The Math of Flight

Before every airplane flight, math is used to ensure that the plane is loaded correctly. Airline employees known as flight dispatchers do weight and balance calculations to make sure that the aircraft’s center of gravity is within it’s tested limits. Every aircraft has a published range of where its center of gravity can be. This range is defined as the distance from a “datum” which is an arbitrary point defined by the manufacturer of the aircraft. A common location of the datum is the tip of the nose. The center of gravity location is calculated by dividing the aircraft’s moment weight by its total weight, which reveals where the center of gravity is located relative to the datum. An objects moment is calculated by multiplying the objects weight by its arm, which is its distance away from the datum. An object 20 inches aft of the datum will have a larger moment then an object with the same weight that is 15 inches aft of the datum. Airplane manufacturers publish diagrams like the one shown below to assist with these calculations. Our diagram is from a Cessna 172, a very common light airplane. Let’s do an example! Say the pilot and front passenger weigh 150 lbs. each, there is one rear passenger who weighs 120 lbs., and there are 30 lbs. of baggage in baggage area 1. The airplane’s empty weight and moment is different for every airplane and is given by the manufacturer, so we’ll use a common estimate for this example. The final variable is fuel, which in this airplane has an arm of 47 inches. Aviation fuel weighs 6 lbs./gal, and for this flight we will need 20 gallons. 20 gal * 6 lbs.=120 lbs.    

                           

	Weight(lbs.)	Arm(in.)	Moment(in.-lb.)
Front Seats	300	37	11,100
Rear Seats	120	73	8760
Baggage Area 1	30	95	2850
Empty Weight	1491	39.13	58347
Fuel	120	47	5640
Total	2,061		89,547

Doing the math gives us a total weight of 2,061 pounds and a total moment of 89,547 in.-lbs. Now, we divide the moment by the weight to give us a center of gravity location of 43.4 inches aft of the datum. We can now use a chart like the one below to determine if the center of gravity is located within the airplane’s limitations. First move along the x-axis until you reach 43.4 inches and then move up the y-axis until you reach the total weight of 2,061 pounds. We can see that the intersection of these two lines is within the normal category limits, so our plane is safely loaded to go flying!

This was a fairly simple example using a small airplane that can only seat a few people. Can you imagine doing this for a massive airliner with hundreds of seats and capacity for thousands of pounds of baggage? Modern day flight dispatchers use computer programs to efficiently accomplish this task, but in the early days of commercial aviation, there were no computers, and this essential computation was made by hand. Center of gravity calculations are just one type of math that keeps planes safe.

The Things You Use Math For

     These are the ways you can use math in your day-to-day life! Math helps you build. Ask any contractor or construction worker and they will tell you just how important math is when it comes to building anything. Creating something that will last and add value to your home out of raw materials requires a broad range of mathematics. Figuring the total amount of bags of concrete needed for a slab, accurately measuring lengths, widths, and angles, and estimating project costs are just a few of the many cases in which math is necessary in real life home improvement projects. Some students may say they do not plan on working in construction and this may be true, but many will own a home at some point in their life. Having the ability to do minor home improvements will save a lot of money. Armed with math, they will also have the ability to check the work and project estimates, ensuring they are getting the best value. Another way you use math in your day-to-day life is baking! Cooking and baking are sciences all their own and can be some of the best ways of introducing children to mathematics. After all, recipes are really just mathematical algorithms or step-by-step sets of operations to be performed. measuring ingredients to follow a recipe, multiplying and dividing fractions to account for more or less than a single batch, converting a recipe from Celsius to Fahrenheit, converting a recipe from metric to US standard units (teaspoon, tablespoon, cups), calculating cooking time per each item and adjusting accordingly, calculating pounds per hour of required cooking time, understanding ratios and proportions, particularly in baking. Cooking is something everyone does and everyone uses math at some point doing it! People also use math while shopping for groceries. Calculating price per unit, weighing produce, figuring percentage discounts, and estimating the final price are ways people use math while shopping. Another way people use math in their day-to-day life is when gardening. Even doing something as mundane as gardening requires basic math skills. If you need to plant or sow new seeds or seedlings you need to make a row or count them out or even make holes. So even without thinking you are doing math. Measuring skills are always needed, and calculations are vital when doing something new in the garden. The final thing (but not the last thing you use math for) is everyday activities. Every outing you plan uses math. Whether you go to the beach or to the zoo. You have to plan your way there use your time wisely, math is your guide that will assist you and help you. When driving you need fuel, oil and water, without it your car will break down. All of these require math.

Math At Home

     In my English class I was assigned a piece on the differences between western parenting and Asian parenting styles. It was called “Why Chinese Mothers Are Superior”, By Amy Chua. It was an incredibly interesting and controversial piece of writing because it said that Western parents are too lackadaisical when it comes to their children’s grades, relationships, and their activities. She argued that Chinese parents are stricter and more helpful when it comes to their children’s schoolwork and lives. A part of her writing that really stuck with me is that Chinese mothers will spend a lot more time helping their children with school work. A Chinese mother will not accept anything below an A. Anything below an A is considered failing in the Chinese household. It is also not unusual for the majority of Chinese children to get As. Chua said that in the American household where B’s are applauded, the low grade is the result of the parent’s fear of hurting their children’s feelings and also lack of serious time to study. Chua argues that there is too much of a focus on sports and extracurricular activities in the West. She says that the time and effort put forth into those activities would be more useful in academic pursuits. I agree with a lot of what Chua says. In the West, more specifically in the United States, most people are raised with the idea that math is really hard to understand and you are never going to use it. In China, there is not as much of a focus on understanding math as there is a focus on memorizing it. The Chinese say that understanding math comes through memorization. In the United States, math is taught through understanding. If parents put more effort into talking about math positively it would make their children think about math more positively. If in the West, like in China, parents sat down with their children and studied until they really understood math, the children would not be as afraid of it. It would just be another subject they had to learn. I think that the Chinese ways of studying could also be a bit intense and create feelings of negativity about math. But the idea of focusing on the subject more often is a good one. The more time you spend with something the more familiar it becomes, and the less afraid of it you are. I think if the west adopted some of the Chinese ideals in studying math, we would be better off.  

"When am I going to use this?"

     “When am I going to use this?” is something most people hear at some point in a math class. Sometimes it is a rhetorical question that is just meant to show the teacher that this information isn’t useful. Sometimes it is a genuine question. I once asked a teacher “When do people use this information?” because I was curious and also, I thought it might help me remember the math. She assumed I was being snarky and told me to stop asking questions like that. I think that it is a good question to ask. But I also think we should respond with real answers instead of telling a student to stop asking questions like that. If we start responding to students who ask the question “When am I going to use this?”, maybe the students will start to see real value in math. A lot of the reason children ask questions like this is because of the attitude surrounding math in the United States. If we raised children to be fearless when it came to math, maybe they would be more interested in learning it. Even if you learn something in math that does not mean a lot to you or it is not something you can use in the future, it is still training your brain. Math will help you stay sharp- minded into old age! Think of math as strength training. If you never exercise, your body will most likely become weaker and less efficient over time. But if you exercise daily whether you are doing a quick warmup, yoga, or a full-blown session, your body is going to be so much stronger! Think of math as strength training for your brain! Another benefit to learning math (even if you are not going to use it ever again), is that it can help you in other subjects too! There are a lot of other subjects that involve a lot of math such as chemistry, physics, biology, and even music. So, if a chemistry student is not interested in learning something in math class, they may not know that the math they are learning can help them in chemistry class! Math is so important. Especially in elementary school and middle school, because it sets the foundation for all other math you will learn in the future. Kids do not usually know what they want to be in elementary school and middle school. But if a kid in elementary school wants to be an astronaut and hates math, a way to get that student motivated could be to teach that student that astronauts have to know a lot of math in order to become an astronaut. So the question “When am I going to use this?” is a wonderful question! We just have to be able to explain to our students that there are a lot of ways we use math!

Music In Math

    

 

     Being in the middle school choir was easy. We learned music rote. Rote means you learn it by ear. The alternative to rote music, is reading music. Reading music is called sight reading. I was so excited for high school chorus. But little did I know that sight reading is the way you read music in high school. Middle school did not prepare me for that. Throughout high school I got better at reading sight music. But there was one thing I couldn’t help but notice. That reading music was closely related to math. I brought this up to my high school choir teacher and she confirmed my observation. She told me that often in college if you are a music major you are required to take several math courses. Who knew? Anyways, she told me about the circle of fifths. The circle of fifths is the relationship among the twelve tones of the chromatic scale, their corresponding key signatures, and the associated major and minor keys. More specifically, it is a geometrical representation of relationships among the 12 pitch classes of the chromatic scale in pitch class space. More simply, the circle of fifths uses intervals, which is something we use in math. An interval in math is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. An interval in the circle of fifths has to do with pitch. Starting at any pitch, ascending by the interval of an equal tempered fifth, one passes all twelve tones clockwise, to return to the beginning pitch class. The circle of fifths is really hard to understand and my chorus teacher barely went into detail about it for that reason. But it uses math! That was fascinating to discover. So, throughout my choir experience I continued to find more examples of how math is a part of music. Subdivisions are a great example of this. Subdivisions are essentially just fractions. Most music is played in 4/4 time. Each beat equals a quarter note, and there are four quarter notes in a whole measure (or bar) of 4/4 music. When one evenly divides the quarter note in two, you get two eight notes. This could be used to explain fractions! When you divide ¼ you get 1/8. When you evenly divide the quarter note in four you get four sixteenth notes! This is the same math we learn in school, only it’s also music. When people ask, “When are we going to use this?” during a math lesson, well, you may use it in music. (If they become a musician of course). Most people don’t think of music as a very mathematical topic, but it truly is!

 

Mathephobia

I remember when I first started to experience mathephobia. It was in 6th grade and I had a math teacher that was really cranky all the time. I remember her creating the perfect environment to develop a fear of math. She would get angry when we didn't get the answer right and it wasn't the kind of anger that came out as understanding with mild frustration. It was actual anger. I was afraid to raise my hand because I might get the question wrong and she would embarrass me in front of the class. Even tests made me afraid. I remember waiting in line to receive my test and when I got there my grade was an 80. I was devastated. Before 6th grade I have 2 memories of math and both are positive. I remember in 2nd grade struggling to understand long division. It was such a frustrating feeling. But I gathered the courage to go up to the teacher and ask him how to do long division. After he explained it I finally got it! I was so proud. The second memory is about 4th grade. My 4th grade teacher gave us a sheet of paper that had multiplication problems on it. We had a minute to do them. I always did so well. She would put our math minute in our math folders and if we got a 100 she would give us a piece of candy along with the math minute. I always got a 95 or above. I felt so comfortable with math until middle school. What changed? I don't know. It could have been in my 6th grade math class with my mean math teacher. But also it seems like everyone began to be afraid of math in middle school. I remember so badly wanting to be in advanced math classes with my friends. I also remember being incredibly jealous of anyone who was great at math. I'm still jealous of anyone who's good at math. My friends are all really good at math. When they complained about math I thought "well at least you're taking AP calculus", where I would be taking college algebra. Math class makes my heart race and my palms sweat. When the teacher announces that there will be a quiz my heart begins to hurt and I feel physically horrible. It makes me want to curl into a ball and cry. I think math is extremely important and I want to understand it at the highest level. But for some reason, I am terrified of math. I have a specific memory from when I was 7 or 8 of my dad asking me mathematical questions during dinner. I can't remember exactly what the question was but it had something to do with multiplication or addition or subtraction. I didn't know how to answer the question let alone the actual answer. Mys dad got so frustrated and so did my mom. My mom would say "You should know this" and my dad something like "are you serious?". For my adult parents the question they asked seemed so easy, but for a kid like me I just didn't know. That memory still makes me feel bad about my math abilities and I was only 7 or 8. Mathephobia is something that affects me everyday. I don't want to be afraid of math. For the first time in a long time I'm taking a math class that is making me think about math in a different way than I have in the past. Hopefully I can end my fear of math, but until then I will actively try not to be afraid.