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# Mathematical Foundations I

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

## Content

### Arithmetic Operations

• 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.

### Measurement and Data

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

### Prime Factorization

• 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.

### Linear Equations and Inequalities

• 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.

### Nonlinear Equations and Inequalities

• 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.

### Polynomials

• 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.

### Functions

• 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.

### Geometry

• 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
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
 1.2.1. Least Common Multiples 1.2.2. Least Common Denominators 1.2.3. Comparing Fractions Using Common Denominators 1.2.4. Adding Fractions With Unlike Denominators Using Models 1.2.5. Subtracting Fractions With Unlike Denominators Using Models 1.2.6. Adding Fractions With Unlike Denominators 1.2.7. 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. Interpreting Multiplication of Fractions and Whole Numbers 1.4.5. Multiplying Mixed Numbers by Whole Numbers 1.4.6. Multiplying Fractions by Unit Fractions Using Models 1.4.7. Multiplying Fractions Using Models 1.4.8. Multiplying Fractions 1.4.9. Multiplying Fractions: Word Problems 1.4.10. Multiplying Mixed Numbers by Fractions 1.4.11. Multiplying Mixed Numbers 1.4.12. Multiplying Mixed Numbers: Word Problems 1.4.13. Connecting Fraction Multiplication and Area 1.4.14. Finding Areas Using Fraction Multiplication
1.5. Dividing Fractions
 1.5.1. Dividing Fractions Using Properties of Division 1.5.2. Dividing Whole Numbers by Unit Fractions Using Models 1.5.3. Dividing Whole Numbers by Unit Fractions 1.5.4. Dividing Unit Fractions by Whole Numbers Using Models 1.5.5. Dividing Unit Fractions by Whole Numbers 1.5.6. Reciprocals 1.5.7. Dividing Whole Numbers by Fractions Using Models 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
27 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. Ratio Tables 2.6.5. Graphing Ratios 2.6.6. Unit Rates 2.6.7. Solving Problems Using Unit Rates 2.6.8. Speed as a Unit Rate
2.7. Working With Units
 2.7.1. Units 2.7.2. Units of Area and Volume 2.7.3. Unit Conversions Using Base Units of Mass 2.7.4. Two-Step Unit Conversions 2.7.5. Unit Conversions Using Base Units of Length 2.7.6. Unit Conversions Using Units of Time 2.7.7. Unit Conversions Using Base Units of Area 2.7.8. Determining Units in Formulas 2.7.9. Selecting Units for Rates of Change 2.7.10. Degrees of Accuracy
2.8. Percentages
 2.8.1. Understanding Percentages Using Models 2.8.2. Converting Between Percentages and Fractions 2.8.3. Converting Between Percentages and Decimals 2.8.4. Finding Part of a Number Given a Whole and a Percentage 2.8.5. Finding Part of a Number Given a Whole and a Percentage: Word Problems 2.8.6. Finding a Percentage Given Two Numbers 2.8.7. Finding a Percentage Given Two Numbers: Word Problems 2.8.8. Finding a Total Given a Part and a Percentage 2.8.9. Percentage Increase, Decrease, and Error
3.
The Number System
22 topics
3.9. Addition and Subtraction of Rational Numbers
 3.9.1. Subtracting a Positive Number From a Smaller Positive Number 3.9.2. Subtracting a Positive Number From a Negative Number 3.9.3. Subtracting a Positive Fraction From a Smaller Fraction 3.9.4. Subtracting a Positive Decimal From a Smaller Decimal 3.9.5. Adding a Positive Number to a Negative Number 3.9.6. Adding a Negative Number 3.9.7. Subtracting a Negative Number 3.9.8. Additive Inverses of Numbers 3.9.9. Solving Problems Using Addition and Subtraction of Rational Numbers 3.9.10. Absolute Value
3.10. Multiplication and Division of Rational Numbers
 3.10.1. Multiplying Positive Numbers With Negative Numbers 3.10.2. Multiplying Negative Numbers 3.10.3. Dividing With Negative Numbers 3.10.4. Dividing With Negative Decimals and Fractions 3.10.5. Reciprocals of Rational Numbers 3.10.6. Multiplying and Dividing With Zero 3.10.7. Fractions of Fractions 3.10.8. Solving Problems Using the Four Operations on Rational Numbers
3.11. Number Theory
 3.11.1. Prime and Composite Numbers 3.11.2. Prime Factorization of Integers 3.11.3. Finding Greatest Common Factors Using Prime Factorization 3.11.4. Finding Lowest Common Multiples Using Prime Factorization
4.
32 topics
4.12. Exponents
 4.12.1. Introduction to Exponents 4.12.2. Evaluating Exponents 4.12.3. Evaluating Larger Exponents 4.12.4. Evaluating Exponents With Fractional Bases 4.12.5. Evaluating Exponents With Decimal Bases 4.12.6. Evaluating Whole Number Expressions Containing Exponents 4.12.7. Squaring Rational Numbers 4.12.8. Cubing Rational Numbers 4.12.9. Exponents With Rational Bases 4.12.10. The Zeroth Power 4.12.11. Powers of Negative One 4.12.12. Negative Exponents
4.13. The Rules of Exponents
 4.13.1. The Product Rule for Exponents 4.13.2. The Quotient Rule for Exponents 4.13.3. The Power Rule for Exponents 4.13.4. The Power of Product Rule for Exponents 4.13.5. The Power of Quotient Rule for Exponents 4.13.6. Combining the Rules of Exponents
4.14. Order of Operations
 4.14.1. Evaluating Rational Number Expressions Without Exponents 4.14.2. Evaluating Rational Number Expressions With Positive Integer Exponents 4.14.3. Evaluating Rational Number Expressions With Integer Exponents
 4.15.1. The Square Root of a Perfect Square 4.15.2. Squaring a Square Root 4.15.3. The Cube Root of a Perfect Cube 4.15.4. Surds 4.15.5. Simplifying Expressions With Surds 4.15.6. Evaluating Numerical Expressions Containing Radicals 4.15.7. Adding and Subtracting Radicals 4.15.8. Rationalizing Denominators
 4.16.1. Radicals as Fractional Exponents 4.16.2. The Product Rule for Radicals 4.16.3. The Quotient Rule for Radicals
6.
Equations & Inequalities
46 topics
6.17. Algebraic Expressions
 6.17.1. Introduction to Algebraic Expressions 6.17.2. Constructing Algebraic Expressions 6.17.3. Evaluating Linear Expressions 6.17.4. Evaluating Rational Expressions 6.17.5. Identifying Terms in Linear Expressions 6.17.6. Identifying Coefficients and Constants in Linear Expressions 6.17.7. The Greatest Common Factor of Two Linear Expressions
6.18. Simplifying Linear Expressions
 6.18.1. Identifying Like Terms in a Linear Expression 6.18.2. Collecting Like Terms in a Linear Expression 6.18.3. Applying the Distributive Law With Linear Expressions 6.18.4. The Distributive Law With Three Terms 6.18.5. Distributing the Negative Sign in a Linear Expression 6.18.6. Simplifying Linear Expressions Using the Distributive Law 6.18.7. Simplifying Linear Expressions When a Group of Terms Is Subtracted 6.18.8. Simplifying Linear Expressions Containing Fractions 6.18.9. Factoring Numerical Expressions 6.18.10. Factoring Linear Expressions
6.19. Solving One-Step Linear Equations
 6.19.1. Solving Linear Equations by Trial and Error 6.19.2. Solving One-Step Addition and Subtraction Equations 6.19.3. Solving One-Step Multiplication and Division Equations
6.20. Solving Multi-Step Linear Equations
 6.20.1. Solving Two-Step Equations 6.20.2. Solving Linear Equations With Fractional Coefficients 6.20.3. Solving Linear Equations With Decimal Coefficients 6.20.4. Solving Linear Equations With Variables on Both Sides 6.20.5. Solving Linear Equations by Clearing a Rational Expression 6.20.6. Solving Linear Equations Using Cross-Multiplication 6.20.7. Solving Two-Variable Equations 6.20.8. Equations With Many Solutions and No Solutions 6.20.9. Solving Linear Equations With Unknown Coefficients 6.20.10. Solving Linear Equations With Unknown Coefficients by Factoring 6.20.11. Solving Many-Variable Equations
6.21. Modeling With Linear Equations
 6.21.1. Interpreting Linear Expressions 6.21.2. Modeling With Linear Equations
6.22. Solving Linear Inequalities
 6.22.1. Representing Inequalities on Number Lines 6.22.2. Representing Compound Inequalities on Number Lines 6.22.3. Verifying Solutions of Linear Inequalities 6.22.4. Solving Linear Inequalities 6.22.5. Solving Compound Inequalities 6.22.6. Expressing Inequalities Using Interval Notation 6.22.7. Unions of Intervals 6.22.8. Intersections of Intervals 6.22.9. Compound OR Inequalities 6.22.10. Compound AND Inequalities
6.23. Nonlinear Equations
 6.23.1. Solving Equations Using the Square Root Method 6.23.2. Solving Equations With Odd Exponents Using the Nth Root Method 6.23.3. Solving Equations With Even Exponents Using the Nth Root Method
7.
Two-Variable Equations
22 topics
7.24. Graphs of Linear Equations
 7.24.1. The Cartesian Coordinate System 7.24.2. The Distance Between Two Points in the Coordinate Plane 7.24.3. Two-Variable Linear Equations and Their Solutions 7.24.4. Graphing Linear Equations 7.24.5. Horizontal and Vertical Lines 7.24.6. Calculating Slopes of Straight Lines 7.24.7. Equations of Lines in Slope-Intercept Form 7.24.8. Finding Properties of Lines Given in Slope-Intercept Form 7.24.9. Equations of Lines in Point-Slope Form 7.24.10. Equations of Lines in Standard Form 7.24.11. Determining Properties of Lines Given in Standard Form
7.25. Modeling With Two-Variable Linear Equations
 7.25.1. Modeling With Linear Equations in Two Variables 7.25.2. Further Modeling With Linear Equations in Two Variables 7.25.3. Analyzing and Interpreting Graphs of Linear Equations
7.26. Systems of Linear Equations
 7.26.1. Introduction to Systems of Linear Equations 7.26.2. Solving Systems of Two Linear Equations Using Substitution 7.26.3. Solving Systems of Two Linear Equations Using Elimination 7.26.4. Systems of Linear Equations With Fractional Coefficients 7.26.5. Systems of Linear Equations With Decimal Coefficients 7.26.6. Solving Systems of Nonlinear Equations Using Graphs 7.26.7. Systems of Equations With No Solutions and Infinitely Many Solutions 7.26.8. Calculating the Intersection of Two Lines
8.
16 topics
 8.27.1. Writing Radical Expressions Using Fractional Exponents 8.27.2. The Square Root of a Perfect Square With Algebraic Expressions 8.27.3. The Square Root of a Perfect Square With Domain Restrictions 8.27.4. The Cube Root of a Perfect Cube With Algebraic Expressions 8.27.5. Simplifying Square Root Expressions Using the Product Rule 8.27.6. Combining Radical Expressions Using the Product Rule 8.27.7. Evaluating Algebraic Radical Expressions 8.27.8. Adding and Subtracting Radical Expressions
8.28. Rational Expressions
 8.28.1. Equivalent Expressions With Fractions 8.28.2. Simplifying Rational Expressions 8.28.3. Simplifying Rational Expressions by Factoring 8.28.4. Splitting Rational Expressions Into Separate Terms
8.29. Rational Equations
 8.29.1. Solving Rational Equations Containing One Fractional Term 8.29.2. Solving Rational Equations Using Cross-Multiplication 8.29.3. Solving Rational Equations Containing Binomials Using Cross-Multiplication 8.29.4. Solving Rational Equations Using the Flip Method
9.
Functions
12 topics
9.30. Functions
 9.30.1. Introduction to Functions 9.30.2. Visual Representations of Functions 9.30.3. Graphs of Functions 9.30.4. The Domain of a Function 9.30.5. Global Extrema of Functions 9.30.6. End Behavior of Functions 9.30.7. The Range of a Function 9.30.8. The Range of a Function: Advanced Cases 9.30.9. The Roots of a Function 9.30.10. Increasing and Decreasing Functions 9.30.11. Piecewise Functions 9.30.12. Modeling With Linear Functions
10.
Absolute Value
6 topics
10.31. Absolute Value Expressions, Equations & Inequalities
 10.31.1. Evaluating Absolute Value Expressions 10.31.2. Rules of Absolute Value 10.31.3. Further Rules of Absolute Value 10.31.4. Absolute Value Equations 10.31.5. Absolute Value Inequalities 10.31.6. Absolute Value Graphs
11.
Exponential Functions
17 topics
11.32. Rules of Exponents
 11.32.1. The Product Rule for Exponents With Algebraic Expressions 11.32.2. The Quotient Rule for Exponents With Algebraic Expressions 11.32.3. The Power Rule for Exponents With Algebraic Expressions 11.32.4. The Power of Product Rule With Algebraic Expressions 11.32.5. The Power of Quotient Rule With Algebraic Expressions
11.33. Exponential Expressions and Equations
 11.33.1. Solving Exponential Equations with Integer Solutions 11.33.2. Creating Exponential Growth Expressions 11.33.3. Creating Exponential Decay Expressions
11.34. Exponential Functions
 11.34.1. Exponential Functions 11.34.2. Modeling Exponential Growth With Functions 11.34.3. Interpreting Exponential Growth 11.34.4. Solving Exponential Growth Problems 11.34.5. Modeling Exponential Decay With Functions 11.34.6. Interpreting Exponential Decay 11.34.7. Solving Exponential Decay Problems 11.34.8. Graphing Exponential Growth Functions 11.34.9. Graphing Exponential Decay Functions
12.
Polynomials
19 topics
12.35. Polynomials
 12.35.1. Introduction to Polynomials 12.35.2. The Degree of a Polynomial 12.35.3. Simplifying Polynomials 12.35.4. The Distributive Law for Polynomials 12.35.5. Adding and Subtracting Polynomials 12.35.6. Monomials, Binomials and Trinomials 12.35.7. Multiplying Binomials 12.35.8. Multiplying Polynomials 12.35.9. Squaring Binomials 12.35.10. Expanding Binomials Using Pascal's Triangle 12.35.11. The Difference of Squares Formula
12.36. Factoring Polynomials
 12.36.1. The Greatest Common Factor of Two Monomials 12.36.2. Factoring Simple Polynomials Using Greatest Common Factors 12.36.3. Factoring Perfect Square Trinomials 12.36.4. Factoring Perfect Square Trinomials With Leading Coefficients 12.36.5. Factoring Differences of Squares 12.36.6. Factoring Trinomials 12.36.7. Factoring Trinomials Using Common Factors 12.36.8. Factoring Trinomials With Leading Coefficients
13.
Geometry Fundamentals
21 topics
13.37. Introduction to Geometry
 13.37.1. Points, Lines, Rays, and Parallel Lines 13.37.2. Segments and Measures of Segments 13.37.3. Congruent Segments 13.37.4. Midpoints 13.37.5. Collinear Points 13.37.6. The Segment Addition Postulate
13.38. Angles
 13.38.1. Angles and Measures of Angles 13.38.2. Congruent Angles 13.38.3. Sums of Angles 13.38.4. Null, Right, Straight, and Full Angles 13.38.5. Vertical Angles 13.38.6. Acute, Obtuse, and Reflex Angles 13.38.7. Solving Problems With Angles