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: Finding the equation of a circle

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Kelsi Kolbe. Her topic, from Precalculus: finding the equation of a circle.

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How can technology be used in order to engage the students on this topic?

A simple Desmos program can be used to see different circles and how the variables affect it. You can write a program on Desmos, where you have to manipulate a given circle to ‘collect all the stars.’ There are stars placed around where the circumference should be. Then the students you a variety of sliders to collect the stars. The sliders can change the radius, and move the circle left to right. I think this simple activity will introduce the parts of a circle equation, like the radius and the center, while the students have fun trying to beat their fellow classmates collect the most stars.

 

 

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

I think a circle themed “Clue” inspired activity could be fun. I would tell the students that there was a crime committed and the students had to use their math skills to figure out what the crime was, who did it, where they did it, and when they did it. The students would get an ‘investigation sheet’ to record their answers. Each group would start off with a question like, ‘Find the equation of a circle that has the center (2,3) and radius 7’. Each table would have an answer to the math questions that corresponds to a clue to answer one of the ‘who, what, where, where’ questions they are trying to figure out, and prompts the next question. Students would continue this process until one team thinks they have it and shouts “EUREKA!” then they say what they think happened and if they are right they win, if they aren’t we keep going until someone does.

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How has this topic appeared in high culture (art, classical music, theatre, etc.)?

Circles are seen in a lot of different Islamic Art. Islamic art is known for its geometrical mosaic art. They had a deep fascination with Euclidean geometry. The circle specifically holds meaning in the Islamic culture. The circle represents unity under a monotheistic God. Their religion is so important it can be seen throughout every aspect of their culture. The repetitiveness also symbolizes god infinite nature. For example, his infinite wisdom and love. Along with circles, the 8-point star is also seen as a very powerful symbol. It represents God’s light spreading over the world. The symbols are very important in the Islamic culture and is shown beautifully in a lot of their art. It’s beautiful how they can pack one art piece with so much geometry and also their beliefs.

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;.

 

Engaging students: Using a truth table

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 Saundra Francis. Her topic, from Geometry: using a truth table.

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

An activity that can be done with truth tables is notice and wonder. You could give students an example of a truth table, say finding the results of p and q, and have them write down what they notice and what they wonder about the table. This can lead to a great discussion about what they notice about the truth table and how they think it works. Some students might be able to figure out what the table represents or it could help display misconceptions before the lesson starts. Students can then present their wonders, which you can make sure to answers by the end of the lesson. This activity is a good way to discover if students have prior knowledge of the concept and you can see what they hope to have learned by the end of the lesson.

 

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How has this topic appeared in pop culture?

In this Alice in Wonderland clip where Alice attends the Mad Hatter’s tea party we notice the importance of truth tables and logic to distinguish what the Mad Hatter, March Hare, and the Dormouse are discussing. Alice is using logic to discover what the Mad Hatter means and argue with him about his logic. Once you have students watch this video you can ask students to share examples of where logic is displayed. You can choose one example to transfer into a truth table later in the lesson to show students that we can use the logic displayed outside of the classroom. This will engage students and display the importance of using logic to distinguish what others mean.

 

 

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

Bertrand Russell, one of the contributors that made truth tables as we use them today, worked within the philosophical branch that combined mathematics and logic. Russell was born in Trelleck, United Kingdom in 1872. He attended Trinity College in Cambridge achieved a First Class distinction in philosophy. In 1903 Russell wrote The Principles of Mathematics with Dr. Allan Whitehead. This book extended the work of Peano and Frege on mathematical logic. This was the book where he contributed to truth tables. In 1918 Russell was imprisoned for six months due to a pacifist article he wrote. While incarcerated he wrote Introduction to Mathematical Philosophy. Russell received the Nobel Prize in literature in 1950 in recognition of his thoughts on freedom of thought and humanitarian ideals. He is known for his humorous and controversial writing.

References
1. https://www.nobelprize.org/nobel_prizes/literature/laureates/1950/russell-bio.html
2. https://www.teacherspayteachers.com/FreeDownload/Notice-and-Wonder-Truth-Table-2125723
3. https://www.youtube.com/watch?v=1oupIOmnLJs

 

Engaging students: Writing if-then statements in conditional form

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 Sarah McCall. Her topic, from Geometry: writing if-then statements in conditional form.

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

For this exercise, I would like to kill two bird with one stone (engage students as well as deepen their understanding) by choosing word problems that are funny and have to do with topics that students are familiar with. For this activity, students could work individually or in groups to construct if then statements out of two given clauses, and then construct two if then statements out of their own clauses: one true and one false. This activity would encourage students to be as funny and outrageous as possible. For example:

1. Construct a logical if then statement using the following:
I do not take a nap today.
I will cry about my homework.
2. Construct a logical if then statement using the following:
Miss. McCall’s dog is the cutest dog on the planet. Pigs can fly.
3. Construct a false if then statement using any two clauses you can think of.
4. Construct a true if then statement using any two clauses you can think of.

Questions 3 and 4 are important because they engage students by requiring them to use their creativity and humor, as well as their logical skills. Students should ask themselves, if I want my entire statement to be false, should my “if” and “then” clauses be true or false? This could also be used as a class project where students present their creations to the class afterwards, which would further facilitate student understanding and involvement.

 

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

To get students engaged in this material, I would like to switch it up from the usual mathematics lesson by introducing students to a whole new side of mathematics: logic! For this activity, students would work in groups to solve the following riddle (citations listed below):

Somewhere on an island far far away, there are two large families quite different from you and I. The people in one of these families, the Truthtellers, tell the truth all the time, even when they’d rather lie. They can’t say something false even by mistake! And the other family, the Liars, always lies, even when they wish they could tell truth. Everything they say is a lie. Now suppose you meet two people on this island: Ashley and Amanda, and Ashley tells you “We are both liars”. What family is Ashley from? What about Amanda?

The goal of this exercise is to get students to use their logical intuition and eventually conclude that Ashley must be a Liar, because if she is a Truthteller, then she cannot say she is a Liar. Hopefully students will be introduced to how to communicate their ideas in a simple if then format and realize that logic is fairly intuitive and natural.

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

Many students often worry (or grumble) that they may never use the topics they are learning in geometry/math in the real world. However, as more and more students are considering jobs in STEM fields, I think it is important to acknowledge that these if then statements are not going away! Not only are if then statements used to develop hypotheses for science experiments, they will also be used in upper level math courses like calculus. Learning how to apply if then logic to everyday situations as well as theorems is a skill that will be helpful to all students who plan to go on to any STEM, medical, scientific or research fields. I believe starting off a lesson on if then logic with these reminders would be helpful in getting students to be engaged in their own learning, because they will see that their learning now will affect them later on.

Reference: http://thinkmath.edc.org/resource/logic-puzzles

 

 

Engaging students: Defining sine, cosine and tangent in a 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 Jessica Williams. Her topic, from Geometry: defining sine, cosine and tangent in a right triangle.

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

I’ve actually had the opportunity to teach this lesson to my 10th graders last semester. It is a difficult concept for the students to understand, however if you teach it in a way the students are actively engaged, it helps extremely. Prior to this lesson, the students knew about the hypotenuse and knew the other sides lengths as “legs.” We started by calling 3 students up to the front to hold up our three triangle posters. (triangle cut outs with the 90 degree angle showing and then there was an agle missing). We asked the students how we could find a missing angle given only one side length. For starters, I demonstrated on one triangle by placing a spray water bottle at the missing angle given, and spray the water across.
I will then ask one student to come up to help me demonstrate on the other two triangles. We asked the student where the water is spraying. All of them said words along the lines of “across, away from the angle.” We eventually got to the word opposite. Then we called two students up to demonstrate with the water bottle to determine which side is opposite. If we always know the hypotenuse is the leg across from 90 degree angle, and the opposite side is the one across from the missing angle, then we discovered the last leg must be the adjacent side. Which adjacent means, “next to” or “beside”. Next, we teacher-lead the students through a SOH-CAH-TOA foldable under the doc cam. This was important because they used to later to answer multiple questions using smart pals. Smart pal questions on the board, allowed for EACH student to have to answer and show their work on their smart pal in order to hold it up once we asked for answers. This allowed for formative assessment for the teachers and for the students to see if they were correctly answering the questions. Next we incorporated a “find someone who” Kagan structure tool, which allowed the students to all be actively engaged and answering questions regarding the task. Then we explained and went over misconceptions as a class. It was a very successful lesson overall, and the students were all actively engaged the entire time!

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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?

Trigonometry was originally developed for the use of sailing as a navigation method. The origins can be traced back to ancient Egypt, the Indus Valley, and Mesopotamia. This was over 4000 years ago. Measuring angles in degrees, minutes, and seconds comes from the Babylonian’s base 60 system of numbers. In 150 B.C.E, Hipparchus made a trigonometric table using sine to solve triangles. Later on, Ptolemy extended the trig calculations in 100 C.E. Also, in interesting fact is the ancient Sinhalese used trig to calculate for water flow. Persian mathematician Abul Wafa introduced the angle addition identities. As you can see, there are MANY different mathematicians who distributed to the topic of trigonometry. A lot of them built upon previous work and discovered new formulas, identities, etc. It’s amazing to see how even trigonometry is used to every day life. You always hear people say, “When will I ever use this is life?” and it bugs me to hear this. However, I always have examples of how math is used in our everyday world and from a past long ago that advanced us to where we are today.

 

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

I would show this video that I found on youtube. I would exclude the first movie example that involves shooting, however the rest are great examples.

Showing movie clips to students is always a great way to grab their attention. Visually showing them that math is a part of movies, and every day life shows them that it is important. This video would also be great to use as practice problem, but blur out one of the side lengths or angles missing. You could play the movie scene then pause it on the part with the triangle and have the students solve for missing angle or side length. It would be a fun activity for the students and involve great practice. You could even make this a homework assignment. It’s engaging to watch and keeps the student’s attention while doing homework. The video shows that math is involved in dancing, buildings, etc. This activity also can excite students to try to find math in the movies or tv shows that the watch. You could assign the students to pay attention to to the next couple of shows or movies the watch and to bring back to class an example or two of how math was incorporated in it. Mathematics goes unnoticed because it is honestly part of our everyday norm/lifestyle.

References: https://www.youtube.com/watch?v=LYNN0OYDUB4

http://www.newworldencyclopedia.org/entry/Trigonometry

 

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 Peter Buhler. His topic, from Geometry: deriving the distance formula.

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

Although the distance formula may be introduced as part of the Geometry curriculum, it also has applications in Algebra and even Pre-calculus. This allows for many possible applications, as it can be used in various ways. One project that students could be assigned to is by modeling something in real life on a coordinate grid, and using the distance formula to calculate various distances within that real life object or place. An example of this could be to take a baseball diamond and use the fact that the bases are 90 feet apart, and calculate the distance between the corners on opposite sides. Another example could be to overlay a map of their town onto a coordinate grid and measure the distance between places that they usually visit. These students can fact check the distances by plugging them in to Google Maps. One aspect of this project to be careful of is to make sure that students are using the distance formula, and not the Pythagorean Theorem. Allowing the students to present their findings could spark curiosity into how mathematics is used in everyday life by city planners, architects, engineers, and in other careers.

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How has this topic appeared in high culture?

The following piece of artwork was created by Mel Bochner and titled, Meditation on the Theorem of Pythagoras.

While immediately this picture appears to be related to the proof of the Pythagorean Theorem, There are also applications to the distance formula. This artwork could be a great engaging activity for students as they come into class, simply by reflecting on what can be seen. A challenging question would be to ask students to guess how many hazelnuts they think the artist used to create this artwork (without counting each piece). It should be noted that each corner of the triangle consists of two corners of the squares, so the answer is not simply 9+16+25, but you must subtract off how many are shared.
We can apply this to the distance formula by asking students how to relate the Pythagorean Theorem with the distance formula. Having students compare and contrast these two mathematical equations could provide excellent discussion. As an instructor, you can also overlay this artwork onto a coordinate grid and have students use the distance formula to calculate the various side lengths and confirm that it works.

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How was this topic adopted by the mathematical community?

The three mathematics who are primarily responsible for what we know as the distance formula are: Euclid, Pythagoras, and Descartes. Euclid stated in his third Axiom that “it is possible to construct a circle with any point as its center and with a radius of any length”. This matters because the distance formula is a corollary of the circle formula. Pythagoras then took this idea, and proceeded to invent the Pythagorean Theorem, which can be easily converted to the distance formula. Later on, Descartes applied this to the coordinate system, in an event consisting of the union of algebra and geometry.
While this material may seem fairly dry to middle school or high school students who are first learning the Pythagorean Theorem, there are certainly some applications that can make the history more appealing. One such application is to ask the students to connect the formula of a circle with the distance formula, and discuss how they are related. This would provide excellent discussion about how Euclid and Pythagoras may have begun their study of the distance formula. Another application could include assigning students to study one of these three mathematicians, and having them provide several interesting facts about the person they chose to study. Consequently, when introducing the distance formula, students will be familiar with those who had a huge impact on the development of the distance formula.

Sources:
http://harvardcapstone.weebly.com/history2.html
http://artgallery.yale.edu/collections/objects/31192