Unit Overview MCV4U: Vectors And Their Operations Bigger Picture: The strategies and rules for performing vector arithmetic can be thought of as an amalgamation and extension of the strategies and rules that students have already developed for performing scalar (standard) arithmetic and for performing geometric analysis (including trigonometry) in the Cartesian plane. Building On: Scalar (standard) arithmetic (addition, subtraction, multiplication and division) Triangle geometry involving trigonometry (eg, using the sine and cosine ratios together with the hypotenuse of a right triangle to determine the lengths of the other two sides) Linear equations and their graphs in the Cartesian plane The ability to visualize objects and shapes in two and three dimensions Big Ideas: Vectors have both magnitude and direction, and can be represented as directed line segments and as ordered pairs of Cartesian coordinates. Vectors can be multiplied by scalars and can be added together; in either case the resultant object is also a vector. Vectors can be multiplied in two ways: the dot product of two vectors yields a scalar and the cross product of two vectors yields a vector. Leading To: Using vectors and their operations to describe lines and planes and to solve related problems. Generalizing vectors and their operations into higher dimensions. Extending from vector arithmetic to matrix arithmetic, thereby opening the door to Linear Algebra. Solving a wide range of problems in related fields such as physics. New Vocabulary/Concepts: New Vocabulary/Concepts (Continued): Vector Orthogonal Scalar Torque Scalar Multiplication Parallelepiped Vector Addition Velocity Dot Product/Scalar Product Work Cross Product/Vector Product Force Projections Acceleration Description of Representative Assessment Tasks/Problems: upon completion of this unit, students should be able to add/subtract vectors graphically (eg, using the triangle law) and using Cartesian coordinates be able to multiply vectors by scalars, and to describe the process in words and/or graphically (eg, when multiplying vectors by positive scalars, we simply change the length; when multiplying vectors by negative scalars we change the length and reverse the direction ) be able to perform the dot product and the cross product, and to describe the difference between the two find the projection of one vector on another using the dot product verify properties of the dot and cross products (eg, are they commutative? associative? distributive?) find the area of a parallelogram using the cross product find the volume of a parallelepiped using the triple product
Unit Plan Specific Expectations Addressed are in red What is a Vector? Geometric and Coordinate L1 definitions. 1.1, 1.2 Scalar Multiplication and Addition of Vectors: L2 Geometrically 2.1 Scalar Multiplication and Addition of Vectors in the L3 Cartesian Plane 1.4, 2.1, 2.2, 2.3 L4 Applications of Addition and Scalar Multiplication 2.1, 2.2, 2.3 Check Point 1: Addition and Scalar Multiplication Activity: Battleship 1.3, 1.4, 2.1 Multiplication of Vectors: L5 The Dot Product 2.4, 2.5, 2.8 Check Point 2: Vector Projection Challenge Activity 2.4, 2.8 Multiplication of Vectors: L6 The Cross Product 2.6, 2.7, 2.8 Applications of the Dot L7 and Cross Product 2.5, 2.7, 2.8 Review L8 All expectations covered in unit Summative Assessment Unit Concept Map Unit Structure
Unit Big Questions Lesson Big Question for the Lesson Mathematical Processes L1 What is a vector? Representing, connecting, communicating L2 How can we add geometric vectors and multiply them by scalars? Connecting, problem solving, representing L3 How can we add Cartesian vectors and multiply them by scalars? How does this compare to the same processes for Geometric vectors? Problem solving, connecting, communicating L4 What are some of the real world applications of addition and scalar multiplication of vectors? Selecting tools and computational strategies, problem solving, communicating, representing Checkpoint 1 Can we apply what we now know about vectors to the game of battleship? Connecting, representing, reflecting L5 What is the dot product and how does it relate to the cosine law? Connecting, reasoning and proving, representing Checkpoint 2 What is meant by the projection of one vector on another and how does this relate to the dot product? Selecting tools and computational strategies, reasoning and proving, problem solving, communicating L6 What is the cross product, and how is it useful in physics? Connecting, problem solving, reasoning and proving L7 What are some applications and properties of the dot and cross products? Problem solving, selecting tools and computational strategies, reasoning and proving Review What have I learned in this unit? Reflecting, representing
Curriculum Analysis Our unit covers Overall Expectations 1 and 2 of Strand C of the course MCV4U. These expectations are as follows: by the end of the unit students will 1) demonstrate an understanding of vectors in two-space and three-space by representing them algebraically and geometrically and by recognizing their applications and 2) perform operations on vectors in two-space and three-space, and use the properties of these operations to solve problems, including those arising from real-world applications. In keeping with these overall expectations, we have identified the following Big Ideas: 1) Vectors have both magnitude and direction, and can be represented as directed line segments and as ordered pairs of Cartesian coordinates; 2) Vectors can be multiplied by scalars and can be added together; in either case the resultant object is also a vector; and 3) Vectors can be multiplied in two ways: the dot product of two vectors yields a scalar and the cross product of two vectors yields a vector. In terms of the specific expectations, we have determined that 1.1-1.4 fall under our first Big Idea, 2.1-2.3 fall under our second Big Idea, and 2.4-2.8 fall under our third Big Idea. Our unit addresses each of these specific expectations in at least one lesson either explicitly (e.g. through specific in-class instruction) or implicitly (e.g. through homework exercises or investigations, or through instruction that touches on several different curriculum expectations). The lessons in which these expectations are addressed are shown in the unit structure template. Our first Big Idea is really the answer to the question What is a vector?. This is a question which is addressed specifically in the first lesson of the unit; however, inasmuch as the entire unit is designed to build the students understanding of vectors, the entire unit addresses this question. Our second Big Idea builds on the first one, by attempting to answer the questions How can we add vectors? and How can we multiply vectors by scalars?. Again, as we explore the answer to these questions, students will continue to build their understanding of what a vector is, and hence their understanding of the first Big Idea will also grow. Our second Big Idea is explicitly addressed by lessons 2-4 in our unit, but once again, it continues to be addressed throughout the remainder of the unit. Our third Big Idea is the focus of the second half of the unit, being explicitly addressed in lessons 5-7. The third Big Idea is essentially the answer to the question How can we multiply vectors?. Through answering this question, students will continue to further their understanding of vector addition, scalar multiplication and vectors in general. Hence all three of our Big Ideas are intimately related, and the instructional sequence of our unit promotes a deep understanding of each of them. The first two Big Ideas are formatively assessed in the first checkpoint (the battleship activity), while the third Big Idea is formatively assessed in the second checkpoint (the vector projection challenge activity). Furthermore, all three Big Ideas are assessed in the end of unit review session (via the open question). The students understandings of each of the Big Ideas are also assessed through a number of informal assessments (homework, activities, worksheets, discussions, etc) throughout the unit. In terms of evaluation, the end of unit problem set contains three problems, each of which corresponds to one of the Big Ideas. The first problem requires students to demonstrate that they understand what a vector is by converting a geometric vector to Cartesian form. The second problem requires students to solve a vector addition problem. The third problem requires students to demonstrate their understanding of the dot product. Moreover, the evaluation activity focuses on students understanding of the cross product. Hence, in conjunction, these two
evaluations cover all the aspects of our three Big Ideas, and therefore cover all of the relevant curriculum expectations.
Assessment and Evaluation Plan Formative Assessment Plan: Following the Growing Success policy document s recommendations on assessment and evaluation, below is a table outlining formative assessment opportunities for teachers in this unit. The assessments might be used to inform or modify instructional practices, might be self-assessment for students, or might be evaluated more formally. Lesson Type of Formative Description Assessment Tool Assessment L1: What is a Vector? Assessment for L2: Scalar Multiplication and Addition of Vectors Geometrically L3: Scalar Multiplication and Addition of Vectors in the Cartesian Plane L4: Applications of Addition and Scalar Multiplication Checkpoint 1: Addition and Scalar Multiplication Activity: Battleship L5: Multiplication of Vectors: The Dot Product Checkpoint 2: Vector Projection Challenge Activity L6: Multiplication of Vectors: The Cross Product L7: Applications of the Dot and Cross Product Assessment for as as as and of and for as While students are using Geometer s SketchPad to investigate, comment on the ways in which they represent and communicate mathematical ideas. Students will complete handout 1 during the class; observe and offer feedback. Homework: Students will complete the Shopping Spree handout for extra practice. During the whole class sharing Graffiti exercise, ask students to help each other understand the key points of the investigation. By comparing and contrasting their responses, students gain a deeper understanding of vector operations. Students will complete worksheet 1:crossing a river and hand in their work Students will complete worksheet 2 as homework Battleship activity: Apply and Investigate addition and scalar multiplication of vectors. Activity: Find the Dot Product for Cartesian Vectors Challenge Activity: Find the Equation for the Projection Investigation: Properties of the Cross Product Practice Problem: Area of a Parallelogram Activity: Volume of a Parallelpiped Observation, anecdotal and/or informal notes. Check-bric, and exit cards. Check-bric
L8: Review as Assessment of Think-Pair-Share: What Have I Learned? Open Question: Demonstration of Learning Check-bric Summative Assessment Plan: Summative assessment in this unit will be comprised of two main tasks: first, there is a problem set which will be taken home by the students and completed. Second, there is an in-class activity which requires them to collaborate in order to come up with a strategy for solving a problem, and then solve it individually. Both of these tasks will be assessed using check-brics. Lesson Summative Assessment Assessment Tool L8: Review (Take Home) Problem Set Check-brics End of Unit (in class) Evaluation Activity Check-bric Justification of Summative Assessment and Evaluation Plan: Assessment is ongoing throughout this unit. There are assessment tools and strategies associated with every single lesson that will be taught. These include individual tasks such as completing worksheets and homework, as well as collaborative activities which require the students to work in pairs or small groups and solve a problem. In particular, there are three main checkpoints during the course of the unit, prior to the final evaluation. These are the battleship activity, the vector projection challenge activity, and the open question in the review session. Each of these activities/tasks is designed to allow the teacher to assess student achievement, and make an informed decision about whether or not it is appropriate to move on. A rational for each of these activities is present in the lesson in which it appears. The Summative assessment is in the form of two tasks. The problem set contains three problems, each of which addresses one of the big ideas in the unit. The students are allowed to take the problems home and work on them so that they will not feel the pressure of a test situation, and will be able to accurately demonstrate their. The evaluation activity is designed to test their ability to integrate what they have learned in the unit and apply it to solving a challenging problem. They will be permitted to discuss the problem and develop a plan for a solution with their peers before solving the problem individually. Taken together, we believe that these two tools will enable us to accurately measure student achievement of the curriculum expectations. Of course, it goes without saying that the formative assessments may also be considered (if the teacher s professional judgment dictates that it is appropriate to do so) for the purposes of informing decisions regarding students final grades.