How long before the cannonball hits the ground? (Parametric Equations) Day 1 of 2
Lesson 1 of 7
Objective: SWBAT define a parametric equation and use the equation to graph.
I start class by giving students are given a projectile motion problem. Although my students have seen this type of problem before, I use this familiar concept to introduce the parameter t.
After working individually, students will then discuss with one another how they determined where the cannonball hit the ground. I have found that some students may forget to set the equation equal 0. In order to help students work through this concept I ask "What is the height of the cannonball when it hits the ground?" A student usually says something about y=0. It is not uncommon for students to be challenged when solving this problem because the problem does not use integers. I try to let my students sit with this difficult concept in order to guide them to realize the best method to solve.
Finally, we review the factoring and finding the value of x.
I now ask the students to determine how long it will take for the cannonball to hit the ground. This slide shows the two parts I use for this problem. To begin, I cover up the second part of the slide when I ask the first question. I want students to think about this question and try to determine the time.
Some students will make a prediction. The most common idea is to take the value they found in the bell work (1989.7 m) and divide that by the initial velocity. I put the prediction on the board. I then ask the students if the beginning velocity will be the velocity through the entire trip. "How do you know the velocity or rate of change will not be constant?" This questions reminds students that we are working with a quadratic. I ask students what will happen to the velocity as the cannonball rises and then falls. For some students this idea initially difficult to understand; thus, we stop to discuss how gravity will slow the cannonball down until it starts to fall and it then increases velocity. Typically students who have already taken Physics will understand this concept, but it is still important to introduce and review this concept to the class given that there may be students who have not yet taken Physics to ensure that everyone understands the context of the problem.
After 3 or 4 minutes of discussion I uncover the second part of the slide. Students are shown the parametric equation that represents the cannonball flight from the bell work. The students will then work to determine the time it takes for the cannonball to hit the ground. In the next lesson the students will be given a formula to find the vertical and horizontal distances, but at this point we just use the formulas we know to answer the questions.
Most students will realize they found the horizontal distance in the bell work. The students take the x(t) equation to determine the time in flight.
By the end of this exercise, I would like students to understand that they could also use the y(t) equation to find the time. I ask the students what is the vertical height when the cannonball lands. We set the y(t) equation equal to zero and show that this equation also gives us the same answer.
We compare this answer to the prediction the students made. We see that the prediction is around 13 while the actual answer is around 15. I ask the students why this is the case, leading students to see that the rate is not constant.
After working with the problem, I put the definition of a plane curve, which includes defining a parametric equation, on the board for students to read. After reading the definition I ask these questions to make sure the student have an adequate understanding of the concept:
- What is a continuous function?
- What is meant by the ordered pair (f(t),g(t))
- How could you find the ordered pairs?
I remind students how the graph is the set of all (f(t),g(t)) points and display a parametric equation to graph. The students will now discuss how to graph this equation in their groups. I look for different ways students work through the problem to share with the class.
My goal is for students to think about how to organize the information. I usually have a student make a table with 3 rows or 3 columns labeled as t, x and y. Other students will just find x and y without identifying the t variable that connects.
I have students put up different examples of the tables, and then I ask a student to graph the points and make a curve. However, I have not yet discussed how we show the direction of the graph when the student is graphing.
I ask "If this graph represented the movement of an object, how could I show how the object is moving and where it is at different t-values?". This guides students to put arrows that represent the direction along the curve and put the value of t and some of the points.
I use this next opportunity to show my student how to use technology to graph. I think it is especially important for the students to see how they can write the conic in parametric form to be graphed. To start, I go through the steps from Graphing parametric equation on a calculator with them. We use the previous problem to see how to graph with the calculator.
I ask students to change the TMIN and TMAX to see how this affects the graph.
Another parametric equation is shared with the students. This is an equation of a circle centered at (0,0) with a radius of 1. I ask "what values of t would ensure that I have a complete graph?" Students should realize that the period of x(t)=cos t is 2pi. Students change the t values in the window and graph. You may also need to have the students square their window to make the graph look like a circle.
I then adjust the circle so that the graph has a radius of 3 and other values. I now ask exploratory questions:
- What do you notice?
- How would the graph change if I switched cosine and sine?
- Could you generalize what you are noticing?
- The center of these circles is always at the origin. What equation would give me a circle with the center at (4, 5) and a radius of 3?
- How could you generalize the formula?
In the next lesson we will convert the equation to rectangular to verify that this is the parametric equation for a circle.
I now want students to think beyond the circle, so I ask "What would we do to make the figure an ellipse?" Some students immediately say we need to make the equations have different coefficients on sine and cosine. I then reinforce the concept with these questions:
- What do the coefficients represent for the graph of an ellipse?
- How could you write an equation to represent an ellipse?
I have found through this activity that students like to see what kind of figures they can make using parametric equations. Therefore, I give my students a few minutes to make graphs from different equations. It is great to have students share some of their drawings.
At the end of class I leave a challenge problem for the students to consider. I ask the students to determine a parametric equation that will produce a hyperbola.
Understandably, this is still a difficult problem for them at this point. However, as we begin converting between parametric and rectangular the students will begin to see patterns with trigonometric identities (Pythagorean Identities) that will help them find the equation.
If students come up with an idea, I have them write it down so they can share it at the start of the next day.