Online Pre-Algebra Course for Grades 6 and 7

The number system and the properties of arithmetic, multi-step integer, fraction and decimal arithmetic, then squares, the laws of exponents and number bases (binary, octal, hex), then factors, primes, GCF, LCM and divisibility.

• Working with Numbers and the Power of StructureThe number system from natural to real numbers, the commutative, associative and distributive properties, and Gauss's pairing method. The same ideas count handshakes and round-robin games and give formulas for sums of odd and even numbers.
• Integers, Fractions and Decimals - Multi-Step ArithmeticMulti-step chains of integers, fractions (including mixed numbers and less-common denominators) and decimals to three places, using the sign rule and the number line. Ends with 2-3 step word problems.
• Exponents - Meaning, Squares and PatternsExponent notation (base, exponent, power), perfect squares through 12² and the four properties of squares. Then the identities for a difference of squares and perfect-square trinomials, used to expand, factor and compute with larger numbers.
• The Laws of ExponentsThe product, quotient and power rules for exponents, with zero and negative exponents and equations solved by matching bases. Also shows why exponentiation is not commutative or associative and why powers of 2 describe computer memory.
• Exponents - Zero, Negative Exponents and Matching BasesReviews the product, quotient and power rules with bigger numbers, then goes further on zero and negative exponents, including negative bases. Also quotients with negative results, n on both sides of an equation, and same exponent with different bases.
• Number Bases and ConversionsPlace value in base 10, then binary, octal and hexadecimal: converting each to and from decimal, with hex digits 0-9 and A-F. Also covers adding in binary and real-world uses of hexadecimal.
• Number Theory - Factors, Primes and GCF-LCMFactors and divisors, prime and composite numbers, and factor trees. Prime factorization is then used to find the GCF and LCM of two or three numbers, ending with real-life GCF and LCM word problems.
• Divisibility, LCM and GCDQuick divisibility tests for 2, 3, 4, 5, 6, 8, 9 and 10 (and 12 by combining tests), proofs about divisors, and 'factor out a common piece' shortcuts for the LCM and GCD of large numbers. Ends with competition-style puzzles.

Mixed numbers and fraction operations, expressions and like terms, then linear equations and word problems: translating phrases, age, coin and consecutive-integer problems, rate, distance and mixtures.

• Advanced Fractions - Mixed Numbers and OperationsProper, improper and mixed fractions, converting between them, and simplifying to lowest terms. Then multiplying and dividing fractions and mixed numbers (cancel first; Keep, Change, Flip), word problems, and quick ways to compare fractions.
• Advanced Fractions - Adding and SubtractingFinding the LCD, then adding and subtracting fractions with unlike denominators, including chains of three. Mixed numbers follow, first with like and then unlike denominators, with carrying and borrowing, plus 2-3 step word problems.
• Expressions - Simplifying and Combining Like TermsParts of an expression (term, coefficient, constant), like and unlike terms, and combining them. Then distributing with negative numbers and nested parentheses, and translating words into one simplified expression.
• EquationsSolving linear equations: one-step, two-step and multi-step, with variables on both sides, no-solution and all-real-numbers cases, and fractions or decimals cleared first. Ends with word problems and checking every answer.
• Word Problems - Translating to EquationsTranslating key phrases into symbols, then one-step and two-step word problems. Then age, consecutive-integer, coin, ticket, percent, overlapping-group and comparison problems, ending with problems that reduce two unknowns to one equation.
• Word Problems - Rate, Time and MixturesRate, time and distance formulas, catch-up problems, and travelers moving toward or away from each other, plus consecutive numbers found with the mean. Mixtures: finding an amount, combining mixtures, adding a pure ingredient, replacing part of a solution.

Decimal arithmetic, rounding and repeating decimals, then ratios, proportions and rates and their word problems, then percent, percent change, and simple and compound interest.

• Arithmetic with DecimalsDecimal place value as powers of 10, then adding, subtracting, multiplying and dividing decimals in multi-step chains. Also rounding (including negatives and halfway cases), decimal-fraction conversion and repeating decimals, including 0.999... = 1.
• Ratios, Proportions and RatesWriting, simplifying and scaling ratios (two-way, multi-way, fraction and decimal), part-to-whole ratios and unit rates. Then proportions solved by cross-multiplying, direct versus inverse proportion, and multi-parameter proportion problems.
• Ratios, Proportions, Rates - Word ProblemsRatio and rate word problems: before-and-after ratios, joint proportions with three interacting quantities, and inverse proportions. Also unit conversions in chains, scale drawings and maps, true average speed and combined work rates.
• PercentConverting between percents, decimals and fractions (including negative and over-100% percents), the three types of percent problem, and mental-math tricks. Ends with discount, markup, tax and tip, and reversing the chain.
• Percent Increase and Decrease, Word ProblemsPercent increase and decrease, the one-step (100 ± n)% shortcut, and finding the original amount by working backward. Then chained and repeated percent changes, and simple and compound interest.

• Radicals - Square Roots and Perfect SquaresSquare roots of perfect squares and the principal root, simplifying with prime factorization, and roots of other degrees. Then combining, multiplying, dividing and rationalizing radicals, approximating, radical equations and fractional exponents.
• Radical Expressions - Simplifying, Operations and RationalizingStarts from a recap of radical basics and the laws of exponents and roots. New: radicals of other degrees with negative radicands (odd versus even index), comparing radicals with different indices, simplest form, and a variable under the radical.
• Radical EquationsSolving radical equations by isolating the radical and raising both sides to a power, checking for extraneous roots, and equations with two radicals. Also solving by factoring, and fractional exponents as another way to write radicals.
• Systems of EquationsA system's solution as the point where two lines cross, then solving by substitution and by elimination (adding, subtracting, or multiplying first). Covers one, no or infinitely many solutions, choosing a method, and word problems.
• Systems of Equations - Word ProblemsAfter a recap of substitution, elimination and the three outcomes, systems word problems: age, consecutive integer, two-digit number, ticket, coin, mixture, investment and rate (boat and current). Includes a system with three variables.

Angles, angle pairs and parallel lines, then polygons, triangles, the Pythagorean theorem and circles, then quadrilateral area and 3-D solids with volume and surface area.

• Angles and LinesNaming, measuring and classifying angles with a protractor, then vertical, complementary and supplementary angles and linear pairs, perpendicular lines and bisectors, and the angle theorems for parallel lines cut by a transversal.
• Plane GeometryInterior and exterior angle sums of polygons, the triangle exterior-angle theorem, triangle types, the triangle inequality, and the Pythagorean theorem. Then special right triangles, triangle area, quadrilaterals, and circles.
• 2D and 3D GeometryArea and perimeter of the parallelogram, rhombus and trapezoid, and composite circle areas. Then 3-D solids, nets, top, front and side views, and volume and surface area of the cube, prisms, cylinder, pyramid and cone.

Inequalities, compound inequalities and absolute value inequalities, then coordinate planes, slope, slope-intercept form, writing equations of lines, midpoint, and parallel and perpendicular lines.

• InequalitiesInequality vocabulary, graphing on a number line, and the properties used to solve inequalities, including the KEY RULE for negative multipliers. Then one-step and multi-step solving, translating word phrases, and real-world uses.
• Compound InequalitiesAND and OR compound inequalities (overlap and union), double-sided inequalities, and the no-solution and all-real-numbers cases, with word problems. Then absolute value, and absolute value equations by the two-case method.
• Absolute Value Inequalities|x| < k and |x| > k read as distances from zero: the less-than case gives an AND (a bounded interval) and the greater-than case an OR (two rays). Also the special cases, multi-step problems and real-world tolerances.
• Graphing Linear Equations - Points and SlopeCoordinate planes, ordered pairs, quadrants and distance between points, then linear equations in standard and slope-intercept form, and graphing with a table, intercepts, or slope and y-intercept. Slope from a graph, two points or a rate.
• Graphing Linear Equations - Slope-Intercept FormSlope-intercept form: reading m and b, graphing, and writing an equation from a graph, a point and slope, or two points. Also point-slope and standard form, midpoint, parallel and perpendicular lines, and modeling real situations.

• StatisticsMean, median, mode and range (with mean word problems), types of data, and data displays: tables, bar, pie and Pareto charts, dot plots, stem-and-leaf displays and histograms. Also scatter plots and time-series plots for two variables.
• Box Plots and ProbabilityQuartiles, the interquartile range, the outlier-fence rule, and building, reading and comparing box plots, plus misleading data displays. Then single-event probability on the 0-to-1 scale with dice, coins, cards, marbles, spinners and frequency tables.

Quadratic equations solved by square roots, factoring, Po-Shen Loh's method and the quadratic formula, then a whole-course review across the earlier topics.

• Quadratic EquationsSolving x² + bx + c = 0 by taking square roots, by factoring with the sum-and-product rule and the Zero-Product Property, and by Po-Shen Loh's average-and-deviation method. Then the quadratic formula, the discriminant and the graph.