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Algebra, grade 8

Eighth-grade algebra.Subjects

The topics, in order:

  • functions: a graph, a table and a formula; increasing and decreasing;
  • the linear function: rate of change, slope, the formula y=mx+b\displaystyle {y}={mx+b};
  • special equations and word problems;
  • systems of equations;
  • inequalities.

There are two kinds of lessons:

  • Learning lessons – an explanation, solved examples and a few questions to try;
  • Practice lessons – questions only.

How the page works: Using a page.

  1. What a function isLearningA rule that gives each x exactly one y: the independent and the dependent variable, examples from everyday life, and a matching that is not a function.
  2. A function's table of valuesLearningA table of x and y pairs: filling it in from a rule, a table that is not a function's, and finding the rule from a table.
  3. A function's graphLearningThe graph as the set of points (x, y), reading values off a graph, a story graph – a bike ride – and the vertical line test: which drawings are graphs of a function.
  4. A function's formulaLearningThe rule as a formula, y = …: substituting in the formula, finding x when y is known, and from a formula to a table and a graph.
  5. Representations of a functionLearningA story, a table, a formula and a graph of the same function: what each one shows, moving from one to another and matching them.
  6. Increasing, decreasing and constant functionsLearningIntervals of increase, decrease and constancy – in words and with < >, maximum and minimum points, from a graph and from a table.
  7. Practice: functionsPracticeFunctions, tables of values, graphs and formulas, moving between representations, increase and decrease, maximum and minimum points.
  8. Linear functions: straight-line graphsLearningThe graphs of y = 2x, y = x + 3 and y = −x + 4: in the table y changes in equal steps, and the graph is straight. Whether a point is on the line – by substituting.
  9. Rate of changeLearningA constant and a varying rate of change: a burning candle and a falling ball, a taxi and a car park. A constant rate of change – a straight graph. The rate of change between two columns of a table.
  10. The slope of a lineLearningThe slope: the change in y over the change in x. Working it out from two points; a positive, a negative and a zero slope; slope and steepness; parallel lines have the same slope.
  11. The formula y = mx + bLearningThe formula of a linear function: m the slope, b where the line meets the y-axis. Drawing a line from its formula, and where it meets the x-axis – by solving an equation.
  12. Finding the formula of a lineLearningThe formula y = mx + b from the slope and a point, from two points and from a graph; a line parallel to a given line through a point.
  13. Practice: the linear functionPracticeA constant rate of change, a point on a line, the slope, the formula y = mx + b, where a line meets the axes, the formula of a line from data and from a graph, parallel lines and a meeting point.
  14. Brackets, fractions and special equationsLearningEquations with brackets on both sides and with fractions – multiplying by a common denominator; 0x=5 and 0x=0: an equation with no solution and one every x solves, and what that means in a story and on a graph.
  15. Word problems: numbers, ages and moneyLearningThe five steps: choose the unknown, express the other quantities with it, write an equation, solve, and answer the question. From words to expressions; problems about numbers, ages now and later, and savings.
  16. Percent problems by an equationLearningx plus 20% of it is 1.2x: sharing in percent, a discount plus a fixed amount, a salary raise, and two prices that became equal – finding the original amount.
  17. Motion problemsLearningDistance = speed × time, and a table of speed, time and distance for each motion: toward each other, one after the other, and there and back.
  18. Word problems: shapes, money and mixturesLearningPerimeter and area, the angles of a triangle, coins and tickets – two quantities with a known total, x and 20 minus x – and simple mixtures.
  19. Practice: equations and word problemsPracticeEquations with brackets and fractions, special equations, and word problems: numbers, ages, money, percent, motion, shapes and mixtures.
  20. Systems of equations on a graphLearningTwo equations in x and y that hold together: the system's solution is an ordered pair, the point where the two lines meet; parallel lines – no solution, the same line – infinitely many.
  21. Solving a system by substitutionLearningSolving a system of equations by substitution: isolate an unknown in one equation and substitute it into the other, solve an equation in one unknown, and check the pair in both equations.
  22. Solving a system by comparison and eliminationLearningTwo more ways to solve a system: set the two expressions for y equal – the meeting point; add or subtract the equations, sometimes multiplying one of them first.
  23. Word problems solved by a system of equationsLearningA problem with two unknowns: a letter for each unknown, an equation for each fact, and the system's solution – prices, numbers, coins, ages, and comparing two companies where their lines meet.
  24. Practice: systems of equationsPracticeChecking a solution, the meeting point on a graph, the number of solutions, substitution, comparison, adding and subtracting equations, and word problems.
  25. InequalitiesLearningA first-degree inequality: its solution set on the x-axis, solving as with an equation – and turning the sign when multiplying or dividing by a negative number, special cases and word problems.
  26. Practice: inequalitiesPracticeA solution set on the x-axis, inequalities in one step and in several, with brackets and fractions, special cases, word problems and inequalities on a graph.
  27. Review: the whole coursePracticeQuestions from every topic: functions as a table, a graph and a formula, the linear function and its slope, equations and word problems, systems of equations and inequalities.
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