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Thomas Edison State College *

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MAT- 105

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Mathematics

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Feb 20, 2024

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docx

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Critical Thinking 5% ● Inductive and deductive reasoning ● Problem solving Sets and Statements 22% ● Set concepts ● Subsets ● Venn diagrams and set operations ● Venn diagrams with three sets ● Application of sets Logic and Systems of Numeration 10% ● Statements and logical connectives ● Truth tables for negation ● Other bases Number Theory and The Real Number System 23% ● Number theory ● Integers ● Rational numbers ● Irrational numbers ● Real numbers and their properties ● Rules of exponents Algebra, Graphs, and Functions 13% ● Order of operations and solving equations ● Formulas ● Linear inequalities ● Graphing linear equations ● Linear inequalities and systems of linear inequalities ● Solving systems of linear equations The Metric System and Consumer Mathematics 17% ● Basic terms and conversions within the metric system ● Length, area and volume ● Dimensional analysis and conversion to and from the metric system ● Percent, personal loans and simple interest ● Compound interest ● Buying a house with a mortgage Probability and Statistics 10% ● Empirical and theoretical probabilities ● Odds ● Expected value (Valuation) ● Tree diagrams ● Sampling techniques and misuses of statistics ● Frequency distributions and statistical graphs

Certainly! Let's break down each topic and provide brief explanations along with examples: Critical Thinking: Inductive and deductive reasoning: - **Inductive reasoning**: Drawing general conclusions based on specific observations or examples. For example, observing that the sun rises every morning leads to the conclusion that the sun will rise tomorrow morning. - **Deductive reasoning**: Drawing specific conclusions from general premises. For example, if all mammals are warm-blooded, and a dog is a mammal, then you can deduce that the dog is warm-blooded. Problem solving: - The process of finding solutions to complex or challenging situations. - Example: Solving a mathematical word problem such as finding the dimensions of a rectangle given its area and one side length. Sets and Statements: Set concepts: - A collection of distinct objects or elements. - Example: Set A = {1, 2, 3, 4}. Subsets: - A set whose elements are all contained within another set. - Example: Set B = {1, 2} is a subset of Set A above. Venn diagrams and set operations: - Venn diagrams visually represent sets and their relationships. - Set operations include union, intersection, difference, and complement. - Example: Given sets A = {1, 2, 3} and B = {3, 4, 5}, the union of A and B is {1, 2, 3, 4, 5}. Venn diagrams with three sets: - Extending Venn diagrams to represent relationships between three sets. - Example: Venn diagram showing the intersection of three sets A, B, and C. Application of sets: - Sets can be applied in various fields, such as probability, statistics, and computer science. - Example: Using sets to represent the outcomes of a dice roll. Logic and Systems of Numeration: Statements and logical connectives: - Statements are declarative sentences that are either true or false. - Logical connectives include AND, OR, NOT, IF-THEN, etc. - Example: Statement A: "It is raining." Statement B: "It is cold." Truth tables for negation:

- Truth tables display all possible truth values for compound statements based on the truth values of their components. - Example: Truth table for the negation of statement A: "¬A". Other bases: - Numeration systems other than base 10, such as binary (base 2) or hexadecimal (base 16). - Example: Converting the decimal number 13 to binary (1101) or hexadecimal (D). Number Theory and The Real Number System: Number theory: - The study of integers and their properties. - Example: Prime factorization of a number. Integers: - Whole numbers and their negatives, including zero. - Example: {..., -3, -2, -1, 0, 1, 2, 3, ...} Rational numbers: - Numbers that can be expressed as a ratio of two integers. - Example: 3/4, -2/5. Irrational numbers: - Numbers that cannot be expressed as a ratio of two integers. - Example: √2, π. Real numbers and their properties: - All rational and irrational numbers together. - Example: 5, -1.75, √3. Rules of exponents: - Rules governing the manipulation and calculation of exponents. - Example: $a^m \times a^n = a^{m+n}$. Let's go through each of these topics and provide brief explanations along with examples: ### Algebra, Graphs, and Functions: #### Order of operations and solving equations: - The sequence in which mathematical operations should be performed. - Example: Solve $2 \times (3 + 4) - 5$. #### Formulas: - Mathematical expressions representing relationships between variables. - Example: The formula for the area of a rectangle is $A = l \times w$. #### Linear inequalities: - Inequalities involving linear expressions. - Example: $3x + 2 < 10$.

#### Graphing linear equations: - Representing linear equations on a coordinate plane. - Example: Graph the equation $y = 2x + 3$. #### Linear inequalities and systems of linear inequalities: - Systems of linear inequalities involve multiple linear inequalities. - Example: Solve the system of inequalities: $2x + y \leq 5$ and $x - y > 3$. #### Solving systems of linear equations: - Finding the values of variables that satisfy multiple linear equations simultaneously. - Example: Solve the system of equations: $2x + y = 5$ and $x - y = 3$. ### The Metric System and Consumer Mathematics: #### Basic terms and conversions within the metric system: - Understanding units of measurement such as meters, grams, and liters. - Example: Convert 2 kilometers to meters. #### Length, area, and volume: - Measurements of one-dimensional, two-dimensional, and three-dimensional space. - Example: Find the volume of a cube with side length 4 meters. #### Dimensional analysis and conversion to and from the metric system: - Converting between different units of measurement using conversion factors. - Example: Convert 500 milliliters to liters. #### Percent, personal loans, and simple interest: - Calculations involving percentages, loans, and simple interest. - Example: Calculate 15% of $200. #### Compound interest: - Interest calculated on the initial principal and also on the accumulated interest from previous periods. - Example: Calculate the compound interest on a $1000 investment with a 5% interest rate over 3 years, compounded annually. #### Buying a house with a mortgage: - Understanding the financial aspects of purchasing a home, including mortgages, down payments, and monthly payments. - Example: Calculate the monthly mortgage payment for a $200,000 loan with a 4% interest rate over 30 years. ### Probability and Statistics: #### Empirical and theoretical probabilities: - Empirical probability is based on observed outcomes, while theoretical probability is based on mathematical analysis. - Example: Rolling a fair six-sided die and calculating the empirical probability of rolling a 3.

#### Odds: - The ratio of the probability of an event occurring to the probability of it not occurring. - Example: The odds of winning a raffle are 1 in 50. #### Expected value (Valuation): - The average value of a random variable over many trials. - Example: Calculate the expected value of rolling a fair six-sided die. #### Tree diagrams: - Visual representations used to calculate probabilities by enumerating all possible outcomes. - Example: Use a tree diagram to find the probability of flipping a coin twice and getting heads both times. #### Sampling techniques and misuses of statistics: - Methods for selecting a subset of individuals or items from a larger population for analysis. - Example: Conducting a random sample survey to estimate the average income of a population. #### Frequency distributions and statistical graphs: - Organizing data into categories and displaying it visually. - Example: Create a histogram to represent the frequency distribution of test scores in a class. These explanations and examples should help you understand the concepts covered in each topic. Practice solving problems related to each area to prepare for your test effectively.

### Multiple Choice Test #### Critical Thinking (5 questions) 1. Which of the following is an example of inductive reasoning? a) All dogs bark, therefore my dog barks. b) If it's raining, then the ground will be wet. c) All squares are rectangles, but not all rectangles are squares. d) Every time I eat peanuts, I sneeze. 2. Deductive reasoning involves: a) Drawing general conclusions from specific observations. b) Drawing specific conclusions from general premises. c) Making educated guesses based on probability. d) None of the above. 3. Which of the following is a problem-solving strategy? a) Making random guesses. b) Breaking a problem down into smaller parts. c) Following the same approach every time. d) Ignoring all other options and sticking with the first idea that comes to mind. 4. In the set {1, 2, 3, 4}, which number is a subset? a) 0 b) 3 c) {1, 2} d) {2, 3, 5} 5. In a Venn diagram representing sets A and B, the intersection represents: a) All elements in set A. b) All elements in set B. c) Elements common to both set A and set B. d) Elements unique to either set A or set B. #### Sets and Statements (22 questions) 6. Which of the following is not a set? a) {apple, orange, banana} b) {2, 4, 6, 8, 10} c) {cat, dog, table, chair} d) 3x - 5 = 10 7. What is the cardinality of the set {apple, banana, orange}? a) 0 b) 1 c) 2 d) 3

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