OER Curriculum

Standard N.CN.8

Extend polynomial identities to the complex numbers.For example, rewrite xi^{2}+ 4 as (x + 2)(x – 2i).Standard N.CN.9

Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. Limit to polynomials with real coefficients.

OER Curriculum

Instructional Tasks

Review Activitiy

Standard A.SSE.1

Interpret polynomial and rational expressions that represent a quantity in terms of its context.^{★}

- Interpret parts of an expression, such as terms, factors, and coefficients.
- Interpret complex expressions by viewing one or more of their parts as a single entity.
For example, examine the behavior ofP(1+r/n)^{nt}asnbecomes large.Standard 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}).

OER Curriculum

Instructional Tasks

Standard A.SSE.4

Understand the formula for the sum of a series and use the formula to solve problems.

- Derive the formula for the sum of an arithmetic series.
- Derive the formula for the sum of a geometric series, and use the formula to solve problems. Extend to infinite geometric series.
For example, calculate mortgage payments.^{★}

OER Curriculum

Instructional Tasks

Review Activitiy

Standard A.APR.1

Understand that all 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.

OER Curriculum

Instructional Tasks

Review Activitiy

Standard A.APR.2

Know and apply the Remainder Theorem: For a polynomialp(x) and a numbera, the remainder on division byx – aisp(a), sop(a) = 0 if and only if (x – a) is a factor ofp(x).Standard 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.

OER Curriculum

Instructional Tasks

Standard 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.Standard A.APR.5

Know and apply the Binomial Theorem for the expansion of (x+y)^{n}in powers ofxandyfor a positive integern, wherexandyare any numbers. For example, with coefficients determined by Pascal’s Triangle.

OER Curriculum

Instructional Tasks

Review Activitiy

Standard A.APR.6

Rewrite simple rational expressions in different forms; write^{a(x)}/_{b(x)}in the formq(x) +^{r(x)}/_{b(x)}, wherea(x),b(x),q(x), andr(x) are polynomials with the degree ofr(x) less than the degree ofb(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.Standard 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.

OER Curriculum

Instructional Tasks

Review Activitiy

Standard 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.Standard A.CED.2

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.Standard 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, maximizing the volume of a box for a given surface area while drawing attention to the practical domain.Standard A.CED.4

Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.For example, rearrange the compound interest formula to solve fort: A = P(1+ r/n)^{nt}.

OER Curriculum

Instructional Tasks

Review Activitiy

Standard A.REI.2

Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

OER Curriculum

Instructional Tasks

Standard A.REI.11

Explain why thex-coordinates of the points where the graphs of the equationsy=f(x) andy=g(x) intersect are the solutions of the equationf(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 wheref(x) and/org(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.^{★}

OER Curriculum

Review Activitiy

Standard 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.^{★}Standard 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 functionh(n)gives the number of person-hours it takes to assemblenengines in a factory, then the positive integers would be an appropriate domain for the function.^{★}Standard 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.^{★}

Instructional Tasks

Standard F.IF.7

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.^{★}

- Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Compare and contrast square root, cubed root, and step functions with all other functions.
- Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
- Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
- Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.
Standard F.IF.8

Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.Standard 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.

OER Curriculum

Review Activitiy

Standard F.BF.1

Write a function that describes a relationship between two quantities.^{★}

- 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.

OER Curriculum

Instructional Tasks

Review Activitiy

Standard F.BF.3

Identify the effect on the graph of replacingf(x) by f(x)+k, k f(x), f(kx), andf(x + k)for specific values ofk(both positive and negative); find the value ofkgiven the graphs. Note the effect of multiple transformations on a single function and the common effect of each transformation across function types. Include functions defined only by a graph. 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.Standard F.BF.4

Find inverse functions.

- Solve an equation of the form
f(x) = cfor a simple functionfthat 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.

OER Curriculum

Instructional Tasks

Review Activitiy

Standard F.LE.3

Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quanitity increasing linearly, quadratically, or (more generally) as a polynomial function.Standard F.LE.4

For exponential models, express as a logarithm the solution toab=^{ct}dwherea,c, anddare numbers and the basebis 2, 10, ore; evaluate the logarithm using technology. Include the relationship between properties of logarithms and properties of exponents, such as the connection between the properties of exponents and the basic logarithm property that logxy=log x+log y.

OER Curriculum

Review Activitiy

Standard F.LE.5

Interpret the parameters in a linear, quadratic, or exponential function in terms of a context.

OER Curriculum

Instructional Tasks

Review Activitiy

Standard F.TF.1

Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.Standard 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.Standard F.TF.3

Use special triangles to determine geometrically the values of sine, cosine, tangent for Π/3, Π/4 and Π6, 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.

OER Curriculum

Instructional Tasks

Review Activitiy

Standard F.TF.5

Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.^{★}Standard F.TF.7

Use inverse functions to solve trignometric equations that arise in modeling context; evaluate the solutions using technology and interpret them in terms of context. Limit solutions to a given interval.^{★}

Standard G.SRT.9

Derive the formulaA= 1/2absin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.Standard G.SRT.10

Prove the Laws of Sines and Cosines and use them to solve problems.Standard G.SRT.11

Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

OER Curriculum

Instructional Tasks

Standard G.GMD.4

Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

OER Curriculum

Instructional Tasks

Review Activitiy

Standard G.MG.1

Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).^{★}Standard G.MG.2

Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).^{★}Standard G.MG.3

Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).^{★}

OER Curriculum

Instructional Tasks

Review Activitiy

Standard 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.

OER Curriculum

Instructional Tasks

Review Activitiy

Standard S.IC.1

Understand that statistics allows inferences to be made about population parameters based on a random sample from that population.Standard S.IC.3

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

OER Curriculum

Instructional Tasks

Review Activitiy

Standard 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.Standard S.IC.6

Evaluate reports based on data.

HONORS - Standard N.CN.3

Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

HONORS - Standard N.CN.4

Represent complex numbers on the complex plane in rectangular form and polar form (including real and imaginary numbers), and explain why the rectangular form of a given complex number represents the same number.HONORS - Standard N.CN.5

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°.HONORS - Standard N.CN.6

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.

HONORS - Standard N.CN.10

Multiply complex numbers in polar form and use DeMoivre's Theorem to find roots of complex numbers.

HONORS - Standard F.IF.7

Graph functions expressed symbolically, and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

- Graph rational functions, identifying zeros, asymptotes, and point discontinuities when suitable factorizations are available, and showing end behavior.
- Define a curve parametrically and draw its graph.

HONORS - Standard F.BF.1

Write a function that describes a relationship between two quantities.

- Compose functions.
For example, ifT(y)is the temperature in the atmosphere as a function of height, andh(t)is the height of a weather balloon as a function of time, thenT(h(t))is the temperature at the location of the weather balloon as a function of time.

HONORS - Standard F.BF.4

Find inverse functions.

- Verify by composition that one function is the inverse of another.
- Read values of an inverse function from a graph or a table, given that the function has an inverse.
- Produce an invertible function from a non-invertible function by restricting the domain.
HONORS - Standard F.BF.5

Understand the inverse relationship between exponents and logarithms, and use this relationship to solve problems involving logarithms and exponents.

HONORS - Standard F.TF.4

Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

HONORS - Standard F.TF.6

Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.HONORS - Standard F.TF.7

Use the inverse functions to solve trigonometric equations that arise in the modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.

HONORS - Standard F.TF.9

Prove the addition and subtraction formulas for sine, cosine, and tangent, and use them to solve problems.

HONORS - Standard G.GMD.2

Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.

OER Curriculum

Instructional Tasks

HONORS - Standard S.CP.9

Use permutations and combinations to compute probabilities of compound events and solve problems.

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