# Pre-Calculus

Curriculum > High > Beijing > Mathematics
sequence
Standard Benchmark DOK CCSS Sequence
7.11 Describe characteristics of graphs, tables, and equations that model families of functions. 　 Stem Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. 1 F.IF.7 1.1.1
7.11 Describe characteristics of graphs, tables, and equations that model families of functions. 　 Stem Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. 1 F.IF.7 1.1.1
7.11 Describe characteristics of graphs, tables, and equations that model families of functions. 　 Stem 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.* 2 F.IF.6 1.1.2
7.11 Describe characteristics of graphs, tables, and equations that model families of functions. 　 Stem 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.* 2 F.IF.6 1.1.2
7.13 Know the difference between a recursive rule and an explicit expression for a function. 　 Stem (+) Compose functions. 1 F.BF.1.c 1.1.2
7.14 Know the difference between a recursive rule and an explicit expression for a function. Continued 7.14.1.C  Combine standard function types using arithmetic operations. 1 F.BF.1.b 1.1.2
7.13 Know the difference between a recursive rule and an explicit expression for a function. 　 Stem (+) Verify by composition that one function is the inverse of another. 1 F.BF.4.b 1.1.3
7.13 Know the difference between a recursive rule and an explicit expression for a function. 　 (+) Read values of an inverse function from a graph or a table, given that the function has an inverse. 2 F.BF.4.c 1.1.4
7.13 Know the difference between a recursive rule and an explicit expression for a function. 　 (+) Produce an invertible function from a noninvertible function by restricting the domain. 2 F.BF.4.d 1.1.5
7.15 Knows that manipulating the parameters of the symbolic rule will result in a predictable transformation of the graph. 7.15.1.A  Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. 2 F.BF.3 1.1.6
7.16 Describe characteristic graph, table, and equation formats for linear, exponential, and quadratic functions. 7.16.1.B  Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two inputoutput pairs (include reading these from a table). 2 F.LE.2 1.1.7
7.17 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle 7.17.1.A  Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle. 2 F.TF.2 1.2.1
7.17 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle 　 STEM (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions. 2 F.TF.4 1.2.2
7.17 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle 7.17.2.A  Prove the Pythagorean identity sin2(θ) + cos2(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. 3 F.TF.8 1.2.3
7.17 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle 7.17.3.A  Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.* 2 F.TF.5 2.1.1
7.17 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle 　 Stem (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the values of sine, cosines, and tangent for x, π+x, and 2π–x in terms of their values for x, where x is any real number. 2 F.TF.3 2.1.2
3.18 Model periodic phenomena with trigonometric functions 　 STEM (+) Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed. 2 F.TF.6 2.1.3
3.18 Model periodic phenomena with trigonometric functions 　 STEM (+) 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. 2 F.TF.7 2.1.3
7.16 Describe characteristic graph, table, and equation formats for linear, exponential, and quadratic functions. 7.16.1.A  Distinguish between situations that can be modeled with linear functions and with exponential functions. 2 F.LE.1 2.2.1
7.16 Describe characteristic graph, table, and equation formats for linear, exponential, and quadratic functions. 7.16.3.A  Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. 2 F.LE.1.b 2.2.2
7.16 Describe characteristic graph, table, and equation formats for linear, exponential, and quadratic functions. 　 Stem Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. 2 F.LE.1.c 2.2.3
7.16 Describe characteristic graph, table, and equation formats for linear, exponential, and quadratic functions. 　 Stem Interpret the parameters in a linear or exponential function in terms of a context. 2 F.LE.5 2.2.4
7.14 Know the difference between a recursive rule and an explicit expression for a function. Continued 　 (+) Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents. 1 F.BF.5 2.2.5
7.16 Describe characteristic graph, table, and equation formats for linear, exponential, and quadratic functions. 7.16.2.A  Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. 3 F.LE.1.a 2.2.6
7.16 Describe characteristic graph, table, and equation formats for linear, exponential, and quadratic functions. 7.16.3.B  For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology. 1 F.LE.4 2.2.7
7.16 Describe characteristic graph, table, and equation formats for linear, exponential, and quadratic functions. 7.16.2.B  Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. 2 F.LE.3 2.2.8
7.17 Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle 7.17.2.A  Prove the Pythagorean identity sin2(θ) + cos2(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. 3 F.TF.8 3.1.1
3.18 Model periodic phenomena with trigonometric functions 　 STEM (+) 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. 2 F.TF.7 3.1.2
3.18 Model periodic phenomena with trigonometric functions 　 STEM (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. 3 F.TF.9 3.1.3
1.5 Represent and model with vector quantities. 　 STEM (+) 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). 1 N.VM.1 3.2.1
1.5 Represent and model with vector quantities. 　 STEM (+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point. 1 N.VM.2 3.2.2
1.5 Represent and model with vector quantities. 　 STEM (+) Solve problems involving velocity and other quantities that can be represented by vectors. 2 N.VM.3 3.2.3
1.6 Perform operations on vectors. 　 STEM (+) Add and subtract vectors. 1 N.VM.4 3.2.4
1.6 Perform operations on vectors. 　 STEM Add vectors end‐to‐end, component‐wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes. 2 N.VM.4.a 3.2.5
1.6 Perform operations on vectors. 　 STEM Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum. 1 N.VM.4.b 3.2.6
1.6 Perform operations on vectors. 　 STEM 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 component‐wise. 2 N.VM.4.c 3.2.7
1.6 Perform operations on vectors. 　 STEM (+) Multiply a vector by a scalar. 1 N.VM.5 3.2.8
1.6 Perform operations on vectors. 　 STEM Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component‐wise, e.g., as c(vx, vy) = (cvx, cvy). 2 N.VM.5.a 3.2.9
1.6 Perform operations on vectors. 　 STEM Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0). 1 N.VM.5.b 3.2.10
1.7 Perform operations on matrices and use matrices in applications. 　 STEM (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.  2 N.VM.6 3.3.1
1.7 Perform operations on matrices and use matrices in applications. 　 STEM (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.    1 N.VM.7 3.3.2
1.7 Perform operations on matrices and use matrices in applications. 　 STEM (+) Add, subtract, and multiply matrices of appropriate dimensions. 1 N.VM.8 3.3.3
1.7 Perform operations on matrices and use matrices in applications. 　 STEM (+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties. 2 N.VM.9 3.3.4
1.7 Perform operations on matrices and use matrices in applications. 　 STEM (+) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of  a square matrix is nonzero if and only if the matrix has a multiplicative inverse. 1 N.VM.10 3.3.5
1.7 Perform operations on matrices and use matrices in applications. 　 STEM (+) 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. 2 N.VM.11 3.3.6
1.7 Perform operations on matrices and use matrices in applications. 　 STEM (+) Work with 2 × 2 matrices as a transformations of the plane, and interpret the absolute value of the determinant in terms of area. 2 N.VM.12 3.3.7
5.3 Perform arithmetic operations with complex numbers. 　 Stem (+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x – 2i). 1 N.CN.8 3.4.1
5.4 Represent complex numbers and their operations on the complex plane. 　 Stem (+) 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. 1 N.CN.4 3.4.2
5.4 Represent complex numbers and their operations on the complex plane. 　 Stem (+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (1 + √3 i)3 = 8 because (1 + √3 i) has modulus 2 and argument 120°. 1 N.CN.5 3.4.3
5.4 Represent complex numbers and their operations on the complex plane. 　 Stem (+) 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. 2 N.CN.6 3.4.4
5.4 Represent complex numbers and their operations on the complex plane. 　 Stem (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex number . 2 N.CN.3 3.4.5
5.4 Represent complex numbers and their operations on the complex plane. 　 Stem (+) Know the Fundamental Theorem of Algebra show that it is true for quadratic polynomials. 2 N.CN.9 3.4.6
9.4 Recognize the purposes of and differences among sample surveys, experiments, and observational studies. 9.4.1.A  Understand statistics as a process for making inferences about population parameters population parameters from that population. 2 S.IC.1 4.1.1
9.4 Recognize the purposes of and differences among sample surveys, experiments, and observational studies. 9.4.2.A  Decide if a specified model is consistent with results from a given datagenerating process, e.g., using simulation.
For 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?
3 S.IC.2 4.1.2
9.4 Recognize the purposes of and differences among sample surveys, experiments, and observational studies. 9.4.3.A  Recognize the purposes of and differences among and differences among experiments, and observational studies; explain how randomization relates to each 3 S.IC.3 4.1.3
9.4 Recognize the purposes of and differences among sample surveys, experiments, and observational studies. 9.4.1.B  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. 3 S.IC.4 4.1.4
9.4 Recognize the purposes of and differences among sample surveys, experiments, and observational studies. 9.4.1.B s e data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant 3 S.IC.5 4.1.5
9.1 Know that Quantitative data can be described in terms of key characteristics: measures of shape, center, and
9.1.1.A  Represent data with plots on the real number line (dot plots, histograms, and box plots). 1 S.ID.1 4.1.6
9.4 Recognize the purposes of and differences among sample surveys, experiments, and observational studies. 9.4.1.B  Evaluate reports based on data. 3 S.IC.6 4.1.6
9.1 Know that Quantitative data can be described in terms of key characteristics: measures of shape, center, and
9.1.2.A  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. 2 S.ID.2 4.1.7
9.1 Know that Quantitative data can be described in terms of key characteristics: measures of shape, center, and
9.1.3.A  Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). 2 S.ID.3 4.1.8
9.2 Know that the shape of a data distribution might be described as symmetric, skewed, flat, or bell shaped, and
it might be summarized by a statistic measuring center (such as mean or median) and a statistic measuring
spread (such as standard deviation or interquartile range).
9.2.1.A  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. 2 S.ID.4 4.1.9
9.2 Know that the shape of a data distribution might be described as symmetric, skewed, flat, or bell shaped, and
it might be summarized by a statistic measuring center (such as mean or median) and a statistic measuring
spread (such as standard deviation or interquartile range).
9.2.2.A  Summarize categorical data for two categories in two‐way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data. 2 S.ID.5 4.1.10
9.2 Know that the shape of a data distribution might be described as symmetric, skewed, flat, or bell shaped, and
it might be summarized by a statistic measuring center (such as mean or median) and a statistic measuring
spread (such as standard deviation or interquartile range).
9.2.3.A  Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. 2 S.ID.6 4.1.11
9.3 Know strategies for fitting a function to a data display and informally assessing the fit. 9.3.1.A  Informally assess the fit of a function by plotting and analyzing residuals. 2 S.ID.6.a 4.1.12
9.3 Know strategies for fitting a function to a data display and informally assessing the fit. 9.3.2.A  Fit a linear function for a scatter plot that suggests a linear association. 2 S.ID.6.c 4.1.13
9.3 Know strategies for fitting a function to a data display and informally assessing the fit. 9.3.3.A  Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. 2 S.ID.7 4.1.14
9.3 Know strategies for fitting a function to a data display and informally assessing the fit. 9.3.4.A  Compute (using technology) and interpret the correlation coefficient of a linear fit. 2 S.ID.8 4.1.15
9.7 Calculate expected values and use them to solve problems 　 Stem (+) 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. 2 S.MD.1 4.1.16
9.5 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities 9.5.1.A  Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, outcomes, or as unions, complements of other events ("or," "and," "not"). 2 S.CP.1 4.2.1
9.5 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities 9.5.2.A  Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are determine if they are 2 S.CP.2 4.2.2
9.5 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities 9.5.3.A  Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the given B is the same as the conditional probability of B conditional probability of B probability of B. 2 S.CP.3 4.2.3
9.5 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities 　 Stem (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model. 2 S.CP.8 4.2.4
9.5 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities 　 Stem (+) Use permutations and combinations to compute probabilities of compound events and solve problems 2 S.CP.9 4.2.5
9.6 Recognize the concepts of conditional probability and independence in everyday language and everyday situations. 9.6.1.A  Construct and interpret two‐way frequency tables of data when two categories are associated with each object being classified. Use the two‐way table as a sample space to decide if events are independent and to approximate conditional probabilities 2 S.CP.4 4.2.6
9.6 Recognize the concepts of conditional probability and independence in everyday language and everyday situations. 9.6.2.A  Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. 2 S.CP.5 4.2.7
9.6 Recognize the concepts of conditional probability and independence in everyday language and everyday situations. 9.6.3.A  Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in terms of the model. 2 S.CP.6 4.2.8
9.6 Recognize the concepts of conditional probability and independence in everyday language and everyday situations. 　 Apply the Addition Rule, P(A or B) = P(A) + P(B) ‐ P(A and B), and interpret the answer in terms of the model. 2 S.CP.7 4.2.9
9.8 Use probability to evaluate outcomes of decisions 　 Stem (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values. a. Find the expected payoff for a game of chance. 2 S.MD.5 4.2.10
9.8 Use probability to evaluate outcomes of decisions 　 Stem (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). 2 S.MD.6 4.2.11
9.8 Use probability to evaluate outcomes of decisions 　 Stem (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game) 2 S.MD.7 4.2.12
7.2 Decompose and recompose algebraic expressions using number properties in the context of solving problems 　 Stem 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.  3 A.SSE.4 4.3.1
7.14 Know the difference between a recursive rule and an explicit expression for a function. Continued 7.14.2.C  Write arithmetic and geometric sequences both recursively and with an explicit formula 2 F.BF.2 4.3.1