Mathematics: Advanced Level—Developmental

Goal Statement

The goal of Advanced Developmental Mathematics is to provide students with sufficient algebra, geometry, and trigonometry to satisfy grade 11 prerequisites for some vocational, career, technical, and/or further academic programs.

Learning Outcomes

1. Operations with Real Numbers

It is expected that learners will be able to:

• write fractions as decimals and repeating decimals as fractions
• add, subtract, multiply and divide rational numbers
• evaluate powers with rational bases and integer exponents
• demonstrate the order of operations with rational numbers
• evaluate radicals with rational radicands and distinguish between exact answers and approximate answers
• simplify, add, subtract, multiply and divide square roots

2. First Degree Equations and Inequalities

It is expected that learners will be able to:

• solve first degree equations, in one variable, including those involving parentheses
• solve formulas for a given variable when other variables are known
• solve formulas for a given variable
• solve first degree inequalities in one variable
• solve practical problems that can be solved using a first degree equation

3. Polynomials

It is expected that learners will be able to:

• distinguish between monomials, binomials, trinomials and other polynomials (in one variable only)
• apply the laws of exponents to variable expressions with integral exponents
• evaluate polynomials by substitution
• add, subtract, and multiply polynomials
• factor polynomials by removing the largest common factor
• factor binomials of the form a²x² – b²y² and trinomials of the form x² + bx + c
• solve quadratic equations using the law of zero products

Optional Outcomes:

• factor trinomials of the form ax² + bx + c

4. Rational Expressions and Equations

It is expected that learners will be able to:

• simplify, by factoring, rational expressions consisting of polynomial numerators and either monomial, binomial, or trinomial denominators
• determine values for which a rational expression is undefined
• multiply and divide rational expressions
• add and subtract rational expressions consisting of monomial and/or binomial denominators
• solve simple rational equations and check solutions

5. Linear Equations

It is expected that learners will be able to:

• graph a linear equation including the forms x = a and y = b
• given a linear equation or its graph, determine its
  • slope
  • x– and y-intercepts
• determine the equation of a line, y = mx + b, given
  • its graph
  • its slope and a point on the line
  • two points on the line

6. Systems of Linear Equations

It is expected that learners will be able to:

• solve a system of first degree equations in two unknowns by graphing, substitution, and elimination methods
• solve practical problems that can be solved using a system of equations

7. Radical Expressions and Equations

It is expected that learners will be able to:

• simplify square roots with variable radicands
• add, subtract, multiply and divide square roots with variable radicands
• solve equations with one square root containing a polynomial radicand and check for extraneous solutions

8. Trigonometry

It is expected that learners will be able to:

• solve right triangles using one or more of
  • the sine ratio
  • the cosine ratio
  • the tangent ratio
  • the Pythagorean theorem
  • the angle sum property of triangles

Optional Outcomes:

• evaluate sine and cosine for angles from 0º to 180º
• solve triangles using the Law of Cosines or the Law of Sines, excluding the ambiguous case

9. Optional Learning Outcomes

Students must complete one of the following four optional topics:

A. The Quadratic Equation

It is expected that learners will be able to:

• solve quadratic equations by factoring
• solve equations of the form x² + bx + c = 0 by completing the square
• solve quadratic equations by using the quadratic formula
• graph y = ax² + bx + c and determine its
• x– and y-intercepts
• vertex
• solve practical problems that can be solved using a quadratic equation

B. Statistics

It is expected that learners will be able to:

• determine the mean, median, mode, range and standard deviation of a set of data
• represent data graphically using broken line graphs and bar graphs
• understand how the normal curve can be used to describe a normally distributed population
• calculate z-scores and determine areas under the normal curve
• use areas under the normal curve to analyze data in terms of the probability of various events

C. Financial Mathematics

It is expected that learners will be able to:

• solve simple interest problems using the formula, i = prt (for any variable)
• solve compound interest problems for A or P using [latex]A=P\left ( 1+\frac{r}{n} \right )^{nt}[/latex]
• find the effective interest rate using [latex]E.R.=\left ( 1+\frac{r}{n} \right )^{n}-1[/latex]
• solve annuity problems using [latex]A=\tfrac{nP[\left ( 1+\frac{r}{n} \right )^{n}-1]}{r}[/latex]  (for A or P only)
• find periodic payment using    [latex]P=\frac{A(\frac{r}{n})}{1-\left ( 1+\frac{r}{n} \right )^{-nt}}[/latex]
• determine the finance charge on a loan
• determine the interest rate on a loan using tables or appropriate technology

D. Geometry

It is expected that learners will be able to:

• classify triangles according to angles and sides
• use the properties of triangles to determine the measure of sides and angles
• determine the measure and/or congruence of angles given a transversal and two parallel lines
• use the triangle congruence theorems in simple guided proofs