Name Withheld
Math 4050
Project, Part 2
20. Inverse, Converse, and Contrapositive
C. How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?
“What do you do with witches?
Burn them!
What do you burn apart from witches?
More witches! … Wood.
So why do witches burn?
Because they’re made of wood?
Good. So how can you tell if she is made of
wood?
Build a bridge out of her!
Ah, but can you not also make bridges out
of stone?
Oh yeah…
Does wood sink in water?
No, it floats.
Ah, and what also floats in water?
Bread, apples, cider, very small rocks… A
duck.
So naturally…
If she weighs the same as a duck she’s made of wood
And therefore…
A witch!”

Monty Python and the Holy Grail is a well-known British comedy film from 1975. In one
scene the townspeople have decided that a young lady is a witch and they want to burn her.
A man incorrectly uses logic to find a way to test if she is a witch. The conditional statement
would be “If she is a witch, then she will burn.” He uses the converse and says that if she
burns she must be a witch. Therefore, since wood burns she must be made out of wood.
Since wood floats and a duck floats, if she is a witch she must weigh the same as a duck.
This scene would be great to use during an engage or an elaborate. It shows how inverses
(or converses, depending on how you look at it) can be misused if you assume they are
equivalent. It’s a funny scene that many students would have seen it before. Even if they
have not, they don’t need to see the whole movie to understand the scene. Students could
go through and write the scene as a list of conditional statements and explain why the logic
is incorrect or the class could discuss it briefly.

Source: Monty Python and the Holy Grail (You can find this scene on youtube by searching
“Monty Python Witch Scene”)
C. How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?
“Hello ladies. Look at your man, now back to me, now at your man, now back to me. Sadly,
he isn’t me, but if he stopped using lady scented body wash and switched to old spice he
could smell like he’s me. Look down, now back up. Where are you? You’re on a boat with the
man your man could smell like. What’s in your hand? Back at me. I am. It’s an oyster with
two tickets to that thing you love. Look again. The tickets are now diamonds. Anything is
possible when you man smells like old spice and not a lady.”

Advertisements often use conditional statements (or at least imply them) to get people to
buy their product. When they use a celebrity to promote their product they are implying
that if you use their product you could be like the celebrity. When a guy uses a certain type
of cologne and girls start chasing after him, advertisers are implying that if you use this
cologne girls will chase after you too. Like in the quote from an old spice commercial listed
above the ad tells ladies that anything is possible if your man uses old spice. Most
commercials use some sort of conditional statement.
Commercials are something that students can directly relate to. Everyone watches TV. It’s
also a real world application for conditional statements other than mathematics. As an
assignment students can find their own commercials or ads that illustrate an inverse,
converse, or contrapositive being used to try and convince people to buy a product or do
something.

Old Spice “The man your man could smell like” commercial. (The commercial can be found
at http://www.youtube.com/watch?v=owGykVbfgUE&feature=related)
C. How has this topic appeared in high culture (art, classical music, theatre, etc.)?
“Then you should say what you mean,” the March Hare went on. “I do,” Alice
hastily replied, “at least I mean what I say—that’s the same thing, you know.”
“Not the same thing a bit!” said the Hatter. “Why, you might just as well say
that ‘I see what I eat’ is the same thing as ‘I eat what I see’!”

Many possible word problems can be created using converses, inverses, and
contrapositives. Taking statements such as “If it is raining, then the grass is
wet,” and finding and testing the converse, inverse, and contrapositive is
easier for most students then starting off with a math statement such as “If a
shape is a square, then it is a quadrilateral,” and testing the converse, inverse,
and contrapositive. Once students do a few everyday statements it’s easier to
apply it to math. It also interests the students a lot more when you throw in real world
statements. If students do several of these then they should also be able to see that the
contrapositive is always equivalent before you prove it using a truth table. Having students
come up with their own conditional statements would also be a good activity for them to
do.
The headline quote is from the book, Alice in Wonderland. With the recent movie that came
out in 2010, using the quote might interest some students. It also illustrates the idea that
the converse of a statement is not equivalent to that statement. The quote would be a good
introduction to the topic, or at least a worksheet.

The quote is from Alice in Wonderland by Lewis Carroll. One possible worksheet can be
found here: http://ritter.tea.state.tx.us/math/training/materials/MTR/912/lessons/geometry1_2.pdf
Name Withheld
Math 4050
Project, Part 4
22. Finding the Equation of a Circle
B. How does this topic extend what your students should have learned in previous courses?
The equation of the circle is nothing more than a straight application of the distance formula, and
consequently, of the Pythagorean Theorem. By definition, a circle is a geometric figure where all
points at its boundary are equidistant from a fixed point, called the center. Then, applying the
Pythagorean Theorem on the right triangle in the picture that (x-a)2 + (y-b)2 = r2, which is the
equation of a circle with radius r and center (a,b).
B. How does this topic extend what your students should have learned in previous courses?
Knowing how to find the equation of a circle will help students
in their future mathematics courses. This topic reappears in precalculus and calculus. The unit circle in trigonometry and some
important identities are derived from the Pythagorean Theorem;
therefore, they are also related to the circle equation.
Identities such as cos2θ+sin2θ=1 describe a circle, more
precisely, the unit circle.
Also, the magnitude of a complex number describe a circle
centered at the origin and with radius |𝑧| = √𝑎2 + 𝑏 2 , where a,b
∈ R.
Finally, polar coordinates are another example of the circle equation.
A. What interesting (i.e., uncontrived) word problems using this topic can your students do now?
During the Transit of Venus on June 5, 2012, the planet
Venus will move across the face of the Sun. This will be
the last time this phenomenon will be visible from Earth
until December 10, 2117!
Depending on your location, the dark disk of Venus will
travel in a straight line across the disk of the Sun, taking
different amounts of time. The figure to the left shows one
such possibility lasting about 7 hours.
The standard equation of a circle centered at (h,k) is given
by:
2
2
2
r = (x - h) + (y-k)
Problem 1 – The diameter of the Sun at that time will be 1890 seconds of arc. Write the equation
for the circular edge of the Sun with this diameter if the origin is at the center of the sun's disk.
Answer: The radius is 1890/2 = 945, so the formula for a center at (0,0) is
2
2
2
2
2
(945) = x + y so x + y = 893,025
Problem 2 – Suppose that from some vantage point on Earth, the start of the transit occurs on
the eastern side of the Sun at the point (-667, +669), and follows a path defined by the equation y
= -0.1326x +580.5. What will be the coordinates of the point it will reach on the opposite,
western edge of the Sun at the end of the transit event?
Answer: We need to find the point where both equations are satisfied at the same time.
2
2
Equation 1) x + y = 893,025 Equation 2) y = -0.1326x +580.5
By substituting Equation 2 into Equation 1 we eliminate the variable ‘y’ and get
2
2
x + (-0.1326x +580.5) = 893,025 which simplifies to:
2
1.0176 X – 153.95 X – 556,045 = 0
Using the quadratic formula to find the two roots, we get x = 76 +/- 743 and the roots x1 = +819
and x2 = -667. We already know the starting point at x=-667, so the desired point must be x1 =
+819 and from the equation for the transit line
Y2 = -0.1326(819)+580.5 = +472.0. so the two endpoints for the transit are (-667, +669) and
(+819, +472).
Problem 3 – If Venus moves across the Sun at a speed of 200 arcseconds/hour, how long with
the transit take based on the endpoints calculated in Problem 2?
Answer: The distance between the transit endpoints can be found from the Pythagorean
Theorem:
2
2 1/2
D = ( (819 + 667) + (472 – 669) ) = 1,499 arcseconds.
At a speed of 200 arcseconds/hour, the transit will take 1,499/200 = 7.5 hours.
Just for fun
Sources:
 Equation of a Circle figure:
http://www.mathsisfun.com/algebra/circle-equations.html

Word Problems
http://spacemath.gsfc.nasa.gov/algebra2.html