Honors Algebra II

Textbook:

            Algebra 2

            Glencoe

General Description:

            Honors Algebra II is the third year of college prep in Math.  The course builds on the evaluation and application skills initiated in Algebra I, and moves towards methods of analyzing functions.  It involves students in the selection of the type of function that fits a given situation, the derivation of the equation of that function and predictions of future events based on their analysis.  Graphing calculators will be used to investigate several concepts.  The main concentration of Algebra II is on Illinois goals six and eight.

Objectives:

1.         Students will be able to recognize and locate rational, irrational and complex numbers. (6A)

2.         Students will be able to understand and use connections between whole numbers, integers, rational numbers, irrational numbers, realnumbers and complex numbers. (6A)

3.         Students will be able to use matrices to organize data. (6A)

4.         Students will be able to work with rational functions using the arithmetic of polynomials and long division. (6B/C)

5.         Students will be able to solve problems that require mastery of the elements of exponential arithmetic, including computations with rational powers and roots. (6B/C)

6.         Students will be able to perform simple computations with matrices. (6B/C)

7.         Students will be able to solve a system of two or more linear equations by several methods including matrices and determinants. (8C/D)

8.         Students will be able to evaluate functions for given domains. (8C/D)

9.         Students will be able to use the definition of function. (8B)

10.       Students will be able to create arithmetic and geometric sequences to fit a given set of conditions and determine the nth term in an arithmetic or geometric sequence. (8A)

11.       For a quadratic equation with real coefficients, students will be able to understand that if a complex number is a solution, then its conjugate is also a solution (8C/D)

12.       Students will be able to use properties of logarithms. (6C/D)

13.       Students will be able to apply connections between equations, graphs, table of values, and properties of common graphs and functions. (8B)

14.       Students will be able to represent quantitative relationships graphically, and interpret the meaning of a specific part of the graph as it relates to the situation represented by the graph. (8B)

Outline:

            Unit 1              Prerequisite Information

            Unit 2              Linear Functions

            Unit 3              Linear Systems of  Equations

            Unit 4              Matrix Operations and Applications

            Unit 5              Polynomials

            Unit 5A           Radicals

            Unit 6              Quadratic Functions

            Unit 7              Rational Algebraic Functions

            Unit 8              Conic Sections

            Unit 9              Exponential Functions

            Unit 10            Sequences and Sums

            Unit 11            Probability

Homework:

            Overnight assignments are due at the beginning of the next day class period.  Absent students will turn in work the day after their return.  Each day’s assignments will be left on the teacher’s voice mail.

Class rules:

            Everyone receives the classroom rules on the first day of school to keep in their journals.  Journals will be collected periodically and checked for the rules.  The class will follow the school-wide final exam policy.

Materials:

            Students should bring the following materials to class each day:

                        Textbook

                        Notebook

                        Pencil

                        Graphing Calculator – not required but strongly recommended

                        Graph Paper

                        Scientific Calculator

Grading Policy:

            The district grading scale will be used in this class.  The grade will be determined in the following manner:

                        Homework                                          30%

                        Journal                                     20%

                        Tests and Quizzes                               50%

Prerequisite course:              Honors Algebra I

Follow up courses:                Trigonometry and Pre Calculus

Unit Plan 

Algebra 2

10^th and 11^th grade

Mrs. Abbett

Unit 6 – Radical Expressions and Equations

15 days                        Feb. 6 – Feb. 23

State Standards:

6A:      1.         Illustrate the relationship between second and third roots and powers of a number.

            2.         Represent numbers in equivalent forms (radical/rational exponents).

6B       1.         Determine an appropriate numerical representation of a problem situation,   including roots and powers, if applicable.

1. Compare and contrast the properties of numbers and number systems, including the complex numbers as solutions to quadratic equations that do not have real solutions.
2. Use the field properties and properties of equalitiy for the set of complex numbers.
3. Determine the opposite, reciprocal, absolute values, and positive integral powers of a complex number.

8A       1.         Create an equation of a line of best fit from a set of ordered pairs or set of data                              points.

8B       1.         Fit an equation to data using a calculator.

Common Core Standards

Standard (N-RN-2)     Rewrite expressions involving radicals and rational exponents using the properties of exponents.

Standard (N-CN.1)     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.

Standard  (N-CN.2)    Use the relation i² = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

Standard (A-REL.2)   Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

Daily Plans

Day 1  Simplifying Radical Expressions

Have students come up to the smart board to fill in any perfect squares, cubes, fourth powers, etc. that they know.  Once they have done all of them that they know, discuss how to find others and fill in a chart with all of the information.  Show how to use the chart to simplify a radical.  Have students draw a problem out of a bowl and put their solutions on the smart board.

Assign:            Simplifying Radicals Worksheet.

Day 2 – Simplifying Radical Expressions With Variables

Recall that square and square root are inverse operations, cube and cube root are inverses, etc.  Review yesterday’s lesson.  Use both methods to simplify a radical that contains one or more variables.  Recall the power rule for exponents and discuss what the inverse would have us do for radical expressions.  Do examples on the smart board for the whole class.

Assign:            Page 248, #34 - 57

Day 3 – Multiplying Radical Expressions

Do calculator investigation on multiplying radical expressions.  Have students share their results.  Discuss how the simplified answer is an exact amount and the calculator answer is an approximation.  Have half of the students simplify, then multiply and the other half multiply, then simplify.  Discuss the advantages and disadvantages of each method.  Do examples in small groups. 

Assign:            Multiplying Radicals Worksheet

Day 4 – Adding and Subtracting Radical Expressions

Have students use calculators to do an investigation involving adding and subtracting radicals.  Discuss results as a whole class.  Decide as a class why we must simplify first before we add or subtract the radicals.  Include cube, fourth fifth, etc. roots as well as square roots.  Show some examples that cannot be added or subtracted  Do examples on the smart board.

Assign:            Adding and Subtracting Radicals Worksheet

Day 5 – Rationalizing Radical Denominators

Define rationalization of a denominator.  Show how to rationalize a square root denominator and a cube root denominator.  Have students do examples in pairs and explain to each other how they got their answers.  Have students extend their methods to determine how to rationalize a fourth root denominator.

Assign:            Page 254, #35 - 48

Day 6 – Binomial Radical Expressions

Have students lead a student at the smart board through the process of multiplying two regular binomial expressions.  Now use their steps to multiply two radical binomial expressions.  Remind students to present their answers in simplest form.  Do the same for rationalizing radical binomial denominators.  Introduce the definition of a conjugate and show how it is used.  Do examples on smart board.

Assign:            Radical Binomial Worksheet

Day 7 – Rational Exponents

Recall the product rule, quotient rule and power rule for exponents.  Define rational exponents and show how to rewrite a rational exponent as a radical expression.  Show both how to evaluate and how to simplify expressions with rational exponents.  Do examples that are written in both forms and give answers both ways.

            Assign:            Page 261, #21 – 50

Day 8 – Rational Equations

Review the difference between an expression and an equation.  Recall the rules for solving an equation.  Demonstrate how to solve a radical equation both with square roots and cube roots.  Then show what to do if there is a radical on both sides of the equation.  Define extraneous solutions and check each solution to see if it is extraneous.  Do examples on smart board.

Assign:            Page 266, #13 – 24

Day 9 – Simplifying Imaginary Numbers

Define i as .  Show how to simplify imaginary expressions both in radical and variable form.  Have students get in small groups to do examples.  Show how to multiply two or more imaginary expressions and simplify the results.

Assign:            Page 274, #18 – 29

Day 10 – Operations on Complex Numbers

Introduce complex numbers in the form a + bi.  Demonstrate how to add, subtract, multiply and divide complex numbers and simplify the solutions.  Recall how to multiply binomial expressions and rationalize binomial denominators.  Use a round robin technique on the example problems.

Assign:            Page 274, #30 – 45

Day 11 – Equations With Complex Solutions

Recall how to solve quadratic equations by taking the square root of both sides.  Discuss how to present the answer when it is the square root of a negative number.  Demonstrate how to find missing parts of complex numbers in an equation with a complex number on both sides.  Do examples as a whole class on the smart board.

Assign:            Page 274, #48 – 61

Day 12 – Unit 6 Review

Day 13 – Unit 6 Test

Unit Plan

Algebra 2

10^th and 11^th grade

Mrs. Abbett

Unit 7 – Quadratic Functions

12 days                        Feb. 24 – Mar. 14 

State Standards:

6B       1.         Solve problems using complex numbers and their various representations.

1. Compare and contrast the properties of numbers and number systems, including the complex numbers as solutions to quadratic equations that do not have real solutions.

8A       1.         Explain the difference between constant and non-constant rate of change.

8B       1.         Interpret the role of the coefficients and constants on the graph of linear and quadratic functions given a set of equations.

1. Relate the situation to the graph and the function values for direct, inverse, and joint variations.

8C       1.         Describe and compare the properties of linear and quadratic functions.

8D.      1.         Solve simple quadratic equations using algebraic or graphical representations.

10A.    1.         Analyze two-variable data for linear or quadratic fit.

Common Core Standards

Standard (N-CN.7) – Solve quadratic equations with real coefficients that have complex solutions.

Standard (N-CN.8) – Extend polynomial identities to the complex numbers.

Standard (A-SSE.3a) – Factor a quadratic expression to reveal the zeros of the function it defines.

Standard (A-SSE.3b) – Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

Standard (A-CED.1) – Create equations and inequalities in one variable and use them to solve problems.

Daily Plans

Day 1  Graphing quadratic Functions With Tables

Students will use the graphing calculators to graph a given quadratic equation.  We will discuss the graph, a parabola, its vertex, intercepts, and direction.  We will use the table function of the calculator to find values and investigate what happens when we change a, b, or c in the equation y = ax² + bx + c.  The assignment will be done with the graphing calculators.

Assign:            Page 291, #14 – 28.

Day 2  Graphing Parabolas by the Vertex and Axis of Symmetry

Students will recall yesterday’s lesson on graphs and discuss the challenges of graphing by a table without a graphing calculator.  Give students the formula h = -b/2a and k = f(h). for finding the vertex.  Show on the smart board how to use the vertex and the y-intercept to graph the parabola.  Find the equation of the axis of symmetry and discuss why it is a vertical line.  Do examples on the smart board.

Assign:            Worksheet

Day 3  Finding a Maximum or Minimum Value

Recall the shape of the graph of a parabola.  Discuss when the vertex would be a maximum and when it would be a minimum.  Do examples on small groups and have a student report out from each group.  Finish with power point flash cards which the students decide either maximum or minimum by looking at the equations.

Assign:            Page 291, #32 – 43, 46, 47

Day 4  Solving a Quadratic Equation by Factoring

Students will recall the methods of factoring a quadratic expression.  They will practice getting a zero on one side of a quadratic equation and then factoring the other side.  We will discuss why we are able to set each factor equal to zero and solve for x.  We will show on the graphing calculators that what we are solving for is the x-intercepts.  Do some examples with one or no x-intercepts.

Assign:            Page 304, #14 – 32 even

Day 5  Finding the Equation of a Quadratic Function Given the Solutions

We will start with the students telling me the steps to solving a problem from yesterday.  I will then reverse the steps they gave me on a new problem given two solutions.  Do other examples including one solution, complex solutions, and rational solutions on the smart board as a whole class.  Check answers on the graphing calculators.

Assign:            Chapter 6 – 3 Study Guide

Day 6  Solving a Quadratic Equation by Completing the Square

Start the lesson by showing students how to solve a quadratic trinomial square by taking the square root of each side.  Next, demonstrate how to find c, given x² + bx so that you have a trinomial square.  Finally, we will show how to solve a trinomial square when a is some value other than 1.  Do all examples on the smart board.

Assign:            Page 311, #14 – 46 even.

Day 7  Solving a Quadratic Equation by the Quadratic Formula and Using the Discriminant

Discuss the pros and cons of solving quadratic equations either by factoring or by completing the square.  Introduce the quadratic formula song.   Have students sing it and point to the words on the smart board.  Show how to use it and how the discriminant came out of the formula.  Do examples in which we first find the discriminant and determine the nature of the roots, and then use the quadratic formula to find those roots, both real and complex.

            Assign:            Page 318, #14 – 38 even.

Day 8  Analyzing Graphs of Quadratic Functions

Start by graphing y = x² on the graphing calculators.  Have students first put positive and negative numbers in for a and graph y = ax² on the same screen.  Discuss the results with the class.  Now, graph y = x² + c and y = (x + c)² and discuss the similarities and differences of the graphs.  Have students make predictions of other examples and them check their predictions on the graphing calculators  Students will use the graphing calculators to do the assignment.

Assign:            Page 321, #4 – 14 even.

Day 9  Writing Equations in Vertex Form

Discuss how we graphed a quadratic equation using the vertex, (h, k).  Show how we can take the vertex and a given point on the graph and find the equation of that graph either in the form y – k = a(x – h)² or in the form y = ax² + bx + c.  Do examples as a whole class on the smart board.

Assign:            Page 326, #16 – 30 even, 40, 42, 44.

Day 10            Graphing Quadratic Inequalities

Recall how we determined where to shade in the graph of a linear inequality.  Expand to include graphing and shading a quadratic inequality.  Do examples both on graph paper and on the graphing calculators.

Assign:            Page 333, #14 – 38 even.

Day 11 Unit 7 Review

Day 12            Unit 7 Test