Engaging students: Solving exponential equations

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 Austin Stone. His topic, from Precalculus: solving exponential equations.

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

Exponential equations can be used in lots of different kinds of word problems. One that is pretty common but is very useful for students involves interest rate. “Megan has $20,000 to invest for 5 years and she found an interest rate of 5%. How much money will she have at the end of 5 years if the interest rate compounds monthly?” I would give them the formula A=P(1+r/n)rt. It is pretty easy to convince students that this is a real-world problem and would get the students engaged about exponential equations. You can also reword the problem to ask for how much Megan started with, what the rate is, or how much time the money was in there. That way students get used to solving equations when the variable is in the exponent and when it is not. This also can lead into or us prior knowledge of natural log to solve for the variable in the exponent.

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

Using the basis of the problem I mentioned above, a teacher could create a Project Based Instruction lesson using this idea. The teacher can set up a scenario where, over the course of a week or two, the students would have to decide which bank to make an investment in by calculating how much money they would profit at each bank. The students would have to research different banks and their interest rate. The teacher could also give each group different scenarios where some groups have more money to invest. Students would have to figure out how long they would like to invest. The teacher would give Do It Yourselves and Workshops that deal with solving exponential equations and also getting used to natural log. They would then make a presentation explaining what bank they have chosen and why. They would also have to explain the math that they would have used.

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How has this topic appeared in the news?

To say that exponential equations have been in the news lately would be an understatement. It has virtually been the news this year. COVID-19 is a virus and viruses spread exponentially. This would get students engaged immediately because the topic would be relatable to their own lives. Doctors and scientists try to figure out different ways to “flatten the curve”, which essentially means to make the spread of the virus not exponential anymore. We have all heard people on the news telling the public how to stop the virus from spreading and how not make people around you at risk of contracting it (contributing the exponential spread). We all have most likely seen a doctor or scientist show a graph of the virus’s spread and their predictions on how it will look in the upcoming weeks. This would give students a chance to see that what they are learning can be applied to very crucial things going on in the world around them.

References

Exponential Functions

Engaging students: Half-life

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 Trenton Hicks. His topic: working with the half-life of a radioactive element.

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

The topic of half life is a direct intersection of math and chemistry. In addition to being a common precalculus problem, we see half life come up in radioactive decay in chemistry. Half life is a concept that extends all the way into upper college chemistry, physics, and even archaeology when it comes to carbon dating. If students use carbon dating to any extent, they can use half life to determine the age of organic material since carbon 14 is radioactive (Wood). Since half life has to due with nuclear chemistry, this can also tie into nuclear power, since half life is crucial in computations related to efficiency and nuclear engineering. Half life is a form of exponential decay. If students have a thorough understanding of half life, they can better understand other natural phenomena that exhibit properties that are consistent with exponential decay. These phenomena include RC circuits, atmospheric pressure, and toxicity.

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In Chernobyl Ukraine, 1986, there was a disaster at a nuclear power plant that has had lasting effects on the environment, people, and culture. The initial explosion was harmful enough, as 2 people lost their lives. Furthermore, radiation leaked into the atmosphere, and it’s speculated that many individuals are suffering the health consequences. When this story first broke, it shook everyone, and scared people away from nuclear power. Lately, there was another documentary that came to light about the incident from HBO. Many people don’t know that the former power plant is still very dangerous to this day. Why? Because the highly radioactive byproducts of the meltdown have half lives that makes them stick around for quite a while. One particularly dangerous isotope, caesium 137, has a half life of about 30 years. This means that in 2016, about half of the caesium decayed. Half of the sheer amount of caesium that was leaked due to the meltdown is still an enormously dangerous amount. News and documentaries report that there’s still a massive constructive effort to contain the radiation. Showing these news stories to students will convey the importance of half life and give them a little bit of insight into how much care should be given to nuclear power.

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Half life began as a model proposed by Rutherford in the late 1800’s and very early 1900’s. Rutherford discovered that radioactive decay would turn one element into another. This change happens at a rate that we recognize as exponential decay, hence the model we use is consistent with that idea. Rutherford’s work would soon earn him a Nobel Prize. Other disciplines have taken the idea of “half life,” and have created convincing arguments for how the universe behaves. For instance, toxicology uses half life to convey how potent a dose of toxin is versus long it takes for the body to metabolize the toxin. Another notable development is the blog post on the fs website (linked below) that discusses half lives in terms of how our brains retain information, as well as the information itself. Relaying that half life isn’t just a chemistry or math topic to students, and providing them with this history might just increase the half life on their retaining of the concept.

References:

Fs blog:

Half Life: The Decay of Knowledge and What to Do About It

Sources:
Author: Rachael Wood
https://theconversation.com/explainer-what-is-radiocarbon-dating-and-how-does-it-work-9690#:
~:text=Radiocarbon%20dating%20works%20by%20comparing,but%20different%20numbers%2
0of%20neutrons.&text=While%20the%20lighter%20isotopes%2012,C%20(radiocarbon)%20is
%20radioactive

Click to access ExponentialDecay.pdf


https://www.greenfacts.org/en/chernobyl/toolboxes/half-life-radioisotopes.htm

Click to access Section_4.5.pdf

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.

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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.

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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.

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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

Engaging students: Solving exponential equations

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

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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.

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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.

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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

Engaging students: Solving exponential equations

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 Andrew Cory. His topic, from Precalculus: solving exponential equations.

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B1. How can this topic be used in your students’ future courses in mathematics or science?

Solving exponential equations is important for students in their future courses. It can apply to mathematics courses in things like finances. Exponential growth is important for figuring out interest rates and how money will grow. It is also important for figuring out the growth rates of bacteria in science classes. This is the most common example used for solving exponential equations and it can help students with science classes they may take in the future.

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

An activity that can be used to get students engaged in a lesson involving exponential equations and exponential growth, can be a quick example of a disease spreading. The teacher can select a student to start out “infected” and they stand up and walk around the classroom and tap a student on the shoulder. Now that student is also “infected.” Now the two students each tap a new person on the shoulder. Then those four people would go “infect” other students. Pretty quickly, the entire class will be standing up, “infected.” This is a good quick activity to get students to understand how the growth of exponential equations increases quickly. It also allows students to get up and move around, which is always good to do with how long students have to sit down during school.

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

Exponential equations have to do with the growth of any populations. One that became very popular recently is the idea of zombies. The idea of exponential growth happens with how rapidly the disease outbreak happens and how quickly the zombie population overtakes the human population. This idea grew in popularity exponentially a few years ago, but has since died out a bit. The idea of how rapidly a disease could spread was intriguing to audiences, but little did they know, they were learning about exponential growth while watching popular TV shows.

 

 

 

Engaging students: Half-life

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 Kerryana Medlin. Her topic: working with the half-life of a radioactive element.

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

Depending on when they take precalculus, this topic may appear earlier or later in chemistry. The following is the list of TEKS for this topic in chemistry.

112.35. Chemistry (12) Science concepts. The student understands the basic processes of nuclear chemistry. The student is expected to:
(A) describe the characteristics of alpha, beta, and gamma radiation;
(B) describe radioactive decay process in terms of balanced nuclear equations; and
(C) compare fission and fusion reactions.

This is likely the most immediate application the students will encounter, but this topic also appears in calculus and, later, in the topic of differential equations, since it involves exponential decay. This topic can also be brought up in environmental science to mention the lifetime of radioactive isotopes. When a student crunches the numbers on the lifetimes of these isotopes, they can see that sometimes a small action has a huge ripple effect, especially for isotopes that humans bring into the picture.

 

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

Ernest Rutherford received a Nobel Prize in Chemistry in 1908 for his discovery of the half-life of radioactive materials and his insistence that we apply this information to find the Earth’s age (Mastin, 2009). This later became more of a reality when Willard Libby started to develop carbon dating in 1946 (Radiocarbon Dating). Since then, carbon dating has been used to find the age of historical artifacts and bones, allowing historians to find more accurate time frames of events.

Carbon is not the only radioactive isotope. There are others which come to mind more readily when the word “radioactive” is used. These are typically the elements used for nuclear reactors. These are elements which readily undergo nuclear fission, which is the splitting of atoms, which releases energy. Uranium and Plutonium are the most common of these isotopes. Uranium-235 is the most commonly used for reactors and bombs (Brain and Lamb, 2000). This is probably the more interesting part of half-lives of elements and can extend the learning to an environmental issue such as nuclear waste, which takes an extremely long time to decay and which the U.S. Government has, in the past, not handled so well. (But I am not going into that, lest I go on a rant).

The last piece of history worth mentioning is fairly recent (and can be seen in real life and in the game mentioned later in this paper) which is that half-lives are not so clear cut. There is definitely a lot of estimating involved in the accepted half-life values. There is an article about this if you are interested (http://iopscience.iop.org/article/10.1088/0026-1394/52/3/S51/pdf), but I will leave it at this: much like most mathematical models, there is error in the half-life model, and the model formed may be a best fit, but there are always outliers for data and while carbon dating and half-lives of Uranium can give great estimates of what we are working with, they are not perfect.

 

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

For this topic, there is an interactive simulation posted on PHET. It lends itself to a guided worksheet which would allow students to use the simulations to create the functions for each half-life.
So the following would be an example of said worksheet without spaces for actual answers:

Radioactive Half-Life of Carbon-14 and Uranium-238

Please access the following website: https://phet.colorado.edu/en/simulation/radioactive-dating-game

Once there, download and run the game.

At the top of the game window are four different tabs: Half Life, Decay Rates, Measurement, and Dating Game. We will be going through each one in that order.

Some information about radioactive isotopes: An isotope is an element which has the same number of protons in its nucleus, but a differing number of neutrons, thus making it radioactive. These elements have lives which are defined by the time it takes to no longer be radioactive.

Part I: Half Life

Select the Carbon-14 atom and start placing the atoms in the white area. (The “add 10” tool is helpful here.) Then observe as each goes to Nitrogen-14 (This means the element is no longer radioactive and the radioactive isotope has run its course.)

What do you observe about the lives of the isotopes?

What time-frame do these lives fall into?

Do the same for Uranium-238 and record the time-frame.

Part II: Decay Rates

This part works by adjusting the slider and allowing the isotopes to run the course of their lives.

What does the graph on the bottom tell us?

How does one read the half-life of an isotope from this graph?

At what percent do we find the first half-life?

What is the half-life of Carbon-14 from this graph? Half-life of Uranium-238?

Part III: Measurement

On this one, you activate two separate events and then take readings of the amount of Carbon-14 and Uranium-238 in the objects.

Which item contains the Carbon-14? The Uranium-238?

Use the pause feature as you are taking the readings to find precise values of the half-lives.

At what percentages should we be reading the half-lives?

Use this data to create a function to model the half-life of both isotopes.

Part IV: Dating Game

Use your functions to estimate the date of two of the items (One C-14 and one U-238) in the dating game. Write down the name of the item and the estimated age of the item.

 

References:

Brain, Marshall and Lamb, Robert. (2000). How Nuclear Power Works. How Stuff Works. Retrieved from
https://science.howstuffworks.com/nuclear-power1.htm
Mastin, Luke. (2009). Important Scientists: Ernest Rutherford (1871-1937). The Physics of the Universe.
Retrieved from http://www.physicsoftheuniverse.com/scientists_rutherford.html
n.a. (2016). Willard Libby and Radiocarbon Dating. The American Chemical Society. Retrieved from
https://www.acs.org/content/acs/en/education/whatischemistry/landmarks/radiocarbon-dating.html
n.a. (n.d). Radioactive Dating Game. PHET Interactive Simulations. Retrieved from
https://phet.colorado.edu/en/simulation/radioactive-dating-game

 

 

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 Megan Termini. Her topic, from Precalculus: exponential growth and decay.

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

A fun and engaging activity for students learning about exponential growth and decay would be a zombie activity. The students will get a scenario about the zombie attacks and they will predict the way the zombie attacks will work. Then to begin, the teacher will be the only one infected and to show the infection, they will have a red dot on their hand. Then they will shut off the lights and turn them back on to indicate a new day. Then the teacher will “infect” one other student by putting a red dot on their hand. Then they will turn the lights off and turn back on for day 2. Then both the teacher and the infected student will both go “infect” one other person. Then it continues day by day until everyone in the class is infected. Then they will put their data in a table, graph it and can see that it is an exponential growth, then write an equation for it (Reference A). This is great way of getting the whole class involved and zombies are very popular with tv shows and movies. It also lets them explore, see the pattern, and try to come up with the equation on their own.

 

 

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

A great use of technology for graphing exponential growth and decay is Desmos. Desmos lets the students take an equation and plug it in to see the graph. They are also able to change the window to see it better. It also will give you the table for the function that you inputted. It’s good for students to graph it on here to see the graph and also, they are able to click anywhere on the graph to see the point they want. This also would be a good program for them to check their work after trying the problem on their own first (Reference B). Another great website is Math Warehouse. This website lets students explore the graph of exponential functions. Students can type in their function and can graph it. It also lets you compare it to y=x, y=x2, and y=x3. It also has the properties for exponential growth and decay. This website is great for students to interact with exponential functions and also explore them (Reference C).

 

 

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

Exponential functions stay with you all through your school career. You use them in many mathematics courses like algebra, algebra 2, pre-calculus, calculus, etc. You also use them in science courses like biology, chemistry, physics, etc. Understanding how to graph exponential growth and decay functions is a very important tool for future courses. For example, in algebra 2 the students will be learning about logarithms and exponentials, and will have to graph both of them and know the difference between them. Another example is in biology, comparing the number of births and the number of deaths of a species. The data may show an exponential growth in the number of births and exponential decay in the number of deaths, and the students would need to know how to plot the data points and graph it. It is also important for them to understand what the graph means and not just how to graph it. These are skills students will need in not only their future mathematics and science courses, but also in their future careers. For example, a biologist who studies a species of animals might have an exponential decay of the animal and would track its progress every week or every day and graph it to show the decrease of the amount of that species. Many students may not realize it now, but graphing exponential growth and decay is an important topic to understand how to do and why it is important to learn.

References:

A. “Zombies: Exploring Exponential Growth.” BetterLesson, betterlesson.com/lesson/460610/zombies-exploring-exponential-growth.
B. “Exponential Growth and Decay.” Desmos Graphing Calculator, http://www.desmos.com/calculator/d7dnmu5cuq.
C. “Interactive Exponential Function Graph/Applet.” Exponential Growth/Decay Graph Applet . Explore graph and equation of exponential functions| Math Warehouse, http://www.mathwarehouse.com/exponential-growth-and-decay/interactive-exponential-graph-applet.php.

 

 

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 Madison duPont. Her topic, from Precalculus: exponential growth and decay.

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

Through an EDSE 4000 assignment (for which we were to find a Higher Level Task,) I found a fantastic activity that demonstrates exponential growth and decay in an exploratory, hands-on manner. The link to the website with the lesson plan as well as the activity can be found below. This activity is beneficial to the students for several reasons. The first is that they use a variety of materials and methods: hands-on manipulatives (M&Ms), technology (graphing calculators), and written work. This provides students with varied learning styles a chance to participate in and understand the concept of exponential growth and decay. Consequently, the students are able to experience how quickly exponential growth and decay occurs as the number of M&Ms they are having to count, collect, shake, and dump on their desk grows or shrinks rapidly. They then are able to see how this real-life phenomenon can be measured mathematically through an equation and represented mathematically in a graph. Another reason why I enjoyed this activity was because the worksheet had them make conjectures, analyze data, and find relationships between factual and actual information. This activity was conducted in my EDSE 4000 class and proved to even interest colleagues because the likelihood of getting an exponential relationship from probability of M&Ms facing a certain way seemed unlikely and intriguing. There were a few tips I took away from conducting the activity in my class that may be helpful to remember when conducting this activity again. First, be sure to instruct students not to eat any of the M&Ms until after they complete both the growth and decay portion. Second, inform students of how to count morphed or faded M&Ms prior to the activity. Third, the students will need to be slightly informed about exponential functions in order to make conjectures or determine theoretical functions as required in the worksheet. Fourth, going over how to use the calculator as directed prior to or during the activity may help the activity run more smoothly. Lastly, skittles do not work as well with this activity because they make a significantly sticky mess as they melt in hands. Overall, the hands-on exploration and intellectual reasoning utilized in this activity makes exponential growth and decay interesting, entertaining, and relatable.

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How has this topic appeared in the news?

Exponential growth and decay is largely recognized in the news media regarding the Exponential Growth in Technology. The links below provide intriguing information about the study of how quickly and steadily technology is growing. Morris’ Law is referenced often to provide some explanation for the startlingly rapid growth of technology and decay of previous forms of technology. Also, provided on these sites are videos of Ray Kurzweil discussing his theories of technology being able to duplicate patterns and behaviors of the human brain even more powerfully than that of a human in the near future due to the exponential pattern of technology’s growth. This would likely be interesting to students as technology is a growing part of their lives, lives that may become even more dependent on technology in this coming generation’s lifetime. All of this plausible reality being convincingly calculated from a simple exponential pattern that can be introduced in a high school classroom is pretty amazing, and possibly even powerful, to the minds of future students that can apply this knowledge to the technology phenomenon (or maybe even in other topics of our society) in their future careers. Another video found on the thatsreallypossible.com site has Dr. Albert Bartlett discussing the relevance and impact of “simple” exponential relationships applied to our global community’s resources and economy that are not just hypothetical, but that have happened, and are likely to happen. Using these sites you not only show students the power and importance of exponential growth and decay, you also inform them as global citizens and expose them to realistic problems and ideas that will need to be solved or explored in their lifetime or near future, which is arguably the essence of teaching.

 

 

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How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? Note: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.

The graphing applet found at mathwarehouse.com (referenced below) is extremely useful in extending student knowledge of the principles of exponential growth and decay. Using this for an activity can help students compare and contrast changing elements of the function without working separate (seemingly unrelated) examples on their own or in groups. Not only is the applet beneficial because you can graph several factors at a time, but you also have clear, graphical representation of the algebraic manipulations along side the algebra. This can be useful for students that learn visually or are ELLs. Activities can be easily carried out by projecting the applet onto a SMART board for full-class evaluation and discussion, having students perform exercises in groups and recording findings for notes, or even just helping students understand differences in homework problems, and hard to understand textbooks notation that are not making sense to students with verbal or written explanations. This being a free website students can access at home on their computer, smart phone, tablet, etc. can be resourceful to students that do not have a graphing calculator and can also be helpful to students as they work through problems independently and try to understand the behaviors of exponential growth and decay outside of the classroom. Because of the applet’s accessibility, aesthetic set up, and ease in manipulation, I recommend this as a useful technology resource both for the teacher and the student as they explore exponential growth and decay.

 

Pleather, D. (n.d.). Precalculus Lesson Plans and Work Sheets. Retrieved November 17, 2016, from http://www.pleacher.com/mp/mlessons/algebra/mm.html.

Document: M&M_GrowthDecayActivity

http://bigthink.com/think-tank/big-idea-technology-grows-exponentially

http://www.thatsreallypossible.com/exponential-growth/

http://www.mathwarehouse.com/exponential-growth-and-decay/interactive-exponential-graph-applet.php