Essential Outcomes for this course based on the Graduation Standards:
Numbers and Quantity:
1) Extend the properties of exponents to rational exponents.
Students should be able to explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.
Students should be able to rewrite expressions involving radicals and rational exponents using the properties of exponents.
2) Use the properties of rational and irrational numbers.
Students should be able to explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
3) Reason quantitatively and use units to solve problems.
Students should be able to use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
4) Perform arithmetic operations with complex numbers.
Students should be able to know there is a complex number i such that i2 = -1, and every complex number has the form a + bi with a and b real numbers.
Students should use the relation i2=-1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
Students should be able to solve quadratic equations with real coefficients that have complex solutions.
Additional Course Required Outcomes:
5) Manipulate and use matrices to solve problems.
Use matrices to represent and manipulate data.
Multiply matrices by scalars to produce new matrices.
Add, subtract, and multiply matrices of appropriate dimensions.
1)Interpret the structure of expressions.
Students should be able to interpret the expressions that represent a quantity in terms of its context, interpret parts of an expression, such as terms, factors, and coefficients. They should also interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P.
Students should use the structure of an expression to identify ways to rewrite it. For example, see X4- Y4 as (X2)2 – (Y2)2 thus recognizing it as a difference of squares that can be factored into the difference of 2 squares (X2-Y2)(X2+Y2)
2) Rewrite rational expressions.
Students should be able to rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.
3) Perform arithmetic operations on polynomials.
Students should be able to add, subtract, multiply, and divide with polynomials
4) Understand the relationship between zeros and factors of a polynomial.
Students should factor a quadratic expression to reveal the zeros of the function it defines.
5) Understand solving equations as a process of reasoning and explain the reasoning.
Students should be able to rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V=IR to highlight resistance R.
Students should be able to explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Student should be able to construct a viable argument to justify a solution method.
6) Solve equations and inequalities with one variable.
Students should be able to solve simple rational and radical equations in one variable and give examples showing how extraneous solutions might arise.
Students should solve linear equations and inequalities in one letter, including equations with coefficients represented by letters.
Students should solve quadratic equations with one variable.
Students should solve quadratic equations by inspection (e.g. for X2=49), taking square roots, complete the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a+bi for real numbers a and b.
7) Solve systems of equations.
Students should be able to solve systems consisting of linear equations exactly and approximately (e.g. graphically) focusing on pairs of linear equations in two variable.
8) Represent and solve equations and inequalities graphically
Students should understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).
Students should be able to explain why the x-coordinates of the points where the graphs of the equations y=f(x) and y=g(x) intersect are the solutions of the equation f(x)=g(x), find the solutions approximately, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
Students should graph the solutions to a linear inequality in two variables as a half plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.
1)Understand the concept of a function and use function notation.
Students understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, the f(x) denotes t eh output of f corresponding to the input x. The graph of f is the graph of the equations y=f(x).
2) Interpret functions that arise in application in terms of the context.
Students will use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.
3) Analyze functions using different representations.
Students will graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.
Students will graph linear and quadratic functions and show intercepts, maxima, and minima.
4) Build a function that models a relationship between two quantities.
Students will write a function that describes a relationship between two quantities.
Students will combine standard function types using arithmetic operations.
Students will compose functions.