## Math of Solar Eclipse

Credits go to: www.stem.org.uk

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Credits go to: www.stem.org.uk

Cool Divisibility Rules

Special Thanks to: Patrick Burnett

You would probably learn the divisibility rules for 2, 3, 4, 5, 6, 8, and 9 in school. But what about the divisibility rule for 7? Well, the divisibility rule for 7 is quite simple, and quite interesting. All you have to do is take off the last digit of the number, multiply it by 2, and subtract that from the rest of the number. Here’s an example: Say you want to know if 469 is divisible by 7. If 469 is divisible by 7, then 46 – 2×9 must also be divisible by 7, and 46 – 18 = 28. Since 28 is divisible by 7, 469 is divisible by 7. That’s a quick way to check divisibility without having to do long division. Here’s another example: You want to know if 999999 is divisible by 7. If 999999 is divisible by 7, then 99999 – 2×9 = 99981 must be divisible by 7, and if 99981 is divisible by 7, then 9998 – 2×1 = 9996 must be divisible by 7, and if 9996 is divisible by 7, then 999 – 2×6 = 987 must be divisible by 7, and if 987 is divisible by 7, 98 – 2×7 = 84 must be divisible by 7, and we know that 84 is divisible by 7. Therefore, 999999 is divisible by 7.

This trick can be generalized to different numbers. For the mathematical minded: If you want to prove if a number 10X + Y is divisible by P, where X is a positive integer, and Y is an integer from 0 to 9 inclusive, and find a K such that 10K + 1 is divisible by P, then X – KY must also be divisible by P.

In layman’s terms: You know that the divisibility rule for 7 involves subtracting 2 times the last digit from the other digits. A similar trick can be applied to other odd numbers. The reason why the last digit is multiplied by 2 is because 21 is the least multiple of 7 that ends in a 1. The divisibility rule for 13 is similar, but you would have to subtract 9 times the second digit from the other digits, as 91 is the least multiple of 13 ending in a 1, and for 17, you would multiply the last digit by 5, as 51 is the least multiple of 17 ending in a 1.

Are you suspicious? I don’t blame you, but see for yourself. Multiply 17 by a large number on a calculator, and try the trick on the large number. For example, you can try 83521. 8352 – 5×1 = 8347, 834 – 5×7 = 799, 79 – 9×5 = 34, and 34 is divisible by 17, so 83521 must also be divisible by 17.

You can do this with any odd ending in 1, 3, 7, or 9, as all odds ending in these numbers will eventually have a multiple ending in 1. Try to find the divisibility rule for 97. Scroll down to see the answer when you find it.

Answer: Subtract 29 times the second digit from the first.

Get out your calculator and type in 331/999 and press enter. What do you get? Something along the lines of 0.331331331…? Now type in 21/9999 and press enter. What do you get? 0.002100210021…? This cool trick works with any fraction less than 1 with all nines in the denominator, which are always of the form 10^{A} – 1, for some number A. Here’s a proof:

- Say x = 0.A
_{1}A_{2}A_{3}…A_{n}…… (Where x is a repeating decimal, and A_{1}A_{2}A_{3}…A_{n }is the string of numbers that is repeated in the decimal, and n is the length of the string) - If we multiply both sides by 10
^{n}, then we would get: x10^{n}= A_{1}A_{2}A_{3}…A_{n}.A_{1}A_{2}A_{3}…A_{n}…, because the decimal point is moved n times to the right - x10
^{n}= A_{1}A_{2}A_{3}…A_{n}.A_{1}A_{2}A_{3}…A_{n}… is the same thing as A_{1}A_{2}A_{3}…A_{n }+ 0.A_{1}A_{2}A_{3}…A_{n}, and since 0.A_{1}A_{2}A_{3}…A_{n}= x, then x10^{n}= A_{1}A_{2}A_{3}…A_{n }+ x - By subtracting both sides of the equation by x, you would get x10
^{n }– x = A_{1}A_{2}A_{3}…A_{n}, or x(10^{n }– 1) = A_{1}A_{2}A_{3}…A_{n} - By dividing both sides by 10
^{n}– 1, you would get… x = (A_{1}A_{2}A_{3}…A_{n})/(10^{n}– 1) - Therefore, by substituting 0.A
_{1}A_{2}A_{3}…A_{n}… for x, you would get… - A
_{1}A_{2}A_{3}…A_{n}… = (A_{1}A_{2}A_{3}…A_{n})/(10^{n}– 1), where A_{1}A_{2}A_{3}…A_{n}is a string of numbers of length n, and 10^{n }– 1 is a string of n 9s, for example 10^{3}– 1 = 1000 – 1 = 999

This proof is a generalization of a trick you learn in middle school. Say you want to convert 0.21… into a fraction. You would set that equal to x to get x = 0.21… and multiply both sides by 100 to get 100x = 21.21…, which is 100x = 21 + 0.21…, or 100x = 21 + x. Next you would subtract x from both sides to get 99x = 21, and x = 21/99. Don’t forget to simplify J.

Note: If you want to do something like 7/99999, add 0s in front of the 7 to make the numerator and denominator have the same number of digits to get 00007/99999, and the decimal of that would be .000070000700007…

**And just like that, another summer is over and a new school year begins! Here are some tips for both parents and students to work together to ensure a successful school year.**

**1. Set goals: Write them out clearly and display them somewhere that you see them everyday**

** ex: I will complete my HW before I watch TV**

**2. Get organized: This includes finding a way to organize papers going back and forth from subject to subject. How are you going to know what your assignments are and when they are due? **

**3. Plan: What HW, tests, and quizzes do you have this week? How will you prepare for them? Make sure you plan out your study time.**

**4. Practice: This is how you learn! Make time each day to practice.**

**5. Get Help: Are you not understanding what you are supposed to be learning? Ask! Get help! Go to your teacher, parent, and of course, MaThCliX! That is what we are here for.**

**1. Make sure that you know how to communicate with your student’s teacher. Know when conferences are and plan to have a presence and be proactive in your student’s academics.**

**2. Check grades! Even if your student is old enough to check their own grades, it never hurts to have a parent checking, too. Know when progress reports and report cards are due. If you see grades dropping, intervene quickly!**

**3. Make sure your student is doing the success tips for students. Ask them how they are doing each one.**

**4. Find out about what student’s are learning each week so that you can help or get help, as needed. Find out about tutorials, teacher websites, and recommended resources.**

**5. Bring your student to MaThCliX! **

The majority of people know what an Advanced Placement (AP) course is, but many do not realize the true importance of taking these higher level classes, as well as the AP Exams. There are many benefits of taking AP classes, including but not limited to: the increased rigor of your overall course load which colleges take into consideration during the application process, an added 0.5 onto your letter grade which counts towards your HOPE GPA (for example, if you make a “B” in an AP class, it counts as a 3.5 instead of a 3.0), and an extra ten points on your net grade point average which class rank is based off of. However, the biggest reason to take an AP class is to succeed on the corresponding AP test in May and receive college credit for the class.

The AP exams are scored on a 1-5 scale; generally 10% of people get 5’s, 20% of people get 4’s, 20% of people get 3’s, and the remaining 50% receive either a 1 or a 2. The College Board, the organization that creates the tests each year, designs the exams and scoring system in an effort to only give approximately half of the people that take the test college credit, which at most universities is a 3 or higher.. However, if you have a good work ethic in the course throughout the year and then study heavily prior to the test, a 3 or higher is definitely attainable.

If you pace yourself with AP classes starting freshman year, it is very possible to come out of high school as a sophomore in college – which would save anywhere between $5,000 to $40,000 depending on the university. A suggested AP course schedule to complete this task is as follows: one AP class freshman year, two sophomore year, 3-4 junior year, and 2-4 senior year. Each exam that you score well on gives you anywhere from 3-6 credit hours – keep in mind a full year of college consists of 30 hours. So this means that to complete a full year from AP exams requires passing about 7-9 of these tests. Although you do have to pay to take the exam – about $95 per test – that is a heck of a lot better than paying thousands for the first year in college.

You may be discouraged from joining the math team because you don’t think you’re not good at it, but like many other things in life, you can get better at math by working hard at it and with the proper attitude. All you need is a love of math, the desire to make yourself better, and a winning attitude. If you don’t like math, then maybe joining math team can change your opinion on math. At math competitions, you commonly get problems that require you to think outside the box and think of creative solutions, unlike traditional high school math. If you are interested or just want to try it out, talk to your school’s math team coach and ask him or her about joining the team. He or she would be more than willing to help you out. If your school doesn’t have a team and you want to start one, take initiative and find a teacher who is willing to sponsor and coach a math team. There are lots of math competitions in Georgia every year, where you get a chance at showing off your math skills, finding opportunities to improve, and simply having fun. If you see a problem you don’t understand, don’t feel discouraged. If you are willing to learn the tricks, you will get better and better at it, and eventually, if you do consistently well at math competitions, you can make it on one of Georgia’s state ARML teams, where the best are selected from all over Georgia to compete in a competition involving teams from all over the US and even from other countries, like Colombia and China. You get to interact with kids who love math just like you from all over Georgia and stay at a dorm at the University of Georgia to #DoMaTh and/or socialize with the kids and compete for a chance for fun and international glory.

Everybody has opinions, and you can even have opinions about math. I, for one, don’t particularly like the method of mathematical induction. Why? Because it doesn’t completely tell you “why”. For example, you can use mathematical induction to prove that 1 + 2 + 3 + 4 + … + n = n(n+1)/2. Great, we know that the formula works, but where does the formula come from? A much more elegant method to prove that formula is derivation. There are many ways to derive that formula, and one method is by writing 1 + 2 + 3 + 4 + … + n forwards and backwards and by adding them together twice. That way, you can see how twice the sum would be n(n+1) (see the visual), so the sum would be n(n+1)/2. By deriving the formula, you see why the formula is the way it is, and you’ll be able to connect the formula to the nature of the series. Also, mathematical induction is not a method to make equations, as it can only be used if the equation is given. With induction, you may prove old equations, but with derivation, you’ll be able to make new equations.

There is a cool property of all odd numbers except one, they can all form Pythagorean triplets. A Pythagorean triplet is any set of positive integers a, b, and c that satisfy the equation a^{2 }+ b^{2 }= c^{2}. For example, a= 3, b= 4, and c= 5 is a Pythagorean triplet because 3^{2 }+ 4^{2 }= 5^{2}. If a is any odd number except 1, it can make a Pythagorean Triplet of the form a= a, b= (a^{2 }– 1)/2, and c= (a^{2} – 1)/2 + 1, or b + 1. You can prove this by substituting b and c with b= (a^{2} – 1)/2 and c= (a^{2} – 1)/2 + 1 in the equation a^{2 }+ b^{2 }= c^{2}. “a” cannot be even for this case because if a were even, (a^{2 }-1)/2 would not be a whole number, and a cannot be one, because then, b would be zero. For all other odd numbers, this would work, and Pythagorean Triplets of this form include…

a= 5, b= 12, c= 13

a= 7, b= 24, c= 25

a= 9, b= 40, c= 41

As you progress through a semester, it is important to continuously assess your progress and set new goals for yourself. This is an important thing to do because it makes you more aware of how you are performing and helps you to create a focused path for the remainder of the term. Furthermore, setting goals helps you to better organize your limited time and resources available as well as improve your motivation

**Assess Your Progress:**

The first step to goal-setting is to assess your current progress. This includes knowing the grades you have received as well as the study habits you used to get them. Make a note of whether you have been doing things like reading the textbook, taking notes in class, completing your homework or thoroughly reviewing before tests. It is also important to assess how well you understand the course material as well as if your grades match your perceived understanding.

**Decide What You Want from the Class:**

Before setting defined goals, it is important to think about what you want to get out of the class. Do you want a good grade to raise your GPA? Do you want a strong foundation on a topic that will be important later in your curriculum? Do you want to be more knowledgeable about a subject that interests you? Knowing what you specifically want from a course will give you the motivation to keep focused and continue to work hard.

**Setting Goals:**

Now that you know how you are doing in your courses and what you want to gain from them, it is time to set goals for yourself.

Set goals for things you would like to see occur. For example, maintaining or improving your grades, study habits or your understanding of certain topics. It is important that your goals are both reasonable and specific to keep you motivated and focused.

In addition to setting goals, you need to create a plan to help you reach them. To get the grades that you want, you must understand what study habits you will need to have. Also, you must assess if you need extra help and find out what resources are available to you. The internet may have additional explanations and examples of topics. Additionally, your teacher may be able to give you additional help outside of class. Don’t forget that MaThCliX is also a great resource!

Once you’ve set goals and created a plan for yourself, you will be on track for a focused and productive term. Following the ideas discussed above will help you stay motivated while providing with the skills and resources to get the most out the courses you are taking.

By Tyler Mathena

Nobody likes every subject in school. Teachers love to pretend that their subject is fascinating to everybody because they like it, but it is pretty clear that is not the case. Many students are passionate about history, others prefer english, and some (like myself) love math and science. As a student, I know how difficult it is to justify learning material in classes that aren’t your favorite, but it is important to learn every subject.

Every subject, regardless of how bizarre or boring, can be applied to real life. Math is everywhere, every job has to write occasionally, and past events are extremely relevant and can help you understand why the world is the way that it is. Also, learning to pay attention and feign interest in these subjects can help you learn how to learn. In your future job, no matter what it is, there will be things you have to learn that you may think are not important. Learning these subjects that you do not like in school makes learning these skills much easier and can make you seem mature to your employer.

Additionally, if you walk into a class convinced that you are going to despise it, then you will. Keep an open mind and maybe you will be surprised. I have a history of disliking social studies classes, so I was not hopeful when I walked into the first day of Microeconomics. I kept an open mind though, and it turned out to be my favorite class of the semester. That single class inspired me to minor in business in college. Keeping an open mind can be tough when you know that you dislike similar classes, but it definitely pays off in the long run.

I know better than anyone that it is hard to stay awake in that one terrible class (we all have one!). Lucky for you, I have learned and stolen some tricks over the years that can make any class bearable. My best advice is to find a study partner. It is better if your partner enjoys the class because often their enthusiasm for the subject can rub off on you. Even if you both aren’t a fan of the class, simply studying with a friend can help. If you absolutely do not get along with anyone in the class, it can sometimes help to talk to the teacher before or after school and get them to honestly explain to you why they enjoy their subject. Again, the enthusiasm may rub off on you, or they may convince you that it is important by bringing something up that you never thought about.

If it is math that is not your thing, do not worry, you have MaThCliX! All of us here love math and it shows. Coming in often, even if you do not enjoy it at first, will make math (or any subject) at least bearable, if not phenomenal.