# A 100-Year old computer for computing Fourier transforms

Many famous machines have been built to do math — like Babbage’s Difference Engine for solving polynomials or Leibniz’s Stepped Reckoner for multiplying and dividing — yet none worked as well as Albert Michelson’s harmonic analyzer. This 19th century mechanical marvel does Fourier analysis: it can find the frequency components of a signal using only gears, springs and levers. We discovered this long-forgotten machine locked in a glass case at the University of Illinois. For your enjoyment, we brought it back to life in this book and in a companion video series — all written and created by Bill Hammack, Steve Kranz and Bruce Carpenter.

A free PDF of their book is available at the above link; the book is also available for purchase. Here are the companion videos for the book.

# New Education Initiative Replaces K-12 Curriculum With Single Standardized Test

As the season of high-stakes testing hits America once again, we have one choice: cry or laugh.

The new test will reportedly cover all topics formerly taught in K-12 classrooms, including algebra, World War I, cursive penmanship, pre-algebra, state capitals, biology, letters of the alphabet, environmental science, civics, French, Newtonian mechanics, parts of speech, and the Cold War. Sources said students will also be expected to demonstrate their knowledge of 19th-century American pioneer life, photosynthesis, and telling time.

Officials said the initiative would also focus on improving teacher performance by tying teachers’ salaries to the test scores of the students they hand the assessment to.

# Student t distribution

Source: http://www.xkcd.com/1347/

# Talking about math to Congress

From MAA Focus:

If you think it’s hard to distill research results into a 15-minute conference presentation, try this: Choose a subject like matrix factorizations or recent progress on the twin prime conjecture. Figure out how to make a nonexpert audience—members of Congress, say—if not fully understand the chosen topic, at least appreciate its significance. Do this in a minute. The clock is ticking.

Jerry McNerney of California’s ninth congressional district has risen to such a challenge more than 10 times in the U.S. House of Representatives, where he has served since 2007. The only current member of the House or Senate to hold a doctorate in mathematics (University of New Mexico, 1981), McNerney has read into the congressional record one-minute expositions of such abstruse subjects as vector bundles, synesthesia, and the Large Synoptic Survey Telescope…

Boiling down complex material into a minute of talking is tricky, McNerney concedes, but he has been pleased with the results. As a member of the Public Face of Mathematics panel at the 2014 Joint Mathematics Meetings, McNerney told listeners that being coaxed into thinking about math has a positive effect on his congressional fellows.

“Instead of all the usual bickering that you get on the House floor, everyone smiles,” he reported. “They say, ‘This is really fun.’”

# MIT Scientists Figured Out How to Eavesdrop Using a Potato Chip Bag

From Gizmodo:

In a scenario straight out of “Enhance, enhance!,” MIT scientists have figured out that the tiny vibrations on ordinary objects like a potato chip bag or a glass of water or even a plant can be reconstructed into intelligible speech. All it takes is a camera and a snappy algorithm. Take a listen.

# Collaborative Mathematics: Challenge 13

I’m a few months late with this, but my colleague Jason Ermer at Collaborative Mathematics has published Challenge 13 on his website: http://www.collaborativemathematics.org/

# Engaging students: Deriving the Pythagorean theorem

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 Delaina Bazaldua. Her topic, from Geometry: introducing proportions.

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

How has this topic appeared in the news?

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

I found a really good blog from a teacher through Pinterest: http://mathequalslove.blogspot.com/2012/04/sugar-packets-and-proportions.html. This website is really great because it is posted from a teacher on a blog who actually tried the lesson. The lesson can be adjusted for a geometry class, but it is really remarkable the way it is without changing a thing especially as an introduction to proportions before going into deeper questions that involve geometry. Like the video above, it can be relatable by the audience of students because of how applicable it is to their life. Likewise, it could also help them eat/drink better! The goal of the lesson is to figure out how many packets of sugar are in a variety of food and drinks using proportions between packets of sugar and grams of sugar! The engage would include the video of someone eating packets of sugar, students brainstorming ideas of how many packets of sugar are in a drink, and then would escalate to students putting the drinks in order of most sugar to least sugar without looking at the nutritional label. After that, students would be given the fact that there are approximately 4 packets of sugar in a gram of sugar. They would also be given the nutritional labels to calculate how many packets are in the drinks using proportions. I think this is a good lesson because it engages the students by allowing them to relate to something that happens in everyday life when they drink/eat things. It is also a good way to introduce proportions with something concrete like bottles before introducing something that is somewhat abstract, such as shapes drawn on a paper which is how geometry is often seen.

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

Perhaps the most famous proportion in history is known as the “Divine Proportion.” Using the research found on the website: http://www.goldennumber.net/golden-ratio-history/, it can help students realize the history behind proportions because, despite popular belief, students need to learn the history of the concept they are being taught to fully grasp the concept of the topic. The website given is really great because it goes through the different names other than divine proportion, such as Golden Ratio and Fibonacci Sequence, and how it was discovered and rediscovered throughout time which is why there are so many unique names that exist now. I also found that the fact that the names that have the words ‘golden’ and ‘divine’ in the name are because of a spiritual background. Understanding divine proportion is important because it is around us every day and it is only a piece of the whole umbrella that engulfs all of probability. It is also applicable to students because it involves them and their physical body along with objects they interact with everyday. I found the topic of divine proportion very interesting and I would hope my students would as well which is why I think this is an extraordinary engage.

References:

http://mathequalslove.blogspot.com/2012/04/sugar-packets-and-proportions.html

http://www.goldennumber.net/golden-ratio-history

# Engaging students: Verifying trigonometric identities

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 Tracy Leeper. Her topic, from Precalculus: verifying trigonometric identities.

Many students when first learning about trigonometric identities want to move terms across the equal sign, since that is what they have been taught to do since algebra, however, in proving a trigonometric identity only one side of the equality is worked at a time. Therefore my idea for an activity to help students is to have them look at the identities as a puzzle that needs to be solved. I would provide them with a basic mat divided into two columns with an equal sign printed between the columns, and give them trig identities written out in a variety of forms, such as $\sin^2 \theta + \cos^2 \theta$ on one strip, and $1$ written on another strip. Other examples would also include having $\tan^2 \theta$ on one, and $\sin^2 \theta/\cos^2 \theta$ on another. The students will have to work within one column, and step by step, change one side to eventually reflect the term on the other side, and each strip has to be one possible representation of the same value. By providing the students with the equivalent strips, they will be able to construct the proof of the identity. I feel that giving them the strips will allow them to see different possibilities for how to manipulate the expression, without leaving them feeling lost in the process, and by dividing the mat into columns, they can focus on one side, and see that the equivalency is maintained throughout the proof. The students would need to arrange the strips into the correct order to prove the left hand side is equivalent to the right hand side, while reinforcing the process of not moving anything across the equal sign.

Trigonometry identities are used in most of the math courses after pre-calculus, as well as the idea of proving an equivalency. If the students learn the concept of proving an equivalency that will help them construct proofs for any future math courses, as well as learning to look at something given, and be able to see it as parts of a whole, or just be able to write it a different way to assist with the calculations. If students learn to see that

$1 = \sin^2 x + \cos^2 x = \sec^2 x - \tan^2 x = \csc^2 x - \cot^2 x$,

their ability to manipulate expressions will dramatically improve, and their confidence in their ability will increase, as well as their understanding of the complexities and relations throughout all of mathematics. The trigonometric identities are the fundamental part of the relationships between the trig functions. These are used in science as well, anytime a concept is taught about a wave pattern. Sound waves, light waves, every kind of wave discussed in science are sinusoidal wave. Anytime motion is calculated, trigonometry is brought into the calculations. All students who wish to progress in the study of science or math need to learn basic trigonometric identities and learn how to prove equivalency for the identities. Since proving trigonometric identities is also a practice in logical reasoning, it will also help students learn to think critically, and learn to defend their conjectures, which is a valuable skill no matter what discipline the student pursues.

For learning how to verify trigonometric identities, I like the Professor Rob Bob (Mr. Tarroy’s) videos found on youtube. He’s very energetic, and very thorough in explaining what needs to be done for each identity. He also gives examples for all of the different types of identities that are used. He is very specific about using the proper terms, and he makes sure to point out multiple times that this is an identity, not an equation, so terms cannot be transferred across the equal sign. He also presents options to use for a variety of cases, and that sometimes things don’t work out, but it’s okay, because you can just erase it and start again. I also like that he uses different colored chalk to show the changes that are being made. He is very articulate, and explains things very well, and makes sure to point out that he is providing examples, but it’s important to remember that there are many different ways to prove the identity presented. I enjoyed watching him teach, and I think the students would enjoy his energy as well.

# 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 Tiffany Wilhoit. Her topic, from Precalculus: finding the equation of a circle.

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

Circles are found everywhere! Everyday, multiple times a day, people come across circles. They are found throughout society. The coins students use to buy sodas are circles. On the news, we hear about crop circles and circular patterns in the fields around the world. One of the first examples of a circle was the wheel. Many logos for large companies involve circles, such as Coca-Cola, Google Chrome, and Target. Even the Roman Coliseum is circular in shape. Since circles are found everywhere, students will be able to identify and be comfortable with the shape (more than say a hexagon). A great way to get the students engaged in the topic of circles would be to have the brainstorm different places they see circles on a normal day. Then have each student pick an example and print or bring a picture of it. Then have the student take their circle (say the Ferris Wheel of the state fair), and place in centered at the origin. The students could then find the equation of their circle. They could do another example where their circle is centered at another point as well. This would allow the students to become more aware of circles around them, and would also allow them some freedom in the assignment.

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

Circles have been an interesting topic for humans since the beginning. We see the sun as a circle in the sky. The ancient Greeks even believed the circle was the perfect shape. Ancient civilizations built stone circles such as Stonehenge, and circular structures such as the Coliseum. The circle led to the invention of the wheel and gears, as well. The study of geometry is focused largely around the study of circles. The study of circles led to many inventions and ideas. Euclid studied circles, and compared them to other polygons. He found ways to create circles that could circumscribe and inscribe polygons. This created a problem called “squaring a circle”. Ancient Greeks tried to construct a circle and square with the same area using only a compass and straightedge. The problem was never solved, but in 1882 it was proved impossible. However, people still tried to solve the problem and were called “circle squarers”. This became an insult for people who attempted the impossible. Borromean Rings is another puzzle involving circles. Circles have been a part of civilization from the beginning, and it is amazing how much they are still prevalent today.

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

The website on www.mathopenref.com/coordgeneralcircle.html is a good site to use when learning to find the equation of a circle. The page contains an applet where the students are able to work with a circle. The circle can be moved so the center is at any point, and the radius can be changed to various sizes. At the top, it shows the equation of the circle shown. This website would allow the students to see how the equation of a circle changes depending on the center and size. This is a good tool to use for the students to explore circles and their equations or to review them before the test. The website also contains some information for the students to read to understand the concept, and there is even an example to try. The website is easy to use, and would not be difficult for students to understand.

Resources:

http://www-history.mcs.st-and.ac.uk/Curves/Circle.html

http://nrich.maths.org/2561

www.mathopenref.com/coordgeneralcircle.html

https://circlesonly.wordpress.com/category/history-of-circles/

# Engaging students: Introducing the number e

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 Nada Al-Ghussain. Her topic, from Precalculus: introducing the number e.

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

Not every student loves math, but almost all students use math in his or her advanced courses. Students in microbiology will use the number e, to calculate the number of bacteria that will grow on a plate during a specific time. Biology or pharmacology students hoping to go into the health field will be able to find the time it takes a drug to lose one-half of its pharmacologic activity. By knowing this they will be able to know when a drug expires. Students going into business and finance will take math classes that rely greatly on the number e. It will help them understand and be able to calculate continuous compound interest when needed. Students who do love the math will get to explore the relation of logarithms and exponentials and how they interrelate. As students move into calculus, they are introduced to derivatives and integrals. The number e is unique, since when the area of a region bounded by a hyperbola y= 1/x, the x-axis, and the vertical lines x=1 and x= e is 1. So a quick introduction to e in any level of studies, reminds the students that it is there to simplify our life!

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

In the late 16th century, a Scottish mathematician named John Napier was a great mind that introduced to the world decimal point and Napier’s bones, which simplified calculating large numbers. Napier by the early 17th century was finishing 20 years of developing logarithm theory and tables with base 1/e and constant 10^7. In doing this, multiplication computational time was cut tremendously in astronomy and navigation. Other mathematicians built on this to make lives easier (at least mathematically speaking!) and help develop the logarithmic system we use today.

Henry Briggs, an English mathematician saw the benefit of using base 10 instead of Napier’s base 1/e. Together Briggs and Napier revised the system to base 10, were Briggs published 30,000 natural numbers to 14 places [those from 1 to 20,000 and from 90,000 to 100,000]! Napier’s became known as the “natural logarithm” and Briggs as the “common logarithm”. This convinced Johann Kepler of the advantages of logarithms, which led him to discovery of the laws of planetary motions. Kepler’s reputation was instrumental in spreading the use of logarithms throughout Europe. Then no other than Isaac Newton used Kepler’s laws in discovering the law of gravity.

In the 18th century Swiss mathematician, Leonhard Euler, figured he would have less distraction after becoming blind. Euler’s interest in e stemmed from the need to calculate compounded interest on a sum of money. The limit for compounding interest is expressed by the constant e. So if you invest $1 at a rate of interest of 100% a year and in interest is compounded continually, then you will have$2.71828… at the end of the year. Euler helped show us many ways e can be used and in return published the constant e. It didn’t stop there but other mathematical symbols we use today like i, f(x), Σ, and the generalized acceptance of π are thanks to Euler.

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

Statistics and math used in the same sentence will make most students back hairs stand up! I would engage the students and ask them if they started a new job for one month only, would they rather get 1 million dollars or 1 penny doubled every day for a month? I would give the students a few minutes to contemplate the question, without using any calculators. Then I would take a toll of the number of the students’ choices for each one. I would show them a video regarding the question and idea of compound interest. Students will see how quickly a penny gets transformed into millions of dollars in a short time. Money and short time used in the same sentence will make students fully alert! I would then ask them another question, how many times do you need to fold a newspaper to get to the moon? As a class we would decide that the thickness is 0.001cm and the distance from the Earth to the moon would be given. I would give them some time to formulate a number and then take votes around the class, which should be correct. The video is then played which shows how high folding paper can go! This one helps them see the growth and compare it to the world around them. After the engaged, students are introduced to the number e and its roll in mathematics.

Money: watch until 2:35:

Paper:

References:

http://mathworld.wolfram.com/e.html

http://betterexplained.com/articles/demystifying-the-natural-logarithm-ln/

http://www.math.wichita.edu/history/men/euler.html

http://www.maa.org/publications/periodicals/convergence/john-napier-his-life-his-logs-and-his-bones-introduction