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School City of Hobart Pre-Calculus 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. |
| Calculus and Pre-Calculus |
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Relations and Functions
The learner will be able to use polynomial, rational, and algebraic functions to write functions and draw graphs to solve word problems, to find composite and inverse functions, and to analyze functions and graphs. They analyze and graph circles, ellipses, parabolas, and hyperbolas.
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| Functions |
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PC.1.1
The learner will be able to recognize and graph various types of functions, including polynomial, rational, algebraic, and absolute value functions. Use paper and pencil methods and graphing calculators.
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PC.1.2
The learner will be able to find domain, range, intercepts, zeros, asymptotes, and points of discontinuity of functions. Use paper and pencil methods and graphing calculators.
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PC.1.3
The learner will be able to model and solve word problems using functions and equations. Example: You are on the committee for planning the prom and need to decide what to charge for tickets. Last year you charged $5.00 and 400 people bought tickets. Earlier experiences suggest that for every 10¢ decrease in price you will sell 50 extra tickets. Use a spreadsheet and write a function to show how the amount of money in ticket sales depends on the number of 10¢ decreases in price. Construct a graph that shows the price and gross receipts. What is the optimum price you should set for the tickets? .
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PC.1.4
The learner will be able to define, find, and check inverse functions.
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PC.1.5
The learner will be able to describe the symmetry of the graph of a function.
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PC.1.6
The learner will be able to decide if functions are even or odd. Example: Is the function tan x even, odd, or neither? Explain your answer. .
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PC.1.7
The learner will be able to apply transformations to functions.
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PC.1.8
The learner will be able to understand curves defined parametrically and draw their graphs.
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PC.1.9
The learner will be able to compare relative magnitudes of functions and their rates of change.
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| Calculus and Pre-Calculus |
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PC.1.10
The learner will be able to write the equations of conic sections in standard form (completing the square and using translations as necessary), in order to find the type of conic section and to find its geometric properties (foci, asymptotes, eccentricity, etc.).
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| Functions |
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Logarithmic and Exponential Functions
The learner will be able to solve word problems involving logarithmic and exponential functions. They draw and analyze graphs, and find inverse functions.
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PC.2.1
The learner will be able to solve word problems involving applications of logarithmic and exponential functions.
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PC.2.2
The learner will be able to find the domain, range, intercepts, and asymptotes of logarithmic and exponential functions. Example: For the function L(x) = log10 (x - 4), find its domain, range, x-intercept, and asymptote. .
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PC.2.3
The learner will be able to draw and analyze graphs of logarithmic and exponential functions. Example: In the last example, draw the graph of L(x). .
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PC.2.4
The learner will be able to define, find, and check inverse functions of logarithmic and exponential functions.
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| Trigonometry |
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Trigonometry in Triangles
The learner will be able to define trigonometric functions using right triangles. They solve word problems and apply the laws of sines and cosines.
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PC.3.1
The learner will be able to solve word problems involving right and oblique triangles. Example: You want to find the width of a river that you cannot cross. You decide to use a tall tree on the other bank as a landmark. From a position directly opposite the tree, you measure 50 m along the bank. From that point, the tree is in a direction at 37º to your 50 m line. How wide is the river? .
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PC.3.2
The learner will be able to apply the laws of sines and cosines to solving problems. Example: You want to fix the location of a mountain by taking measurements from two positions 3 miles apart. From the first position, the angle between the mountain and the second position is 78º. From the second position, the angle between the mountain and the first position is 53º. How far is the mountain from each position? .
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PC.3.3
The learner will be able to find the area of a triangle given two sides and the angle between them. Example: Calculate the area of a triangle with sides of length 8 cm and 6 cm enclosing an angle of 60º. .
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Trigonometric Functions
The learner will be able to define trigonometric functions using the unit circle and use degrees and radians. They draw and analyze graphs, find inverse functions, and solve word problems.
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PC.4.1
The learner will be able to define sine and cosine using the unit circle. Example: Find the acute angle A for which sin 150º = sin A. .
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PC.4.2
The learner will be able to convert between degree and radian measures. Example: Convert 90º, 45º, and 30º to radians. .
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PC.4.3
The learner will be able to learn exact sine, cosine, and tangent values for 0, ~/2, ~/3, ~/4, ~/6, and multiples of ~. Use those values to find other trigonometric values.
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PC.4.4
The learner will be able to solve word problems involving applications of trigonometric functions. Example: In Indiana, the day length in hours varies through the year in a sine wave. The longest day of 14 hours is on Day 175 and the shortest day of 10 hours is on Day 355. Sketch a graph of this function and find its formula. Which other day has the same length as July 4? .
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PC.4.5
The learner will be able to define and graph trigonometric functions (i.e., sine, cosine, tangent, cosecant, secant, cotangent). Example: Graph y = sin x and y = cos x, and compare their graphs. .
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PC.4.6
The learner will be able to find domain, range, intercepts, periods, amplitudes, and asymptotes of trigonometric functions. Example: Find the asymptotes of tan x and find its domain. .
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PC.4.7
The learner will be able to draw and analyze graphs of translations of trigonometric functions, including period, amplitude, and phase shift.
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PC.4.8
The learner will be able to define and graph inverse trigonometric functions.
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PC.4.9
The learner will be able to find values of trigonometric and inverse trigonometric functions.
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PC.4.10
The learner will be able to know that the tangent of the angle that a line makes with the x-axis is equal to the slope of the line. Example: Use a right triangle to show that the slope of a line at 135º to the x-axis is -1. .
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PC.4.11
The learner will be able to make connections between right triangle ratios, trigonometric functions, and circular functions.
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Trigonometric Identities and Equations
The learner will be able to prove trigonometric identities, solve trigonometric equations, and solve word problems.
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PC.5.1
The learner will be able to know the basic trigonometric identity cos2x + sin2x = 1 and prove that it is equivalent to the Pythagorean Theorem.
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PC.5.2
The learner will be able to use basic trigonometric identities to verify other identities and simplify expressions.
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PC.5.3
The learner will be able to understand and use the addition formulas for sines, cosines, and tangents.
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PC.5.4
The learner will be able to understand and use the half-angle and double-angle formulas for sines, cosines, and tangents.
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PC.5.5
The learner will be able to solve trigonometric equations.
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PC.5.6
The learner will be able to solve word problems involving applications of trigonometric equations. Example: In the example about day length in Standard 4, for how long in winter is there less than 11 hours of daylight? .
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Polar Coordinates and Complex Numbers
The learner will be able to define polar coordinates and complex numbers and understand their connection with trigonometric functions.
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PC.6.1
The learner will be able to define polar coordinates and relate polar coordinates to Cartesian coordinates.
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PC.6.2
The learner will be able to represent equations given in rectangular coordinates in terms of polar coordinates.
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PC.6.3
The learner will be able to graph equations in the polar coordinate plane.
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PC.6.4
The learner will be able to define complex numbers, convert complex numbers to trigonometric form, and multiply complex numbers in trigonometric form. Example: Write 3 + 3i and 2 - 4i in trigonometric form and then multiply the results. .
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PC.6.5
The learner will be able to state, prove, and use De Moivre's Theorem.
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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, understand the concept of a limit, and solve word problems.
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PC.7.1
The learner will be able to understand and use summation notation.
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PC.7.2
The learner will be able to find sums of infinite geometric series.
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PC.7.3
The learner will be able to prove and use the sum formulas for arithmetic series and for finite and infinite geometric series.
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PC.7.4
The learner will be able to use recursion to describe a sequence.
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PC.7.5
The learner will be able to understand and use the concept of limit of a sequence or function as the independent variable approaches infinity or a number. Decide whether simple sequences converge or diverge.
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PC.7.6
The learner will be able to solve word problems involving applications of sequences and series. Example: You put $100 in your bank account today, and then each day put half the amount of the previous day (always rounding to the nearest cent). Will you ever have $250 in your account? .
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| Data Interpretation |
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Data Analysis
The learner will be able to model data with linear and non-linear functions.
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PC.8.1
The learner will be able to find linear models using the median fit and least squares regression methods. Decide which model gives a better fit. Example: Measure the wrist and neck size of each person in your class and make a scatter plot. Find the median fit line and the least squares regression line. Which line is a better fit? Explain your reasoning. .
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PC.8.2
The learner will be able to calculate and interpret the correlation coefficient. Use the correlation coefficient and residuals to evaluate a "best-fit" line. Example: Calculate and interpret the correlation coefficient for the linear regression model in the last example. Graph the residuals and evaluate the fit of the linear equation. .
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PC.8.3
The learner will be able to find a quadratic, exponential, logarithmic, power, or sinusoidal function to model a data set and explain the parameters of the model. Example: Drop a ball and record the height of each bounce. Make a graph of the height (vertical axis) versus the bounce number (horizontal axis). Find an exponential function of the form y = a.bx that fits the data and explain the implications of the parameters a and b in this experiment. .
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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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PC.9.1
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, examining simpler problems, and working backwards. Example: The half-life of carbon-14 is 5,730 years. The original concentration of carbon-14 in a living organism was 500 grams. How might you find the age of a fossil of that living organism with a carbon-14 concentration of 140 grams? .
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PC.9.2
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 about 10,000 years. Is his answer reasonable? Why or why not? .
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PC.9.3
The learner will be able to decide if a given algebraic statement is true always, sometimes, or never (statements involving rational or radical expressions, trigonometric, logarithmic or exponential functions). Example: Is the statement sin 2x = 2 sinx cosx true always, sometimes, or never? Explain your answer. .
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PC.9.4
The learner will be able to use the properties of number systems and order of operations to justify the steps of simplifying functions and solving equations.
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PC.9.5
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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PC.9.6
The learner will be able to define and use the mathematical induction method of proof. Example: Prove De Moivre's Theorem using mathematical induction. .
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