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Mathematics - High School Mathematics III (LA and STEM)

Math III LA course does not include the (+) standards.
Math III STEM course includes standards identified by (+) sign
Math III TR course (Technical Readiness) includes standards identified by (*)
Math IV TR course (Technical Readiness) includes standards identified by (^)

Math III Technical Readiness and Math IV Technical Readiness are course options (for juniors and seniors) built for the mathematics content of Math III through integration of career clusters. These courses integrate academics with hands-on career content. The collaborative teaching model is recommended based at our Career and Technical Education (CTE) centers. The involvement of a highly qualified Mathematics teacher and certified CTE teachers will ensure a rich, authentic and respectful environment for delivery of the academics in "real world" scenarios.

All West Virginia teachers are responsible for classroom instruction that integrates content standards and mathematical habits of mind. Students in this course will make connections and applications the accumulation of learning that they have from their previous courses, with content grouped into four critical units. Students will apply methods from probability and statistics to draw inferences and conclusions from data. They will expand their repertoire of functions to include polynomial, rational and radical functions and their study of right triangle trigonometry to include general triangles. Students will bring together their experiences with functions and geometry to create models and solve contextual problems. Mathematical habits of mind, which should be integrated in these content areas, include: making sense of problems and persevering in solving them, reasoning abstractly and quantitatively; constructing viable arguments and critiquing the reasoning of others; modeling with mathematics; using appropriate tools strategically; attending to precision, looking for and making use of structure; and looking for and expressing regularity in repeated reasoning. Students will continue developing mathematical proficiency in a developmentally-appropriate progressions of standards.

Inferences and Conclusions from Data

Summarize, represent, and interpret data on a single count or measurement variable.

M.3HS.1 (*)

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.

Understand and evaluate random processes underlying statistical experiments.

M.3HS.2 (*)

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

M.3HS.3 (*)

Decide if a specified model is consistent with results from a given data-generating process, for example, using simulation. (e.g., 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?)

Make inferences and justify conclusions from sample surveys, experiments, and observational studies.

M.3HS.4 (*,^)

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

M.3HS.5 (*,^)

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.

M.3HS.6 (*,^)

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

M.3HS.7 (*,^)

Evaluate reports based on data.

Use probability to evaluate outcomes of decisions.

M.3HS.8 (+, ^)

Use probabilities to make fair decisions (e.g., drawing by lots or using a random number generator).

M.3HS.9 (+, ^)

Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, and/or pulling a hockey goalie at the end of a game).

Polynomial, Rational and Radical Relationships

Use complex numbers in polynomial identities and equations.

M.3HS.10 (+)

Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x – 2i).

M.3HS.11 (+)

Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

Interpret the structure of expressions.

M.3HS.12 (*)

Interpret expressions that represent a quantity in terms of its context.

  1. Interpret parts of an expression, such as terms, factors, and coefficients.
  2. Interpret complicated expressions by viewing one or more of their parts as a single entity. (e.g., Interpret P(1 + r)n as the product of P and a factor not depending on P.)
M.3HS.13 (*)

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.

M.3HS.14 (^)

Derive the formula for the sum of a geometric series (when the common ratio is not 1), and use the formula to solve problems. (e.g., Calculate mortgage payments.)

Perform arithmetic operations on polynomials.

M.3HS.15 (*)

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.

Understand the relationship between zeros and factors of polynomials.

M.3HS.16 (*)

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).

M.3HS.17 (*)

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.

M.3HS.18 (^)

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.

M.3HS.19 (+,^)

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.

Rewrite rational expressions.

M.3HS.20 (*)

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.

M.3HS.21 (+)

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.

Understand solving equations as a process of reasoning and explain the reasoning.

M.3HS.22 (*)

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.

M.3HS.23 (*,^)

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.

Analyze functions using different representations.

M.3HS.24 (*,^)

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. Graph polynomial functions, identifying zeros when suitable factorizations are available and showing end behavior.

Trigonometry of General Triangles and Trigonometric Functions

Apply trigonometry to general triangles.

M.3HS.25 (+,^)

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.

M.3HS.26 (+, ^)

Prove the Laws of Sines and Cosines and use them to solve problems.

M.3HS.27 (+,^)

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 and/or resultant forces).

Extend the domain of trigonometric functions using the unit circle.

M.3HS.28 (*)

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

M.3HS.29 (*)

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.

Model periodic phenomena with trigonometric functions.

M.3HS.30 (*)

Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

Mathematical Modeling

Create equations that describe numbers or relationships.

M.3HS.31 (*,^)

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.

M.3HS.32 (*,^)

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

M.3HS.33 (*,^)

Represent constraints by equations or inequalities and by systems of equations and/or inequalities and interpret solutions as viable or non-viable options in a modeling context. (e.g., Represent inequalities describing nutritional and cost constraints on combinations of different foods.)

M.3HS.34 (*,^)

Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. (e.g., Rearrange Ohm’s law V = IR to highlight resistance R.)

Interpret functions that arise in applications in terms of a context.

M.3HS.35 (*)

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.

M.3HS.36 (*)

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. (e.g., 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.)

M.3HS.37 (*)

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.

M.3HS.38 (*,^)

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.
  2. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline and amplitude.
M.3HS.39 (*,^)

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

M.3HS.40 (*,^)

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). (e.g., Given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.)

Build a function that models a relationship between two quantities.

M.3HS.41 (*)

Write a function that describes a relationship between two quantities. Combine standard function types using arithmetic operations. (e.g., 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.

M.3HS.42 (*)

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.

M.3HS.43 (*)

Find inverse functions. 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. (e.g., f(x) = 2 x3 or f(x) = (x+1)/(x-1) for x ≠ 1.)

Construct and compare linear, quadratic and exponential models and solve problems.

M.3HS.44 (*)

For exponential models, express as a logarithm the solution to a bct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

Visualize relationships between two dimensional and three-dimensional objects.

M.3HS.45 (*,^)

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

Apply geometric concepts in modeling situations.

M.3HS.46 (*,^)

Use geometric shapes, their measures and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

M.3HS.47 (*,^)

Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile or BTUs per cubic foot).

M.3HS.48 (*,^)

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