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Massachusetts: Model Advanced Quantitative Reasoning [AQR] Math Standards

31 standards · 10 domains

ALGEBRA - ARITHMETIC WITH POLYNOMIALS AND RATIONAL EXPRESSIONS

AQR.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.

ALGEBRA - REASONING WITH EQUATIONS AND INEQUALITIES

AQR.A-REI.C.8 (+) Represent a system of linear equations as a single matrix equation in a vector variable.
AQR.A-REI.C.9 (+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 x 3 or greater).

FUNCTIONS - TRIGONOMETRIC FUNCTIONS

AQR.F-TF.A.3 (+) Use special triangles to determine geometrically the values of sine, cosine, tangent for π∕3, π∕4 and π∕6, 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.
AQR.F-TF.A.4 (+) Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.
AQR.F-TF.B.7 (+) Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.
AQR.F-TF.C.9 (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

GEOMETRY - CIRCLES

AQR.G-C.A.4 (+) Construct a tangent line from a point outside a given circle to the circle.

GEOMETRY - GEOMETRIC MEASUREMENT AND DIMENSION

AQR.G-GMD.A.2 (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures.

GEOMETRY - EXPRESSING GEOMETRIC PROPERTIES WITH EQUATIONS

AQR.G-GPE.A.3.a (+) Use equations and graphs of conic sections to model real-world problems.

GEOMETRY - SIMILARITY, RIGHT TRIANGLES, AND TRIGONOMETRY

AQR.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).

NUMBER AND QUANTITY - VECTOR AND MATRIX QUANTITIES

AQR.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).
AQR.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.
AQR.N-VM.A.3 (+) Solve problems involving velocity and other quantities that can be represented by vectors.
AQR.N-VM.C.6 (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
AQR.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.
AQR.N-VM.C.8 (+) Add, subtract, and multiply matrices of appropriate dimensions.
AQR.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.
AQR.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.
AQR.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.
AQR.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.

STATISTICS AND PROBABILITY - CONDITIONAL PROBABILITY AND THE RULES OF PROBABILITY

AQR.S-CP.B.8 (+) 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.
AQR.S-CP.B.9 (+) Use permutations and combinations to compute probabilities of compound events and solve problems.

STATISTICS AND PROBABILITY - USING PROBABILITY TO MAKE DECISIONS

AQR.S-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.
AQR.S-MD.A.2 (+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.
AQR.S-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.
AQR.S-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?
AQR.S-MD.B.5.a (+) 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.
AQR.S-MD.B.5.b (+) 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.
AQR.S-MD.B.6 (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
AQR.S-MD.B.7 (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game and replacing the goalie with an extra skater).