Subjects › Algebra 1
Algebra 1: learn it by talking through the problems
This Algebra 1 course follows the Common Core traditional pathway, from operations with signed numbers through functions and data analysis. You'll work with expressions and polynomials, solve linear, quadratic, radical, and absolute-value equations and inequalities, graph and build linear, quadratic, exponential, and piecewise functions, solve systems of equations, and describe one- and two-variable data. A few enrichment topics - the Binomial Theorem, matrices, logarithms, and function composition - are included as optional extensions beyond the core standards.
With a Socratic tutor, you don't get handed a worked example to copy. The tutor asks you to try a step, listens to what you actually did, and asks the next question based on that - whether that means pushing you further or backing up to rebuild something shaky. Over time the tutor checks that ideas you explained last month, like why squaring both sides of an equation can introduce extraneous solutions, still hold up when they resurface in a new problem.
Who it is for
- Students taking or about to take Algebra 1 in high school
- Students who know middle school math (fractions, integers, percent, one- and two-step equations) but haven't done polynomial algebra
- Parents comparing how a course will actually teach, not just what it covers
- Teachers looking for a tool that checks both explanation and execution, not just final answers
- Students reviewing before Algebra 2 who want to firm up specific weak spots
What you will be able to do
- Simplify expressions and operate on polynomials and rational expressions, including factoring
- Solve linear, quadratic, absolute-value, radical, and rational equations and inequalities, and explain why each step is valid
- Graph and interpret linear, quadratic, exponential, square root, absolute value, and piecewise functions
- Solve systems of linear equations and systems of a linear and a quadratic equation, algebraically and graphically
- Build and compare linear, quadratic, and exponential models from tables, graphs, or word problems
- Display and describe one-variable and two-variable data, including fitting a line and reading residuals
A taste of a lesson on Solving quadratic equations by factoring, square roots, and the quadratic formula
The tutor asks, you answer, and it works from what you actually said. No videos to sit through.
The 38 topics in Algebra 1
Each topic builds on the ones before it. The tutor checks what you already know and starts there.
Operations with signed numbers and rational numbers
Add, subtract, multiply, and divide integers and fractions/decimals, including negatives. Understand the number line, absolute value, and order of operations (PEMDAS). Recognize that sums and products of rationals are rational and that rational plus irrational is irrational. Real understanding means explaining why a negative times a negative is positive using pattern or number-line reasoning.
Integer exponents, rational exponents, and radicals
Laws of exponents (product, quotient, power, zero and negative exponents). Define rational exponents by extending integer exponent properties, e.g., 5^(1/3) is the cube root of 5, and rewrite radicals with rational exponents and back. Simplify expressions using these properties. Understanding means explaining why the definition must be 5^(1/3) rather than something else.
Builds on Operations with signed numbers and rational numbers
Interpreting expressions and defining quantities
Read algebraic expressions as descriptions of contexts: identify terms, factors, coefficients, and constants; view a complicated part as a single entity (e.g., in P(1+r)^n, treat (1+r) as one factor). Define variables and quantities with appropriate units, track units through multi-step problems, and choose sensible scales and origins for graphs. Choose appropriate accuracy when reporting measured quantities.
Builds on Operations with signed numbers and rational numbers
Adding, subtracting, and multiplying polynomials
Polynomials are closed under addition, subtraction, and multiplication. Combine like terms, distribute, and multiply binomials and larger polynomials (FOIL and the general distributive method). Understand degree and standard form. Real understanding includes seeing why polynomials behave like integers under these operations.
Builds on Operations with signed numbers and rational numbers, Interpreting expressions and defining quantities
Polynomial identities
Prove and use identities such as (a+b)^2 = a^2+2ab+b^2, (a-b)^2, the difference of squares a^2-b^2=(a-b)(a+b), and the sum/difference of cubes. Use identities to compute and describe numerical relationships, e.g., generating Pythagorean triples from (x^2+y^2)^2 = (x^2-y^2)^2 + (2xy)^2. Understanding means justifying an identity by expanding both sides.
Builds on Adding, subtracting, and multiplying polynomials
Factoring quadratic and other polynomial expressions
Factor out greatest common factors, factor quadratics of the form x^2+bx+c and ax^2+bx+c, factor difference of squares and perfect square trinomials, and factor by grouping. Recognize structure in an expression (e.g., x^4-y^4 as a difference of squares) to choose the factoring. Understand factoring as reversing expansion and check results by multiplying out.
Builds on Polynomial identities
Simplifying and operating on rational expressions
Simplify rational expressions by factoring and canceling common factors, noting excluded values. Add, subtract, multiply, and divide rational expressions, finding common denominators. Rewrite a(x)/b(x) as q(x) + r(x)/b(x) by polynomial long division with degree of r less than degree of b. Understand that rational expressions are closed under these operations (division by a nonzero expression).
Builds on Factoring quadratic and other polynomial expressions
Remainder Theorem, zeros, and rough graphs of polynomials
Divide a polynomial by x - a and state the remainder equals p(a); conclude p(a)=0 iff (x-a) is a factor. Identify zeros from factored forms and use zeros, end behavior, and degree to sketch a rough graph of a polynomial function. Understanding includes why each zero corresponds to an x-intercept.
Builds on Factoring quadratic and other polynomial expressions
Binomial Theorem and Pascal's Triangle
Build Pascal's Triangle row by row and use it to expand (x+y)^n for positive integer n, with coefficients given by binomial coefficients. Understand why the coefficients count ways to choose terms. This is an enrichment topic (advanced) for students extending beyond the core.
Builds on Polynomial identities
Solving linear equations and inequalities in one variable
Solve linear equations including those with letter coefficients, justifying each step as preserving equality. Solve linear inequalities, reversing the inequality sign when multiplying or dividing by a negative. Solve equations with rational/fractional coefficients and detect no-solution or all-real-solution cases. Rearrange formulas to isolate a chosen variable (e.g., V=IR solved for R, or A=P(1+rt) solved for r).
Builds on Operations with signed numbers and rational numbers, Interpreting expressions and defining quantities
Solving absolute value equations and inequalities
Solve equations of the form |ax+b| = c and inequalities |ax+b| < c or |ax+b| > c in one variable, interpreting |x - a| as distance on the number line. Recognize no-solution cases when c is negative, split into two linear cases, and graph solution sets on a number line. Check solutions in the original equation.
Builds on Operations with signed numbers and rational numbers, Solving linear equations and inequalities in one variable
Solving radical and rational equations; extraneous solutions
Solve simple equations containing square roots or rational expressions by isolating the radical or clearing denominators. Explain why squaring or multiplying by an expression containing the variable can introduce extraneous solutions and check all candidates in the original equation.
Builds on Integer exponents, rational exponents, and radicals, Simplifying and operating on rational expressions, Solving linear equations and inequalities in one variable
Justifying solution steps
Explain each step in solving an equation as following from equality asserted at the previous step, starting from the assumption that a solution exists. Construct a viable argument (a short logical chain) that a proposed value is or is not a solution, and identify where an invalid step (e.g., dividing by zero, squaring without checking) breaks the argument.
Builds on Solving linear equations and inequalities in one variable
Solving quadratic equations by factoring, square roots, and the quadratic formula
Solve quadratics in one variable by inspection, taking square roots (e.g., x^2=49), factoring, and the quadratic formula, choosing the method suited to the form. Recognize when the quadratic formula gives complex or no real solutions and when the discriminant indicates this. Real understanding includes knowing which method to pick and why the zero-product property justifies factoring.
Builds on Integer exponents, rational exponents, and radicals, Factoring quadratic and other polynomial expressions, Solving linear equations and inequalities in one variable
Creating and solving equations and inequalities to model problems
Translate word problems into one-variable linear, quadratic, simple rational, and exponential equations or inequalities; solve them; and interpret solutions as viable or nonviable in context. Includes constraints (e.g., nutrition or budget limits) and checking reasonableness of answers.
Builds on Simplifying and operating on rational expressions, Solving linear equations and inequalities in one variable, Solving quadratic equations by factoring, square roots, and the quadratic formula
Completing the square and deriving the quadratic formula
Transform any quadratic ax^2+bx+c into the form (x-p)^2 = q by completing the square, solve it, and derive the quadratic formula from this process. Use completing the square to find the vertex and maximum or minimum value of a quadratic function, and to rewrite the function in vertex form.
Builds on Polynomial identities, Solving quadratic equations by factoring, square roots, and the quadratic formula
Factoring and completing the square to reveal function features
Use factored form to find zeros, completed-square (vertex) form to find maximum or minimum, and both to find the axis of symmetry. Interpret these features in context (e.g., maximum height of a projectile, break-even points of a profit model). Choose the most useful form for the question.
Builds on Remainder Theorem, zeros, and rough graphs of polynomials, Completing the square and deriving the quadratic formula
Graphing linear equations and functions; the coordinate plane
Understand that the graph of an equation in two variables is the set of all its solutions in the coordinate plane, often a curve (a line for linear equations). Graph lines from slope-intercept, point-slope, and standard form, with labeled axes and appropriate scales. Create two-variable linear equations from context, identify intercepts, and explain slope as rate of change.
Builds on Interpreting expressions and defining quantities, Solving linear equations and inequalities in one variable
Solving systems of linear equations exactly and approximately
Solve pairs of linear equations in two variables by graphing, substitution, and elimination; interpret the solution as the intersection point. Identify no-solution (parallel) and infinitely-many-solution (same line) cases. Estimate solutions from graphs when exact values are not needed. Use systems to solve word problems.
Builds on Solving linear equations and inequalities in one variable, Graphing linear equations and functions; the coordinate plane
Why elimination works: equivalent systems
Prove that replacing one equation in a two-variable system with the sum of that equation and a multiple of the other produces a system with the same solutions. Understand this as the logical foundation for elimination and for matrix row operations.
Builds on Justifying solution steps, Solving systems of linear equations exactly and approximately
Linear inequalities in two variables and systems of inequalities
Graph the solution to a linear inequality in two variables as a half-plane, using a dashed boundary for strict inequalities and solid for non-strict. Graph the solution set of a system of linear inequalities as the intersection of half-planes. Represent real constraints (e.g., nutrition, budget, time) with systems and interpret feasible regions as viable options.
Builds on Creating and solving equations and inequalities to model problems, Solving systems of linear equations exactly and approximately
Matrices: representing and solving systems (enrichment)
Represent a system of linear equations as a single matrix equation Ax = b in a vector variable. Find the inverse of a 2x2 matrix when it exists (determinant nonzero) and use it to solve the system; understand that 3x3 and larger inverses are computed with technology.
Builds on Why elimination works: equivalent systems
Functions: definition, notation, domain and range
Define a function as an assignment from each domain element to exactly one range element. Use function notation f(x), evaluate functions at inputs in the domain, and interpret statements like f(3)=10 in context. Identify domain restrictions from context and relate them to the graph. Recognize sequences as functions with domain a subset of integers, including recursively defined sequences such as Fibonacci.
Builds on Interpreting expressions and defining quantities, Graphing linear equations and functions; the coordinate plane
Key features of function graphs and average rate of change
For functions modeling relationships, interpret intercepts, intervals of increase/decrease, maxima/minima, symmetry, end behavior, and periodicity in terms of the quantities. Sketch a graph from a verbal description. Calculate and interpret average rate of change over an interval from a formula, table, or graph, and estimate rate of change from a graph.
Builds on Functions: definition, notation, domain and range
Linear models: slope, intercept, and building linear functions
Construct linear functions from a graph, a verbal description, or two input-output pairs (including from tables). Interpret the slope as constant rate of change and the intercept as the starting value, in context. Interpret parameters of a linear function in context. Distinguish situations modeled by linear functions (constant difference per equal interval) from those modeled by exponential functions (constant ratio).
Builds on Key features of function graphs and average rate of change
Distinguishing linear and exponential growth
Prove that linear functions change by equal differences over equal intervals and exponential functions change by equal factors over equal intervals. Recognize constant-rate (linear) versus constant-percent-rate (exponential) situations in context. Observe with tables and graphs that exponential growth eventually exceeds linear, quadratic, or polynomial growth.
Builds on Integer exponents, rational exponents, and radicals, Linear models: slope, intercept, and building linear functions
Transformations of function graphs
Describe the effect on a graph of replacing f(x) by f(x)+k, kf(x), f(kx), and f(x+k) for positive and negative k. Given a graph, find the value of k that produces it. Experiment with families such as quadratics, absolute value, and square roots to explain the transformation rules.
Builds on Functions: definition, notation, domain and range, Key features of function graphs and average rate of change
Square root, absolute value, step, and piecewise functions
Graph square root, cube root, absolute value, step, and piecewise-defined functions, identifying key points, domain, and range. Evaluate piecewise functions at given inputs and interpret piecewise models in context (e.g., tax brackets, shipping charges). Connect absolute value graphs to the distance interpretation of |x - a| from absolute value equations.
Builds on Solving absolute value equations and inequalities, Transformations of function graphs
Quadratic functions: graphs, features, and models
Graph quadratic functions in standard, factored, and vertex forms, showing intercepts, vertex (maximum or minimum), and axis of symmetry. Model projectile motion and area problems with quadratics and interpret features in context. Compare quadratic growth with linear growth.
Builds on Factoring and completing the square to reveal function features, Transformations of function graphs
Systems of a linear and a quadratic equation
Solve a system with one linear and one quadratic equation algebraically (substitution leading to a quadratic) and graphically (line and parabola or circle), finding up to two intersection points, and recognize when there are none. Example: intersection of y = -3x with x^2 + y^2 = 3.
Builds on Solving quadratic equations by factoring, square roots, and the quadratic formula, Why elimination works: equivalent systems, Quadratic functions: graphs, features, and models
Exponential functions: graphs, growth and decay, and properties of exponents
Write and graph exponential functions y = ab^x, interpret the initial value a and growth/decay factor b in context, and identify percent rate of change (e.g., 1.02^t is 2% growth). Use exponent properties to rewrite expressions, e.g., 1.15^t = (1.15^(1/12))^(12t) ≈ 1.012^(12t) to find equivalent monthly rates. Graph exponential functions showing intercepts and end behavior.
Builds on Integer exponents, rational exponents, and radicals, Key features of function graphs and average rate of change, Distinguishing linear and exponential growth
Arithmetic and geometric sequences: recursive and explicit forms
Write arithmetic and geometric sequences both recursively (a_{n+1} = a_n + d or r·a_n) and with explicit formulas, translate between forms, and use them to model situations. Derive and use the formula for the sum of a finite geometric series with ratio not equal to 1, e.g., for mortgage payments or savings.
Builds on Solving linear equations and inequalities in one variable, Functions: definition, notation, domain and range, Exponential functions: graphs, growth and decay, and properties of exponents
Building functions: combining, composing, and modeling
Write explicit, recursive, or step-by-step functions describing a relationship from a context. Combine standard function types by arithmetic operations (e.g., a constant plus a decaying exponential for cooling). Compose functions, e.g., T(h(t)), as an enrichment extension, and interpret the composition in context.
Builds on Transformations of function graphs, Quadratic functions: graphs, features, and models, Arithmetic and geometric sequences: recursive and explicit forms
Inverse functions
Find the inverse of a simple invertible function by solving y = f(x) for x and swapping variables, e.g., f(x)=2x^3 or f(x)=(x+1)/(x-1). Verify by composition that two functions are inverses. Read inverse values from a graph or table. Restrict a domain (e.g., x^2 for x≥0) to produce an invertible function.
Builds on Solving radical and rational equations; extraneous solutions, Building functions: combining, composing, and modeling
Logarithms as inverses of exponentials
Understand logarithms as the inverse relationship to exponentials, solve ab^(ct)=d by expressing the solution as a logarithm, and evaluate logarithms with base 2, 10, or e using technology. Use logarithms and exponents together to solve problems. Enrichment, usually taken in Algebra 2 but included here as an extension of exponentials.
Builds on Exponential functions: graphs, growth and decay, and properties of exponents, Inverse functions
Displaying and describing one-variable data
Represent data on a number line with dot plots, histograms, and box plots. Describe shape, center, and spread, and choose median/IQR for skewed data or mean/standard deviation for symmetric data. Compare two or more data sets by center and spread, accounting for outliers. Understand standard deviation as typical distance from the mean and use the normal distribution to estimate percentages when appropriate, recognizing when it is not.
Builds on Operations with signed numbers and rational numbers, Interpreting expressions and defining quantities
Two-way frequency tables
Summarize categorical data for two categories in two-way tables. Compute joint, marginal, and conditional relative frequencies and interpret them in context. Recognize possible associations between categories from differences in conditional relative frequencies.
Builds on Displaying and describing one-variable data
Scatter plots, linear fit, residuals, and correlation
Represent bivariate data on a scatter plot and describe the relationship (form, direction, strength, outliers). Fit a linear function to data suggesting linear association, and fit other functions (e.g., exponential, quadratic) when appropriate; use fitted functions to predict. Assess fit by plotting and analyzing residuals. Interpret the slope and intercept of a linear model in context. Compute and interpret the correlation coefficient r. Distinguish correlation from causation.
Builds on Linear models: slope, intercept, and building linear functions, Displaying and describing one-variable data
Practice questions in Algebra 1
A few of the kinds of questions you will work through, with answers.
Evaluate: -3 + 4 × (-2) - (-6)
See the answer
-5
By order of operations, multiply first: 4 × (-2) = -8. Then -3 + (-8) - (-6) = -3 - 8 + 6 = -5.
Factor: x² - 9x + 14
See the answer
(x - 2)(x - 7)
Need two numbers multiplying to 14 and adding to -9: -2 and -7 work.
Solve for x: 3(x - 2) + 5 = 2x + 1
See the answer
x = 0
Distribute to get 3x - 6 + 5 = 2x + 1, so 3x - 1 = 2x + 1, giving x = 2... recheck: 3x - 1 = 2x + 1 → x = 2.
Solve: x² = 49
See the answer
x = 7 or x = -7
Taking the square root of both sides gives two solutions since both 7² and (-7)² equal 49.
A population grows according to P(t) = 500(1.04)^t, where t is in years. What is the annual percent growth rate, and what does 500 represent?
See the answer
4% growth per year; 500 is the initial population
In y = ab^x, b = 1 + r gives the growth factor, so 1.04 means r = 0.04, or 4%, and a is the starting value.
A scatter plot shows points scattered fairly evenly above and below a fitted line, with no obvious curve or widening pattern. What does this suggest about the linear fit?
See the answer
The linear model is a reasonable fit
A residual pattern with no systematic curve or spread change suggests the linear model captures the relationship reasonably well.
Questions about this subject
Do I need to know any algebra before starting?
No prior algebra with polynomials is assumed. You should be comfortable with fractions, decimals, negative numbers, percent, order of operations, and simple one- and two-step equations from middle school.
Are the enrichment topics like matrices and logarithms required?
No. The Binomial Theorem, matrices, logarithms, and function composition are marked as enrichment and go beyond the core Algebra 1 standards; they're there if you want to go further, but the course works without them.
Does this course cover trigonometry or complex numbers?
No. The course stops before trigonometry, complex numbers, and full logarithm laws, which belong to Algebra 2.
How does the tutor handle a topic I get wrong?
The tutor asks a follow-up question based on your specific answer, often stepping back to an earlier idea the current topic builds on, rather than just marking it wrong and moving on.
Will this prepare me for Algebra 2?
The course covers the full Common Core Algebra 1 scope, including functions, systems, and quadratics, which are the main prerequisites for Algebra 2.