Sample 5.3.B.2 Complete
|Domain: Vector Quantities and Matrices |
|Cluster: Precalculus |
|Standards: CCSS.Math.Content.HSN-VM.A.1 CCSS.Math.Content.HSN-VM.A.2 CCSS.Math.Content.HSN-VM.A.3 CCSS.Math.Content.HSN-VM.B.4 CCSS.Math.Content.HSN-VM.B.4a CCSS.Math.Content.HSN-VM.B.4b |
|CCSS.Math.Content.HSN-VM.B.4c CCSS.Math.Content.HSN-VM.B.5 CCSS.Math.Content.HSN-VM.B.5a CCSS.Math.Content.HSN-VM.B.5b CCSS.Math.Content.HSN-VM.C.6 CCSS.Math.Content.HSN-VM.C.7 |
|CCSS.Math.Content.HSN-VM.C.11 CCSS.Math.Content.HSN-VM.C.9 CCSS.Math.Content.HSN-VM.C.10 CCSS.Math.Content.HSN-VM.C.8 CCSS.Math.Content.HSA-REI.C.9 CCSS.Math.Content.HSA-REI.C.8 |
|Essential Questions |Enduring Understandings |Activities, Investigation, and Student Experiences |
|How do we find the magnitude of a vector? |Use the component form of a vector for finding | |
|How do we find the components of a vector? What do they represent? |magnitude and direction. Express the importance of a |Investigation Tools |
|Give an example of adding and subtracting of two vectors. |unit vector where magnitude is one unit and the same |1. TI-SmartView Program |
|Show adding, subtracting, and scalar multiplication of two vectors |direction as a given nonzero vector. |2. WolframAlpha Website |
|graphically. |CCSS.Math.Content.HSN-VM.A.1 |3. Textbook Activities |
|Which discipline use vectors more often and what does it represent? |The component form of a vector’s initial point |
|Give real life examples of the usefulness of matrices. |coordinates are subtracted from terminal points. |3e/students/sso.html |
|Explain the importance of dimensions when we add, subtract and |CCSS.Math.Content.HSN-VM.A.2 |
|multiply two matrices. |Represent and model with vector quantities. |ins_resources/ap.html |
|Matrix multiplication of square matrices satisfies which two |Applications of vectors such as velocity and angle |
|properties? Give an example. |between the horizon and vector. |limits-a-graphing-approach-5th-edition/ |
|How do we know if a matrix has a multiplicative inverse? |CCSS.Math.Content.HSN-VM.A.3 | |
|How do we find the multiplicative inverse of a matrix? |To add two vectors geometrically, position them | |
|What is the identity matrix and why do we need it? |(without changing their lengths and directions) so that| |
|Explain the shortcut of finding the determinant of 2x2 matrices. |the initial point of one coincides with the terminal | |
|Explore your graphing calculator in relation to matrices. |point of the other which is called Parallelogram Law |Example Student Experiences |
|List all the functions a graphing calculator can perform involving |and it proves that adding two vectors are not equal to |1. Consider two forces of equal magnitude acting on a point. |
|matrices. |adding magnitudes of these two vectors. |If the magnitude of the resultant is the sum of the magnitudes of the|
| |CCSS.Math.Content.HSN-VM.B.4a |two forces, make a conjecture about the angle between the forces. |
| |Apply vector addition. Find the magnitude and direction|If the resultant of the forces is zero, make a conjecture about the |
| |of the sum. CCSS.Math.Content.HSN-VM.B.4b |angle between the forces. |
| |Apply vector subtraction component-wise and show it |Can the magnitude of the resultant be greater than the sum of the two|
| |geometrically. CCSS.Math.Content.HSN-VM.B.4c |forces? Explain. |
| |Apply scalar multiplication. Geometrically, the |2. Use vectors to prove that the diagonals of a rhombus are |
| |product of a vector v and a scalar k is the vector that|perpendicular. |
| |is abs (k) times as long as v. |3. When solving a system of equations, how do you recognize that the|
| |CCSS.Math.Content.HSN-VM.B.5a |system has no solution? |
| |Compute the magnitude of a scalar multiple. |4. Briefly explain whether or not it is possible for a consistent |
| |CCSS.Math.Content.HSN-VM.B.5b |system of linear equations to have exactly two solutions. |
| |Apply matrices in real life cases. |5. When using Gaussian elimination to solve a system of linear |
| |CCSS.Math.Content.HSN-VM.C.6 |equations, explain how you can recognize that the system has no |
| |Scalar multiplication of matrices. |solution. Give an example that illustrates your answer. |
| |CCSS.Math.Content.HSN-VM.C.7 |6. In your own words, describe the difference between a matrix in |
| |Perform operations on matrices and use properties of |row-echelon form and a matrix in reduced row-echelon form. |
| |matrices operations. CCSS.Math.Content.HSN-VM.C.8 |7. If a and b are real numbers such that ab=0, then a=0 or b=0. |
| |Multiplication of square matrices satisfies the |However, if A and B are matrices such that AB=0, prove that it is not|
| |associative and distributive properties but it does not|necessarily true that A=0 or B=0. |
| |satisfy the commutative property. Prove it with |8. Write the steps of finding the determinant and inverse of the 2x2|
| |different examples. CCSS.Math.Content.HSN-VM.C.9 |and 3x3 matrices. (Use examples) |
| |(+) Understand that the zero and identity matrices play|9. Write an argument that explains why the determinant of a 3x3 |
| |a role in matrix addition and multiplication similar to|triangular matrix is the product of its main diagonal entries. |
| |the role of 0 and 1 in the real numbers. |10. Use your school's library, the Internet, or some other |
| |CCSS.Math.Content.HSN-VM.C.10 |references to research a few current real-life uses of cryptography. |
| |Multiplication of matrices and vectors to produce |Write a short summary of these uses. |
| |another vector. CCSS.Math.Content.HSN-VM.C.11 | |
| |Use 2x2 matrices as a transformation of the plane and | |
| |use absolute value of the determinant in terms of area.| |
| |CCSS.Math.Content.HSN-VM.C.12 | |
| |Solve systems of equations by using matrices. | |
| |CCSS.Math.Content.HSA-REI.C.8 | |
| |Find the inverse of the matrix to solve systems of | |
| |equations. CCSS.Math.Content.HSA-REI.C.9 | |
|Content Statements | | |
| | | |
|Students will be able to represent vectors as directed line segments.| | |
|Students will be able to write component forms of vectors. | | |
|Students will be able to perform basic vector operations and | | |
|represent them graphically. | | |
|Students will be able write vectors as linear combinations of unit | | |
|vectors. | | |
|Students will be able find the direction angles of vectors. | | |
|Students will be able to use vectors to model and solve real-life | | |
|problems. | | |
|Students will be able to recognize vector quantities as having both | | |
|magnitude and direction. | | |
|Students will be able to represent vector quantities as directed line| | |
|segments and use appropriate symbols for vectors and their | | |
|magnitudes. | | |
|Students will be able to find the components of a vector by | | |
|subtracting the coordinates of an initial point from the coordinates | | |
|of a terminal point. | | |
|Students will be able to solve problems involving velocity and other | | |
|quantities that can be represented by vectors. | | |
|Students will be able to add and subtract vectors. | | |
|Students will be able to add vectors end-to-end, component –wise, and| | |
|by the parallelogram rule. | | |
|Students will be able to 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. | | |
|Students will be able to multiply a vector by a scalar. | | |
|Students will be able to represent scalar multiplication graphically | | |
|by scaling vectors and possibly reversing their direction; perform | | |
|scalar multiplication component-wise, e.g., as c(vx, vy) = (cvx, | | |
|cvy). | | |
|Students will be able to compute the magnitude of a scalar multiple | | |
|cv using ||cv|| = |c|v and 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). | | |
|Students will be able to use matrices to represent and manipulate | | |
|data, e.g., to represent payoffs or incidence relationships in a | | |
|network. | | |
|Students will be able to multiply matrices by scalars to produce new | | |
|matrices, e.g., as when all of the payoffs in a game are doubled. | | |
|Students will be able to add, subtract, and multiply matrices of | | |
|appropriate dimensions. | | |
|Students will be able to understand that the magnitude of a sum of | | |
|two vectors is typically not the sum of the magnitudes. | | |
|Students will be able to understand that, unlike multiplication of | | |
|numbers, matrix multiplication for square matrices is not a | | |
|commutative operation, but still satisfies the associative and | | |
|distributive properties. | | |
|Students will be able to 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. | | |
|Students will be able to multiply a vector (regarded as a matrix with| | |
|one column) by a matrix of suitable dimensions to produce another | | |
|vector, and work with matrices as transformations of vectors. | | |
|Students will be able to work with 2 × 2 matrices as transformations | | |
|of the plane, and interpret the absolute value of the determinant in | | |
|terms of area. | | |
| | | |
| | | |
|Assessments | |
|Do-Now problems | |
|Warm-up questions | |
|Group/Partner Activities | |
|Oral Questioning Assessments | |
|Worksheets | |
|Student Interactive Handheld Devices | |
|Exit Cards | |
|Daily Homework | |
|Quizzes on concepts | |
|Unit Tests | |
|Equipment Needed: |Teacher Resources: |
|Promethean Board |Safari Montage: Math’s Cool and Algebra’s Cool video series |
|Blackboard | |
|ActiView Camera |Math Warehouse |
|Computer | |
| | |
| |Kutasoftware worksheets |
| | |
| | |
| |National Library of Virtual Manipulatives |
| | |
| | |
| |Algebra Lessons and PowerPoint Activities |
| | |
| | |
| |Classzone Interactive Games |
| | |
| | |
| |Virtual Math Lab |
| | |
| | |
| |Khan Academy Math Videos |
| | |
| | |
| |Center of Teaching and Learning |
| | |
| | |
| |Larson Textbook Activities |
| |
| |3e/students/sso.html |
| |
| |ins_resources/ap.html |
| |
| |limits-a-graphing-approach-5th-edition/ |
| | |
| | |
Represent and model with vector quantities.
• CCSS.Math.Content.HSN-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).
• CCSS.Math.Content.HSN-VM.A.2 (+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
• CCSS.Math.Content.HSN-VM.A.3 (+) Solve problems involving velocity and other quantities that can be represented by vectors.
Perform operations on vectors.
• CCSS.Math.Content.HSN-VM.B.4 (+) Add and subtract vectors.
o CCSS.Math.Content.HSN-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.
o CCSS.Math.Content.HSN-VM.B.4b Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.
o CCSS.Math.Content.HSN-VM.B.4c 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.
• CCSS.Math.Content.HSN-VM.B.5 (+) Multiply a vector by a scalar.
o CCSS.Math.Content.HSN-VM.B.5a Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(vx, vy) = (cvx, cvy).
o CCSS.Math.Content.HSN-VM.B.5b 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).
Perform operations on matrices and use matrices in applications.
• CCSS.Math.Content.HSN-VM.C.6 (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
• CCSS.Math.Content.HSN-VM.C.7 (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.
• CCSS.Math.Content.HSN-VM.C.8 (+) Add, subtract, and multiply matrices of appropriate dimensions.
• CCSS.Math.Content.HSN-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.
• CCSS.Math.Content.HSN-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.
• CCSS.Math.Content.HSN-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.
• CCSS.Math.Content.HSN-VM.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.
• CCSS.Math.Content.HSA-REI.C.8 (+) Represent a system of linear equations as a single matrix equation in a vector variable.
• CCSS.Math.Content.HSA-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 × 3 or greater)
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