Guest Post: A Very Merry Mathy Christmas

Today’s guest post comes from Madison Hodson, an instructor at AcerPlacer who recently earned her BS in mathematics from Weber State University. Congratulations, Maddy! We are all very proud of you!

Merry Christmas everybody, it’s Maddy from AcerPlacer! Today I would like to talk about the Mathematics of Christmas. This article is based on the book The Indisputable Existence of Santa Claus by Dr. Hannah Fry and Dr. Thomas Evans. Today I’ll be talking about a few of the most important aspects of Christmas such as wrapping the Christmas tree in garland, buying presents, and – of course – the existence of Santa Claus.

If you are thinking of adorning your tree in garland, but are unsure how much garland you’ll need to buy – have no fear, there is a formula for you to use that will provide you with the exact length of garland needed for your Christmas tree! Let’s first make a little sense of the equation before I present it to you. If h represents the height of your Christmas tree, r is the radius of the base, and n is the number of loops around the tree you want your garland to make, knowing all of those numbers can help you plug them into this equation and presto – you’ll get out the perfect length for your garland!

Happy wrapping!

Moving on to the presents, it’s likely that most of us want to maximize our gift giving and receiving during a gift exchange. Please note, this doesn’t always mean giving and receiving to most expensive gifts you can find! What we will use in this next equation is the value of the gift you will give, the value of the gift you will receive, and those warm fuzzy feelings you get while giving the gift. These variables will provide us with a much simpler equation than the one we just looked at:

Total Utility = Value of the gift received – value of gift offered + beneft of offering gift

Now it might be hard to quantify the benefit of offering a gift (those fuzzy feelings) so to make it simpler just assume that the benefit is equal to have to half of the gift’s value. Now our equation looks like this:

There may be a few challenges in attaining maximized gift giving and receiving, such as determining the value of the gift before you actually receive it (which would ruin all surprises), but hey – the possibility is out there!

Last but definitely not least, let’s talk about the existence of Santa Claus. In the book the authors make two claims:

  1. An existing Santa exists.
  2. An existing Santa does not exist.

We know that one of these statements is true, but statement 2 seems to contradict itself so let’s take a closer look at it. Can an existing Santa not exist? By definition, something that exists “has reality, being, or lives,” so then an existing Santa must be alive and must be real. This means the second claim is false, leaving the first claim to be true!

There you have it, folks – Santa is real, maximized gift giving is attainable, and your Christmas tree will be Instagram worthy with its perfectly wrapped garland.

Until next Christmas, may your math be merry and bright!

Guest Post: An Ode to Victor Borge and a Story

Jodie Larsen has a BS in applied mathematics from BYU-Idaho with a minor in biology. In addition to teaching at AcerPlacer, she tutors students on several topics in math.

Børge Rosenbaum (1909 – 2000), known professionally as Victor Borge, was a Danish and American comedian, conductor, and pianist (he was a prodigy) who achieved great popularity in radio and television in the United States and Europe. His blend of music and comedy earned him humorous nicknames such as “The Clown Prince of Denmark”.

One of his most famous bits was called “Inflationary Language,” in which he added one to every number or homophone of a number in the words he spoke. For example: “once upon a time” becomes “twice upon a time,” “wonderful” becomes “twoderful,” and “anyone for tennis” becomes “anytwo five elevennis,”. Since prices keep going up, he reasoned, why shouldn’t language go up too (three)?

I used to watch his routines and incredible musical talent growing up with my siblings and my parents (my mom is a piano prodigy herself!) on gasp VHS tapes. Teaching math as I do, I like to talk about this sketch as a humorous side note when math is becoming a bit intense (if that’s possible?!). Every time I read or create a passage in inflationary language, it gives me the giggles – I can’t help it – I love punny, silly humor! The following is a story Victor Borge was fond of reading in inflationary language – see if you can pick up each instance of inflation!

Jack and the Twoderful Beans

Twice upon a time there lived a boy named Jack in the twoderful land of Califivenia. Two day Jack, a double-minded lad, decided three go fifth three seek his fivetune.

After making sure that Jack nine a sandwich and drank some Eight-Up, his mother elevenderly said, “Threedeloo, threedeloo. Try three be back by next Threesday.” Then she cheered, “Three, five, seven, nine. Who do we apprecinine? Jack, Jack, yay!”

Jack set fifth and soon met a man wearing a four-piece suit and a threepee. Fifthrightly Jack asked the man, “I’m a Califivenian. Are you two three?”

“Cerelevenly,” replied the man, offiving the high six. “Anytwo five elevennis?”

“Not threeday,” answered Jack inelevently. “But can you help me three locnine my fivetune?”

“Sure,” said the man. “Let me sell you these twoderful beans.”

Jack’s inthreeition told him that the man was a three-faced triple-crosser. Elevensely Jack shouted, “I’m not behind the nine ball. I’m a college gradunine, and I know what rights our fivefathers crenined in the Constithreetion. Now let’s get down three baseven about these beans.”

The man tripled over with laughter. “Now hold on a third,” he responded. “There’s no need three make such a three-do about these beans. If you twot, I’ll give them three you.”

Well, there’s no need three elabornine on the rest of the tale. Jack oned in on the giant and two the battle for the golden eggs. His mother and he lived happily fivever after — and so on, and so on, and so fifth.

© Copyright 2000 Annals of Improbable Research (AIR)

Since this article is being posted around Thanksgiving, I leave you with this… I sincerely hope everytwo is able three go fiveth three see family and friends and eat a twoderful feast of fascinineting goodies. I hope your turkeys are elevender and full of magnifidollar flavor. I hear there is an inelevense storm coming so go fiveth with care and arrive in two pieces….. Hang on a third… maybe I should stop befive I go three far…..

Cones in Space!

In geometry, there are a group of shapes called the conic sections: the circle, the ellipse, the parabola, and the hyperbola. Why are they called conic? It’s because each can be generated by cutting a cone a certain way:

Conic_Sections

Figure 1: Source

It sometimes comes as a surprise to students why so much emphasis is placed on these shapes. Of all the curves that can be made, why spend so much time discussing and learning about four of them?

Honestly, it’s because these shapes have a large number of applications for the world in which we live, and I’ll be returning to a few in my upcoming posts. However, one use of conic section is literally how the world goes ’round. Conic sections are seen with nearly every object in space to some degree, because conic sections are the shapes of orbits.

Conic_section_orbits

Figure 2: Circular orbit: yellow, Elliptical orbits: red, Parabolic trajectory: blue, Hyperbolic trajectory: green. Source

Circular Orbits

circular orbit is a type of orbit where the orbiting body (like a moon or satellite) is always the same distance from the body it is orbiting  (such as a star or planet). Despite them being the most popular orbits to show in many pictures of orbits, there aren’t any completely circular orbits in real-life. To have a perfectly circular orbit, the velocity of the object has to be exactly  right, and there is always a small degree of error. However, one example that comes close to being completely circular are geostationary satellites used for communications and weather satellites. It is important for satellites like these to always be above the same spot on Earth, and the only way to do this is to have a circular orbit directly on the equator. However, it takes fuel to maintain that circular orbit, so these satellites aren’t able to maintain these orbits indefinitely.

Elliptical Orbits

Basically everything that has an orbit has an elliptical orbit (shown in red in Figure 2). Without getting too far into the math, the body being orbited is found in one of the foci (pronounced foe’-sigh; the singular version would be focus).

Foci

Figure 3: Source

Every planet orbits the sun in an ellipse, which means that they move closer or further way at different times of year. For the eight major planets, this difference is very small. For the outer dwarf planets like Pluto and Eris, their orbits can be highly elliptical.

sedna_orbits2

Figure 4: The orbits of a few outer dwarf planets. Source

The closer an object is to what it is orbiting, the faster it moves. The further away it is, the slower it moves. The Russians took advantage of this when solving a problem they had. The geostationary satellites that were mentioned when we talked about circular orbits? Their main downside is that they must be over the equator. This doesn’t work well for Russia in the Arctic Circle. As a result, they invented the Molniya orbit to help fix the problem. They put their communications satellites in a highly elliptical orbit so that they would be above Russia for 10 out of every 12 hours. This is just one example how these conic sections can help provide creative solutions to difficult problems.

Parabolic and Hyperbolic Trajectories

In order for something to escape the gravitational pull of a bigger body, it has to be moving fast enough. The minimum speed to escape is called the escape velocity. If an object moves at exactly the escape velocity, their trajectory will form a parabola (shown in blue in Figure 2). Like circular orbits, perfectly parabolic orbits do not truly exist, but most of our spacecraft will typically be as close to they can. Using the minimum amount of fuel to definitely escape is one of the main challenges to engineering. You don’t get to carry an extra can of rocket fuel with you into space, just in case! Naturally, most escape trajectories will be hyperbolic in nature (shown in green in Figure 2), which is when the speed is above the escape velocity.

The next time you look into the night sky and see the moon, or a planet, or even the stars, hopefully you’ll remember how powerful an understanding of the conic shapes are in being able to describe what you see. And maybe, just maybe, the next time your math teacher asks you to find the focus, focusing will not longer be an issue.

Guest Post: Spooky Math

Today’s guest post comes from Madison Hodson, an instructor at AcerPlacer who studies mathematics at Weber State University.

What do you get when you divide the circumference of a pumpkin by it’s diameter..? Pumpkin Pie! It’s Maddy here, and since it’s Halloween, I thought I would open up this little article with a Halloween math joke – I’ll try to keep the theme of connecting Halloween and math going throughout the article. Thanks to the internet, I was able to do a little research and find out some weird, creepy, and spooky facts about some numbers!

Vampire Numbers

A number v = xy with an even number of n digits formed by multiplying a pair of  n/2 digit numbers (where the digits are taken from the original number in any order) x and y together. If v is a vampire number then x and y are called its “fangs.”

  • 21 × 60 = 1260
  • 41 × 35 = 1435
  • 15 × 93 = 1395
  • 30 × 51 = 1530

 

Tombstone

Also known as the halmos symbol, the tombstone ▮ indicates the end of a proof.

 

Napier’s Bones

An abacus created by John Napier used to calculate the product and quotients of numbers.

 

Devil’s Staircase

The Devil’s Staircase, also known as the Cantor Function, is an example of a function that is continuous, but not absolutely continuous.

 

Witch of Agnesi

A curve studied by Maria Agnesi in 1748 in her book Instituzioni analitiche ad uo della gioventù italiana (the first surviving mathematical work written by a woman). The Cartesian equation is 


Guest Post: Why Some People Actually Enjoy Math and How You Can Too

Jaden Steele is a recent recruit to the AcerPlacer crew. He is studying education at Weber State University.

Throughout my life, I’ve always been intrigued by mathematics. This has caused people to question my normality. How could I enjoy something that basically everyone else hated? To answer this, we must consider what many people enjoy, and question what makes them enjoy those things. I will list some reasons that I believe people find pleasure in certain things using examples of enjoyable activities. Then I will use those same examples comparing them to why I, and many other mathematicians, enjoy math.

1. People like feeling that they are good at something.

Many people are good at a lot of things, including sports, playing music, or any number of unique talents. Because they are good at those things, they tend to enjoy doing them. It makes them feel good knowing that they have talent, and it builds their confidence and self worth. Most likely, they will continue to build on those talents for the rest of their lives.

One of the most famous researchers of the learning process is Edward Thorndike. To paraphrase, he states that any behavior that is followed by positive reinforcement will increase in likelihood. On the flip side, any behavior that is followed by some kind of punishment will decrease is likelihood. This basically means that when we do a good job at something, and someone lets us know that, we feel good about ourselves and continue to perform exceptionally well. Also, if we don’t do well at something, and others make us feel like we are poor at it, we’ll likely never want to try that thing again.

For me, my talent was always math. From the time I was two years old, I learned to count, and I would literally count myself to sleep every night. I could count as high as any normal adult could count and would normally fall asleep by the time I got into the three or four hundreds. My parents noticed my weird and unique talent and they praised me for it. They helped me to have a higher understanding of numbers that lead to me develop a love for math. As I went through school and brought home tests and quizzes all with a 100% and a smiley face drawn on the top of the page, my parents always let me know just how proud they were of me. Their encouragement only lead me to thrive in all things math.

Unfortunately, not everyone was born a math wizard, and even worse, most people believe they are terrible at math. This belief not only comes from impatient parents, teachers, and tutors who make them feel inferior, but people telling themselves that they are horrible at math, when in reality they’re not. This punishment causes people to not find any sort of joy while doing math. If you wish to start enjoying math, the first step is to stop believing that you have no math skills. As your skills improve, math will become much more enjoyable.

 

2. People like what other people like.

A word that I always use as a joke with my family is “sheeple.” We’ll use that word when someone does something just because they saw somebody else doing it. They act like sheep, following the actions of the shepherd. This isn’t a bad thing, it’s just how most people’s brains work. Before they do something, they like to feel that others are also doing what they are doing. It feels good to fit in, especially if what you’re doing is what the “cool kids” are doing.

Many people develop interests from the interests of other people. If one of your role models listens to a certain type of music, then you might decide that you like that music too. If your dad likes a certain sports team, you’ll probably end up liking that team as well. We like what other people also like. This principle has been coined as “joining the bandwagon.”

Math isn’t something that people generally like. It definitely doesn’t make you cool to like math. In fact, it is cool to be bad at math, which is why some people pretend like they can’t do math. The world has made fun of people who like math. They call mathematicians nerds, and people don’t want to be associated with math geeks. If you wish to enjoy math, you’re going to have to care less about being viewed as “cool” and about being part of the bandwagon.

 

3. People like what they believe to be important, good, or useful.

People value things in life very differently. What people value high, they also enjoy. Almost everyone cares deeply about their family, and they enjoy being with them. A great example that shows how people enjoy more valuable things would be a dinner date. If a couple went to a super nice restaurant with a fancy environment and expensive food, they would enjoy it much more than going to Burger King and ordering cheap, two dollar burgers.

Sometimes people can be misled into believing that one thing is more valuable than something nearly identical to it. The price of an object can sometimes make people believe that the more expensive thing must be more valuable – like soda, for example. A two liter thing of Coke from a plastic bottle is the same exact product as a smaller coke from a glass bottle, but the glass bottle is more expensive, and you look cooler drinking from a glass bottle, so people consider it more valuable.

Math teachers all over have always heard students ask, “When am I ever going to use this?” Although they may be right that they probably won’t use math on a daily basis, this attitude really causes the student to dislike math. The reality is that they probably won’t ever need to remember historical facts, or the chemical makeup of molecules, or how to construct Shakespearean literature on the daily either, but you hardly hear students complain about the importance of those subjects.

If you wish to begin enjoying math, stop telling yourself that it isn’t important, and maybe start thinking of ways that math skills can help your life. Practicing problem solving will help your brain to figure out ways to solve real world problems that you’ll encounter later in life.

 

4. People like what they understand.

Perhaps the biggest factor on why people enjoy or don’t enjoy something is the level of understanding they have of that particular thing. This is true in sports, music, art, or any aspect of culture. An athlete might look at someone who enjoys comics and question why in the world they enjoy such a weird thing, and the comic nerd might ask the same question about the athlete. They understand what they enjoy, which is why they enjoy it so much. They don’t however understand other people’s interests.

The more understanding I gain of mathematics, the more I enjoy doing it. I promise as you begin to understand math, you’ll start to enjoy it as well.

 

These points will not only help you enjoy math, but they can help you to enjoy any new thing that you didn’t previously enjoy. Try to understand the topic, find reasons why it is important or good, this will help you relate to others who enjoy that topic, and finally, once you’ve mastered that new topic, you’ll feel really good about it.

Guest Post: Math With Purpose

Chris Allen is a new addition to the AcerPlacer family. He is working towards a degree in engineering at Weber State University.

Often times the study of math is derided as a secondary subject, essentially a means to another subject such as physics or engineering. We sometimes see our math classes as merely an object to overcome to get to another goal. It’s somewhat rare for a student to stop and really consider, “Why even all the fuss about these?” In an age that we can simply take 15 seconds and look up a formula for anything we could ever want then magically we get an output that’s presumably correct, why bother with learning these archaic methods that can be thousands of years old (read: out of date)? Sure, the people who program the black boxes that feed us the right answers should/need to know this stuff, but I don’t.

The answer to that lies in part simply to build an intuition for when the black box might be feeding us a helping of bovine excrement. Take the story of an engineer who was running analysis on a trailer. This was a completely enclosed trailer and no part stuck outside of the trailer’s base. After putting a model of the trailer into the computer, it gave the center of mass about 3 feet outside of the trailer, which is physically impossible. However, this engineer trusted the computer all the way to the next project meeting, much to the engineer’s embarrassment.

Building this intuition for how the math outputs should look like is only part of what we learn as we learn different methods and approaches in math. The real gold in a strong mathematical education is not expecting 2+2=4, but by teaching us a diverse way of thinking about the world around us. How we can use a few basic rules and a bit of thinking out side of the box to solve nearly any puzzle. It’s a field that, despite appearances, creativity in mathematics is the most rewarded attribute.

We sometimes are told it’s important to learn math because it teaches us how to think, or some might say that that it teaches us different ways to think. But looking at the vast diversity in the mathematical fields of thought, I think the real reason why we should all study mathematics is that it teaches and reminds us that we can think.

Guest Post: The Most Common Math Mistakes and How to Fix Them

Jodie Larsen has a BS in applied mathematics from BYU-Idaho with a minor in biology. In addition to teaching at AcerPlacer, she tutors students on several topics in math.

After many years of teaching / tutoring, I have helped and observed an enormous amount of students and have come to recognize the types of mistakes students most often make which has led me to realize what the most common ones are, and it’s not generally as simple as not knowing how to do a problem. Here are the most common mistakes I’ve observed and some suggested remedies.

1. Trying to memorize versus really understanding

I feel this is the most critical problem to correct in order to be accurate in all things math. Math isn’t a subject of memorization. It won’t be mastered simply by knowing verbiage and formulas. Many people have eidetic memories – a trait I always wish I possessed – however, that isn’t enough. Math is beautiful in so many ways, but in large part (in my opinion) because concepts can be combined in new, and endless, ways.

The key to fixing this problem is to ASK QUESTIONS. Students need to be voracious in wanting to know the WHYS in everything they are doing and in understanding the whys, students can then apply them in new situations with much more ease and accuracy. In order for this to be effective, though, the students need good teachers who are willing to teach them the reasons and meanings behind what is being done rather than just talking at them.

2. Not understanding the fundamentals before moving on

For the same reasons you shouldn’t build a house on sand, a person should make sure to learn the fundamental rules (learn and UNDERSTAND them) before trying to use said rules in more complicated applications. If students struggle with exponents, for example, but then try to do intense factoring, the factoring will be much more difficult. This mistake goes along with the mistake listed above because when learning the processes and rules which will continue to be used and applicable, a student must make sure they have a firm hold on the rules and will be able to apply them whenever needed.

The remedy to this problem is to be very vocal with your instructor / tutor and let them know that you’d like more practice and clarification until you feel you’ve mastered each rule or concept.

3. Poor handwriting and disorganization

This one seems perhaps a bit obvious, but it’s actually amazing how often this can throw people off and will cause major frustration when, for example, x’s look like y’s and 5’s look like s’s. Students will often say such things as, ‘I could get it right if I could read my own writing!’ and though they know it and can laugh about it, they don’t often work on it. This may be the hardest mistake to fix, but it just takes care and really slowing down.

Along this same topic, students sometimes aren’t properly taught how to organize their work, or they may just not take heed when they are. I often see cramming in tight spaces, overwriting to avoid rewriting, etc., and thus it becomes a sort of ‘Where’s Waldo?’ situation whenever I need to help students error-check a problem.

One of my biggest recommendations is to get a notebook of graph paper to do work in. There are built-in columns which make it very visually easy to place one number in each box and have everything nicely spaced out. As an added bonus, this can drastically help students who may suffer any form of dyscalculia – I witnessed this with a student years ago and it helped her (and her daughter) tremendously.

4. Using pen

I know, I know, some students are die-hard pen enthusiasts and though I understand the reasons why, it often really hinders them in math. If any mistakes are made, the student often tends to scribble them out or write over them instead of just doing the problem over.

Mistakes are made much more often with these types of actions and so I always recommend working in pencil. If students don’t like the scratching sound of a pencil, then I recommend a pen such as a FriXion pen which ‘erases’ with friction instead of an eraser (we all know how well those don’t work!) which can help avoid the mess.

5. Skipping steps (head math)

Though head math can be impressive, it’s also very error prone. I understand that students want to go as quickly as possible in order to get homework done sooner, but if mistakes are made, then the problem needs to be redone anyway, rendering the step-skipping rather useless.

I firmly believe that writing more steps out yields more correct answers and higher retention overall. A phrase I often use is, “When in doubt… write it out!” If you are working on a problem and you get it wrong, I will recommend you restart the problem and SHOW ME the steps while talking them out. Oftentimes, you will find your own mistake, and that’s empowering!


If I can summarize in one sentence, it would be this: Always ask WHY and practice extra material until concepts are mastered, use a pencil to write out ALL steps neatly, and graph paper is your friend. It’s as simple as that and will greatly increase your chance of success in math (and life) overall!