Mathematics from Scratch Course

Lesson 1 • Mathematical Symbols and Vocabulary

Covers core symbols: equality, inequality, operations, and grouping. Provides the notation literacy required to interpret any mathematical expression.

Lesson 2 • Understanding Number Systems

Introduces natural numbers, whole numbers, integers, and their relationships. Establishes the vocabulary needed for all subsequent arithmetic work.

Lesson 3 • Place Value and Decimal Notation

Explains positional value from ones through billions and into decimal places. Connects place value to reading and writing large and small numbers accurately.

Lesson 1 • Arithmetic Properties and Mental Math

Formalizes commutative, associative, and identity properties for efficient calculation. Students use these properties to simplify and check arithmetic mentally.

Lesson 2 • Addition and Subtraction in Depth

Covers carrying, borrowing, and signed-number rules for addition and subtraction. Links these operations to real-world contexts like totals and differences.

Lesson 3 • Multiplication Techniques

Develops fluency with single- and multi-digit multiplication, including decimals. Introduces the distributive property as a mental-math and algebraic tool.

Lesson 4 • Division Techniques

Teaches long division, decimal quotients, and interpreting remainders. Connects division to fractions and ratio concepts introduced later.

Lesson 1 • Decimal Arithmetic and Conversion

Extends decimal notation to arithmetic operations and fraction-decimal conversion. Reinforces place value understanding from Chapter 1 in computational contexts.

Lesson 2 • Fraction Fundamentals

Defines numerator, denominator, proper, improper, and mixed fractions. Builds the conceptual foundation for all fraction arithmetic that follows.

Lesson 3 • Percentages and Their Applications

Defines percent as parts per hundred and links it to fractions and decimals. Students calculate percent of a quantity, percent change, and reverse percentages.

Lesson 1 • Ratio Concepts and Notation

Defines ratio as a comparison of two quantities and introduces colon and fraction notation. Connects ratio to equivalent fractions established in Chapter 3.

Lesson 2 • Proportions and Cross-Multiplication

Establishes proportional relationships and the cross-multiplication solving method. Students verify proportionality and solve for unknown quantities in equations.

Lesson 3 • Rates and Unit Rates

Distinguishes rates from ratios and focuses on unit-rate calculation and comparison. Applies rate reasoning to speed, pricing, and efficiency problems.

Lesson 1 • Perimeter, Area, and Circumference

Derives and applies formulas for perimeter and area of common 2-D shapes and circles. Connects formula use to proportional reasoning from Chapter 4.

Lesson 2 • Surface Area and Volume

Extends 2-D measurement to 3-D solids including prisms, cylinders, cones, and spheres. Students select and apply the correct formula for each solid type.

Lesson 3 • Points, Lines, and Angles

Defines geometric primitives and classifies angles by measure and relationship. Establishes the vocabulary and notation used throughout all geometry sections.

Lesson 4 • Quadrilaterals and Polygons

Identifies properties of parallelograms, rectangles, squares, trapezoids, and regular polygons. Students calculate interior angle sums for any polygon.

Lesson 1 • Solving Multi-Step Equations

Extends equation solving to two or more steps, including distributive property use. Students solve equations with variables on both sides of the equal sign.

Lesson 2 • Variables, Expressions, and Terms

Introduces variables, constants, coefficients, and the structure of algebraic expressions. Builds on arithmetic properties from Chapter 2 to simplify expressions.

Lesson 3 • Word Problems and Algebraic Modeling

Translates verbal problem statements into equations and solves them systematically. Integrates skills from Chapters 1–5 within algebraic problem-solving contexts.

Lesson 4 • Solving One-Step Equations

Applies inverse operations to isolate a variable in one-step equations. Reinforces the balance model of equality as the foundation for all equation solving.

Lesson 1 • Systems of Linear Equations

Solves two-equation systems by graphing, substitution, and elimination methods. Students interpret solutions as intersection points and model real scenarios.

Lesson 2 • Functions and Function Notation

Defines a function, domain, range, and f(x) notation using mapping diagrams and tables. Distinguishes functions from non-functions using the vertical line test.

Lesson 3 • The Coordinate Plane

Reviews the Cartesian coordinate system, quadrants, and ordered-pair plotting. Connects coordinate geometry to the geometric concepts from Chapter 5.

Lesson 4 • Slope and Rate of Change

Defines slope as rise over run and connects it to unit-rate reasoning from Chapter 4. Students calculate slope from two points, a graph, and a table.

Lesson 1 • Measures of Central Tendency

Calculates mean, median, and mode and selects the appropriate measure for a dataset. Connects arithmetic skills from Chapter 2 to statistical computation.

Lesson 2 • Measures of Spread and Variability

Introduces range, interquartile range, and standard deviation as measures of data spread. Students interpret variability to compare datasets meaningfully.

Lesson 3 • Data Collection and Organization

Covers data types, sampling methods, frequency tables, and tally charts. Establishes data literacy as the prerequisite for all statistical analysis in this chapter.

Lesson 4 • Basic Probability

Defines probability as a ratio between 0 and 1 and calculates simple and compound event probabilities. Applies fraction and ratio skills from Chapters 3 and 4.