Three Strategies to Conquer MaThCliX Digits of Pi Contest
On March 14th, MaThCliX will be hosting our third annual Pi Day, which is filled with a variety of activities for students of all ages. The most anticipated event of the day is the “Digits of Pi” Contest. The rules are simple: whoever wants to participate merely has to recite as many digits of Pi, the famous irrational number, as they can (in order, of course), in other words, master the digits of Pi. The person that says the most digits of Pi accurately wins a Pi Day t-shirt and a pie/cake! Good luck to everyone competing; I hope you find these tips useful! (P.S. make sure you have the correct digits of Pi pulled up on your phone or computer while attempting to memorize it.)
One way to memorize the digits of Pi effectively is through auditory learning. Look at the first 5 digits of Pi and say each one of them out loud. Repeat the process four more times while still looking at the correct form of Pi to guide you. Then look away and try to say the five digits by memory. If you get it correct the first time, then repeat it four more times while looking away. However, if you get it wrong the first time, look back at the correct form and repeat the five digits five times while looking again. Next, attempt to say it five times without looking (successfully this time, hopefully). Repeat these steps until you feel like you have those five digits glued to your brain. If you can use this strategy every day for 10 days before the contest, you will have memorized the first 50 digits of Pi!
Carry around a piece of paper with Pi written on it. Whenever you have a minute to spare either in the classroom or at home, take out the piece of paper and begin writing the digits of Pi by memory, as many as you can do. Then look at the correct form of Pi and assess how you did. Next, write it again, maybe this time adding one or two digits on to the end. If you make this a habit for a week or two before the contest, you are bound for success.
This final strategy is based off the idea that it is easier to remember numbers that have a purpose rather than a random sea of numbers. What you do is assign phone numbers to each set of ten digits in Pi and then attempt to memorize each phone number. It helps to set patterns within the phone numbers to better remember them: make the first letter of the name for the first phone number an “A”, the first letter of the name for the second phone number a “B”, etc. Also, try making the numbers of letter in each name correspond with the first number in that phone number. Try memorizing one phone number every 2 days, and in 10 days you will know 50 digits.
Everyone is different, so a technique that works for one person might not work for another. Experiment with different memorization techniques and find which one works best for YOU. Also, just a reminder: last year’s winner recited 108 digits of Pi. Good luck, and we’ll see you on March 14th!
What is a Math Person Anyway?
“I’m just not a math person!” A phrase often said multiple times by students, but one that is quite false. But, what is a math person anyway? While math ability may be genetic to some, there are many different factors that could assist you in becoming a better math student all around. To showcase this, I dive into two factors that could help you become a more successful math student in the future:
One of the keys used to achieve the highest level of potential in a certain area of study is preparation. Athletes practice their craft everyday in order to become the best they can be. Musicians prepare their music and practice for numerous hours before ever thinking about stepping on to a live stage in front of their fans. The same principle applies when it comes to the study of mathematics. The most common quote could not be more true for math as practice truly does make perfect. Math requires being exposed to numerous problems in order to fully grasp all the ways a question could be asked to you. Without proper preparation, you are only hindering yourself from the success you could be seeing. Preparation helps take down the surprise effect whenever you’re sitting in class for a test and running through the questions. On top of that, preparation makes you feel more confident, which in turn could lead to more engagement in class or even with a tutor if you’re seeing one. This boosted confidence is just a reassurance to yourself that you’re starting to understand the concepts better and next thing you know, you yourself are slowly becoming that almighty “math person”!
This principle is definitely overlooked, but ties in hand-in-hand with the skill of preparation. Students should attempt to become an organized person, as it helps in the studying process for tests and quizzes they will foresee in math. For example, a highly organized person always knows where to look back for practice problems, notes, and even homework assignments when preparing for upcoming tests and finals. This can come in handy as they will be able to not only follow along with their notes, but they’ll be able to utilize them in order to create practice tests and problems for a complete review. Organization is one of the key factors tied with any successful person in any industry, and so it certainly matters in the learning of mathematics!
All in all, I want it to be known that anybody can achieve the status of being a “math person”. These two factors are certainly not the only ones that can help benefit you in the long run, but they can be seen as the building blocks to beginning your successful math journey. It all starts with some preparation and organization and the future of your math career is all in your hands! So next time you see that student next to you make that high math grade, remember, it’s more likely that he or she was prepared and organized and worked for that grade and is not just a “math person”.
Math is a pretty hard subject to grasp when it doesn’t come naturally to you, but there is always a way to learn. Here are a few methods that will get you through the struggle:
- DO Math
Time and time again we see students come in at the last minute to study for an exam. This is not a very effective way of learning. In order to truly understand math, you have to DO it. And I don’t mean just the homework, do extra! If you didn’t understand the homework fully by the time you got to the end, that is an indication that you need to search for some extra practice. If you have a textbook, this is the easiest way to find extra practice. Typically teachers do not assign you all of the problems in a certain section. Do the unassigned problems, and if the answers aren’t at the end of the book: use Google! I promise you can always find some way to check your work. If you don’t have a textbook, search the subject plus the word “worksheet” in Google. A lot of times you’ll find something that comes up and has the answers! If all else fails, ask your teacher where you could find some extra practice. They may just provide it for you!
- Take Notes
I know you are thinking, well duh, but some students don’t know how to take notes in math. If you aren’t already math-minded, you may not understand just an example of a problem without explanations between steps, so write down the example with the steps in words! (If you teacher is moving too fast, ask him/her politely to slow down, trust me, other students in your class with thank you for it.) Here is an example of what I mean:
Factor: 2x^2 + 4x + 2
Step 1: Look for a common factor, we see that it is 2, so divide every term by 2
2(x^2 + 2x + 1)
Step 2: For trinomials with a leading coefficient of 1, we can find two numbers that multiply to the last number and add to the middle number.
Step 3: These numbers become the factors!
Personalize these to fit your needs! Experiment with different methods and find what the best one for you is. If you miss something in class, make sure to ask your teacher after class or during your lunch/study hall. And if you can: email them! They are there to help you, so take advantage of all of the time you have.
- Keep Organized!
It helps a lot to keep your notebook and your work organized. Keep section/chapters in order with the homework, so it is easier to look back if you are reviewing. Also, organize your work by keeping steps clear and logical. If you can’t write all of your work neatly and understandable on the worksheet provided, use extra paper! It will be easier for you to understand when looking back.
- Review for Tests
If your teacher does not provide a review for you, just ask! Most teachers will at least give you a list of all topics that will be on the test. Take a look back at your notes, and look back on the homeworks. Choose 4 to 5 questions from each section to practice. If you get them all correct, yay! If not, redo the whole assignment. Don’t wait until the last minute to start studying for the test. Start a week before and go section by section. This will ensure that you will have time to ask the teacher questions if you are having trouble with a certain sections.
I know that math is not an easy subject for everyone, but the best way to beat a problem is by working. Never give up, and keep pushing through, one example at a time.
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.
The Cool Thing about Repeating Decimals
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 10A – 1, for some number A. Here’s a proof:
- Say x = 0.A1A2A3…An…… (Where x is a repeating decimal, and A1A2A3…An 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 10n, then we would get: x10n = A1A2A3…An.A1A2A3…An…, because the decimal point is moved n times to the right
- x10n = A1A2A3…An.A1A2A3…An… is the same thing as A1A2A3…An + 0.A1A2A3…An, and since 0.A1A2A3…An = x, then x10n = A1A2A3…An + x
- By subtracting both sides of the equation by x, you would get x10n – x = A1A2A3…An, or x(10n – 1) = A1A2A3…An
- By dividing both sides by 10n – 1, you would get… x = (A1A2A3…An)/(10n – 1)
- Therefore, by substituting 0.A1A2A3…An… for x, you would get…
- A1A2A3…An… = (A1A2A3…An)/(10n – 1), where A1A2A3…An is a string of numbers of length n, and 10n – 1 is a string of n 9s, for example 103 – 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…