Arithmetic Operations With Matrices Overview Number of instruction days: 6 8 (1 day = 53 minutes) Content to Be Learned Use matrices to represent and manipulate data. Perform arithmetic operations with matrices. Conceptually understand zero and identity matrices. Multiply matrices for use with vectors and transformations of vectors. Find the determinant and use it to find the inverse of a square matrix. Understand when you can use commutative, associative, and distributive properties in matrix operations. Conceptually understand matrices and their uses as tools for solving various real-world applications. Mathematical Practices to Be Integrated 4 Model with mathematics. Use matrices to represent and solve real-world problems. 6 Attend to precision. Carefully consider the placement of elements when performing operations with matrices. 7 Look for and make use of structure. Express the determinant of a 3x3 matrix as the sum of the determinants of three 2x2 matrices multiplied by their respective cofactors. Essential Questions What does a matrix represent? How do you transform a vector by using matrices? Where are matrices used in the real world? What real number properties can be used with matrices? Providence Public Schools D-83
Arithmetic Operations with Matrices (6 8 days) Standards Common Core State Standards for Mathematical Content Number and Quantity Vector and Matrix Quantities N-VM 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. 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. Common Core State Standards for Mathematical Practice 4 Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. 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. In the elementary grades, students give carefully formulated D-84 Providence Public Schools
Arithmetic Operations with Matrices (6 8 days) Precalculus, Quarter 3, Unit 3.4 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. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 8 equals the well remembered 7 5 + 7 3, in preparation for learning about the distributive property. In the expression x 2 + 9x + 14, older 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. Clarifying the Standards Prior Learning Students have classified and used arithmetic properties since third grade. They have been using these properties as well as arithmetic operations since kindergarten. Adding, subtracting, multiplying, and dividing rational numbers is the culmination of numerical work with the four basic operations. Students continued to develop understanding of the number system in Grade 8; with the introduction of irrational numbers, they expanded their understanding to real numbers. With the introduction of imaginary numbers in high school, students understanding of the number system expanded to include complex numbers. Because there are no specific standards for rational number arithmetic in later grades, and because so much other work in Grade 7 depended on rational numbers, fluency with rational number arithmetic should have been mastered in Grade 7. Students have solved systems of equations since eighth grade. Current Learning Students use matrices to represent and manipulate data. They carry out operations such as addition, subtraction, scalar multiplication, and matrix multiplication. They investigate properties involving zero and identity matrices. They also find and use the determinant to get the inverse in order to solve realworld problems. Future Learning Students will use matrices in linear algebra. Cryptography makes use of matrices to write messages in code. Matrix multiplication can be used to encode and decode messages. Additional Findings There are no additional findings for this unit. Providence Public Schools D-85
Arithmetic Operations with Matrices (6 8 days) Assessment When constructing an end-of-unit assessment, be aware that the assessment should measure your students understanding of the big ideas indicated within the standards. The CCSS for Mathematical Content and the CCSS for Mathematical Practice should be considered when designing assessments. Standards-based mathematics assessment items should vary in difficulty, content, and type. The assessment should comprise a mix of items, which could include multiple choice items, short and extended response items, and performance-based tasks. When creating your assessment, you should be mindful when an item could be differentiated to address the needs of students in your class. The mathematical concepts below are not a prioritized list of assessment items, and your assessment is not limited to these concepts. However, care should be given to assess the skills the students have developed within this unit. The assessment should provide you with credible evidence as to your students attainment of the mathematics within the unit. Represent and manipulate data using matrices. Add, subtract, and multiply matrices to solve real world problems. Multiply matrices by a scalar. Understand and apply the associative and distributive properties to matrix operations. Identify the similarities and differences of the associative, distributive and commutative properties on matrices and real numbers. Use inverse and identify matrices to solve real world problems. Instruction Learning Objectives Students will be able to: Use matrices to represent and manipulate data. Add and subtract matrices. Perform scalar multiplication with matrices and find the product of two matrices. Evaluate determinants of square matrices, and use it to find the corresponding inverse matrix. Understand and apply zero and identity matrices. Review and demonstrate knowledge of important concepts and procedures related to matrices. D-86 Providence Public Schools
Arithmetic Operations with Matrices (6 8 days) Precalculus, Quarter 3, Unit 3.4 Resources Advanced Mathematical Concepts: Precalculus with Applications, Glencoe, 2006, Teacher Edition and Student Edition Section 2-3 (pp. 78-86) Section 2-5 (pp. 98-105) TeacherWorks All-In-One Planner and Resource Center CD-ROM Exam View Assessment Suite Note: The district resources may contain content that goes beyond the standards addressed in this unit. See the Planning for Effective Instructional Design and Delivery and Assessment sections for specific recommendations. Materials TI-Nspire graphing calculators Instructional Considerations Key Vocabulary array of terms column dimension element matrix row square matrix scalar multiplication Planning for Effective Instructional Design and Delivery Reinforced vocabulary taught in previous grades or units: scalar, sum, difference, and product. Have students use a nonlinguistic (kinesthetic) representations approach to discovery of the rows and columns of a matrix. Set the stage in your classroom by having desks or chairs in a row x column arrangement. Call out a position in the matrix and have students name who is in that place. This is an introduction, reinforcement, and mastery level for Precalculus students. Be sure to have students set up matrices using word problems involving organization of data and arithmetic operations between matrices. Students might have difficulty setting up matrices when solving problems. Make sure to emphasize the importance of both the dimensions and the labels of the rows and columns of matrices. When working with multiple matrices, have students find the common labels and line them up in the subsequent matrices to help them set up the matrices properly. Have students make the connection between the coordinates of the Cartesian coordinate system and the rows and columns of a matrix. The name of a point on a grid system is (x, y) or (horizontal, vertical). The name of the element of the matrix is (row, column) or (horizontal, vertical). Use graphing calculators to enhance performing all operations with matrices. Use real-world examples in solving systems of equations with matrices. Matrices are used in almost all businesses to find the breakeven point of cost and sale price of a product given many different variables. Be sure to have students set up matrices using word problems, for solving equations and systems of equations. Providence Public Schools D-87
Arithmetic Operations with Matrices (6 8 days) Have students use a nonlinguistic (kinesthetic) representations approach to discovery of multiplication of matrices. Set up students as the elements of each matrix. Be sure to have some matrices that will not multiply together because of wrong dimensions. Set the stage in your classroom by moving all the furniture out of the way and having students become the element. Use card stock or some other inventive way for students to identify the elements of the matrix. Have one student be the caller, one student be the recorder, and one student be the verifier of the information. Switch students on each task until everyone in the class has been an element. Notes D-88 Providence Public Schools