Morris School District 31 Hazel Street Morristown, NJ 07960

Morris School District Linear Algebra Curriculum

Dr. Thomas Ficarra, Superintendent Date: September, 2012

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Course Rationale and Philosophical Tenets: In order to prepare for global competition and high expectations, Morris School District students must have increased opportunities for mathematical experiences that extend critical thinking and reasoning. Specifically, access to higher mathematics is essential. Linear Algebra is one of three courses essential to higher mathematics in college: Linear Algebra, Differential Equations, and Multivariable Calculus. Linear Algebra is a highly applicable field in mathematics that is useful in mathematics, engineering, chemistry, physics, biology, economics, and computer science. Students will build an understanding of vector spaces and subspaces, solving large systems of equations, and connecting geometric and algebraic interpretations of complex problems to further their ability to reason abstractly and generalize when appropriate. In teaching and learning Linear Algebra, it is important for teachers and students to comprehend the following Big Ideas and Enduring Understandings and to establish connections and applications of the individual skills and concepts to these broad principles as the critical goals and objectives of the course for each unit: ● ● ● ● ● ●

Matrices and Linear Systems Vector Spaces Linear Transformations Determinants Eigenvectors and Eigenvalues Applications of Linear Algebra

Additionally, there are applicable state and national standards with which to conform.

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COURSE: Linear Algebra Time Frame

MSD CURRICULUM MAP

GRADE LEVEL(S): 11-12

Observable Proficiencies/Skills Content/Topic

Performance Benchmarks/Assessments

NJCCS

Materials Used

Unit 1: Matrices and Linear Systems (8-10 class periods) 2-3 class periods

Matrix Computations: How are matrix operations similar and different from operations over the real numbers?

• Students will use proper matrix notation and correctly compute sums and products of matrices and scalars. • Students will provide examples of when the properties of matrix computation differ from the real number system.

N-VM.7. N-VM.8. N-VM.9.

Assessment of fluency through exercises on homework as well as traditional quizzes and proofs (prose or symbolic). Students will also be asked to verbally explain how they multiply matrices.

Anton text Worksheets Traditional pencil and paper assessments

Formative assessment of problems presented at the board. Students will be graded on showing steps as well as the accuracy of work. In addition, homework and quiz problems will be assigned and evaluated. Assessment of fluency through exercises on homework as well as

Strang text Anton text TI-89 Calculator

N-VM.10. N-VM.11.

2 class periods

2-3 class periods +

Using Matrices to Solve Systems of Linear Equations: How can matrices help us solve many equations of many variables at once?

• Students will write systems of equations as a matrix equation. •

Solving Matrix Equations: Students will use properties of inverses to solve

• Students will be able to verify through multiplication that two matrices are

A-REI.8. N-VM.6.

Students will be able to show their work using step by step Gaussian elimination to solve systems of linear equations and write the solution in parametric form.

A-REI.9.

TI 89 Calculator Paper and pencil

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2 class periods synthesis and unit assessment

matrix equations of the form Ax=b.

inverses of each other. A-REI.1. • Students will be able to show their work and compute inverses of square matrices by using elementary matrices. • Students will correctly notate the steps to solving a matrix equation.

traditional quizzes. Students will be quiz graded on showing steps as well as Anton text the accuracy of work. Worksheet

Unit 2: Vector Spaces (14-16 class periods) 2 class periods

Vector Computations: What are the properties of vectors, which make up matrices?

• Students will represent vector addition, vector subtraction, and scalar multiplication by drawing 2 dimensional vector quantities. • Students will numerically calculate the sum, difference, inner product and norm of vectors of 2 or 3 dimensions • Students will calculate the angle between two given vectors.

N-VM.1. N-VM.2. N-VM.3. N-VM.4.

Students will display graphical work on the board or document camera for assessment. Homework problems will be assigned for numerical and symbolic representation and will be graded on accuracy.

Rulers Graph paper Strang text

For homework, matrix notation must denote specific combinations of vectors to illustrate linear dependence. Students will show all steps in determining if vectors are linearly dependent. Justification for solutions and solution space of a particular equation will be evaluated for logic, clarity, and mathematical accuracy through presentations at the board as well as on homework or extended test

Strang, text Anton text Traditional paper and pencil assessments TI 89 Calculator

N-VM.5. N-VM.11.

4-5 class periods

Solving Ax=b: Does the system of equations have solutions? What are all solutions that can be satisfied for a given coefficient matrix A?

• Students will demonstrate when one vector is a combination of other vectors in a collection by identifying the combination. • Students will be able to determine if a collection of vectors is linearly independent. • Students will use transformations and show their work to put matrices in row echelon form. • Students will describe all possible combinations using either symbolic

A-REI.1. A-REI.9.

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notation or prose. • Students will determine when a matrix equation has no solutions, infinitely many solutions, or a solution by identifying the rank of a matrix and how it affects the solution space of a matrix equation. • Given a consistent equation of the form Ax = b, students will be able to describe possible values for b as a linear combination of the columns of A.

problems.

2-3 class period(s)

Vector Spaces: What makes a space (or system) complete? Do vector spaces have all of the attributes of the real number system?

• Students will be list and test the properties of a vector space and subspace. • Given a subset of a given vector space V, students will explain why it is or is not a subspace of V. • Students will give examples of vector spaces.

Students will write all properties Worksheet from memory using correct Strang text mathematical notation during an inclass quiz. Students will correctly identify and justify whether a collection of vectors and operations is a subspace on homework and traditional quizzes.

4 class periods + 2 class periods synthesis and unit assessment

Spanning and Basis: What is necessary and sufficient to generate a specific vector space?

• Given a set S of vectors from a vector space V, students will explain why S is or is not a basis for V. • Given a particular vector space V, students will be able to find a basis for V. • Students will explain why linearly dependent vectors do not affect the dimension of a space. • Students will determine the dimension of the solution space for Ax=b.

Justification for why vectors do or Strang text do not constitute a basis for a vector space will be evaluated for logic and mathematical accuracy. Homework and traditional quizzes will be used to assess ability to find a basis for a vector space.

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Unit 3: Linear Transformations (11 class periods) 5 class periods

Functions: What kind of rules transform one infinite or finite set into another?

• Students will describe functions as N-VM.11. injective, surjective, or bijective. • Students will provide examples of linear N-VM.12. transformations. • Given a mapping f from a vector space V to another vector space W, students will explain why f is or is not a linear transformation. • Students will provide examples of rotation and translation matrices in Rn. • Students will compose linear transformations and represent them as one matrix. • Given a vector x in a vector space V with basis U, find the coordinates of x with respect to a different basis W. • Given a vector space V with bases U and W, write the matrix for change of basis from U to W and vice versa as a linear transformation.

Justifications will be evaluated for Strang text logic, clarity, and mathematical Mathematica accuracy. Homework and software traditional quizzes will be used daily to assess procedural capabilities. Students will be asked to verbally explain solution methods and justifications for the class.

4 class periods + 2 class periods synthesis and unit assessment

Vector Spaces from Transformations: What are the connections between the vector spaces of a linear transformation?

• Students will identify the domain, range, image, and inverse image for a given linear transformation. • Given a linear transformation L from a vector space V to a vector space W, find the range of L and kernel of L. • State the Rank Nullity Theorem and provide an example to illustrate it.

Formative assessment of problems presented at the board. Students will be graded on showing steps as well as the accuracy of work. In addition, homework and quiz problems will be assigned and evaluated.

Strang text

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Unit 4: Determinants (11-12 class periods) 5 class periods

Finding Derivatives: How can we find a determinant? What uses does a determinant have?

• Students will compute the determinant of a 2x2 and 3x3 matrix using the formal definition as well as faster algorithms. • Students will compute the determinant of a square matrix of n x n size. • Students will be able to solve a system of linear equations using determinants and Cramer’s Rule. • Students will compute area and volume by using determinants.

N-VM.12.

Formative assessment made of problems presented at the board. Students will be graded on showing steps as well as the accuracy of work. Homework assignments will focus on applications of matrices and determinants after the computational skill of finding the determinant of a matrix.

Strang text Anton text TI 89 Calculator Mathematica software

4-5 class periods + 2 class periods synthesis and unit assessment

Properties of Determinants: What information does the existence of a non-zero determinant tell us about a matrix or system of equations?

• Given a matrix and its determinant, students will be able to solve for the determinant of the inverse and transpose matrix. •Students will be able to illustrate with examples how a nonzero determinant is equivalent to having independent columns and guarantees an inverse. • Similarly, students will be able to illustrate with examples how a zero determinant is equivalent to having dependent columns and no inverse.

N-VM.10.

Assessment of fluency will be made through exercises on homework as well as traditional timed quizzes and proofs (prose or symbolic). Students will also be asked to generate new examples during timed and untimed assessments (homework and traditional quizzes) to illustrate concepts practiced in class.

Strang text TI 89 Calculator Mathematica software

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3-4 class periods

Introduction to Eigenvalues: What are eigenvalues and how do we find them? How can eigenvalues help us solve differential equations?

4 class periods + 2 class periods synthesis and unit assessment

Properties and Applications of Eigenvalues: If we can find one eigenvalue, what good are all of the eigenvectors? How can diagonalization help us simplify matrices?

Unit 5: Eigenvectors and Eigenvalues (9-10 class periods) • Students will state the definition of an Formative assessment of problems eigenvalue and eigenvector. presented at the board. Students • Students will find the characteristic will be graded on showing steps as equation and compute the eigenvalues and well as the accuracy of work. In eigenvectors of a square matrix. addition, homework and quiz problems will be assigned and evaluated. • Students will show with symbols and Justification for finding eigenvalues illustration in the 2 or 3 dimensional plane and eigenvectors of a particular that multiplying an eigenvector by a equation will be evaluated for logic, matrix A results in a vector parallel to the clarity, and mathematical accuracy original vector. through presentations at the board • Students will describe the eigenspaces of as well as on homework or a matrix and show that this space is a extended test problems. vector space. • Students will define and illustrate with examples a diagonalized matrix. • Students will explain and show step by step how to diagonalize a matrix, as well as explain when it is or is not possible.

Strang text Mathematica software

Strang text TI 89 Calculator Mathematica software

Unit 6: Applications of Linear Algebra (variable class periods) Note: Lessons from this unit may be presented as applicable throughout the course at the discretion of the teacher and available speakers. • Students will identify applications of Students will write brief reports on Available speakers Who Studies Linear Algebra? linear algebra in various fields of study. each speaker and how he or she from various fields • Students will prepare questions for and uses mathematics in his or her field. record responses from professionals in Additionally, students will correctly various fields to understand the use vocabulary to describe mathematics used for entrance into the applications of linear algebra field as well as daily use by the referenced by the presenter. professional.

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Specific Applications of Linear Algebra

• Students will identify how transformations can be used in various artistic applications. • Students will demonstrate how to use transformations for image compression. • Students will encrypt and decode simple messages by hand using an encoding matrix.

Personal Applications of Linear Algebra

• Students will find applications of linear algebra relevant to a field they are interested in. • Students will demonstrate which concepts of linear algebra are necessary to solve problems in a chosen field.

Students will show step by step work of transformations on particular matrices. Additionally, students will visually illustrate and verbally describe the effects of transformations for art. Students will prepare a portfolio of problems which have been solved with linear algebra. Students will present on personally relevant applications of linear algebra. Presentations will be assessed on the detail of problems addressed as well as the accuracy and applicability of linear algebra methods. Problems presented may also be submitted with portfolio.

Internet search engines Laptops Mathematica

Mathematica Powerpoint

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Appendix: The Common Core Mathematical Practices The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and their connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report “Adding It Up:” adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy). The Common Core Mathematical Practice Standards are: 1. Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 2. Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. 3. Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases and can recognize and use counterexamples. They justify

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their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. 4. Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. Engineers will often minimize volume of material in order to reduce cost. In addition concepts of volume can be applied to real world situations like the consumption of a volume of material per unit time. 5. Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. 6. Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. Initially, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. 7. Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Students will see 7 × 8 equals the well-remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, algebra students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. 8. Look for and express regularity in repeated reasoning.

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Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, algebra students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

Though Linear Algebra is a college level course, the foundations of Linear Algebra are covered in the Common Core. Most notably, students in Linear Algebra must be able to perform operations with vectors and matrices, express situations with symbols, in this case with matrices, and understand solving equations as a process of reasoning and be able to explain the reasoning, though not with equations of one variable, but with equations involving matrices. High School: Number & Quantity: Vector & Matrix Quantities Represent and model with vector quantities. 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). N-VM.2. (+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point. N-VM.3. (+) Solve problems involving velocity and other quantities that can be represented by vectors. Perform operations on vectors. N-VM.4. (+) Add and subtract vectors. o 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 Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum. o 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. N-VM.5. (+) Multiply a vector by a scalar. o 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 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. N-VM.6. (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network. N-VM.7. (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled. N-VM.8. (+) Add, subtract, and multiply matrices of appropriate dimensions. 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.

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

High School: Algebra: Reasoning with Equations & Inequalities Understand solving equations as a process of reasoning and explain the reasoning. A-REI.1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Solve systems of equations. A-REI.8. (+) Represent a system of linear equations as a single matrix equation in a vector variable. A-REI.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).

MASTERY OBJECTIVES (NJCCCS) All courses of study must include the following, which replace the Workplace readiness standards: Career Education and Consumer, Family, and Life Skills Career and Technical Education: All students will develop career awareness and planning, employability skills, and foundational knowledge necessary for success in the workplace. Consumer, Family, and Life Skills: All students will demonstrate critical life skills in order to be functional members of society. Scans Workplace Competencies Effective workers can productively use: Resources: They know how to allocate time, money, materials, space and staff. Interpersonal Skills: They can work on teams, teach others, serve customers, lead, negotiate, and work well with people from culturally diverse backgrounds. Information: They can acquire and evaluate data, organize and maintain files, interpret and communicate, and use computers to process information. Systems: They understand social, organizational, and technological systems; they can monitor and correct performance; and they can design or improve systems. Technology: They can select equipment and tools, apply technology to specific tasks, and maintain and troubleshoot equipment. SCANS Foundations Skills Competent workers in the high-performance workplace need: Basic Skills: reading, writing, arithmetic, and mathematics, speaking and listening. Thinking Skills – the ability to learn, reason, think creatively, make decisions, and solve problems. Personal Qualities – individual responsibility self-esteem and self-management, sociability, integrity, and honesty.

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Units of Study Student Outcomes: Unit 1: Matrices and Linear Systems Matrix operations and properties; Special matrices; Linear systems of equations; The inverse matrix; LU-decompositions of matrices Students will be able to use and understand matrix notation, addition, scalar multiplication, matrix multiplication, and matrix transposition. Students will be able to compute with matrices and describe when these properties differ from the real number system. Students will be able to use Gaussian elimination to solve systems of linear equations and write the solution in parametric form. Students will be able to compute inverses of square matrices using elementary matrices. Students will be able to convert systems of equations to a matrix equation. Unit 2: Vector Spaces Operations and properties of vector spaces; Vectors; Subspaces; Linear independence; Basis and Dimension; Row space; Column space; Rank; Rank-Nullity Theorem; Students will be able to use and understand vector notation, addition, the dot product, and norm of a vector. Students will be able to state the Cauchy-Schwarz theorem. Students will be able to normalize vectors, obtain vectors of a given length in a given direction, and explain how to tell if two vectors are orthogonal. Students will be able to determine if a collection of vectors is linearly independent. Students will be able to show when vectors are linear combinations of one another. Given a consistent equation of the form Ax = b, students will be able to describe b as a linear combination of the columns of A. Students will be able to list and test the properties of a vector space and subspace. Given a subset of a given vector space V, explain why it is or is not a subspace of V. Give examples of vector spaces. Explain the defining properties of a basis for a vector space. Given a particular vector space V, students will be able to find a basis for V. Students will be able to identify the rank of a matrix and how it affects the solution space of a matrix equation. Given a set S of vectors from a vector space V, explain why S is or is not a basis for V. Unit 3: Linear Transformations Functions; Linear transformations; Matrix representations; Change of basis; Properties of linear transformations; Kernel and image Students will know the vocabulary of functions: domain, range, image, inverse image, kernel, injective (one-to-one), surjective (onto), and bijective. Students will be able to state the definition of a linear transformation L from a vector space V to another vector space W. Students will be able to give examples of linear transformations. Given a mapping f from a vector space V to another vector space W, explain why f is or is not a linear transformation. Given a vector x in a vector space V with basis U, find the coordinates of x with respect to a different basis W.

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Given a vector space V with bases U and W, write the matrix for change of basis from U to W and vice versa as a linear transformation. Given a linear transformation L from a vector space V to a vector space W, find the range of L, kernel of L. State the definition of similarity of square matrices. Give examples of similar matrices.

Unit 4: Determinants Computation and properties of determinants; Minors and cofactors; Geometry and determinants; Products, inverses, transposes, and determinants Students will be able to compute the determinant of a square matrix of n x n size. Students will be able to illustrate with examples how a nonzero determinant is equivalent to having independent columns and guarantees an inverse. Similarly, students will be able to illustrate with examples how a zero determinant is equivalent to having dependent columns and no inverse. Unit 5: Eigenvectors and Eigenvalues Definitions and examples of eigenvectors and eigenvalues; Computational methods for finding eigenvectors and eigenvalues; Properties of eigenvectors and eigenvalues; Diagonalization of matrices

Students will be able to state the definition of an eigenvalue and eigenvector. Students will be able to find its characteristic equation and compute the eigenvalues and eigenvectors of a square matrix. Students will be able to describe the eigenspaces of a matrix and show that this space is a vector space. Students will be able to explain how to diagonalize a matrix, as well as explain when it is or is not possible.

Unit 6: Applications of Linear Algebra. Students will find applications of linear algebra in other fields such as computer science, economics, biology, or engineering. Students will demonstrate which concepts of linear algebra are necessary to solve problems in these other fields.

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Assessment and Testing Strategies Sound and productive classroom assessments are built on a foundation of the following five key dimensions (Stiggins et al, 2006): Key 1: Clear and appropriate purpose. Did the teacher specify users and uses, and are these appropriate? Key 2: Valued achievement targets. Has the teacher clearly specified the achievement targets to be reflected in the exercises? Do these represent important learning outcomes? Key 3: Design. Does the selection of the method make sense given the goals and purposes? Is there anything in the assessment that might lead to misleading results? Key 4: Communication. Is it clear how this assessment helps communication with others about student achievement? Key 5: Student Involvement. Is it clear how students are involved in the assessment as a way to help them understand achievement targets, practice hitting those targets, see themselves growing in their achievement, and communicate with others about their success as learners? The Linear Algebra course will include a variety of assessment tools for the effective teaching and learning of mathematics. Indicators of Sound Classroom Assessment Practice will consist of both formative and summative assessments that may include, but are not limited to: ● Observation ● Interviews ● Paper-and-pencil tests/quizzes ● Performance Tasks ● Journals/Self-Reflections

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Texts and Materials Student Text: Strang, G. (2009). Introduction to Linear Algebra. 4th ed. Wellesley, Massachusetts: Wellesley- Cambridge Press. Teacher Materials and Resources: ○ Anton, H. and Chris Rorres. (2000). Elementary Linear Algebra. 8th ed. John Wiley & Sons, Inc. ○ Cullen, C. (1990). Matrices and Linear Transformations. 2nd ed. New York: Dover Publications. ○ Greenes, C., & Rubenstein, R. (2008). Algebra and Algebraic Thinking in School Mathematics, Seventieth Yearbook. Reston, VA: National Council of Teachers of Mathematics. ○ Massachusetts Institute of Technology. (2010). MITOpenCourseWare: Linear Algebra. ○ http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/ ○ National Council of Teachers of Mathematics. (2006). Navigating Through Mathematical Connections in Grades 9-12. Reston, VA: NCTM. Technology/Computer Software ○ Geometer’s Sketchpad: Key Curriculum Press ○ Fathom: Key Curriculum Press ○ Mathematica: Wolfram Research ○ TI-89 Calculators & TI-Nspire Calculators ○

REFERENCES: Anton, H. and Chris Rorres. (2000). Elementary Linear Algebra. 8th ed. John Wiley & Sons, Inc. Cullen, C. (1990). Matrices and Linear Transformations. 2nd ed. New York: Dover Publications. Daggett, Bill and McNulty, Ray. Rigor and Relevance Framework: International Center for Leadership in Education (www.leadered.com). Greenes, C., & Rubenstein, R. (2008). Algebra and Algebraic Thinking in School Mathematics, Seventieth Yearbook. Reston, VA: National Council of Teachers of Mathematics. Killion, Joellen P. (2008). Collaborative Professional Learning in School and Beyond: A Toolkit for New Jersey Educators. Trenton, NJ: New Jersey Department of Education, the New Jersey Professional Teaching Standards Board, and the National Development Council. Massachusetts Institute of Technology. (2010). MITOpenCourseWare: Linear Algebra. http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/ National Council of Teachers of Mathematics (2009). Focus in High School Mathematics: Reasoning and Sense Making. Reston, VA: National Council of Teachers of Mathematics. National Council of Teachers of Mathematics (2000). Principles and Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics. New Jersey Department of Education (2010). New Jersey Common Core Curriculum Standards 2010. www.njcccs.org. Stiggins, Rick, Arter, Judith, Chappuis, Jan, and Chappuis, Steve (2006). Classroom Assessment for Student LearningSUPPLEMENTARY MATERIAL. Portland, OR: Educational Testing Service. The Secretary’s Commission on Achieving Necessary Skills. (1992). Learning a Living: A Blueprint for High Performance. A SCANS Report for America 2000. Executive Summary. Washington, DC: U.S. Department of Labor.

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Morris School District Linear Algebra Curriculum

Dr. Thomas Ficarra, Superintendent Date: September, 2012

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Course Rationale and Philosophical Tenets: In order to prepare for global competition and high expectations, Morris School District students must have increased opportunities for mathematical experiences that extend critical thinking and reasoning. Specifically, access to higher mathematics is essential. Linear Algebra is one of three courses essential to higher mathematics in college: Linear Algebra, Differential Equations, and Multivariable Calculus. Linear Algebra is a highly applicable field in mathematics that is useful in mathematics, engineering, chemistry, physics, biology, economics, and computer science. Students will build an understanding of vector spaces and subspaces, solving large systems of equations, and connecting geometric and algebraic interpretations of complex problems to further their ability to reason abstractly and generalize when appropriate. In teaching and learning Linear Algebra, it is important for teachers and students to comprehend the following Big Ideas and Enduring Understandings and to establish connections and applications of the individual skills and concepts to these broad principles as the critical goals and objectives of the course for each unit: ● ● ● ● ● ●

Matrices and Linear Systems Vector Spaces Linear Transformations Determinants Eigenvectors and Eigenvalues Applications of Linear Algebra

Additionally, there are applicable state and national standards with which to conform.

2

COURSE: Linear Algebra Time Frame

MSD CURRICULUM MAP

GRADE LEVEL(S): 11-12

Observable Proficiencies/Skills Content/Topic

Performance Benchmarks/Assessments

NJCCS

Materials Used

Unit 1: Matrices and Linear Systems (8-10 class periods) 2-3 class periods

Matrix Computations: How are matrix operations similar and different from operations over the real numbers?

• Students will use proper matrix notation and correctly compute sums and products of matrices and scalars. • Students will provide examples of when the properties of matrix computation differ from the real number system.

N-VM.7. N-VM.8. N-VM.9.

Assessment of fluency through exercises on homework as well as traditional quizzes and proofs (prose or symbolic). Students will also be asked to verbally explain how they multiply matrices.

Anton text Worksheets Traditional pencil and paper assessments

Formative assessment of problems presented at the board. Students will be graded on showing steps as well as the accuracy of work. In addition, homework and quiz problems will be assigned and evaluated. Assessment of fluency through exercises on homework as well as

Strang text Anton text TI-89 Calculator

N-VM.10. N-VM.11.

2 class periods

2-3 class periods +

Using Matrices to Solve Systems of Linear Equations: How can matrices help us solve many equations of many variables at once?

• Students will write systems of equations as a matrix equation. •

Solving Matrix Equations: Students will use properties of inverses to solve

• Students will be able to verify through multiplication that two matrices are

A-REI.8. N-VM.6.

Students will be able to show their work using step by step Gaussian elimination to solve systems of linear equations and write the solution in parametric form.

A-REI.9.

TI 89 Calculator Paper and pencil

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2 class periods synthesis and unit assessment

matrix equations of the form Ax=b.

inverses of each other. A-REI.1. • Students will be able to show their work and compute inverses of square matrices by using elementary matrices. • Students will correctly notate the steps to solving a matrix equation.

traditional quizzes. Students will be quiz graded on showing steps as well as Anton text the accuracy of work. Worksheet

Unit 2: Vector Spaces (14-16 class periods) 2 class periods

Vector Computations: What are the properties of vectors, which make up matrices?

• Students will represent vector addition, vector subtraction, and scalar multiplication by drawing 2 dimensional vector quantities. • Students will numerically calculate the sum, difference, inner product and norm of vectors of 2 or 3 dimensions • Students will calculate the angle between two given vectors.

N-VM.1. N-VM.2. N-VM.3. N-VM.4.

Students will display graphical work on the board or document camera for assessment. Homework problems will be assigned for numerical and symbolic representation and will be graded on accuracy.

Rulers Graph paper Strang text

For homework, matrix notation must denote specific combinations of vectors to illustrate linear dependence. Students will show all steps in determining if vectors are linearly dependent. Justification for solutions and solution space of a particular equation will be evaluated for logic, clarity, and mathematical accuracy through presentations at the board as well as on homework or extended test

Strang, text Anton text Traditional paper and pencil assessments TI 89 Calculator

N-VM.5. N-VM.11.

4-5 class periods

Solving Ax=b: Does the system of equations have solutions? What are all solutions that can be satisfied for a given coefficient matrix A?

• Students will demonstrate when one vector is a combination of other vectors in a collection by identifying the combination. • Students will be able to determine if a collection of vectors is linearly independent. • Students will use transformations and show their work to put matrices in row echelon form. • Students will describe all possible combinations using either symbolic

A-REI.1. A-REI.9.

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notation or prose. • Students will determine when a matrix equation has no solutions, infinitely many solutions, or a solution by identifying the rank of a matrix and how it affects the solution space of a matrix equation. • Given a consistent equation of the form Ax = b, students will be able to describe possible values for b as a linear combination of the columns of A.

problems.

2-3 class period(s)

Vector Spaces: What makes a space (or system) complete? Do vector spaces have all of the attributes of the real number system?

• Students will be list and test the properties of a vector space and subspace. • Given a subset of a given vector space V, students will explain why it is or is not a subspace of V. • Students will give examples of vector spaces.

Students will write all properties Worksheet from memory using correct Strang text mathematical notation during an inclass quiz. Students will correctly identify and justify whether a collection of vectors and operations is a subspace on homework and traditional quizzes.

4 class periods + 2 class periods synthesis and unit assessment

Spanning and Basis: What is necessary and sufficient to generate a specific vector space?

• Given a set S of vectors from a vector space V, students will explain why S is or is not a basis for V. • Given a particular vector space V, students will be able to find a basis for V. • Students will explain why linearly dependent vectors do not affect the dimension of a space. • Students will determine the dimension of the solution space for Ax=b.

Justification for why vectors do or Strang text do not constitute a basis for a vector space will be evaluated for logic and mathematical accuracy. Homework and traditional quizzes will be used to assess ability to find a basis for a vector space.

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Unit 3: Linear Transformations (11 class periods) 5 class periods

Functions: What kind of rules transform one infinite or finite set into another?

• Students will describe functions as N-VM.11. injective, surjective, or bijective. • Students will provide examples of linear N-VM.12. transformations. • Given a mapping f from a vector space V to another vector space W, students will explain why f is or is not a linear transformation. • Students will provide examples of rotation and translation matrices in Rn. • Students will compose linear transformations and represent them as one matrix. • Given a vector x in a vector space V with basis U, find the coordinates of x with respect to a different basis W. • Given a vector space V with bases U and W, write the matrix for change of basis from U to W and vice versa as a linear transformation.

Justifications will be evaluated for Strang text logic, clarity, and mathematical Mathematica accuracy. Homework and software traditional quizzes will be used daily to assess procedural capabilities. Students will be asked to verbally explain solution methods and justifications for the class.

4 class periods + 2 class periods synthesis and unit assessment

Vector Spaces from Transformations: What are the connections between the vector spaces of a linear transformation?

• Students will identify the domain, range, image, and inverse image for a given linear transformation. • Given a linear transformation L from a vector space V to a vector space W, find the range of L and kernel of L. • State the Rank Nullity Theorem and provide an example to illustrate it.

Formative assessment of problems presented at the board. Students will be graded on showing steps as well as the accuracy of work. In addition, homework and quiz problems will be assigned and evaluated.

Strang text

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Unit 4: Determinants (11-12 class periods) 5 class periods

Finding Derivatives: How can we find a determinant? What uses does a determinant have?

• Students will compute the determinant of a 2x2 and 3x3 matrix using the formal definition as well as faster algorithms. • Students will compute the determinant of a square matrix of n x n size. • Students will be able to solve a system of linear equations using determinants and Cramer’s Rule. • Students will compute area and volume by using determinants.

N-VM.12.

Formative assessment made of problems presented at the board. Students will be graded on showing steps as well as the accuracy of work. Homework assignments will focus on applications of matrices and determinants after the computational skill of finding the determinant of a matrix.

Strang text Anton text TI 89 Calculator Mathematica software

4-5 class periods + 2 class periods synthesis and unit assessment

Properties of Determinants: What information does the existence of a non-zero determinant tell us about a matrix or system of equations?

• Given a matrix and its determinant, students will be able to solve for the determinant of the inverse and transpose matrix. •Students will be able to illustrate with examples how a nonzero determinant is equivalent to having independent columns and guarantees an inverse. • Similarly, students will be able to illustrate with examples how a zero determinant is equivalent to having dependent columns and no inverse.

N-VM.10.

Assessment of fluency will be made through exercises on homework as well as traditional timed quizzes and proofs (prose or symbolic). Students will also be asked to generate new examples during timed and untimed assessments (homework and traditional quizzes) to illustrate concepts practiced in class.

Strang text TI 89 Calculator Mathematica software

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3-4 class periods

Introduction to Eigenvalues: What are eigenvalues and how do we find them? How can eigenvalues help us solve differential equations?

4 class periods + 2 class periods synthesis and unit assessment

Properties and Applications of Eigenvalues: If we can find one eigenvalue, what good are all of the eigenvectors? How can diagonalization help us simplify matrices?

Unit 5: Eigenvectors and Eigenvalues (9-10 class periods) • Students will state the definition of an Formative assessment of problems eigenvalue and eigenvector. presented at the board. Students • Students will find the characteristic will be graded on showing steps as equation and compute the eigenvalues and well as the accuracy of work. In eigenvectors of a square matrix. addition, homework and quiz problems will be assigned and evaluated. • Students will show with symbols and Justification for finding eigenvalues illustration in the 2 or 3 dimensional plane and eigenvectors of a particular that multiplying an eigenvector by a equation will be evaluated for logic, matrix A results in a vector parallel to the clarity, and mathematical accuracy original vector. through presentations at the board • Students will describe the eigenspaces of as well as on homework or a matrix and show that this space is a extended test problems. vector space. • Students will define and illustrate with examples a diagonalized matrix. • Students will explain and show step by step how to diagonalize a matrix, as well as explain when it is or is not possible.

Strang text Mathematica software

Strang text TI 89 Calculator Mathematica software

Unit 6: Applications of Linear Algebra (variable class periods) Note: Lessons from this unit may be presented as applicable throughout the course at the discretion of the teacher and available speakers. • Students will identify applications of Students will write brief reports on Available speakers Who Studies Linear Algebra? linear algebra in various fields of study. each speaker and how he or she from various fields • Students will prepare questions for and uses mathematics in his or her field. record responses from professionals in Additionally, students will correctly various fields to understand the use vocabulary to describe mathematics used for entrance into the applications of linear algebra field as well as daily use by the referenced by the presenter. professional.

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Specific Applications of Linear Algebra

• Students will identify how transformations can be used in various artistic applications. • Students will demonstrate how to use transformations for image compression. • Students will encrypt and decode simple messages by hand using an encoding matrix.

Personal Applications of Linear Algebra

• Students will find applications of linear algebra relevant to a field they are interested in. • Students will demonstrate which concepts of linear algebra are necessary to solve problems in a chosen field.

Students will show step by step work of transformations on particular matrices. Additionally, students will visually illustrate and verbally describe the effects of transformations for art. Students will prepare a portfolio of problems which have been solved with linear algebra. Students will present on personally relevant applications of linear algebra. Presentations will be assessed on the detail of problems addressed as well as the accuracy and applicability of linear algebra methods. Problems presented may also be submitted with portfolio.

Internet search engines Laptops Mathematica

Mathematica Powerpoint

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Appendix: The Common Core Mathematical Practices The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and their connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report “Adding It Up:” adaptive reasoning, strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy). The Common Core Mathematical Practice Standards are: 1. Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 2. Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. 3. Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases and can recognize and use counterexamples. They justify

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their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. 4. Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. Engineers will often minimize volume of material in order to reduce cost. In addition concepts of volume can be applied to real world situations like the consumption of a volume of material per unit time. 5. Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. 6. Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. Initially, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. 7. Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Students will see 7 × 8 equals the well-remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, algebra students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. 8. Look for and express regularity in repeated reasoning.

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Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, algebra students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

Though Linear Algebra is a college level course, the foundations of Linear Algebra are covered in the Common Core. Most notably, students in Linear Algebra must be able to perform operations with vectors and matrices, express situations with symbols, in this case with matrices, and understand solving equations as a process of reasoning and be able to explain the reasoning, though not with equations of one variable, but with equations involving matrices. High School: Number & Quantity: Vector & Matrix Quantities Represent and model with vector quantities. 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). N-VM.2. (+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point. N-VM.3. (+) Solve problems involving velocity and other quantities that can be represented by vectors. Perform operations on vectors. N-VM.4. (+) Add and subtract vectors. o 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 Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum. o 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. N-VM.5. (+) Multiply a vector by a scalar. o 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 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. N-VM.6. (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network. N-VM.7. (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled. N-VM.8. (+) Add, subtract, and multiply matrices of appropriate dimensions. 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.

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

High School: Algebra: Reasoning with Equations & Inequalities Understand solving equations as a process of reasoning and explain the reasoning. A-REI.1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Solve systems of equations. A-REI.8. (+) Represent a system of linear equations as a single matrix equation in a vector variable. A-REI.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).

MASTERY OBJECTIVES (NJCCCS) All courses of study must include the following, which replace the Workplace readiness standards: Career Education and Consumer, Family, and Life Skills Career and Technical Education: All students will develop career awareness and planning, employability skills, and foundational knowledge necessary for success in the workplace. Consumer, Family, and Life Skills: All students will demonstrate critical life skills in order to be functional members of society. Scans Workplace Competencies Effective workers can productively use: Resources: They know how to allocate time, money, materials, space and staff. Interpersonal Skills: They can work on teams, teach others, serve customers, lead, negotiate, and work well with people from culturally diverse backgrounds. Information: They can acquire and evaluate data, organize and maintain files, interpret and communicate, and use computers to process information. Systems: They understand social, organizational, and technological systems; they can monitor and correct performance; and they can design or improve systems. Technology: They can select equipment and tools, apply technology to specific tasks, and maintain and troubleshoot equipment. SCANS Foundations Skills Competent workers in the high-performance workplace need: Basic Skills: reading, writing, arithmetic, and mathematics, speaking and listening. Thinking Skills – the ability to learn, reason, think creatively, make decisions, and solve problems. Personal Qualities – individual responsibility self-esteem and self-management, sociability, integrity, and honesty.

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Units of Study Student Outcomes: Unit 1: Matrices and Linear Systems Matrix operations and properties; Special matrices; Linear systems of equations; The inverse matrix; LU-decompositions of matrices Students will be able to use and understand matrix notation, addition, scalar multiplication, matrix multiplication, and matrix transposition. Students will be able to compute with matrices and describe when these properties differ from the real number system. Students will be able to use Gaussian elimination to solve systems of linear equations and write the solution in parametric form. Students will be able to compute inverses of square matrices using elementary matrices. Students will be able to convert systems of equations to a matrix equation. Unit 2: Vector Spaces Operations and properties of vector spaces; Vectors; Subspaces; Linear independence; Basis and Dimension; Row space; Column space; Rank; Rank-Nullity Theorem; Students will be able to use and understand vector notation, addition, the dot product, and norm of a vector. Students will be able to state the Cauchy-Schwarz theorem. Students will be able to normalize vectors, obtain vectors of a given length in a given direction, and explain how to tell if two vectors are orthogonal. Students will be able to determine if a collection of vectors is linearly independent. Students will be able to show when vectors are linear combinations of one another. Given a consistent equation of the form Ax = b, students will be able to describe b as a linear combination of the columns of A. Students will be able to list and test the properties of a vector space and subspace. Given a subset of a given vector space V, explain why it is or is not a subspace of V. Give examples of vector spaces. Explain the defining properties of a basis for a vector space. Given a particular vector space V, students will be able to find a basis for V. Students will be able to identify the rank of a matrix and how it affects the solution space of a matrix equation. Given a set S of vectors from a vector space V, explain why S is or is not a basis for V. Unit 3: Linear Transformations Functions; Linear transformations; Matrix representations; Change of basis; Properties of linear transformations; Kernel and image Students will know the vocabulary of functions: domain, range, image, inverse image, kernel, injective (one-to-one), surjective (onto), and bijective. Students will be able to state the definition of a linear transformation L from a vector space V to another vector space W. Students will be able to give examples of linear transformations. Given a mapping f from a vector space V to another vector space W, explain why f is or is not a linear transformation. Given a vector x in a vector space V with basis U, find the coordinates of x with respect to a different basis W.

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Given a vector space V with bases U and W, write the matrix for change of basis from U to W and vice versa as a linear transformation. Given a linear transformation L from a vector space V to a vector space W, find the range of L, kernel of L. State the definition of similarity of square matrices. Give examples of similar matrices.

Unit 4: Determinants Computation and properties of determinants; Minors and cofactors; Geometry and determinants; Products, inverses, transposes, and determinants Students will be able to compute the determinant of a square matrix of n x n size. Students will be able to illustrate with examples how a nonzero determinant is equivalent to having independent columns and guarantees an inverse. Similarly, students will be able to illustrate with examples how a zero determinant is equivalent to having dependent columns and no inverse. Unit 5: Eigenvectors and Eigenvalues Definitions and examples of eigenvectors and eigenvalues; Computational methods for finding eigenvectors and eigenvalues; Properties of eigenvectors and eigenvalues; Diagonalization of matrices

Students will be able to state the definition of an eigenvalue and eigenvector. Students will be able to find its characteristic equation and compute the eigenvalues and eigenvectors of a square matrix. Students will be able to describe the eigenspaces of a matrix and show that this space is a vector space. Students will be able to explain how to diagonalize a matrix, as well as explain when it is or is not possible.

Unit 6: Applications of Linear Algebra. Students will find applications of linear algebra in other fields such as computer science, economics, biology, or engineering. Students will demonstrate which concepts of linear algebra are necessary to solve problems in these other fields.

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Assessment and Testing Strategies Sound and productive classroom assessments are built on a foundation of the following five key dimensions (Stiggins et al, 2006): Key 1: Clear and appropriate purpose. Did the teacher specify users and uses, and are these appropriate? Key 2: Valued achievement targets. Has the teacher clearly specified the achievement targets to be reflected in the exercises? Do these represent important learning outcomes? Key 3: Design. Does the selection of the method make sense given the goals and purposes? Is there anything in the assessment that might lead to misleading results? Key 4: Communication. Is it clear how this assessment helps communication with others about student achievement? Key 5: Student Involvement. Is it clear how students are involved in the assessment as a way to help them understand achievement targets, practice hitting those targets, see themselves growing in their achievement, and communicate with others about their success as learners? The Linear Algebra course will include a variety of assessment tools for the effective teaching and learning of mathematics. Indicators of Sound Classroom Assessment Practice will consist of both formative and summative assessments that may include, but are not limited to: ● Observation ● Interviews ● Paper-and-pencil tests/quizzes ● Performance Tasks ● Journals/Self-Reflections

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Texts and Materials Student Text: Strang, G. (2009). Introduction to Linear Algebra. 4th ed. Wellesley, Massachusetts: Wellesley- Cambridge Press. Teacher Materials and Resources: ○ Anton, H. and Chris Rorres. (2000). Elementary Linear Algebra. 8th ed. John Wiley & Sons, Inc. ○ Cullen, C. (1990). Matrices and Linear Transformations. 2nd ed. New York: Dover Publications. ○ Greenes, C., & Rubenstein, R. (2008). Algebra and Algebraic Thinking in School Mathematics, Seventieth Yearbook. Reston, VA: National Council of Teachers of Mathematics. ○ Massachusetts Institute of Technology. (2010). MITOpenCourseWare: Linear Algebra. ○ http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/ ○ National Council of Teachers of Mathematics. (2006). Navigating Through Mathematical Connections in Grades 9-12. Reston, VA: NCTM. Technology/Computer Software ○ Geometer’s Sketchpad: Key Curriculum Press ○ Fathom: Key Curriculum Press ○ Mathematica: Wolfram Research ○ TI-89 Calculators & TI-Nspire Calculators ○

REFERENCES: Anton, H. and Chris Rorres. (2000). Elementary Linear Algebra. 8th ed. John Wiley & Sons, Inc. Cullen, C. (1990). Matrices and Linear Transformations. 2nd ed. New York: Dover Publications. Daggett, Bill and McNulty, Ray. Rigor and Relevance Framework: International Center for Leadership in Education (www.leadered.com). Greenes, C., & Rubenstein, R. (2008). Algebra and Algebraic Thinking in School Mathematics, Seventieth Yearbook. Reston, VA: National Council of Teachers of Mathematics. Killion, Joellen P. (2008). Collaborative Professional Learning in School and Beyond: A Toolkit for New Jersey Educators. Trenton, NJ: New Jersey Department of Education, the New Jersey Professional Teaching Standards Board, and the National Development Council. Massachusetts Institute of Technology. (2010). MITOpenCourseWare: Linear Algebra. http://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/ National Council of Teachers of Mathematics (2009). Focus in High School Mathematics: Reasoning and Sense Making. Reston, VA: National Council of Teachers of Mathematics. National Council of Teachers of Mathematics (2000). Principles and Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics. New Jersey Department of Education (2010). New Jersey Common Core Curriculum Standards 2010. www.njcccs.org. Stiggins, Rick, Arter, Judith, Chappuis, Jan, and Chappuis, Steve (2006). Classroom Assessment for Student LearningSUPPLEMENTARY MATERIAL. Portland, OR: Educational Testing Service. The Secretary’s Commission on Achieving Necessary Skills. (1992). Learning a Living: A Blueprint for High Performance. A SCANS Report for America 2000. Executive Summary. Washington, DC: U.S. Department of Labor.

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