Today, the bell problem is a simple parametric equation. This problem should challenge students to think about replacing the t with the x variable to eliminate the parameter. I find that by starting with a more simple and straightforward problem, students are able to gain more confidence in their decisions around these problems.
I know some students will question their reasoning. The students think "it can't be that easy", but I respond with questions like "Did you eliminate the parameter (t)? or Does your final equation have more than 2 variables?". This helps to reinforce their confidence in their initial reasoning.
Students perk up when they are able to quickly do a problem; I find that they are usually more willing to persevere through more difficult concepts as we move on to more challenging problems.
I bring back the Cannon ball equations from yesterday. The students are asked to verify the parametric equation is the same as the rectangular equation.
I let students look and think about the process for 1 to 2 minutes. When I see that the students have either came up with an idea or seem stuck, I step in and look back at the bell problem. Questions I use include: "Why were you able to replace the t with x in this problem?" "Is there anyway we could do a similar process with the parametric equations here?"
Students should think about solving the x(t) equation for t and then substituting the result into the y(t) equation. We then simplify the expression. Due to rounding the numbers may be a little off. We discuss why the numbers are not exactly the same. I always remind students about the affect of rounding on answers.
After working with polynomials I now bring equations for circles that we worked with the last lesson. I put a parametric equation for a circle on the board. I let the students consider what we might do with this problem. After a few minutes. I ask the students if we have any identities that involve sine and cosine. If necessary I have the students find their formula sheet for the identities unit.
I want students to remember the Pythagorean Identity. In order to guide them towards this reasoning, I ask if we can solve each piece of the parametric equation for the trigonometric function. The students replace sine and cosine in the Pythagorean Identity and then simplify.
Once the students have seen a specific example I give the students the equations we developed in the previous lesson x=r cos(t)+h, y=r sin(t)+k for the students to rewrite. The amazed look on the students' faces when they see the connection is always great. The students are amazed and how this works and that they are able to do the problem.
I now give the students an ellipse to find the standard equation. This again produces looks of amazement as the students see the relationship.
After working with the circle and the ellipse I ask the students about the problem I posed in the last lesson. "What parametric equation will produce a hyperbola?" Many students have found a hyperbola by using x=tan t and y=cot t. This does work but the hyperbola is rotated from what we did with the conic unit. I now ask the students to give me the other Pythagorean Identities. We discuss how 1+(tan t)^2=(sec t)^2 can be rearranged. The students begin to realize that we now have an equation equal to 1 that is a difference. The students put x=tan t and y=sec t to see that this makes a hyperbola.
If you have started to notice a pattern, I begin all my lessons on parametric equations with the cannonball problem. I believe that projectile motion is a great application of parametric equations. Therefore, I give my students the Parametric Equation Applications worksheet to help them practice this concept and application.
This worksheet begins by giving students a formula for how to find the vertical and horizontal distance. This idea is developed from vectors, where the initial velocity is the resultant. I briefly review how we can find x and y when we know a vector's angle and resultant. The formula we consider assumes there is no gravity. However, I have a student read aloud how the formula changes when gravity is considered. We then write out the formula using gravity so that the students have the correct formula to use for the problems.
I display the Cannonball bell problem from the previous lesson on the board. I have the students determine the parametric equation for this problem. When students verify my equations it helps them to see exactly how to use the equations.
As class ends I assign the problems on the worksheet problems in addition to p. 774, # 8, 10, 12, 22, 33-36 from Larson "Precalculus with Limits, 2nd edition." The problems assigned ask students to sketch graphs, eliminate parameters, and find parametric equations. Building upon these skills is necessary for students who will take classes such as calculus in the future.