Engaging students: Introducing proportions

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 again comes from my former student Deborah Duddy. Her topic, from Pre-Algebra: introducing proportions.

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

Proportions, in the form a/b = c/d, is a middle school math topic. The introduction of proportions builds upon the students’ understanding of fractions and ability to solve simple equations. This topic is used in the students’ future Geometry and Statistics courses. The use of proportions is used in Geometry to identify similar polygons which are defined as having congruent corresponding angles and proportional corresponding sides. The use of similar triangles and proportions are used to perform indirect measurements. In Statistics, proportions are used throughout measures of central tendency. Additionally, statistics uses sampling proportions including the proportion of successes.

The ability to use proportions for indirect measurements is also included in the study of Physics, Chemistry and Biology. Chemistry uses proportions to determine based upon the chemical structure of a compound, the number of atoms pertaining to each element of the compound. The study of Anatomy also uses many proportions including leg length/stature or the sitting height ratio (sitting heigh/stature x 100).

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

In art, proportions are expressed in terms of scale and proportion. Scale is the proportion of 2 different size objects and proportion is the relative size of parts within the whole. An example of proportion is Michelangelo’s David. The proportions within the body are based on an ancient Greek mathematical system which is meant to define perfection in the human body. Da Vinci’s Vitruvian Man is also an example of art based upon proportions or constant rates of fractal expansion. The music of Debussy has been studied to show that several piano pieces are built precisely and intricately around proportions and the two ratios of Golden Section and bisection so that the music is organized in various geometrical patterns which contribute substantially to its expansive and dramatic impact.

The use of proportions is also a constant within Greek and Roman classical architecture. Many classical architecture buildings such as the Parthenon illustrate the use of proportions through the building. Additionally, classical architecture uses specific proportions to determine roof height and length plus the placement of columns.

How has this topic appeared in the news?

Proportions are constantly in the news even though they may not be presented in a/b=c/d format. However, the concept of proportion is used throughout news reporting and even advertising. The current news topic is the upcoming Presidential election. Daily, we are provided with new and different poll results. These results are derived via a proportion. For example, 100 people are polled, these results are then derived via proportional concepts to provide a percentage voting for each candidate. Percentage is a specific type of the a/b = c/d proportion. Daily news uses proportions when reporting growth trends for national debt, crime and even new housing starts in DFW. Today, proportions were used when discussing the new Samsung Note7 and its ability to explode. During the winter, proportions are used to tell us how many inches of rain would result from 2 inches of snow. Sports broadcasters also use proportions when discussing the potential of athletes. If the athlete can hit 10 homeruns in 20 games, then he will potentially hit 50 homeruns in 100 games. Proportions even appear in advertising for new medicines detailing the data associated with the medicine trial.

References:

Debussy in Proportion: A Musical Analysis, Dr Roy Howat

Michelangelo’s David

Click to access MOEFranklin.pdf

http://www.brightstorm.com

Engaging students: Combinations

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

This student submission again comes from my former student Heidee Nicoll. Her topic, from probability: combinations.

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

As a teacher, I would give my students an activity where, with a partner, they would be in charge of creating an ice cream shop. Each ice cream shop has large cones, which can hold two scoops of ice cream, and six different flavors of ice cream. Each shop would be required to make a list of all the different cone options available. (Note: cones with two scoops of the same flavor are not allowed.) The groups would calculate the total number of combinations, and try to find any patterns in their work. I would ask them how to calculate the number of options for 7 flavors of ice cream, and then ask them to find a general rule or pattern for calculating the total for n flavors, and have them try their formula a few times to see if it gives them the correct answer. As a bonus, I would also ask them how many flavors of ice cream they would need to be able to advertise at least 100 different cone combinations.

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

Historia Mathematica, a scientific journal, has an article called “The roots of combinatorics,” which describes records of ancient civilizations’ work in combinations and permutations. I would share with my students the first part of this description of the medical treatise of Susruta, without reading the last sentence that gives the answers:

“It seems that, from a very early time, the Hindus became accustomed to considering questions involving permutations and combinations. A typical example occurs in the medical treatise of Susruta, which may be as old as the 6th century B.C., although it is difficult to date with any certainty. In Chapter LX111 of an English translation [Bishnagratna 19631] we find a discussion of the various kinds of taste which can be made by combining six basic qualities: sweet, acid, saline, pungent, bitter, and astringent. There is a systematic list of combinations: six taken separately, fifteen in twos, twenty in threes, fifteen in fours, six in fives, and one taken all together” (Biggs 114).

I would ask them to estimate the number of combinations of any size group within those “six basic qualities” without doing any actual calculations. Once they had all made their estimates, as a class we would do the calculations and comment on the accuracy of our earlier estimates.

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

Sonic commercials boast that their fast food restaurant offers more than 168,000 drink combinations. This commercial shows a man trying to calculate the total number of options after buying a drink:

I would show my students the commercial, as well as images of Sonic menus and advertisements for their drinks, such as the following:

The Wall Street Journal also has an article about the accuracy of the company’s claim to 168,000 drink options, found at http://blogs.wsj.com/numbers/counting-the-drink-combos-at-a-sonic-drive-in-230/. The author talks about the number of base soft drinks and additional flavorings available, and says that according to the math, Sonic’s number should be well over 168,000 and closer to 700,000. He describes the claim of a publicist who works for Sonic that 168,000 was the number of options available for no more than 6 add-ins, which the company deemed a reasonable number. The article also notes the difference between reasonable combinations and literally all combinations, which could spur a good discussion in the classroom about context and its importance in real world problems.

References

Biggs, N.l. “The Roots of Combinatorics.” Historia Mathematica 6.2 (1979): 109-36. Web. 08 Sept. 2016.

Carl Bialik. “Counting the Drink Combos at a Sonic Drive-In.” The Wall Street Journal. N.p., 27 Nov. 2007. Web. 08 Sept. 2016.

http://www.youtube.com/channel/UC9fSZEMOuJjptiXVsYf8SqA. “TV Commercial Spot – Sonic Drive In Sonic Splash Sodas – Calculator Phone – This Is How You Sonic.” YouTube. YouTube, 29 Oct. 2014. Web. 08 Sept. 2016.

Engaging students: Adding and subtracting decimals

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 Daniel Herfeldt. His topic, from Pre-Algebra: adding and subtracting decimals.

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

A great engage activity that I have thought about as a teacher would be to have the students add and subtract money. For this activity I would provide the students with play money (dollar bills, quarters, dimes, nickels, and pennies) needed to proceed. I would then ask the students to show me what 65 cents looks like. Most outcomes will probably look similar with two quarters, a dime, and a nickel while in fact there are many ways to show what 65 cents looks like. Some students might come up with a quarter and four dimes, or 13 nickels. After the students finish with their first example, I would ask them if they could find another way to add up the coins to get 65 cents. This is a very simple activity that refreshes the students’ knowledge on how to add decimals. The activity also shows the teacher which students have a harder time with the topic.

How was this topic appeared in pop culture?

The concept of adding and subtracting decimals is all over the world. It is used for everyday things, such as sports. One of the most watched things on television is the summer or winter Olympics. People from all over the world compete in several events and get scores. For example, gymnasts compete for the highest score in the specific event they are doing and then add it to their total score to be declared the winner. After the first event, one gymnast may have the highest score of 16.543 while the person below her has a score of 15.785. Then in the second event, the person that previously had a higher score only scored 12.400, while the person that was behind her scored a 15.115. To declare the winner of the two, you would have to sum up both scores and see which of the two competitors had the higher score. You would get the total of 28.943 for the first gymnast, and 30.900 for the second. From here the winner would be the second gymnast.

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

http://www.coolmath.com/prealgebra/02-decimals/decimals-cruncher/addition

This game would be a great tool to refresh a students’ memory on how to add decimals if you are planning to have a test. To start, you would hand out every student their own small whiteboard and marker. You would then put the game on the projector screen so that all of the students can see it. Start out with clicking the easy button so that you don’t start with a difficult problem. Ex: 31 + .4. This should be a problem that all students can answer. Have all the students write down on their own white board what they think the answer would be. After the students finish, ask them to put their white board face down. Once all of the students finish, have everyone raise up their answers. Afterwards plug in the answer that is most common amongst the students to see if the majority was correct. If most are correct, proceed to the next difficulty, which is medium, and repeat the steps that you did for the easy problem. If the majority of the class gets it right, then go to the final and hardest difficulty and repeat the steps one more time. If the majority of the students get the answer wrong for any difficulty, do the problem on the board to show the steps and try another problem of the same difficulty. The students will then remember the steps and have a higher chance of being correct. When the students get the hard problems correct, keep doing the hard problems until you feel the students have grasped the concept.

Engaging students: Rational and Irrational Numbers

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

This student submission again comes from my former student Daniel Adkins. His topic, from Pre-Algebra: rational and irrational numbers.

What interesting (i.e., uncontrived) word problems using this topic can your students do now? (You may find resources such as http://www.spacemath.nasa.gov to be very helpful in this regard; feel free to suggest others.)

The largest hurdle to overcome in mathematics, is the introduction to foreign, and new concepts. Quite often, individuals are stuck with their “old math”. When a new object appears before them, they won’t play with it, or recognize that even though it’s new, it still works with what we already know. This is especially true when it comes to introducing concepts of new sets of numbers, such as the imaginary numbers, or irrational numbers. More often than not, when it comes to irrational numbers, students freeze up. I believe that the best way to prevent this, is to show that students already know a lot about this set. By taking it step by step and reminding students what they already know about rational numbers, you can show them they have known about irrational numbers in some form or fashion for quite some time.

A simple two step project would be to first introduce the concept of an irrational number, then the instructor can draw a circle with a marked radius, and say, this is my pizza pie. Now I want a piece of pizza pie, but when it comes to pieces of pizza pie, I’m particular. I want to proficiently partake of my pizza pie by partitioning it perfectly, to where each piece is equally cut. If all I know is the radius though, how can I know where to cut it? Eventually students will point out that by finding the circumference, and then dividing the circumference by how many pieces you want, you can make sure they’re all equal. At this point, point out that you have a ratio of pieces to circumference, but how did the students get to the circumference in the first place? 2*pi*r so that means the radius of a circle is in a ratio to its circumference right? So we can right pi as some sort of fraction correct? If the students are aware that this isn’t possible, then the digging isn’t necessary, but if they aren’t ask them to try and write it as a fraction.

The second part of this exercise would be to emphasize nested sets. Divide the students up into 2-4 groups, and have a several Natural, whole, integer, rational, and irrational numbers written on pieces of colored paper (with each team having 1 color). Students will line up in front of “nestable” baskets spread out in front of them labeled by the different sets of numbers as listed above, and will one at a time aim for the smallest set that their number can fall in. After all the papers have been thrown, the papers will be collected and compared as a class, and each paper made in the correct basket will count as a point for that team. At the end of it all, put the numbers back in their baskets and show how the baskets can all fit inside of each other, except that the irrational and rational baskets are the same size, and so they can’t nest inside of each other. This can be emphasized by drawing it on the board. This exercise reminds students of what it means for sets to share qualities, and that irrational numbers don’t have the same qualities that rational numbers do.

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

Throughout my high school career, it was never brought to my attention that there were conflicts within the history of math, or in fact that there even was a history of math. In fact, it wasn’t till my collegiate years that my classmates and I came to learn such things were at one point a problem, that math could have different viewpoints.

The individual who is credited with discovering irrational numbers is Hipassus of Metapontum. He was a philosopher who studied Pythagorean based concepts, and while trying to use the Pythagorean Theorem to solve for a ratio between a unit square’s side length and its diagonal, he learned that there wasn’t such a thing. At the time, the other Pythagorean philosophers believed that only positive rational numbers existed. So when Hipassus introduced his discovery to them, they weren’t exactly happy. The story varies, and no one may ever know what truly happened to him, but some of the more versed stories range from the other Pythagoreans simply killing him, to Pythagoras himself ostracized him, and upon the Gods discovering the abomination he discovered, they had him drowned by the sea to hide it away.

Regardless of the validity of these stories, it shows how discoveries like these can often cause turmoil in time periods.

How was this topic adopted by the mathematical community?

Hipassus’s discovery caused such a drastic response because of two reasons; first off, it contradicted the core belief of Pythagoreans that Mathematics and geometry were indefinitely correlated, as in they were completely inseparable. But it also raised another problem that would eventually be brought up by another philosopher named Zeno. The problem was in the discrete vs. continuous argument, and how geometry couldn’t solve it. All in all, when Hipassus introduced this concept, it was met with malice. Many individuals would write this off as simply how things were “back then”, but a closer examination at something like imaginary numbers will reveal a similar pattern. It wasn’t until the Middle Ages when Middle Eastern mathematicians introduced concepts of algebra that irrational numbers became fully accepted within the mathematical community.

All in all, the stories behind things like irrational and imaginary numbers should be shared within schools much more often. Not only is it extremely interesting, and can convince students to do their own research, but it also shows that people were afraid to learn new thing, that these foreign concepts that are terrifying now, were terrifying to the people who discovered them too. It teaches students that instead of ostracizing others for bizarre concept, but instead to analyze them themselves. Because those bizarre concepts, may become commonplace. It shows students that Hipassus was on the right side of history, even though he was alone for quite a while.

References:

https://brilliant.org/wiki/history-of-irrational-numbers/

http://www-history.mcs.st-andrews.ac.uk/HistTopics/Arabic_mathematics.html

https://www.algebra.com/algebra/homework/Problems-with-consecutive-odd-even-integers/Problems-with-consecutive-odd-even-integers.faq.question.580533.html

http://tulyn.com/8th-grade-math/irrational-numbers/wordproblems

Engaging students: Absolute value

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

This student submission again comes from my former student Dalia Rodriguez. Her topic, from Pre-Algebra: absolute value.

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

Get a deck of cards and take the Ace cards out, as students walk through the door give them a card. The red cards will represent a negative number, the black will represent a positive number, Jacks will represent the number 11, Queen will represent the number 12, and King will represent the number 13. From the student roster call out two students at a time and ask them their number, the two students will then decide which card has the highest value, do this until all students are called. Then ask the students, “What if I told you that the red cards represented a negative number?” This will engage the student because they will feel confident about their answer until they hear that the red cards represented a negative integer. Then the student will start thinking and coming up with conceptions on how a negative number affects which integer is higher. The teacher can then ask another pair of students what their numbers are, and follow up by asking which integer is higher. The students will most likely answer incorrectly so this would be a time to ask other students what their thoughts are. All the students will be participating and thinking. The last question the teacher would ask before beginning the lesson would be, “What if I told you that the color of your card does not matter, or affect the number on the car?” Allowing all the students to participate by calling on them, at the beginning, will break at least a small barrier and open the doors for them to share their opinion. Also, asking scaffolding question to let the students start thinking about properties of absolute value will let the students remember the activity and acknowledge that even thought the number is negative or positive absolute values is the distance away from zero and it will always be positive.

In Finding Dory, her parents laid out sea shells on the ocean floor that lead to her parents. The sea shells were spread out in lines going around the house, the distance from the beginning shell to the house is always positive, even though they are in the left side (negative side). The teacher can tell the student that each sea shell represents 1 unit, as they see the length of the sea shells lines the students will think of these lines as positive numbers, no matter what direction the sea shells are coming from.

Students should have already learned about positive, negative integers, and distances. You can engage your student by asking them question and having a class discussion. Questions like:

“What is a positive integer?”

“What is a negative integer?”

“How do you measure distance?”

“Can distance be negative?”

These types of questions will scaffold student to get a base line idea of what absolute value is, but also allow them to remember what they already have learned. Allowing students to realize that their connections from past knowledge to new knowledge will let them better understand what they are learning. Having a class discussion on their previous knowledge will allow a teacher to see where there might be misconceptions and also see a base line where the students are at, or what they might need help at. A small review lesson from the teacher, after a discussion, will then clear up any final misconceptions and allow the class to move forward from the same starting position.

Engaging students: Solving one-step algebra problems

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

This student submission again comes from my former student Anna Park. Her topic, from Algebra: solving one-step algebra problems.

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

Bingo game:

The teacher will create a bingo sheet with a free space in the middle, and integers in the other spaces. These integers represent the answers to the word problems that the teacher will be putting on the board or projector screen. Each word problem will either be a one-step equation or a two-step equation. A one step equation involves only one step to solve for the variable, this means only one operation will be done on the equation. The goal is to have the variable by itself on the left side of the equal sign and the numbers on the right side of the equal sign. A two-step equation is similar to a one step equation. A two-step equation is where it takes only two steps to solve for the variable in the equation that has more than one operation. The goal is the same as a one-step equation.

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

In future courses students will need to know how to isolate a variable in an equation to receive its value. They will need to know how to graph equations and inequalities in future mathematics courses. From Algebra and on students will need to know how to solve for the value of a variable.
Students will also need to know how to create an equation given to them in word problems. Some of the classes that this will be needed for is Physics, geometry, algebra II, Pre-Calculus, Calculus, college courses..etc. Algebra is a tool for problem solving, and critical thinking. Word problems give real life examples of algebra and students will be able to apply this knowledge to real life situations and understand the problems given to them in future classes.

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

Kolumath is a great youtube math channel that explains how to do certain math operations with great visual examples and clear explanations. The speaker talks clearly and is easy to understand, and the examples he uses ties in information the students have learned in previous courses. His visual examples allow students who struggle with picturing math functions to connect to the lesson.
This channel also gives definitions over the topic and any definition relatable to the operations done in the video.
Listed below are examples he uses on how to solve one-step and two-step equations. (References)

Solving one step equations: https://www.youtube.com/watch?v=Ot-KSERw8Gc

Solving two step equations: https://www.youtube.com/watch?v=m7acIUcQ-7E

Engaging students: Venn diagrams

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 Amber Northcott. Her topic, from Probability: Venn diagrams.

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

There are a few activities you can do with Venn diagrams. One idea is for the first day of class you can put up a big poster with a Venn diagram on it or you can draw one on the board. One circle can be ‘loves math’, while the other is ‘do not like math’. Then of course the center where the two circles intertwine will be the students who love math, but yet don’t like it. When your students come into the room you can have them put their name where it seems fit. This way you can get to better know your students on the topic of math. Another idea is that when you get to a topic, for instance arithmetic and geometric sequences, you can create a giant poster Venn diagram or draw it on the board. Then you can have your students write one thing that either arithmetic has or geometric has or both of them have. Once each student has put up just one thing on the Venn diagram, you can start a class discussion on the Venn diagram. While the discussion goes on you may fix a couple things here and there or even add to it. At the end each student will have their own Venn diagram to fil out, so they can have it in their notes.

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

Venn diagrams are an easier way to compare and contrast two topics. It can help differentiate between the two topics. For example, how are geometric and arithmetic sequences different? Do they have anything in common? What do they have in common? This helps students identify the topics more thoroughly and helps them get a better understanding about each topic.

How has this topic appeared in the news.

Not too long ago Hillary Clinton posted a Venn diagram about gun control on twitter. In response she was getting mocked and criticized. A short article on thehill.com goes into the mockery by showing pictures of people’s tweets to Hillary Clinton. Some had two circles separate from each other with one stating people who know how to make Venn diagrams and the other one stating Hillary’s graphic design staff. The other article from the Washington Post actually goes through her Venn diagram and fixes errors. These errors include the data in the Venn diagram.

Letting students see this, would definitely cause a discussion. I think we can turn the discussion into whether or not we think the Venn diagram was wrong. By having this discussion, we can learn more about what the students know about Venn diagrams and shed more light on how we can use the Venn diagrams in many different ways for many different topics.

References

https://www.washingtonpost.com/news/the-fix/wp/2016/05/20/we-fixed-hillary-clintons-terrible-venn-diagram-on-gun-control/

http://thehill.com/blogs/ballot-box/presidential-races/280706-clinton-mocked-for-misuse-of-venn-diagram

What I Learned by Reading “Gamma: Exploring Euler’s Constant” by Julian Havil: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post.

When I was researching for my series of posts on conditional convergence, especially examples related to the constant $\gamma$ , the reference Gamma: Exploring Euler’s Constant by Julian Havil kept popping up. Finally, I decided to splurge for the book, expecting a decent popular account of this number. After all, I’m a professional mathematician, and I took a graduate level class in analytic number theory. In short, I don’t expect to learn a whole lot when reading a popular science book other than perhaps some new pedagogical insights.

Boy, was I wrong. As I turned every page, it seemed I hit a new factoid that I had not known before.

In this series, I’d like to compile some of my favorites along with the page numbers in the book — while giving the book a very high recommendation.

Part 1: The smallest value of $n$ so that $1 + \frac{1}{2} + \dots + \frac{1}{n} > 100$ (page 23).

Part 2: Except for a couple select values of $m<n$ , the sum $\frac{1}{m} + \frac{1}{m+1} + \dots + \frac{1}{n}$ is never an integer (pages 24-25).

Part 3: The sum of the reciprocals of the twin primes converges (page 30).

Part 4: Euler somehow calculated $\zeta(26)$ without a calculator (page 41).

Part 5: The integral called the Sophomore’s Dream (page 44).

Part 6: St. Augustine’s thoughts on mathematicians — in context, astrologers (page 65).

Part 7: The probability that two randomly selected integers have no common factors is $6/\pi^2$ (page 68).

Part 8: The series for quickly computing $\gamma$ to high precision (page 89).

Part 9: An observation about the formulas for $1^k + 2^k + \dots + n^k$ (page 81).

Part 10: A lower bound for the gap between successive primes (page 115).

Part 11: Two generalizations of $\gamma$ (page 117).

Part 12: Relating the harmonic series to meteorological records (page 125).

Part 13: The crossing-the-desert problem (page 127).

Part 14: The worm-on-a-rope problem (page 133).

Part 15: An amazingly nasty formula for the $n$ th prime number (page 168).

Part 16: A heuristic argument for the form of the prime number theorem (page 172).

Part 17: Oops.

Part 18: The Riemann Hypothesis can be stated in a form that can be understood by high school students (page 207).

The Running Nerd: The US Marathoner Who Is Also a Statistics Professor

I loved these articles about Jared Ward, an adjunct professor of statistics at BYU who also happens to be a genuine and certifiable jock… he finished the 2016 Olympic marathon in 6th place with a time of 2:11:30.

Ward started teaching at his alma mater after graduating from BYU with a master’s degree in statistics in April 2015…

Ward wrote his master’s thesis on the optimal pace strategy for the marathon. He analyzed data from the St. George Marathon, and compared the pace of runners who met the Boston Marathon qualifying time to those who did not.

The data showed that the successful runners had started the race conservatively, relative to their pace, and therefore had enough energy to take advantage of the downhill portions of the race.

Ward employs a similar pacing strategy, refusing to let his adrenaline trick him into running a faster pace than he can maintain.

And, in his own words,

[A]t BYU, on our cross-country field, on the guys side, there were maybe 20 guys on the team; half of them were statistics or econ majors. There was one year when we thought if we pooled together all of the runners from our statistics department, we could have a stab with just that group of guys at being a top-10 cross-country team in the nation…

To be a runner, it’s a very internally motivated sport. You’re out there running on the road, trying to run faster than you’ve ever run before, or longer than you’ve ever gone before. That leads to a lot of thinking and analyzing.We’re out there running, thinking about what we’re eating, what we need to eat, energy, weightlifting, how our body feels today, how it’s going to feel tomorrow with how much we run today. We’re gauging all of these efforts based on how we feel and trying to analyze how we feel and how we can best get ourselves ready for a race. As opposed to all the time on a soccer field, you’re listening to do a drill that your coach tells you to do, and then you go home.

I think we have a lot of time to think about what we are doing and how it impacts our performance. And statistics is the same way. It’s thinking about how numbers and data lead to answers to questions.

Yes, I think there’s probably some sort of connection there to nerds and runners.

Sources: http://www.nbcolympics.com/news/running-nerd-us-marathoner-who-also-statistics-professor and http://www.chronicle.com/article/Trading-One-Marathon-for/237595?utm_source=Sailthru&utm_medium=email&utm_campaign=Issue:%202016-08-29%20Higher%20Ed%20Education%20Dive%20Newsletter%20%5Bissue:7064%5D&utm_term=Education%20Dive:%20Higher%20Ed

Computing e to Any Power: Index

I’m doing something that I should have done a long time ago: collecting a series of posts into one single post. The following links comprised my series examining one of Richard Feynman’s anecdotes about mentally computing $e^x$ for three different values of $x$ .

Part 1: Feynman’s anecdote.

Part 2: Logarithm and antilogarithm tables from the 1940s.

Part 3: A closer look at Feynman’s computation of $e^{3.3}$ .

Part 4: A closer look at Feynman’s computation of $e^{3}$ .

Part 5: A closer look at Feynman’s computation of $e^{1.4}$ .