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Algebra, grade 9

Ninth-grade algebra.Subjects

The topics, in order:

  • the laws of powers and square roots;
  • the extended distributive law and the short multiplication formulas;
  • factoring and quadratic equations;
  • algebraic fractions and equations with the unknown in the denominator;
  • the properties of a function;
  • the quadratic function and its graph, the parabola;
  • the quadratic formula and quadratic inequalities;
  • systems of equations of the second degree, and word problems.

Two kinds of lessons:

  • learning lessons – an explanation, examples and a few questions to try;
  • practice lessons – questions only.

Go in order.

How the page works: Using a page.

  1. The laws of powersLearningMultiplying and dividing powers with the same base, a power of a power, of a product and of a fraction, the exponent 0 – with numbers and with variables.
  2. The laws of square rootsLearningThe root of a product and of a fraction, the square of a root and the root of a square, simplifying a root, adding like roots.
  3. Practice: powers and rootsPracticeThe laws of powers with numbers and with variables, the exponent 0, the laws of square roots, simplifying roots and adding like roots.
  4. The extended distributive lawLearningMultiplying two sums: every term by every term, with areas and with negative numbers; brackets with three terms; simplifying expressions.
  5. The short multiplication formulasLearningThe square of a sum, the square of a difference and the difference of squares: derived with the extended distributive law, shown with areas, used for mental arithmetic and for simplifying.
  6. Equations and areas with the distributive lawLearningEquations in which x² cancels once the brackets are opened, and comparing areas: which is bigger, by how much, and when they are equal.
  7. Practice: algebraic techniquePracticeThe extended distributive law and the short multiplication formulas: opening brackets, simplifying, mental arithmetic, equations, comparing areas and a proof.
  8. Factoring: a common factorLearningFactoring is the distributive law backwards: the greatest common factor of numbers and of variables, taken out of the brackets, and a check by opening them.
  9. Factoring a trinomialLearningx² + bx + c = (x + m)(x + n): two numbers whose sum is b and whose product is c; their signs; a common factor first.
  10. Factoring by the short multiplication formulasLearningThe short multiplication formulas read backwards: a difference of squares and a perfect square trinomial; a common factor first, then a formula.
  11. Practice: factoringPracticeA common factor, a trinomial, the formulas and all of them together; choosing the method, complete factoring, mental arithmetic, a rectangle and an even number.
  12. Quadratic equations by factoringLearningA product is zero only when one of its factors is zero. Solving x² = 9, x² − 5x = 0 and x² − 5x + 6 = 0 by factoring – after moving everything to one side.
  13. Algebraic fractionsLearningA fraction with a variable in its denominator: its domain – the denominator is never 0 – and reducing it: factor the numerator and the denominator, then cancel a common factor.
  14. Multiplying and dividing algebraic fractionsLearningAs with number fractions: factor, cancel, then multiply; dividing is multiplying by the reciprocal. A number that makes the numerator of the divisor 0 is left out of the domain too.
  15. Equations with the unknown in the denominatorLearningThe domain first; factoring the denominators; multiplying both sides by the common denominator; checking each solution against the domain – a solution outside it is rejected.
  16. Practice: algebraic fractions and equationsPracticeQuadratic equations by factoring, the domain of a fraction, reducing, multiplying and dividing algebraic fractions, and equations with the unknown in the denominator.
  17. The properties of a functionLearningRead from a graph: the intersection points with the axes, the positive and negative domains, the extremum points – local and absolute – and the increasing and decreasing domains.
  18. Equations, the graph, shifts and reflectionLearningSolving f(x)=k from the graph; moving a graph up and down, right and left; reflecting it in the x-axis; which properties a shift keeps.
  19. The function y = ax²LearningA table and a graph of x², −x² and 2x²: the parabola, its vertex and its axis of symmetry; a > 0 – a minimum, a < 0 – a maximum; the properties of y = ax².
  20. Shifting the parabolaLearningy = ax² + k, y = a(x − b)² and y = a(x − b)² + k: the vertex (b, k), the axis of symmetry, and the properties of the parabola from this form, intercepts included.
  21. The product form and the standard formLearningy = a(x − x₁)(x − x₂): the x-intercepts and the vertex halfway between them; y = ax² + bx + c: the y-intercept c and the vertex at x = −b/(2a); from one form to the other, and sketching a parabola.
  22. The quadratic formulaLearningEvery quadratic equation ax²+bx+c=0 solved by one formula; what is under the root tells how many solutions there are; choosing the shortest way to solve.
  23. Quadratic inequalitiesLearningA quadratic inequality asks where a parabola is above or below the x-axis: find the roots, sketch, read the domain.
  24. Systems of equations of the second degreeLearningA pair (x, y) that satisfies two equations is an intersection point of their graphs: a parabola and a line, two parabolas – solved by substitution.
  25. Word problemsLearningWord problems solved by a quadratic equation or a system: numbers, areas, the Pythagorean theorem, motion and percentages – and a check that the solution makes sense.
  26. Review: the whole coursePracticeEverything together: powers and roots, the extended distributive law and the formulas, factoring, algebraic fractions, equations, functions and the parabola, inequalities, systems and word problems.
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