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Wisconsin Standards for Mathematics: Algebra 1 Wisconsin Standards for Mathematics: Algebra 1
Wisconsin Standards for Mathematics: Algebra 2 Wisconsin Standards for Mathematics: Algebra 2
Wisconsin Standards for Mathematics: All High School Pathways Wisconsin Standards for Mathematics: All High School Pathways
Wisconsin Standards for Mathematics: Fourth Courses Wisconsin Standards for Mathematics: Fourth Courses
Wisconsin Standards for Mathematics: Geometry Wisconsin Standards for Mathematics: Geometry
Wisconsin Standards for Mathematics: Mathematical Practices Wisconsin Standards for Mathematics: Mathematical Practices
Wisconsin Standards for Mathematics: Mathematics I Wisconsin Standards for Mathematics: Mathematics I
Wisconsin Standards for Mathematics: Mathematics II Wisconsin Standards for Mathematics: Mathematics II
Wisconsin Standards for Mathematics: Mathematics III Wisconsin Standards for Mathematics: Mathematics III
Wisconsin Standards for Mathematics: Precalculus Wisconsin Standards for Mathematics: Precalculus

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N-RN The Real Number System

N.RN.A Extend the properties of exponents to rational exponents.
- M.N.RN.A.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents.
- M.N.RN.A.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.
- M.N.RN.A.3 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
N.RN.B Use properties of rational and irrational numbers.
- M.N.RN.B.4 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

N-Q Quantities

N.Q.A Reason quantitatively and use units to solve problems. (M)
- M.N.Q.A.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
- M.N.Q.A.2 Define appropriate quantities for the purpose of descriptive modeling.
- M.N.Q.A.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

N-CN The Complex Number System

N.CN.A Perform arithmetic operations with complex numbers.
- M.N.CN.A.1 Know there is a complex number i such that i² = -1, and every complex number has the form a + bi with a and b real. Understand why complex numbers exist.
  - Introduction to complex numbers (A2-K.1)
- M.N.CN.A.2 (+) Use the relation i² = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
- M.N.CN.A.3 (+) Find the conjugate of a complex number; use conjugates to find moduli (absolute values) and quotients of complex numbers.
N.CN.B Represent complex numbers and their operations on the complex plane.
- M.N.CN.B.4 (+) 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.
- M.N.CN.B.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 + v3i²)³ = 8 because (-1 + v3i²) has modulus 2 and argument 120°.
- M.N.CN.B.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.
  - Midpoints in the complex plane (PC-W.7)
  - Distance in the complex plane (PC-W.8)
N.CN.C Use complex numbers in polynomial identities and equations.
- M.N.CN.C.7 Solve quadratic equations with real coefficients that have complex solutions. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.
- M.N.CN.C.8 (+) Extend polynomial identities to the complex numbers. For example, rewrite x² + 4 as (x + 2i)(x - 2i).
- M.N.CN.C.9 (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
  - Fundamental Theorem of Algebra (A2-Q.8)
  - Fundamental Theorem of Algebra (PC-F.7)

N-VM Vector and Matrix Quantities

N.VM.A Represent and model with vector quantities. (M)
- M.N.VM.A.1 (+) 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).
- M.N.VM.A.2 (+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
- M.N.VM.A.3 (+) Solve problems involving velocity and other quantities that can be represented by vectors.
- M.N.VM.C.6 (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
- M.N.VM.C.7 (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.
- M.N.VM.C.8 (+) Add, subtract, and multiply matrices of appropriate dimensions.
- M.N.VM.C.9 (+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.
- M.N.VM.C.10 (+) 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.
- M.N.VM.C.11 (+) 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.
- M.N.VM.C.12 (+) Work with 2 x 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.
N.VM.B Perform operations on vectors.
- M.N.VM.B.4 (+) Add and subtract vectors.
  - M.N.VM.B.4a 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.
  - M.N.VM.B.4b Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
    - Find the magnitude and direction of a vector sum (PC-Y.9)
  - M.N.VM.B.4c 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.
- M.N.VM.B.5 (+) Multiply a vector by a scalar.
  - M.N.VM.B.5a 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).
    - Multiply a vector by a scalar (PC-Y.10)
    - Scalar multiples of three-dimensional vectors (PC-Z.4)
  - M.N.VM.B.5b 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).
    - Find the magnitude of a vector scalar multiple (PC-Y.11)
    - Determine the direction of a vector scalar multiple (PC-Y.12)
N.VM.C Perform operations on matrices and use matrices in applications. (M)

G-CO Congruence

G.CO.A Experiment with transformations in the plane.
- M.G.CO.A.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
  - Angle vocabulary (G-C.1)
- M.G.CO.A.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
- M.G.CO.A.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
  - Transformations that carry a polygon onto itself (G-H.15)
- M.G.CO.A.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
  - Rotate polygons about a point (G-H.7)
- M.G.CO.A.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
G.CO.B Understand congruence in terms of rigid motion.
- M.G.CO.B.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
- M.G.CO.B.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
- M.G.CO.B.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.
G.CO.C Prove geometric theorems.
- M.G.CO.C.9 Prove theorems about lines and angles. Theorems should include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.
- M.G.CO.C.10 Prove theorems about triangles. Theorems should include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
- M.G.CO.C.11 Prove theorems about parallelograms. Theorems should include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
G.CO.D Make geometric constructions.
- M.G.CO.D.12 Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
- M.G.CO.D.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

G-SRT Similarity, Right Triangles, and Trigonometry

G.SRT.A Understand similarity in terms of similarity transformations.
- M.G.SRT.A.1 Verify experimentally the properties of dilations given by a center and a scale factor:
  - M.G.SRT.A.1a A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
    - Dilations: graph the image (G-I.1)
    - Dilations and parallel lines (G-I.6)
  - M.G.SRT.A.1b The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
- M.G.SRT.A.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
- M.G.SRT.A.3 Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
  - Similarity rules for triangles (G-M.8)
  - Similar triangles and indirect measurement (A1)
G.SRT.B Prove theorems involving similarity.
- M.G.SRT.B.4 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
- M.G.SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
G.SRT.C Define trigonometric ratios and solve problems involving right triangles. (M)
- M.G.SRT.C.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
- M.G.SRT.C.7 Explain and use the relationship between the sine and cosine of complementary angles.
- M.G.SRT.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
G.SRT.D Apply trigonometry to general triangles.
- M.G.SRT.D.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.
- M.G.SRT.D.10 (+) Prove the Laws of Sines and Cosines and use them to solve problems.
- M.G.SRT.D.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).

G-C Circles

G.C.A Understand and apply theorems about circles.
- M.G.C.A.1 Identify and describe relationships among inscribed angles, radii, and chords. Prove properties of angles for a quadrilateral inscribed in a circle. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
G.C.B Find arc lengths and areas of sectors of circles.
- M.G.C.B.2 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

G-GPE Expressing Geometric Properties

G.GPE.A Translate between the geometric description and the equation for a conic section.
- M.G.GPE.A.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.
- M.G.GPE.A.2 (+) Derive the equation of a parabola given a focus and directrix.
- M.G.GPE.A.3 (+) Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.
G.GPE.B Use coordinates to prove simple geometric theorems algebraically.
- M.G.GPE.B.4 Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).
  - Classify shapes on the coordinate plane: justify your answer (G-O.13)
- M.G.GPE.B.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).
- M.G.GPE.B.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
- M.G.GPE.B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles (e.g., using the distance formula).

G-GMD Geometric Measurement and Dimension

G.GMD.A Explain volume formulas and use them to solve problems. (M)
- M.G.GMD.A.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.
- M.G.GMD.A.2 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
G.GMD.B Visualize relationships between two-dimensional and three-dimensional objects.
- M.G.GMD.B.3 Identify three-dimensional objects generated by rotations of two-dimensional objects.
  - Cross sections of three-dimensional figures (G-T.4)
  - Solids of revolution (G-T.5)
G.GMD.C Apply geometric concepts in modeling situations. (M)
- M.G.GMD.C.4 Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).
  - Area and perimeter: word problems (A1-D.1)
  - Pythagorean theorem (G-P.1)
- M.G.GMD.C.5 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
  - Calculate density, mass, and volume (G-V.9)
- M.G.GMD.C.6 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).

S-ID Interpreting Categorical and Quantitative Data

SP.ID.A Summarize, represent, and interpret data on a single count or measurement variable. (M)
- M.SP.ID.A.1 Represent data with plots on the real number line (dot plots, histograms, and box plots).
- M.SP.ID.A.2 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.
- M.SP.ID.A.3 Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).
- M.SP.ID.A.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 statistical packages calculators, spreadsheets, and tables to estimate areas under the normal curve.
SP.ID.B Summarize, represent, and interpret data on two categorical and quantitative variables. (M)
- M.SP.ID.B.5 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 as examples of proportionality and disproportionality). Recognize possible associations and trends in the data.
  - Find probabilities using two-way frequency tables (A1-KK.3)
  - Find conditional probabilities using two-way frequency tables (A1-KK.4)
- M.SP.ID.B.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
  - M.SP.ID.B.6a Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize appropriate families of functions to model.
  - M.SP.ID.B.6b Informally assess the fit of a function by plotting and analyzing residuals
    - Interpret a scatter plot (A1-JJ.1)
  - M.SP.ID.B.6c Fit a linear function for a scatter plot that suggests a linear association.
    - Write equations for lines of best fit (A1-JJ.5)
SP.ID.C Interpret linear models (M)
- M.SP.ID.C.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.
- M.SP.ID.C.8 Use technology to create a correlation coefficient for a linear fit and then interpret its meaning for the model.
- M.SP.ID.C.9 Distinguish between correlation and causation. For example, cities with a higher number of fast food restaurants tend to have more hospitals. While there is a clear correlation (likely driven by population), we cannot conclude that fast food restaurants are causing more hospitals to open. There is a relationship between height and reading level in elementary school children. This is not because changes in height cause better reading ability. Rather as children get older, they get both taller and improve in their reading skills.

S-IC Making Inferences and Justifying Conclusions

SP.IC.A Understand and evaluate random processes underlying statistical experiments. (M)
- M.SP.IC.A.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population.
- M.SP.IC.A.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?
SP.IC.B Make inferences and justify conclusions from sample surveys, experiments, and observational studies. (M)
- M.SP.IC.B.3 Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.
  - Experiment design (A2-RR.8)
  - Experiment design (PC-FF.8)
- M.SP.IC.B.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.
- M.SP.IC.B.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.
  - Analyze the results of an experiment using simulations (A2-RR.9)
  - Analyze the results of an experiment using simulations (PC-FF.9)
- M.SP.IC.B.6 Evaluate reports based on data.

S-CP Conditional Probability and the Rules of Probability

SP.CP.A Understand independence and conditional probability and use them to interpret data. (M)
- M.SP.CP.A.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ("or," "and," "not").
- M.SP.CP.A.2 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 independent.
- M.SP.CP.A.3 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 probability of A, and the conditional probability of B given A is the same as the probability of B.
- M.SP.CP.A.4 Represent data from two categorical variables using two-way frequency tables and/or venn diagrams. Interpret the representation when two categories are associated with each object being classified. Use the representation as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.
- M.SP.CP.A.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.
SP.CP.B Use the rules of probability to compute probabilities of compound events in a uniform probability model.
- M.SP.CP.B.6 Use a representation such as a two-way table or venn diagram to 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.
- M.SP.CP.B.7 Use a representation such as a two-way table or venn diagram to 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.
- M.SP.CP.B.8 (+) Use a representation such as a tree diagram to 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.
- M.SP.CP.B.9 (+) Use permutations and combinations to compute probabilities of compound events and solve problems.

S-MD Using Probability to Make Decisions

SP.MD.A Calculate expected values and use them to solve problems. (M)
- M.SP.MD.A.1 (+) 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.
- M.SP.MD.A.2 (+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.
  - Expected values of random variables (A2-PP.5)
  - Expected values of random variables (PC-DD.5)
- M.SP.MD.A.3 (+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes.
- M.SP.MD.A.4 (+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households?
SP.MD.B Use probability to evaluate outcomes of decisions. (M)
- M.SP.MD.B.5 (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.
  - M.SP.MD.B.5a Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant.
    - Expected values for a game of chance (A2-PP.9)
    - Expected values for a game of chance (PC-DD.9)
  - M.SP.MD.B.5b Evaluate and compare strategies on the basis of expected values. For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.
    - Choose the better bet (A2-PP.10)
    - Choose the better bet (PC-DD.10)
- M.SP.MD.B.6 Use probabilities to make fair decisions (e.g., drawing for a party door prize where attendees earn one entry to the drawing for each activity they complete, using an electronic spinner to pick a team spokesperson at random from a group, flip a coin to decide which of two friends gets to choose the movie, using a random number generator to select people to include in a sample for an experiment).
- M.SP.MD.B.7 Analyze decisions and strategies using probability concepts (e.g., balancing expected gains and risk, medical product testing, choosing an investment option, deciding when to kick an extra point vs. two point conversion after a touchdown in football).
  - Choose the better bet (A2-PP.10)
  - Choose the better bet (PC-DD.10)

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N-RN The Real Number System

N.RN.A Extend the properties of exponents to rational exponents.

M.N.RN.A.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents.

M.N.RN.A.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.

M.N.RN.A.3 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

N.RN.B Use properties of rational and irrational numbers.

M.N.RN.B.4 Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

N-Q Quantities

N.Q.A Reason quantitatively and use units to solve problems. (M)

M.N.Q.A.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

M.N.Q.A.2 Define appropriate quantities for the purpose of descriptive modeling.

M.N.Q.A.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

N-CN The Complex Number System

N.CN.A Perform arithmetic operations with complex numbers.

M.N.CN.A.1 Know there is a complex number i such that i² = -1, and every complex number has the form a + bi with a and b real. Understand why complex numbers exist.

M.N.CN.A.2 (+) Use the relation i² = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

M.N.CN.A.3 (+) Find the conjugate of a complex number; use conjugates to find moduli (absolute values) and quotients of complex numbers.

N.CN.B Represent complex numbers and their operations on the complex plane.

M.N.CN.B.4 (+) 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.

M.N.CN.B.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 + v3i²)3 = 8 because (-1 + v3i²) has modulus 2 and argument 120°.

M.N.CN.B.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.

N.CN.C Use complex numbers in polynomial identities and equations.

M.N.CN.C.7 Solve quadratic equations with real coefficients that have complex solutions. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

M.N.CN.C.8 (+) Extend polynomial identities to the complex numbers. For example, rewrite x² + 4 as (x + 2i)(x - 2i).

M.N.CN.C.9 (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

N-VM Vector and Matrix Quantities

N.VM.A Represent and model with vector quantities. (M)

M.N.VM.A.1 (+) 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).

M.N.VM.A.2 (+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

M.N.VM.A.3 (+) Solve problems involving velocity and other quantities that can be represented by vectors.

M.N.VM.C.6 (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.

M.N.VM.C.7 (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.

M.N.VM.C.8 (+) Add, subtract, and multiply matrices of appropriate dimensions.

M.N.VM.C.9 (+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.

M.N.VM.C.10 (+) 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.

M.N.VM.C.11 (+) 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.

M.N.VM.C.12 (+) Work with 2 x 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.

N.VM.B Perform operations on vectors.

M.N.VM.B.4 (+) Add and subtract vectors.

M.N.VM.B.4a 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.

M.N.VM.B.4b Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.

M.N.VM.B.5 (+) Multiply a vector by a scalar.

M.N.VM.B.5a 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).

M.N.VM.B.5b 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).

N.VM.C Perform operations on matrices and use matrices in applications. (M)

A-SSE Seeing Structure in Expressions

A.SSE.A Interpret the structure of expressions. (M)

M.A.SSE.A.1 Interpret expressions that represent a quantity in terms of its context.

M.A.SSE.A.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).

A.SSE.B Write expressions in equivalent forms to solve problems. (M)

M.A.SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

M.A.SSE.B.3a Factor a quadratic expression to reveal the zeros of the function it defines.

M.A.SSE.B.3b Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

M.A.SSE.B.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, calculating mortgage payments or tracking the amount of an antibiotic in the human body when prescribed for an infection.

A-APR Arithmetic with Polynomials and Rational Expressions

A.APR.A Perform arithmetic operations on polynomials.

M.A.APR.A.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.

A.APR.B Understand the relationship between zeros and factors of polynomials.

M.A.APR.B.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).

M.A.APR.B.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.

A.APR.C Use polynomial identities to solve problems.

M.A.APR.C.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.

A.APR.D Rewrite rational expressions.

M.A.APR.D.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.

A-CED Creating Equations

A.CED.A Create equations that describe numbers or relationships. (M)

M.A.CED.A.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.

M.A.CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

M.A.CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange the formula C = 5/9 (F-32) so you solve for F.

A-REI Reasoning with Equations and Inequalities

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

M.A.REI.A.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

M.A.REI.A.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

A.REI.B Solve equations and inequalities in one variable.

M.A.REI.B.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

A.REI.C Solve systems of equations.

M.A.REI.C.5 Justify that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

M.N.CN.B.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 + v3i²)³ = 8 because (-1 + v3i²) has modulus 2 and argument 120°.

M.A.SSE.A.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).

M.A.APR.C.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.

M.F.LE.A.4 For exponential models, express as a logarithm the solution to abc^t = d where a, c, and d are numbers and the base b is 2,10, or e; evaluate the logarithm using technology.