Wayfinder GED — Algebra Tier Map

Foundations, Expressions, Equations, Inequalities, Functions — all tiers with prerequisites.

PurposeComplete prerequisite-ordered curriculum map for GED Algebra.

StatusActive Development — real, complete pedagogical content; not yet run with real students

Superseded by—

EvidenceFull document reproduced below, unedited from source.

GED TIER MAP: ALGEBRA 1

Complete Curriculum Structure — Integer Operations Through Functions

Version: 1.3 — May 2026
Source: Lacefield Adaptive Learning System
Status: Ready for assignment generation and concept sequencing

PART 1: ALGEBRA 1 STRUCTURE

Five main sections: 1. Foundations — integers, order of operations, equals sign, variables, properties 2. Expressions & Equations — combining like terms, solving one-step through multi-step equations 3. Linear Functions — slope, intercepts, graphing, writing equations 4. Systems — solving systems of two equations 5. Polynomials & Quadratics — factoring, solving quadratic equations, functions

Load-bearing concepts (master these first): - The equals sign as balance (concept ALG-F-05) - Variables as unknowns vs. varying quantities (ALG-F-08 and ALG-F-09) - Inverse operations (ALG-F-07) - Order of operations (ALG-F-03)

Misconceptions cluster (most common): - Integer sign operations (negative × negative = positive) - Fraction operations (especially LCD) - Order of operations (do left-to-right instead of respecting priority) - Literal equations (solving for a variable in terms of others) - Solving equations (adding to same side instead of moving to opposite side)

PART 2: FOUNDATIONS (ALG-F)

Tier 1: Integer Operations & Number Systems

Concept ALG-F-01: Integer Operations ★ CORE

What it is: Addition, subtraction, multiplication, and division of positive and negative integers. The number line as a model for signed quantity. Absolute value as distance from zero.

Sign rules: - Same signs → positive result (3 + 5 = 8; −3 + −5 = −8; 3 × 5 = 15; −3 × −5 = 15) - Different signs → negative result (3 − 5 = −2; 3 × −5 = −15)

Absolute value: Distance from zero; always non-negative. |−5| = 5; |5| = 5

Misconceptions: - MIS-ALG-01: “Negative × negative = negative” — Wrong. Negative × negative = positive. - MIS-ALG-02: “Subtraction of negatives is confusing” — Subtracting a negative is the same as adding the opposite: 5 − (−3) = 5 + 3 = 8 - MIS-ALG-03: “Absolute value makes everything positive” — |−5| = 5, but |−5| ≠ negative anything. It’s always non-negative distance.

Prerequisite: None — foundational skill

Concept ALG-F-02: Rational Numbers & the Number Line

What it is: Fractions, decimals, and percentages as representations of the same quantity. Placement on the number line. Density of rationals (between any two rationals is another rational). Distinction between rational and irrational.

Examples: - 1/4 = 0.25 = 25% (all represent the same quantity) - Between 1/3 and 1/2 is 5/12 (density property) - √2 ≈ 1.414… is irrational (non-repeating, non-terminating)

Prerequisite: ALG-F-01

Tier 1: Conventions & Properties

Concept ALG-F-03: Order of Operations ★ CORE

What it is: The conventional priority ordering for evaluating expressions: 1. Parentheses (and other grouping symbols) 2. Exponents 3. Multiplication/Division (left to right) 4. Addition/Subtraction (left to right)

Mnemonic: PEMDAS

Misconceptions: - MIS-ALG-04: “Always do addition before subtraction” — No. Left to right for addition and subtraction at the same level. 10 − 3 + 2 = 7 + 2 = 9 (not 10 − 5) - MIS-ALG-05: “Multiplication always comes before division” — No. Left to right: 12 ÷ 3 × 2 = 4 × 2 = 8 (not 12 ÷ 6) - MIS-ALG-06: “Exponents apply to the sign” — (−3)² = 9, but −3² = −9. Parentheses matter.

Prerequisite: ALG-F-01

Concept ALG-F-04: Exponents & Powers

What it is: Repeated multiplication notation. Base and exponent. x³ = x · x · x

Key properties: - Zero exponent: a⁰ = 1 (for any nonzero a) - Negative exponent: a⁻ⁿ = 1/aⁿ - Same base multiplication: aᵐ · aⁿ = aᵐ⁺ⁿ - Quotient: aᵐ ÷ aⁿ = aᵐ⁻ⁿ

Misconceptions: - MIS-ALG-07: “2³ = 2 · 3” — No. 2³ = 2 · 2 · 2 = 8 - MIS-ALG-08: “2⁻³ is negative” — No. 2⁻³ = 1/8 (positive) - MIS-ALG-09: “(−2)⁴ = −16” — No. (−2)⁴ = 16 (even exponent makes result positive)

Prerequisite: ALG-F-03

Tier 2: The Algebra Foundation

Concept ALG-F-05: The Equals Sign as Balance ★ CORE

What it is: The equals sign asserts that two expressions represent the same quantity — it is a relational symbol, not a command to “compute” or “move things around.” An equation is a scale in perfect balance: both sides weigh the same.

This is the most load-bearing concept in all of algebra. If students do not understand this, every subsequent solving technique is magical nonsense to them.

Examples: - 3 + 5 = 8 means (3 + 5) and 8 are the same quantity - 2x + 3 = 11 means (2x + 3) and 11 are the same quantity - The equation is balanced; do anything to both sides to maintain balance

Misconceptions: - MIS-ALG-10: “The equals sign means do what’s on the left and write the answer on the right” — No. The equals sign says both sides are identical in value. - MIS-ALG-11: “You can only move things to the other side of the equals sign” — You can do anything to an equation AS LONG AS you do it to both sides equally.

Prerequisite: ALG-F-01

Concept ALG-F-06: Substitution Property of Equality

What it is: If two expressions are equal, one may be replaced by the other anywhere it appears without changing the truth of any statement containing it.

If a = b, then b can replace a in any expression.

Why it matters: This makes solving by substitution, evaluating expressions, and replacing variables with values all logically valid operations.

Prerequisite: ALG-F-05

Concept ALG-F-07: Inverse Operations ★ CORE

What it is: An inverse operation undoes another operation and returns to the starting value.

Addition and subtraction are inverses
Multiplication and division are inverses
Squaring and square-rooting are inverses (with sign considerations)

This is the mechanism behind every algebraic solving technique — isolating a variable means systematically undoing the operations applied to it, in reverse order.

Example: If 2x + 3 = 11: 1. Subtract 3 from both sides (undo addition): 2x = 8 2. Divide by 2 (undo multiplication): x = 4

Prerequisite: ALG-F-03, ALG-F-05

Concept ALG-F-08: Variables as Unknown Quantities ★ CORE

What it is: A variable is a placeholder for a specific unknown value. In a given equation, x has one value (or a specific set of values). The variable represents what we do not yet know.

Example: In 2x + 3 = 11, x is the unknown. It’s not “any number” — it’s one specific number that makes the equation true (x = 4).

Misconceptions: - MIS-ALG-12: “x can be any number” — In an equation, x is one specific value (or a specific set of values) - MIS-ALG-13: “x stands for a general number” — No. In this context, x is an unknown with a specific value

Prerequisite: ALG-F-05

Concept ALG-F-09: Variables as Varying Quantities

What it is: In a function or formula context, a variable can represent a quantity that changes — not an unknown with a fixed answer, but a placeholder for any input value.

Example: y = 2x + 1 describes a relationship between x and y for all values of x. As x varies, y varies accordingly.

Misconceptions: - MIS-ALG-14: “This is the same as solving an equation” — No. This is conceptually distinct. Here, x is varying; in an equation, x is fixed.

Prerequisite: ALG-F-08

Concept ALG-F-10: Mathematical Prose Comprehension

What it is: The ability to read a word problem with sufficient precision to extract: 1. What is being asked? 2. What information is given? 3. Which information is relevant? 4. What relationship is the problem describing?

This is a reading skill, not a math skill — but it is the primary bottleneck for applied mathematics.

Prerequisite: None (but depends on English reading ability)

Concept ALG-F-11: Mathematical Lexical Precision

What it is: Correct mathematical interpretation of high-ambiguity terms — words that have everyday meanings that conflict with their mathematical meaning.

Examples: - “Difference” = subtraction in math, but might mean “variation” in everyday language - “Product” = multiplication in math, but means “result of a process” in everyday language - “Sum” = addition in math, but “summary” in everyday language

Misconceptions: Systematic errors that look like conceptual failures but are vocabulary failures.

Prerequisite: ALG-F-10

PART 3: EXPRESSIONS & EQUATIONS (ALG-EX, ALG-EQ)

Tier 2: Algebraic Expressions

Concept ALG-EX-01: Algebraic Expressions ★ CORE

What it is: A combination of numbers, variables, and operations that represents a quantity. Unlike an equation, an expression has no equals sign — it is not a statement that can be true or false. It can be evaluated (given a value for the variable) or simplified, but not “solved.”

Examples: - Expression: 3x + 5 (no equals sign; represents a quantity) - Equation: 3x + 5 = 11 (has equals sign; can be true or false)

Misconceptions: - MIS-ALG-15: “Expressions can be solved” — No. Only equations can be solved. Expressions can be evaluated or simplified.

Prerequisite: ALG-F-08

Concept ALG-EX-02: Like Terms & Combining ★ CORE

What it is: Terms with identical variable parts (same variable, same exponent) can be combined by adding or subtracting their coefficients.

3x + 5x = (3 + 5)x = 8x

This is an application of the distributive property in reverse.

Terms that cannot combine: - 3x + 5y (different variables) - 3x + 5x² (same variable, different exponents)

Misconceptions: - MIS-ALG-16: “3x + 5 = 8x” — No. 3x and 5 have different variable parts (x vs. constant). Cannot combine. - MIS-ALG-17: “3x + 5x² = 8x²” — No. Same variable, different exponents. Cannot combine.

Prerequisite: ALG-F-08, ALG-EX-01

Concept ALG-EX-03: Evaluating Expressions ★ CORE

What it is: Substituting a specific numerical value for a variable and computing the result.

Example: If x = 3, evaluate 2x + 5: 2(3) + 5 = 6 + 5 = 11

Requires: Correct application of order of operations after substitution. The expression itself does not change — only the representation changes when we substitute.

Prerequisite: ALG-F-03, ALG-EX-01

Concept ALG-EX-04: The Distributive Property ★ CORE

What it is: a(b + c) = ab + ac

Multiplication distributes over addition.

Why: This follows from what multiplication means. Distribution runs in both directions: - Expanding (left to right): 3(x + 2) = 3x + 6 - Factoring (right to left): 3x + 6 = 3(x + 2)

Every factoring technique in algebra is the distributive property in reverse.

Misconceptions: - MIS-ALG-18: “You can’t distribute negative signs” — You can: −(x − 3) = −x + 3 - MIS-ALG-19: “Distribution only applies to parentheses” — It’s the fundamental property; parentheses just make it visible

Prerequisite: ALG-F-05, ALG-EX-02

Tier 3: Solving Equations

Concept ALG-EQ-01: Solving One-Step Equations ★ CORE

What it is: Finding the value of a variable that makes an equation true by applying one inverse operation to both sides.

Logic: Whatever preserves balance is valid. The goal: isolate the variable.

Examples: - x + 5 = 12 → x = 12 − 5 = 7 - 3x = 12 → x = 12 ÷ 3 = 4

Misconceptions: - MIS-ALG-20: “Move the number to the other side and change the sign” — This works but misses the logic. You’re applying the inverse operation to both sides to maintain balance.

Prerequisite: ALG-F-05, ALG-F-07

Concept ALG-EQ-02: Solving Multi-Step Equations ★ CORE

What it is: Equations requiring multiple operations to isolate the variable.

Standard sequence: 1. Distribute (if needed) 2. Combine like terms 3. Move variable terms to one side 4. Move constants to other side 5. Divide by coefficient

This sequence is a heuristic, not a law — students should understand why each step works, not just the order.

Example: 2x + 3 = 11 1. Subtract 3 from both sides: 2x = 8 2. Divide by 2: x = 4

Misconceptions: - MIS-ALG-21: “Always move the positive number first” — The sequence depends on the equation. What matters is understanding the inverse operations.

Prerequisite: ALG-EQ-01, ALG-EX-04

Concept ALG-EQ-03: Equations with Fractions & Decimals

What it is: Clearing denominators by multiplying both sides by the LCD before solving. Multiplying both sides by a power of 10 to clear decimals.

These are applications of the multiplicative property of equality, not tricks.

Example: (x/3) + 2 = 5 1. Multiply both sides by 3: x + 6 = 15 2. Subtract 6: x = 9

Prerequisite: ALG-EQ-02, understanding of fractions (ALG-F-02)

Concept ALG-EQ-04: Literal Equations & Formulas

What it is: Solving for one variable in terms of others.

The process is identical to solving for a numerical variable — apply inverse operations to isolate the target variable — but the result is an expression, not a number.

Example: Solve for h in A = (1/2)bh: 1. Multiply both sides by 2: 2A = bh 2. Divide by b: h = 2A/b

Reveals that “solving” is about isolation, not computation.

Misconceptions: - MIS-ALG-22: “Literal equations are different from regular equations” — Same logic; the result is an expression instead of a number.

Prerequisite: ALG-EQ-02

Concept ALG-EQ-05: No Solution & Infinitely Many Solutions

What it is: A linear equation has exactly one solution, no solution, or infinitely many solutions — these are the only three possibilities.

No solution: The equation reduces to a false statement (e.g., 3 = 7). The variable has disappeared and left something untrue.

Infinitely many solutions: The equation reduces to a true statement (e.g., 0 = 0). The variable has disappeared and left something always true — every value of x satisfies the original equation.

Both outcomes are valid mathematical conclusions, not errors.

Example: - x + 3 = x + 5 → 3 = 5 (false, no solution) - 2x + 4 = 2(x + 2) → 2x + 4 = 2x + 4 → 0 = 0 (true, infinitely many solutions)

Prerequisite: ALG-EQ-02

PART 4: INEQUALITIES (ALG-IN)

Concept ALG-IN-01: Linear Inequalities in One Variable ★ CORE

What it is: Like equations, but the solution is a range of values rather than a single value.

Properties (mostly same as equations with ONE exception): - Add/subtract the same number to both sides ✓ - Multiply/divide both sides by a positive number ✓ - Multiply/divide both sides by a negative number? Reverse the inequality direction ⚠

Why reverse for negative? The number line: multiplying by −1 reflects all points across zero, which reverses the direction.

Example: −2x > 6 1. Divide both sides by −2: x < −3 (direction reversed!)

Prerequisite: ALG-EQ-02

Concept ALG-IN-02: Compound Inequalities

What it is: Two inequalities joined by “and” or “or.”

“And” inequalities (intersection): Both conditions must be true. −2 < x < 5 means x > −2 AND x < 5
“Or” inequalities (union): At least one condition must be true. x < 0 OR x > 10

Graphed as overlapping or separate regions on a number line.

Prerequisite: ALG-IN-01

PART 5: LINEAR FUNCTIONS & GRAPHING (ALG-LF)

Concept ALG-LF-01: The Coordinate Plane ★ CORE

What it is: A two-dimensional representation system using two perpendicular number lines (axes). Every point represents an ordered pair (x, y) — a specific x-value and a specific y-value. The order matters: (3, 5) ≠ (5, 3).

Quadrant signs: - Quadrant I: (+, +) - Quadrant II: (−, +) - Quadrant III: (−, −) - Quadrant IV: (+, −)

Prerequisite: ALG-F-02 (number line familiarity)

Concept ALG-LF-02: Slope as Rate of Change ★ CORE

What it is: Slope measures how much y changes for every one-unit change in x. It is a rate — a ratio of vertical change to horizontal change.

m = (y₂ − y₁) / (x₂ − x₁)

Interpretations: - Positive slope: y increases as x increases (line goes up left to right) - Negative slope: y decreases as x increases (line goes down left to right) - Slope of 0: horizontal line (y doesn’t change) - Undefined slope: vertical line (x doesn’t change; division by 0)

Misconceptions: - MIS-ALG-23: “Slope is always positive” — No. Slope can be positive, negative, zero, or undefined. - MIS-ALG-24: “Steeper line = higher slope” — Steeper line does have higher |slope|, but negative slopes can be just as steep as positive slopes.

Prerequisite: ALG-LF-01, ALG-F-09

Concept ALG-LF-03: Slope-Intercept Form y = mx + b ★ CORE

What it is: The equation of any non-vertical line where: - m = slope - b = y-intercept (the y-value when x = 0)

This form makes reading the slope and y-intercept immediate.

Understanding: Every point (x, y) that satisfies the equation lies on the line; every point on the line satisfies the equation. The equation IS the line.

Example: y = 2x − 3 has slope 2 and y-intercept −3

Misconceptions: - MIS-ALG-25: “The y-intercept is the first point on the line” — No. It’s where the line crosses the y-axis.

Prerequisite: ALG-LF-02, ALG-EX-03

Concept ALG-LF-04: Point-Slope Form & Standard Form

What it is: - Point-slope form: y − y₁ = m(x − x₁) — useful when slope and one point are known - Standard form: Ax + By = C — useful for identifying intercepts quickly

All three forms describe the same line; form selection is strategic, not mathematical.

Prerequisite: ALG-LF-03

PART 6: SYSTEMS OF EQUATIONS (ALG-SY)

Concept ALG-SY-01: What a System Is ★ CORE

What it is: A system of equations is a set of equations that must be simultaneously satisfied — all true at the same time for the same values of the variables.

The solution is the ordered pair (or set) that makes every equation in the system true.

Geometrically: The intersection point of the lines.

Example: - y = 2x + 1 - y = −x + 4 - Solution: (1, 3) — the point where the lines intersect

Prerequisite: ALG-LF-03

Concept ALG-SY-02: Solving by Substitution ★ CORE

What it is: Solve one equation for one variable, substitute that expression into the other equation. This reduces the system to a single equation in one variable.

Logic: If y = 2x + 1, then anywhere y appears can be replaced by 2x + 1 without changing the relationships (Substitution Property of Equality).

Prerequisite: ALG-SY-01, ALG-EQ-02

Concept ALG-SY-03: Solving by Elimination

What it is: Adding or subtracting equations to eliminate one variable.

Valid because: Adding equal quantities to both sides of an equation preserves equality — and both sides of a true equation are equal quantities.

Often requires: Multiplying one or both equations by a constant first to make coefficients align.

Requires understanding why this works, not just the steps.

Prerequisite: ALG-SY-01

PART 7: POLYNOMIALS & QUADRATICS (ALG-PO, ALG-QA)

Concept ALG-PO-01: Polynomials — Structure & Vocabulary ★ CORE

What it is: A polynomial is a sum of terms where each term is a product of a coefficient and a variable raised to a whole number exponent.

Vocabulary: - Degree: Highest exponent - Leading coefficient: Coefficient of the highest-degree term - Naming: Monomial (1 term), binomial (2 terms), trinomial (3 terms)

Example: 3x² − 5x + 2 has degree 2, leading coefficient 3, is a trinomial

Prerequisite: ALG-EX-01

Concept ALG-PO-02: Polynomial Operations ★ CORE

What it is: - Addition/subtraction: Combine like terms - Multiplication: Distribute each term of the first polynomial to each term of the second

FOIL is a mnemonic for binomial × binomial only — not a general method. The general method is repeated distribution.

Example: (x + 2)(x + 3) = x(x + 3) + 2(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6

Prerequisite: ALG-EX-04

Concept ALG-PO-03: Factoring as Reverse Distribution ★ CORE

What it is: Factoring finds the product form of an expression. It is the distributive property applied in reverse.

Three main types: 1. GCF factoring: Factor out the greatest common factor from all terms 2. Trinomial factoring (ax² + bx + c): Find two binomials whose product is the trinomial 3. Difference of squares (a² − b²): Equals (a + b)(a − b)

Verification: Distribute back to confirm.

Misconceptions: - MIS-ALG-26: “Factoring is a process for finding answers” — No. It’s a way of rewriting an expression to reveal its structure. - MIS-ALG-27: “Sum of squares (a² + b²) factors” — No. Sum of squares does NOT factor over real numbers.

Prerequisite: ALG-PO-02, ALG-EX-04

Concept ALG-QA-01: Quadratic Equations — What They Are ★ CORE

What it is: An equation where the highest power of the variable is 2.

Standard form: ax² + bx + c = 0

Key property: A quadratic has at most two solutions (roots), because a parabola crosses a horizontal line at most twice. The solutions are where the parabola crosses the x-axis — where the function value equals zero.

Prerequisite: ALG-PO-01

Concept ALG-QA-02: Zero Product Property ★ CORE

What it is: If the product of two factors equals zero, then at least one of the factors must equal zero:

If ab = 0, then a = 0 or b = 0

This follows from a fundamental property of zero — zero is the only number that produces zero when multiplied by anything.

Critical: The equation must equal zero for this property to apply. This is not a trick; it is a logical consequence of what zero is.

Prerequisite: ALG-QA-01

Concept ALG-QA-03: Solving by Factoring ★ CORE

What it is: 1. Set equation equal to zero (if not already) 2. Factor the quadratic 3. Apply Zero Product Property: set each factor equal to zero 4. Solve

This method works only when the equation is set equal to zero and is factorable.

Prerequisite: ALG-QA-02, ALG-PO-03

Concept ALG-QA-04: The Quadratic Formula

What it is: x = (−b ± √(b² − 4ac)) / 2a

Works for any quadratic, not just factorable ones.

The discriminant (b² − 4ac) tells you the number and type of solutions: - Positive → two real roots - Zero → one real root (double root) - Negative → no real roots

The formula is derivable by completing the square — understanding this prevents the formula from being a pure memorization task.

Prerequisite: ALG-QA-03

PART 8: FUNCTIONS (ALG-FU)

Concept ALG-FU-01: What a Function Is ★ CORE

What it is: A function is a rule that assigns exactly one output to each input. For every x there is exactly one f(x).

This is the defining characteristic — not a formula, not a graph, not an equation.

A function can be represented as: - A table - A graph - An equation - A verbal description

Vertical line test: If any vertical line crosses the graph more than once, it’s not a function.

Prerequisite: ALG-F-09 (variables as varying quantities)

Concept ALG-FU-02: Domain & Range ★ CORE

What it is: - Domain: All valid input values (x-values) - Range: All resulting output values (y-values)

For functions in Algebra 1, domain restrictions arise from: - Division by zero (denominator can’t be 0) - Square roots of negatives (under the radical must be ≥ 0)

Identifying from a graph: - Domain is the horizontal extent - Range is the vertical extent

Prerequisite: ALG-FU-01

PART 9: PREREQUISITE CHAINS

Master Chain: Algebra 1

ALG-F-01 (Integer Operations)
  ↓
ALG-F-03 (Order of Operations) & ALG-F-02 (Rational Numbers)
  ↓
ALG-F-05 (Equals Sign as Balance) ← FOUNDATION 1
  ↓
ALG-F-07 (Inverse Operations) & ALG-F-08 (Variables as Unknowns)
  ↓
ALG-EQ-01 (One-Step Equations)
  ↓
ALG-EQ-02 (Multi-Step Equations) ← FOUNDATION 2
  ↓
ALG-LF-02 (Slope) → ALG-LF-03 (y = mx + b) ← FOUNDATION 3
  ↓
ALG-SY-01 (Systems) & ALG-SY-02 (Substitution)
  ↓
ALG-PO-02 (Polynomial Operations) & ALG-PO-03 (Factoring)
  ↓
ALG-QA-01 through ALG-QA-04 (Quadratics)
  ↓
ALG-FU-01 & ALG-FU-02 (Functions)

Red flags: - If ALG-F-05 is weak, do NOT advance - If ALG-EQ-02 is weak, do NOT advance - If ALG-LF-03 is weak, do NOT advance to systems - Factoring (ALG-PO-03) must be solid before quadratics

End of GED Algebra 1 Tier Map

All prerequisites are firm — breaking them leads to foundation gaps that compound downstream.