Topic: Algebra II |
Common Core Mathematics 9-12 |
| Cluster Perform arithmetic operations with complex numbers. |
| | Grade 9-12 |
| | | N.CN.1. | Know there is a complex number i such that i^{2} = − 1, and every complex number has the form a + bi with a and b real. |
| | | N.CN.2. | Use the relation i^{2} = −1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. |
| Cluster Use complex numbers in polynomial identities and equations. |
| | Grade 9-12 |
| | | N.CN.7. | Solve quadratic equations with real coefficients that have complex
solutions. |
| | | N.CN.8. | (+) Extend polynomial identities to the complex numbers. For example,
rewrite x^{2} + 4 as (x + 2i)(x − 2i). |
| | | N.CN.9. | (+) Know the Fundamental Theorem of Algebra; show that it is true for
quadratic polynomials. |
| Cluster Interpret the structure of expressions |
| | Grade 9-12 |
| | | A.SSE.1. | Interpret expressions that represent a quantity in terms of its context. |
| | | A.SSE.1.a. | Interpret parts of an expression, such as terms, factors, and
coefficients. |
| | | A.SSE.1.b. | Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)^{n} as the product of P and a factor not depending on P. |
| | | A.SSE.2. | Use the structure of an expression to identify ways to rewrite it. For
example, see x^{4} − y^{4} as (x^{2})^{2} − (y^{2})^{2}, thus recognizing it as a difference of squares that can be factored as (x^{2} − y^{2})(x^{2} + y^{2}). |
| Cluster Write expressions in equivalent forms to solve problems |
| | Grade 9-12 |
| | | A.SSE.4. | 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. For
example, calculate mortgage payments. |
| Cluster Perform arithmetic operations on polynomials |
| | Grade 9-12 |
| | | A.APR.1. | Understand that polynomials form a system analogous to the integers,
namely, they are closed under the operations of addition, subtraction,
and multiplication; add, subtract, and multiply polynomials. |
| Cluster Understand the relationship between zeros and factors of polynomials |
| | Grade 9-12 |
| | | A.APR.2. | Know and apply the Remainder Theorem: For a polynomial p(x) and a
number a, the remainder on division by x – a is p(a), so p(a) = 0 if and
only if (x – a) is a factor of p(x). |
| | | A.APR.3. | Identify zeros of polynomials when suitable factorizations are
available, and use the zeros to construct a rough graph of the function
defined by the polynomial. |
| Cluster Use polynomial identities to solve problems |
| | Grade 9-12 |
| | | A.APR.4. | Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x^{2} + y^{2})^{2} = (x^{2} – y^{2})^{2} + (2xy)^{2} can be used to generate Pythagorean triples. |
| | | A.APR.5. | (+) 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] |
| Cluster Rewrite rational expressions |
| | Grade 9-12 |
| | | A.APR.6. | Rewrite simple rational expressions in different forms; write a(x)/b(x)
in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are
polynomials with the degree of r(x) less than the degree of b(x), using
inspection, long division, or, for the more complicated examples, a
computer algebra system. |
| | | A.APR.7. | (+) Understand that rational expressions form a system analogous
to the rational numbers, closed under addition, subtraction,
multiplication, and division by a nonzero rational expression; add,
subtract, multiply, and divide rational expressions. |
| Cluster Create equations that describe numbers or relationships |
| | Grade 9-12 |
| | | A.CED.1. | Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. |
| | | A.CED.2. | Create equations in two or more variables to represent relationships
between quantities; graph equations on coordinate axes with labels
and scales. |
| | | A.CED.3. | Represent constraints by equations or inequalities, and by systems of
equations and/or inequalities, and interpret solutions as viable or nonviable
options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. |
| | | A.CED.4. | Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R. |
| Cluster Understand solving equations as a process of reasoning and explain the reasoning |
| | Grade 9-12 |
| | | A.REI.2. | Solve simple rational and radical equations in one variable, and give
examples showing how extraneous solutions may arise. |
| Cluster Represent and solve equations and inequalities graphically |
| | Grade 9-12 |
| | | A.REI.11. | Explain why the x-coordinates of the points where the graphs of
the equations y = f(x) and y = g(x) intersect are the solutions of the
equation f(x) = g(x); find the solutions approximately, e.g., using
technology to graph the functions, make tables of values, or find
successive approximations. Include cases where f(x) and/or g(x)
are linear, polynomial, rational, absolute value, exponential, and
logarithmic functions. |
| Cluster Interpret functions that arise in applications in terms of the context |
| | Grade 9-12 |
| | | F.IF.4. | 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. |
| | | F.IF.5. | Relate the domain of a function to its graph and, where applicable, to
the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. |
| | | F.IF.6. | 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. |
| Cluster Analyze functions using different representations |
| | Grade 9-12 |
| | | F.IF.7.b. | Graph square root, cube root, and piecewise-defined functions,
including step functions and absolute value functions. |
| | | F.IF.7.c. | Graph polynomial functions, identifying zeros when suitable
factorizations are available, and showing end behavior. |
| | | F.IF.7.e. | Graph exponential and logarithmic functions, showing intercepts
and end behavior, and trigonometric functions, showing period,
midline, and amplitude. |
| | | F.IF.8. | Write a function defined by an expression in different but equivalent
forms to reveal and explain different properties of the function. |
| | | F.IF.8.b. | Use the properties of exponents to interpret expressions for
exponential functions. For example, identify percent rate of change
in functions such as y = (1.02)^{t}, y = (0.97)^{t}, y = (1.01)^{12t}, y = (1.2)^{t/10}, and classify them as representing exponential growth or decay. |
| | | F.IF.9. | Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. |
| Cluster Build a function that models a relationship between two quantities |
| | Grade 9-12 |
| | | F.BF.1.b. | Combine standard function types using arithmetic operations. For
example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. |
| Cluster Build new functions from existing functions |
| | Grade 9-12 |
| | | F.BF.3. | 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. |
| | | F.BF.4.a. | Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For
example, f(x)=2 x^{3} or f(x) = (x+1)/(x-1) for x≠ 1. |
| Cluster Construct and compare linear, quadratic, and exponential models and solve problems |
| | Grade 9-12 |
| | | F.LE.4. | For exponential models, express as a logarithm the solution to
ab^{ct} = d where a, c, and d are numbers and the base b is 2, 10, or e;evaluate the logarithm using technology. |
| Cluster Extend the domain of trigonometric functions using the unit circle |
| | Grade 9-12 |
| | | F.TF.1. | Understand radian measure of an angle as the length of the arc on the
unit circle subtended by the angle. |
| | | F.TF.2. | 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. |
| Cluster Model periodic phenomena with trigonometric functions |
| | Grade 9-12 |
| | | F.TF.5. | Choose trigonometric functions to model periodic phenomena with
specified amplitude, frequency, and midline. |
| Cluster Prove and apply trigonometric identities |
| | Grade 9-12 |
| | | F.TF.8. | 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. |
| Cluster Summarize, represent, and interpret data on a single count or
measurement variable |
| | Grade 9-12 |
| | | S.ID.4. | 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. |
| Cluster Understand and evaluate random processes underlying statistical experiments |
| | Grade 9-12 |
| | | S.IC.1. | Understand statistics as a process for making inferences about
population parameters based on a random sample from that
population. |
| | | S.IC.2. | Decide if a specified model is consistent with results from a given
data-generating 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? |
| Cluster Make inferences and justify conclusions from sample surveys, experiments, and observational studies |
| | Grade 9-12 |
| | | S.IC.3. | Recognize the purposes of and differences among sample surveys,
experiments, and observational studies; explain how randomization
relates to each. |
| | | S.IC.4. | 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. |
| | | S.IC.5. | Use data from a randomized experiment to compare two treatments;
use simulations to decide if differences between parameters are
significant. |
| | | S.IC.6. | Evaluate reports based on data. |