**Math Topics**

**Learning Support**

**Professional**

Select the cluster for resources on this page:

**Standards**:

- 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*). - VM-A.2. (+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
- VM-A.3. (+) Solve problems involving velocity and other quantities that can be represented by vectors.

**Technology-enhanced investigations:**

Mathwords.com: Math Dictionary: Key vocabulary for this domain. Use with VM-A, VM-B, VM-C:

CK-12 Flexbook: Trigonometry Concepts: Ch. 5: Triangles and Vectors: concepts, practice problems for learners, and videos included. Complete the following sections:

- 5.13: Directed Line Segments: aligns with VM-A.1.
- 5.19: Unit Vectors and Components: aligns with VM-A.2.
- 5.16: Resultant of Two Displacements: aligns with VM-A.3

GeoGebra: Vector Addition applet for viewing 2D vectors and their sum.

Mathispower4u: Trigonometry Videos is an extensive collection on nearly every topic in trig from James Sousa and includes a section on vectors and their operations. Aligns with VM-A and VM-B.

PhET Interactive Simulations: Vector Addition and Vector Addition: Equations. Aligns with VM-A.1, VM-A.2, VM-B.4, and VM-B.5.

PhysicsClassroom.com: Vectors -- Motion and Forces in Two Dimensions includes three lessons, each with multiple sections, animations, and embedded problems for learners to test their understanding. Lesson 1: Vector fundamentals and operations; Lesson 2: Projectile motion; and Lesson 3: Forces in two dimensions. Aligns with VM-A and VM-B.

TED-Ed Lessons: What is a vector? In this YouTube video (approx. 5 minutes), educator David Huynh "explains how vectors are a prime example of the elegance, beauty, and fundamental usefulness of mathematics" and begins by reminding viewers that "Physicists, air traffic controllers, and video game creators all have at least one thing in common: vectors." Learn what vectors are and why they matter. Aligns with VM-A.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives. Note: Within the Wolfram Demonstration Project are 13 manipulatives addressing VM-A.1, 8 manipulatives addressing VM-A.2, and 4 manipulatives addressing VM-A.3. Among those are:

- Basics of two dimensional vectors: Aligns with VM-A.1, VM-A.2, VM-B.4.
- 2D Vector addition: Aligns with VM-A.1, VM-A.2, VM-B.4.
- Head-to-toe vector addition: Aligns with VM-A.1, VM-A.2, VM-B.4.
- Addition of N vectors in 2D: Aligns with VM-A.1, VM-A.2, VM-B.4.
- Vector addition is commutative: Aligns with VM-A.1, VM-B.4.
- Resultant of a vector: Airplane flies while wind deflects it from its path. Aligns with VM-A.1, VM-A.3, VM-B.4.
- Airplanes and crosswinds: Aligns with VM-A.1, VM-B.4.
- Mechanical work: Aligns with VM-A.3.

**Multiple Choice:**

Khan Academy: Precalculus Vectors unit. Practice questions with videos.

- Equivalent vectors: Aligns with VM-A.1.
- Components of vectors: Aligns with VM-A.2.

**Constructed-response:**

Algebra Lab:

- Introduction to Vectors: Lesson with practice problems on vectors and their basic components. Vectors in two dimensions are defined and quantified. Aligns with VM-A.1.
- Determining the direction of a vector: Eight additional practice problems. Aligns with VM-A.1.
- Word Problems: Vectors - Right Triangle Relationships. Note: Learners should know "A typical problem involving vectors and right triangles gives us information about its components." Aligns with VM-A.2 and VM-A.3.
- Vector Dot Products: Lesson and practice problems on dot products, which can be used to find the angle between two vectors.
- Word Problems: Vectors - Non-Right Triangle Relationships. Lesson with practice problems. Note: Learners should know "A typical problem involving vectors can be solved by adding and/or subtracting vectors, using a dot product, or applying the Law of Sines or Law of Cosines to give us information about vectors that form a non-right triangle." Aligns with VM-A.2 and VM-A.3.

Khan Academy: HSN-N-VM includes HSN-VM.A.1 through VM.A.3 Common Core aligned multiple choice and constructed response problems.

Also see the Precalculus Vectors unit with practice questions with videos.

- Magnitude of vectors: Aligns with VM-A.1.
- Vector word problems: Aligns with VM-A.3.

**Performance tasks:**

Georgia Standards of Excellence Framework: Pre-Calculus Unit 7: Vectors. This unit contains ten tasks, including a culminating task: Putting It All Together. Tasks align with VM-A.1, VM-A.2, VM-A.3, VM-B.4, VM-B.5, and VM-C.11 and number standards CN-A.3, CN-B.4, CN-B.5, CN-B.6.

**Standards**:

- VM-B.4. (+) Add and subtract vectors.
- 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.
- VM-B.4b. Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
- VM-B.4c. Understand vector subtraction
–*v*as*w*+ (–*v*), 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.*w*

- VM-B.5. (+) Multiply a vector by a scalar.
- VM-B.5a. Represent scalar multiplication graphically by
scaling vectors and possibly reversing their direction; perform
scalar multiplication component-wise, e.g., as
*c*(*v*_{x},*v*_{y}) = (*cv*_{x},*cv*_{y}). - VM-B.5b. Compute the magnitude of a scalar multiple
*c*using ||*v**c*|| = |*v**c*|. Compute the direction of*v**c*knowing that when |*v**c*|≠ 0, the direction of*v**c*is either along*v*(for*v**c*> 0) or against(for*v**c*< 0).

- VM-B.5a. Represent scalar multiplication graphically by
scaling vectors and possibly reversing their direction; perform
scalar multiplication component-wise, e.g., as

**Technology-enhanced investigations:**

A. Dendane: Analyze Math:

- Vector Addition: This HTML 5 applet helps learners to explore addition of vectors. Enter two vectors and see the resultant vector. Vectors are mathematical quantities used to represent concepts such as force or velocity which have both a magnitude and a direction. The applet can also be used for subtraction, as the subtraction of vectors A - B may be changed into an addition A + (- B).
- Vector Addition and Scalar Multiplication: Explanation, examples, and HTML 5 applet.
- Vector Subtraction: This HTML 5 applet helps learners to explore subtraction of vectors.
- Magnitude and direction of a vector calculator: Enter the components of vector v as real numbers and press "enter". The magnitude || v || and direction in degrees are displayed. This is a good tool to also help learners check their work.
- Dot product of two vectors calculator: Enter the components of the two vectors as real numbers and press "Dot*". The answer appears as a scalar. Learners can also get an explanation of the dot product and see examples.

Academo.org: 3D Vector Plotter. Enter two vectors of the form (x, y, z) and see the sum, difference, and cross-product of the vectors. Clicking on the end of a vector will reveal its components. You can rotate the diagram and zoom in and out. When z=0, the interactive is useful for viewing vectors in 2D. Aligns with VM-B.4.

CK-12 Flexbook: Trigonometry Concepts: Ch. 5: Triangles and Vectors: concepts, practice problems for learners, and videos included. Complete the following sections:

- 5.14: Vector Addition: aligns with VM-B.4.
- 5.15: Vector Subtraction: aligns with VM-B.4.
- 5.17: Vector Multiplied by a Scalar: aligns with VM-B.5.
- 5.20: Resultant as the Sum of Two Components: aligns with VM-B.4.
- 5.21: Resultant as Magnitude and Direction: aligns with VM-B.4.

Interactive Mathematics: Introduction to Vectors includes 1-D, 2-D, and 3-D vector exploration. Aligns with VM-B.4.

Mathinsight.org: Introduction to Vectors, including operations on vectors( +, -, scalar multiplication) and applets to explore magnitude and direction, vector addition. Aligns with VM-B.4 and VM-B.5.

Mathispower4u: Trigonometry Videos is an extensive collection on nearly every topic in trig from James Sousa and includes a section on vectors and their operations. Aligns with VM-A and VM-B.

PhET Interactive Simulations:

- Vector Addition. Aligns with VM-A.1, VM-A.2, VM-B.4, and VM-B.5.
- Vector Addition: Equations. Aligns with VM-B.4.c and VM-B.5.

PhysicsClassroom.com: Vectors -- Motion and Forces in Two Dimensions includes three lessons, each with multiple sections, animations, and embedded problems for learners to test their understanding. Lesson 1: Vector fundamentals and operations; Lesson 2: Projectile motion; and Lesson 3: Forces in two dimensions. Aligns with VM-A and VM-B.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives. Note: Within the Wolfram Demonstration Project are 12 manipulatives addressing VM-B.4. Among those are:

- Basics of two dimensional vectors: Aligns with VM-A.1, VM-A.2, VM-B.4.
- 2D Vector addition: Aligns with VM-A.1, VM-A.2, VM-B.4.
- Head-to-toe vector addition: Aligns with VM-A.1, VM-A.2, VM-B.4.
- Addition of N vectors in 2D: Aligns with VM-A.1, VM-A.2, VM-B.4.
- Vector addition is commutative: Aligns with VM-A.1, VM-B.4.
- Resultant of a vector: Airplane flies while wind deflects it from its path. Aligns with VM-A.1, VM-A.3, VM-B.4.
- Airplanes and crosswinds: Aligns with VM-A.1, VM-B.4.

**Multiple Choice:**

Khan Academy: Linear-algebra: Vectors and spaces unit and Precalculus Vectors unit. Two units with practice questions with videos.

- Add vectors: magnitude and direction to component: Aligns with VM-B.4.
- Graphically adding and subtracting vectors: Aligns with VM-B.4.
- Scalar multiplication: Aligns with VM-B.5.

**Constructed-response:**

Algebra Lab:

- Operations with Vectors: Lesson with practice problems on vector addition and subtraction, including a description of scalar multiplication.

Khan Academy: HSN-N-VM includes HSN-VM.B.4 and VM.B.5 Common Core aligned multiple choice and constructed response problems.

Linear-algebra: Vectors and spaces unit and Precalculus Vectors unit. Two units with Practice questions with videos.

- Add and subtract vectors: Aligns with VM-B.4.
- Unit vectors: Aligns with VM-B.5.
- Add vectors: magnitude and direction: Aligns with VM-B.5.

**Performance tasks:**

**Standards**:

- VM-C.6. (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
- VM-C.7. (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.
- VM-C.8. (+) Add, subtract, and multiply matrices of appropriate dimensions.
- VM-C.9. (+) Understand that, unlike multiplications of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.
- 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.
- VM-C.11. (+) Multiply a vector (regarded as a matrix with one column) by matrix suitable dimensions to produce another vector. Work with matrices as transformations of vectors.
- VM-C.12. (+) Work with 2x2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.

**Technology-enhanced investigations:**

A. Dendane: Analyze Math:

- Find Inverse of 2 by 2 Matrix Calculator
- Find Inverse of 3 by 3 Matrix Calculator
- Find Deteminant of 2 by 2 Matrix Calculator
- Find Deteminant of 3 by 3 Matrix Calculator

CK-12: Algebra: Matrices The unit in algebra for matrices begins with Introduction to Matrices: concepts, practice problems for learners, and videos included, which aligns with VM-C.6. The unit proceeds with matrix algebra, adding/subtracting/multiplying matrices including multiplying by a scalar and limitations of matrix multiplication, determinants, Cramer's Rule, matrix equations and solving those, solving systems of linear equations with matrices, inverse matrices, and applications of matrices. Aligns with VM-C.

CK-12 Flexbook: Algebra II with Trigonometry: Ch. 4: Matrices. This chapter includes operations on matrices, multiplying matrices, determinants and Cramer's rule, identity and inverse matrices, and solving linear systems using inverse matrices. Content also includes practice exercises for learners and videos. Aligns with VM-C.

Mathinsight.org: Introduction to Matrices is comprehensive, including multiplying matrices and vectors with examples, how to transpose a matrix, matrices and determinants, the relationship between determinants and area or volume, determinants and linear transformations, and more. Aligns with VM-C.

TED-Ed Lessons: How to organize, add and multiply matrices. In this YouTube video (approx. 5 minutes), educator Bill Shillito introduces the concept of matrices. He shows how to work with matrices, including tips for adding, subtracting, and multiplying them. He begins by noting that matrices date back to ancient China. They are everywhere, used in business, economics, physics, cryptography, electronics, and 3D graphics. The video is followed by an six question quiz. Aligns with VM-C.

Interactive Mathematics: The following align with VM-C.7 and VM-C.8:

- Simple online matrix calculator (2×2 ). "This matrix calculator allows you to enter your own 2×2 matrices and it will add and subtract them, find the matrix multiplication (in both directions) and the inverses for you. It shows you the steps for obtaining the answers."
- Add, subtract, and multiply matrices. This interactive "will help you to learn how addition, subtraction, scalar multiplication and multiplication of matrices work. It will generate many different sized (up to 5 by 5) matrices with different random numbers each time. You can use it to see plenty of examples of matrix operations. You can step through each calculation involved." Test your understanding by solving the problem first.

Math is Fun: Matrix Calculator enables the user to work with matrices up to 10 x 10. Operations include addition, subtraction, multiplication, finding a determinant, an inverse, squaring a matrix, and more. Aligns with VM-C.8.

MatrixCalc.org: Matrix Calculator helps the user to find the determinant of a matrix, the rank, raise a matrix to a power, find sum and multiplication of matrices, calculate the inverse matrix. Enter the matrix elements and click a button. You can also solve systems of equations using matrices. Aligns with VM-C.8.

Matrix Calculator developed by PJ in Belgium enables the user to work with matrices up to 10 x 10. The calculator will add, subtract, multiply, divide, transpose, find an inverse, find a determinant, and raise to a power. The beauty of this calculator is that each answer comes with a step-by-step, easy to understand, solution. Users can show instructions for each operation, which are clearly written and help to understand the math. Aligns with VM-C.8.

Meta-Calculator.com: With the Matrix Calculator you can add, subtract, multiply, transpose, find the inverse, determinant, and more. Aligns with VM-C.8.

Purple Math:

- Matrix Definitions: Augmented & coefficient matrices, matrix size, notation & types, matrix equality
- Matrix Addition and Subtraction
- Scalar and Matrix Multiplicatoin
- Matrix Row Operations
- Matrix Inversion: Finding the inverse of a matrix
- Minors and Cofactors: technique for finding determinants of square matrices 4 x 4 or larger.
- Cramer's Rule
- Determinants

Stat Trek: Tutorials: Matrix Algebra. Lessons delve into matrix types, operations (addition, multiplication, vector multiplication, elementary operations), Echelon matrices, properties (vector dependence, matrix rank, determinants), the matrix inverse, applications, matrix theorems, notations. Most lessons include sample problems. Aligns with VM-C.

Study.com: Common Core HS Math: What is a Matrix? and How to Take the Determinant of a Matrix. These video lessons also include sample problems and a short quiz. Aligns with VM-C.6.

YouTube videos: Videos on solving systems with matrices and working with matrices.

Wolfram Demonstrations Project: Download the free Wolfram CDF player to interact with the following manipulatives. Note: Within the Wolfram Demonstration Project are 1 manipulative addressing VM-C.8, 4 manipulatives addressing VM-C.11, and 8 manipulatives addressing VM-C.12. Among those are:

- Matrix multiplication: Aligns with VM-C.8, VM-C.11.
- Matrix transformations: "F": Aligns with VM-C.11, VM-C.12.
- Reflection matrix in 2D: Aligns with VM-C.11.
- Linear transformations of a polygon: Aligns with VM-C.11, VM-C.12.
- Rotation about a point in the plane: Aligns with VM-C.12.
- Area of a triangle using a determinant: Aligns with VM-C.12.
- Determinants seen geometrically: Aligns with VM-C.12.

**Multiple Choice:**

Khan Academy: Practice questions with videos.

- Precalculus Matrices unit: Learn what matrices are and about their various uses: solving systems of equations, transforming shapes and vectors, and representing real-world situations. Learn how to add, subtract, multiply matrices and find inverses. Aligns with VM-C.
- Matrices as Transformations: Aligns with VM-C.12.

**Constructed-response:**

Khan Academy: HSN-N-VM includes HSN-VM.C.6 through HSN-VM.C.8 and HSN-VM.C.10 through HSN-VM.C.12 Common Core aligned problems.

Also see the Precalculus Matrices unit with practice questions with videos.

- Multiply matrices by scalars: Aligns with VM-C.7.
- Add and subtract matrices: Aligns with VM-C.8.
- Multiply two matrices: Aligns with VM-C.8.
- Transform vectors using matrices: Aligns with VM-C.8, VM-C.11.

Zweigmedia.com: Tutorials and examples using fill-in or multiple choice to test understanding. Application examples of concepts are included, as well as additional review exercises and topic true-false quizzes. Learners can also choose to do game versions of some topics. Use with VM-C:

- Matrix addition and scalar multiplication
- Matrix multiplication
- Matrix inversion: finding the inverse of a matrix
- Using matrices to solve systems of equations

**Performance tasks:**

Georgia Standards of Excellence Framework:

- Pre-Calculus Unit 5: Matrices. This unit contains five tasks: Central High Booster Club, Walk Like a Mathematician, Candy? What Candy?, An Okefenoke Food Web, and a Culminating Task: Vacationing in Georgia. Tasks align with VM-C.6, VM-C.7, VM-C.8, VM-C.9, VM-C.10, VM-C.12 and algebra standards REI-C.8, REI-C.9.
- Pre-Calculus Unit 7: Vectors. This unit contains ten tasks, including a culminating task: Putting It All Together. Tasks align with VM-A.1, VM-A.2, VM-A.3, VM-B.4, VM-B.5, and VM-C.11 and number standards CN-A.3, CN-B.4, CN-B.5, CN-B.6.

Common Core Math:
Intro | HS Number and Quantity Domain:
RN | Q | CN | **VM** |