# 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. 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. 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.

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.

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/

# Engaging students: Vectors in two dimensions

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 Fidel Gonzales. His topic, from Precalculus: vectors in two dimensions. How can this topic be used in your students’ future courses in mathematics?

When a student learns about vectors in two dimensions, they worry about the magnitude of the vector and the direction that it goes. The direction is kept within its limitations which are up, down, left, and right. A student might be curious as to how this topic can be extended further. The way it extends further is by extending vectors into higher dimensions. It is even possible to extend vectors to the sixth dimension! However, for the sake of showing how vectors in two dimensions extend to future courses in math, we will stick to three-dimensions. Learning about vectors in the second dimension creates groundwork to learn about vectors in the third dimension. With the third dimension, vectors could be seen from our point of view compared to seeing it in the two dimensions on paper. The new perspective of the third dimension in vectors includes up, down, left, right, forward, and backwards. Having the new dimension to account for will give students a bigger tie into how mathematics applies into the real world. How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

Vectors in the two dimension is used all around our everyday life and we as people rarely notice it. The most common use of vectors in our culture is a quantity displaying a magnitude and direction. This is normally done on a x and y graph. Now you might be asking yourself, I do not play any types of games that sound like this. I am here to tell you that you do. One game that iPhone users play without noticing this would be a game on gamepigeon called knockout. The game appears to be an innocent game of knocking out your friends’ penguins while keeping yours in the designated box. However, math is involved, and you probably didn’t notice. First you must anticipate where the enemy is going. Then you must decide how strong you want to launch your penguin troopers without making them fall out of the ring. Does that sound familiar? Having to apply a force (magnitude) and direction to a quantity. Congratulations, you have now had fun doing math. Next time you are playing a game, try to see if there is any involvement of vectors in two dimensions involved. How could you as a teacher create an activity or project that involves your topic?

Vectors in two dimensions has many ways to be incorporated in the classroom. A way to do so while connecting to the real world would be having an activity where the students tell a robot where to go using vectors. The students will have a robot that can walk around and in need of directions. The students will be given maps and asked to create a path for the robot to end up in its destination. Essentially, programming the robot to navigate though a course solely using vectors. If the robot falls or walks too far, then the student will realize that either the magnitude was wrong or the direction. Some students might seem to think this would be impractical to the real world, however, there is always a way to show relevance to students. Towards the end of the activity, the students will be asked to guide me to around the class using vectors. Then to sweeten the deal, they will also be asked to show me on a map being projected to them how to get to McDonald’s. Students will realize that vectors in the second dimension could be used to give directions to somewhere and can be applied to everyday life.  They will walk outside of the classroom seeing math in the real world from a different perspective.

References:

# Engaging students: Compound interest

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 Lydia Rios. Her topic, from Precalculus: compound interest. How did people’s conception of this topic change over time?

While this concept is tied with business which is something that started rapidly changing in the early nineteen hundreds, we have understand that there is an accrued interest on loans long before then. People would loan out seeds or cattle and the interest would be paid after a harvest or with the young of the cattle. Of course now we use this concept mathematically but the concept still holds. We understand that there is a base fee and you must return that fee along with a little more. We then started using this with loose change and then as our currency changed from the gold standard we adapted to a new understanding of compound interest. Today we use the equation $A = P \left(1 + \frac{r}{n} \right)^{nt}$ , where $A$ is the amount accumulated, $P$ is the principal, $r$ is interest rate, $n$ is the compound period and $t$ is the number of periods.

Compound Interest Is Responsible for Modern Civilization (businessinsider.com) What are the contributions of various cultures to this topic?

We have all experienced trade over the years. Native Americans would trade corn for other goods and offered payment plus interest with their corn harvest. The Silks Roads was a network of trading routes where China and other countries would trade textiles and other materials. They established the concept of payment and interest for purchases. Banks in America and other countries also have a set principal and a interest, whether this be in reference to your savings account or the billed interest on your credit card purchases. Even the invention of cars played a part on this and how our interest can decrease with the deterioration of the car. Over the years your interest payment can go down as the worth of the car goes down. How have different cultures throughout time used this topic in their society?

# Engaging students: Deriving the double angle formulas for sine, cosine, and tangent

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 Morgan Mayfield. His topic, from Precalculus: deriving the double angle formulas for sine, cosine, and tangent. How could you as a teacher create an activity or project that involves your topic? I want to provide some variety for opportunities to make this an engaging opportunity for Precalculus students and some Calculus students. Here are my three thoughts: IDEA 1: For precalculus students in a regular or advanced class, have them derive this formula in groups. After students are familiar with the Pythagorean identities and with angle sum identities, group students and ask them to derive a formula for double angles Sin(2θ), Cos(2θ), Tan(2θ). Let them struggle a bit, and if needed give them some hints such as useful formulas and ways to represent multiplication so that it looks like other operations. From here, encourage students to simplify when they can and challenge students to find the other formulas of Cos(2θ). Ask students to speculate instances when each formula for Cos(2θ) would be advantageous. This gives students confidence in their own abilities and show how math is interconnected and not just a bunch of trivial formulas. Lastly, to challenge students, have them come up with an alternative way to prove Tan(2θ), notably Sin(2θ)/Cos(2θ). This would make an appropriate activity for students while having them continue practicing proving trigonometric identities. IDEA 2: This next idea should be implemented for an advanced Precal class, and only when there is some time to spare. Euler was an intelligent man and left us with the Euler’s Formula: $e^{ix}=\cos x + i \sin x$. Have Precalculus students suspend their questions about where it comes from and what it is used for. This is not something they would use in their class. Reassure them that for what they will do, all they need to understand is imaginary numbers, multiplying imaginary numbers, and laws of exponents. Have them plug in x = A + B and simplify the right-hand side of the equation so that we get: $\cos(A+B)+i\sin(A+B)= a + bi$ where $a$ and $b$ are two real numbers. The goal here is to get $\cos(A+B)+i\sin(A+B)= \cos \theta \cos \theta - \sin \theta \sin \theta + (\sin \theta \cos \theta + \cos \theta \sin \theta)i$. All the steps to get to this point is Algebra, nothing out of their grasp. Now, the next part is to really get their brains going about what meaning we can make of this. If they are struggling, have them think about the implications of two imaginary numbers being equal; the coefficient of the real parts and imaginary parts must be equal to each other. Lastly, ask them if these equations seem familiar, where are they from, and what are they called…the angle sum formulas. From here, this can lead into what if x=2A? Students will either brute force the formula again, and others will realize x = A + A and plug it in to the equation they just derived and simplify. This idea is a 2-in-1 steal for the angle sum formulas and double angle formulas. It’s biggest downside is this is for Sin(2θ) and Cos(2θ).   IDEA 3: Take IDEA 2, and put it in a Calculus 2 class. Everything that the precalculus class remains, but now have the paired students prove the Euler’s Formula using Taylor Series. Guide them through using the Taylor Series to figure out a Taylor Series representation of $e^x$, $sin x$,  and $cos x$. Then ask students to find an expanded Taylor Series of to 12 terms with ellipses, no need to evaluate each term, just the precise term. Give hints such as $i^2= -1$ and to consider $i^3=i^2 \cdot i = -i$ and other similar cases. Lastly, ask students to separate the extended series in a way that mimics $a + bi$ using ellipses to shows the series goes to infinity. What they should find is something like this: Look familiar? Well it is the addition of two Taylor Series that represent Sin(x) and Cos(x). This is the last connection students need to make. Give hints to look through their notes to see why the “a” and “b” in the imaginary number look so familiar. This, is just one way to prove Euler’s Formula, then you can continue with IDEA 2 until your students prove the angle sum formulas and double angle formulas. How does this topic extend what your students should have learned in previous courses? Students in Texas will typically be exposed to the Pythagorean Theorem in 8th grade. At this stage, students use $a^2+b^2=c^2$ to find a missing side length. Students may also be exposed to Pythagorean triples at this stage. Then at the Geometry level or in a Trigonometry section, students will be exposed to the Pythagorean Identity. The Identity is $\sin^2 \theta + \cos^2 \theta = 1$.  I think that this is not fair for students to just learn this identity without connecting it to the Pythagorean Theorem. I think it would be a nice challenge student to solve for this identity by using a right triangle with hypotenuse c so that Sin (θ) = b/c and cos (θ) = a/c, one could then show either $c^2 \sin^2 \theta + c^2 \cos^2 \theta = c^2$ and thus $c^2(\sin^2 \theta + \cos^2 \theta) = c^2$ or one could show $(a/c)^2 + (b/c)^2 = (c/c)^2 = 1$ (using the Pythagorean theorem). From here, students learn about the angle addition and subtraction formulas in Precalculus. This is all that they need to derive the double angle formulas. $\sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta$ $\sin(\alpha - \beta) = \sin \alpha \cos \beta - \cos \alpha \sin \beta$ $\cos(\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta$ $\cos(\alpha - \beta) = \cos \alpha \cos \beta + \sin \alpha \sin \beta$ $\tan(\alpha + \beta) = \displaystyle \frac{\tan \alpha + \tan \beta}{1 - \tan \alpha \tan \beta}$ $\tan(\alpha - \beta) = \displaystyle \frac{\tan \alpha - \tan \beta}{1 + \tan \alpha \tan \beta}$

This would be a good challenge exercise for students to do in pairs. Sin(2θ) = Sin(θ + θ), Cos(2 θ) = Cos(θ + θ), Tan(2θ) = Tan(θ + θ). Now we can apply the angle sum formula where both angles are equal: Sin(2θ) = sin(θ)cos(θ) + cos(θ)sin(θ) = 2sin(θ)cos(θ) Cos(2θ) = cos(θ)cos(θ) – sin(θ)sin(θ) =  (We use a Pythagorean Identity here) Tan(2θ) = $\displaystyle \frac{\tan \theta + \tan \theta}{1 - \tan^2 \theta} = \frac{2 \tan \theta}{1-\tan^2 \theta}$ Bonus challenge, use Sin(2θ) and Cos(2θ) to get Tan(2θ). Well, if $\tan \theta = \displaystyle \frac{\sin \theta}{\cos \theta}$, then $\tan 2\theta = \displaystyle \frac{\sin 2\theta}{\cos 2\theta}$ $= \displaystyle \frac{2 \sin \theta \cos \theta}{\cos^2 \theta - \sin^2 \theta}$ $= \displaystyle \frac{ \frac{2 \sin \theta \cos \theta}{\cos^2 \theta} }{ \frac{\cos^2 \theta - \sin^2 \theta}{\cos^2 \theta} }$ $= \displaystyle \frac{2 \tan \theta}{1 - \tan^2 \theta}$

The derivations are straight forward, and I believe that many students get off the hook by not being exposed to deriving many trigonometric identities and taking them as facts. This is in the grasp of an average 10th to 12th grader. What are the contributions of various cultures to this topic? I have included four links that talk about the history of Trigonometry. It seemed that ancient societies would need to know about the Pythagorean Identities and the angles sum formulas to know the double angle formulas. Here is our problem, it’s hard to know who “did it first?” and when “did they know it?”. Mathematical proofs and history were not kept as neatly written record but as oral traditions, entertainment, hobbies, and professions. The truth is that from my reading, many cultures understood the double angle formula to some extent independently of each other, even if there was no formal proof or record of it. Looking back at my answer to B2, it seems that the double angle formula is almost like a corollary to knowing the angle sum formulas, and thus to understand one could imply knowledge of the other. Perhaps, it was just not deemed important to put the double angle formula into a category of its own. Many of the people who figured out these identities were doing it because they were astronomers, navigators, or carpenters (construction). Triangles and circles are very important to these professions. Knowledge of the angle sum formula was known in Ancient China, Ancient India, Egypt, Greece (originally in the form of broken chords theorem by Archimedes), and the wider “Medieval Islamic World”. Do note that that Egypt, Greece, and the Medieval Islamic World were heavily intertwined as being on the east side of the Mediterranean and being important centers of knowledge (i.e. Library of Alexandria.) Here is the thing, their knowledge was not always demonstrated in the same way as we know it today. Some cultures did have functions similar to the modern trigonometric functions today, and an Indian mathematician, Mādhava of Sangamagrāma, figured out the Taylor Series approximations of those functions in the 1400’s. Greece and China for example relayed heavily on displaying knowledge of trigonometry in ideas of the length of lines (rods) as manifestations of variables and numbers. Ancient peoples didn’t have calculators, and they may have defined trigonometric functions in a way that would be correct such as the “law of sines” or a “Taylor series”, but still relied on physical “sine tables” to find a numerical representation of sine to n numbers after the decimal point. How we think of Geometry and Trigonometry today may have come from Descartes’ invention of the Cartesian plane as a convenient way to bridge Algebra and Geometry. References: https://www.mathpages.com/home/kmath205/kmath205.htm https://en.wikipedia.org/wiki/History_of_trigonometry https://www.ima.umn.edu/press-room/mumford-and-pythagoras-theorem

# Engaging students: Computing the cross product of two vectors

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 Chi Lin. Her topic, from Precalculus: computing the cross product of two vectors. How could you as a teacher create an activity or project that involves your topic?

I found one of the real-life examples of the cross product of two vectors on a website called Quora. One person shares an example that when a door is opened or closed, the angular momentum it has is equal to $r \times p$, where $p$ is the linear momentum of the free end of the door being opened or closed, and $r$ is the perpendicular distance from the hinges on which the door rotates and the free end of the door. This example gives me an idea to create an example about designing a room. I try to find an example that closes to my idea and I do find an example. Here is the project that I will design for my students. “If everyone here is a designer and belongs to the same team. The team has a project which is to design a house for a client. Your manager, Mr. Johnson provides a detail of the master room to you and he wants you to calculate the area of the master room to him by the end of the day. He will provide every detail of the master room in three-dimension design paper and send it to you in your email. In the email, he provides that the room ABCD with $\vec{AB} = \langle -2,2,5 \rangle$ and $\vec{AD} = \langle 5,6,3 \rangle$. Find the area of the room  (I will also draw the room (parallelogram ABCD) in three dimensions and show students).”

Reference:

https://www.quora.com/What-are-some-daily-life-examples-of-dot-and-cross-vector-products

https://www.nagwa.com/en/videos/903162413640/ How does this topic extend what your students should have learned in previous courses?

This topic is talking about computing cross product of two vectors in three dimensions. First, students should have learned what a vector is. Second, students should know how to represent vectors and points in space and how to distinguish vectors and points. Notice that when students try to write the vector in space, they need to use the arrow. Next, since we are talking about how to distinguish the vectors and the points, here students should learn the notations of vectors and what each notation means. For example, $\vec{v} = 1{\bf i} + 2 {\bf j} + 3 {\bf k}$. Notice that $1{\bf i} + 2 {\bf j} + 3 {\bf k}$ represents the vectors in three dimensions. After understanding the definition of the vectors, students are going to learn how to do the operation of vectors. They start with doing the addition and scalar multiplication, and magnitude. One more thing that students should learn before learning the cross product which is the dot product. However, students should understand and master how to do the vector operation before they learn the dot product since the dot product is not easy. Students should have learned these concepts and do practices to make sure they are familiar with the vector before they learn the cross products.

References: How did people’s conception of this topic change over time?

Most people have the misconception that the cross product of two vectors is another vector. Also, the majority of calculus textbooks have the same misconception that the cross product of two vectors is just simply another vector. However, as time goes on, mathematicians and scientists can explain by starting from the perspective of dyadic instead of the traditional short‐sighted definition. Also, we can represent the multiplication of vectors by showing it in a geometrical picture to prove that encompasses both the dot and cross products in any number of dimensions in terms of orthogonal unit vector components. Also, by using the way that the limitation of such an entity to exactly a three‐dimensional space does not allow for one of the three metric motions (reflection in a mirror). We can understand that the intrinsic difference between true vectors and pseudo‐vectors.

Reference:

https://www.tandfonline.com/doi/abs/10.1080/0020739970280407

# Engaging students: Exponential Growth and Decay

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 Ashlyn Farley. Her topic, from Precalculus: exponential growth and decay. The most current example of exponential growth and decay is with the global pandemic, Covid 19. One example is that The Washington Post wrote an article stating that “The spread of coronavirus boils down to a simple math lesson.” The article goes on to explain what exponential growth is and how that applies to Covid 19.  Another website, ourworldindata.org, has a graph of the daily new cases of Covid 19. This graph allows one to see the information for multiple countries, and starts on January 28th 2020 until Today, whatever day that you may be viewing it. Many other news sources also have graphs and information on the growth, and decay in some cases, of the pandemic situation. Teachers can use this information to easily make a connection from math class to the real world. One idea of teaching graphing exponential functions so that it is engaging is to use a project over the zombie apocalypse. The spread of a disease is a common and great example of exponential functions, so although this disease is pretend, the idea can be applied in the real world, like with a global pandemic. Three examples of projects are:

• News Reporters
• This project has the students analyzing data they received to best report to the people who are dealing with the outbreak. It allows students to learn how to read the graphs of exponential functions, understand the functions, integrate technology into the class by creating news reports, and practice an actual career.
• Government Officials
• This project has the students running a simulation of their city. They are to use the statistics of a city to see what the impact of a zombie outbreak would be. After finding the best and worst case scenario, they are to write a letter to the mayor of the city that explains the scenarios so that government can implement plans to keep the outbreak to a minimum. This allows the students a chance to practice analyzing exponential functions, modifying exponential functions, and informing others of the meaning of the functions and modifications.
• Scientists
• This project has the students predicting the outcome of a zombie outbreak, finding a cure, and determining at what point is the zombie population controlled. The students will get practice with the exponential functions, making changes to the functions, finding the point of “control”, as well as creating an action plan.

Each of these projects can be used separately or can be combined to create one major project to learn about exponential functions and their graphs. The goal is to get students excited about learning math instead of dreading it. Math is used daily, even if the students don’t realize it, so the understanding of real-life implications is very important for a teacher to bring into the classroom. Of the many websites, one key website for educators trying to make lessons engaging is YouTube. YouTube has songs, such as the Exponential Function Music Video, explanatory videos, such as from Kahn Academy, and allows students to create their own videos about the topics. Explanatory videos may help students get a specific idea they didn’t quite understand in class, music is very catchy allowing quick memorization of information, and creating videos shows that the students truly have an understanding of the material. By giving the students multiple types of representation of the material, allows all types of learners a chance to understand the material. Multiple representations is very important in keeping students engaged in the class and having them truly learn the material.

Resources:

https://www.teacherspayteachers.com/Product/Zombie-Apocalypse-Exponential-Function-Pandemics-21st-Century-Math-Project-767712?epik=dj0yJnU9UnRuNHVLLUxrV0JkTVJQc1ZFY0szb3JJNXRyenQwb2omcD0wJm49aEQ2UjFHVUcyYm5FakE1ZXhSXzhpQSZ0PUFBQUFBR0ZnTWRB

https://medium.com/innovative-instruction/math-mini-project-idea-the-zombie-apocalypse-5ddd0e6af389#.sph1x08k8

https://www.teacherspayteachers.com/Product/Exponential-Growth-and-Decay-Activity-Exponential-Functions-Zombie-Apocalypse-2609226

https://ourworldindata.org/coronavirus/country/united-states

https://www.washingtonpost.com/weather/2020/03/27/what-does-exponential-growth-mean-context-covid-19/

# Engaging students: Finding the domain and range of a function

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 Sydney Araujo. Her topic, from Precalculus: finding the domain and range of a function. How can this topic be used in your students’ future courses in mathematics or science?

Expanding on finding the domain, this topic is frequently seen in calculus classes. Students need to understand the domain to understand and find limits of functions. Continuity directly expands on domain & range and how it works. We also see domain and range when students are exploring projectile motion. This makes since because when we think about projectile motion, we think about parabolas. With projectile motion there is a definite start, end, and peak height of the projectile. So we can use the domain to show how far the projectile travels and the range to show how high it travels. Looking even further ahead when students start to explore different functions and sets, they start to learn about a codomain and comparing it to the range which is a very valuable concept when you start to learn about injective, surjective, and bijective functions. How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

Desmos is a great website for students to use when exploring domain and ranges. Desmos has premade inquiry-based lessons for students to explore different topics. Teachers also have the option of creating their own lessons and visuals for their students to interact with. Desmos can also animate functions by showing how they change with a sliding bar or actually animate and show it move. This would be a great tool to use for students to visually understand domain and ranges as well as how they are affected when asymptotes and holes appear. This would also be great for ELLs because instead of focusing on just math vocabulary, they can actually visually see how it connects to the graph and the equation. For example, https://www.desmos.com/calculator/vz4fjtugk9, this ready-made desmos activity actually shows how restricting the domain and range effects the graph and what parts of the graph are actually included with the given domain and range. How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

Like I discussed earlier, domain and range is directly used in calculus. In the movie Stand and Deliver, they directly discuss the domain and range of functions. The movie Stand and Deliver is about a Los Angeles high school teacher, Jaime Escalante, who takes on a troublesome group of students in a not great neighborhood and teaches them math. He gets to the point where he wants to teach them calculus so they can take the advanced placement test. If they pass the advanced placement test then they get college credit which would motivate them to actually go to college and make a better life for themselves. However through great teaching and intensive studying, the students as a whole ace the exam but because of their backgrounds they are accused of cheating and must retake the exam. There is a few scenes, but one in particular where the students are finally understanding key concepts in calculus and Mr. Escalante is having them all say the domain of the function repeatedly.

# Engaging students: Deriving the distance formula

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 Geometry: deriving the distance formula. How does this topic extend what your students should have learned in previous courses?

In previous courses, it’s safe to say around 8th grade, students learn the Pythagorean Theorem ( $a^2+b^2=c^2$). This deals with the sides and length of a triangle. The Distance Formula is the same concept but with coordinate values and finding the length of a so-called “distance”. We could go as far as to say that the formula can use earthly coordinates, such as North, South, East, West, and all that fall in between. Since the students are familiar with the Pythagorean Theorem, introducing the Distance Formula is a small step up. Another concept that is extended is building on the idea of coordinate points and understanding word problems. As stated earlier, the Distance Formula uses point on a coordinate graph and this can be transformed into a mapping concept, with compass directions. With this topic, students must extend their knowledge on word problems talking about “45 degrees south of east” and “30 degrees north of west” and how to apply this to coordinates. What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

Wow, the people who contributed to the discovery and development of the Distance Formula might as well be some of the biggest nerds Math majors know. A man by the name of Euclid (known as the father of Geometry) is who started the foundation for this formula. Euclid, as stated in his third Axiom, said it is “possible to construct a circle with any point as its center and with a radius of any length” (also Postulate 3 in “Euclid’s Elements: Book I”). Comparing the Distance Formula to a circle may seem a little confusing but let me challenge you to think again. Look at the standard form of the equation of a circle below: $r^2 = (x-h)^2+(y-k)^2$

Now look at the Distance Formula: $d = \sqrt{(x_1-x_0)^2+(y_1-y_0)^2}$

There are some similarities, right? Pretty close similarities too! A traveler, scientist, and philosopher by the name of Pythagoras took this idea from Euclid and ran with it, essentially being the man who invented the Distance Formula, or what is called the “Pythagorean Theorem. What interests me the most about this man is that he was a traveler, and he created the “Distance Formula” (get it, because he traveled distances…I thought that was ironic). Lastly, we must recognize Renee DesCartes (he developed the coordinate system which is connected to geometry and the Distance Formula uses these coordinates). Euclid, Pythagoras, and DesCartes contributed to the discovery of the Distance Formula and the development was so exemplifying that many, many, many occupations use it 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?

I believe technology is close to vital resource when teaching the Distance Formula to students. I say this because the topic is having to do with “going/finding a certain distance” per say. Having access to visuals helps the students put an idea to a tangible concept they experience every day, traveling. The resource below from Desmos is a prime example of how teachers can use technology to teach a lesson and make it interactive. Khan Academy also has some videos in which students can watch and follow along. Even more so, Khan Academy took a scenario from an athlete perspective and answered his question using the Pythagorean Theorem and Distance Formula. Having real life scenarios is what draws students to be engaged. If a student walks into a lesson not knowing the “why”, why are they going to want to sit through your class with a topic they see as useless? Therefore, I think technology, especially visuals (such as Desmos) and the Khan Academy example, would be beneficial for teachers to use in their classrooms when teaching the Distance Formula.

Reference(s):

http://harvardcapstone.weebly.com/history2.html

https://mathcs.clarku.edu/~djoyce/elements/bookI/post3.html

https://www.desmos.com/calculator/s7blqjtusy

https://teacher.desmos.com/activitybuilder/custom/5600a868e795241d06683511

https://www.chilimath.com/lessons/intermediate-algebra/derivation-of-distance-formula/

# Engaging students: Deriving the proportions of a 45-45-90 right triangle

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 Morgan Mayfield. His topic, from Geometry: deriving the proportions of a 45-45-90 right triangle. How could you as a teacher create an activity or project that involves your topic?

I included a lesson plan from Virgina Lynch of Oklahoma Panhandle State University. In her lesson plan, she includes a section where students draw a 45-45-90 triangle, or right-isosceles triangles, and then uses the variable x for the leg lengths to prove the proportion for students. Then, she uses a section where she has students cut out actual 45-45-90 triangles with 4-in leg lengths. Each student measures their hypotenuse to some degree of accuracy and reports their length. Lastly, Ms.Lynch averages the lengths and has students divide the average by root 2 on a calculator to show that the answer is incredibly close to 4.

My likes: These are two different styles of proving the 1:1:root 2 proportions of a triangle for students: one mathematical and the other more deductive after knowing the mathematical proof. This provides students with an auditory, tactile, and visual way to understand the proportion of the side lengths. I think that the tactile part can be the biggest thing for students. Rarely do we end up building a triangle and measuring its sides to show that this relationship makes rough sense in the real world.

My adaptation: In a geometry class, I would find the mathematical proof to be a fun exercise for students to flex their understanding of algebra, geometry, and the Pythagorean theorem. I would group students up and probably help them start connecting the algebra portion by giving them the leg length “x” and saying I want to know the length of the hypotenuse in exact terms. Group members can collaborate and use their collective knowledge to apply the understanding that a 45-45-90 triangle is isosceles and right, then use the Pythagorean theorem to find the length of the hypotenuse in terms of x.

Then, I would have some groups cut out 45-45-90 triangles of some leg length and other groups cut out 45-45-90 triangles of some other leg length to have more variety, but still show the root 2 proportion in our physical environment. How can this topic be used in your students’ future courses in mathematics or science?

45-45-90 triangles are very helpful in understanding the unit circle. This may be taught at the geometry level or in precalculus. In a unit circle, our radius is 1, so when we want to know the sine or cosine of 45 degrees or $45+ \pi/2$, then we can apply the relationship that we already know about 45-45-90 triangles. So, on the unit circle, build a right triangle where the hypotenuse connects the center to the circumference of the circle at a 45-degree angle from the x-axis. Since the triangle is both right and has one 45-degree angle, we know the other angle is 45 degrees as well. This should immediately invoke the sacred root 2 ratio, but this time we only know the length of the hypotenuse, which is 1, which is the radius. Thus, we divide the radius, 1, by root 2, and then get rid of the root 2 from the denominator to get $\sqrt{2}/2$ for both legs. Lastly, we apply our knowledge of sine and cosine to understand that sine of an angle in a right triangle, that is not the right angle, is the “length of the opposite side over the hypotenuse”, which is just $\sqrt{2}/2$ because we have the convenience of being in a unit circle. How does this topic extend what your students should have learned in previous courses?

The basis for understanding a 45-45-90 triangle takes its understanding from 8th grade math when students are introduced to the Pythagorean theorem and the beginning of the geometry course when students cover identities of isosceles triangles, mostly from a Euclid perspective. Even before that, students learn other basic things about triangles such as the interior angles add up to 180 degrees and that a right triangle has a 90-degree angle.

This is how students connect the three Euclid book I propositions: 5, 6, and 47. Students learn that from propositions 5 and 6 in a geometry class, isosceles triangles have two sides of equal length which imply the angles between those equal sides and the third sides are equal and vice-versa. So, a 45-45-90 triangle implies that it has two equal sides, which are the legs of the right triangle. Now, we apply proposition 47, the Pythagorean theorem because this is a right triangle, to then show algebraically the hypotenuse is $x\sqrt{2}$ where $x$ is the length of one of the legs. How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

I find the topic of “Dynamic Rectangles” and “Dynamic Symmetry” very fascinating. This is frequently used in art, usually in drawing, painting, and photography. Jay Hambridge formalized the idea that classical art used Dynamic Symmetry which includes the ratio of 1:. This ratio is usually built inside of a rectangle or square to give very interesting, symmetrical focal points within a piece that could not be achieved within just any regular rectangle. The photothunk blog below details how the diagonals of the dynamic rectangles and the perpendiculars to the diagonals form a special symmetry that is lost when used in a rectangle that doesn’t have the 1: $\sqrt{x}$ ratio. For example, I’ve included a piece of art by Thomas Kegler and a Youtube analysis of the piece of art that uses Root 2 Dynamic Symmetry.

What does this mean for the 45-45-90 triangles? Well, to build these dynamic rectangles, we must start off with a square. Think about the diagonal of a square. When we form this diagonal, we form a right triangle with two 45-degree angles. All squares are two 45-45-90 triangles. Now, using the length of the diagonal, which we know mathematically to be $x\sqrt{2}$ where $x$ is the length of one of the legs, we can build our dynamic rectangle and then build other dynamic rectangles because $1^2 + (\sqrt{x})^2 = x+1$ . I’ve included a diagram I made in Geogebra to show off a way to build the root 2 dynamic rectangle using just circles and lines.

Starting with a square ABCD, we can place two circles with centers C and D and radii AC. Why AC? This is because AC is the diagonal of the square, which we know to be $x\sqrt{2}$ where $x$ is the length of one of the sides of the square. Now, we know our radii is equal to $x\sqrt{2}$. We can extend the sides of our square CB and DA to find the intersection points of the circles and the extended lines E and F. Now, all we must do is connect E to F and voila, we have a root 2 dynamic rectangle FECD.

How have different cultures throughout time used this topic in their society?

This answer will be my most speculative answer using concepts of the 45-45-90 triangles. First, I must ask the reader to suspend the round world belief and act that we live on a relatively flat plane of existence. Our societies have been build around organizing land into rectangular and square shaped pieces of land. I will talk about the “Are” system which has shaped a lot of Western Europe and the Americas due to colonization by the European powers. You may have heard the term “hectare”, which is still popular in the United States. It is literally a mash up of the words “hecto-”, coming from Greek and meaning one-hundred and “are”, coming from Latin and meaning area. So, this is 100 ares, which is a measure of land that is 10 meters x 10 meters. That means a hectare is 100 meters x 100 meters.

Well, one would imagine that with Greek, Latin, and Western European obsession with symmetry, we would want to split these square pieces of land in half with many different diagonals, so it must have been useful to understand the proportions of the 45-45-90 triangle to makes paths and roads that travel from one end of the hectare to the other end efficiently while also utilizing the space and human travel within the hectare efficiently. Again, this is my speculation, but knowing that two 45-45-90 triangles form a square means that all squares and symmetry involve using this 1: $\sqrt{2}$ ratio; they are inseparable.

References:

https://ipoxstudios.com/the-simplicity-and-beauty-of-dynamic-symmetry-visual-glossary/

http://photothunk.blogspot.com/2016/03/dynamic-symmetry-and-jay-hambridge.html

http://www.opsu.edu/www/education/MATH-ESE%204%20ALL/Virginia%20Lynch/Special%20Right%20Triangles-%20Lesson%20Plan.pdf

https://en.wikipedia.org/wiki/Hectare

# Engaging students: Midpoint formula

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 again comes from my former student Chi Lin. Her topic, from Geometry: deriving the midpoint formula. What interesting (i.e., uncontrived) word problems using this topic can your students do now? (You may find resources such as http://www.spacemath.nasa.gov to be very helpful in this regard; feel free to suggest others.)

To let students engage in the topic, as teachers, we want to create some good examples for students to let them interested in doing it. We need to know what students are interested in or students realize they can use this knowledge in the real life. For example, if students like eating pizza, then I will create some examples about pizza or some delicious food and using pizza representation to raise their attention. In this topic, since we are going to talk about the midpoint formula, one of the real-world problems that I can come up with is using Google Maps. I will show a big Google map of the US in the class, and I will ask students question that “Miss Lin is planning a road trip from Dallas to Arizona on Thanksgiving. However, she wants to split the driving into two days. Now Miss Lin needs your help to figure out what is the middle city (midpoint) between Texas (Dallas) to Arizona.” After students talk with their groupmates, I will invite students to come to the map and circle the city that their think is the middle city between Texas (Dallas) to Arizona and explain their thoughts as well. How does this topic extend what your students should have learned in previous courses?

Khan Academy shows that what students show know before we learn how to derive the midpoint formula. It gives some details which help the teacher to prepare the lesson. First, students should know points in the coordinate plane. Students should require describing every point on the plane with an ordered pair in the form  correctly. Second, students have learned how to use addition, subtraction, and square with negative numbers. Students need to know the distance and slope between points on the coordinate plane, how to represent points on the left or below the original point. Third, students have learned the distance and displacement between points to calculate the slope. Students need to understand what absolute value is as well. The last thing I think students should have learned in the previous class is the slope and square root.

Reference: 