Vector and Matrix Quantities · • Connect the Pythagorean Theorem and distance formula to the calculation of magnitude using component form. ... Row, column, dimension, matrix

Vector and Matrix Quantities Core Guide Secondary I Honors

IH.N.VM.1 Page | 1

Related Standards: Current Course Related Standards: Future Courses I.N.VM.2, I.N.VM.3, I.N.VM.4, I.N.VM.5, I.N.VM.11, I.G.CO.1,I.G.CO.4, I.G.CO.12, I.G.GPE.4, I.G.GPE.7

II.F.TF.8, II.G.SRT.8, II.G.GPE.4, II.G.GPE.6, P.N.CN.3, P.N.CN.4,P.N.CN.5, P.N.CN.6, P.N.CN.10

Support for Teachers

Critical Background Knowledge (Access Background Knowledge)

• Draw points, lines, line segments, rays, and angles (4.G.1) and measure angles in whole number degrees (4.MD.6)

• Draw geometric shapes, paying attention to angles and sides (7.G.2)

• Use the Pythagorean Theorem to solve for unknown side lengths of right triangles (8.G.7) and to find the distance betweentwo points (8.G.8)

Academic Vocabulary Vector, magnitude, direction

Resources

Curriculum Resources: https://www.uen.org/core/core.do?courseNum=5600#78795

Represent and model with vector quantities (Standards N.VM.1-3) Standard (Honors) N.VM.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).

Concepts and Skills to Master • Recognize vector quantities as having both magnitude and direction.• Represent vector quantities by directed line segments, and use appropriate symbols for vectors (v) and their

magnitudes (e.g., |v|,||v||, v).• Connect the Pythagorean Theorem and distance formula to the calculation of magnitude using component form.

https://www.uen.org/core/core.do?courseNum=5600#78796


























IH.N.VM.2 Page | 2

Related Standards: Current Course Related Standards: Future Courses

I.N.VM.1, I.N.VM.3, I.N.VM.4, I.N.VM.5, I.N.VM.11, I.G.CO.1,I.G.GPE.4, I.G.GPE.5

II.G.GPE.4, II.G.GPE.6, P.N.CN.6


Critical Background Knowledge (Access Background Knowledge) • Graph points in all four quadrants of the coordinate plane (6.NS.8)• Explain why the slope is the same between any two points on a non-vertical line (8.EE.6)

Academic Vocabulary Vector, components, initial point, terminal point

Resources


Represent and model with vector quantities (Standards N.VM.1-3) Standard (Honors) N.VM.2: Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

Concepts and Skills to Master

• Relate initial and terminal points of a vector to the coordinate system.

• Find the horizontal and vertical components of a vector by subtracting the coordinates of an initial point from thecoordinates of a terminal point (relate to finding slope between points).
















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I.SMP.4, I.N.VM.1, I.N.VM.2, I.N.VM.4, I.N.VM.5, I.N.VM.11,I.G.CO.12, I.G.GPE.4, I.G.GPE.5, I.G.GPE.7

II.G.SRT.8, II.G.GPE.4, II.G.GPE.6, III.F.TF.3



• Understand the magnitude and direction of a vector as well as graph and use vector notation (I.N.VM.1)

• Draw points, lines, line segments, rays, and angles (4.G.1) and measure angles in whole number degrees (4.MD.6)

• Draw geometric shapes, paying attention to angles and sides (7.G.2)

Academic Vocabulary Vector, direction, magnitude, velocity, force

Resources Curriculum Resources: https://www.uen.org/core/core.do?courseNum=5600#78797

Represent and model with vector quantities (Standards N.VM.1-3) Standard (Honors) N.VM.3: Solve problems involving velocity and other quantities that can be represented by vectors.


• Represent real world contexts with geometric vector models.• Solve contextual problems involving velocity and other quantities that can be represented by vectors (e.g. science, sports,

medicine).





















IH.N.VM.4 Page | 4


I.N.VM.1, I.N.VM.2, I.N.VM.3, I.N.VM.5, I.N.VM.11, I.G.CO.2,I.G.CO.4, I.G.CO.5, I.G.CO.6, I.G.CO.7, I.G.CO.12, I.G.GPE.4

II.G.GPE.4, II.G.GPE.6, II.G.CO.9, II.G.CO.11, P.N.CN.5


Critical Background Knowledge (Access Background Knowledge) • Vector notation, magnitude, and direction (I.N.VM.1, I.N.VM.2)• Verify the property of translations: lines are taken to lines, angles are taken to angles (8.G.1)

Academic Vocabulary vector, end-to-end, component-wise, parallelogram rule, magnitude, resultant vector

Resources


Perform operations on vectors (Standards N.VM.4-5) Standard (Honors) N.VM.4: Add and subtract vectors.

a. Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of twovectors is typically not the sum of the magnitudes.

b. Given two vectors, determine the magnitude of their sum.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.


• Find the sum of vectors by translating vectors on the coordinate plane to form end-to-end pairings.• Add and subtract vectors using components.• Use the parallelogram rule to find the sum of two vectors.• Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.• Understand vector subtraction as adding the additive inverse (for example, v – w as v + (–w), where –w is the additive

inverse of w)























IH.N.VM.5 Page | 5

Related Standards: Current Course Related Standards: Future Courses I.N.VM.1, I.N.VM.2, I.N.VM.3, I.N.VM.4, I.N.VM.7, I.N.VM.11,I.G.CO.1, I.G.CO.2, I.G.CO.4, I.G.CO.5, I.G.CO.12

II.F.BF.3, II.G.SRT.1, II.G.SRT.2, II.G.SRT.3, II.G.SRT.5, II.G.GPE.6,II.G.CO.9, III.F.BF.3, P.N.CN.5



• Represent vectors as directed line segments (I.N.VM.1)• Solve problems involving scale drawings (7.G.1)• Create similar figures through dilations (8.G.4)

Academic Vocabulary Scalar


Perform operations on vectors (Standards N.VM.4-5) Standard (Honors) N.VM.5: Multiply a vector by a scalar.

a. Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalarmultiplication component-wise, e.g., as c(vx, vy) = (cvx, cvy).

b. 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).


• Dilate vectors graphically and with scalar multiplication (component-wise).• Compute the magnitude of a scalar multiple of a vector.• Understand the direction of a scalar multiple of a vector is the same direction as the original vector or in its opposite

direction.


























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I.SMP.4, I.N.VM.7, I.N.VM.8, I.N.VM.9, I.N.VM.10, I.N.VM.11,I.N.VM.12, I.N.VM.13, I.A.REI.6, I.A.REI.11

II.S.ID.5, III.A.REI.11



• Organize data in a table (8.SP.4)• Identify important quantities to construct a mathematical model (SMP.4)

Academic Vocabulary Row, column, dimension, matrix, element

Resources


Perform operations on matrices and use matrices in applications (Standards N.VM.6-13) Standard (Honors) N.VM.6: Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.

Concepts and Skills to Master • Use matrices to represent, organize, and manipulate data.• Interpret data in a matrix by understanding how each element of the matrix corresponds to the context.• Identify and name matrix attributes (e.g. dimensions, rows, columns, etc.)• Recognize and use matrix notation.

















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I.N.VM.5, I.N.VM.6, I.N.VM.7, I.N.VM.8, I.N.VM.9, I.N.VM.10,I.N.VM.11, I.N.VM.12, I.N.VM.13, I.A.REI.5, I.G.CO.2, I.G.CO.12

II.F.BF.3, II.G.SRT.1, II.G.SRT.2, II.G.GMD.3, III.F.BF.3, P.N.CN.5,P.A.REI.8, P.A.REI.9


Critical Background Knowledge (Access Background Knowledge) • Apply properties of operations (distributive) to multiply (3.OA.5, 5.OA.2, 6.EE.3)

Academic Vocabulary Scalar


Perform operations on matrices and use matrices in applications (Standards N.VM.6-13) Standard (Honors) N.VM.7: Multiply matrices by scalars to produce new matrices, e.g., as when all of the pay-off in a game are doubled.

Concepts and Skills to Master • Connect the procedure for scalar multiplication to a contextual framework.• Multiply each element of the matrix by the scalar to form a new matrix.


























IH.N.VM.8 Page |8


I.N.VM.5, I.N.VM.6, I.N.VM.7, I.N.VM.9, I.N.VM.10, I.N.VM.11,I.N.VM.12, I.N.VM.13, I.A.REI.5, I.A.REI.6, I.G.CO.2

P.A.REI.8, P.A.REI.9



• Apply operations (add, subtract, multiply) to solve problems

Academic Vocabulary Row, column, matrix

Resources


Perform operations on matrices and use matrices in applications (Standards N.VM.6-13) Standard (Honors) N.VM.8: Add, subtract, and multiply matrices of appropriate dimensions.


• Recognize the necessary conditions for matrix operations.• Add, subtract and multiply matrices.
















IH.N.VM.9 Page | 9

Related Standards: Current Course Related Standards: Future Courses I.N.VM.5, I.N.VM.6, I.N.VM.7, I.N.VM.9, I.N.VM.10, I.N.VM.11,I.N.VM.12, I.N.VM.13, I.G.CO.2




• Matrix multiplication, properties of real numbers (I.N.VM.8)• Understand the properties of operations (3.OA.5, 6.EE.3)

Academic Vocabulary Associative, commutative, distributive, square matrix

Resources


Perform operations on matrices and use matrices in applications (Standards N.VM.6-13) Standard (Honors) N.VM.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.

Concepts and Skills to Master • Understand that multiplication of matrices is not commutative.• Justify and provide counter-examples for why matrix operations have or do not have certain properties.• Connect dimensional analysis of matrices to specific matrix operations.

















IH.N.VM.10 Page | 10

Related Standards: Current Course Related Standards: Future Courses I.N.VM.5, I.N.VM.6, I.N.VM.7, I.N.VM.9, I.N.VM.10, I.N.VM.11,I.N.VM.12, I.N.VM.13, I.A.REI.1, I.A.REI.5, I.A.REI.6, I.G.CO.2




• Add, subtract and multiply matrices (I.N.VM.8)• Understand additive inverse (7.NS.1)• Understand the relationship between multiplication and division as inverse operations (3.OA.6)

Academic Vocabulary Identity, inverse, determinant

Resources


Perform operations on matrices and use matrices in applications (Standards N.VM.6-13) Standard (Honors) N.VM.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. Concepts and Skills to Master

• Describe the use of zeros and ones and their role in forming the zero matrix and identity matrix for certain operations.• Verify the identity matrix for addition and subtraction holds by performing matrix operations.• Explore the relationship of inverse matrices and the identity matrix.• Calculate the determinant of a matrix to verify it does or does not have an inverse.• Use the determinant of a matrix to calculate the inverse of a 2 x 2 matrix.






















I.N.VM.1, I.N.VM.5, I.N.VM.6, I.N.VM.7, I.N.VM.8, I.N.VM.9,I.N.VM.10, I.N.VM.12, I.N.VM.13, I.G.CO.2, I.G.CO.3, I.G.CO.5



• Represent vector quantities (I.N.VM.1)• Multiply matrices (I.N.VM.8)

Academic Vocabulary Vector, matrix, transformation, column matrix

Resources


Perform operations on matrices and use matrices in applications (Standards N.VM.6-13) Standard (Honors) N.VM.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.

Concepts and Skills to Master • Define and represent a vector as a matrix with one column.• Recognize that multiplication of a vector (v) by a matrix (A) is calculated as Av.• Understand that a matrix is a representation of a function where v is the input, and the product of A and v is the output.• Transform a vector using a matrix.


















Related Standards: Current Course Related Standards: Future Courses I.N.VM.9, I.N.VM.10, I.N.VM.11, I.N.VM.12, I.N.VM.13, I.A.REI.1,I.A.REI.5, I.A.REI.6, I.G.CO.2, I.G.CO.3

III.G.SRT.9, III.G.GMD.4



• Find the area of a rectangle (3.MD.7), triangles (6.G.1) and other polygons (6.G.1, 7.G.6 , 8.G.9)• triangle, ordered pairs, definition of a function, determinant, matrix operations, absolute value

Academic Vocabulary Matrix, determinant, transformation

Resources


Perform operations on matrices and use matrices in applications (Standards N.VM.6-13) Standard (Honors) N.VM.12: Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.

Concepts and Skills to Master • Recognize matrix transformations as a function.• Transform geometric figures using 2X2 matrices.• Find the area of a triangle using determinants.• Discover the relationship between the determinant of a matrix and its resulting area after the transformation.• Connect the matrix representation of a polygon to its representation graphically.• Find the area of a polygon on the coordinate plane using varied approaches.• Use the procedures of finding the determinant of a matrix and matrix multiplication to connect the dilation of area after

performing matrix multiplication for transformation.






















I.N.VM.6, I.N.VM.7, I.N.VM.8, I.N.VM.9, I.N.VM.10, I.N.VM.11,I.N.VM.12, I.N.VM.13, I.A.REI.1, I.A.REI.5, I.A.REI.6, I.A.REI.11

II.A.REI.7, P.A.REI.8, P.A.REI.9



• Solve systems of two linear equations in two variables graphically (8.EE.8)• Solve systems of linear equations (I.A.REI.6)

Academic Vocabulary Matrix row reduction


Perform operations on matrices and use matrices in applications (Standards N.VM.6-13) Standard (Honors) N.VM.13: Solve systems of linear equations up to three variables using matrix row reduction.


• Solve systems of linear equations.• Apply matrix row reduction to solve systems of linear equations.



















Documents

Vector and Matrix Quantities · • Connect the Pythagorean Theorem and distance formula to the calculation of magnitude using component form. ... Row, column, dimension, matrix