Although the notion of number changes, the four operations stay the same in important ways. The commutative, associative, and distributive properties extend the properties of operations to the integers, rational numbers, real numbers, and complex numbers. Extending the properties of exponents leads to new and productive notation; for example, since the properties of exponents suggest that (5^{1/3})^{3} = 5^{(1/3)·3} = 5^{1} = 5, we define 5^{1/3} to be the cube root of 5.
Calculators are useful in this strand to generate data for numerical experiments, to help understand the workings of matrix, vector, and complex number algebra, and to experiment with noninteger exponents.
In their work in measurement up through Grade 8, students primarily measure commonly used attributes such as length, area, and volume. In high school, students encounter a wider variety of units in modeling, e.g. acceleration, currency conversions, derived quantities such as personhours and heating degree days, social science rates such as percapita income, and rates in everyday life such as points scored per game or batting averages. They also encounter novel situations in which they themselves must conceive the attributes of interest. For example, to find a good measure of overall highway safety, they might propose measures such as fatalities per year, fatalities per year per driver, or fatalities per vehiclemile traveled. Such a conceptual process might be called quantification. Quantification is important for science, as when surface area suddenly “stands out” as an important variable in evaporation. Quantification is also important for companies, which must conceptualize relevant attributes and create or choose suitable measures for them.Code 
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HSNRN.A.1 
Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 5^{1/3 }to be the cube root of 5 because we want (5^{1/3})^{3} = 5^{(1/3)3} to hold, so (5^{1/3})^{3 }must equal 5. 
Multiply radicals


HSNRN.A.2 
Rewrite expressions involving radicals and rational exponents using the properties of exponents. 
Simplify Expressions with Exponents 
Manipulate fractional exponents 

HSNRN.B.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. 
Recognize rational and irrational expressions 
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HSNQ.A.1 
Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. 
Reasonable units 

HSNQ.A.2 
Define appropriate quantities for the purpose of descriptive modeling. 

Define appropraite units 

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

Measurement precision 

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HSNCN.A.1 
Know there is a complex number i such that i^{2} = –1, and every complex number has the form a + bi with a and b real. 
Powers of i 


HSNCN.A.2 
Use the relation i^{2} = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. 
Add Complex Numbers 

HSNCN.A.3 
(+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. 


HSNCN.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. 
The Complex Plane 

HSNCN.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 + √3 i)^{3} = 8 because (1 + √3 i) has modulus 2 and argument 120°. 
Operations on the Complex Plane (include DeMoivre's Theorem) 


HSNCN.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. 

Distance on the Complex Plane 

HSNCN.C.7 
Solve quadratic equations with real coefficients that have complex solutions. 
Quadratic formula with complex solutions 


HSNCN.C.8 
(+) Extend polynomial identities to the complex numbers. For example, rewrite x^{2} + 4 as (x + 2i)(x – 2i). 
Expressions with complex numbers 


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

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HSNVM.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). 
Recognize vector quantities 


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


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


HSNVM.B.4 
(+) Add and subtract vectors. 
See Below 

HSNVM.B.4a 
Add vectors endtoend, componentwise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes. 
Add and subtract vectors in rectangular form 

HSNVM.B.4c 
Understand vector subtraction v – w as v + (–w), where –w is the additive inverse ofw, 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 componentwise. 


HSNVM.B.5 
(+) Multiply a vector by a scalar. 

HSNVM.C.6 
(+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network. 
Represent relationships with matrices 


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

HSNVM.C.8 
(+) Add, subtract, and multiply matrices of appropriate dimensions. 
Defined & undefined matrix operations 

HSNVM.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. 
Properties of matrix multiplication 


HSNVM.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. 


HSNVM.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. 

HSNVM.C.12 
(+) Work with 2 × 2 matrices as a transformations of the plane, and interpret the absolute value of the determinant in terms of area. 
Geometric transformation with matrix multiplication 

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