# My Favorite One-Liners: Part 88

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

In the first few weeks of my calculus class, after introducing the definition of a derivative,

$\displaystyle \frac{dy}{dx} = y' = f'(x) = \lim_{h \to 0} \displaystyle \frac{f(x+h) - f(x)}{h}$,

I’ll use the following steps to guide my students to find the derivatives of polynomials.

1. If $f(x) = c$, a constant, then $\displaystyle \frac{d}{dx} (c) = 0$.
2. If $f(x)$ and $g(x)$ are both differentiable, then $(f+g)'(x) = f'(x) + g'(x)$.
3.  If $f(x)$ is differentiable and $c$ is a constant, then $(cf)'(x) = c f'(x)$.
4. If $f(x) = x^n$, where $n$ is a nonnegative integer, then $f'(x) = n x^{n-1}$.
5. If $f(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0$ is a polynomial, then $f'(x) = n a_n x^{n-1} + (n-1) a_{n-1} x^{n-2} + a_1$.

After doing a few examples to help these concepts sink in, I’ll show the following two examples with about 3-4 minutes left in class.

Example 1. Let $A(r) = \pi r^2$. Notice I’ve changed the variable from $x$ to $r$, but that’s OK. Does this remind you of anything? (Students answer: the area of a circle.)

What’s the derivative? Remember, $\pi$ is just a constant. So $A'(r) = \pi \cdot 2r = 2\pi r$.

Does this remind you of anything? (Students answer: Whoa… the circumference of a circle.)

Generally, students start waking up even though it’s near the end of class. I continue:

Example 2. Now let’s try $V(r) = \displaystyle \frac{4}{3} \pi r^3$. Does this remind you of anything? (Students answer: the volume of a sphere.)

What’s the derivative? Again, $\displaystyle \frac{4}{3} \pi$ is just a constant. So $V'(r) = \displaystyle \frac{4}{3} \pi \cdot 3r^2 = 4\pi r^2$.

Does this remind you of anything? (Students answer: Whoa… the surface area of a sphere.)

By now, I’ve really got my students’ attention with this unexpected connection between these formulas from high school geometry. If I’ve timed things right, I’ll say the following with about 30-60 seconds left in class:

Hmmm. That’s interesting. The derivative of the area of a circle is the circumference of the circle, and the derivative of the area of a sphere is the surface area of the sphere. I wonder why this works. Any ideas? (Students: stunned silence.)

This is what’s known as a cliff-hanger, and I’ll give you the answer at the start of class tomorrow. (Students groan, as they really want to know the answer immediately.) Class is dismissed.

If you’d like to see the answer, see my previous post on this topic.

# Math Maps The Island of Utopia

Under the category of “Somebody Had To Figure It Out,” Dr. Andrew Simoson of King University (Bristol, Tennessee) used calculus to determine the shape of the island of Utopia in the 500-year-old book by Sir Thomas More based on the description of island given in the book’s introduction.

Paper by Dr. Simoson: http://archive.bridgesmathart.org/2016/bridges2016-65.html

# A natural function with discontinuities (Part 3)

This post concludes this series about a curious function:

In the previous post, I derived three of the four parts of this function. Today, I’ll consider the last part ($90^\circ \le \theta \le 180^\circ$).

The circle that encloses the grey region must have the points $(R,0)$ and $(R\cos \theta, R \sin \theta)$ on its circumference; the distance between these points will be $2r$, where $r$ is the radius of the enclosing circle. Unlike the case of $\theta < 90^\circ$, we no longer have to worry about the origin, which will be safely inside the enclosing circle.

Furthermore, this line segment will be perpendicular to the angle bisector (the dashed line above), and the center of the enclosing circle must be on the angle bisector. Using trigonometry,

$\sin \displaystyle \frac{\theta}{2} = \frac{r}{R}$,

or

$r = R \sin \displaystyle \frac{\theta}{2}$.

We see from this derivation the unfortunate typo in the above Monthly article.

# A natural function with discontinuities (Part 2)

Yesterday, I began a short series motivated by the following article from the American Mathematical Monthly.

Today, I’d like to talk about the how this function was obtained.

If $180^\circ \le latex \theta \le 360^\circ$, then clearly $r = R$. The original circle of radius $R$ clearly works. Furthermore, any circle that inscribes the grey circular region (centered at the origin) must include the points $(-R,0)$ and $(R,0)$, and the distance between these two points is $2R$. Therefore, the diameter of any circle that works must be at least $2R$, so a smaller circle can’t work.

The other extreme is also easy: if $\theta =0^\circ$, then the “circular region” is really just a single point.

Let’s now take a look at the case $0 < \theta \le 90^\circ$. The smallest circle that encloses the grey region must have the points $(0,0)$, $(R,0)$, and $(R \cos \theta, R \sin \theta)$ on its circumference, and so the center of the circle will be equidistant from these three points.

The center must be on the angle bisector (the dashed line depicted in the figure) since the bisector is the locus of points equidistant from $(R,0)$ and $(R \cos \theta, R \sin \theta)$. Therefore, we must find the point on the bisector that is equidistant from $(0,0)$ and $(R,0)$. This point forms an isosceles triangle, and so the distance $r$ can be found using trigonometry:

$\cos \displaystyle \frac{\theta}{2} = \displaystyle \frac{R/2}{r}$,

or

$r = \displaystyle \frac{R}{2} \sec \frac{\theta}{2}$.

This logic works up until $\theta = 90^\circ$, when the isosceles triangle will be a 45-45-90 triangle. However, when $\theta > 90^\circ$, a different picture will be needed. I’ll consider this in tomorrow’s post.

# Engaging students: Radius, Diameter, and Circumference of a Circle

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Zacquiri Rutledge. His topic, from Geometry: radius, diameter, and circumference of a circle.

There are many ideas about how to introduce students and have them study the relationships between the radius, diameter and circumference of a circle. However, one of my favorites has always been the month long project assigned to students at the beginning of class. On the very first day of class, the teacher is to assign the students their project. The instructions of this project are for each of the students to find and measure ten different round or circular objects around their home. The students will need to measure the length around the object (the circumference) using a piece of string and a ruler (the teacher might explain to the students or give an example so they know how to do this), the length from one side of the object to the other side passing through the middle (diameter), and the length from the center of the object to the outside (radius). If the students already know what these terms are called that is okay. However, the teacher should avoid explaining these terms until later.

Then a month later, the students are to bring their findings to class. At this point during the class the teacher will have begun her segment of lessons about circles and the various properties of circles. By now the students should have a good idea what the terms radius, diameter, and circumference mean. So the day the students bring in their work, they will be given the following chart, originally designed by the University of Illinois. From here students will slowly begin to fill in their charts with the information they gathered. Once completed students will then begin finding the ratios between diameter-radius and circumference-diameter and recording them. Finally at the bottom, students will find the average of their ratios from the last two columns. Once all of this data is completed, the students should have found that the diameter and radius share a ratio of 2-1 since the diameter is twice the radius. The last column should have produced something close to an average of 3.14159265359 or better known as pi (). Not only will this help students understand that pi is not just a number, but it will also help them to know where it comes from and its importance. From here the teacher would be able to lead into a lesson about some of the other uses of pi and how they all relate back to the relationships between radius, diameter and circumference.

Radius, diameter, and circumference are very important in many topics beyond their definitions. For instance, later on in the geometry course students will talk about the area of circles. Even though the students might have learned how to find the area of simple polygons such as triangles and quadrilaterals, finding the area of a circle is different because of the use of pi. To find the area of a circle, students have to recall the relationships between the radius, diameter, and circumference in order to understand how the area of a circle uses those relationships. Another example of how they are used is in pre-calculus. In pre-calculus students will talk about the unit circle, a circle with a fixed radius of 1 unit. Using the fixed radius of 1, students will discover that the length around (circumference) the unit circle is 2π. This 2π is important because it can be broken into pieces, called radians, and used to help measure Sine, Cosine, and Tangent at certain radians around the circle. Learning about sine, cosine, and tangent opens up even more things for the students, such as trigonometry and calculus. However, no matter how advanced the mathematics become, they always relate back to the simple concepts of the radius, diameter and circumference of a circle and their relationships.

Radius, diameter, and circumference is a topic that has been talked about and used dating back to 2000 B.C. But, what has it actually been used for all this time? How about architecture? Think about massive constructs such as the Theatre of Ephesus in Rome, Italy. Even though the theatre is not a full circle, look at how each of the seats are evenly placed from the stage. This is because when it was designed, the architect likely used the radius and circumference to accurately plot how far each seat needed to be placed in order to be the same exact distance from the stage as everyone else in their row. Even though only half a circle was used for this theatre, the circumference and radius would have been used to find the ratio pi in order to get the area of how much space was allowed for seating.

Another great example of circumference being used is in the invention of the clock. The clock originated as a sun dial, which would use the sun to cast a shadow, which would tell the time of day. These sun dials date back as early as 3500 B.C. However, in 1583 Galileo found a way to use a pendulum to create a clock that always followed the same length of time (Clock). This is important because not long after the first clock was born, so was the circular face of a clock. The face of a clock has the numbers 1-12 on it, each one evenly spaced around the edge of the clock. By using the circumference of any size of circle, the person building the clock would know just how far to space out each of the numbers, giving each hour the same amount of time between them. If even one of the numbers were off on the clock, the time would be off. Also, it can be seen that on modern clocks, the minute hand always stretches the radius of the clock. By stretching out the minute hand on the clock, the designer of the clock can create evenly spaced notches on the face using the circumference, in order to have the minute hand indicate the minute of the hour.

One final example is the use of radius in war, or more specifically the invention of the radar. Radar was originally being experimented with by German physicist Heinrich Hertz in 1887. He had discovered that certain materials allowed radio waves to pass through them, while others reflected them. In 1890, Nikola Tesla realized that large objects could reflect large enough radio waves to be detected. By harnessing this idea, pulse radar would come to be introduced into United States in 1925, and later used in the British Air Force to defend against German air raids during WWII (Science). The reason radar works, however, is because the system has a set radius in which it can detect radio waves. Once the radar system sends out a radio wave, if it does not reflect back within the radius of the detection system, then the radar will not pick up on anything. The system measures the distance by measuring how long it takes for the radio wave to return to the system after it is sent out and comparing that time to radius of detection. This allowed not only military to defend against air attacks, but it was commonly used during naval combat to defend against submarines as Germany used their U-Boats to attack several American and British naval ships during WWII, as well as WWI before the invention of radar.

References:

“Circumference and Pi.” Circumference and Pi. N.p., n.d. Web. 08 Oct. 2015.

“Clock a History – Timekeepers.” Clock a History – Timekeepers. N.p., n.d. Web. 08 Oct. 2015.

“Science Explorations: Journey Into Space: Radar and Sonar | Scholastic.com.” Science Explorations: Journey Into Space: Radar and Sonar | Scholastic.com. N.p., n.d. Web. 08 Oct. 2015.

# Engaging students: Finding the area of a circle

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Joe Wood. His topic, from Geometry: finding the area of a circle.

The students would be greeted with Lion King’s “Circle of Life” song. While the song has nothing to do with area of a circle, it would create a different and exciting buzz in the classroom that wouldn’t always be offered in this form. (Plus, who doesn’t want to hear a little Lion King music?)

C1. How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

A1. What interesting (i.e., uncontrived) word problems using this topic can your students do now?

A great activity I found on the Mathbits website tackles both questions C1 and A1.
The two-page worksheet below is based off a scene from the movie Castaway. In the scene, Tom Hanks calculates the area of a circle to figure out the likelihood of his rescue. He then compares his calculated area to the area of Texas (which for young students who are all about Texas like I was, this is another attention getter on its own). I would show the clip (having sent a permission slip home since Tom Hanks is shirtless) which can be seen at https://www.youtube.com/watch?v=y89VE9_2Cig so that students can have a good laugh and also understand the scene described on the worksheet. While most, if not all students will never be stranded on a deserted island, this would be an interesting real world problem for the “survivalist” kid in the class.

The worksheet is great because it starts off asking if Tom’s calculations were even correct. It then has several example problems for area of a circle so they can practice, but it also brings in linear speed calculations, and a circumference problem which is great review (and a good warm up if you were maybe moving into angular speed later).

A1. What interesting (i.e., uncontrived) word problems using this topic can your students do now?

Another interesting real world problem can be found at http://spacemath.gsfc.nasa.gov/geometry.html. The problem deals with solar energy on satellites (or solar panels in general). It talks about how much energy is needed to operate a satellite, tells the student how much energy is provided by solar cells per square centimeter, gives them different shaped solar panels, and ask is the solar panel can produce enough energy.

This specific worksheet only uses half of a circle on one problem, so it should be revised by the teacher to include more circles; however, once again, I think keeping all the different shapes is a great review for students. I also think having the semicircular shaped panel is a great idea to keep the students on their toes.

A2. How could you as a teacher create an activity or project that involves your topic?

If one stereotype can be made about middle/high school students (especially the boys), it is that they love to eat! And, what do they like to eat? PIZZA! There are several ways this next idea could be carried out (pun intended), but for the purposes of this assignment I will call it a class project that ends in a pizza party.
The idea is that each pair of student will be assigned a pizza restaurant in the area, and they will do a presentation on why we should order pizza from this pizza place specifically. They will have find all the pizza sizes (small, medium, large, etc.) , their  prices, their diameters, the areas of  each pizza, the price per square inch of each of the pizzas, and the best buy. They can talk about anything else they want (such as quality vs price or customer service or whatever) so long as they are trying to sway the class on why the pizza should be purchased from this specific place. Finally, the students will need to provide some kind of proof of their work (menus, calculations, etc) in an organized fashion: PowerPoint, poster board, or some other method.

After the project is complete, the teacher can select the place to buy from, or hold it to a class vote, and have a pizza party during lunch hour or after school or in class.

# The worst math education video on YouTube

We now have a winner for the worst math education video on YouTube:

My personal favorite part is demonstrating that 140*9 is a multiple of 9 by casting out nines.

Why is this so awful? There are two essential ideas that make this work:

1. Humans have chosen a convention that there are 360 degrees in a circle. There’s nothing particularly magical about 360; that’s just the number that humanity has chosen for measuring angles with degrees. Notice that 360 happens to be a multiple of 9.
2. In base-10 arithmetic, one can check an integer is divisible by 9 by checking if the sum of the digits is a multiple of 9.

The first part of the video shows that, when 360 degrees is successively bisected, the digits of the resulting angle still sum to 9. That’s because dividing by 2 is the same as multiplying by 5 and then dividing by 10. Dividing by 10 is unimportant for the purpose of adding digits, so the only operation that’s important is multiplying by 5. And of course, if a multiple of 9 is multiplied by 5, the product is still a multiple of 9.

Notice that’s important that the angles are successively bisected. If the angles were trisected instead, this would fail (360/3 = 120, which is not a multiple of 9.)

The second part of the video notes that the sum of the angles in a convex polygon is a multiple of 9. That’s because the sum of the angles is (in degrees) $180(n-2)$, which of course is a multiple of 9. Furthermore, this formula is a consequence of the human convention of choosing 360 degrees to measure a complete rotation. From this number, the measure of a straight angle is 180 degrees. From this, the sum of the angles in a triangle is determined to be 180 degrees, and from this the sum of the angles in a convex polygon is found to be $180(n-2)$ degrees. All this to say, there’s nothing mystical about this. The second part of the video is a logical consequence of choosing 360 degrees for measuring circles.

The third part of the video is utter nonsense.

# Engaging students: Finding the equation of a circle

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Tiffany Wilhoit. Her topic, from Precalculus: finding the equation of a circle.

How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)? How has this topic appeared in high culture (art, classical music, theatre, etc.)?

Circles are found everywhere! Everyday, multiple times a day, people come across circles. They are found throughout society. The coins students use to buy sodas are circles. On the news, we hear about crop circles and circular patterns in the fields around the world. One of the first examples of a circle was the wheel. Many logos for large companies involve circles, such as Coca-Cola, Google Chrome, and Target. Even the Roman Coliseum is circular in shape. Since circles are found everywhere, students will be able to identify and be comfortable with the shape (more than say a hexagon). A great way to get the students engaged in the topic of circles would be to have the brainstorm different places they see circles on a normal day. Then have each student pick an example and print or bring a picture of it. Then have the student take their circle (say the Ferris Wheel of the state fair), and place in centered at the origin. The students could then find the equation of their circle. They could do another example where their circle is centered at another point as well. This would allow the students to become more aware of circles around them, and would also allow them some freedom in the assignment.

What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

Circles have been an interesting topic for humans since the beginning. We see the sun as a circle in the sky. The ancient Greeks even believed the circle was the perfect shape. Ancient civilizations built stone circles such as Stonehenge, and circular structures such as the Coliseum. The circle led to the invention of the wheel and gears, as well. The study of geometry is focused largely around the study of circles. The study of circles led to many inventions and ideas. Euclid studied circles, and compared them to other polygons. He found ways to create circles that could circumscribe and inscribe polygons. This created a problem called “squaring a circle”. Ancient Greeks tried to construct a circle and square with the same area using only a compass and straightedge. The problem was never solved, but in 1882 it was proved impossible. However, people still tried to solve the problem and were called “circle squarers”. This became an insult for people who attempted the impossible. Borromean Rings is another puzzle involving circles. Circles have been a part of civilization from the beginning, and it is amazing how much they are still prevalent today.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

The website on www.mathopenref.com/coordgeneralcircle.html is a good site to use when learning to find the equation of a circle. The page contains an applet where the students are able to work with a circle. The circle can be moved so the center is at any point, and the radius can be changed to various sizes. At the top, it shows the equation of the circle shown. This website would allow the students to see how the equation of a circle changes depending on the center and size. This is a good tool to use for the students to explore circles and their equations or to review them before the test. The website also contains some information for the students to read to understand the concept, and there is even an example to try. The website is easy to use, and would not be difficult for students to understand.

Resources:

http://www-history.mcs.st-and.ac.uk/Curves/Circle.html

http://nrich.maths.org/2561

www.mathopenref.com/coordgeneralcircle.html

https://circlesonly.wordpress.com/category/history-of-circles/

# Circumference

Source: http://www.xkcd.com/1184/

Further comments, from Nicholas Vanserg, “Mathmanship,” The American Scientist, Vol. 46, No. 3 (1958):

In an article published a few years ago, the writer intimated with befitting subtlety that since most concepts of science are relatively simple (once you understand them), any ambitious scientist must, in self-preservation, prevent his colleagues from discovering that his ideas are simple too…

The object of… Mathmanship is to place unsuspected obstacles in the way of the pursuer until he is obliged, by a series of delays and frustrations, to give up the chase and concede his mental inferiority to the author…

[U]se a superscript as a key to a real footnote. The knowledge seeker reads that $S$ is $-36.7^{14}$ calories and thinks, “Gee what a whale of a lot of calories,” until he reads to the bottom of the page, finds footnote 14 and says, “oh.”