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High school students (9 th or 10 th graders) in Algebra II or Pre-calculus Requires previous math knowledge (up to Algebra II) Students generally interested in learning Any socioeconomic level Ability to complete assignment with study materials Learning Environment Action Buttons

Access to a computer Access to Internet, class notes, book, etc. Quiet or noisy setting depending on learner’s preference Work is individual Lesson moves at learner’s own pace Target Audience Objectives

Given a PowerPoint presentation of information and review and practice, students should: – Be able to recognize different types of graphs and draw graphs on polar coordinate planes in 100% accuracy on the quiz. – Be able to plot points and find the function to double check their work and receive 100% accuracy on the quiz. – Be able to compare and contrast the different graphs in an “A” essay given Word processing. Learning Environment

History Circles Spirals Lemnis- cates Limacons Roses Quiz Revie w Practice Modern Use

Do you remember the Polar Coordinate System?? pole polar axis Θ (polar angle) radius point More Review

Circular grid based off a central fixed origin and ray A point is graphed based on the length (r) from the origin and bond angle theta (θ) in relation to fixed ray (r,θ) exists as coordinates and location of the point (r, θ) More Review Review

Symmetry (r, -θ) = (-r, -πθ) Sine: symmetric to vertical axis Cosine: symmetric to horizontal axis Graphing on calculator! **Only to be used in emergencies** 1. 2nd FORMAT (ZOOM) RectGC  PolarGC 2. MODE Func  Pol 3. Y= r1= (enter equation) HistoryReview

Pythagoras: octave ratio 2:1, chord Archimedes: spiral (r=a+bθ) Hipparchus: Worked off Archimedes spiral and Pythagoras’ theorems to create a table of chord, to determine given length of a chord for each angle Modern Use Review

Calculus! (Differential and Integral) Finding Arc length Flight Navigation Surveying Physics Spirals : Parker spiral of solar wind, Catherine’s wheel of fireworks Spirals History

r= aθ For smaller values a and b, the spiral is tighter. For larger values a and b, the spiral is wider. Circles Modern Use

r= asinθ or r= acosθ r= diameter Remember! – Sin: symmetric to y – Cos: symmetric to x r= 3sinθ Limacons Spirals

r= a+bcosθ 1.a>2b: convex Limacon 2.a>b: Limacon w/ dimple 3.a=b: Cardioid (heart shape) 4.a<b: Limacon w/ loop 1234 For cosine:  Length left of y axis: a-b  Length right of y axis: a+b Lemnis- cates Circles

r 2 = a 2 cos2θ – a= length of each loop – cosθ indicates symmetry around x-axis – sinθ indicates symmetry around y-axis RosesLimacons

r= asin (nθ) a= length of petals n= determines # of petals n=even  2n petals n=odd  n petals Cos: aligns on x-axis, or all axes when n is even Sin: aligns on y-axis, or between axes when n is even r=cos4θ r= -4.5 sinθ Practice Lemniscates

Here are 3 problems for you to try on your own! 1.Draw the polar coordinate graph (a picture is given on the next slide) on a piece of paper. 2.Analyze the different parts of the function and decide what each tells you about the graph. 3.Draw the graph! Proceed to Practice Problems! Roses

1.Graph r= 2cosθ S#1Instructions

Watch me work out Problem #1 here!here! – Please note this link will take you out of the presentation. After viewing the solution, please click back into the presentation and continue. P#2 P#1

Graph r= 2cos(3θ) S#2 S#1

Watch me work out Problem #2 here!here! – Please note this link will take you out of the presentation. After viewing the solution, please click back into the presentation and continue. P#3 P#2

Graph r= 2- 2sinθ S#3 S#2

Watch me work out Problem #3 here!here! – Please note this link will take you out of the presentation. After viewing the solution, please click back into the presentation and continue. QUIZ P#3

Go home at any time to review material! Warning! Returning Home during quiz will not save your place! Quiz Practice

What is the polar graph of r= 2cosθ? Circle of radius _____ centered at _____. 2, x axis 1, y axis 4, x axis 2, y axis

What does cos(θ) indicate? What does the value “a” represent in the equation r= a cosθ ? Try Again! or Review Material! or Go Home!

The answer is A: Cos (θ) indicates the equation lies on the x axis A= length (diameter)= 2

What is correct about the number of petals on a rose? n petals if n is even, 2n if n is odd 2n petals if n is even, n if n is odd 2n petals if n is even, 4n if n is odd 4n petals if n is even, n if n is odd

A rose has the equation r= acos(nθ). What occurs in the graph when n is even or odd? Try Again! or Review Material! or Go Home!

The answer is B: A rose has n petals if n is odd and 2n petals if n is even!

What is the polar graph of r= 2-sinθ ?

Does the negative sign effect the graph in any way? Where does θ=0? Try Again! or Review Material! or Go Home!

The answer is D: Because sinθ has a negative sign, the graph points down. The graph intersects the x axis at 3.

Which Greek philosopher developed the table of chord? Archimedes Donatello Hipparchus Socrates

Think back to the people discussed in the History section. Hint: He’s not a ninja turtle! Try Again! or Review Material! or Go Home!

The answer is C: Hipparchus discovered the table of chord! – Archimedes discovered the spiral – Socrates was a Greek philosopher. – Donatello was an Italian artist and sculptor (also a ninja turtle!)

What shape does the graph r= 6-4cosθ make? Lemniscate Limacon with loop Cardioid Limacon with dimple

Limacons have the equation r= a-bcosθ. What is the relationship between a and b? Try Again! or Review Material! or Go Home!

The answer is D: a>b, in the equation r= a-bcosθ so the limacon has a dimple!

What is the graph of r=3sin4θ?

In a rose equation r= asin(nθ), what does the value “a” represent? “n”? How does sinθ affect the graph? Try Again! or Review Material! or Go Home!

The answer is B: In the rose equation r=asin(nθ), – a=3, the length of the petals – n=4, which is even, so there are 2n or 8 petals total Sinθ gives symmetry to the y-axis

What does the equation r 2 = a 2 sin2θ represent? Circle Limacon Rose Lemniscate

Which graph has an r 2 value in its general equation? Try Again! or Review Material! or Go Home!

The answer is D: Lemniscates are the only polar graphs with an r 2 value in their general equation!

Which is NOT a way polar graphing is used today? Differential/ Integral Calculus Physics and Arc Length Flight and Navigation All of the above are uses of polar graphing.

Remember polar graphing has many uses! Try Again! or Review Material! or Go Home!

The answer is D: Polar graphing has many real world applications, and that is why we are taking the time to learn it!

In a general spiral equation r=aθ, a spiral is tighter for _______ “a” values and wider for ______ “a” values? larger, smaller even, odd smaller, larger odd, even

It is the size of the number “a” that shrinks or widens the spiral. Try Again! or Review Material! or Go Home!

The answer is C: Just as you would think, smaller values shrink the graph and larger values widen it!

What shape does the graph y=sin(θ)cos(3θ) make? Spider Fish Butterfly Flower *Hint: You may need to use your calculator!

Did you switch your calculator to polar coordinates? Try Again! or Review Material! or Go Home!

The answer is C: It’s a (sideways) butterfly!!

Number of Questions Correct Eskimo Status 0-2 Eskimo Faux- You need to brush up on some material and retake the quiz! 3-5 Eskimo Slow- You should review the material and retake the quiz! 6-7 Average Eskimo Joe- You should still review the material but you’re on your way! 8-10 Eskimo Pro- Review the material before the test, but you’re well prepared! Check out these resources for more information!

Anderson, Dawn Leigh. “Assignment 11: Polar Equations.” The University of Georgia. 23 June “Graphing in Polar Coordinates.” Sparknotes. < parametricequationsandpolarcoordinates/section3.rhtml Leathrum, Tom. “Graphing in Polar Coordinates” Java Applet. Addison-Wesley Materials Web. 11 Nov < / a-happy-cartoon-polar-bear-jumping-and-smiling.jpg caption-2 Now that you’re done, go take a nice polar bear snooze!