SWBAT review the important concepts of parametric equations and polar coordinates.

The review worksheet will reinforce the important concepts of this unit.

10 minutes

I start class today by spending a few minutes going over the homework from yesterday's lesson on polar form of conic sections. There will inevitably be a few questions on this topic so I want to clear up any misconceptions before we start reviewing for our assessment.

Usually the questions will center around a few concepts that can be elusive to some students. Here is a list of things that usually come up after the assignment:

**The focus is always located at the pole**. Students may forget this. If they must write an equation of a conic in polar form, this can be a key piece of information that is needed.**Positives and negatives can be tricky**. One simple sign can make the equation right or wrong. It is important that students realize what the addition or subtraction sign represents for these formulas.**We can always plug in points that we know are on the graph.**This is something that is often overlooked at all levels of math! If we know one point on a conic and need to find the*p*value, for example, we can plug in the point we know and use it to solve for*p*just like we did in Algebra to find the y-intercept of a linear function.

10 minutes

In an attempt to solidify the knowledge of conic sections, **I assign question #1** of the review worksheet and give student about 10 minutes to work on it. I want to do this question together before I let them loose on the review for the rest of the hour since this is a good summary problem that connects what we worked on in this polar unit with our work with conic sections in the previous unit. Students will** work on this question with their table groups**. If they finish early, I instruct them that they can continue working on the rest of the review since it will be due tomorrow.

Question #1 is difficult for students because students must **move fluidly between rectangular and polar coordinate systems**. Students can usually graph it pretty easily, but it is difficult for them to find the rectangular equation of the ellipse. I think students can get "stuck" and think that if the equation is in polar form then they cannot consider any properties of rectangular form. For example, for #1 students know that (1.5, 90°) and (3, 270°) are the endpoints of the major axis of the ellipse, but may not know how to find the center of the ellipse. It is a conceptual jump to switch these points to (0, 1.5) and (0, -3) on the rectangular coordinate plane and to just find the midpoint in order to know the center.

30 minutes

There are** two main methods that students may use** to come up with the rectangular equation of the ellipse. I will try to choose students to show both ways. If one method does not come up, you can introduce it to students and finish it up as a class.

**Method 1**

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**Method 2**

Another way to find the rectangular equation is to just start with the polar form and use the conversions in order to get an equation with* x* and *y*. Some clever algebra may be needed but one way is to cross multiply so you have 6*r* + 2*r*sin*θ* = 12. Now *r* can be written as sqrt(*x*^2 + *y*^2) and *r*sin*θ* = *y*. The form that it is written in may not be recognizable as an ellipse, so method 1 may be more useful.

After going through this problem, it is time to keep working on the rest of the unit review and finish it for homework. This will be a good time for students to recap all of the important concepts that we worked on in this unit.