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.

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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!

Guest Post: My Path Through Math

There is an administrative assistant at AcerPlacer who loves sharing the story of her journey through math with students. Today, she was kind enough to write her experience so that everyone could see it.

If you think back to your very first day of college and the first classroom that you walked in to, how did you feel? Excited, overwhelmed,​ ​or amazed you made it to the right classroom? As you searched for what will be your unspoken assigned seat and looked around the classroom, what did you notice first? For me, I realized on the first day that there is nothing like the feeling of dread when you realize the class has 30+ students all in one space. How is the teacher going to help you if you have questions about the material, or just need some extra help?

While this may not have been your first college classroom experience, it was for me. I double-checked and even toured the campus before the semester started to make sure I knew where I was going (still got lost), I bought all my books early, and was ready to start my classes. I ended up in the engineering building and wandered into an advanced math class that made me run for the hills like the room was on fire. After what felt like an eternity, I eventually found my first class of the day — math. As if that wasn’t alarming enough, I walked in late and had to pick a random seat next to a stranger. At least my best friend was three seats down and looked just as panicked as I felt.

His expression and my feeling of alarm seemed justified. Just about everyone we knew had enrolled into a similar class or the class just one level higher. “Two-thirds of the students at community colleges, and 4 in 10 of those at four-year institutions take remedial courses. Math is a much bigger sand trap than English: Far more postsecondary students fall into remedial math than reading, and fewer move on to credit-bearing courses” (Gewertz, 2018).

Knowing that I wasn’t a mathematician, I thought to myself, “Here we go.” I was enrolled in the lowest level of math offered on campus. Could I do this? I could do this, right? As I sat down and unpacked my new school supplies, I looked around the room. I had an idea of what the college classrooms looked like, and it didn’t look like this room. What alarmed me the most about my new math class was that there were easily 35 students. All in the same room. With one teacher. In a math class.

Adelman (1999) states: “Of all pre-college curricula, the highest level of mathematics one studies in secondary school has the strongest continuing influence on bachelor’s degree completion. Finishing a course beyond the level of Algebra-2 (e.g. trigonometry or pre-calculus) more than doubles the odds that a student who enters postsecondary education will complete a bachelor’s degree” (p.vii).

Based on the study referenced above I was in big trouble. Math in high school was easy for me because I wasn’t required to take it during my junior or senior year. The last class that I was required to take was Algebra 2, and to be honest, I had no idea what was going on during the entire class. It was a miracle that I passed. Starting out in a new class, I felt that I could finish my math and avoid the so-called “sand trap”. I was going to do it. At least that was what I thought. I was in for a very rough semester.

Math was proving to be very difficult for me, and it was the class that I found myself dreading. In the upcoming semesters, I tried everything from traditional classes, computer-based classes, and even sought help from resources offered at my university. I just could not wrap my brain around this math thing. I had amazing professors who would help when they could, but I started to feel like a burden. I just could not understand what they were telling me or why we were moving “x” to the right side of the equation. What was factoring and why is this 3 all the sudden a negative number? I stopped asking questions. I came to the conclusion that I hated math. I hated all the rules, classes, material. All of it. It was the class I hated to attend, and even register for.

I learned that anytime I asked my family or friends for help that it only caused me more confusion and frustration. I found that not everyone who is wonderful at doing math can actually teach math. After a handful of math classes (13 to be exact), I found myself with a degree that was completed but out of reach because of my math requirements. How could I enjoy and pass higher level courses but not pass my math classes?! I felt defeated and hated to admit that math was again a class I had to repeat.

“Large numbers of students have been prevented from pursuing careers they’re interested in because of the math,” said Briars, a math consultant who was the president of the National Council of Teachers of Mathematics from 2014 to 2016. “They’re underprepared, but they’re put into the typical course sequence anyway. And we’ve done this at the expense of other mathematics, like quantitative literacy, or statistics, that is vitally important, and maybe more important for some careers” (Gewertz, 2018).

When you reflect on your previous classes, what made the class enjoyable? What made you successful in the class? Was there something in particular that stood out? For me, that answer is simple. I needed a small class that allowed me to ask questions and receive personalized help. I needed to be one of a handful of students, not one of 30+. I needed a class that had an uplifting, positive feel to it that encouraged mistakes and provided hands-on learning with an instructor who was invested not only in the topic, but also my success.

After what seemed like a never-ending nightmare of failed math classes, I had a degree that was one class away from being 100% completed and a job that only offered advancement if you possessed a degree. I had no idea what to do. I felt that I was out of options. I did the best thing I could have ever done for my math education. I discovered a new way of learning and really understanding math! So long, YouTube tutorials!

I was able to jump into a class that offered small classes, personalized help, out of class resources, and teachers who had the time to invest that had a real interest in my personal success and struggles. It is amazing how my view of math changed because I was finally able to get a grasp of what was actually happening. Why “x” moved to the right side of the equation, why that 3 becomes negative, and even how to read the trig wheel. Commonly I hear from students looking in to AcerPlacer, “Now I know that you work there, so you have to tell me that this program works, but will this program really help me test out and understand math?” I love that I can say, “Believe me, I know first hand that math can be a very difficult educational hurdle, but you are in the right place!”

AcerPlacer instructors have math-loving hearts of gold. They take the time and are truly invested in getting to know your learning style, your educational goals, and are always a great math support. They provide encouragement, comfort, math jokes, and bring not only their math experience, but also teaching methods that can unlock and help students grasp concepts. Each class is capped at 8 students per room so that it was easy to get the help I needed while in class. I could ask my instructor to repeat the material, say it a different way, and associate it with a story. The best part was that I never felt like a burden and I never felt out of place asking questions. I was completely comfortable admitting my wrong answers and thought process. For myself, it was the invested staff of instructors and the small personalized classes that helped me unlock so many math doors.

This program was the change that myself and many struggling students need! AcerPlacer was a game changer for me, and I love that I get to be a part of a team that helps students finish their college math requirements! So as the AcerPlacer t-shirts say… “Math is nothing to b² of”!


References:

  • Adelman, Clifford. (1999, June). Answers in the Tool Box. Academic Intensity, Attendance Patterns, and Bachelor’s Degree Attainment. Education Publications Center (ED Pubs). Retrieved from https://files.eric.ed.gov/fulltext/ED431363.pdf
  • Gewertz, Catherine. (2018). Avoiding a Remedial-Math Roadblock to a Degree. Education Week, 37(32), 14–15.

Additional Reading:

Guest Post: Valid Grading Measurements Continued

Today’s guest post comes from Madison Hodson, an instructor at AcerPlacer who studies mathematics/statistics education at Utah State University. This is a continuation of her previous article, which can be found here.

As described in my last article, I discussed the two principles necessary to produce a valid knowledge measurement (question on assignment or exam). This article will go into more detail and will provide examples for reference.

The validity of a measurement is broken up into two parts; one of those parts is measurement relevance. For a measurement to be relevant, it must reflect the unit goal or objective. That means for a question to be relevant it must pertain to the correct mathematical content and learning levels for each specific goal. If a measurement is relevant, then educated decisions and evaluations can take place based on the results.

In this example, let’s assume that the unit objective is, “When confronted with a real-life problem, the student determines whether or not computing the area of a surface will help solve the problem.” Listed below are three separate questions that could be included in the assignment. We need to determine which question is the most relevant.


Question 1.
Computing a surface area will help you solve one of the following three problems. Which one is it? (Circle the letter in front of your answer.)

  1. We have a large bookcase we want to bring into our classroom. Our problem is to determine if the bookcase can fit through the doorway.
  2. As part of a project to fix up our classroom, we want to put stripping along the crack where the walls meet the floor. Our problem is to decide how much stripping to buy.
  3. As part of a project to fix up our classroom, we want to install new carpet on the floor. Our problem is to decide how much carpet to buy.

Question 2.
What is the surface area of one side of the sheet of paper from which you are now reading? Use your ruler and calculator to help answer the question. (Circle the letter in front of your answer.)

  1. 93.5 square inches
  2. 93.5 inches
  3. 20.5 square inches
  4. 20.5 inches
  5. 41.0 square inches
  6. 41.0 inches

Question 3.
As part of our project for fixing up the classroom, we need to buy some paint for the walls. The paint we want comes in two different size cans. A 5-liter can costs $16.85, and a 2-liter can costs $6.55. Which one of the following would help us decide which size can is the better buy? (Circle the letter in front of your answer.)

  1. Compare 5 x $16.85 to 2 x $6.55
  2. Compare $16.85/5 to $6.55/2
  3. Compare $16.85 – $6.55 to 2/5

In the example given only the first question is relevant to the objective of, “When confronted with a real-life problem, the student determines whether or not computing the area of a surface will help solve the problem.” It is relevant because the students were faced with various real-life problems and had to determine when finding the surface area would actually be helpful. Based on the class’ answers to this question, the teacher could make an informed decision on whether or not the students achieved the objective.

The other two questions were focused on calculations that were not relevant to surface area. The results from these measurements would provide no feedback on whether or not the students had achieved the objective regarding when to calculate surface area.

As math instructors it is important for us to not only teach how compute numbers and algorithms but to teach logic and reasoning. Many individuals today struggle with the application of mathematics in real life. Perhaps had they experienced relevant math questions (to unit objectives and to life), these feelings would decrease. Creating mathematic contact that is both relevant and reliable takes a large time investment— however, as educators, it is worth our time to properly educate and evaluate our students.