Course Overview

MA1040 Math Fundamentals I is designed to provide students with a solid foundation in arithmetic, algebra, linear equations, graphing, right triangle trigonometry, exponents, and polynomials. This course prepares students for further studies in mathematics by developing essential skills and problem-solving strategies.

1. Review of Arithmetic

1.1 Fractions

• Fraction: A number that represents a part of a whole, written as a/b where a is the numerator and b is the denominator.

• Reducing Fractions: Simplifying fractions to their lowest terms by dividing numerator and denominator by their greatest common factor.

• Equivalent Fractions: Different fractions that represent the same value.

• Types of Fractions:

  • Proper Fraction: Numerator is less than denominator.

  • Improper Fraction: Numerator is greater than or equal to denominator.

  • Mixed Number: Combination of a whole number and a proper fraction.

• Conversions: Convert between improper fractions and mixed numbers as needed.

• Least Common Denominator (LCD): The smallest common multiple of denominators for adding/subtracting fractions.

• Operations: Add, subtract, multiply, and divide fractions and mixed numbers.

• Order of Operations: Follow PEMDAS/BODMAS rules, especially with fractions and mixed numbers.

• Application Problems: Solve real-world problems involving fractions and mixed numbers.

1.2 Decimals

• Operations: Add, subtract, multiply, and divide decimals.

• Exact vs. Approximate Values: Understand rounding and estimation.

• Significant Figures: Digits in a number that contribute to its precision.

  • Rules for identifying significant figures and applying them in calculations.

• Scientific Notation: Expressing numbers as for easier computation with very large or small values.

• Application Problems: Use decimals and scientific notation in practical contexts.

1.3 Percents, Ratios, and Proportions

• Percent: A ratio out of 100; convert between fractions, decimals, and percents.

• Percent Calculations: Find percent of a number, what percent one number is of another, and the base number given a percent.

• Ratios: Comparison of two quantities, written as a:b or a/b.

• Proportions: Equation stating two ratios are equal; solve for unknowns in proportional relationships.

• Unit Conversions: Convert between SI and imperial units using proportions or conversion factors.

2. Introduction to Algebra

2.1 Real Numbers and Operations

• Number Line: Visual representation of numbers in order.

• Positive and Negative Numbers: Numbers to the right (positive) and left (negative) of zero.

• Subsets of Real Numbers: Natural, whole, integer, rational, and irrational numbers.

• Operations: Add, subtract, multiply, and divide real numbers, considering sign rules.

• Distributive Property:

• Exponents: Write numbers in exponent form and expanded form.

• Order of Operations: Apply rules to simplify expressions, including nested grouping symbols.

2.2 Algebraic Expressions

• Like and Unlike Terms: Terms with the same variable and exponent.

• Evaluating Expressions: Substitute values for variables and simplify.

• Operations with Variables: Add, subtract, multiply, and divide expressions with variables.

3. Linear Equations, Formulae, and Inequalities

3.1 Solving Linear Equations

• Addition and Multiplication Principles: Isolate variables using inverse operations.

• Equation Forms: Solve equations of the form , with variables on both sides, with parentheses, and involving fractions.

• Multi-step Equations: Combine multiple operations to solve for the variable.

• Formulas: Solve for an indicated variable and evaluate for given values.

• Translating Words to Algebra: Convert English phrases into algebraic expressions or equations.

• Application Problems: Solve problems involving comparisons, periodic charges, percent increase/decrease, simple interest, and coin problems.

• Geometry Applications: Calculate area and perimeter of 2D objects, and volume and surface area of 3D objects.

3.2 Inequalities

• Number Line Representation: Show solutions to inequalities graphically.

• Solving Linear Inequalities: Use similar steps as equations, but reverse the inequality when multiplying/dividing by a negative.

• Translating Words to Inequalities: Express verbal statements as inequalities.

• Application Problems: Solve real-world problems involving inequalities.

4. Graphing Linear Functions

4.1 The Cartesian Coordinate System

• Definition: A plane defined by x (horizontal) and y (vertical) axes.

• Plotting Points: Locate points using ordered pairs (x, y).

• Variables: Distinguish between dependent (y) and independent (x) variables.

4.2 Graphing Linear Equations

• Table of Values: Generate points to plot a line.

• x- and y-intercepts: Find where the line crosses the axes.

• Vertical and Horizontal Lines: Recognize equations of the form x = a and y = b.

• Slope: Measure of steepness, calculated as .

• Equation of a Line: Slope-intercept form .

• Parallel and Perpendicular Lines: Parallel lines have equal slopes; perpendicular lines have slopes that are negative reciprocals.

• Graphing Linear Inequalities: Shade the solution region on the coordinate plane.

4.3 Relations and Functions

• Relation: A set of ordered pairs.

• Function: A relation where each input has exactly one output.

• Domain and Range: Set of possible inputs (domain) and outputs (range).

• Identifying Functions: Use mapping diagrams or the vertical line test on graphs.

• Function Notation: represents the output for input .

5. Right Triangle Trigonometry

5.1 Trigonometric Ratios

• Primary Ratios: Sine, cosine, and tangent for right triangles.

• Calculator Use: Find values and angles using trigonometric functions.

• Solving Triangles: Find unknown sides or angles using ratios.

• Angles of Elevation and Depression: Angles measured from the horizontal up (elevation) or down (depression).

• Contextual Problems: Apply trigonometry to real-world scenarios, including multiple triangles.

6. Exponents and Polynomials

6.1 Exponents

• Laws of Exponents: Rules for simplifying expressions with exponents, including rational exponents.

6.2 Polynomials

• Definition: An algebraic expression with one or more terms, each consisting of a variable raised to a non-negative integer power.

• Types:

  • Monomial: One term

  • Binomial: Two terms

  • Trinomial: Three terms

• Degree: Highest exponent in the polynomial.

• Operations: Add, subtract, and multiply polynomials (including special products like perfect square trinomials).

• Division: Divide polynomials by monomials, binomials (long division), and linear binomials (synthetic division).

• Special Forms: Distinguish between , , and .