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 Angelica Albarracin. Her topic, from Precalculus: exponential growth and decay.

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

During my freshman year of high school, my school offered AP Human Geography. One of the most important figures you learn about in this class is Thomas Malthus, who was an English economist and demographer during the late 1700s and early 1800s. Malthus was most known for his theory that population growth would outpace the world’s food supply. His argument was that since population grows at an exponential rate, and food supply at the time was increasing at a linear rate, then the world would run out of food in a short amount of time. Of course, today we know that Malthus’s theory was incorrect because it did not account for the profound effect that the industrial revolution would have on agriculture. However, if this theory were to be explained to a group of people who may not know what the difference between a linear and exponential function is, the usage of a graph as a visual aid would be extremely helpful.

Given this premise, students may be asked to create a graph with given coordinates to compare the difference between a linear and exponential graph, allowing students to see for themselves why this theory may have been extremely alarming to people during this time. After this, the students may be presented with several different scenarios such as “Graph a constant population of 1 billion vs. a rapidly declining food supply due to locust swarms” or “Graph a sudden population boom 5 years prior to a boom in food supply that increases at twice the rate of the population”. Students could be asked questions such as “Will the population have enough food to survive?” or “How many years will it take for there to be enough food to feed the entire population?”. I think this would be an extremely engaging activity for students as the premise behind it is an interesting piece of mathematical history and students’ imaginations can be engaged during the different scenarios.

How can this this topic be used in your students’ future courses in mathematics or science?

Exponential growth functions are commonly used to model the population growth of a species in Environmental Science. An important concept in Environmental Science is carrying capacity, which is the largest population a habit can support without degradation. Due to the carrying capacity, we typically see S-curves in the population models in Environmental Science as opposed to the normal J-curves. When students are familiar with the rapid rate in which exponential functions can grow, it provides intuitive reasoning for why carrying capacity exists in nature as habits very clearly have a finite amount of resources that cannot possibly support an infinitely growing population.

The concept of radioactive decay and half-lives is also very important in Chemistry. A half-life is a measure of the amount of time it takes for half of a radioactive isotope to decay. While not all isotopes are radioactive, the ones that are decay at an exponential rate. Having knowledge of an isotopes half-life enables scientists to handle such material safely. Typically, scientists wait to handle such radioactive material until it has decayed below detection limits, which occurs around 10 half-lives. Beyond this, doctors must also use their knowledge of half-lives when using radioactive isotopes to help treat patients. For a radioactive isotope to be useful in this manner, its radioactivity must be active enough to treat the condition, but not too long as to harm healthy cells.

How has this topic appeared in the news?

Historically, exponential growth and decay graphs have been used to model the spread of epidemics/pandemics. Recently, with the advent of the Covid-19 epidemic, we are constantly seeing such graphs all over the news and agency websites such as the CDC. In the graph depicted below, we can see exponential growth in the number of cases around March, a small decline, and then another bout of exponential growth around June. Of course, in the real world, very few data follow an exact mathematical form so using the phrase “exponential growth” is an approximation. However, this exponential trend demonstrates just how contagious this virus is as we can see how thousands of people can be affected in a short amount of time.

This image has an empty alt attribute; its file name is covid.png

During the Australian bushfires that occurred during January 2020, many articles began to attribute this disaster with climate change due to human activity. Though the causes of wildfires are highly variable and difficult to track, many scientists felt that Australia’s record warmth and dryness during the previous year, at the very least, allowed the fires to spread much quicker. In the graph below, we can see a slight trend between the climate change seen in Australia (as recorded by the Australian Bureau of Meteorology (BOM)) versus the average climate change seen around the world by 41 models. A line of best fit has been drawn through the graph of 41 climate models, though hard to see, allows us to see more clearly that this data set increases at an exponential rate. While it is still difficult to determine whether this climate change can be directly attributed to the wildfires, we can still see our risk for such disasters increase as time goes on.

This image has an empty alt attribute; its file name is wildfire.png

References:

https://www.britannica.com/biography/Thomas-Malthus

https://opentextbc.ca/introductorychemistry/chapter/half-life-2/#:~:text=An%20interesting%20and%20useful%20aspect,initial%20amount%20of%20that%20isotope.

https://www.dummies.com/education/science/chemistry/nuclear-chemistry-half-lives-and-radioactive-dating/

https://covid.cdc.gov/covid-data-tracker/#trends_dailytrendscases

http://www.bom.gov.au/climate/change/index.shtml#tabs=Tracker&tracker=timeseries&tQ=graph%3Dtmax%26area%3Daus%26season%3D0112%26ave_yr%3D0

https://www.nytimes.com/2020/03/04/climate/australia-wildfires-climate-change.html

Predicate Logic and Popular Culture (Part 234): Linkin Park

Let $p$ be the statement “They turn down the lights,” and let $q$ be the statement “I hear my battle symphony.” Translate the logical statement

$p \Rightarrow q$ .

This matches part of the chorus of “Battle Symphony” by Linkin Park.

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.

Engaging students: Solving exponential equations

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 Jesus Alanis. His topic, from Precalculus: solving exponential equations.

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

An activity for solving exponential equations is Bingo. If you know how to play Bingo, you know that there are many ways to win. You could either have five in a row, blackout, in an X and 4 corners. In the regular Bingo game, you have a free space, but it is up to you if you want to have a free space or add an extra problem on there. The way I would do the bingo cards is use all the spaces so that means I must create 25 equations with graphs. I am using this website as a reference to get some ideas on how to setup and may even borrow some graphs and equations. The way I would set it up is on the bingo card to have a mix of both equations and graphs. I would also create like a class set and place them in sheet protectors so the students can use expo markers. Since students cannot write on the bingo card, give the students scratch paper so that the students are able to work it out. Once students have solved their Bingo cards, we would start the game, and this would make students not have to worry about a time limit. Students could just play and check their work as well since the students will have the same graphs and equations. During the game, you as the teacher could go over the question and this would be a good time to teach students or show students how the problem will be solved and the answer. This will also give students the how and why the answer is the answer.

How has this topic appeared in the news?

The way exponential equations have appeared in the news is in our current times we are in a pandemic. The coronavirus pandemic to be specific. When the pandemic first started and quarantine had been placed, the news was talking about the number of cases that were being reported. The news had displayed a graph of the number of cases that had happen in a few days. Now the graph has changed to months and the graph is an example of an exponential function. The coronavirus has been a very contagious disease that has taken deaths and sadly there is a graph for this to and it is exponential. The graphs that are being displayed are of exponential function and sadly they are exponential growth functions. This is also a real-world connection of exponential equations and why they are used.

How can technology be used to effectively engage students with this topic?

The way technology can be used to effectively engage students to exponential equations is to show or make students hear the song Billionaire with Bruno Mars. Using the song will make students wake up and be ready for class. It is up to you how long you want to play the song, or you could have it as background music while having these questions posted either on your whiteboard or projector. The question is “Would you rather be given million dollars right now or be given one penny today and each day be given double what you were given the day before for thirty days?”. This question will make students think and start to do math. The question talks about the penny and double each previous day’s amount. The value earned is exponential growing. This could also introduce the lesson and reference it to businesses and how they work. This could also be a life lesson about being patient and how things take time to become successful.

Reference

Foresta, Stephanie. “Exponential Equations Bingo.” Forestamath.com, 2013, http://www.forestamath.com/previews/Preview_Game_ExpoEquationBingo.pdf
Mars, Bruno and Travie McCoy, directors. Billionaire. YouTube, Vidx, 2011, https://www.youtube.com/watch?v=MQyl2ObYKaM

Predicate Logic and Popular Culture (Part 233): Panic! At The Disco

Let $F(x)$ be the statement “ $x$ feels good,” let $H(x)$ be the statement “ $x$ tastes good,” let $M(x)$ be the statement “ $x$ is mine,” and let $H$ be the set of all things. Translate the logical statement

$\forall x \in H( (F(x) \land H(x)) \Rightarrow M(x))$ .

This matches a line from “Emperor’s New Clothes” by Panic! At The Disco.

Engaging students: Finding the area of a right triangle

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 Trenton Hicks. His topic, from Geometry: finding the area of a right triangle.

As an engagement activity, give students the following problem: “A rectangle has dimensions of base: b and height: h. How many different ways can you cut this rectangle in half with a straight line? How many shapes can you make from the different ways you cut the rectangle in half? Now out of those, which shapes are not rectangles?”

This will leave only two different ways to cut the rectangle in half, yielding identical triangles on either side of the line. Now, ask the students, “From what we’ve already learned about rectangles, what would be the area of this rectangle?” After confirming the area is base times height, wait a few moments before saying anything else. Now that the students are thinking about the base, they will now start to make predictions about the triangles that we’ve just made and their areas. Have them write their guesses for the formula of the triangle down on a piece of paper, and keep them to the side through the lesson. From here, we can break them up into groups and give them 3 right triangles to solve for the area, and one equilateral triangle to solve for the area. Go through the answers together and compare groups’ answers, as well as their predictions on what the area of a triangle is. The odds are, many groups will be stuck once they get to the equilateral triangle. If so, you may want to send them back to their groups to try and find the area, giving them the hint, that they may have to make new shapes, just as they did with our rectangle at the beginning. This lesson assumes that the students understand the pythagorean theorem so they can solve for the height of an equilateral triangle by making a new triangle. This way, the students can explore the phenomena of triangles’ area, and see if they can recognize that the height isn’t always a side of the triangle, but rather something they may have to solve for.

The students should be able to use similar techniques to find the area of a parallelogram, trapezoids, and other shapes, as these shapes are partially composed of triangles. As students progress to more complex 2-dimensional shapes, you can derive formulas as you go. As you move onto 3-dimensional shapes, we actually see lots of different triangles appear in the shapes’ respective nets. For instance, when computing the surface area of a triangular prism, we need to know how to compute the area of the base. We also see this same idea in computing volumes of triangular prisms, where we need to know the area of the triangular base. This is also applicable to pyramids, tetrahedrons, and octahedrons. Finally, these ideas are brought up again later in trigonometry where we can determine different parts of the formula with trigonometric ratios and functions and whenever the students learn Heron’s formula.

This concept of finding the area of a triangle expands many things that the students may already know. This won’t be the students’ first time seeing a triangle, nor will it be the first time they compute the area. Overall, this content should be a refresher and not new to the students. However, this may be the first time that the students are presented a rectangle and told to make a triangle out of it. From that point, they are told to make conclusions about the triangle’s area based on the rectangles area. As students think through this, they are using logic and reasoning to argue what makes geometric sense to one another. This further develops their mathematical reasoning skills, which may be a bit rusty since we far too often focus on the “what” and not the “why” and “how.”

Predicate Logic and Popular Culture (Part 232): Limp Bizkit

Let $B(x)$ be the statement “ $x$ knows what it’s like to be the bad man,” let $H(x)$ be the statement “ $x$ knows what it’s like to be hated,” and let $P$ be the set of all people. Translate the logical statement

$\forall x \in P(\lnot B(x) \land \lnot H(x))$ .

This matches the opening lines of “Behind Blue Eyes” by Limp Bizkit.

Engaging students: Defining the terms complementary angles, supplementary angles, and vertical angles

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 Mason Maynard. His topic, from Geometry: defining the terms complementary angles, supplementary angles, and vertical angles.

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 found this website that is an interactive game that students can play when learning about this topic. With this type of topic, the main thing that you want your students to remember is vocabulary. The website that I found uses a game to get kids identify the missing angle in degrees and each angle is either complementary, supplementary or vertical. I really like this game because our students need to develop that muscle memory of seeing an angle and knowing whether it is a complementary, supplementary or vertical set. Once you can get the students to see it and immediately identify it, they can then transition into finding the specific degrees. I also think that anytime you can put something into a game format, students will try harder. Everyone is competitive so why not channel that into learning. The game on the website is very straight forward with the students so you can count on it not causing any misconceptions.

https://www.mathgames.com/skill/8.85-complementary-supplementary-vertical-and-adjacent-angles

How has this topic appeared in high culture?

The article that I found touches on angles but I feel like you could use in throughout the entire unit and just touch on in during every topic you cover. Overall, the article refers to the history of the Geometric Abstract Art Movement. It mainly focuses on the use of lines and shapes and angles but I really feel like you could connect this to the students in your classroom. Within a lot of these paintings or sculptures during this period, you will find all three angle types. These artists needed these angles to make the piece balanced and have harmony. Other needs to use a specific angle to demonstrate contrast. That is really the most beautiful thing about mixing art with math. The artist has the power to use it in a way that conveys their feelings and allows for expression. This is really a way to go beyond the scope of math and show students that we are learning real life and important topics.

https://www.kooness.com/posts/magazine/the-history-of-geometric-abstract-art

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

An activity that I found online was that you just give students papers and have them fold them in specific patterns and then they use the protractor to measure out the degree of angles. I really like this because it is simple but yet you can branch out with it in many ways. With students folding papers, you will get many different folds from the students and this allows them to do some investigation on their own and then afterward, you can allow them to share their findings with their classmates. This cooperative learning allows for all of them to pounce ideas of one another and for the teacher, it can show you who is struggling with anything specific. The really cool thing about it is that if you fold the paper twice then you can setup the scenario of them finding adjacent angles. Then this could potentially lead them to discovering opposite angles on their own for future lessons.

https://www.oercommons.org/courseware/lesson/3311/overview

Predicate Logic and Popular Culture (Part 231): Aristocats

Let $C(x)$ be the statement “ $x$ wants to be a cat,” and let $P$ be the set of all people. Translate the logical statement

$\forall x in P(C(x))$ .

This matches the opening line of “Everyone Wants to be a Cat” from the movie “The Aristocats.”

Engaging students: Using a truth table

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 Jonathan Chen. His topic, from Geometry: using a truth table.

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

There are many kinds word problems that the students will be able to understand using truth tables. Truth tables are very common and appear in everyone’s life. Some of these problems may not even appear in a math class. Statements such as, “Sarah has a cat and the Sarah’s cat is a tabby” can be broken down on a truth table and see if the statement is true or false. While the topic of truth tables is very basic, the concept of math helping students getting better at English and understanding statements is truly shocking and revolutionary to students. The misconception that math cannot help a student’s ability to better understand or speak English is not true because concepts such as truth tables have the students look closely at the sentences to determine if a statement is true or false. This can help students better understand and connect how math can build upon a student’s skill to better understand a language.

How can this topic be used in your student’ future courses in mathematics or science?

This topic reappears when doing any kind of proof, especially proofs that involve proof by negation or proof by contrapositive. Understanding the wording of a statement is very important when trying to prove that a statement is true. The proof of a statement can depend on whether an “and” or an “or” is used in the statement trying to be proven. Mathematicians can take the negation of a statement and prove that the negation is impossible to prove that the original statement is true because the negation of a statement being false means the original statement is true. Mathematicians can take the contrapositive of a statement and prove that the contrapositive is true to prove that the original statement is true because the contrapositive of a statement provide the same result as the original statement. Truth tables also help students prepare for Venn diagrams, specifically with the idea of union and interception. Union in Venn diagrams have a similar effect and design as “or” in a statement on a truth table, and interception in Venn diagrams have a similar effect and design to “and” in a statement on a truth table.

How was this topic adopted by the mathematical community?

Truth tables have been around to help mathematicians provide and solve all kinds of proofs, specifically involving “if-then” statements. Through verbal rules and word choices, truth tables can be used to help mathematicians learn which statements are true or false. With this information, proving theorems, lemmas, corollaries, and more become much easier and possible. Some statements can only work or are easier to prove when the proof begins with the backwards from their original statement. This helped build a draft of the words and order mathematicians use to create their proofs. More specifically, it helped mathematicians create a language that help other mathematicians better understand how they got their conclusion. Many important theorems have been proven because the concept of truth tables have provided statements with alternative methods to solve or show how the theorem can be proven. This can be shown when mathematicians use the concept of negation and contrapositive to prove that their original statement is considered true. Truth tables can also make it visible to understand how two parts, that are either true or false, can create a true or false statement depending on the two parts given. This concept is similar to union and intersection in Venn Diagrams.

References:

Lodder, J. (n.d.). Deduction through the Ages: A History of Truth. Retrieved from Mathematical Association of America: https://www.maa.org/press/periodicals/convergence/deduction-through-the-ages-a-history-of-truth

Sets and Venn Diagrams. (n.d.). Retrieved from Math is Fun: https://www.mathsisfun.com/sets/venn-diagrams.html