Compacted Math 6+/7+ Syllabus: Structured Study Guide for Prealgebra Foundations

Unit 1 - Ratios, Rates, and Proportional Reasoning

Ratios and Proportional Relationships

This unit introduces the concept of ratios, rates, and proportional reasoning, which are fundamental for understanding relationships between quantities.

• Ratio: A comparison of two quantities by division. Expressed as "a to b," "a:b," or $\frac{a}{b}$.

• Equivalent Ratios: Ratios that represent the same relationship, often found by scaling both terms.

• Unit Rate: A rate with a denominator of 1. Example: If 120 miles are driven in 2 hours, the unit rate is $\frac{120}{2} = 60$ miles per hour.

• Percent: A rate per 100. Example: 45% means 45 out of 100.

• Constant of Proportionality: The constant $k$ in $y = kx$ for proportional relationships.

Example: If a recipe uses 2 cups of flour for every 3 cups of sugar, the ratio is 2:3.

Unit 2 - Number System and Rational Numbers

Understanding Rational Numbers

This unit covers positive and negative numbers, absolute value, and the ordering of rational numbers, including fractions and decimals.

• 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. $|{-5}| = 5$.

• Factors and Multiples: Factors are numbers that divide another number exactly; multiples are products of a number and an integer.

• GCF and LCM: Greatest Common Factor and Least Common Multiple.

• Signed-Number Rules: Rules for addition, subtraction, multiplication, and division of positive and negative numbers.

Example: $-3 + 7 = 4$; $-3 \times 7 = -21$.

Unit 3 - Numerical and Algebraic Expressions

Variables and Expressions

This unit explores variables, exponents, and the order of operations, as well as how to evaluate and simplify expressions.

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

• Order of Operations: PEMDAS: Parentheses, Exponents, Multiplication/Division, Addition/Subtraction.

• Exponents: $a^n$ means $a$ multiplied by itself $n$ times.

• Combine Like Terms: Add or subtract terms with the same variable and exponent.

Example: Simplify $3x + 2x = 5x$.

Unit 4 - Equations and Inequalities

Solving Equations and Inequalities

This unit focuses on solving one-step, two-step, and multi-step equations and inequalities, including those with variables on both sides.

• Equation: A mathematical statement that two expressions are equal. Example: $2x + 3 = 7$.

• Inequality: A statement comparing two expressions using $<, >, \leq, \geq$.

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

• Graphing Solutions: Representing solutions on a number line.

Example: Solve $x + 5 = 12$; $x = 7$.

Unit 5 - Coordinate Plane and Quantitative Relationships

Coordinate Plane and Relationships

This unit introduces the coordinate plane, ordered pairs, and how to analyze relationships using tables, graphs, and equations.

• Coordinate Plane: A two-dimensional surface defined by the x-axis and y-axis.

• Ordered Pair: $(x, y)$ represents a point's location.

• Independent/Dependent Variables: The independent variable is manipulated; the dependent variable responds.

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

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

Area and Volume

This unit covers calculating area, surface area, and volume for various shapes, including triangles, quadrilaterals, polygons, and three-dimensional figures.

• Area of Rectangle: $A = l \times w$

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

• Volume of Rectangular Prism: $V = l \times w \times h$

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

• Area of Circle: $A = \pi r^2$

Example: Find the area of a triangle with base 8 cm and height 5 cm: $A = \frac{1}{2} \times 8 \times 5 = 20$ cm2.

Unit 7 - Scale Drawings and Geometric Relationships

Scale and Angles

This unit explores scale drawings, scale factors, and relationships between angles.

• Scale Factor: The ratio of the length in the drawing to the actual length.

• Complementary Angles: Two angles whose measures add up to 90°.

• Supplementary Angles: Two angles whose measures add up to 180°.

• Vertical Angles: Opposite angles formed by two intersecting lines; always equal.

Example: If one angle is 30°, its complement is 60°.

Unit 8 - Statistics

Statistical Measures and Data Representation

This unit introduces statistical questions, measures of central tendency, and data representation.

• Mean: Average; sum of values divided by number of values.

• Median: Middle value when data is ordered.

• Mode: Most frequently occurring value.

• Range: Difference between highest and lowest values.

• Box Plot: Visual representation of data distribution.

Example: Data: 2, 4, 4, 5, 7. Mean = $\frac{2+4+4+5+7}{5} = 4.4$.

Unit 9 - Probability

Probability Concepts

This unit covers theoretical and experimental probability, sample spaces, and compound events.

• Probability: Likelihood of an event, between 0 and 1. $P = \frac{\text{number of favorable outcomes}}{\text{total outcomes}}$

• Sample Space: All possible outcomes.

• Compound Event: Combination of two or more events.

Example: Probability of rolling a 3 on a six-sided die: $P = \frac{1}{6}$.

Unit 10 - Irrational Numbers, Square Roots, and Radicals

Rational and Irrational Numbers

This unit distinguishes between rational and irrational numbers and introduces square roots and radicals.

• Irrational Number: Cannot be written as $\frac{a}{b}$; decimal expansion is non-repeating and non-terminating.

• Square Root: $\sqrt{a}$ is a number that, when multiplied by itself, gives $a$.

• Cube Root: $\sqrt[3]{a}$ is a number that, when cubed, gives $a$.

Example: $\sqrt{16} = 4$; $\sqrt{2}$ is irrational.

Unit 11 - Integer Exponents and Scientific Notation

Exponents and Scientific Notation

This unit covers the laws of exponents and how to use scientific notation for very large or small numbers.

• Exponent Laws: $a^m \times a^n = a^{m+n}$; $a^0 = 1$; $a^{-n} = \frac{1}{a^n}$

• Scientific Notation: Express numbers as $a \times 10^n$.

Example: $3 \times 10^4 = 30,000$.

Unit 12 - Transformations, Congruence, and Similarity

Transformations and Similarity

This unit introduces translations, reflections, rotations, dilations, and the concepts of congruence and similarity.

• Translation: Sliding a figure without rotating or flipping.

• Reflection: Flipping a figure over a line.

• Rotation: Turning a figure around a point.

• Dilation: Resizing a figure by a scale factor.

• Congruent Figures: Same shape and size.

• Similar Figures: Same shape, proportional size.

Example: A triangle rotated 90° remains congruent to its original.

Unit 13 - Linear Relationships and Slope

Linear Relationships and Slope

This unit covers recognizing linear relationships, calculating slope, and interpreting y-intercepts.

• Slope: $m = \frac{\Delta y}{\Delta x}$; measures steepness of a line.

• y-intercept: Where the line crosses the y-axis.

• Linear Equation: $y = mx + b$

Example: For points (2, 3) and (4, 7): $m = \frac{7-3}{4-2} = 2$.

Unit 14 - Linear Equations and Systems

Solving Linear Equations and Systems

This unit explores complex linear equations and systems of equations, including graphical and algebraic solutions.

• System of Equations: Two or more equations with the same variables.

• One Solution, No Solution, Infinite Solutions: Types of system outcomes.

Example: Solve $x + y = 5$ and $x - y = 1$.

Unit 15 - Functions

Understanding Functions

This unit defines functions, explores their representations, and distinguishes between linear and nonlinear functions.

• Function: A relation where each input has exactly one output.

• Linear Function: $y = mx + b$; constant rate of change.

• Nonlinear Function: Rate of change is not constant.

Example: $f(x) = 2x + 3$ is linear; $f(x) = x^2$ is nonlinear.

Unit 16 - Pythagorean Theorem

Pythagorean Theorem and Applications

This unit introduces the Pythagorean Theorem for right triangles and its applications in finding distances.

• Pythagorean Theorem: $a^2 + b^2 = c^2$ for right triangles.

• Converse: If $a^2 + b^2 = c^2$, then the triangle is right.

Example: If legs are 3 and 4, hypotenuse is $\sqrt{3^2 + 4^2} = 5$.

Unit 17 - Three-Dimensional Geometry

Volume of 3D Figures

This unit covers the volume formulas for cylinders, cones, and spheres.

• Volume of Cylinder: $V = \pi r^2 h$

• Volume of Cone: $V = \frac{1}{3} \pi r^2 h$

• Volume of Sphere: $V = \frac{4}{3} \pi r^3$

Example: Find the volume of a cylinder with $r = 2$, $h = 5$: $V = \pi \times 4 \times 5 = 20\pi$.

Unit 18 - Bivariate Data

Bivariate Data and Associations

This unit introduces scatter plots, associations, and lines of best fit for analyzing relationships between two variables.

• Bivariate Data: Data involving two variables.

• Scatter Plot: Graph showing relationship between two variables.

• Line of Best Fit: Line that best represents the data trend.

Example: Plot height vs. weight for a group of students.

Mathematical Practices

Mathematical Reasoning and Problem Solving

Throughout all units, students are encouraged to develop strong mathematical reasoning, model real-world situations, and critique arguments.

• 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 use them strategically.

• Attend to precision.

• Recognize mathematical structure and repeated reasoning.

Mastery and Review Requirements

Assessment and Remediation

Mastery is assessed through unit tests, cumulative reviews, and targeted remediation. Students must achieve at least 90% mastery before advancing.

• Mastery Levels: Below 75%: reteach; 75-84%: additional practice; 85-89%: proficient; 90-94%: strong mastery; 95-100%: advanced mastery.

• Cumulative Review: After every three units, review all previous content.

• Remediation Plan: Target weak skills for improvement.

Formula Reference Sheet (Sample Table)

Key Formulas for Prealgebra

Concept	Formula	Example
Area of Rectangle	$A = l \times w$	$A = 5 \times 3 = 15$
Area of Triangle	$A = \frac{1}{2} b h$	$A = \frac{1}{2} \times 6 \times 4 = 12$
Volume of Rectangular Prism	$V = l \times w \times h$	$V = 2 \times 3 \times 4 = 24$
Circumference of Circle	$C = 2 \pi r$	$C = 2 \pi \times 5 = 10\pi$
Pythagorean Theorem	$a^2 + b^2 = c^2$	$3^2 + 4^2 = 9 + 16 = 25 = 5^2$
Slope	$m = \frac{\Delta y}{\Delta x}$	$m = \frac{7-3}{4-2} = 2$

Additional info: This syllabus covers all foundational and advanced prealgebra topics, including number systems, equations, geometry, statistics, probability, and functions, preparing students for high school algebra and beyond.