Mathematics, Grade 9  12, Precalculus, 2013


1.) (+) Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number. [NCN4]

2.) (+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. [NCN5]

3.) (+) Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints. [NCN6]



4.) Determine numerically, algebraically, and graphically the limits of functions at specific values and at infinity. (Alabama)
a. Apply limits in problems involving convergence and divergence. (Alabama)



5.) (+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, v, v, v). [NVM1]

6.) (+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point. [NVM2]

7.) (+) Solve problems involving velocity and other quantities that can be represented by vectors. [NVM3]



8.) (+) Add and subtract vectors. [NVM4]
a. (+) Add vectors endtoend, componentwise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes. [NVM4a]
b. (+) Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum. [NVM4b]
c. (+) Understand vector subtraction v  w as v + (w), where w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction componentwise. [NVM4c]

9.) (+) Multiply a vector by a scalar. [NVM5]
a. (+) Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication componentwise, e.g., as c(v_{x}, v_{y}) = (cv_{x}, cv_{y}). [NVM5a]
b. (+) Compute the magnitude of a scalar multiple cv using cv = cv. Compute the direction of cv knowing that when cv ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0). [NVM5b]



10.) (+) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors. [NVM11]

11.) (+) Work with 2 x 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area. [NVM12]



12.) Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.* (Extend to infinite geometric series.) [ASSE4] (Alabama)
Example: Calculate mortgage payments.



13.) (+) Know and apply the Binomial Theorem for the expansion of (x + y)^{n} in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined, for example, by Pascal's Triangle. (The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.) [AAPR5]



14.) (+) Represent a system of linear equations as a single matrix equation in a vector variable. [AREI8]



15.) Create graphs of conic sections, including parabolas, hyperbolas, ellipses, circles, and degenerate conics, from seconddegree equations. (Alabama)
Example: Graph x^{2}  6x + y^{2}  12y + 41 = 0 or y^{2}  4x + 2y + 5 = 0.
a. Formulate equations of conic sections from their determining characteristics. (Alabama)
Example: Write the equation of an ellipse with center (5, 3), a horizontal major axis of length 10, and a minor axis of length 4.



16.) For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. (Key features include intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. Determine odd, even, neither.)* [FIF4] (Alabama)

17.) Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.* [FIF6]



18.) Graph functions expressed symbolically, and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* [FIF7]
a. Graph square root, cube root, and piecewisedefined functions, including step functions and absolute value functions. [FIF7b]
b. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. [FIF7c]
c. (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. [FIF7d]
d. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. [FIF7e]



19.) (+) Compose functions. [FBF1c]
Example: If T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.



20.) Determine the inverse of a function and a relation. (Alabama)

21.) (+) Verify by composition that one function is the inverse of another. [FBF4b]

22.) (+) Read values of an inverse function from a graph or a table, given that the function has an inverse. [FBF4c]

23.) (+) Produce an invertible function from a noninvertible function by restricting the domain. [FBF4d]

24.) (+) Understand the inverse relationship between exponents and logarithms, and use this relationship to solve problems involving logarithms and exponents. [FBF5]

25.) Compare effects of parameter changes on graphs of transcendental functions. (Alabama)
Example: Explain the relationship of the graph y = e^{x2} to the graph y = e^{x}.



26.) Determine the amplitude, period, phase shift, domain, and range of trigonometric functions and their inverses. (Alabama)

27.) Use the sum, difference, and halfangle identities to find the exact value of a trigonometric function. (Alabama)

28.) Utilize parametric equations by graphing and by converting to rectangular form. (Alabama)
a. Solve applicationbased problems involving parametric equations. (Alabama)
b. Solve applied problems that include sequences with recurrence relations. (Alabama)



29.) (+) Use special triangles to determine geometrically the values of sine, cosine, and tangent for ^{π}/_{3}, ^{π}/_{4}, and ^{π}/_{6}, and use the unit circle to express the values of sine, cosine, and tangent for π  x,
π + x, and 2π  x in terms of their values for x, where x is any real number. [FTF3]

30.) (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. [FTF4]



31.) (+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed. [FTF6]

32.) (+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.* [FTF7]



33.) Prove the Pythagorean identity sin^{2}(θ) + cos^{2}(θ) = 1, and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle. [FTF8] (Alabama)

34.) (+) Prove the addition and subtraction formulas for sine, cosine, and tangent, and use them to solve problems. [FTF9]





35.) (+) Derive the formula A = (^{1}/_{2})ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. (Apply formulas previously derived in Geometry.) [GSRT9] (Alabama)



36.) (+) Derive the equations of a parabola given a focus and directrix. [GGPE2]

37.) (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant. [GGPE3]



38.) (+) Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures. [GGMD2]



39.) Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. (Focus on increasing rigor using standard deviation). [SID2] (Alabama)

40.) Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). (Identify unifrom, skewed, and normal distridutions in a set of data. Determine the quartiles and interquartile range for a set of data.) [SID3] (Alabama)

41.) Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve. [SID4]



42.) Compute (using technology) and interpret the correlation coefficient of a linear fit. [SID8]

43.) Distinguish between coorelation and causation. [SID9]



44.) Understand statistics as a process for making inferences about population parameters based on a random sample from that population. [SIC1]

45.) Decide if a specified model is consistent with results from a given datagenerating process, e.g., using simulation. [SIC2]
Example: A model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model'



46.) Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. [SIC3]

47.) Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling. [SIC4]

48.) Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant. [SIC5]

49.) Evaluate reports based on data. [SIC6]



50.) (+) Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions. [SMD1]

51.) (+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution. [SMD2]

52.) (+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. [SMD3]
Example: Find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiplechoice test where each question has four choices, and find the expected grade under various grading schemes.

53.) (+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. [SMD4]
Example: Find a current data distribution on the number of television sets per household in the United States, and calculate the expected number of sets per household. How many television sets would you expect to find in 100 randomly selected households'



54.) (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values. [SMD5]
a. Find the expected payoff for a game of chance. [SMD5a]
Examples: Find the expected winnings from a state lottery ticket or a game at a fastfood restaurant.
b. Evaluate and compare strategies on the basis of expected values. [SMD5b]
Example: Compare a highdeductible versus a lowdeductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.
