TITLE POLAR COORDINATES NAME ELAINE HEBERT DATE OF LESSON

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Name:

Title: Polar Coordinates


Name: Elaine Hebert


Date of Lesson: Tuesday of Week 1


Description of Class: High School Geometry


Grade Level: High School Geometry


Length of Lesson: 50 minutes


TEKS addressed:


Algebra:

b.1 (A): The student describes independent and dependent quantities in functional relationships.


Geometry:

d.2 (A): The student uses one and two-dimensional coordinate systems to represent points, lines and line segments.


I. Overview:


Using a java script applet, students will explore a polar coordinate system. They discover differences and similarities between polar and Cartesian coordinate systems.


II. Learner Objectives:


Students will be able to:

- describe the differences and similarities between polar and Cartesian coordinate systems.

- plot points on a polar coordinate system.

- explain relationships between the values of x, y, r, and phi on the applet.

- use the java applet to answer corresponding questions.



III. Materials:


-Student logbooks

-Computer (one per student) with java script and internet access

-Calculators (one per student) if needed

-Attached handout

-Geometer’s Sketchpad software (optional)


Engagement:


I want everyone to take out their logbooks and draw a circle. Beside your circle, write how many degrees there are in a circle. How many degrees are there in a quarter of a circle? What about an eighth of a circle? Talk to your group members about these numbers. What is special about these numbers: 360, 90, and 45? Keep in mind that their units are degrees.” Draw a circle on the board with the center at the origin of a coordinate system. “So if zero degrees is here (point to x axis), where is 90 degrees? What about 360 degrees? If you do not know the answer, today’s activity will hopefully help. We are going to be exploring the beauty of a polar coordinate system. Yesterday we went over Cartesian coordinates, but sometimes we plot things on a polar coordinate system, which deals a lot with circles and angles. The term radius will be used so don’t freak out. We all know what a radius is. By the end of the activity we should be able to tell the differences and similarities between polar and Cartesian coordinate systems.”


Exploration:


Teacher Does

Expected Student Response

Ongoing Evaluation

“Today we are going to be working with an applet for polar coordinates. On your worksheet, you have the web address and you will click on the box called ‘Polar Coordinates.’”

Students listening.


“You can drag the point, P, around. Play around with the point and see the values of x, y, r and phi change. You can also make grid lines.”

Students listening.

“What do x, y, r and phi represent? How are these gridlines different than on a Cartesian coordinate system?”

“When you are done exploring on your own, answer the questions on your worksheet. Some of them require you to do calculations, like 3a. A calculator might be needed.”

Students are exploring the applet.

“How is this applet similar to the Unit Circle? What is the unit circle?”

I walk around helping students with the applet and worksheet questions.

Students work on the worksheet.

“Be thinking about the circumstances for which it might be better to graph things on a polar coordinate system.”

“When you are done, I want you to write in your logbooks about the lesson. Write 1 difference and 1 similarity between polar and Cartesian coordinate systems.”

As students finish with the web-based activity, they go back to their seats and write in their logbooks.


“If you are done with the logbooks, get on GSP and see if you can create an applet similar to the one we were working on.”

The finished students work on GPS.

“How are you going to get the x, y, r and phi values on GPS?”

When most of the students are done with the applet activity: “Ok, let’s go back to our seats and talk about what we did today.” (see explanation)

Students go back to seats.




Explanation:


Teacher Does

Expected Student Response

Ongoing Evaluation

“Through the applet, we discovered that polar coordinates use an ordered pair just like a Cartesian coordinate system. Talk in your groups for a couple minutes to discuss how the two ordered pairs are different.”

Students talk about how the polar system uses a radius and an angle.


“On a blank sheet in your log books, draw the line from the origin to the point (4, 200) on a polar coordinate system. Don’t look on your neighbors’. Remember the problem about the angles within each quadrant.”

Students draw the line.


“Now talk to your group members to see if they all look the same. Discuss which one was correct and why. When you come to an agreement, one person from the group can come up to the board and draw it.”

Students discuss, and then one from each group comes up to the board to draw it. Hopefully they will all look the same.


Talk about the drawings and discuss why each one is right or not right. If there is time left over, go over some of the main points from the worksheet, like the relationships between x, y, r and phi.


“Those who have not written in your logbooks may now do so. We are writing about 1 difference and 1 similarity between Cartesian and polar coordinate systems.”














































Name: Date:



Polar Coordinate System Worksheet


Go to this website: http://www.univie.ac.at/future.media/moe/galerie/zeich/zeich.html and then click on the box that says “Polar Coordinates”. In a separate window the applet will open up. Explore this applet on your own for a few minutes then answer the questions below. Make sure the unit is in degrees. You can change this at the bottom of the screen.

Note: A polar coordinate is given in terms of a radius and an angle, (r, phi). This is the point called P in the applet.



1. What do the values of x, y, r and phi represent?









2. Are phi and r ever negative? Why or why not?







3a. Without using the applet, if you are given x= 2 and y= 3, find the value of r. Show your work below.











3b. Now move the point P to get x = 2 and y = 3 on the graph (or as close as you can). Was your answer right?







4. What are the ranges of the angles in the first, second, third and fourth quadrants?








5a. Move P along the x-axis. What do you notice about the values of x, y, r and phi? Explain why this happens.






5b. Move P along the y-axis. What do you notice about the values of x, y, r, and phi? Explain why this happens.







6. What is phi when the values of x and y are the same and in the first quadrant? Check a few values to make sure this is always the case. (Hint: this is one of the special angles we talked about at the beginning of class)







7. What are the polar coordinates (r, phi) at the origin? Why do you think that is?





8. Do you think graphing with a radius and an angle is harder or easier than graphing with an x and y value? Why?




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