Slightly Incorrect Ugly Mathematical Christmas T-Shirts: Part 2

This was another T-shirt that I found in my search for the perfect ugly mathematical Christmas sweater: https://www.amazon.com/Pascals-Triangle-Math-Christmas-shirt/dp/B07KJS5SM2/I love the artistry of this shirt; the “ornaments” at the corners of the hexagons and the presents under the tree are nice touches.

There’s only one small problem:

\displaystyle {8 \choose 3} = \displaystyle {8 \choose 5} = \displaystyle \frac{8!}{3! \times 5!} = 56.

Oops.

Slightly Incorrect Ugly Mathematical Christmas T-Shirts: Part 1

This year, I thought I’d surprise my family with matching ugly mathematical Christmas sweaters. Admittedly, I didn’t look very hard, but I couldn’t find a sweater that I liked both artistically and mathematically. However, I did stumble upon this T-shirt: https://www.amazon.com/Christmas-Math-Teacher-Tshirt-Lovers/dp/B077X14254/

I gave one to my wife and daughter, and it was a big hit.

However, I made the mistake of not inspecting the merchandise closely enough. About a minute after receiving her shirt, my daughter pointed at a digit in the sixth row of the decimal expansion and asked, “Shouldn’t this be a 5? Or maybe I’m mis-remembering.”

In that moment, I remembered that, a few years ago, she had memorized the first few dozen digits of \pi for her elementary school’s talent show. Somehow, she had retained that bit of trivia all these years later. I didn’t miss the irony: I did not remember that she could remember the first few dozen digits of \pi.

As I’ve learned not to daughter my daughter’s memory, I checked two different references (https://www.piday.org/million/ and https://www.wolframalpha.com/input/?i=N%5BPi,1000%5D), and, sure enough, she was right.

The shirt correctly wrote the first 47 digits of \pi after the decimal point. But things went haywire after that. Not only did was the T-shirt’s 48th digit incorrect, but it skipped a few hundred digits in the decimal expansion of \pi before picking it up again! Furthermore, after completing the “tree,” a few thousand more digits were skipped before constructing the base of the tree. And these latter digits were used twice!

The first 4,000 digits of \pi are shown below (in blocks of 10 digits). The ones that appear on the T-shirt are marked in boldface and are underlined.

3.
1415926535 8979323846 2643383279 5028841971 6939937
510 5820974944
5923078164 0628620899 8628034825 3421170679 8214808651 3282306647
0938446095 5058223172 5359408128 4811174502 8410270193 8521105559
6446229489 5493038196 4428810975 6659334461 2847564823 3786783165
2712019091 4564856692 3460348610 4543266482 1339360726 0249141273
7245870066 0631558817 4881520920 9628292540 9171536436 7892590360
0113305305 4882046652 1384146951 9415116094 3305727036 5759591953
0921861173 8193261179 3105118548 0744623799 6274956735 1885752724
8912279381 8301194912 9833673362 4406566430 8602139494 6395224737
1907021798 6094370277 0539217176 2931767523 8467481846 7669405132
0005681271 4526356082 7785771342 7577896091 7363717872 1468440901
2249534301 4654958537 1050792279 6892589235 4201995611 2129021960
8640344181 5981362977 4771309960 5187072113 4999999837 2978049951
0597317328 1609631859 5024459455 3469083026 4252230825 3344685035
2619311881 7101000313 7838752886 5875332083 8142061717 7669147303
5982534904 2875546873 1159562863 8823537875 9375195778 1857780532
1712268066 1300192787 6611195909 2164201989 3809525720 1065485863
2788659361 5338182796 8230301952 0353018529 6899577362 2599413891
2497217752 8347913151 5574857242 4541506959 5082953311 6861727855
8890750983 8175463746 4939319255 0604009277 0167113900 9848824012
8583616035 6370766010 4710181942 9555961989 4676783744 9448255379
7747268471 0404753464 6208046684 2590694912 9331367702 8989152104
7521620569 6602405803 8150193511 2533824300 3558764024 7496473263
9141992726 0426992279 6782354781 6360093417 2164121992 4586315030
2861829745 5570674983 8505494588 5869269956 9092721079 7509302955
3211653449 8720275596 0236480665 4991198818 3479775356 6369807426
5425278625 5181841757 4672890977 7727938000 8164706001 6145249192
1732172147 7235014144 1973568548 1613611573 5255213347 5741849468
4385233239 0739414333 4547762416 8625189835 6948556209 9219222184
2725502542 5688767179 0494601653 4668049886 2723279178 6085784383
8279679766 8145410095 3883786360 9506800642 2512520511 7392984896
0841284886 2694560424 1965285022 2106611863 0674427862 2039194945
0471237137 8696095636 4371917287 4677646575 7396241389 0865832645
9958133904 7802759009 9465764078 9512694683 9835259570 9825822620
5224894077 2671947826 8482601476 9909026401 3639443745 5305068203
4962524517 4939965143 1429809190 6592509372 2169646151 5709858387
4105978859 5977297549 8930161753 9284681382 6868386894 2774155991
8559252459 5395943104 9972524680 8459872736 4469584865 3836736222
6260991246 0805124388 4390451244 1365497627 8079771569 1435997700
1296160894 4169486855 5848406353 4220722258 2848864815 8456028506
0168427394 5226746767 8895252138 5225499546 6672782398 6456596116
3548862305 7745649803 5593634568 1743241125 1507606947 9451096596
0940252288 7971089314 5669136867 2287489405 6010150330 8617928680
9208747609 1782493858 9009714909 6759852613 6554978189 3129784821
6829989487 2265880485 7564014270 4775551323 7964145152 3746234364
5428584447 9526586782 1051141354 7357395231 1342716610 2135969536
2314429524 8493718711 0145765403 5902799344 0374200731 0578539062
1983874478 0847848968 3321445713 8687519435 0643021845 3191048481
0053706146 8067491927 8191197939 9520614196 6342875444 0643745123
7181921799 9839101591 9561814675 1426912397 4894090718 6494231961
5679452080 9514655022 5231603881 9301420937 6213785595 6638937787
0830390697 9207734672 2182562599 6615014215 0306803844 7734549202
6054146659 2520149744 2850732518 6660021324 3408819071 0486331734
6496514539 0579626856 1005508106 6587969981 6357473638 4052571459
1028970641 4011097120 6280439039 7595156771 5770042033 7869936007
2305587631 7635942187 3125147120 5329281918 2618612586 7321579198
4148488291 6447060957 5270695722 0917567116 7229109816 9091528017
3506712748 5832228718 3520935396 5725121083 5791513698 8209144421
0067510334 6711031412 6711136990 8658516398 3150197016 5151168517
1437657618 3515565088 4909989859 9823873455 2833163550 7647918535
8932261854 8963213293 3089857064 2046752590 7091548141 6549859461
6371802709 8199430992 4488957571 2828905923 2332609729 9712084433
5732654893 8239119325 9746366730 5836041428 1388303203 8249037589
8524374417 0291327656 1809377344 4030707469 2112019130 2033038019
7621101100 4492932151 60
84244485 9637669838 9522868478 3123552658
2131449576 8572624334 4189303968 6426243410 7732269780 2807318915
4411010446 8232527162 0105265227 2111660396…

I can understand getting a digit or two wrong on the T-shirt, but I have no idea how anybody could have possibly made a mistake like this.

Upon discovering this, my first reaction reflected my inner mathematician: “I want a refund.” After all, \pi has been known to 47 decimal places since the 1700s, long before the advent of modern computers. However, upon further reflection, I decided that being able to tell this story of a Christmas \pi T-shirt that incorrectly printed the digits of \pi — and especially the story of how this error was brought to my attention — was by itself well worth the price of the shirt.

Christmas ornament

Last year, one of my former students gave me this Christmas ornament. I couldn’t wait to hang it on our family’s Christmas tree.

Engaging students: Using Pascal’s 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 Rachel Delflache. Her topic, from Precalculus: using Pascal’s triangle.

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How does this topic expand what your students would have learned in previous courses?

In previous courses students have learned how to expand binomials, however after (x+y)^3 the process of expanding the binomial by hand can become tedious. Pascal’s triangle allows for a simpler way to expand binomials. When counting the rows, the top row is row 0, and is equal to one. This correlates to (x+y)^0 =1. Similarly, row 2 is 1 2 1, correlating to (x+y)^2 = 1x^2 + 2xy + 1y^2. The pattern can be used to find any binomial expansion, as long as the correct row is found. The powers in each term also follow a pattern, for example look at (x+y)^4:

1x^4y^0 + 4x^3y^1 + 6x^2y^2 + 4x^1y^3 + 1x^0y^4

In this expansion it can be seen that in the first term of the expansion the first monomial is raised to the original power, and in each term the power of the first monomial decreases by one. Conversely, the second monomial is raised to the power of 0 in the first term of the expansion, and increases by a power of 1 for each subsequent term in the expansion until it is equal to the original power of the binomial.

 

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Sierpinski’s Triangle is triangle that was characterized by Wacław Sieriński in 1915. Sierpinski’s triangle is a fractal of an equilateral triangle which is subdivided recursively. A fractal is a design that is geometrically constructed so that it is similar to itself at different angles. In this particular construction, the original shape is an equilateral triangle which is subdivided into four smaller triangles. Then the middle triangle is whited out. Each black triangle is then subdivided again, and the patter continues as illustrated below.

Sierpinski’s triangle can be created using Pascal’s triangle by shading in the odd numbers and leaving the even numbers white. The following video shows this creation in practice.

 

 

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What are the contributions of various cultures to this topic?

The pattern of Pascal’s triangle can be seen as far back as the 11th century. In the 11th century Pascal’s triangle was studied in both Persia and China by Oman Khayyam and Jia Xian, respectively. While Xian did not study Pascal’s triangle exactly, he did study a triangular representation of coefficients. Xian’s triangle was further studied in 13th century China by Yang Hui, who made it more widely known, which is why Pascal’s triangle is commonly called the Yanghui triangle in China. Pascal’s triangle was later studies in the 17th century by Blaise Pascal, for whom it was named for. While Pascal did not discover the number patter, he did discover many new uses for the pattern which were published in his book Traité du Triangle Arithméthique. It is due to the discovery of these uses that the triangle was named for Pascal.

Reference:
https://en.wikipedia.org/wiki/Pascal%27s_triangle
http://mathforum.org/workshops/usi/pascal/images/fill.comb.gif
https://www.britannica.com/biography/Blaise-Pascal#toc445406main
https://en.wikipedia.org/wiki/Sierpinski_triangle

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 Precalculus: 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 know of a good project/activity for the students to do that will be extremely engaging. You could either do this for an elaborate activity for your students or maybe an opening activity for day 2 of a lesson. For my class, I would get a square cookie cake, and have the slices cut into right triangles. I would allow each student to have a piece (but not eat it just yet). The students will be provided with rulers and a protractor. The students will each measure the hypotenuse of their cookie cake and the degree of whichever angle you would like them to measure, however each student should be measuring the same parts so do this unanimously). As a class, decide on an average for the measurements for everyone to use so that the data is not off. Then take the supplies away from the students and ask the students to find the rest of the missing sides and angles of their piece of cookie cake. They will also be provided with a worksheet to go along with this activity. This is a good review activity or al elaborate activity to allow further practice of real world application of right triangle trigonometry. Then go over as a class step by step how they solved for their missing angles and side lengths and make each group be accountable for sharing one of them. This allows the students to all be actively participating. Through out the lesson, make sure to tell the kids as long as they are all participating they will get to eat their slice when the lesson is done. Lastly, allow the students to eat their slice of cookie cake.

 

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How does this topic extend what your students should have learned in previous courses?

Prior to learning about right triangle trigonometry the students will know how to use the Phythagorean Theorem to find how long the missing side length is of a right triangle. The students know basic triangle information such as, the sum of the angles in a triangle is 180 degrees. The students already know the difference between the hypotenuse and the other two legs. The students know that hypotenuse will be the longest leg and the leg across from the 90 degree angle. The students will also know the meaning of a fraction or ratio. The students may need some refreshing of memory on some parts of prior knowledge, but as teachers we know this is an extremely important part of a lesson plan. Even as teacher we tend to forget things and require a jog of memory. A simple activity such as headbands or a kahoot with vocabulary would be an excellent idea for accessing the students prior knowledge. This allows the students to formally assess themselves and where they stand with the knowledge. Also, it allows the teacher to formally assess the students and see what they remember or parts they are struggling on. This allows the teacher to know what things to spend more time on.

 

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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?

Technology is always an amazing aspect of the classroom. Like stated above, a vocab review using headbands or kahoot would be a good idea for this type of lesson that DEFINITELY needs prior knowledge to be applied in order to succeed. Also, showing the students how to plug in sine, cosine, and tangent is crucial. They have seen these buttons on the calculator but they do not know what they mean or how to use them. Using an online TI on display for the class is great. I had to do this with my 10th grade students to make sure they understood how to use the 3 buttons. Also, when using arcsin, arccos, and arctan it can be confusing. Using technology to show the class as a whole is the best route to go. Also, technology can used as review for a homework assignment or even extra credit for the students. It benefits them by getting extra review and extra credit points. I found a website called http://www.learnalberta.ca/content/mejhm/index.html?l=0&ID1=AB.MATH.JR.SHAP&ID2=AB.MATH.JR.SHAP.TRI&lesson=html/object_interactives/trigonometry/use_it.html , which is a golf game that requires review of triangles and trigonometry. It allows the students to practice the ratios of SOH-CAH-TOA using a given triangle.

 

 

 

Happy Fibonacci Day!

Today is 11/23, and 1, 1, 2, 3 are the first four terms of the Fibonacci sequence.

Courtesy Kris McCoy’s Pointless Math Fact of the Day.

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/