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: 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 Michelle Contreras. Her topic, from Precalculus: compound interest.

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

Compound interest can be something difficult to understand sometimes. That’s why before I even start refreshing my future pre-calculus class about the general formulas they are going to be working with, I would like to start the lesson with a “game”/ activity. Starting class with this activity can be beneficial in the long run because they are going to be more willing to pay attention the rest of class. The game is my own little twist of what we know is the marshmallow game. In the marshmallow game the teacher hands a marshmallow to one of her students challenging him/her to just hold on to it for about 10 minutes and not eat it. If the student managed to hold on and not ingest the marshmallow then the student would get another extra marshmallow. The teacher then ups the reward to two marshmallows more if the student manages to not eat any of the two marshmallows already in their possession.

My own twist in this game is instead of handing one of my students a marshmallow and challenging him/her to not eat it, I would give the student a fun sized M&M’s baggy and challenge him/her with that particular candy. I would then tell my student if he/her manages to not eat the baggy of M&M’s for a minute I would give them another baggy at the end of the minute. While I’m waiting for this minute to be over I would instruct half of the class to give a 30 second argument of why he/she should eat the chocolate right then and there. Then I’ll instruct the other half of the class to make an argument against eating the chocolate for 30 seconds, making the choice for him/her even more difficult. If the student manages to not eat the M&M’s then I will hand him the other baggy of chocolates as promised, then ask the student to wait another minute and not eat the candy’s and this time he/she will get 2 more baggies. What I hope the students are taking from this activity is that they see the connection between waiting a period of time to get more of the desired item. I would explain at the end of the activity that compound interest works in similar ways. When you decided to leave some money untouched in a savings account for a certain amount of time, the compensation for leaving your money alone will be making more money overtime.

 

 

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

There has been a 360 degree change in the way we view compound interest today than how people/communities viewed it long time ago. There has been evidence in texts from the Christian and Islamic faith that talk about how compound interest is a sin or a usury. Back then the people thought if you lend money to a person there should be no interest being added to the loan because that would not be morally right to do to someone in need. Things have changed drastically since those times. We consider someone “smart” or being successful if you earn an interest in whatever it is they are doing. There was also talk about a Roman law where having interest on a loan was illegal. I believe many people changed their view or simply saw compound interest rate as something that would be beneficial financially because of what Albert Einstein once said. There’s speculation that he said “Compound Interest is the eighth wonder of the world. He who understands it, earns it…he who doesn’t….pays it.”

 

 

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

While searching online about compound interest I ran upon a really cool video clip from one of the episodes from the animated T.V show Futurama. In this video clip it talks about Fry, the main character in the T.V show, trying to find out how much money he has in his bank account after being accidently frozen for 1,000 years. The video clip itself is pretty interesting and funny so I believe it would capture the kiddo’s attention. I would probably start with this video the following class day after starting the compound interest lesson. Before showing the video clip to my students, I would explain to them the situation that Fry is in and will ask my kiddos to make a guess of how much money he has in his bank account just by letting them know he was frozen for 1,000 years. I would then proceed to show them the video clip and leave out the part where the lady say’s the amount of money currently in his bank account and have the kiddos calculate the amount themselves with the given principal, interest rate, and amount of time. After giving the kids 2 minutes I would reveal the answer by playing the full video.

 

References:

“The Marshmallow Game” https://blog.kasasa.com/2016/04/marshmallow-game-compound-interest/
“Usury: a Universal Sin” http://www.giveshare.org/BibleStudy/050.usury.html
“Albert Einstein” https://www.goodreads.com/quotes/76863-compound-interest-is-the-eighth-wonder-of-the-world-he
Futurama; http://threeacts.mrmeyer.com/frysbank/

 

Engaging students: Graphing an ellipse

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 Peter Buhler. His topic, from Precalculus: graphing an ellipse.

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

One project that could be assigned to students during the unit on conic sections could be to challenge students to either find or make an ellipse. This could be with a household object, a computer simulated object, or it could be something such as the movement of the planets around the sun. Students would be expected to visually display their object(s) of choice, as well as provide an equation for the ellipse. For example, if the student chose to use a deflated basketball or football, students would use the actual units found when measuring the object and then create an equation for that ellipse. Of course, students would also be expected to graph the ellipse using the appropriate equation, and then check the graph with the actual object (if possible). This project would allow students to be creative in choosing something of ellipse form, and would allow them to further explore the graphing and equation-building of an ellipse.

 

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

While graphing an ellipse is a topic within the Pre-Calculus curriculum, it also has applications within other topics as well. One of these is the unit circle, which is also taught in most Pre-Calculus courses. The unit circle is simply an ellipse where both major and minor axis are of length 1, as well as the center at (0,0). Students can be encouraged to draw comparisons between the two topics. Not only can they rewrite the equation of an ellipse to fit the unit circle, but students can also use the distance formula to calculate sine and cosine values on the unit circle. They can then use the distance formula on various forms of an ellipse, and compare and contrast between the two.

Later on in a students’ mathematical career, some students may encounter ellipse used in three dimensions in Calculus III, in an engineering course, or even in an astronomy course. Ellipses have many applications, and students may benefit from you (as the teacher) perhaps mentioning some of these applications when going over the unit on conic sections.

 

 

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How has this topic appeared in high culture?

One particularly intriguing application of an ellipse (among many applications) is in the design of a whispering gallery. This is essentially a piece of architecture that is designed in the shape of an ellipse so that when someone is standing at one focus of the ellipse, they can clearly hear someone whispering from the exact location of the other focus. Some of examples of these “whispering rooms include St. Paul’s Cathedral, the Echo Wall in Beijing, and in the U.S. Capitol building. It has been commonly noted that President John Quincy Adams would eavesdrop on others while standing in the Capitol, simply due to the physics of sound waves traveling inside an ellipse shaped building.

On a more personal business, I can remember multiple visits to the Science Museum in Fair Park, where various forms of sciences were displayed in formats that children (and adults!) could interact with. There was one exhibit that was set up for several years that also incorporated this ellipse-shaped architecture. I remember it clearly, due to the fact that I was so fascinated with how I could stand 30 yards from someone and be able to hear their whisper clearly. This could also be a class project or even a class trip that would allow students to hypothesize why this works the way it does. It can be noted that this would work for both Physics and Math classes, as it has applications to both.

Sources:
http://www.twc.edu/twcnow/blogs/student/10179/fun-fact-capitol-hill
http://thedistrict.com/sightseeing/other-washington-d-c-attractions/u-s-capitol/

Engaging students: Finding the area of a square or rectangle

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 Trent Pope. His topic, from Geometry: finding the area of a square or rectangle.

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

On this website I saw that the Kelso High School GIC students went out and built a home as a class project. They were able to get a $13,000 grant from Lowe’s Home Improvement and blueprints from Fleetwood Homes to go out and build a physical house. I like the idea of having students use geometry in a real world application, as a teacher I would bring this idea to paper. Students would design and create blueprints for their dream house using squares and rectangles. I would start by giving them the total area their house will be. For example, I would tell them to make the blueprints for a 400 square foot house. They could have anywhere from 5 rooms to 20 rooms in their house. They will be responsible for showing the measurements for each room. After creating the layout of the house and calculating the areas of each room, students will be given a set amount of money to spend on flooring. They will then calculate the cost to put either carpet, wood, or tile in each room. This is to have students decide if they would have enough money to have a large room and if so what flooring would be best. There are other aspects you can add to this project to make it more personalized, but as teachers we just want to make sure we are having students find the area of square and rectangles.

http://tdn.com/news/local/geometry-in-construction-class-finishes-building-first-home/article_74143492-d7a9-11e2-995a-001a4bcf887a.html

 

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How has this topic appeared in the news?

There are many instances of where area made the news. I found multiple websites that talk about how schools are having building projects for geometry and construction classes. These students are building homes from 128 square feet to 400 square feet. Teachers are having students make these homes so that they can see that geometry is in the real world. By having a range of sizes, students have to adjust their calculations. When creating a house or mobile home you need to accommodate for walking space in each room. In order for students to know if there is enough space, they must find the area of each room. Teachers are using this project because blueprints for houses only use squares and rectangles, making it easier for students to practice solving for area of these shapes. This is just the start of teachers making the concept of area more applicable.

http://design.northwestern.edu/projects/profiles/tiny-house.html

 

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

The topic of finding area of squares and rectangles is used throughout many cultures. In the Native American culture along with todays, we see it in growing crops. A farmer must know how big their crop is so they can figure out how much food they will have at harvest. An instance used by many cultures is creating monuments in the shape of square pyramids. In order to build it the right way, you must know the area of the bottom base to build on top of that. A final use of it in our culture is in construction, when we decide how we want to build a building. The concept of area is something that many cultures use today because of how easy it is to calculate. This creates a great way for cultures that are less educated to become familiar with the same concepts as other cultures.

References

Geometry in Construction class finishes building first home. n.d. <http://tdn.com/news/local/geometry-in-construction-class-finishes-building-first-home/article_74143492-d7a9-11e2-995a-001a4bcf887a.html&gt;.
Tiny House Project. n.d. 6 10 2017. <http://design.northwestern.edu/projects/profiles/tiny-house.html&gt;.