with thanks from Lynnette Cardenas), 2005 4 • State the definition of subtraction of whole numbers (using addition). • Use a comparison model to illustrate subtraction of whole numbers. • Use a number-line model to illustrate subtraction of whole numbers. Section 2-4: Multiplication and Division of Whole Numbers • Use a repeated-addition model to illustrate multiplication of whole numbers. • Use an array model to illustrate multiplication of whole numbers. • State the definition of multiplication of whole numbers (using addition). • Use a Cartesian-product model to illustrate multiplication of whole numbers. • State the alternate definition of multiplication of whole numbers (using Cartesian product). • Identify the factors and product in a multiplication problem. • Identify and use the properties of whole-number