Mathematical Foundations I is the first course in a sequence specially crafted for adults seeking the most direct path to prepare for university math courses. The course starts with fraction arithmetic and covers all the way up through the algebra of linear equations. Upon completing the course, students will be prepared for Mathematical Foundations II, the second step of the journey towards university-level math.

Solidify your arithmetic, learn about variables and graphs, level up your algebra, and learn the essentials of geometry.

- Use number lines to visualize addition, subtraction, and negative numbers.
- Leverage equivalent fractions as a strategy to add and subtract fractions.
- Generalize previous understandings of multiplication and division to multiply and divide fractions.
- Interpret exponents as repeated multiplication and radicals as the opposite of exponents.
- Understand the relationship between fractional exponents and radicals.
- Apply the order of operations to evaluate arithmetic expressions.

- Convert between units of measurement and compute unit rates.
- Understand the relationship between whole, part, and percentage.

- Express a composite integer as the product of its prime factors.
- Leverage prime factorization as a strategy to compute greatest common factors and least common multiples.

- Understand the concept of a variable as a placeholder in a mathematical expression or verbal phrase.
- Create equations that describe relationships between quantities.
- Understand rules of algebra as a process of reasoning and use them to solve linear equations, inequalities, and systems.
- Relate the solutions of two-variable equations and inequalities to graphs in the coordinate plane.
- Set up and solve systems of linear equations in real-world modeling contexts.

- Leverage multiplication as a strategy to transform rational equations into linear equations.
- Extend knowledge of radicals and solving linear equations to handle cases when the variable is raised to a power.
- Relate absolute value to distance on the number line and apply this understanding to solve absolute value equations and inequalities.

- Leverage the distributive property and an understanding of like terms to manipulate polynomial expressions.
- Understand the relationship between multiplication and factoring and apply a systematic method to factor trinomials with leading coefficients.

- Identify, interpret, and build functions that model relationships between two quantities.
- Represent and reason about functions using formal notation.
- Visualize functions in the coordinate plane by graphing.

- Classify triangles and quadrilaterals based on their properties.
- Use geometric relationships to solve for unknown angle measures and side lengths.
- Compute area and perimeter of circles, triangles, and various types of quadrilaterals.
- Calculate side lengths and areas of special right triangles using geometric reasoning and the Pythagorean theorem.

1.

Fractions
41 topics

1.1. Adding and Subtracting Fractions and Whole Numbers

1.1.1. | Adding Fractions and Whole Numbers Using Models | |

1.1.2. | Adding Fractions and Whole Numbers | |

1.1.3. | Subtracting Fractions and Whole Numbers Using Models | |

1.1.4. | Subtracting Fractions and Whole Numbers |

1.2. Adding and Subtracting Fractions

1.2.1. | Least Common Multiples | |

1.2.2. | Least Common Denominators | |

1.2.3. | Comparing Fraction Models Using Common Denominators | |

1.2.4. | Comparing Fractions Using Common Denominators | |

1.2.5. | Adding Fractions With Unlike Denominators Using Models | |

1.2.6. | Subtracting Fractions With Unlike Denominators Using Models | |

1.2.7. | Adding Fractions With Unlike Denominators | |

1.2.8. | Subtracting Fractions With Unlike Denominators |

1.3. Adding and Subtracting Mixed Numbers

1.3.1. | Adding Mixed Numbers With Unlike Denominators | |

1.3.2. | Subtracting Mixed Numbers With Unlike Denominators | |

1.3.3. | Subtracting Mixed Numbers With Unlike Denominators Using Improper Fractions |

1.4. Multiplying Fractions

1.4.1. | Multiplying Unit Fractions by Whole Numbers Using Models | |

1.4.2. | Multiplying Fractions by Whole Numbers Using Models | |

1.4.3. | Multiplying Fractions by Whole Numbers | |

1.4.4. | Multiplying Fractions by Whole Numbers: Word Problems | |

1.4.5. | Strategies to Simplify Whole Number and Fraction Multiplication | |

1.4.6. | Multiplying Mixed Numbers by Whole Numbers | |

1.4.7. | Multiplying Fractions by Unit Fractions Using Models | |

1.4.8. | Multiplying Fractions Using Models | |

1.4.9. | Multiplying Fractions | |

1.4.10. | Multiplying Fractions: Word Problems | |

1.4.11. | Strategies to Simplify Fraction Multiplication | |

1.4.12. | Multiplying Mixed Numbers by Fractions | |

1.4.13. | Multiplying Mixed Numbers | |

1.4.14. | Multiplying Mixed Numbers: Word Problems |

1.5. Dividing Fractions

1.5.1. | Dividing Whole Numbers by Unit Fractions Using Models | |

1.5.2. | Dividing Whole Numbers by Unit Fractions | |

1.5.3. | Dividing Unit Fractions by Whole Numbers Using Models | |

1.5.4. | Dividing Unit Fractions by Whole Numbers | |

1.5.5. | Reciprocals | |

1.5.6. | Dividing Whole Numbers by Fractions Using Models: Whole Number Results | |

1.5.7. | Dividing Whole Numbers by Fractions Using Models: Fractional Results | |

1.5.8. | Dividing Whole Numbers by Fractions | |

1.5.9. | Dividing Fractions by Whole Numbers Using Models | |

1.5.10. | Dividing Fractions by Whole Numbers | |

1.5.11. | Dividing Fractions Using Models | |

1.5.12. | Dividing Fractions |

2.

Ratios & Percentages
40 topics

2.6. Ratios

2.6.1. | Introduction to Ratios | |

2.6.2. | Equivalent Ratios | |

2.6.3. | Writing Ratios Using Fractions | |

2.6.4. | Working With Equivalent Ratios | |

2.6.5. | More on Equivalent Ratios | |

2.6.6. | Ratio Tables | |

2.6.7. | Graphing Ratios | |

2.6.8. | Unit Rates | |

2.6.9. | Solving Problems Using Unit Rates | |

2.6.10. | Speed as a Unit Rate |

2.7. Proportional Relationships

2.7.1. | Understanding Proportional Relationships Using Tables | |

2.7.2. | Understanding Proportional Relationships Using Graphs | |

2.7.3. | Understanding Proportional Relationships From Descriptions | |

2.7.4. | Modeling with Direct Variation | |

2.7.5. | Inverse Variation | |

2.7.6. | Modeling With Inverse Variation |

2.8. Working With Units

2.8.1. | Units of Length | |

2.8.2. | Units of Mass | |

2.8.3. | Units of Time | |

2.8.4. | Units of Volume | |

2.8.5. | Units of Area and Volume | |

2.8.6. | Unit Conversions Using Base Units of Mass | |

2.8.7. | Two-Step Unit Conversions | |

2.8.8. | Unit Conversions Using Base Units of Length | |

2.8.9. | Unit Conversions Using Units of Time | |

2.8.10. | Converting Units of Area to Smaller Units | |

2.8.11. | Converting Units of Area to Larger Units | |

2.8.12. | Converting Between Mixed Units | |

2.8.13. | Determining Units in Formulas | |

2.8.14. | Selecting Units for Rates of Change | |

2.8.15. | Degrees of Accuracy |

2.9. Percentages

2.9.1. | Understanding Percentages Using Models | |

2.9.2. | Converting Between Percentages and Fractions | |

2.9.3. | Converting Between Percentages and Decimals | |

2.9.4. | Finding Part of a Number Given a Whole and a Percentage | |

2.9.5. | Finding Part of a Number Given a Whole and a Percentage: Word Problems | |

2.9.6. | Finding a Percentage Given Two Numbers | |

2.9.7. | Finding a Percentage Given Two Numbers: Word Problems | |

2.9.8. | Finding a Total Given a Part and a Percentage | |

2.9.9. | Percentage Increase, Decrease, and Error |

3.

The Number System
27 topics

3.10. Addition and Subtraction of Rational Numbers

3.10.1. | Subtracting a Positive Number From a Smaller Positive Number | |

3.10.2. | Subtracting a Positive Number From a Negative Number | |

3.10.3. | Subtracting a Positive Fraction From a Smaller Fraction | |

3.10.4. | Subtracting a Positive Decimal From a Smaller Decimal | |

3.10.5. | Adding a Positive Number to a Negative Number | |

3.10.6. | Adding a Negative Number | |

3.10.7. | Subtracting a Negative Number | |

3.10.8. | Additive Inverses of Numbers | |

3.10.9. | Solving Problems Using Addition and Subtraction of Rational Numbers | |

3.10.10. | Absolute Value |

3.11. Multiplication and Division of Rational Numbers

3.11.1. | Multiplying Positive Numbers With Negative Numbers | |

3.11.2. | Multiplying Negative Numbers | |

3.11.3. | Dividing With Negative Numbers | |

3.11.4. | Equivalent Fractions With Negative Numbers | |

3.11.5. | Dividing With Negative Decimals and Fractions | |

3.11.6. | Reciprocals of Rational Numbers | |

3.11.7. | Multiplying and Dividing With Zero | |

3.11.8. | Fractions of Fractions | |

3.11.9. | Solving Problems Using the Four Operations on Rational Numbers |

3.12. Number Theory

3.12.1. | Divisibility of Integers by 2, 5, 10, 4, and 8 | |

3.12.2. | Divisibility of Integers by 3, 9, 6, and 7 | |

3.12.3. | Prime and Composite Numbers | |

3.12.4. | Prime Factors | |

3.12.5. | The Fundamental Theorem of Arithmetic | |

3.12.6. | Finding Greatest Common Factors Using Prime Factorization | |

3.12.7. | Finding Lowest Common Multiples Using Prime Factorization | |

3.12.8. | Natural Numbers, Integers, and Rational Numbers |

4.

Exponents & Radicals
32 topics

4.13. Exponents

4.13.1. | Introduction to Exponents | |

4.13.2. | Evaluating Exponents | |

4.13.3. | Evaluating Larger Exponents | |

4.13.4. | Evaluating Exponents With Fractional Bases | |

4.13.5. | Evaluating Exponents With Decimal Bases | |

4.13.6. | Evaluating Whole Number Expressions Containing Exponents | |

4.13.7. | Squaring Rational Numbers | |

4.13.8. | Cubing Rational Numbers | |

4.13.9. | Exponents With Rational Bases | |

4.13.10. | The Zeroth Power | |

4.13.11. | Powers of Negative One | |

4.13.12. | Negative Exponents |

4.14. The Rules of Exponents

4.14.1. | The Product Rule for Exponents | |

4.14.2. | The Quotient Rule for Exponents | |

4.14.3. | The Power Rule for Exponents | |

4.14.4. | The Power of Product Rule for Exponents | |

4.14.5. | The Power of Quotient Rule for Exponents | |

4.14.6. | Combining the Rules of Exponents |

4.15. Order of Operations

4.15.1. | Evaluating Rational Number Expressions Without Exponents | |

4.15.2. | Evaluating Expressions With Positive Integer Exponents | |

4.15.3. | Evaluating Expressions With Integer Exponents |

4.16. Radicals

4.16.1. | The Square Root of a Perfect Square | |

4.16.2. | Squaring a Square Root | |

4.16.3. | The Cube Root of a Perfect Cube | |

4.16.4. | Surds | |

4.16.5. | Simplifying Expressions With Surds | |

4.16.6. | Evaluating Numerical Expressions Containing Radicals | |

4.16.7. | Adding and Subtracting Radicals | |

4.16.8. | Rationalizing Denominators |

4.17. The Rules of Radicals

4.17.1. | Radicals as Fractional Exponents | |

4.17.2. | The Product Rule for Radicals | |

4.17.3. | The Quotient Rule for Radicals |

5.

Equations & Inequalities
49 topics

5.18. Algebraic Expressions

5.18.1. | Introduction to Algebraic Expressions | |

5.18.2. | Constructing Algebraic Expressions | |

5.18.3. | Evaluating Linear Expressions | |

5.18.4. | Evaluating Rational Expressions | |

5.18.5. | Identifying Terms in Linear Expressions | |

5.18.6. | Identifying Coefficients and Constants in Linear Expressions | |

5.18.7. | The Greatest Common Factor of Two Linear Expressions |

5.19. Simplifying Linear Expressions

5.19.1. | Identifying Like Terms in a Linear Expression | |

5.19.2. | Collecting Like Terms in a Linear Expression | |

5.19.3. | Applying the Distributive Law With Linear Expressions | |

5.19.4. | The Distributive Law With Three Terms | |

5.19.5. | Distributing the Negative Sign in a Linear Expression | |

5.19.6. | Simplifying Linear Expressions Using the Distributive Law | |

5.19.7. | Simplifying Linear Expressions When a Group of Terms Is Subtracted | |

5.19.8. | Simplifying Linear Expressions Containing Fractions | |

5.19.9. | Factoring Linear Expressions |

5.20. Solving One-Step Linear Equations

5.20.1. | Solving Linear Equations by Trial and Error | |

5.20.2. | Solving One-Step Addition and Subtraction Equations | |

5.20.3. | Solving One-Step Multiplication and Division Equations |

5.21. Solving Multi-Step Linear Equations

5.21.1. | Solving Two-Step Equations | |

5.21.2. | Solving Linear Equations With Fractional Coefficients | |

5.21.3. | Solving Linear Equations With Decimal Coefficients | |

5.21.4. | Solving Linear Equations With Variables on Both Sides | |

5.21.5. | Solving Linear Equations by Clearing a Rational Expression | |

5.21.6. | Solving Linear Equations Using Cross-Multiplication | |

5.21.7. | Solving Two-Variable Equations | |

5.21.8. | Equations With Many Solutions and No Solutions | |

5.21.9. | Solving Linear Equations With Unknown Coefficients | |

5.21.10. | Solving Linear Equations With Unknown Coefficients by Factoring | |

5.21.11. | Solving Many-Variable Equations |

5.22. Modeling With Linear Equations

5.22.1. | Interpreting Linear Expressions | |

5.22.2. | Modeling With Linear Equations |

5.23. Solving Linear Inequalities

5.23.1. | Inequalities | |

5.23.2. | Compound Inequalities | |

5.23.3. | Verifying Solutions of Linear Inequalities | |

5.23.4. | Solving One-Step Linear Inequalities | |

5.23.5. | Solving Two-Step Inequalities | |

5.23.6. | Representing Solutions to Inequalities on Number Lines | |

5.23.7. | Further Solving Linear Inequalities | |

5.23.8. | Solving Compound Inequalities | |

5.23.9. | Interval Notation | |

5.23.10. | Unbounded Intervals | |

5.23.11. | Unions of Intervals | |

5.23.12. | Intersections of Intervals | |

5.23.13. | Compound OR Inequalities | |

5.23.14. | Compound AND Inequalities |

5.24. Nonlinear Equations

5.24.1. | Solving Equations Using the Square Root Method | |

5.24.2. | Solving Equations With Odd Exponents Using the Nth Root Method | |

5.24.3. | Solving Equations With Even Exponents Using the Nth Root Method |

6.

Two-Variable Equations
29 topics

6.25. Graphs of Linear Equations

6.25.1. | The Cartesian Coordinate System | |

6.25.2. | Distances Between Points in the Coordinate Plane | |

6.25.3. | Two-Variable Linear Equations and Their Solutions | |

6.25.4. | Graphing Linear Equations | |

6.25.5. | Horizontal and Vertical Lines | |

6.25.6. | Calculating Slopes of Straight Lines | |

6.25.7. | Equations of Lines in Slope-Intercept Form | |

6.25.8. | Finding Properties of Lines Given in Slope-Intercept Form | |

6.25.9. | Equations of Lines in Point-Slope Form | |

6.25.10. | Equations of Lines in Standard Form | |

6.25.11. | Determining Properties of Lines Given in Standard Form | |

6.25.12. | Parallel Lines in the Coordinate Plane |

6.26. Modeling With Two-Variable Linear Equations

6.26.1. | Modeling With Linear Equations in Two Variables | |

6.26.2. | Further Modeling With Linear Equations in Two Variables | |

6.26.3. | Analyzing and Interpreting Graphs of Linear Equations | |

6.26.4. | Distance-Time Graphs | |

6.26.5. | Calculating Acceleration From a Speed-Time Graph | |

6.26.6. | Calculating Distance From a Speed-Time Graph |

6.27. Systems of Linear Equations

6.27.1. | Introduction to Systems of Linear Equations | |

6.27.2. | Solving Systems of Equations by Substitution | |

6.27.3. | Introduction to the Elimination Method | |

6.27.4. | Solving Systems of Linear Equations Using Elimination: One Transformation | |

6.27.5. | Solving Systems of Linear Equations Using Elimination: Two Transformations | |

6.27.6. | Systems of Linear Equations With Fractional Coefficients | |

6.27.7. | Systems of Linear Equations With Decimal Coefficients | |

6.27.8. | Solving Systems of Nonlinear Equations Using Graphs | |

6.27.9. | Systems of Equations With No Solutions and Infinitely Many Solutions | |

6.27.10. | Consistency and Dependency in Linear Systems | |

6.27.11. | Calculating the Intersection of Two Lines |

7.

Radical & Rational Expressions
16 topics

7.28. Radical Expressions

7.28.1. | Writing Radical Expressions Using Fractional Exponents | |

7.28.2. | The Square Root of a Perfect Square With Algebraic Expressions | |

7.28.3. | The Square Root of a Perfect Square With Domain Restrictions | |

7.28.4. | The Cube Root of a Perfect Cube With Algebraic Expressions | |

7.28.5. | Simplifying Square Root Expressions Using the Product Rule | |

7.28.6. | Combining Radical Expressions Using the Product Rule | |

7.28.7. | Evaluating Algebraic Radical Expressions | |

7.28.8. | Adding and Subtracting Radical Expressions |

7.29. Rational Expressions

7.29.1. | Equivalent Expressions With Fractions | |

7.29.2. | Simplifying Rational Expressions | |

7.29.3. | Simplifying Rational Expressions by Factoring | |

7.29.4. | Splitting Rational Expressions Into Separate Terms |

7.30. Rational Equations

7.30.1. | Solving Rational Equations Containing One Fractional Term | |

7.30.2. | Solving Rational Equations Using Cross-Multiplication | |

7.30.3. | Solving Rational Equations Containing Binomials Using Cross-Multiplication | |

7.30.4. | Solving Rational Equations Using the Flip Method |

8.

Functions
12 topics

8.31. Functions

8.31.1. | Introduction to Functions | |

8.31.2. | Visual Representations of Functions | |

8.31.3. | Graphs of Functions | |

8.31.4. | The Domain of a Function | |

8.31.5. | Global Extrema of Functions | |

8.31.6. | End Behavior of Functions | |

8.31.7. | The Range of a Function | |

8.31.8. | The Range of a Function: Advanced Cases | |

8.31.9. | The Roots of a Function | |

8.31.10. | Increasing and Decreasing Functions | |

8.31.11. | Piecewise Functions | |

8.31.12. | Modeling With Linear Functions |

9.

Absolute Value
11 topics

9.32. Absolute Value Expressions, Equations & Inequalities

9.32.1. | Absolute Value Expressions | |

9.32.2. | Rules of Absolute Value | |

9.32.3. | Further Rules of Absolute Value | |

9.32.4. | Absolute Value Equations | |

9.32.5. | Further Absolute Value Equations | |

9.32.6. | Absolute Value Inequalities | |

9.32.7. | Absolute Value Graphs | |

9.32.8. | Vertical Reflections of Absolute Value Graphs | |

9.32.9. | Stretches of Absolute Value Graphs | |

9.32.10. | Combining Transformations of Absolute Value Graphs | |

9.32.11. | Domain and Range of Absolute Value Functions |

10.

Exponential Functions
17 topics

10.33. Rules of Exponents

10.33.1. | The Product Rule for Exponents With Algebraic Expressions | |

10.33.2. | The Quotient Rule for Exponents With Algebraic Expressions | |

10.33.3. | The Power Rule for Exponents With Algebraic Expressions | |

10.33.4. | The Power of Product Rule With Algebraic Expressions | |

10.33.5. | The Power of Quotient Rule With Algebraic Expressions |

10.34. Exponential Expressions and Equations

10.34.1. | Solving Exponential Equations | |

10.34.2. | Creating Exponential Growth Expressions | |

10.34.3. | Creating Exponential Decay Expressions |

10.35. Exponential Functions

10.35.1. | Exponential Functions | |

10.35.2. | Modeling Exponential Growth With Functions | |

10.35.3. | Interpreting Exponential Growth | |

10.35.4. | Solving Exponential Growth Problems | |

10.35.5. | Modeling Exponential Decay With Functions | |

10.35.6. | Interpreting Exponential Decay | |

10.35.7. | Solving Exponential Decay Problems | |

10.35.8. | Graphing Exponential Growth Functions | |

10.35.9. | Graphing Exponential Decay Functions |

11.

Polynomials
20 topics

11.36. Polynomials

11.36.1. | Introduction to Polynomials | |

11.36.2. | The Degree of a Polynomial | |

11.36.3. | Simplifying Polynomials | |

11.36.4. | The Distributive Law for Polynomials | |

11.36.5. | Adding and Subtracting Polynomials | |

11.36.6. | Monomials, Binomials and Trinomials | |

11.36.7. | Multiplying Binomials | |

11.36.8. | Multiplying Polynomials | |

11.36.9. | Squaring Binomials | |

11.36.10. | Expanding Binomials Using Pascal's Triangle | |

11.36.11. | The Difference of Squares Formula |

11.37. Factoring Polynomials

11.37.1. | The Greatest Common Factor of Two Monomials | |

11.37.2. | Factoring Simple Polynomials Using Greatest Common Factors | |

11.37.3. | Factoring Perfect Square Trinomials | |

11.37.4. | Factoring Perfect Square Trinomials With Leading Coefficients | |

11.37.5. | Factoring Differences of Squares | |

11.37.6. | Factoring Trinomials | |

11.37.7. | Factoring Trinomials Using Common Factors | |

11.37.8. | Factoring Trinomials With Leading Coefficients | |

11.37.9. | Further Factoring Trinomials With Leading Coefficients |

12.

Geometry Fundamentals
26 topics

12.38. Introduction to Geometry

12.38.1. | Points, Lines, Rays, and Segments | |

12.38.2. | Parallel and Perpendicular Lines | |

12.38.3. | Measures of Segments | |

12.38.4. | Congruent Segments | |

12.38.5. | Midpoints | |

12.38.6. | Collinear Points | |

12.38.7. | The Segment Addition Postulate |

12.39. Angles

12.39.1. | Angles and Measures of Angles | |

12.39.2. | Measuring Angles Using a Protractor | |

12.39.3. | Sums of Angles | |

12.39.4. | Right, Straight, Full, and Null Angles | |

12.39.5. | Acute, Obtuse, and Reflex Angles | |

12.39.6. | Congruent Angles | |

12.39.7. | Vertical Angles | |

12.39.8. | Solving Problems With Angles |

12.40. Advanced Angles

12.40.1. | Complementary Angles | |

12.40.2. | Supplementary Angles | |

12.40.3. | Corresponding Angles | |

12.40.4. | Alternate Angles | |

12.40.5. | Consecutive Angles | |

12.40.6. | Angle Bisectors | |

12.40.7. | Segment Bisectors |

12.41. Circles

12.41.1. | Circles | |

12.41.2. | Arcs, Segments, and Sectors of Circles | |

12.41.3. | The Circumference of a Circle | |

12.41.4. | Areas of Circles |

13.

Polygons
21 topics

13.42. Triangles

13.42.1. | Interior Angles of Triangles | |

13.42.2. | Exterior Angles of Triangles | |

13.42.3. | Classifying Triangles | |

13.42.4. | The Triangle Inequality | |

13.42.5. | The Isosceles Triangle Theorem | |

13.42.6. | Heights of Triangles | |

13.42.7. | The Pythagorean Theorem | |

13.42.8. | The 45-45-90 Triangle | |

13.42.9. | The 30-60-90 Triangle |

13.43. Polygons

13.43.1. | Polygons | |

13.43.2. | Interior Angles of Polygons | |

13.43.3. | Exterior Angles of Polygons | |

13.43.4. | Congruent Polygons | |

13.43.5. | Regular Polygons | |

13.43.6. | The Perimeter of a Polygon | |

13.43.7. | Classifying Quadrilaterals | |

13.43.8. | Properties of Rectangles and Squares |

13.44. Area and Perimeter

13.44.1. | Areas of Rectangles and Squares | |

13.44.2. | Areas of Triangles | |

13.44.3. | Areas of Trapezoids | |

13.44.4. | Areas and Perimeters of Composite Shapes |