## Senior Mathematics Courses Scope & Sequence

These courses are made up from a combination of ACE courses.

Algebra II are any PACEs numbered from 1121 to 1132.

Trigonometry is 1133

Senior Maths have the designation SM and are numbered from 1 – 22

**Introduction to Advanced Mathematics (Level 2)**

**1133 **To review properties of angles and triangles Learn definitions of the six trigonometric functions for acute angles. Find the values of the trigonometric functions using a graphing calculator Establish and use basic trigonometric identities Show that the values of the trigonometric functions depend only on the angle Solve problems involving right triangles Begin a study notebook for trigonometry Study the eternal nature of God and His plan for mankind Memorize Revelation 1:8; 2 Corinthians 4:18 and Hebrews 5:9

**SM 3 **Applications of Trigonometric Ratios: The study of directions and bearings, and contour lines and maps, using geometrical facts, angles of elevation and depression, directions and bearings, contour lines and maps, and average slope

**SM 5 **Univariate Data: Working with data – variables, the collection of data, categorical data and numerical data.

**SM 6 **Indices and Logarithmic Functions: incorporating the index form. Tarional indices, exponential functions, and indicial equations

**SM 8 **Data descriptors: The study of the mean of grouped and ungrouped data. The advantages and disadvantages of the use of the mean and its standard deviation. The study of the median, the mode and skewed data.

**SM 9 **Relations and functions: define and describe functions and relations. Define table values, domain, rand and odd and even functions. Sketch graphs to test and to model different functions and relationships. Find the permissible x and y values for a variety of functions.

**SM 10 **Periodic functions: the radian measure. Geometrical definitions of the tangent function, specific periodic functions, reciprocal trigonometric functions and special angles.

**1122 **Arithmetic Sequences and Series: Arithmetic Sequences; The nth Term; Arithmetic Means; Arithmetic Series; Summation Notation. Geometric Sequences and Series: Geometric Sequences; The nth Term; Geometric Means; Geometric Series; Infinite Geometric Series. Open Sentences in One Variable: Algebraic Terminology; Adding Polynomials; Subtracting Polynomials; Combining Polynomials; Solving the Open Sentence; Inequalities.

*1*** 123 **Linear Equations: Graphing Ordered Pairs; Graphing Linear Equations; Slope Formula; Slope of the Line Ax + By =C. Graphs and Linear Equations: Graphing by Using the Slope Method; Point-Slope Equation; Two-Point Equation; Slope-Intercept Equation; Parallel and Perpendicular Lines.

Systems of Linear Equations: Classifications of Two Linear Equations; Solving Simultaneous Equations by Using the Linear Combinations Method; Inconsistent and Dependent Systems of Equations.

**1****124 **Reviewing Polynomials: Laws of Exponents; Product or Quotient of a Monomial and a Polynomial; Product of Two Binomials; Squares of Binomials; Product of a Binomial and a Trinomial. Factoring: Integers and Monomial Factoring; Factoring Monomials from Polynomials; Reviewing Trinomial Factoring; Group Factoring; Factoring the Difference of Squares; Factoring Perfect Square Trinomials; Factoring the Sum and Difference of two Cubes. Solving Algebraic Expressions: Fractional Exponents; Equalities; Inequalities; Division of Polynomials

**1****125 **Fractions and Operations: Negative Exponents; Rational Numbers; Simplifying Fractional Expressions; Multiplication of Fractions; Division of Fractions; Least Common Denominator; Adding, and Subtracting Fractions. Fractions and Equations: Complex Fractions; Synthetic Division; Fractional Equations; Fractional Inequalities; Applied Problems. Rational Numbers as Decimals: Decimals; Scientific Notation; Approximations

**1****126 **First-Degree Functions: Relations; Functions; Inverse of a Relation; Linear Equations; Relations and Slope; Linear Inequalities; Direct Variation. Second-Degree Functions: Quadratic Functions; Axis of Symmetry and the Vertex; Minimum and Maximum Points; Completing the Square; Axis of Symmetry and the Vertex from y = a (x – h )2 + k. Further Considerations

**Level 3 Mathematics with Calculus**

*1*** 127 **Radicals: Square Roots: Roots of Radicals; Rational and Irrational Numbers. Operating with Radicals: Products and Quotients; Sums and Differences; Simplification of Radicals; Rationalizing Denominators; Radicals and Exponents; Radicals and Equations. Radicals Within Radicals; Complex Numbers: Pure Imaginary Numbers; imaginary and Real Numbers; Complex Conjugates and Division

*1*** 128 **Quadratic Equations: Solving Quadratics by Factoring; Fractional Equations and Quadratics; Solving Quadratics by Completing the Square. Quadratic Solutions: The Quadratic Formula; The Discriminant and Solutions; Quadratic Coefficients. Polynomial Functions: Evaluating, Polynomial Functions; Synthetic Substitution; Remainder Theorem; Factor Theorem

*1*** 129 **Quadratic Relations: Distance Formula; Circles: Centre at Origin; Circles: Centre at (h, k). The Parabola, Ellipse, and Hyperbola: Parabolas: Vertex at Origin; Sketching the Graph of a Parabola; Parabolas: Vertex at (h, k); Quadratic Equations in Parabolic Form; Ellipses: Centre at Origin; Hyperbolas: Centre at Origin. Quadratic Systems: Quadratic-Linear Systems; Quadratic-Quadratic Systems

** 1130 **Exponential Functions: Review of Laws of Exponents; Irrational Exponents; Inverse of y = ax. Logarithms: Meaning of a Logarithm; Logarithmic Properties; Common Logarithms; Mantissa and Characteristic; Interpolation; Antilogarithms. Computations and Logarithms: Computations; Logarithmic Equations

**SM 11 **Introduction to differentiation: focusing on tangents and secants, limits and continuity, theorems and points of discontinuity. The derivative of a function and differentiation from first principles. Differentiation rules relating to a constant, a power, a constant times a function, the sum of two functions and the difference of two functions.

**SM 12 **Sketching Graphs: Students will work with linear functions including simultaneous equations with practical examples, modeling linear functions including break even analysis, polynomial functions including quadric functions and modeling them, higher order simultaneous equations.

A study of cubic functions will be covered, including their modeling, quartic functions, and rational functions, including rectangular hyperbola. The unit concludes with inverse functions and their graphs.

**SM 13 **Trigonometric Graphs: This unit covers trigonometric functions including angles in different quadrants; trigonometric functions with a negative angle including sine, cosine and tangent. Section 2 covers graphing trigonometric functions, the unit circle and the tangent, changing the parameters including dilation, translation, and reflection. The student will also sketch trigonometric functions, including phase shift, identifying functions, domain and range and applications of this work.

**SM 14 **Applications of Derivatives: This unit involves interpretation and application of the derivative both algebraically and geometrically. Particle motion in a straight line is examined with attention given to position, velocity and acceleration. Application of differentiation and of rates of change is treated. Students study the equation of the tangent, the equation of the normal and the angle between two curves. The unit also covers increasing and decreasing functions, the nature of stationary values including the second derivative test, greatest and least values and points of inflection and applications of the derivative.

**SM 15 **Differentiation Rules: Relevant concepts of continuity limits are dealt with including convergent and divergent sequences, the limit of a function, including limit theorems, the derivative of a function including polynomials, rules for differentiation including the second derivative, and piecewise-defined functions. The chain rule, product rule, and quotient rule assist with further differentiation of more complex functions.

**SM 16 **Antidifferentiation: The student will study antidifferentiation including indefinite integrals and the family of curves, integrating the sum of difference of functions, definite integrals including properties of definite integrals, areas bounded by a curve including the mid-point and trapezoid rules, and the calculation of area including the area between a curve and the y-axis.

**SM 19 **Trigonometric, Exponential and Logarithmic Functions: This unit deals with trigonometric functions including special angles and trigonometric identities, trigonometric derivatives including product and quotient rule and applications of trigonometric derivatives, derivatives of exponential functions, and the exponential function. The student also deals with indices and logarithms, derivatives of logarithmic functions, the integration of trigonometric and exponential functions and the integration of 1/x.

**SM 20 **Applications of the Integral: Attention will be given to areas between two curves, volumes of revolution including rotations about the x and y axis, rates of changes including total changes from given rates of changes and related rates and differential equations. Attention will also be given to growth and decay and applications of exponential derivatives.

**Level 3 Mathematics with Statistics**

*1*** 127 **Radicals: Square Roots: Roots of Radicals; Rational and Irrational Numbers. Operating with Radicals: Products and Quotients; Sums and Differences; Simplification of Radicals; Rationalizing Denominators; Radicals and Exponents; Radicals and Equations. Radicals Within Radicals; Complex Numbers: Pure Imaginary Numbers; imaginary and Real Numbers; Complex Conjugates and Division

*1*** 128 **Quadratic Equations: Solving Quadratics by Factoring; Fractional Equations and Quadratics; Solving Quadratics by Completing the Square. Quadratic Solutions: The Quadratic Formula; The Discriminant and Solutions; Quadratic Coefficients. Polynomial Functions: Evaluating, Polynomial Functions; Synthetic Substitution; Remainder Theorem; Factor Theorem

*1*** 129 **Quadratic Relations: Distance Formula; Circles: Centre at Origin; Circles: Centre at (h, k). The Parabola, Ellipse, and Hyperbola: Parabolas: Vertex at Origin; Sketching the Graph of a Parabola; Parabolas: Vertex at (h, k); Quadratic Equations in Parabolic Form; Ellipses: Centre at Origin; Hyperbolas: Centre at Origin. Quadratic Systems: Quadratic-Linear Systems; Quadratic-Quadratic Systems

*1*** 130 **Exponential Functions: Review of Laws of Exponents; Irrational Exponents; Inverse of y = ax. Logarithms: Meaning of a Logarithm; Logarithmic Properties; Common Logarithms; Mantissa and Characteristic; Interpolation; Antilogarithms. Computations and Logarithms: Computations; Logarithmic Equations

**SM 11 **Introduction to differentiation: focusing on tangents and secants, limits and continuity, theorems and points of discontinuity. The derivative of a function and differentiation from first principles. Differentiation rules relating to a constant, a power, a constant times a function, the sum of two functions and the difference of two functions.

**SM 12 **Sketching Graphs: Students will work with linear functions including simultaneous equations with practical examples, modeling linear functions including break even analysis, polynomial functions including quadric functions and modeling them, higher order simultaneous equations.

A study of cubic functions will be covered, including their modeling, quartic functions, and rational functions, including rectangular hyperbola. The unit concludes with inverse functions and their graphs.

**SM 13 **Trigonometric Graphs: This unit covers trigonometric functions including angles in different quadrants; trigonometric functions with a negative angle including sine, cosine and tangent. Section 2 covers graphing trigonometric functions, the unit circle and the tangent, changing the parameters including dilation, translation, and reflection. The student will also sketch trigonometric functions, including phase shift, identifying functions, domain and range and applications of this work.

**SM 17 **Probability: This unit examines the introduction to set notation and probability theory, the probability of an event, developing sample spaces including lattice and tree diagrams, complementary and mutually exclusive events, and the addition law of probability. The student will also study independent events, the multiplication law of probability, and conditional probability.

**SM 18 **Discrete Probability Distributions: Students will focus on random variables including a review of probability, probability distribution of discrete random variables, graphing probability distributions, expected values including commercial ones, and expected value theorems including variance and standard deviation. Focus will also be on the binomial probability distribution, parameters and their effects, and expected values and standard deviation.

**SM 21 **The Normal Distribution: This study incorporates continuous probability distributions, the normal distribution including the properties of the normal distribution curve, probabilities less than 0.5, applications, solving the impossible and confidence limits.

**SM 22 **Hypothesis Testing: This final unit deals with identifying a hypothesis, the statistical hypothesis, testing the null hypothesis and the sign test. In the testing of a hypothesis the student will deal with acceptable level of error, levels of significance and binomial distribution. Confidence intervals and upper and lower confidence limits are also treated.