Algebra 1 Online Course for Grades 7 and 8

Signed numbers and the order of operations, then the laws of exponents with scientific notation, then radicals, higher-degree roots and fractional exponents.

• Working with Numbers and Order of OperationsSigned-number arithmetic, the number system, properties of arithmetic and the order of operations (PEMDAS) with exponents and absolute value. Also expanding and factoring with the distributive property and multiplying fractions by canceling.
• Exponents and the Laws of PowersPowers, base and exponent, and perfect squares, then the laws of exponents: same-base products and quotients, negative and zero exponents, a power of a power, and the same power with different bases. Ends with scientific notation.
• Radicals and Fractional ExponentsSquare roots and radicands, simplifying radicals with the product and quotient rules, and adding and subtracting radicals. Then cube roots, higher-degree roots, fractional exponents and converting between radical and exponential form.

Divisibility, primes, LCM, GCD and factorials; then polynomial expressions, factoring and special products with Pascal's Triangle; then linear, factorable and radical equations with word problems.

• Number TheoryMultiples, divisors and factors, the divisibility tests for 2 through 10, primes and even/odd reasoning. Then prime factorization, LCM and GCD with word problems, and the factorial n!.
• Expressions and Special ProductsPolynomial vocabulary, simplifying and expanding, factoring by common factors and by grouping, then the special products: difference of squares, difference and sum of cubes, perfect-square trinomials. Also factorials and Pascal's Triangle.
• EquationsLinear equations (with decimals, fractions and variables in denominators), including the no-solution and all-real-numbers cases, literal equations, equations solved by factoring, and radical equations with extraneous roots. Ends with word problems.

Sets, functions, domain and range, and graph features; the Pythagorean Theorem, distance, slope and lines; parallel and perpendicular lines and transformations.

• Sets and FunctionsSets (subsets, union, intersection, complement), then relations and functions with domain, codomain and range, mapping diagrams and the Vertical Line Test. Graph features follow: increasing, decreasing, symmetry, and intercepts of a line.
• The Pythagorean Theorem, Distance, and Linear FunctionsThe Pythagorean Theorem with special right triangles and triples, then the distance and midpoint formulas. Linear equations in slope-intercept and standard form, graphing from intercepts, and slope with its four types.
• Linear Functions, Transformations, Parallel and Perpendicular LinesComparing steepness by |slope|, writing a line from a point and slope or from two points (point-slope form), parallel and perpendicular lines, and translating, reflecting and rotating linear functions, alone and in sequence.
• Cumulative Review and SAT-Style PracticeA cumulative SAT-style review: a short reminder, problems and a worked solution per topic. Topics: order of operations, exponents, radicals, factoring, sets, functions, solving equations, lines, distance and equations by factoring.

Systems of linear equations by graphing, substitution and elimination; inequalities in one and two variables and systems of inequalities; absolute value equations, inequalities and functions.

• Systems of Linear EquationsWhat a system of linear equations is and what its solution means on a graph, then three solving methods: graphing, substitution and elimination (by addition, subtraction and multiplication). Covers one, no or infinitely many solutions and word problems.
• InequalitiesOne-step, multi-step and compound inequalities (flip the sign when needed), graphed on a number line and written in interval notation, with word phrases. Then two-variable inequalities and systems of inequalities on the coordinate plane.
• Absolute Value Equations and InequalitiesAbsolute value as distance. Equations |expr| = k split into two cases, then equations with a variable on both sides (four-case method) and extraneous roots. Then < and > inequalities, special cases, multi-step and tolerance problems.
• Absolute Value Functions and GraphingThe parent function f(x) = |x| and the form f(x) = a|x − h| + k: vertex, axis of symmetry, domain, range, shifts, stretch and compression. Also x-intercepts, and a proof that |x − 2| and |−x + 2| give the same graph.

Factoring, completing the square, the quadratic formula and discriminant, parabolas and vertex form, then the three forms of a quadratic function and word problems.

• Quadratic Equations and FactoringSpecial products with area models, simple equations x² = k and the Zero Product Property, then factoring x² + bx + c with the r, s method and ax² + bx + c with the AC method. Quadratic equations are solved by factoring.
• Special Factors and Completing the SquareA recap of solving by factoring, with Vieta's sum and product of roots, then perfect-square trinomials, differences of squares, and sums and differences of cubes. Then completing the square for a = 1 and a ≠ 1, and choosing the fastest method.
• The Quadratic Formula and DiscriminantDeriving the quadratic formula by completing the square, using the discriminant to predict the number of real solutions, and applying the formula to word problems. Also Po-Shen Loh's method and extraneous roots from radical equations.
• Quadratic Function - ParabolaStandard form of a quadratic function, graphing a parabola from a table, opening direction, vertex, axis of symmetry, zeros, domain and range. Then vertex form, how a, h and k transform the graph, and applications such as maximum height and revenue.

Rational expressions and functions; adding, subtracting, multiplying and dividing them; complex fractions, polynomial long division, rationalizing denominators, radical equations and quadratic-in-form equations.

• Rational ExpressionsRational expressions and rational functions: excluded values, domain, range, holes and asymptotes, including the parent function 1/x and its shifts. Then simplifying by factoring and canceling, and multiplying and dividing.
• Rational Expressions - Add and SubtractThe least common denominator of monomials and polynomials, adding and subtracting rational expressions with like and unlike denominators, complex fractions, and rational equations. Polynomial long division, with a remainder, also appears.
• Rational Expressions - Polynomial DivisionMore complex fractions, then polynomial long division with the result written as Q(x) + remainder/divisor. Also rationalizing single-term and binomial (conjugate) denominators, and radical equations checked for extraneous solutions.
• Rational and Radical ExpressionsFour topics: more practice with radical equations, dense complex rational expressions with a fraction inside, radical identities built on perfect-square binomials, and graphing a quadratic from vertex form, then quadratic-in-form equations.

• Fractions, Decimals, and ReviewConverting between fractions and terminating decimals using powers of 10, repeating decimals to fractions, and fraction and ratio algebra.
• Arithmetic SequencesSequence notation, the common difference and the general term aₙ = a₁ + d(n − 1), the arithmetic mean, and the sum formula with Gauss's pairing trick. Also how many terms give a target sum, and word problems modeled by sequences.
• Geometric SequencesΣ notation and an arithmetic recap, then geometric sequences: the common ratio, the general term, the sum formula and the geometric mean. Ends with infinite geometric series that converge and word problems such as rice on a chessboard.
• Word ProblemsTranslating words into equations: digit, age, consecutive-integer, coin and mixture problems, then percent problems (markup, discount, growth and decay) and rate-time-distance problems (catching up, meeting, moving apart).
• Word Problems (Ratios and Proportions)Mixture (percent-of-solution) problems including dilution, then rate and proportion problems: multi-way ratios, currency exchange, direct, inverse and joint proportion, and unit conversion. Average speed and combined work rate close the week.
• Word Problems (Work Rate)Work-rate problems: rate = work ÷ time, workloads with several machines or workers, and combined work rate by adding the rates of two or three workers. Average speed and three-way unit conversion return, with a mixed-practice challenge.

• Exponents and LogarithmsThe properties of exponents and exponential equations (same base, quadratic-in-disguise, nested exponents), then logarithms: definition, converting between forms, evaluating, common and natural logs, change of base and logarithmic equations.
• Data and ProbabilitySamples and sampling methods (biased or unbiased), mean, median and mode, frequency tables and histograms. Then tree diagrams, simple, compound and mutually exclusive or inclusive probability, and permutations and combinations for counting.
• SAT QuestionsA practice week: two SAT-style tests (50 and 25 questions) covering the year's algebra, functions, exponents, radicals, quadratics, sequences, geometry, probability and counting. Every answer has a fully worked explanation.