Contextual note: See reflection Students demonstrating operations
Class begins with students working on two questions. We have not discussed operations of vectors. I want students to think about what could be the most logical way for us to add and subtract.
I move around the room and record what students are thinking. After a couple of minutes I put the answers on the board. (The students in Physics will know how to do add vectors. Depending on the class I may ask the Physics students to stay quiet so the other students can reason through the process) I ask students to explain the different ideas.
Once we have the reasoning of the students I begin developing the the operations we will work with addition, subtraction and scalar multiplication.
By the time we discuss the different thoughts many students have determined the most logical way to add vectors is to add like components. I want students to understand why we add the components to add vectors.
I begin by asking "What is meant by adding vectors? How could we use a diagram to show the adding of vectors?" I then go through the process of drawing a diagram to show the addition of the 2 vectors.
I then ask the students to draw the diagram for subtraction. As students consider how to draw the diagram I ask "If you start at 4 on a number line how would you show subtraction of 3 on the number line? What about subtraction -2?" This discussion helps remind students that subtraction is adding the opposite so if you subtract 3 you move to the left on a number line. Once students think about subtraction in this way they can make a diagram for subtraction.
I use these diagrams to show how we can draw both vectors in standard position then make a parallelogram the is the same as the vector we found in the drawings.
Once students understand how to add and subtract and that subtraction is adding the opposite I define the vector that results from the addition as the resultant.
I pose a problem that shows scalar multiplication. I let the students discuss how to determine the answer. Students reason quickly to multiply each component by 3. I ask why. The students use structure to explain that 3u means u+u+u. I define this operation as scalar multiplication. I remind students of the scalar multiplication of matrices. I define a scalar as a number without units.
I next ask how do you think the magnitude of 3u will compare to the magnitude of u. Students think about this for a little while. Students use the example to see what compare the original magnitude to the 3u magnitude. Once students see that it is 3 times more I ask if it will always be the the constant times more than the original magnitude. I help the students generalize what they have seen with the example.
Most books have many rules and formulas that students want to memorize. By having the students reason numerically, then abstractly, the rules will make more sense.
After working with the basic vector operations I want students to do operations with vectors written as a linear combination of vectors i and j. Using the standard unit vectors to write a vectors is used in physics so this is an important ideas for students to understand.
I introduce the standard unit vectors asking "What is meant by the term unit vector? What would be the simplest unit vector to write?" I am hoping students says <0,1> or <1,0>. If students do not give these I will begin a vector like <0, ?> and ask "what does the y component be for this to be a unit vector?"
I then label <1,0> as i and <0,1> as j. I call these the standard unit vectors. I ask how we could take the vector <3,8> and write it as a sum using the standard unit vectors. Students share ideas and until we determine that the result would be 3i+8j
As a final question, I tell students to discuss with those around them how to find a unit vector in the same direction as a given vector. This question let's me analyze if the students can see how to use the magnitude to find a specific vector.
As students are discussing I am listening to the conversation to see which students are thinking about using the magnitude. After a little discussion I tell students to put an answer on a piece of paper. I allow those that discussed the problem to share give one answer before they leave. This question will lead into tomorrow lesson.