Comprehensive Study Guide: Prealgebra and Early Algebra Concepts (WCPSS Compacted Math 6 PLUS / Math 7 PLUS Syllabus)

Unit 1 — Ratios, Rates, and Proportional Reasoning

Concept Overview

This unit introduces the foundational concepts of ratios, rates, and proportional relationships, which are essential for understanding relationships between quantities in mathematics and real-world contexts.

• Ratio: A comparison of two quantities by division. Written as "a to b," "a:b," or $a \div b$.

• Rate: A special ratio comparing two quantities with different units (e.g., miles per hour).

• Unit Rate: A rate with a denominator of 1 (e.g., 60 miles per 1 hour).

• Proportional Relationship: Two quantities that have a constant ratio or rate.

• Percent: A rate per 100, written as "%".

Key Formulas and Rules

• Equivalent Ratios: $\frac{a}{b} = \frac{ka}{kb}$ for any nonzero $k$.

• Unit Rate: $\text{Unit Rate} = \frac{a}{b}$ where $b = 1$.

• Percent Equation: $\text{Part} = \text{Percent} \times \text{Whole}$

• Constant of Proportionality: $y = kx$

Examples

• Example 1: If 3 apples cost $6, what is the unit price?$\frac{6}{3} = 2$ dollars per apple.

• Example 2: 40% of 50 is $0.4 \times 50 = 20$.

Unit 2 — Number System and Rational Numbers

Concept Overview

This unit covers positive and negative numbers, rational numbers, and their operations, including representation on the number line and the coordinate plane.

• Rational Number: Any number that can be written as $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$.

• Absolute Value: The distance of a number from zero on the number line, denoted $|a|$.

• Greatest Common Factor (GCF): The largest integer that divides two numbers.

• Least Common Multiple (LCM): The smallest multiple common to two numbers.

Key Formulas and Rules

• Signed Number Rules: $(-a) + (-b) = -(a + b)$; $(-a) \times (-b) = ab$

• Absolute Value: $|a| = a$ if $a \geq 0$; $|a| = -a$ if $a < 0$

Examples

• Example 1: $|-7| = 7$

• Example 2: $\frac{2}{3} + \frac{1}{6} = \frac{4}{6} + \frac{1}{6} = \frac{5}{6}$

Unit 3 — Numerical and Algebraic Expressions

Concept Overview

This unit introduces variables, expressions, exponents, and the properties of operations, as well as how to evaluate and manipulate algebraic expressions.

• Variable: A symbol (often a letter) representing an unknown value.

• Numerical Expression: An expression containing only numbers and operations.

• Algebraic Expression: An expression containing variables, numbers, and operations.

• Order of Operations: The rules for evaluating expressions: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction (PEMDAS).

• Properties: Commutative, Associative, Distributive.

Key Formulas and Rules

• Distributive Property: $a(b + c) = ab + ac$

• Combine Like Terms: $2x + 3x = 5x$

Examples

• Example 1: Simplify $3(x + 4) = 3x + 12$

• Example 2: Evaluate $2^3 + 5 = 8 + 5 = 13$

Unit 4 — Equations and Inequalities

Concept Overview

This unit focuses on solving equations and inequalities, including those with variables on both sides, and interpreting solutions in context.

• Equation: A mathematical statement that two expressions are equal.

• Inequality: A statement that compares two expressions using <, >, ≤, or ≥.

• Solution: A value that makes an equation or inequality true.

Key Formulas and Rules

• Solving One-Step Equations: $x + a = b \implies x = b - a$

• Solving Two-Step Equations: $ax + b = c \implies ax = c - b \implies x = \frac{c - b}{a}$

• Inequality Sign Reversal: When multiplying or dividing both sides by a negative, reverse the inequality sign.

Examples

• Example 1: Solve $2x + 3 = 11$. $2x = 8 \implies x = 4$

• Example 2: Solve $-3x < 12$. $x > -4$ (inequality sign reverses)

Unit 5 — Coordinate Plane and Quantitative Relationships

Concept Overview

This unit explores the coordinate plane, ordered pairs, and how to represent and analyze relationships between variables using tables, graphs, and equations.

• Coordinate Plane: A two-dimensional surface defined by an x-axis (horizontal) and y-axis (vertical).

• Ordered Pair: A pair of numbers $(x, y)$ representing a point's location.

• Independent Variable: The variable that represents input values (often x).

• Dependent Variable: The variable that represents output values (often y).

Key Formulas and Rules

• Distance between points: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

Examples

• Example 1: Plot (3, -2) on the coordinate plane.

• Example 2: If $y = 2x + 1$, then for $x = 3$, $y = 7$.

Unit 6 — Geometry: Area, Surface Area, and Volume

Concept Overview

This unit covers the calculation of area, surface area, and volume for various two- and three-dimensional figures, including composite shapes and circles.

• Area of Triangle: $A = \frac{1}{2}bh$

• Area of Rectangle: $A = lw$

• Surface Area of Rectangular Prism: $SA = 2lw + 2lh + 2wh$

• Volume of Rectangular Prism: $V = lwh$

• Area of Circle: $A = \pi r^2$

• Circumference of Circle: $C = 2\pi r$

Examples

• Example 1: Area of a triangle with base 8 and height 5: $A = \frac{1}{2} \times 8 \times 5 = 20$

• Example 2: Volume of a box with $l=3$, $w=4$, $h=5$: $V = 3 \times 4 \times 5 = 60$

Unit 7 — Scale Drawings and Geometric Relationships

Concept Overview

This unit introduces scale factors, scale drawings, and angle relationships, as well as geometric constructions and cross-sections of three-dimensional figures.

• Scale Factor: The ratio of any two corresponding lengths in two similar geometric figures.

• Complementary Angles: Two angles whose measures add up to $90^\circ$.

• Supplementary Angles: Two angles whose measures add up to $180^\circ$.

• Vertical Angles: Opposite angles formed by two intersecting lines; they are congruent.

Examples

• Example 1: If a scale drawing uses a scale of 1 cm : 2 m, then 5 cm represents $5 \times 2 = 10$ m.

• Example 2: If angle A and angle B are complementary and angle A is $35^\circ$, angle B is $55^\circ$.

Unit 8 — Statistics

Concept Overview

This unit covers statistical questions, data collection, measures of center and variability, and graphical representations of data.

• Mean: The average of a data set.

• Median: The middle value when data are ordered.

• Mode: The value that appears most often.

• Range: The difference between the highest and lowest values.

• Interquartile Range (IQR): The range of the middle 50% of data.

• Mean Absolute Deviation (MAD): The average distance between each data value and the mean.

Examples

• Example 1: Data: 2, 4, 4, 7. Mean: $(2+4+4+7)/4 = 4.25$

• Example 2: Range: $7 - 2 = 5$

Unit 9 — Probability

Concept Overview

This unit introduces probability as a measure of likelihood, including theoretical and experimental probability, probability models, and compound events.

• Probability: A number from 0 to 1 that describes the likelihood of an event.

• Theoretical Probability: $P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$

• Experimental Probability: $P(E) = \frac{\text{Number of times event occurs}}{\text{Total trials}}$

• Sample Space: The set of all possible outcomes.

Examples

• Example 1: Probability of rolling a 3 on a fair die: $P(3) = \frac{1}{6}$

• Example 2: If you flip a coin 10 times and get 7 heads, experimental probability of heads is $\frac{7}{10}$

Unit 10 — Exponents and Polynomials

Concept Overview

This unit introduces the laws of exponents and basic polynomial operations, including addition, subtraction, and multiplication of polynomials.

• Exponent: Indicates how many times a number (the base) is multiplied by itself.

• Polynomial: An expression consisting of variables and coefficients, involving only addition, subtraction, and multiplication.

Key Laws of Exponents

• $a^m \times a^n = a^{m+n}$

• $\frac{a^m}{a^n} = a^{m-n}$

• $(a^m)^n = a^{mn}$

• $a^0 = 1$ (for $a \neq 0$)

Examples

• Example 1: $2^3 \times 2^2 = 2^{3+2} = 2^5 = 32$

• Example 2: $(x + 2) + (3x - 5) = 4x - 3$

Mathematical Practices

Overview

• Make sense of problems and persevere in solving them.

• Reason abstractly and quantitatively.

• Construct mathematical arguments and critique reasoning.

• Model real-world situations mathematically.

• Select appropriate mathematical tools and attend to precision.

• Recognize mathematical structure and patterns.

Assessment and Mastery

Mastery Levels

Score (%)	Mastery Level
Below 75%	Reteach the unit
75–84%	Additional targeted practice required
85–89%	Proficient
90–94%	Strong mastery
95–100%	Advanced mastery

Target: At least 90% mastery before progressing to the next unit.

Additional info:

• This syllabus covers all major Prealgebra topics and extends into early Algebra and Geometry, preparing students for high school mathematics.

• Each unit should include concept summaries, vocabulary, key formulas, worked examples, practice questions, and mastery assessments as described.