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Mathematical Foundations I

Overview

Outcomes

Content

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

Upon successful completion of this course, students will have mastered the following:

Arithmetic Operations

Measurement and Data

Prime Factorization

Linear Equations and Inequalities

Nonlinear Equations and Inequalities

Polynomials

Functions

Geometry

1.
Fractions
40 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
1.5.7. Dividing Whole Numbers by Fractions
1.5.8. Dividing Fractions by Whole Numbers Using Models
1.5.9. Dividing Fractions by Whole Numbers
1.5.10. Dividing Fractions Using Models
1.5.11. 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 Rational Number Expressions With Positive Integer Exponents
4.15.3. Evaluating Rational Number 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
27 topics
6.25. Graphs of Linear Equations
6.25.1. The Cartesian Coordinate System
6.25.2. The Distance Between Two 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
6.26. 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 Two Linear Equations Using 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. 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
25 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. The Perpendicular Bisector Theorem
12.41. Circles
12.41.1. Introduction to 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