# Massachusetts - Mathematics: Number and Quantity

## Curriculum Frameworks | Adopted: 2017

### N-RN: : The Real Number System

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

N-RN.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.

Exponents and Power Rules

Choose the correct steps to simplify expressions with exponents using the rules of exponents and powers. Use feedback to diagnose incorrect steps. 5 Minute Preview

### N-CN: : The Complex Number System

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

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.

Points in the Complex Plane

Identify the imaginary and real coordinates of a point in the complex plane. Drag the point in the plane and investigate how the coordinates change in response. 5 Minute Preview

Roots of a Quadratic

Find the root of a quadratic using its graph or the quadratic formula. Explore the graph of the roots and the point of symmetry in the complex plane. Compare the axis of symmetry and graph of the quadratic in the real plane. 5 Minute Preview

N-CN.A.2: : Use the relation ??² = –1 and the Commutative, Associative, and Distributive properties to add, subtract, and multiply complex numbers.

Points in the Complex Plane

Identify the imaginary and real coordinates of a point in the complex plane. Drag the point in the plane and investigate how the coordinates change in response. 5 Minute Preview

N-CN.A.3: : Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

Points in the Complex Plane

Identify the imaginary and real coordinates of a point in the complex plane. Drag the point in the plane and investigate how the coordinates change in response. 5 Minute Preview

Roots of a Quadratic

Find the root of a quadratic using its graph or the quadratic formula. Explore the graph of the roots and the point of symmetry in the complex plane. Compare the axis of symmetry and graph of the quadratic in the real plane. 5 Minute Preview

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

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.

Points in the Complex Plane

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.

Points in the Complex Plane

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.

Points in the Complex Plane

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

N-CN.C.7: : Solve quadratic equations with real coefficients that have complex solutions.

Points in the Complex Plane

Roots of a Quadratic

Find the root of a quadratic using its graph or the quadratic formula. Explore the graph of the roots and the point of symmetry in the complex plane. Compare the axis of symmetry and graph of the quadratic in the real plane. 5 Minute Preview

### N-VM: : Vector and Matrix Quantities

N-VM.A: : Represent and model with vector quantities.

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).

Adding Vectors

Move, rotate, and resize two vectors in a plane. Find their resultant, both graphically and by direct computation. 5 Minute Preview

Vectors

Manipulate the magnitudes and directions of two vectors to generate a sum and learn vector addition. The x and y components can be displayed, along with the dot product of the two vectors. 5 Minute Preview

N-VM.A.3: : Solve problems involving velocity and other quantities that can be represented by vectors.

Adding Vectors

Move, rotate, and resize two vectors in a plane. Find their resultant, both graphically and by direct computation. 5 Minute Preview

N-VM.B: : Perform operations on vectors.

N-VM.B.4: : Add and subtract vectors.

N-VM.B.4.a: : 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.

Adding Vectors

Move, rotate, and resize two vectors in a plane. Find their resultant, both graphically and by direct computation. 5 Minute Preview

Vectors

Manipulate the magnitudes and directions of two vectors to generate a sum and learn vector addition. The x and y components can be displayed, along with the dot product of the two vectors. 5 Minute Preview

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

Vectors

Manipulate the magnitudes and directions of two vectors to generate a sum and learn vector addition. The x and y components can be displayed, along with the dot product of the two vectors. 5 Minute Preview

N-VM.B.4.c: : 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.

Adding Vectors

Vectors

N-VM.B.5: : Multiply a vector by a scalar.

Dilations

Dilate a figure and investigate its resized image. See how scaling a figure affects the coordinates of its vertices, both in

N-VM.B.5.a: : Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as ??(???, ?? subscript ??) = (?????, ???? subscript ??).

Dilations

Dilate a figure and investigate its resized image. See how scaling a figure affects the coordinates of its vertices, both in

Vectors

N-VM.B.5.b: : Compute the magnitude of a scalar multiple ???? using ||????|| = |??|??. Compute the direction of ???? knowing that when |??|?? ? 0, the direction of ???? is either along ?? (for ?? > 0) or against ?? (for ??

Vectors

N-VM.C: : Perform operations on matrices and use matrices in applications.

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.

Dilations

Dilate a figure and investigate its resized image. See how scaling a figure affects the coordinates of its vertices, both in

N-VM.C.8: : Add, subtract, and multiply matrices of appropriate dimensions.

Translations

Translate a figure horizontally and vertically in the plane and examine the matrix representation of the translation. 5 Minute Preview

N-VM.C.12: : Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.

Dilations

Translations

Translate a figure horizontally and vertically in the plane and examine the matrix representation of the translation. 5 Minute Preview

Correlation last revised: 9/16/2020

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