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School City of Hobart Algebra II In this technological age, mathematics is more important than ever. When students leave school, they are more and more likely to use mathematics in their work and everyday lives - operating computer equipment, planning timelines and schedules, reading and interpreting data, comparing prices, managing personal finances, and completing other problem-solving tasks. What they learn in mathematics and how they learn it will provide an excellent preparation for a challenging and ever-changing future. |
| Functions |
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Relations and Functions
The learner will be able to graph relations and functions and find zeros. They use function notation and combine functions by composition. They interpret functions in given situations.
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A2.1.1
The learner will be able to recognize and graph various types of functions, including polynomial, rational, and algebraic functions.
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A2.1.2
The learner will be able to use function notation. Add, subtract, multiply, and divide pairs of functions. Example: Let f(x) = 7x + 2. Find the value of f(3). .
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A2.1.3
The learner will be able to understand composition of functions and combine functions by composition.
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A2.1.4
The learner will be able to graph relations and functions with and without graphing technology.
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A2.1.5
The learner will be able to find the zeros of a function. Example: In the last example, find the zeros of the function; i.e., find x when y = 0. .
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A2.1.6
The learner will be able to solve an inequality by examining the graph. .
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A2.1.7
The learner will be able to graph functions defined piece-wise. Example: Draw the graph of the function that relates the weight of a letter to the cost of postage. .
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A2.1.8
The learner will be able to interpret given situations as functions in graphs, formulas, and words. Example: You and your parents are going to Boston and want to rent a car at Logan International Airport on a Monday morning and drop the car off in downtown Providence, Rhode Island on the following Wednesday. Find the rates from two national car companies and plot out the costs on a graph. Decide which company offers the best deal. Explain your answer. .
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| Algebraic Concepts |
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Linear and Absolute Value Equations
The learner will be able to solve systems of linear equations and inequalities and use them to solve word problems. They model data with linear equations.
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A2.2.1
The learner will be able to graph absolute value equations and inequalities. Example: Draw the graph of y = 2x - 5 and use that graph to draw the graph of y = |2x - 5|. .
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A2.2.2
The learner will be able to use substitution, elimination, and matrices to solve systems of two or three linear equations in two or three variables. Example: Solve the system of equations: x - 2y + 3z = 5, x + 3z = 11, 5y - 6z = 9.z .
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A2.2.3
The learner will be able to use systems of linear equations and inequalities to solve word problems. Example: Each week you can work no more than 20 hours altogether at the local bookstore and the drugstore. You prefer the bookstore and want to work at least 10 more hours there than at the drugstore. Draw a graph to show the possible combinations of hours that you could work. .
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A2.2.4
The learner will be able to find a linear equation that models a data set using the median fit method and use the model to make predictions. Example: You light a candle and record its height in centimeters every minute. The results recorded as (time, height) are (0, 20), (1, 18.3), (2, 16.5), (3, 14.8), (4, 13.2), (5, 11.5), (6, 10.0), (7, 8.2), (9, 4.9), and (10, 3.1). Find the median fit line to express the candle's height as a function of the time and state the meaning of the slope in terms of the burning candle. .
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Quadratic Equations and Functions
The learner will be able to solve quadratic equations, including the use of complex numbers. They interpret maximum and minimum values of quadratic functions. They solve equations that contain square roots.
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A2.3.1
The learner will be able to define complex numbers and perform basic operations with them. Example: Multiply 7 - 4i and 10 + 6i. .
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A2.3.2
The learner will be able to understand how real and complex numbers are related, including plotting complex numbers as points in the plane. Example: Plot the points corresponding to 3 - 2i and 1 + 4i. Add these complex numbers and plot the result. How is this point related to the other two? .
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A2.3.3
The learner will be able to solve quadratic equations in the complex number system.
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A2.3.4
The learner will be able to graph quadratic functions. Apply transformations to quadratic functions. Find and interpret the zeros and maximum or minimum value of quadratic functions.
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A2.3.5
The learner will be able to solve word problems using quadratic equations. Example: You have 100 feet of fencing to make three sides of a rectangular area using an existing straight fence as the fourth side. Construct a formula in a spreadsheet to determine the area you can enclose and use the spreadsheet to make a conjecture about the maximum area possible. Prove (or disprove) your conjecture by solving an appropriate quadratic equation. .
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A2.3.6
The learner will be able to solve equations that contain radical expressions.
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A2.3.7
The learner will be able to solve pairs of equations, one quadratic and one linear, or both quadratic.
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Conic Sections
The learner will be able to write equations of conic sections and draw their graphs.
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A2.4.1
The learner will be able to write the equations of conic sections (circle, ellipse, parabola, and hyperbola). Example: Write an equation for a parabola with focus (2, 3) and directrix y = 1. .
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A2.4.2
The learner will be able to graph conic sections. Example: Graph the parabola in the last example. .
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Polynomials
The learner will be able to use the binomial theorem, divide and factor polynomials, and solve polynomial equations.
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A2.5.1
The learner will be able to understand the binomial theorem and use it to expand binomial expressions raised to positive integer powers.
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A2.5.2
The learner will be able to divide polynomials by others of lower degree.
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A2.5.3
The learner will be able to factor polynomials completely and solve polynomial equations by factoring.
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A2.5.4
The learner will be able to use graphing technology to find approximate solutions for polynomial equations.
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A2.5.5
The learner will be able to use polynomial equations to solve word problems. Example: You want to make an open-top box with a volume of 500 square inches from a piece of cardboard that is 25 inches by 15 inches. Find the possible dimensions of the box. .
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A2.5.6
The learner will be able to write a polynomial equation given its solutions. Example: Write an equation that has solutions x = 2, x = 5i and x = -5i. .
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A2.5.7
The learner will be able to understand and describe the relationships among the solutions of an equation, the zeros of a function, the x-intercepts of a graph, and the factors of a polynomial expression.
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Algebraic Fractions
The learner will be able to use negative and fractional exponents. They simplify algebraic fractions and solve equations involving algebraic fractions. They solve problems of direct, inverse, and joint variation.
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A2.6.1
The learner will be able to understand and use negative and fractional exponents.
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A2.6.2
The learner will be able to add, subtract, multiply, divide, and simplify algebraic fractions.
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A2.6.3
The learner will be able to simplify complex fractions.
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A2.6.4
The learner will be able to solve equations involving algebraic fractions.
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A2.6.5
The learner will be able to solve word problems involving fractional equations. Example: Two students, working independently, can complete a particular job in 20 minutes and 30 minutes, respectively. How long will it take to complete the job if they work together? .
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A2.6.6
The learner will be able to solve problems of direct, inverse, and joint variation. Example: One day your drive to work takes 10 minutes and you average 30 mph. The next day the drive takes 15 minutes. What is your average speed that day? .
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| Functions |
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Logarithmic and Exponential Functions
The learner will be able to graph exponential functions and relate them to logarithms. They solve logarithmic and exponential equations and inequalities. They solve word problems using exponential functions.
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A2.7.1
The learner will be able to graph exponential functions.
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A2.7.2
The learner will be able to prove simple laws of logarithms.
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A2.7.3
The learner will be able to understand and use the inverse relationship between exponents and logarithms.
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| Algebraic Concepts |
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A2.7.4
The learner will be able to solve logarithmic and exponential equations and inequalities.
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A2.7.5
The learner will be able to use the definition of logarithms to convert logarithms from one base to another.
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A2.7.6
The learner will be able to use the properties of logarithms to simplify logarithmic expressions and to find their approximate values.
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A2.7.7
The learner will be able to use calculators to find decimal approximations of natural and common logarithmic numeric expressions. Example: Find a decimal approximation for ln 500.
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| Functions |
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A2.7.8
The learner will be able to solve word problems involving applications of exponential functions to growth and decay. .
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| Numeration |
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Sequences and Series
The learner will be able to define and use arithmetic and geometric sequences and series.
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A2.8.1
The learner will be able to define arithmetic and geometric sequences and series. Example: What type of sequence is 10, 100, 1,000, 10,000, ...? .
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A2.8.2
The learner will be able to find specified terms of arithmetic and geometric sequences. Example: Find the 10th term of the arithmetic sequence 3, 7, 11, 15, ... .
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A2.8.3
The learner will be able to find partial sums of arithmetic and geometric series. Example: In the last example, find the sum of the first ten terms. .
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A2.8.4
The learner will be able to solve word problems involving applications of sequences and series. Example: You have on a Petri dish 1 square millimeter of a certain mold that doubles in size each day. What area will it cover after a week? .
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| Probability/Statistics |
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Counting Principles and Probability
The learner will be able to use fundamental counting principles to compute combinations, permutations, and probabilities.
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A2.9.1
The learner will be able to understand and apply counting principles to compute combinations and permutations. Example: There are five students who work in a bookshop. If the bookshop needs three people to operate, how many days straight could the bookstore operate without the same group of students working twice? .
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A2.9.2
The learner will be able to use the basic counting principle, combinations, and permutations to compute probabilities. Example: You are on a chess team made up of 5 players. What is the probability that you will be chosen if a three-man team is selected at random? .
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| Problem Solving |
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Mathematical Reasoning, Problem Solving
The learner will be able to use a variety of strategies to solve problems.
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A2.10.1
The learner will be able to use a variety of problem-solving strategies, such as drawing a diagram, guess-and-check, solving a simpler problem, writing an equation, and working backwards. Example: The swimming pool at Roanoke Park is 24 feet long and 18 feet wide. The park district has determined that they have enough money to put a walkway of uniform width, with a maximum area of 288 square feet, around the pool. How could you find the maximum width of a new walkway? .
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A2.10.2
The learner will be able to decide whether a solution is reasonable in the context of the original situation. Example: John says the answer to the problem in the first example is 20 feet. Is that reasonable? .
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| Algebraic Concepts |
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A2.10.3
The learner will be able to decide if a given algebraic statement is true always, sometimes, or never (statements involving rational or radical expressions, logarithmic or exponential functions).
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A2.10.4
The learner will be able to use the properties of number systems and the order of operations to justify the steps of simplifying functions and solving equations.
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| Problem Solving |
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A2.10.5
The learner will be able to understand that the logic of equation solving begins with the assumption that the variable is a number that satisfies the equation, and that the steps taken when solving equations create new equations that have, in most cases, the same solution set as the original. Understand that similar logic applies to solving systems of equations simultaneously.
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A2.10.6
The learner will be able to use counterexamples to show that statements are false. Example: Show by an example that this statement is false: the product of two complex numbers is always a complex number. .
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