Foundations of Algebra

Summary

This chapter introduces the fundamental building blocks of algebra that serve as the foundation for all future mathematical study. Students will learn to identify and work with variables, constants, coefficients, terms, and expressions while mastering the essential order of operations. By the end of this chapter, students will understand how algebraic language represents mathematical relationships and be prepared to manipulate algebraic expressions with confidence.

Concepts Covered

This chapter covers the following 17 concepts from the learning graph:

Number
Variable
Constant
Coefficient
Term
Expression
Equation
Inequality
Order of Operations
Evaluating Expressions
Substitution
Like Terms
Combining Like Terms
Simplifying Expressions
Expanding Expressions
Monomial
Prime Factorization

Prerequisites

This chapter assumes only the prerequisites listed in the course description.

Introduction to Algebra

Algebra is the language of mathematics. Just as you learned to read and write English to communicate ideas, you'll learn to read and write algebraic expressions to communicate mathematical relationships. In this chapter, you'll discover the fundamental building blocks that make algebra work.

Think of algebra like learning a new language. First, you need to understand the alphabet and basic words before you can write sentences. In algebra, numbers, variables, and operations are like the alphabet. Once you understand these basics, you can build more complex expressions and solve real-world problems.

What Makes Algebra Different?

Arithmetic uses specific numbers: "3 + 5 = 8". Algebra uses variables to represent unknown or changing values: "$x + 5 = 8$, what is $x$?" This simple difference opens up endless possibilities for solving problems and understanding patterns.

Numbers: The Foundation

A number is a mathematical object used to count, measure, and label quantities. You've worked with numbers your entire life, but let's review the types of numbers you'll encounter in algebra.

Types of Numbers

The following table shows the main types of numbers used in Algebra I:

Number Type	Description	Examples
Natural Numbers	Counting numbers starting from 1	1, 2, 3, 4, 5, ...
Whole Numbers	Natural numbers plus zero	0, 1, 2, 3, 4, ...
Integers	Whole numbers and their negatives	..., -2, -1, 0, 1, 2, ...
Rational Numbers	Numbers that can be written as fractions	$\frac{1}{2}$, $-\frac{3}{4}$, 0.5, 2
Irrational Numbers	Numbers that cannot be written as fractions	$\pi$, $\sqrt{2}$, $\sqrt{3}$

Understanding these number types helps you know what kind of solutions to expect when solving equations. For example, the equation $2x = 7$ has a rational solution ($x = \frac{7}{2}$ or $3.5$), while $x^2 = 2$ has an irrational solution ($x = \sqrt{2}$).

Variables: The Power of the Unknown

A variable is a symbol (usually a letter) that represents an unknown or changing value. Variables are what make algebra powerful—they let us work with values we don't know yet or values that can change.

Why Use Variables?

Imagine you're planning a party and need to buy pizza. Each pizza costs $12. If you know you're buying 3 pizzas, you can calculate: $12 \times 3 = 36$ dollars. But what if you don't know how many pizzas you need yet? You can write:

Cost = $12 × number of pizzas

Or using algebra: $C = 12n$

where:

$C$ is the total cost
$n$ is the number of pizzas

Now you have a formula that works for any number of pizzas! This is the power of variables.

Common Variable Names

The following are common conventions for variable names:

$x$, $y$, $z$ - commonly used for unknown values
$a$, $b$, $c$ - often used for coefficients (numbers multiplying variables)
$n$ - frequently used for counting numbers
$t$ - often represents time
$d$ - commonly represents distance
$r$ - frequently used for rate or radius

You can use any letter as a variable, but these conventions help communicate meaning clearly.

Diagram: Variable Types Interactive Infographic

<summary>Variable Types Interactive Infographic</summary>
Type: infographic

Purpose: Create an interactive visual guide showing different types of variables and their uses in real-world contexts

Layout: 2x2 grid showing four common variable categories

Categories:
1. Position Variables ($x$, $y$, $z$)
   - Icon: Coordinate grid with point marked
   - Hover text: "Used to represent positions, locations, or unknown quantities in equations"
   - Example shown: "Finding x: $2x + 5 = 13$"

2. Time Variables ($t$)
   - Icon: Clock or timeline
   - Hover text: "Represents time in motion problems, growth models, and sequences"
   - Example shown: "Distance after t hours: $d = 55t$"

3. Counting Variables ($n$)
   - Icon: Stack of items with counter
   - Hover text: "Used for discrete quantities like items, steps, or people"
   - Example shown: "Cost of n pizzas: $C = 12n$"

4. Rate Variables ($r$, $m$)
   - Icon: Speedometer or slope indicator
   - Hover text: "Represents rates of change, speed, or slopes"
   - Example shown: "Simple interest: $I = Prt$"

Interactive elements:
- Hover over each category to see expanded description
- Click to reveal 2-3 additional real-world examples
- Color coding: Blue for position, green for time, orange for counting, purple for rates

Visual style: Modern card-based layout with icons and clear typography
Color scheme: Use distinct colors for each category as specified above

Implementation: HTML/CSS/JavaScript with hover states and click-to-expand functionality

Constants: Values That Stay the Same

A constant is a value that doesn't change. In algebraic expressions, constants are usually numbers written by themselves.

Examples of Constants

In everyday life and mathematics, you encounter many constants:

The speed of light: approximately 299,792,458 meters per second
Pi ($\pi$): approximately 3.14159... (the ratio of a circle's circumference to its diameter)
The number of days in a week: 7
The price of an item (at a specific time): $15

In the expression $3x + 7$, the numbers 3 and 7 are constants because they don't change, while $x$ is a variable because its value can vary.

Mathematical Constants

Some constants appear so frequently in mathematics that they have special names:

Constant	Symbol	Approximate Value	Meaning
Pi	$\pi$	3.14159...	Ratio of circumference to diameter
Euler's number	$e$	2.71828...	Base of natural logarithms
Golden ratio	$\phi$	1.61803...	Special ratio found in nature

Coefficients: Numbers That Multiply Variables

A coefficient is a number that multiplies a variable. Coefficients tell us "how many" of a variable we have.

Identifying Coefficients

Let's look at the expression $5x + 3y - 2$

The coefficient of $x$ is 5 (we have 5 x's)
The coefficient of $y$ is 3 (we have 3 y's)
The number -2 is a constant (no variable)

Special Coefficients

Sometimes coefficients are not written explicitly:

When you see $x$ by itself, the coefficient is 1 (we write $x$ instead of $1x$)
When you see $-x$, the coefficient is -1 (we write $-x$ instead of $-1x$)

Understanding coefficients helps you combine like terms and solve equations efficiently.

Terms: Building Blocks of Expressions

A term is a single number, variable, or the product of numbers and variables. Terms are separated by plus (+) or minus (-) signs.

Anatomy of a Term

Consider the term $-4x^2$

This term has three parts:

Sign: negative (-)
Coefficient: 4
Variable part: $x^2$

Counting Terms

How many terms are in each expression?

$5x + 3$ has 2 terms
$2x - 7y + 9$ has 3 terms
$x^2 + 3x - 4x + 8$ has 4 terms
$15$ has 1 term (called a monomial)

Understanding terms helps you organize expressions and perform operations correctly.

Run the Expression Explorer Fullscreen

Diagram: Expression Explorer MicroSim

<summary>Expression Explorer MicroSim</summary>
Type: MicroSim

Learning objective: Help students identify and understand the different parts of an algebraic expression interactively

Canvas layout (800x600px):
- Top section (800x100): Title and instructions
- Left side (600x450): Main visualization area
- Right side (200x450): Control panel
- Bottom (800x50): Current analysis display

Visual elements:
- Display a randomly generated algebraic expression in large font
- Color-code different parts:
  - Coefficients in blue
  - Variables in green
  - Constants in orange
  - Operation signs in gray
- Highlight current selection with yellow background

Interactive controls:
- Button: "New Expression" (generates new random expression)
- Dropdown: "Expression Complexity" (Simple, Medium, Complex)
- Checkbox options to show/hide:
  - "Show Coefficients"
  - "Show Variables"
  - "Show Constants"
  - "Show Terms"
- Button: "Check My Understanding" (quiz mode)

Default parameters:
- Complexity: Simple
- All checkboxes checked
- Starting expression: $3x + 5y - 7$

Behavior:
- User can hover over any part to see its classification
- Click on checkboxes to highlight only selected element types
- "New Expression" button generates expressions based on complexity:
  - Simple: 2-3 terms, single variables, small coefficients
  - Medium: 3-5 terms, may include exponents, larger coefficients
  - Complex: 4-6 terms, multiple variables, exponents, negative terms
- "Check My Understanding" mode: Shows expression without colors, user clicks to identify parts, system provides feedback

Display area shows:
- Number of terms
- List of coefficients found
- List of variables found
- List of constants found

Implementation notes:
- Use p5.js for rendering
- Store expression as parsed components
- Generate random expressions using arrays of coefficients, variables, and operations
- Track user interactions for assessment

Expressions: Putting It All Together

An expression is a mathematical phrase that can contain numbers, variables, operators, and grouping symbols. Unlike equations, expressions don't have an equals sign—they represent a value but don't make a statement about equality.

Examples of Expressions

The following are valid algebraic expressions:

$5$
$x + 3$
$2a - 7b + 9$
$\frac{1}{2}x^2 + 3x - 4$
$(2x + 1)(x - 3)$

Parts of an Expression

Every expression contains:

Terms: Parts separated by + or - signs
Coefficients: Numbers multiplying variables
Variables: Letters representing unknown values
Constants: Numbers standing alone

Understanding the structure of expressions helps you manipulate them correctly and solve problems.

Equations: Making Mathematical Statements

An equation is a mathematical statement that two expressions are equal. Equations always contain an equals sign (=).

Equations vs. Expressions

The key difference:

Expression: $3x + 5$ (represents a value)
Equation: $3x + 5 = 17$ (states that two things are equal)

Equations make claims that you can verify or solve. When you solve an equation, you find the value(s) of the variable that make the equation true.

Examples of Equations

Here are some equations you'll work with:

$x + 7 = 12$ (simple linear equation)
$2y - 3 = y + 5$ (equation with variables on both sides)
$x^2 = 16$ (quadratic equation)
$\frac{x}{4} + 2 = 9$ (equation with fractions)

Each equation is asking the question: "What value makes this statement true?"

Inequalities: Greater Than and Less Than

An inequality is a mathematical statement that compares two expressions using inequality symbols instead of an equals sign.

Inequality Symbols

The following symbols are used to write inequalities:

Symbol	Meaning	Example	Read As
$<$	Less than	$x < 5$	"$x$ is less than 5"
$>$	Greater than	$x > 3$	"$x$ is greater than 3"
$\leq$	Less than or equal to	$x \leq 10$	"$x$ is less than or equal to 10"
$\geq$	Greater than or equal to	$x \geq 0$	"$x$ is greater than or equal to 0"
$\neq$	Not equal to	$x \neq 7$	"$x$ is not equal to 7"

Real-World Inequalities

Inequalities appear frequently in real situations:

"You must be at least 13 years old to have a social media account": $\text{age} \geq 13$
"The speed limit is 65 mph": $\text{speed} \leq 65$
"The store needs more than 50 customers today to meet its goal": $\text{customers} > 50$

Unlike equations (which usually have one or a few specific solutions), inequalities typically have infinitely many solutions forming a range of values.

Order of Operations: The Rules of the Game

The Order of Operations is a set of rules that tells us the correct sequence for performing calculations in expressions with multiple operations.

PEMDAS: A Memory Tool

The acronym PEMDAS helps remember the order:

Parentheses
Exponents
Multiplication and Division (left to right)
Addition and Subtraction (left to right)

Some people remember this as "Please Excuse My Dear Aunt Sally."

Why Order Matters

Consider the expression: $3 + 4 \times 2$

Wrong approach (left to right): $3 + 4 = 7$, then $7 \times 2 = 14$
Correct approach (PEMDAS): $4 \times 2 = 8$, then $3 + 8 = 11$

The order of operations ensures everyone gets the same answer!

Examples of Order of Operations

Let's evaluate: $2 + 3^2 \times 4 - 5$

Following PEMDAS:

Exponents: $3^2 = 9$, giving us $2 + 9 \times 4 - 5$
Multiplication: $9 \times 4 = 36$, giving us $2 + 36 - 5$
Addition and Subtraction (left to right): $2 + 36 = 38$, then $38 - 5 = 33$

Answer: 33

Diagram: Order of Operations Challenge MicroSim

<summary>Order of Operations Challenge MicroSim</summary>
Type: microsim

Learning objective: Practice applying the order of operations (PEMDAS) through interactive step-by-step problem solving

Canvas layout (1000x700px):
- Top section (1000x150): Problem display area with large expression
- Middle section (700x400): Step-by-step work area showing calculation stages
- Right side (300x400): PEMDAS reference guide and controls
- Bottom (1000x150): Input area and feedback

Visual elements:
- Large expression display with color-coded operations
- Step-by-step visualization showing transformation at each stage
- Animated arrows showing which operation is being performed
- Visual PEMDAS reminder chart
- Progress indicator showing current step
- Score tracker and streak counter

Interactive controls:
- Dropdown: "Difficulty Level" (Easy, Medium, Hard)
- Button: "New Problem"
- Button: "Show Next Step" (reveals next calculation step)
- Button: "Show Answer" (reveals full solution)
- Input field: "Your Answer"
- Button: "Check Answer"
- Checkbox: "Step-by-step mode" (requires identifying next operation)

Default parameters:
- Difficulty: Easy
- Step-by-step mode: Off
- Starting expression: $5 + 3 \times 2$

Difficulty levels:
- Easy: 3-4 operations, parentheses, simple exponents (like $2^2$)
  Example: $10 - 2 \times 3 + 4$
- Medium: 4-6 operations, nested parentheses, exponents up to $3^3$
  Example: $(8 + 2) \times 3 - 4^2 \div 2$
- Hard: 6-8 operations, multiple nested parentheses, fractions, negative numbers
  Example: $3 + 4 \times (2 + 3)^2 \div 5 - 2 \times 3$

Behavior:
- Display expression with operations color-coded by PEMDAS level
- In step-by-step mode, highlight the next operation to perform
- User selects which operation should be done next
- System provides immediate feedback (correct/incorrect)
- "Show Next Step" reveals and animates the next calculation
- Expression simplifies with each step until final answer
- Track accuracy and time for each problem
- Provide encouraging feedback and explanations for mistakes

Visual styling:
- Parentheses: Red highlight
- Exponents: Purple highlight
- Multiplication/Division: Blue highlight
- Addition/Subtraction: Green highlight
- Current operation: Yellow animated pulse

PEMDAS reference panel shows:
- Acronym with full words
- Memory aid: "Please Excuse My Dear Aunt Sally"
- Icon for each operation type
- Note: "Multiply/Divide left to right, Add/Subtract left to right"

Implementation notes:
- Use p5.js for rendering
- Parse expressions into operation tree
- Animate transitions between steps
- Store problem bank for each difficulty level
- Track student progress and common mistakes

Evaluating Expressions: Finding the Value

Evaluating an expression means finding its numerical value by substituting specific numbers for variables and performing the calculations using the order of operations.

The Evaluation Process

To evaluate an expression:

Substitute the given values for all variables
Apply the order of operations (PEMDAS)
Simplify step by step to find the final value

Example: Evaluating an Expression

Evaluate $3x + 5$ when $x = 4$

Step 1: Substitute $3(4) + 5$

Step 2: Multiply $12 + 5$

Step 3: Add $17$

Answer: 17

Multiple Variables Example

Evaluate $2a - 3b + 7$ when $a = 5$ and $b = 2$

Step 1: Substitute $2(5) - 3(2) + 7$

Step 2: Multiply (left to right) $10 - 6 + 7$

Step 3: Subtract and Add (left to right) $4 + 7 = 11$

Answer: 11

Substitution: Replacing Variables with Values

Substitution is the process of replacing variables in an expression or equation with specific numerical values. You've already seen substitution in action when evaluating expressions.

Why Substitution Matters

Substitution allows us to:

Evaluate expressions for specific cases
Check if a value is a solution to an equation
Test formulas with real-world data
Simplify complex expressions by replacing one variable at a time

Using Substitution to Check Solutions

Is $x = 6$ a solution to the equation $2x - 3 = 9$?

Substitute $x = 6$: $2(6) - 3 = 9$

Evaluate left side: $12 - 3 = 9$

Check: $9 = 9$ ✓ True!

Yes, $x = 6$ is a solution.

Substitution with Formulas

The area of a triangle is given by the formula:

Area of a Triangle

$A = \frac{1}{2}bh$

where:

$A$ is the area
$b$ is the base
$h$ is the height

Find the area when $b = 10$ and $h = 6$:

Substitute: $A = \frac{1}{2}(10)(6)$

Multiply: $A = \frac{1}{2}(60)$

Simplify: $A = 30$

The area is 30 square units.

Like Terms: Terms That Match

Like terms are terms that have exactly the same variable parts (same variables raised to the same powers). Only the coefficients can be different.

Identifying Like Terms

Look at these examples:

Like terms: - $3x$ and $7x$ (both have the variable $x$) - $-5y^2$ and $2y^2$ (both have $y^2$) - $4ab$ and $-ab$ (both have $ab$)

NOT like terms: - $3x$ and $3x^2$ (different exponents) - $5x$ and $5y$ (different variables) - $2xy$ and $2x$ (different variable combinations)

Why Like Terms Matter

You can only combine terms that are like terms. Think of it like fruit: you can add 3 apples + 5 apples = 8 apples, but you can't directly combine 3 apples + 5 oranges into a single number of one fruit.

Combining Like Terms: Simplifying by Adding

Combining like terms means adding or subtracting the coefficients of like terms while keeping the variable part unchanged. This is one of the most important skills in algebra for simplifying expressions.

How to Combine Like Terms

Step 1: Identify like terms Step 2: Add or subtract their coefficients Step 3: Keep the variable part the same

Example: Combining Like Terms

Simplify: $5x + 3 + 2x - 7$

Identify like terms: - $x$ terms: $5x$ and $2x$ - Constant terms: $3$ and $-7$

Combine $x$ terms: $5x + 2x = 7x$

Combine constants: $3 - 7 = -4$

Final answer: $7x - 4$

Multiple Variables Example

Simplify: $4x + 3y - 2x + 5y - 1$

Group like terms: - $x$ terms: $4x - 2x = 2x$ - $y$ terms: $3y + 5y = 8y$ - Constants: $-1$

Final answer: $2x + 8y - 1$

Diagram: Like Terms Matching Game MicroSim

<summary>Like Terms Matching Game MicroSim</summary>
Type: microsim

Learning objective: Help students practice identifying and combining like terms through an interactive matching and simplification game

Canvas layout (1000x700px):
- Top section (1000x100): Instructions and score display
- Main area (1000x500): Two-column matching interface or expression simplification area
- Bottom section (1000x100): Feedback area and controls

Game modes (selectable):
1. **Matching Mode**: Drag-and-drop matching of like terms
2. **Simplification Mode**: Combine like terms in expressions
3. **Challenge Mode**: Timed expression simplification

Visual elements for Matching Mode:
- Left column: 8-10 terms in boxes (e.g., $3x$, $5y$, $-2x$, $7$, $4y$, $x^2$, $-8$, $2x^2$)
- Right column: Empty "buckets" labeled "x terms", "y terms", "x² terms", "constants"
- Terms can be dragged to appropriate buckets
- Correct matches turn green, incorrect turn red with shake animation
- Connecting lines show which terms can combine

Visual elements for Simplification Mode:
- Display an unsimplified expression: $4x + 3 - 2x + 5 + x - 1$
- Color-code like terms with matching highlight colors
- Show work area where terms can be grouped
- Input field for final simplified answer
- Step-by-step verification available

Interactive controls:
- Dropdown: "Game Mode" (Matching, Simplification, Challenge)
- Dropdown: "Difficulty" (Easy, Medium, Hard)
- Button: "New Problem"
- Button: "Show Hint" (highlights one set of like terms)
- Button: "Check Answer"
- Button: "Show Solution"
- Timer display (for Challenge Mode)
- Score and accuracy tracker

Default parameters:
- Mode: Matching
- Difficulty: Easy
- Time limit (Challenge): 60 seconds

Difficulty levels:
- Easy: 2 variables (x, y), constants, 6-8 terms total
- Medium: 3 variables, exponents ($x^2$), 8-12 terms, some negative coefficients
- Hard: 4+ variables, various exponents, 10-15 terms, negative coefficients, fractions

Behavior - Matching Mode:
- User drags terms to buckets
- Immediate feedback on correct/incorrect placement
- Once all terms correctly sorted, show combined results
- Visual animation of coefficient addition
- Confetti or celebration on completion

Behavior - Simplification Mode:
- Display expression with terms in random order
- User can click terms to highlight/group them
- Color coding shows which terms are like terms
- Input simplified expression
- System checks coefficient addition and final form
- Provide specific feedback on errors

Behavior - Challenge Mode:
- Present 5 expressions to simplify within time limit
- Increasing difficulty with each correct answer
- Point multiplier for speed
- Streak bonuses for consecutive correct answers
- Leaderboard showing personal best

Visual styling:
- Draggable terms: Cards with shadows and hover effects
- Like terms: Matching background colors (blue, green, orange, purple)
- Correct answers: Green glow animation
- Incorrect answers: Red shake animation
- Clean, modern interface with clear typography

Feedback messages:
- Correct: "Great job! $5x + 2x = 7x$"
- Incorrect: "Not quite. Remember, $x$ and $x^2$ are not like terms."
- Hint: "Look for terms with the same variable and exponent."
- Completion: "Excellent! You simplified the expression correctly!"

Implementation notes:
- Use p5.js for rendering and interaction
- Implement drag-and-drop with mouse/touch support
- Generate random expressions with controlled complexity
- Parse expressions and identify like terms programmatically
- Store terms as objects with coefficient, variable, and exponent properties
- Track timing, accuracy, and completion metrics

Simplifying Expressions: Making Them Cleaner

Simplifying an expression means rewriting it in the simplest possible form by combining like terms, performing operations, and reducing complexity while keeping the same mathematical value.

Why Simplify?

Simplified expressions are:

Easier to read and understand
Faster to evaluate
Less prone to errors in later calculations
Better for identifying patterns

The Simplification Process

To simplify an expression:

Remove parentheses using the distributive property (if needed)
Combine like terms
Write terms in standard order (usually highest to lowest exponent)

Example: Simplifying with Multiple Steps

Simplify: $3(x + 2) + 4x - 5$

Step 1: Distribute $3x + 6 + 4x - 5$

Step 2: Identify like terms - $x$ terms: $3x$ and $4x$ - Constants: $6$ and $-5$

Step 3: Combine like terms - $3x + 4x = 7x$ - $6 - 5 = 1$

Final simplified form: $7x + 1$

Order Matters for Readability

Standard form typically lists:

Terms from highest to lowest exponent
Variables in alphabetical order
Constant term last

For example, write $5 + 3x^2 - 2x$ as $3x^2 - 2x + 5$

Expanding Expressions: Removing Parentheses

Expanding an expression means removing parentheses by applying the distributive property and multiplying terms. Expanding is the opposite of factoring.

The Distributive Property

The distributive property states:

Distributive Property

$a(b + c) = ab + ac$

where:

$a$ is the factor being distributed
$b$ and $c$ are terms inside parentheses

This property allows us to multiply a number or variable by everything inside parentheses.

Expanding Examples

Example 1: Expand $4(x + 3)$

$4(x + 3) = 4x + 12$

Example 2: Expand $-2(3y - 5)$

$-2(3y - 5) = -6y + 10$

(Note: $-2 \times (-5) = +10$)

Example 3: Expand $x(x + 7)$

$x(x + 7) = x^2 + 7x$

When to Expand vs. Factor

Expand when you need to combine like terms or simplify further
Factor when you need to solve equations or identify common factors

Both skills are important and complement each other.

Diagram: Distributive Property Visualizer MicroSim

<summary>Distributive Property Visualizer MicroSim</summary>
Type: microsim

Learning objective: Visualize the distributive property using area models and step-by-step algebraic expansion

Canvas layout (900x700px):
- Top (900x100): Problem display and mode selector
- Left side (450x500): Visual area model representation
- Right side (450x500): Algebraic step-by-step solution
- Bottom (900x100): Controls and input area

Visual elements - Area Model:
- Rectangle divided into sections representing multiplication
- For $3(x + 5)$:
  - Vertical side labeled "3"
  - Horizontal side divided into two sections: "x" and "5"
  - Rectangle divided showing two areas: "3x" and "15"
- Color coding: Variable areas in blue, constant areas in orange
- Dimensions labeled clearly
- Areas labeled with expressions

Visual elements - Algebraic:
- Step-by-step expansion shown with arrows
- Original expression at top
- Intermediate step showing distribution
- Final expanded form at bottom
- Color coding matching area model

Interactive controls:
- Input: "Factor outside parentheses" (e.g., 3, -2, x)
- Input: "First term inside" (e.g., x, 2y, 3)
- Input: "Second term inside" (e.g., 5, -2, x)
- Button: "Generate Random Problem"
- Dropdown: "Problem Type" (Numeric, Single Variable, Two Variables)
- Slider: "Animation Speed" (slow to fast)
- Button: "Show Next Step" (step through expansion)
- Button: "Animate Full Solution"
- Checkbox: "Show Area Model"

Default parameters:
- Problem: $3(x + 5)$
- Problem Type: Single Variable
- Animation Speed: Medium
- Show Area Model: Checked

Problem types:
- Numeric: $4(3 + 2)$ → area model with numbers only
- Single Variable: $3(x + 5)$, $-2(y - 4)$ → one variable
- Two Variables: $2(3x + 4y)$, $x(x + 7)$ → multiple variables or exponents

Behavior - Area Model:
- Draw rectangle with animated dimensions
- Divide rectangle according to terms
- Fill each section with color and label
- Animate the "sweeping" of distributed factor across terms
- Show area calculations for each section
- Highlight how areas add to total

Behavior - Algebraic Steps:
- Display original expression: $3(x + 5)$
- Show distribution arrows: $3 \times x$ and $3 \times 5$
- Animate each multiplication
- Display intermediate: $3x + 15$
- Highlight final answer

Behavior - Animation:
- Sync visual and algebraic representations
- Pulse/glow effect on current operation
- Smooth transitions between steps
- Option to pause/play at any point

Special cases to demonstrate:
- Positive factor: $3(x + 2) = 3x + 6$
- Negative factor: $-2(x + 5) = -2x - 10$
- Subtraction inside: $4(x - 3) = 4x - 12$
- Variable factor: $x(x + 4) = x^2 + 4x$

Practice mode:
- Show problem and area model
- Student fills in blanks for expanded form
- Immediate feedback on each term
- Visual confirmation in area model

Visual styling:
- Clean geometric area models with clear labels
- Color-coded sections (blue for variables, orange for constants)
- Large, readable algebraic notation
- Animated arrows showing distribution
- Grid background for area model

Implementation notes:
- Use p5.js for rendering
- Draw dynamic rectangles based on expression complexity
- Parse input expressions to generate appropriate models
- Implement smooth animations with easing
- Support negative values with special visual treatment (different colors/patterns)

Monomials: Single-Term Expressions

A monomial is an algebraic expression with only one term. Monomials are the simplest type of polynomial.

Characteristics of Monomials

A monomial can be:

A constant: $7$, $-3$, $\frac{1}{2}$
A variable: $x$, $y$
A product of constants and variables: $5x$, $-3y^2$, $2xy$, $4x^2y^3$

What is NOT a Monomial

These are not monomials because they have more than one term:

$x + 5$ (binomial - two terms)
$2x - 3y + 7$ (trinomial - three terms)

Identifying Monomials

The following table shows examples and non-examples:

Expression	Monomial?	Reason
$8$	Yes	Single constant
$x^3$	Yes	Single variable with exponent
$-5xy$	Yes	Single product of coefficient and variables
$3x + 2$	No	Two terms separated by +
$\frac{4}{x}$	No	Variable in denominator (not a polynomial)
$7x^2y^3$	Yes	Single product with multiple variables

Understanding monomials helps you recognize more complex polynomials and perform operations like multiplication and division.

Prime Factorization: Breaking Numbers Apart

Prime factorization is the process of breaking a number down into its prime factors—the prime numbers that multiply together to give the original number.

What are Prime Numbers?

A prime number is a number greater than 1 that has exactly two factors: 1 and itself.

Prime numbers: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, ...

Composite numbers have more than two factors: 4, 6, 8, 9, 10, 12, ...

Finding Prime Factorization

Use a factor tree to break down numbers systematically:

Example: Find the prime factorization of 24

Prime factorization of 24: $2 \times 2 \times 2 \times 3 = 2^3 \times 3$

Why Prime Factorization Matters in Algebra

Prime factorization helps you:

Find greatest common factors (GCF)
Simplify fractions
Factor algebraic expressions
Solve certain equations

Diagram: Prime Factorization Tree Builder MicroSim

<summary>Prime Factorization Tree Builder MicroSim</summary>
Type: microsim

Learning objective: Practice building factor trees to find prime factorization of composite numbers through interactive tree construction

Canvas layout (900x700px):
- Top (900x100): Number to factor, instructions, and score
- Main area (900x500): Interactive factor tree workspace
- Bottom (900x100): Controls, prime number reference, and feedback

Visual elements:
- Root node at top showing the number to factor
- Branching tree structure growing downward
- Circular nodes for each number in factorization
- Connecting lines between parent and child nodes
- Prime numbers highlighted in gold/green
- Composite numbers in blue (can be split further)
- Final prime factorization displayed at bottom

Interactive controls:
- Input field: "Enter a number to factor" (range 4-200)
- Button: "Start New Problem"
- Button: "Random Number"
- Dropdown: "Difficulty" (Easy: 4-50, Medium: 50-100, Hard: 100-200)
- Button: "Give Hint" (highlights one composite number that can be split)
- Button: "Check Tree" (verifies if factorization is complete)
- Button: "Show Answer"
- Display: Prime number reference list (primes up to 20)

Default parameters:
- Number: 24
- Difficulty: Easy
- Tree partially built or blank (selectable)

Behavior:
- User clicks on a composite number node
- Input boxes appear below asking for two factors
- User enters two factors that multiply to the number
- System checks if factors are correct
- If correct: creates two child nodes with those factors
- If incorrect: shakes and shows error message
- Prime numbers automatically highlighted and cannot be split
- When all nodes are prime, tree is complete

Visual feedback:
- Correct factors: Smooth animation creating child nodes
- Incorrect factors: Red shake animation, error message
- Prime numbers: Gold border and fill color
- Composite numbers: Blue with "click to factor" cursor
- Completed tree: Celebration animation

Tree visualization:
- Hierarchical layout with proper spacing
- Nodes arranged to avoid overlap
- Animated growth as factors are added
- Lines connecting parent to children
- Node size based on number magnitude (optional)

Final answer display:
- Show prime factorization in exponential form
- Example: "$24 = 2^3 \times 3$"
- List all prime factors with multiplicity
- Option to copy prime factorization

Hint system:
- Level 1: Highlights one composite number
- Level 2: Shows that number is divisible by 2, 3, or 5
- Level 3: Shows one factor
- Level 4: Shows both factors

Game modes:
1. **Free Build**: User factors any number step by step
2. **Guided Mode**: System suggests next number to factor
3. **Challenge Mode**: Timed factorization race
4. **Quiz Mode**: Factor 5 numbers, track accuracy

Prime reference panel:
- List of primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
- Divisibility rules reminder:
  - 2: Even numbers
  - 3: Sum of digits divisible by 3
  - 5: Ends in 0 or 5

Special features:
- Compare different factorization paths (different factor pairs lead to same prime factorization)
- Show multiple factor trees for same number side by side
- Verify uniqueness of prime factorization

Visual styling:
- Clean tree diagram with smooth curves
- Color-coded by number type (prime vs composite)
- Clear, large fonts for readability
- Hover effects showing factor possibilities
- Celebration effects on completion (confetti, glow)

Implementation notes:
- Use p5.js for rendering
- Calculate tree layout dynamically to prevent overlap
- Store tree structure as nested objects
- Implement prime checking algorithm
- Track student's factorization path
- Provide specific feedback on common mistakes (e.g., non-factors, missed primes)

Putting It All Together: Chapter Summary

In this chapter, you've learned the fundamental vocabulary and concepts of algebra. Let's review the key ideas:

Core Concepts Mastered

You now understand:

Numbers: The different types (natural, whole, integers, rational, irrational) and their properties
Variables: Symbols representing unknown or changing values
Constants: Fixed values that don't change
Coefficients: Numbers that multiply variables
Terms: Building blocks of expressions
Expressions: Mathematical phrases combining numbers, variables, and operations
Equations: Statements of equality between two expressions
Inequalities: Comparisons using <, >, ≤, ≥, or ≠

Essential Skills Developed

You can now:

Apply the Order of Operations (PEMDAS) correctly
Evaluate expressions by substituting values for variables
Use substitution to test solutions
Identify like terms and combine them correctly
Simplify expressions by combining like terms
Expand expressions using the distributive property
Recognize monomials as single-term expressions
Find prime factorization using factor trees

Looking Ahead

These foundational skills will be used throughout your algebra journey. In upcoming chapters, you'll:

Solve equations and inequalities
Work with functions and graphs
Master polynomials and factoring
Explore quadratic and exponential relationships
Apply algebra to real-world problems

The vocabulary and techniques you've learned in this chapter are the language you'll use to communicate mathematically in all future topics. Practice these skills regularly—they're the foundation for everything that follows!

Key Takeaways

Remember these essential ideas:

Algebra uses variables to represent unknown values, making it possible to solve problems generally
The Order of Operations (PEMDAS) ensures consistent calculations
Like terms can be combined; unlike terms cannot
Simplifying makes expressions cleaner and easier to work with
Every algebraic concept builds on the ones before it—master the basics first!

Continue to the next chapter to begin solving equations and applying these algebraic foundations to real problems.

References

Order of Operations - PEMDAS - Math is Fun - Interactive explanation of PEMDAS with visual examples and common mistakes, perfect for understanding why operation order matters in evaluating algebraic expressions.
Simplifying Algebraic Expressions - 2024 - Mathematics LibreTexts - Comprehensive guide to the distributive property and combining like terms with extensive worked examples progressing from basic to complex expressions.

Symbol	Meaning	Example	Read As
\(<\)	Less than	\(x < 5\)	"\(x\) is less than 5"
\(>\)	Greater than	\(x > 3\)	"\(x\) is greater than 3"
\(\leq\)	Less than or equal to	\(x \leq 10\)	"\(x\) is less than or equal to 10"
\(\geq\)	Greater than or equal to	\(x \geq 0\)	"\(x\) is greater than or equal to 0"
\(\neq\)	Not equal to	\(x \neq 7\)	"\(x\) is not equal to 7"

Constant	Symbol	Approximate Value	Meaning
Pi	\(\pi\)	3.14159...	Ratio of circumference to diameter
Euler's number	\(e\)	2.71828...	Base of natural logarithms
Golden ratio	\(\phi\)	1.61803...	Special ratio found in nature

Expression	Monomial?	Reason
\(8\)	Yes	Single constant
\(x^3\)	Yes	Single variable with exponent
\(-5xy\)	Yes	Single product of coefficient and variables
\(3x + 2\)	No	Two terms separated by +
\(\frac{4}{x}\)	No	Variable in denominator (not a polynomial)
\(7x^2y^3\)	Yes	Single product with multiple variables