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ALGEBRA 1 Version 2009
Revised and highly enlarged.
ALGEBRA 2
ALGEBRA 3
ALGEBRA 4
GEOMETRY 1
ELECTRICITY 1
ELECTRICITY 2
ELECTRICITY 1&2
OPTICS 1
OPTICS 2
MECHANICS 3
 
ALGEBRA 2
Identities. Inequalities. Linear and Quadratic Equations and Functions
Middle school 15-17 years


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   1. Order relation on the set of integers Z.
  •            1. Order relation on the set N.
  •            2. Comparing two natural numbers.
  •            3. Extension of the order relation to the set Z.
  •            4. Comparing two integers.
  •            5. Subsets of Z defined by an inequality.
  •            6. Representing integers that satisfy an inequality.
  •            7. Subsets of Z defined by a two-sided inequality.
  •            8. Illustrating a two-sided inequality and counting the number of integers it contains.
  •            9. Free activity.
   2. Ordering the set of rational numbers Q. Translation invariance.
  •            1. Comparison of two positive rational numbers.
  •            2. Comparison of two rational numbers with arbitrary signs.
  •            3. The order of two numbers depends on the sign of their difference.
  •            4. Ordering two decimal numbers by calculating their difference.
  •            5. Comparison of two fractions.
  •            6. Ordering two fractions by calculating the sign of their difference.
  •            7. To regroup similar terms in inequalities using transpositions.
   3. Effects of multiplication / division of both sides of an inequality by the same number.
  •            1. Multiplication / division of both sides of an inequality by a positive number.
  •            2. Effects of modifying an inequality by changing the signs of both sides.
  •            3. Multiplication / division of both sides of an inequality by a negative number.
  •            4. Solving reduced linear inequalities and representing the solutions on an axis.
  •            5. Linear inequalities with coefficients in the set Z.
  •            6. Linear inequalities with coefficients in the set Q.
   4. Setting and solving inequalities.
  •            1. Renting a car.
  •            2. A worthwhile offer ?
  •            3. Which monthly consumption rate of electricity to choose ?
   5. Operations permitted between two inequalities in the same direction.
  •            1. Term-by-term addition of two inequalities in the same direction.
  •            2. Term-by-term multiplication of two inequalities between positive numbers.
  •            3. Term-by-term multiplication of two inequalities between negative numbers.
  •            4. Framing a number.
  •            5. Upper and lower bounds for the sum of two numbers.
  •            6. Framing of the perimeter of a rectangle.
  •            7. Upper and lower bounds for a product of positive numbers.
  •            8. Framing the area of a rectangle.
  •            9. Upper and lower bounds for the difference of two numbers.
   6. Two elementary algebraic identities.
  •            1. The square of a sum in a non-commutative algebra.
  •            2. The square of a sum of two numbers in the current commutative algebra.
  •            3. One formula yielding countless others.
  •            4. Using the identity ( a + b ) 2 = a 2 + 2ab + b 2.
  •            5. The product of the sum and difference of the same two numbers.
  •            6. Using the identity ( a + b ) ( a - b ) = a 2 - b 2 .
  •            7. Combination of two identities.
   7. Using elementary identities to factor algebraic expressions.
  •            1. Factoring a monomial.
  •            2. Factoring a binomial.
  •            3. Factoring based on the identity a2 + 2 a b + b2 = ( a + b )2 .
  •            4. Factoring based on the identity a2 - b2 = ( a + b ) ( a - b ) .
  •            5. Factoring after grouping terms.
  •            6. Factoring using two identities.
   8. Applications of factoring : roots of a polynomial. Sign of a polynomial.
  •            1. Product of factors nil.
  •            2. Definition of a polynomial of degree n.
  •            3. Roots of a polynomial.
  •            4. Roots of first-degree polynomials.
  •            5. Roots of factored polynomials.
  •            6. Factoring polynomials and determining their roots.
  •            7. Factoring using identities and determining the degree and roots of polynomials.
  •            8. Method of factoring second-degree polynomials.
  •            9. Factoring simple second-degree polynomials.
  •            10. Sign of a first degree polynomial. Sign of a factored polynomial.
  •            11. High school exercises. Factoring polynomials and determining their signs.
   9. Square roots. Irrational numbers.
  •            1. The square of a rational number written as an irreducible fraction.
  •            2. Factoring squares of rational numbers.
  •            3. The square root of a rational number.
  •            4. The square root of a rational number is not generally a rational number.
  •            5. Recognizing whether a square root is a rational number.
  •            6. Geometric representation of irrational numbers.
  •            7. Using Lagrange's Theorem to represent Rac(n#.
  •            8. Algebraic definition of irrational number. The notion of Dedekind cuts.
  •            9. The algebra of real numbers.
   10. Mastering the basic notions of algebra.
  •            1. Right words.
   11. Definition and properties of absolute value.
  •            1. Definition of the absolute value of a real number.
  •            2. Absolute values of products and quotients.
  •            3. Solving equations of the form | l x | = a.
  •            4. Absolute value and distance on the R axis.
  •            5. The absolute value verifies the triangle inequality.
  •            6. Determining points on the R axis that are at distance d from point A.
  •            7. Solving equations that express the value of a distance.
  •            8. The abscissa of the middle of a segment. Defining an interval by an absolute value.
  •            9. Determining and sketching an interval defined by absolute value.
  •            10. Equality of two absolute values.
  •            11. Equation involving two absolute values.
   12. Factorization properties of square roots of products.
  •            1. Theorem: factoring of the square root of a product of factors, of a square, of a quotient.
  •            2. To reduce the square root of a natural number by factoring it into prime numbers.
  •            3. Square roots of positive and negative powers of 10.
  •            4. To simplify expressions by regrouping similar terms.
  •            5. To simplify by effecting products of factors.
  •            6. Simplifying by using the identity (a + b)2 = a2 + 2 ab + b2.
  •            7. Simplifying by using the identity (a + b) (a - b) = a2 - b2.
  •            8. Rationalizing square roots in the denominator of a fraction.
  •            9. Rationalizing combinations of square roots in the denominator of a fraction.
   13. Calculating and simplifying expressions with square roots.
  •            1. Simplifying a product containing square roots.
  •            2. To reduce and regroup a sum containing square roots.
  •            3. Calculating square roots of decimal numbers by comparison.
  •            4. Simplifying by effecting product of factors.
  •            5. Expressions of the form ( a + b )2.
  •            6. Expressions of the form ( a + b ) ( a - b ).
  •            7. Simplifying square roots in quotients.
  •            8. To eliminate combinations of square roots from the denominator.
   14. Relation between two real variables. Linear functions.
  •            1. Example of a linear relation between two variables.
  •            2. Filling up with gas and finding the cost.
  •            3. Definition and graphical representation of a linear mapping.
  •            4. Examples of linear mappings.
  •            5. Using linearity to determine a missing value.
  •            6. Calculating missing values of a linear function.
  •            7. General definition of the slope.
  •            8. Positive and negative slopes.
  •            9. Linear functions passing through given points.
   15. Linear function with an additive constant. Graphical representation.
  •            1. Example of a linear function with an additive constant.
  •            2. Geometric significance of the parameters a and b of a linear function.
  •            3. Drawing a line passing through two points and determining its slope.
  •            4. Point-slope formula for a straight line.
  •            5. Determining and drawing a line with known slope, passing through a given point.
  •            6. Equation of a line passing through two given points.
  •            7. Determining the density of a hydrocarbon.
   16. Getting to know the properties of linear functions and their representations.
  •            1. To draw the graph of a linear function. Finding the missing coordinate.
  •            2. Verifying whether given points are collinear.
  •            3. A line passing through a given point and parallel to a given line.
  •            4. Determining the point of intersection of two lines.
  •            5. Study and graph of the functions absolute value of x, |x|, and y=|ax+b|.
  •            6. An even function composed by the sum of absolute values.
  •            7. An odd function composed by the difference of absolute values.
   17. Lines in the plane and their representation. Parallelism and orthogonality.
  •            1. Lines in the plane represented by linear functions.
  •            2. Generalization of linear functions.
  •            3. Relationship between the slopes of orthogonal lines : aa' = -1.
  •            4. Points to remember.
  •            5. Equation of the perpendicular from a given point to a line.
  •            6. Coordinates of the midpoint of a segment. Distance between two points.
  •            7. Equation of the perpendicular bisector of a line segment [AB].
  •            8. Determining the perpendicular bisector of a segment by two methods and drawing it.
   18. Allowable operations involving two equals fractions. Diagonal transpositions.
  •            1. Diagonal transpositions involving two equals fractions.
  •            2. Solving linear equations using transposition and cross-multiplication.
  •            3. Solving for unknowns given equalities between fractions.
  •            4. To determine the coordinates of the intersection of two rational functions' graphs.
  •            5. Solving equations that can be brought to linear equations.
   19. Linear equations in two unknowns. Analytical and graphical solutions.
  •            1. Linear equation in two unknowns and its graphical representation. Line passing through 2 points.
  •            2. Solving a system of two linear equations in two unknowns.
  •            3. Solution of systems of two equations in two unknowns by elimination.
  •            4. A system of two equations in two unknowns with no solution.
  •            5. Condition for the existence of a unique solution : determinant of coefficients non-zero.
  •            6. Solution formula using determinants.
  •            7. Solving systems using determinants.
  •            8. Solving two linear equations by comparison or substitution.
  •            9. Solution of two linear equations by comparison or substitution.
   20. Setting up and solving equations in two unknowns.
  •            1. Sum and difference.
  •            2. Sum and ratio.
  •            3. At the flower shop.
  •            4. Past and future age ratios.
  •            5. Encircling four identical coins.
  •            6. Two durations for cycling the same distance.
   21. First-degree inequalities in one variable.
  •            1. The sign of a first-degree polynomial.
  •            2. To display the sign of a polynomial on a table.
  •            3. Solving first-degree inequalities.
  •            4. Comparing two first-degree polynomials.
  •            5. Graphical solution of two inequalities involving two first degree polynomials.
  •            6. Choosing a rental car company.
   22. Linear inequalities in two variables.
  •            1. Signs of first degree polynomials in two variables.
  •            2. Solving linear inequalities in two variables.
  •            3. Subsets of R2 satisfying two simultaneously linear inequalities.
  •            4. Solving two simultaneous linear inequalities graphically.
  •            5. Defining the set of interior points of a triangle by inequalities.
  •            6. Plane region defined by two inequalities involving absolute values.
  •            7. Plane region defined by an inequality involving absolute values.
   23. Quadratic polynomials. Translated or reduced form. Symmetry and graph. Roots.
  •            1. Studying the function y = x2. Parity, increase and decrease variation.
  •            2. General properties of parabolas with equation y = a x2 for a>0 and a<0.
  •            3. Standard form for the second degree polynomial. Translated function and its graph.
  •            4. Writing second degree polynomials in Standard form and drawing their graphs.
  •            5. Brief history of the second degree polynomial's roots.
  •            6. Existence condition of real roots for a second degree polynomial. General formula and examples.
  •            7. To find the roots of second degree polynomials.
  •            8. Intersection of a line and a parabola. Study of the case where the line is tangent to the parabola.
  •            9. Factorization and sign of the second degree polynomial. Second degree inequalities.
  •            10. To solve quadratic inequalities or inequalities involving absolute values.
  •            11. Points to remember.
   24. Sum and product of the quadratic equation's roots. Biquadratic equations. The ellipse.
  •            1. Definition and formula of the golden number.
  •            2. Sum and product of the roots of the quadratic equation.
  •            3. Determining two numbers when their sum S and product P are known.
  •            4. Equation involving fractions, which reduces to a quadratic equation.
  •            5. Irrational equation brought to a quadratic. Trapezoid perimeter.
  •            6. Biquadratic equations and inequalities.
  •            7. Biquadratic equations with two or four roots. Signs of the corresponding polynomials.
  •            8. Optimization: to find the maximum area.
  •            9. Equation of the ellipse in Cartesian coordinates.