OpenStax Algebra and Trigonometry

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry, Chapter 1: Prerequisites, students learn to classify real numbers as natural, whole, integer, rational, or irrational numbers. The lesson also covers performing calculations using order of operations, applying properties of real numbers such as commutative, associative, distributive, inverse, and identity properties, and simplifying algebraic expressions. These foundational concepts establish the prerequisite knowledge students need for further study of algebra and trigonometry.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry, students learn the rules of exponents, including the product rule, quotient rule, power rule, zero exponent rule, and negative exponent rule, along with how to simplify exponential expressions. The lesson then applies these rules to scientific notation, showing students how to work with very large and very small numbers using powers of ten. This material is covered in Lesson 1.2 of Chapter 1: Prerequisites.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry, Chapter 1, students learn to work with radical expressions by evaluating square roots, applying the product and quotient rules to simplify radicands, and adding or subtracting square roots. The lesson also covers rationalizing denominators and using rational exponents to express roots in exponential form. Real-world applications, such as using the Pythagorean Theorem to find ladder lengths, help students connect these skills to practical problem-solving.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry, students learn to work with rational expressions — quotients of two polynomial expressions. The lesson covers simplifying rational expressions by factoring and canceling common factors, as well as multiplying, dividing, and adding or subtracting them. Students also practice simplifying complex rational expressions, building essential skills for working with polynomial fractions.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry, students learn to plot ordered pairs in the Cartesian coordinate system, identify x- and y-coordinates, and locate points across the four quadrants. The lesson also covers graphing equations by plotting points, finding x- and y-intercepts, and applying the distance and midpoint formulas. It is part of Chapter 2: Equations and Inequalities, which builds foundational skills for working with linear and quadratic equations.

In this Grade 7 lesson from OpenStax Algebra and Trigonometry, students learn to solve linear equations in one variable using algebraic operations, including how to identify identity, conditional, and inconsistent equations. The lesson also covers solving rational equations, finding linear equations, and determining whether two lines are parallel or perpendicular based on their equations. Real-world problem-solving contexts, such as calculating work hours needed to reach a savings goal, are used to ground the algebraic concepts throughout the chapter.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry, students learn how to set up and solve linear equations to model real-world applications, including problems involving fixed costs, variable costs, and unknown quantities. Using examples like car rental fees, cell phone plans, and test score calculations, students practice translating verbal expressions into mathematical equations. The lesson covers identifying known quantities, assigning variables to unknowns, and applying a step-by-step method to solve practical problems from Chapter 2: Equations and Inequalities.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry, students learn to work with complex numbers by expressing square roots of negative numbers as multiples of the imaginary unit i and writing results in standard form a + bi. The lesson covers adding, subtracting, multiplying, and dividing complex numbers, simplifying powers of i, and plotting complex numbers on the complex plane. Students build on their knowledge of real numbers to understand how imaginary and complex numbers extend the number system.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry, students learn four methods for solving quadratic equations: factoring, the square root property, completing the square, and the quadratic formula. The lesson covers key concepts including standard form, the zero-product property, and factoring expressions with a leading coefficient of 1. Part of Chapter 2: Equations and Inequalities, this lesson builds the foundational algebraic skills needed to solve real-world problems involving second-degree polynomials.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry (Chapter 2), students learn how to solve several advanced equation types, including equations with rational exponents, radical equations, absolute value equations, and polynomial equations solved through factoring. Key techniques include raising both sides of an equation to a reciprocal exponent to isolate a variable and applying the zero factor theorem after factoring out the lowest-exponent term. This lesson builds on prior knowledge of linear, rational, and quadratic equations to expand students' algebraic problem-solving toolkit.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry, students learn how to express and solve linear inequalities and absolute value inequalities using interval notation, set-builder notation, and number lines. The lesson covers key concepts such as using brackets and parentheses to represent inclusive and exclusive bounds, applying properties of inequalities, and solving inequalities in one variable algebraically. Part of Chapter 2: Equations and Inequalities, this lesson builds the foundational skills students need for higher-level algebra and trigonometry topics.

Grade 7 students explore functions and function notation in Lesson 3.1 of OpenStax Algebra and Trigonometry, learning to distinguish relations from functions by identifying domain, range, input, and output values. The lesson covers how to determine whether a relation represents a function, evaluate function values, apply the vertical line test, and recognize one-to-one functions. Students build foundational skills for understanding how dependent and independent variables relate across mathematical and real-world contexts.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry, students learn how to find the domain and range of functions defined by equations, including how to identify and exclude values that cause division by zero or even roots of negative numbers. The lesson also introduces interval notation, teaching students to express sets of numbers using brackets and parentheses to indicate inclusive and exclusive endpoints. Students additionally explore how to graph piecewise-defined functions within the broader context of Chapter 3: Functions.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry, students learn how to calculate the average rate of change of a function using the formula delta-y over delta-x, or the change in output divided by the change in input. Students also use graphs to identify where a function is increasing, decreasing, or constant, and to locate local and absolute maxima and minima. The lesson is part of Chapter 3: Functions and applies these concepts to real-world contexts like gasoline price changes over time.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry, students learn how to combine functions using algebraic operations — including addition, subtraction, multiplication, and division of functions — and how to create new functions through composition, written in the form C(T(d)). Students practice evaluating composite functions, finding the domain of a composite function, and decomposing a composite function back into its component parts. This lesson builds essential skills for understanding how functions interact and depend on one another within Chapter 3: Functions.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry, students learn how to transform functions by applying vertical and horizontal shifts, reflections across the x-axis and y-axis, and compressions and stretches to their graphs. Using notation such as g(x) = f(x) + k, students practice identifying how changes to a function's equation affect its graph and determine whether a function is even, odd, or neither. The lesson is part of Chapter 3: Functions and builds students' ability to combine multiple transformations to model real-world scenarios.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry, students learn how to graph absolute value functions by identifying the corner vertex and applying transformations such as vertical stretches, horizontal shifts, and reflections. The lesson also covers how to write and solve absolute value equations, including expressing real-world constraints like manufacturing tolerances and test score ranges using absolute value notation. This is Lesson 3.6 in Chapter 3: Functions.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry, students learn how to verify, find, and evaluate inverse functions using function notation such as f⁻¹(x). The lesson covers determining the domain and range of inverse functions, restricting domains to make functions one-to-one, and using the graph of a one-to-one function to graph its inverse on the same axes. Part of Chapter 3: Functions, this lesson builds students' understanding of how inverse functions "undo" each other through function composition.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry, students learn to represent and interpret linear functions using word form, function notation, tabular form, and graphical form. The lesson covers key concepts including slope as a rate of change, the slope-intercept form f(x) = mx + b, and how to determine whether a linear function is increasing, decreasing, or constant. Students also explore how to write equations for parallel and perpendicular lines, using real-world contexts like train motion and plant growth to ground the algebra.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry, students learn to build linear function models from verbal descriptions by identifying initial values, rates of change, and slope-intercept form. Using real-world scenarios, they write equations in the form M(t) = mt + b, find horizontal axis intercepts by setting output to zero, and determine reasonable domain restrictions. The lesson is part of Chapter 4: Linear Functions and develops core skills in modeling constant rate of change situations.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry (Chapter 5), students learn to identify and analyze quadratic functions by recognizing key characteristics of parabolas, including the vertex, axis of symmetry, zeros, y-intercept, and how the coefficient a in both general form f(x) = ax² + bx + c and standard form f(x) = a(x - h)² + k determines whether a parabola opens upward or downward. Students also practice finding a quadratic function's minimum or maximum value and apply these concepts to real-world problems involving area and projectile motion.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry, students explore the graphs of polynomial functions, learning to identify key characteristics such as zeros and their multiplicities, end behavior, turning points, and continuity. Students practice using factoring to find x-intercepts and apply the Intermediate Value Theorem to analyze polynomial behavior. The lesson builds graph-reading and function analysis skills within the context of Chapter 5: Polynomial and Rational Functions.

In this Grade 7 lesson from OpenStax Algebra and Trigonometry, students learn how to divide polynomials using both long division and synthetic division. The lesson covers the Division Algorithm, showing how any polynomial dividend can be expressed as the product of a divisor and quotient plus a remainder. These skills are applied to real-world problems involving polynomial expressions, building on students' existing understanding of arithmetic long division.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry, students learn how to find the zeros of polynomial functions using key theorems including the Remainder Theorem, the Factor Theorem, the Rational Zero Theorem, and Descartes' Rule of Signs. The lesson also introduces the Linear Factorization Theorem and guides students through using synthetic division to evaluate and fully factor polynomial expressions. These skills are applied to real-world problems, such as solving cubic equations to determine the dimensions of a rectangular solid.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry (Chapter 5), students learn to work with rational functions — functions that have variables in the denominator — by finding domains, identifying vertical and horizontal asymptotes, and using arrow notation to describe end behavior. Students also practice graphing rational functions and apply these concepts to real-world problems such as calculating average cost functions. The lesson builds directly on prior work with polynomial functions and toolkit functions like the reciprocal function f(x) = 1/x.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry, students learn how to find the inverse of invertible polynomial functions and how to restrict the domain of a polynomial function to make it one-to-one before finding its inverse. The lesson covers key concepts including radical functions, square root functions as inverses of quadratic functions, and the horizontal line test, using real-world contexts like cone volume and parabolic troughs to illustrate the ideas. Part of Chapter 5: Polynomial and Rational Functions, this lesson builds students' understanding of how inverse functions reverse input-output relationships in polynomial and rational expressions.

In this Grade 7 lesson from OpenStax Algebra and Trigonometry, students learn to solve direct variation, inverse variation, and joint variation problems using the constant of variation and equations of the form y = kxⁿ. The lesson covers identifying the constant of variation, writing variation equations, and applying them to real-world contexts such as sales commissions and ocean temperature. Students practice all three variation types with worked examples and guided exercises from Chapter 5 on Polynomial and Rational Functions.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry, students learn how to graph exponential functions of the form f(x) = b^x, identifying key characteristics such as the horizontal asymptote, domain, range, and constant ratio. The lesson covers both exponential growth (b > 1) and exponential decay (0 < b < 1), and extends to graphing exponential functions using transformations. Students practice plotting points, drawing smooth curves, and analyzing how changes to the base affect the shape and behavior of the graph.

In this Grade 7 lesson from OpenStax Algebra and Trigonometry, students learn how to convert between logarithmic and exponential form, evaluate logarithms, and work with common (base-10) and natural logarithms. The lesson introduces the definition of the logarithmic function log_b(x) = y as the inverse of the exponential function b^y = x, using real-world examples like the Richter Scale to build understanding. Students practice applying these concepts to solve problems involving base-ten logarithmic scales and exponential relationships.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry, students learn key logarithmic properties including the product rule, quotient rule, power rule, and the change-of-base formula. The lesson covers how to expand and condense logarithmic expressions by applying these rules, drawing on the connection between logarithms and exponents. Real-world context such as the pH formula is used to illustrate how logarithmic properties apply in science and everyday situations.

In this Grade 7 lesson from OpenStax Algebra and Trigonometry, students learn how to solve exponential and logarithmic equations using techniques such as the one-to-one property of exponential functions, rewriting terms with a common base, and applying logarithms to isolate unknowns. The lesson covers both like-base and unlike-base exponential equations, as well as logarithmic equations solved through the definition of a logarithm and its one-to-one property. Real-world applications, including exponential population growth, are used to show how these equation-solving methods apply to practical problems.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry, students learn to draw angles in standard position on the coordinate plane, identify initial and terminal sides, and distinguish between positive and negative angles. The lesson covers converting between degrees and radians, finding coterminal angles, and calculating arc length using angle measures. Students also apply linear and angular speed to describe motion along a circular path.

In this Grade 7 lesson from OpenStax Algebra and Trigonometry, students learn to define and evaluate all six trigonometric functions — sine, cosine, tangent, secant, cosecant, and cotangent — using the ratios of sides in a right triangle, including the mnemonic SohCahToa. The lesson covers finding exact function values for special angles such as 30°, 45°, and 60°, and applying cofunction relationships for complementary angles. Students also use right-triangle trigonometry to solve real-world problems, such as measuring heights and distances.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry, students learn how to define sine and cosine functions using the unit circle, where the x- and y-coordinates of a point on the circle correspond directly to cosine and sine values of the central angle. The lesson covers finding exact trigonometric function values for 30°, 45°, and 60°, identifying the domain and range of sine and cosine, and using reference angles to evaluate trigonometric functions across all four quadrants. This foundational unit circle approach builds on students' earlier work with right triangle trigonometry and prepares them for analyzing periodic and circular motion.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry, students learn to define and evaluate the four remaining trigonometric functions — tangent, secant, cosecant, and cotangent — using unit circle coordinates and reciprocal relationships with sine and cosine. The lesson covers finding exact values at common angles such as π/3, π/4, and π/6, using reference angles, and applying even and odd function properties. Students also practice using fundamental identities and a calculator to evaluate all six trigonometric functions.

In this Grade 7 lesson from OpenStax Algebra and Trigonometry, students learn to graph the sine and cosine functions by connecting unit circle values to coordinates on a coordinate plane. The lesson covers key properties such as amplitude, period, and phase shifts of y = sin(x) and y = cos(x). Students practice interpreting and creating variations of these graphs, building foundational skills in periodic functions.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry, students explore the graphs of the tangent, secant, cosecant, and cotangent functions, analyzing key features such as vertical asymptotes, periodicity, and odd/even function behavior. Learners graph variations of y = tan x, y = sec x, y = csc x, and y = cot x by identifying transformations including shifts and stretches. The lesson is part of Chapter 8 on Periodic Functions and builds on prior knowledge of sine and cosine graphs.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry, students learn how to understand and apply the inverse sine, cosine, and tangent functions — also known as arcsine, arccosine, and arctangent — to find an angle measure from a given ratio of sides. The lesson covers how restricting the domains of trigonometric functions makes them invertible, and guides students through finding exact values of inverse trig expressions and evaluating composite functions. Students also practice using a calculator to evaluate inverse trigonometric functions in the context of Chapter 8's study of periodic functions.

In this Grade 7 lesson from OpenStax Algebra and Trigonometry, students learn how to verify and apply fundamental trigonometric identities, including the Pythagorean identities, reciprocal identities, quotient identities, and even-odd identities. The lesson covers algebraic techniques such as factoring, expanding expressions, and finding common denominators to simplify trigonometric expressions. Students practice rewriting complex trigonometric expressions by transforming one side of an equation into an equivalent form using these core identities.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry, students learn the sum and difference identities for cosine, sine, and tangent, including how these formulas are derived using the unit circle and the distance formula. Students practice applying the sum and difference formulas to find exact trigonometric values of angles by rewriting them as combinations of known special angles, and extend this to cofunction identities. The lesson also covers using these identities to verify and simplify complex trigonometric expressions and equations.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry, students learn how to solve trigonometric equations involving sine and cosine, including linear equations, quadratic-form equations, and equations with multiple angles. The lesson covers techniques such as using the unit circle, applying fundamental identities, and adding periodic solutions in the form 2πk to express all possible answers. Students also practice solving right triangle problems using a calculator, building on the trigonometric identities studied earlier in Chapter 9.

In this Grade 7 lesson from OpenStax Algebra and Trigonometry, students learn how to apply the Law of Sines to solve oblique triangles using the ASA, AAS, and SSA configurations. The lesson covers how the sine ratios of angles to their opposite sides are equal across all three vertices, and students use this relationship to find unknown side lengths and angle measurements. Real-world applications, such as determining aircraft altitude from two radar stations, demonstrate how the Law of Sines extends trigonometry beyond right triangles.

In this Grade 7 lesson from OpenStax Algebra and Trigonometry, students learn to apply the Law of Cosines to solve oblique triangles in SAS and SSS configurations where the Law of Sines falls short. The lesson covers deriving the three Law of Cosines equations from the Generalized Pythagorean Theorem, solving for unknown sides and angles, and using Heron's formula to calculate triangle area. Real-world applied problems, such as finding distances in navigation scenarios, help students connect these formulas to practical contexts.

In this Grade 7 lesson from OpenStax Algebra and Trigonometry, students learn to plot points using polar coordinates (r, θ) on a polar grid and convert between polar and rectangular coordinate systems using the relationships x = r cosθ and y = r sinθ. The lesson also covers transforming equations between polar and rectangular forms and graphing polar equations. It is part of Chapter 10: Further Applications of Trigonometry.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry, students learn how to test polar equations for symmetry with respect to the line θ = π/2, the polar axis, and the pole by substituting coordinate pairs such as (−r, −θ) and (r, −θ). Students also practice graphing polar equations by plotting points on the polar grid using coordinates in the form (r, θ). The lesson is part of Chapter 10: Further Applications of Trigonometry and builds students' understanding of how polar coordinate systems represent curves differently from Cartesian equations.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry (Chapter 10), students learn to convert complex numbers between rectangular and polar form using the modulus, argument, and trigonometric conversion formulas. The lesson covers plotting complex numbers in the complex plane, finding absolute values using the formula |z| = √(x² + y²), and applying De Moivre's Theorem to find products, quotients, powers, and roots of complex numbers in polar form. These skills build a foundation for working with complex numbers analytically and geometrically.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry, students learn how to work with parametric equations by expressing x and y as separate functions of an independent parameter t. The lesson covers how to parameterize a curve, eliminate the parameter, convert between parametric and rectangular equations, and use parametric equations to describe motion and direction along a path. Drawing from Chapter 10 on Further Applications of Trigonometry, students gain tools for representing curves that cannot be expressed as standard functions of the form y = f(x).

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry (Chapter 10), students learn how to graph plane curves described by parametric equations by plotting points and constructing tables of t, x(t), and y(t) values. The lesson covers sketching graphs of both polynomial and trigonometric parametric equations, including identifying features such as vertex location and curve direction based on increasing values of t. Real-world applications like projectile motion in baseball are used to show how parametric equations model the path of a moving object.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry, students explore vectors as directed line segments, learning key concepts including magnitude, direction, standard position, and component form. Building on trigonometry foundations from Chapter 10, students practice vector addition, scalar multiplication, unit vectors, and the dot product using i and j notation. Real-world applications such as calculating an airplane's ground speed and bearing show how vector operations apply outside the classroom.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry, students learn to solve three-by-three systems of linear equations using Gaussian elimination and back-substitution to find ordered triple solutions. The lesson covers how to achieve upper triangular form, identify inconsistent systems with no solution, and express solutions for dependent systems with infinitely many solutions. Real-world investment scenarios are used to ground the algebraic concepts in practical context.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry, students learn the fundamentals of matrices, including how to identify matrix dimensions, locate entries using row-column notation such as a₂₃, and classify types like square, row, and column matrices. Students then practice finding the sum and difference of two matrices by adding or subtracting corresponding entries, as well as computing scalar multiples and products of matrices. This lesson builds the foundation for using matrix operations to organize and solve real-world data problems and systems of equations.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry, students learn how to evaluate 2×2 and 3×3 determinants and apply Cramer's Rule to solve systems of two and three linear equations. The lesson covers the formula det(A) = ad - bc for a 2×2 matrix and demonstrates how determinants are used to find unique solutions, identify inconsistent systems, and understand matrix invertibility. Part of Chapter 11: Systems of Equations and Inequalities, this lesson builds on prior methods such as substitution and Gaussian elimination by introducing a determinant-based approach.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry, students explore the hyperbola as a conic section, learning to identify its key features including vertices, foci, the transverse axis, conjugate axis, and asymptotes. Students practice writing equations of hyperbolas in standard form and graphing them both at and away from the origin. Real-world applications such as sonic booms and cooling towers show how hyperbolas model phenomena involving the difference of distances from two foci.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry, students learn to graph parabolas and write their equations in standard form, exploring key features such as the vertex, focus, directrix, axis of symmetry, and latus rectum. The lesson covers parabolas with vertices at the origin as well as vertices at other points in the coordinate plane, including the derivation of the parabola equation using the distance formula. Students also apply these concepts to real-world problems involving parabolic reflectors, satellite dishes, and orbital calculations.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry (Chapter 12: Analytic Geometry), students learn how to identify, graph, and write polar equations of conic sections — ellipses, parabolas, and hyperbolas — using eccentricity and the focus-directrix relationship. Students practice converting polar equations to standard form and determining the type of conic by comparing the eccentricity value to 1. Real-world applications, such as modeling planetary and comet orbits in polar coordinates, help students connect these algebraic concepts to physical phenomena.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry, students learn how to write and interpret sequences using both explicit formulas and recursive formulas, as well as how to apply factorial notation. The lesson covers key concepts such as identifying terms of a sequence, using subscript notation like a_n, and evaluating the nth term of a sequence from its formula. It is part of Chapter 13, which introduces sequences alongside probability and counting theory.

In this Grade 7 math lesson from OpenStax Algebra and Trigonometry, students learn to identify and work with geometric sequences by finding the common ratio, listing terms, and applying both recursive and explicit formulas. The lesson covers how to determine whether a sequence is geometric by dividing consecutive terms, and how to generate new terms by multiplying by a constant factor. Real-world contexts like annual salary growth are used to connect the concept of geometric sequences to exponential change.

In this Grade 7 lesson from OpenStax Algebra and Trigonometry, students learn how to construct probability models, compute probabilities of equally likely outcomes using the formula P(E) = n(E)/n(S), and apply the complement rule and counting theory to find probabilities. The lesson introduces key terms such as experiment, sample space, outcomes, and events, building a foundation for understanding the likelihood of real-world scenarios. Students practice these concepts through examples like rolling a number cube and tossing a fair coin.