# Math - Secondary Math III

Resources by Strand

Strand: MATHEMATICAL PRACTICES (MP)
The Standards for Mathematical Practice in Secondary Mathematics III describe mathematical habits of mind that teachers should seek to develop in their students. Students become mathematically proficient in engaging with mathematical content and concepts as they learn, experience, and apply these skills and attitudes (Standards MP.1-8).

Strand: NUMBER AND QUANTITY - The Complex Number System (N.CN)
Use complex numbers in polynomial identities and equations. Build on work with quadratic equations in Secondary Mathematics II (Standards N.CN.8–9).
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Standard N.CN.8
Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(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.

Strand: ALGEBRA - Seeing Structures in Expressions (A.SSE)
Interpret the structure of expressions. Extend to polynomial and rational expressions (Standards A.SSE.1–2)
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Standard A.SSE.1
Interpret polynomial and rational expressions that represent a quantity in terms of its context.

1. Interpret parts of an expression, such as terms, factors, and coefficients.
2. Interpret complex expressions by viewing one or more of their parts as a single entity. For example, examine the behavior of P(1+r/n)nt as n becomes large.

Standard A.SSE.2
Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).

Write expressions in equivalent forms to solve problems (Standard A.SSE.4).
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Standard A.SSE.4
Understand the formula for the sum of a series and use the formula to solve problems.

1. Derive the formula for the sum of an arithmetic series.
2. 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.

Strand: ALGEBRA - Arithmetic With Polynomials and Rational Expressions (A.APR)
Perform arithmetic operations on polynomials, extending beyond the quadratic polynomials (Standard A.APR.1).
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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.

Understand the relationship between zeros and factors of polynomials (Standards A.APR.2–3)
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Standard 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).

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.

Use polynomial identities to solve problems (Standards A.APR.4–5)
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Standard A.APR.4
Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 – y2)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 of x and y for a positive integer n, where x and y are any numbers. For example, with coefficients determined by Pascal’s Triangle.

Rewrite rational expressions (Standards A.APR.6–7).
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Standard 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.

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.

Strand: ALGEBRA - Creating Equations (A.CED)
Create equations that describe numbers or relationships, using all available types of functions to create such equations (Standards A.CED.1–4).
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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 for t: A = P(1+ r/n)nt.

Strand: ALGEBRA - Reasoning with Equations and Inequalities (A.REI)
Understand solving equations as a process of reasoning and explain the reasoning (Standard A.REI.2)
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Standard A.REI.2
Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

Represent and solve equations and inequalities graphically (Standard A.REI.11).
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Standard 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.

Strand: FUNCTIONS - Interpreting Functions (F.IF)
Interpret functions that arise in applications in terms of a context (Standards F.IF.4–6).
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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 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.

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.

Analyze functions using different representations (Standards F.IF.7–9).
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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.

1. 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.
2. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.
3. Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.
4. 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.

Strand: FUNCTIONS - Building Functions (F.BF)
Build a function that models a relationship between two quantities. Develop models for more complex or sophisticated situations (Standards F.BF.1).
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Standard F.BF.1
Write a function that describes a relationship between two quantities.

1. 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.
Build new functions from existing functions (Standards F.BF.3–4).
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Standard 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. 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.

1. 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 x3 or f(x) = (x+1)/(x–1) for x ≠ 1.

Strand: FUNCTIONS - Linear, Quadratic, and Exponential Models (F.LE)
Construct and compare linear, quadratic, and exponential models and solve problems (Standards F.LE.3–4).
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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 to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; 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 log xy = log x + log y.

Interpret expressions for functions in terms of the situation it models. Introduce f(x) = ex as a model for continuous growth (Standard F.LE.5).
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Standard F.LE.5
Interpret the parameters in a linear, quadratic, or exponential function in terms of a context.

Strand: FUNCTIONS - Trigonometric Functions (F.TF)
Extend the domain of trigonometric functions using the unit circle (Standards F.TF.1–3).
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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.

Model periodic phenomena with trigonometric functions (Standards F.TF.5–7).
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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.

Strand: GEOMETRY - Similarity, Right Triangles, and Trigonometry (G.SRT)
Apply trigonometry to general triangles. With respect to the general case of the Laws of Sines and Cosines, the definitions of sine and cosine must be extended to obtuse angles (Standards G.SRT.9–11).
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Standard G.SRT.9
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.

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).

Strand: GEOMETRY - Geometric Measurement and Dimension (G.GMD)
Visualize relationships between two-dimensional and three-dimensional objects (Standards G.MD.4).
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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.

Strand: GEOMETRY - Modeling With Geometry (G.MG)
Apply geometric concepts in modeling situations (Standards G.MG.1–3).
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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).

Strand: STATISTICS - Interpreting Categorical and Quantitative Data (S.ID)
Summarize, represent, and interpret data on a single count or measurement variable. While students may have heard of the normal distribution, it is unlikely that they will have prior experience using it to make specific estimates. Build on students’ understanding of data distributions to help them see how the normal distribution uses area to make estimates of frequencies (which can be expressed as probabilities). Emphasize that only some data are well described by a normal distribution (Standard S.ID.4).
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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.

Strand: STATISTICS - Making Inferences and Justifying Conclusions (S.IC)
Understand and evaluate random processes underlying statistical experiments (Standard S.IC.1).
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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.

Draw and justify conclusions from sample surveys, experiments, and observational studies. In earlier grades, students are introduced to different ways of collecting data and use graphical displays and summary statistics to make comparisons. These ideas are revisited with a focus on how the way in which data is collected determines the scope and nature of the conclusions that can be drawn from that data. The concept of statistical significance is developed informally through simulation as meaning a result that is unlikely to have occurred solely as a result of random selection in sampling or random assignment in an experiment. For S.IC.4, focus on the variability of results from experiments - that is, focus on statistics as a way of dealing with, not eliminating, inherent randomness (Standards S.IC.3-4, 6).
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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 - Strand: NUMBER AND QUANTITY - Complex Number System (N.CN)
Perform arithmetic operations with complex numbers (Standard N.CN.3)
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HONORS - Standard N.CN.3
Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

Represent complex numbers and their operations on the complex plane (Standard N.CN.4–6).
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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.

Use complex numbers in polynomial identities and equations (Standard N.CN.10).
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HONORS - Standard N.CN.10
Multiply complex numbers in polar form and use DeMoivre's Theorem to find roots of complex numbers.

HONORS - Strand: FUNCTIONS - Interpreting Functions (F.IF)
Analyze functions using different representations (Standard F.IF.7, d and f).
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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.

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

HONORS - Strand: FUNCTIONS - Building Functions (F.BF).
Build a function that models a relationship between two quantities (Standard F.BF.1.c).
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HONORS - Standard F.BF.1
Write a function that describes a relationship between two quantities.

1. Compose functions. For 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.
Build new functions from existing functions (Standards F.BF.4, b,c,d–5).
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HONORS - Standard F.BF.4
Find inverse functions.

1. Verify by composition that one function is the inverse of another.
2. Read values of an inverse function from a graph or a table, given that the function has an inverse.
3. 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 - Strand: FUNCTIONS - Trigonometric Functions (F.TF)
Extend the domain of trigonometric functions using the unit circle (Standard T.FT.4).
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HONORS - Standard F.TF.4
Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

Model periodic phenomena with trigonometric functions (Standards T.FT.6–7).
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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.

Prove and apply trigonometric identities (Standard T.FT.9).
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HONORS - Standard F.TF.9
Prove the addition and subtraction formulas for sine, cosine, and tangent, and use them to solve problems.

HONORS - Strand: GEOMETRY - Geometric Measurement and Dimension (G.GMD)
Explain volume formulas and use them to solve problems (Standard G.GMD.2).
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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.

HONORS - Strand: STATISTICS AND PROBABILITY - Conditional Probability and the Rules of Probability (S.CP)
Use the rules of probability to compute probabilities of compound events in a uniform probability model (Standard S.CP.9).
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