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 Emma White. Her topic, from Precalculus: finding the equation of a circle.

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

Ironically, this morning on the way to class I received a notification saying Coldplay dropped a new album called “ Music of the Spheres” and I couldn’t help but look into it more! Although we are talking about circles, as mathematicians (or other people who came across this blog), we realize that circles and spheres are related in some ways. Although that is a discussion for another time, I want to focus on this album and how it relates to our world. Circles are used in various ways when it comes to the “circle of life” or “time on a ticking clock”. One song talks about “Humankind” and how we’re designed. This is a continuous cycle as humans pass away and are born and the cycle continues. While this may be a more serious thing to think about, life happens and cycles (we also see this in history and cycles of conflicts, wars, and much more). Furthermore (and maybe on a more lighthearted feel), we see the concept of circle in “The Circle of Life” as seen in “The Lion King”. I encourage you to look at the lyrics below:

“From the day we arrive on the planet

And, blinking, step into the sun

There’s more to see than can ever be seen

More to do than can ever be done

There’s far too much to take in here

More to find than can ever be found

But the sun rolling high

Through the sapphire sky

Keeps great and small on the endless round

It’s the circle of life

And it moves us all

Through despair and hope

Through faith and love

‘Til we find our place

On the path unwinding

In the circle

The circle of life.”

Source: LyricFind

Songwriters: Elton John / Tim Rice

Circle of Life lyrics © Walt Disney Music Company

 Whatever your background may be, we can agree that much in life happens in cycles (think of cells as well!) and that is done in a metaphorical circular motion. The moon rotates around the sun, the planets rotate around the sun, and so forth. Many songs capture the concept of “circling” or time (think of the Sundial), and I bet if we took the time to really dig deep, we could find more songs with this concept more than we think.

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

According to many articles, the discovery of the circle goes way back before recorded history. It started with the Egyptians (the inventors of Geometry) who invented the wheel. I find this intriguing that the people following the Egyptians “investigated” a simple man made tool, the wheel, to go about finding the equation of a circle. I want to emphasize this point because there is so much in life relating to math if only we stop to look and/or think about it more in depth! Furthermore, Euclid (naturally), contributed to the finding of the properties of the circle and “problems of inscribing polygons” (“Circle”, n.d.). Around 650 BC, Thales, a mathematical philosopher who contributed to various elementary geometry theorems, contributed to the theorems regarding circles. Nearly 400 years later, Apollonius, “a Greek mathematician known as ‘The Great Geometer’”, also contributed to the finding of the equation for a circle, specifically the equation itself (J J O’Connor and E F Roberts).  He founded the bipolar equation “mr^2 + nr'^2=c^2 represent[ing] a circle whose centre divides the line segment between the two fixed points of the system in the ratio n  to m” (“Circle”, n.d.). Needless to say, the people who helped create this equation were years apart and it’s pretty cool to see how their work built off of each other over time.

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

When it comes to the equation of a circle, using technology would be a great way to visually show students what is happening and understand where the equation comes from. KhanAcademy is a great resource for students to work through problems and furthermore, Desmos could be a resource for students to use at home for homework to check their work and understand how different values for ‘x’ and ‘y’ change the circle. A beneficial video to share/watch with your students would be “Lesson Video: Equation of a Circle”, for it provides a visual representation of how to derive the equation (I think exposing students to how to derive the equation will make the equation easier to understand and how the equation formulated). Giving your students technological resources is beneficial and I bet the students appreciate having multiple resources to help them become more understanding of the subject matter.

Resources: http://jwilson.coe.uga.edu/EMT668/EMAT6680.F99/Kim/emat6690/instructional%20unit/circle/Circle/Circle.htm

http://britanica.com

http://mathworld.wolfram.com/Circle.html

https://mathshistory.st-andrews.ac.uk/Curves/Circle/

https://mathshistory.st-andrews.ac.uk/Biographies/Apollonius/

https://www.nagwa.com/en/videos/370167476508/

Predicate Logic and Popular Culture (Part 253): Brad Paisley

Let p be the proposition “I love her,” and let q be the statement “I love to fish.” Translate the logical statement

p \land q

This matches the opening words of Brad Paisley’s “I’m Gonna Miss Her,” providing a light-hearted example of how the conjunction but is nevertheless translated as \land.

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Context: This semester, I taught discrete mathematics for the first time. Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.

Predicate Logic and Popular Culture (Part 252): The Two Towers

Let p be the statement “I fear death,” and let q be the statement “I fear pain.” Translate the logical statement

\lnot p \land \lnot q

This matches one of the great lines of \acute{\rm{E}}owyn, shieldmaiden of Rohan, in the book The Two Towers.

Context: Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.

Predicate Logic and Popular Culture (Part 251): Animaniacs

“Survey Ladies” is one of the classics shorts from the 90s cartoon Animaniacs. While none of the survey questions can be stated in predicate logic (after all, they’re questions), there are many, many silly and somewhat repetitive statements that can be motivated by this cartoon:

Let P be the set of all people, let M(x) be the statement “x is watching a movie,” let B(x) be the statement “x is eating beans,” and let G(x) be the statement “x is with George Wendt.” Translate the following into symbolic logic:

  • Nobody is eating beans
  • Somebody is with George Wendt.
  • Somebody is not watching a movie.
  • Everyone watching a movie is eating beans.
  • Nobody watching a movie is with George Wendt.
  • Somebody is watching a movie but is not with George Wendt.
  • Nobody is both eating beans and is with George Wendt.
  • Everyone is watching a movie and is eating beans.

I’ll also share this for anyone who doesn’t remember the greatness of George Wendt:

Context: Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.

Predicate Logic and Popular Culture (Part 250): Marvin Gaye

Let T be the set of all things, let M(x) be the statement “x is a mountain,” let V(x) be the statement “x is a valley,” let R(x) be the statement “x is a river,” let H(x) be the statement “x is high enough to keep me from getting to you, baby,” let L(x) be the statement “x is low enough to keep me from getting to you, baby,” and let W(x) be the statement “x is wide enough to keep me from getting to you, baby.” Translate the logical statement

\sim \exists x \in T ((M(x) \land H(x)) \land (V(x) \land L(x)) \land (R(x) \land W(x)))

This matches the chorus of the timeless “Ain’t No Mountain High Enough” by Marvin Gaye, which has increased in popularity in recent years thanks to the Marvel Cinematic Universe.

Context: Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.

Predicate Logic and Popular Culture (Part 249): Billy Joel

Let p be the statement “We started the fire,” let q be the statement “We lit the fire,” and let r be the statement “We tried to fight the fire.” Translate the logical statement

\sim p \land \sim q \land r

This matches part of the chorus of “We Didn’t Start the Fire” by Billy Joel.

Context: Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.

Predicate Logic and Popular Culture (Part 248): The Fellowship of the Ring

Let T be the set of all things, let P be the set of all people, let G(x) be the statement “x is made of gold,” let B(x) be the statement “x glitters,” let W(x) be the statement “x wanders,” and let L(x) be the statement “x is lost.” Translate the logical statement

\sim \forall x \in T(G(x) \Longrightarrow B(x)) \land \sim \forall x \in P(W(x) \Longrightarrow L(x))

This matches the opening two lines of the poem “All That Is Gold Does Not Glitter” in the book The Fellowship of the Ring.

Context: Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.

Predicate Logic and Popular Culture (Part 247): The Fellowship of the Ring

Let p be the statement “One simply walks into Mordor.” Translate the logical statement

\sim p

Of course, this matches one of the most famous lines, which spawned countless memes, from the movie The Fellowship of the Ring.

Context: Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.

Predicate Logic and Popular Culture (Part 246): Rihanna

Let T be the set of all times, and let D(t) be the statement “I want to do this at time t.” Translate the logical statement

\forall t \in T(\sim D(t))

This matches the opening line of the chorus from “Unfaithful” by Rihanna.

Context: Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.

Predicate Logic and Popular Culture (Part 245): Gibson guitars

Let T be the set of all things, and let G(x) be the statement “x is good enough.” Translate the logical statement

G(\hbox{Gibson}) \land \forall x in T(x \ne \hbox{Gibson} \Longrightarrow \sim G(x))

This existence and uniqueness example matches an old advertising logo for Gibson guitars.

Context: Part of the discrete mathematics course includes an introduction to predicate and propositional logic for our math majors. As you can probably guess from their names, students tend to think these concepts are dry and uninteresting even though they’re very important for their development as math majors.

In an effort to making these topics more appealing, I spent a few days mining the depths of popular culture in a (likely futile) attempt to make these ideas more interesting to my students. In this series, I’d like to share what I found. Naturally, the sources that I found have varying levels of complexity, which is appropriate for students who are first learning prepositional and predicate logic.

When I actually presented these in class, I either presented the logical statement and had my class guess the statement in actual English, or I gave my students the famous quote and them translate it into predicate logic. However, for the purposes of this series, I’ll just present the statement in predicate logic first.