Review Sheet with Example Problems Click Here.

Example Solutions. Click Here.

Home Assignments: Click Here.

Assignment Solutions. Click Here.

Procedure

1) Scalar vs. Vector: Write a table on the board that gives examples of both scalars and vectors in the left column. Title the two right columns as "Magnitude = Size" and "Direction." Then, as a class, either mark "Y" for yes and "N" in the two right columns as needed (i.e. a scalar will get a "Y" in the "Magnitude" column and an "N" in the direction column) as you move through the list of examples (i.e. Temperature, displacement, velocity, speed, position, acceleration, force). Ask the students how they would give directions to someone visiting their home, this can only be done by using vectors. Tell them how GPS, Cartography and navigation are only possible by using vectors.

2) Vector Representation: Drawing a plane and show the students the difference between the Polar form (magnitude and angle) and the Cartesian form (x-y coordinates). Make sure to mention what a scale is and how it is used in drawing vectors (i.e. the vector <2,4> should be twice the length of the vector <1,2> in Cartesian form). Give in class problems for the student to complete on the board so that they can demonstrate their attainment of knowledge. The magnitude of a vector that has Cartesian form is found via Pythagorean Theorem. If the vector is in Polar form, then the magnitude is given by the r-value.This is where a lesson on trigonometry might be needed so that the conversion between Polar and Cartesian can be performed.

3) Trig Mini-Lesson: This is where a mini-lesson on trigonometry can be placed if the students need it. It is most likely that they will. Start with some right triangles with two sides labeled with a length. Work with Pythagorean Theorem to determine the third (don't always leave the hypotenuse non-labeled because the students need to be know how to work through such a problem). "How can we determine one of the non-90 degree angles?" This is where sine and cosine and tangent come in. Draw the graphs for each function and explain SOHCAOTOA. Work with some more triangle to convince the students that these functions really do work.

4) Vector Manipulation: Make sure the students understand that (x,y) is a point on the plane while <x,y> is a vector from the origin the the point (x,y)!! Vector manipulation is key to developing the students concepts. Without doing problems and playing with vectors, they will never truly know the material. DO NOT use Polar form for algebraic manipulation. Start graphically. Scaling is vector means you just make it longer or shorter by a certain factor. If they have taken Geometry, it is the same as when they drew sides of similar triangles. Scaling is even easier in Cartesian form: a*V = a*<Vx, Vy> = <a*Vx, a*Vy>. Vector Addition. Show students the "Parallelogram Method" where the two vectors have both their tails in the same coordinate and a parallelogram is formed and the diagonal from the common endpoint is the resultant vector. Another graphical method is the "Head-to-Tail Method" where the tail of one vector is placed on the head of another and the line connecting the non-common points is the resultant vector. Algebraic vector manipulation is the easiest with Cartesian representation since everything adds component-wise. Given two vectors V1 = <V!x, V!y> and V2 = <V2x, V2y> you get V = <V1x + V2x, V1y + V2y>. Scaling is even easier in Cartesian form: a*V = a*<Vx, Vy> = <a*Vx, a*Vy>. Resolving Vectors. A vector that is written in Cartesian form is already naturally resolved into an x-only vector plus a y-only vector. To challenge things, see if the students can resolve a vector into two vectors that are not simply the x,y vectors. If the vector is written or draw in Polar form, they trigonometry is needed.

5) Applications: This is a good time to get into some related material. Projectiles and Trajectories are the primary example of how the x-y coordinates are not dependent on each other. A soccer ball is kicked and only the y-component of velocity changes with time since gravity only act in the y-direction. Airplane and Boat problems are great i.e. A plane travels 100 m/s N and there is a crosswind of 10 m/s from the NW. What is the plane's displacement after 1 hour? Force Balance is also a great extension that shows the students how to use vectors to solve problems i.e. A gymnast hangs from the rings with their arms at 45 degrees and has a mass of 80 kg. How much force is acting on the gymnast's arms?

6) Problem Day: This can be done in conjunction with Options or solely mathematical problems can be solved. I think that by working with problems, students will start to understand the concepts better. Working with word problems are the best. Make a game out of it. Break the room up into teams of 5 (or so). State problems and which even team answers first gets a point. The team with the most points get something (I gave a free-homework pass - they get a 100% on the next assignment).

7) Test Day: Each student get photo copies of nautical charts of an area. They also receive a sheet of paper that describe the currents and winds for that day. Then, they have to give appropriate speeds and directions (polar form vectors) in order to move as stated in the problems. (i.e. Go from "A" to "B" in 20 minutes). See Chart Test Link.

I utilized CPO Physics for this lesson, but any high school physics text will provide a number of problems and similar outline. I also utilized Professor Duffy's (Boston University, Physics Department) Physlets (Physics-Applets) for some graphical needs.