# 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: