Openstax Prealgebre 2E

In this lesson from OpenStax Prealgebra 2E, students learn to identify and distinguish counting numbers and whole numbers, understand place value, and use place value to read, write, and round whole numbers. The lesson also introduces the number line, the concept of the origin and coordinates, and how digits change value based on their position in a number. Designed for prealgebra learners reviewing foundational math, it builds the vocabulary and number sense needed for all subsequent arithmetic and algebra topics.

In this OpenStax Prealgebra 2E lesson, students learn to use addition notation, identify addends and sums, and model addition of whole numbers using base-10 blocks. The lesson also covers adding whole numbers without models, translating word phrases into math expressions, and solving real-world addition problems. Designed for prealgebra students, it builds foundational arithmetic skills needed for more advanced math topics.

In this lesson from OpenStax Prealgebra 2E, Chapter 1, students learn to subtract whole numbers using subtraction notation, base-10 block models, and standard algorithms. The lesson covers translating word phrases into math expressions, understanding the concept of difference, and applying subtraction skills to real-world problems such as calculating remaining quantities. Though the grade level is not specified, this content is appropriate for students building foundational prealgebra skills.

In this lesson from OpenStax Prealgebra 2E, students learn to multiply whole numbers using multiplication notation (×, ·, and parentheses), counters to model repeated addition, and standard multiplication facts. Learners practice translating word phrases into math notation and applying multiplication to real-world problems. This foundational prealgebra lesson is suitable for middle school students building fluency with whole number operations.

In this lesson from OpenStax Prealgebra 2E, students learn to divide whole numbers using division notation (÷, /, and fraction bar), model division with counters, and identify key vocabulary including dividend, divisor, and quotient. Students also practice translating between word phrases and math notation and apply division to real-world problems. This foundational lesson connects division to repeated subtraction and its inverse relationship with multiplication.

In this lesson from OpenStax Prealgebra 2e, Chapter 2, students learn the foundational language of algebra, including how to identify variables and constants, interpret algebraic symbols for the four basic operations, and distinguish between expressions and equations. Students also practice simplifying expressions with exponents and applying the order of operations. The lesson builds algebraic literacy by translating between symbolic notation and written words using terms like sum, difference, product, and quotient.

In this lesson from OpenStax Prealgebra 2E, students learn to evaluate algebraic expressions by substituting given values for variables and applying the order of operations. The lesson also covers identifying terms, coefficients, and like terms, simplifying expressions by combining like terms, and translating word phrases into algebraic expressions. These foundational algebra skills are taught through worked examples involving expressions with exponents, multiplication, and multi-step simplification.

In this lesson from OpenStax Prealgebra 2e, Chapter 2, students learn how to solve equations using the Subtraction and Addition Properties of Equality, including how to determine whether a number is a solution to an equation by substituting values for variables. Students also practice translating word phrases into algebraic equations and solving real-world problems using these foundational equation-solving techniques.

In this lesson from OpenStax Prealgebra 2E, students learn to identify multiples of numbers, apply divisibility tests for 2, 5, and 10, find all factors of a number, and distinguish between prime and composite numbers. The lesson uses patterns in digit endings and multiplication to build number sense skills essential for prealgebra. It covers key vocabulary and methods that support future work with fractions, ratios, and algebraic reasoning.

In this lesson from OpenStax Prealgebra 2E, students learn how to find the prime factorization of composite numbers using the factor tree method and express results in exponential form. The lesson also covers finding the least common multiple (LCM) of two numbers, building on prior knowledge of divisibility, factors, and prime versus composite numbers. These foundational skills support more advanced work with fractions and number theory throughout the prealgebra course.

In this lesson from OpenStax Prealgebra 2E, Chapter 3, students are introduced to integers by learning to locate positive and negative numbers on the number line, order and compare integers, and find opposites. The lesson also covers simplifying expressions with absolute value and translating word phrases into integer expressions. Real-world contexts like temperature, elevation, and bank balances are used to build conceptual understanding of negative numbers.

In this lesson from OpenStax Prealgebra 2E, students learn how to add integers, including both positive and negative numbers, using two-color counter models and number line strategies. The lesson builds skills in simplifying expressions with integers, evaluating variable expressions, and translating word phrases into algebraic expressions. Real-world applications such as temperature, elevation, and banking help reinforce the concept of integer addition in context.

In this lesson from OpenStax Prealgebra 2E, Chapter 3, students learn how to subtract integers, including subtracting negative numbers using a counter model. The lesson covers simplifying integer expressions, evaluating variable expressions with integers, and translating word phrases into algebraic expressions. Students also apply integer subtraction to real-world problems, building a concrete foundation for working with signed numbers.

In this lesson from OpenStax Prealgebra 2E, Chapter 3, students learn to multiply and divide integers by applying the rules for signed numbers, including why the product of two negatives is positive and two numbers with different signs yields a negative result. Students also practice simplifying expressions with integers, evaluating variable expressions, and translating word phrases into algebraic expressions. Real-world applications such as temperature changes and financial scenarios reinforce how integer multiplication and division work in context.

In this lesson from OpenStax Prealgebra 2E, Chapter 3, students learn how to solve equations with integer solutions using the Addition, Subtraction, and Division Properties of Equality. Key skills include determining whether an integer is a solution by substitution, applying inverse operations to isolate variables, and translating word phrases into algebraic equations. The lesson builds on earlier whole-number equation work by extending solution sets to include negative integers.

In this lesson from OpenStax Prealgebra 2E, Chapter 4, students learn the meaning of fractions by identifying numerators and denominators, modeling improper fractions and mixed numbers, and finding equivalent fractions. Students also practice locating fractions and mixed numbers on a number line and ordering them by value. The lesson builds foundational fraction skills essential for algebra and everyday problem-solving contexts.

In this lesson from OpenStax Prealgebra 2E, students learn how to simplify fractions by identifying and removing common factors using the Equivalent Fractions Property, multiply fractions, find reciprocals, and divide fractions. The lesson covers key techniques such as prime factorization to find common factors, canceling those factors from the numerator and denominator, and applying the same process to negative and improper fractions. This foundational prealgebra content builds the skills students need to work confidently with fractions in more advanced math.

Students learn how to multiply and divide mixed numbers by first converting them to improper fractions, then applying standard fraction multiplication and division rules including using reciprocals. The lesson also covers translating verbal phrases involving quotients and ratios into fractional expressions, simplifying complex fractions, and evaluating expressions with fraction bars. This content is from Lesson 4.3 of OpenStax Prealgebra 2e, Chapter 4.

In this prealgebra lesson from OpenStax Prealgebra 2E, students learn how to add and subtract fractions with common denominators by combining or removing same-size pieces and writing the result over the shared denominator. The lesson uses fraction circle models and visual diagrams to build conceptual understanding before moving to the standard procedure. Students also practice simplifying the resulting fractions as part of each calculation.

In this lesson from OpenStax Prealgebra 2E, students learn how to add and subtract fractions with different denominators by finding the least common denominator (LCD) and converting fractions to equivalent forms. The lesson covers using the least common multiple (LCM) to determine the LCD, applying the order of operations to simplify complex fraction expressions, and evaluating variable expressions with fractions. This is a foundational prealgebra skill covered in Chapter 4 of the fractions unit.

In this prealgebra lesson from OpenStax Prealgebra 2E, Chapter 4, students learn how to add and subtract mixed numbers, including cases with common denominators and different denominators. The lesson covers modeling mixed number operations, converting between mixed numbers and improper fractions, and evaluating variable expressions with fractions. Practice problems range from straightforward computation to real-world applications like measuring fabric and combining baking ingredients.

Students learn how to solve equations with fractions in Lesson 4.7 of OpenStax Prealgebra 2E, covering how to determine whether a fraction is a solution of an equation and how to apply the Addition, Subtraction, Division, and Multiplication Properties of Equality to isolate variables. The lesson also includes translating word sentences into equations and solving them with fractional solutions. This prealgebra lesson builds directly on prior skills with integers and fraction operations.

In this lesson from OpenStax Prealgebra 2E, students learn the fundamentals of decimal notation, including how to name, write, and read decimals using place value concepts such as tenths, hundredths, and thousandths. Students also practice converting decimals to fractions or mixed numbers, locating decimals on a number line, ordering decimals, and rounding to a given place value. Real-world examples like money and gas prices help make the connection between decimal notation and everyday math clear and accessible.

In this prealgebra lesson from OpenStax Prealgebra 2E, students learn to add, subtract, multiply, and divide decimals by aligning decimal points and applying place value principles. The lesson also covers using decimal operations in real-world money applications such as calculating sales tax and total costs. Designed for middle school prealgebra students, this section builds fluency with decimal computation across all four operations.

In this prealgebra lesson from OpenStax Prealgebra 2E, Chapter 5, students learn how to convert fractions to decimals by dividing the numerator by the denominator, including cases that produce repeating decimals. The lesson also covers ordering decimals and fractions, applying the order of operations to expressions with decimals, and finding the circumference and area of circles. Real-world money applications, such as calculating earnings from multiple jobs, reinforce decimal arithmetic skills.

Students learn how to solve equations with decimals by applying the Properties of Equality — including the Addition, Subtraction, Multiplication, and Division Properties — to isolate variables in equations involving decimal values. The lesson also covers how to determine whether a given decimal is a solution by substituting it into an equation and checking both sides. This material is from Lesson 5.4 of OpenStax Prealgebra 2E and connects decimal operations to real-world contexts such as money and budgeting.

In this prealgebra lesson from OpenStax Prealgebra 2E, Chapter 5, students learn how to calculate the mean, median, and mode of a set of numbers and apply the basic definition of probability. The lesson covers computing the arithmetic mean by dividing the sum of values by the count of values, identifying the median as the middle value in an ordered data set, and determining the mode as the most frequently occurring value. Real-world contexts such as test score averages and temperature data help students connect these statistical concepts to everyday situations.

In this lesson from OpenStax Prealgebra 2E, students learn how to write ratios and rates as fractions, find unit rates and unit prices, and translate real-world phrases into fractional expressions. The lesson covers simplifying ratios involving whole numbers and decimals, including techniques for clearing decimals using the Equivalent Fractions Property. Part of the Chapter 5 Decimals unit, this material builds essential skills for understanding proportional relationships in everyday contexts such as pricing and debt-to-income comparisons.

In this prealgebra lesson from OpenStax Prealgebra 2E, students learn to simplify expressions with square roots, identify perfect squares, and work with square roots of both positive numbers and variable expressions. The lesson covers estimating and approximating square roots and applying them to real-world problems. It is designed for middle school students building foundational skills before algebra.

In this OpenStax Prealgebra 2E lesson, students learn the definition of percent as a ratio with a denominator of 100 and practice using the percent symbol to represent real-world data. The lesson covers how to convert percents to fractions by writing them over 100 and simplifying, as well as converting between percents, fractions, and decimals. This foundational lesson prepares students for applying percent concepts to practical problems involving money, statistics, and everyday situations.

In this lesson from OpenStax Prealgebra 2E, Chapter 6, students learn to translate and solve basic percent equations by identifying the amount, base, and percent, and applying algebraic methods to real-world problems such as calculating tips and interpreting percents greater than 100%. The lesson also covers finding percent increase and percent decrease. It is designed for middle school prealgebra students building on their skills with fractions, decimals, and equation solving.

In this lesson from OpenStax Prealgebra 2E, Chapter 6, students learn to solve real-world percent applications including sales tax, commission, discount, and mark-up problems. Using a step-by-step algebraic strategy, students practice translating word problems into percent equations of the form Sales Tax = Tax Rate × Purchase Price and solving for unknowns such as total cost or commission earned. This lesson builds practical math skills applicable to everyday financial situations.

In this lesson from OpenStax Prealgebra 2e, Chapter 6, students learn to apply the simple interest formula I = Prt to real-world financial scenarios involving principal, rate, and time. Students practice solving for unknown variables such as interest earned, principal invested, and interest rate by substituting known values into the formula and using decimal equivalents of percents. The lesson also reinforces percent concepts like discount, sales tax, and commission through writing exercises and self-assessment checklists.

In this lesson from OpenStax Prealgebra 2E, students learn how to define and write proportions, solve for unknown values using cross products, and apply proportional reasoning to real-world rate and ratio problems. The lesson also covers how to translate percent equations into proportion form and solve percent proportions. It is designed for middle school prealgebra students building foundational skills in ratios, rates, and percent calculations.

In this lesson from OpenStax Prealgebra 2E, Chapter 7, students learn to identify and classify rational and irrational numbers, including how to express integers, fractions, and decimals in the form p/q to determine whether they are rational. The lesson builds on prior knowledge of counting numbers, whole numbers, and integers to introduce the broader set of real numbers. It is designed for prealgebra students preparing to transition into foundational algebra concepts.

In this prealgebra lesson from OpenStax Prealgebra 2E, students learn the commutative and associative properties of addition and multiplication, including why these properties do not apply to subtraction or division. Students practice using these properties to rewrite and simplify expressions involving real numbers. The lesson builds foundational algebraic reasoning skills that support solving equations.

In this prealgebra lesson from OpenStax Prealgebra 2E, students learn to simplify and evaluate algebraic expressions using the Distributive Property, applying the rule a(b + c) = ab + ac to expand terms such as 3(x + 4) into 3x + 12. Practice problems progress from basic distribution to multi-term expressions involving decimals, fractions, and negative numbers. Real-world word problems, such as calculating stamp costs and counting cash, reinforce how the property works in everyday contexts.

In this OpenStax Prealgebra 2E lesson, students learn to identify and apply the identity properties of addition and multiplication, recognizing that 0 is the additive identity and 1 is the multiplicative identity. Students also practice using the inverse properties, finding additive inverses (opposites) and multiplicative inverses (reciprocals) of real numbers, including fractions and decimals. The lesson builds toward simplifying expressions by applying these properties of identities, inverses, and zero.

This lesson from OpenStax Prealgebra 2e teaches students how to make unit conversions within the U.S. customary system, within the metric system, and between both systems, including Fahrenheit and Celsius temperature conversions. Students apply the identity property of multiplication using unit fractions to convert measurements of length, weight, volume, and time. The lesson covers practical skills such as converting inches to feet, pounds to kilograms, and working with mixed units of measurement.

In this OpenStax Prealgebra 2E lesson from Chapter 8, students learn to solve linear equations by applying the Subtraction and Addition Properties of Equality to isolate variables and find solutions. The lesson covers how to verify solutions by substitution, simplify equations before solving, and translate real-world word problems into algebraic equations. These foundational skills prepare students for more advanced equation-solving techniques introduced later in the chapter.

In this lesson from OpenStax Prealgebra 2E, Chapter 8, students learn to solve one-variable linear equations by applying the Division and Multiplication Properties of Equality, including cases where the variable is multiplied by a coefficient or divided by a number. Learners practice isolating the variable by performing inverse operations on both sides of the equation, then simplifying equations that require multiple steps before solving. The lesson also covers interpreting negative coefficients such as recognizing that negative r equals negative one times r.

In this lesson from OpenStax Prealgebra 2E, students learn to solve linear equations that have variables and constants on both sides by applying the Addition, Subtraction, Division, and Multiplication Properties of Equality to isolate the variable. The lesson guides learners through a step-by-step general strategy for transforming equations into the form ax = b, covering cases with constants on both sides, variables on both sides, and mixed combinations. This foundational algebra skill is suitable for middle school math students working through Chapter 8 of the prealgebra curriculum.

In this lesson from OpenStax Prealgebra 2E, Chapter 8, students learn how to solve linear equations with fraction and decimal coefficients using two key techniques: clearing fractions by multiplying both sides by the least common denominator (LCD), and clearing decimals by multiplying by a power of 10. Students apply the Multiplication Property of Equality and the General Strategy for Solving Linear Equations to simplify and solve these types of equations efficiently.

In this lesson from OpenStax Prealgebra 2E, Chapter 9, students learn how to apply a structured problem-solving strategy to translate word problems into algebraic equations and solve them. The lesson covers approaching word problems with a positive mindset, identifying variables, and working through number problems step by step. It serves as the foundation for the geometry and math modeling topics explored throughout the rest of the chapter.

In this OpenStax Prealgebra 2E lesson, students learn to solve coin word problems and ticket and stamp word problems by setting up algebraic equations using the number-times-value-equals-total-value relationship. Learners practice organizing information in tables and writing variable expressions to represent unknown quantities before solving multi-step equations. This lesson builds on prior skills in decimal operations, the distributive property, and solving equations with variables on both sides.

Students learn to identify and solve problems involving supplementary and complementary angles, properties of triangles, and the Pythagorean Theorem in this lesson from Chapter 9 of OpenStax Prealgebra 2e. The lesson introduces key angle concepts such as vertex, degrees, and angle measurement, then applies a structured problem-solving strategy to geometry applications. These foundational geometry skills build on prior algebra techniques to prepare students for real-world math modeling.

In this lesson from OpenStax Prealgebra 2E, Chapter 9, students learn to distinguish between linear, square, and cubic measure and apply the geometric properties of rectangles, triangles, and trapezoids to solve real-world problems. Topics include calculating perimeter and area using formulas for each shape, as well as identifying appropriate units of measure for length, area, and volume. This section is designed for middle school prealgebra students building foundational skills in geometry and measurement.

Students learn how to apply the properties of circles — including the circumference formula C = 2πr and area formula A = πr² — and how to find the area of irregular figures by breaking them into familiar shapes. The lesson is part of Chapter 9 of OpenStax Prealgebra 2E and guides students through a structured seven-step problem-solving strategy for geometry applications. Real-world contexts such as circular sandboxes are used to build understanding of these core measurement concepts.

In this OpenStax Prealgebra 2E lesson from Chapter 9, students learn how to calculate the volume and surface area of three-dimensional figures, including rectangular solids, spheres, cylinders, and cones. Using formulas such as V = LWH and surface area equations, students apply a step-by-step problem-solving strategy to real-world geometry applications. The lesson also reinforces finding the area of irregular figures and circles using different approximations of pi.

Students learn to identify and classify polynomials as monomials, binomials, and trinomials, determine the degree of polynomial terms, and add and subtract polynomials. This lesson is from Chapter 10 of OpenStax Prealgebra 2e, covering foundational polynomial concepts including evaluating polynomial expressions for a given value. It is designed for middle school prealgebra students building toward more advanced algebraic operations.

In this lesson from OpenStax Prealgebra 2E, Chapter 10, students learn to simplify expressions with exponents using the Product Property, Power Property, and Product to a Power Property of exponents. They practice evaluating exponential notation, distinguishing expressions like (-3)^4 from -3^4, and multiplying monomials by applying multiple exponent rules. This lesson builds essential algebraic fluency for working with polynomial expressions.

In this lesson from OpenStax Prealgebra 2E, Chapter 10, students learn how to multiply polynomials using the Distributive Property, covering three key skills: multiplying a polynomial by a monomial, multiplying a binomial by a binomial, and multiplying a trinomial by a binomial. Students practice applying the Distributive Property to expressions such as multiplying monomials like -2x across a trinomial or using the FOIL-style distribution to expand binomial products like (x + 6)(x + 8). The lesson builds on prior knowledge of exponent properties and combining like terms to simplify polynomial expressions.

In this lesson from OpenStax Prealgebra 2E, Chapter 10, students learn to divide monomials by applying the Quotient Property of Exponents, the zero exponent rule, and the Quotient to a Power Property. Learners practice simplifying algebraic expressions by subtracting exponents when dividing terms with the same base, including cases where the larger exponent appears in the numerator or denominator. The lesson builds on multiplication properties of exponents to develop a complete toolkit for working with monomial expressions.

In this lesson from OpenStax Prealgebra 2E, Chapter 10, students learn to work with integer exponents, including applying the definition of a negative exponent to rewrite expressions such as a⁻ⁿ = 1/aⁿ and simplifying expressions that contain negative or zero exponents. The lesson also covers converting numbers between standard decimal notation and scientific notation, as well as multiplying and dividing values written in scientific notation. Special attention is given to correctly identifying the base when negative signs and exponents interact, using the order of operations to avoid common errors.

In this lesson from OpenStax Prealgebra 2E, Chapter 10, students learn how to find the greatest common factor (GCF) of two or more expressions and use it to factor polynomials by reversing the multiplication process. The lesson introduces factoring as the process of breaking a product down into its factors, starting with numerical examples before applying the same method to algebraic expressions. Suited for middle school prealgebra students, it builds on prior skills in prime factorization and polynomial multiplication.

In this OpenStax Prealgebra 2E lesson, students learn to plot and identify points on the rectangular coordinate system using ordered pairs, x-coordinates, and y-coordinates. The lesson also covers the four quadrants, verifying solutions to equations in two variables, and completing tables of solutions to linear equations. This foundational graphing content prepares pre-algebra students for working with linear equations and slope in the chapters that follow.

In this lesson from OpenStax Prealgebra 2E, students learn how to graph linear equations in two variables by plotting ordered pairs on a rectangular coordinate system and connecting them to form a straight line. Key skills include completing a table of solutions, verifying whether a given point satisfies a linear equation, and graphing special cases such as vertical and horizontal lines. Students also explore the fundamental relationship between the solutions of a linear equation and the points that make up its graph.

In this lesson from OpenStax Prealgebra 2E, Chapter 11, students learn how to identify x-intercepts and y-intercepts on a graph, find intercepts directly from a linear equation, and use those intercepts to graph a line on the coordinate plane. The lesson also covers graphing vertical and horizontal lines and choosing the most convenient method to graph a given linear equation. It is designed for prealgebra students building foundational skills in linear graphing.

In this OpenStax Prealgebra 2E lesson from Chapter 11, students learn to understand and calculate the slope of a line using the formula m = rise/run, where rise measures vertical change and run measures horizontal change. Using geoboard models and coordinate graphs, learners explore how slope determines whether a line slants up or down and how steep it appears. The lesson builds foundational skills for identifying positive, negative, zero, and undefined slopes in real-world and algebraic contexts.