Mathematical Induction

Why I Don’t Like Mathematical Induction

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.

Pythagorean Triplets

Cool Thing About Pythagorean Triplets

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 a2 + b2 = c2. For example, a= 3, b= 4, and c= 5 is a Pythagorean triplet because 32 + 42 = 52. If a is any odd number except 1, it can make a Pythagorean Triplet of the form a= a, b= (a2 – 1)/2, and c= (a2 – 1)/2 + 1, or b + 1. You can prove this by substituting b and c with b= (a2 – 1)/2 and c= (a2 – 1)/2 + 1 in the equation a2 + b2 = c2. “a” cannot be even for this case because if a were even, (a2 -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

Setting Goals

Setting Goals for the Semester

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.



Keeping Focused in Classes You Don’t Enjoy

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.

Study Habits 

“I don’t write it down to remember it later, I write it down to remember it now.” Umberto Eco

Here are a few of my ideas to help you study, take notes, and learn.

  1. Take copious and detailed notes, especially in math and science. The more detailed and clear, with diagrams, the better. The process of making the drawing/diagrams and taking the notes will cement the concepts in your brain. It is almost like you are writing on a whiteboard in your brain that you can refer back to when you need it. The reason this works is because you are using multiple senses to learn and you are focusing to do it correctly.
  2. If you have questions that arise, write them down in your notes and make sure you ask the questions as soon as possible. Interaction with the teacher will help you learn the material and how important the subject is to the teacher. If you can ask the questions during the class lecture time, do so: you will not be the only person with the question, and it will encourage others to ask helpful questions. It is my opinion that the only bad question is the one that is not asked.
  3. Dedicate a specific time and place for studying. A special quiet place without distractions is best. MaThCliX provides such a place and many of our best students take advantage of the environment to do their work. For many, even if they intend on doing their math work at home, there are a host of distractions awaiting in that familiar environment. In college, we called the distraction of our bed the ‘rack monster’, so we were more productive in the library away from our familiar spaces.
  4. This is controversial, but I believe, and many studies show, that music causes enough distraction to defocus and make you less productive in studying. It can be used when you take a break to recharge your brain, but during intense study, absolute quiet is best.
  5. You may not know this, but your attitude have a huge effect on your success in learning a subject/topic. If you go into a topic with eagerness, wanting to learn and master it, you will be a lot more successful than if you do it begrudgingly, not wanting to learn it, or hating the subject. You will do a lot better if you trick yourself into finding something to enjoy in the subject. Finding some other subject that is related can help you find a mental link to hang the new facts on.

There are a lot more suggestions, but practicing these will help you as you study.

Stay Sharp Over Breaks

Stay Sharp Over Breaks

Every teacher’s worst nightmare is to come back to a classroom of students in desperate need of review material after having a long break.  Likewise, no student wants to be left behind after returning from their break because they have forgotten things learned in the fall.  Here are some tips to help you stay sharp over your breaks to avoid an unpleasant return to the classroom.

  • Spend at least 10 minutes each day redoing problems from your finals study guides until you have completed it again.
    • These study guides are reusable!  Try making an extra copy before completing it the first time, or simply cover the answers to work through it again.
  • Make flashcards of important vocabulary, formulas, or rules (hint: it’s on your final, it will likely be important for the rest of the school year)!
    • These can be made on index cards, or using an app such as Quizlet, to have in an easy, on-the-go format.
  • Read over your textbook or course material for 10 minutes each day to stay on top of the information.
    • This won’t take long and will keep the relevant material fresh in your mind.
  • Involve your friends and classmates with your reviews!
    • Everything is more fun with a friend, so get someone to quiz you to insure you’re at the top of your game.
  • Make it a competition!
    • A little friendly competition between friends is a great motivator! Form teams and race for the answer, or compete head to head and see who can get the most right answers in a session!
  • Come visit your MaThCliX family!
    • We are here to support you in all your academic needs.  We’ll run flashcards with you, help you complete your reviews, understand textbook concepts, and quiz you to get you ready for your return to class.

“You will either step forward into growth, or you will step backward into safety.”
-Abraham Maslow

In order to learn and grow, you have to step out of your comfort zone.  To help you “step forward into growth” your MaThCliX family will be beside you to cheer you on and support you in every step you take.  Don’t step backward into safety this break or any break, use these tips and become the best you you can be!

Study Tips for Midterms and Finals

As midterms/finals approach, both students and teachers become very stressed. It seems that the only thing on anyone’s mind is how much work they have ahead of them, yet how little time to do the work there is. Because of this, students tend to shut down to the pressure and give up. They tell themselves they’ll pick their grades up next year and let whatever happens, happen. This is the wrong mindset.

Despite how it may seem, finals do not have to be all that stressful if you have the right approach. If you start with these simple tips and tricks, finals will not be as bad as everyone makes them out to be.

  • Start by reviewing the topics you have covered throughout the year and make some notes on your weak areas that way you can come back to them.
  • Based on your weak areas, make your own personalized study guide.
  • Be sure to write down any questions you come across on a separate piece of paper so you can ask the teachers quick and concise questions as their time is just as limited as their students!
  • Be sure to go in for any tutoring sessions available to you! More often than not, your teachers will teach during these final/midterm tutoring sessions to the very material on the test itself.
  • Be sure to start the studying process early. As the time gets closer to finals you do NOT want to be overwhelmed studying for 4-6 finals at a time.
  • Get together with some friends and study together. Often, any questions you have one of your friends may be able to help you with and vice versa.  
  • Possibly one of the most important, take breaks! Do not overload yourself with too much information at once. Every hour or so watch TV or relax for 15-20 minutes. NO LONGER. If the break is too extensive you will lose focus and desire to return to studying.  
  • Be sure to stay well-nourished and well-rested. A healthy body and brain will result in better focus and concentration.
  • Believe in yourself! You made it this far and as long as you taken the necessary steps in studying for finals, you will do great!

Although this can be a stressful time, make sure you do not become overwhelmed. Even though finals are a large portion of the final grade, worrying will not make anything any better. Make sure you study and prepare to the best of your ability and then there is nothing more to worry about. And after you have test, enjoy your break with friends and family!

I Can’t Do Math

I Can't Do MathI know very intelligent people who for whatever reason believe that they just can’t do math.  While there may be some people who have suffered a brain injury of some sort where that part of the brain is damaged, for normal people, the brain is quite capable of learning and doing even complex math.

Brain researchers have discovered a section of the brain that is wired specifically for math. Studies show that part is activated when a person concentrates on math problems much like the speech center reacts when speaking or reading. See the references at the end.

So why is it that many people believe they cannot do math? I think it is partially due to mental laziness because learning math takes time and effort. Like learning to do any complex task, doing math requires practice and focus. Another reason people believe that they cannot do math is because they have been told by others (parents, teachers, friends) that mastering math is only for “special math people”, who have the talent of math. You have probably heard people remark that they can’t do math because they are not a math person, perpetuating the myth even further.

Another reason some don’t want to believe that they can do math is because they think that those who do math well are socially inept nerds. The stigma of being a smart person (in math, science or any other academic subject) is promoted in the media and is “uncool” among middle school and high school students. How many of us remember such a character in TV and or movies? That is a stereotype that needs to be demolished. I believe it is part of the belief that if you work hard to excel that you will miss out of fun that you would otherwise have. I have discovered a wise saying from a friend, “If you take the easy way out of life, life gets harder, if you take the hard way out, life gets easier”.. That’s why I take exception to the phrase “take it easy”.

Resist the temptation to excuse yourself from learning math. You are born with the ability, all you have to do is develop it. While you may not eventually do higher math in your vocation, studies have shown that exercising the math part of your brain strengthens the decision making part of your brain (prefrontal cortex) which will help you in mature relationships and the wise decisions you have to make.

References: About being bad at math:–pmPog#t=192.26306

Brain on math:


Innumeracy –

Should Students Memorize Multiplication Facts?

multiplication-factsShould Students Memorize Multiplication Facts?

I have been struck recently on how many students (and adults) struggle  with multiplication. This skill is so basic to math that lack of mastery makes progress in math very difficult, at best.  I  am not even concerned so much about multiplication of positive and negative numbers as knowing what 6×7 is, for example, because it is hindering many students in higher math.

I don’t believe that a person needs to know “what” multiplication is, for example 6×7 is 6 groups of 7. The multiplication facts are more like the basic phonemes or words of math. When you learn to speak your native language as a child, you don’t really understand the meaning of the words, you are learning and mimicking the sounds and patterns.

As a young pre-teen, I was determined to learn the Morse code in order to become a Ham Radio Operator. I spent time memorizing the code, like a is dot-dash, b is dash-dot-dot-dot, and so forth. I found that I was limited in the speed I could decode the sounds because of the way I was learning the code. I would hear dot-dash and count the dots and dashes.

Unfortunately the requirement for the General Class license was 13 words a minutes, well beyond the mere 5 words a minute I could muster by counting the dots and dashes. My breakthrough came when an older Ham told me, “you can’t just count the dots and dashes, you have to hear the patterns”.

So I started listening for the patterns, so that a was not simply dot-dash, it was a short beep followed by a longer beep. In listening for the patterns I was able to boost my speed up to 18 words per minute, well above the required speed. After a while, you tend to hear whole words and ending as a single unit, so you aren’t just decoding single letters but whole words, much like we learn to hear a language.

So what does all this have to do with learning to multiply? I propose that it is easier to learn the multiplication facts by seeing the patterns and learning the patterns without trying to understand what they mean. That will become clear as you use them. Young children are like sponges and have no trouble learning repetitious and sometime annoying things. Why not teach them the multiplication tables? In the past, memorization devices like catechisms were used to teach children profound truths that they didn’t understand but became part of their knowledge and understanding when they were older.

For older children, repetition is the key. Use all the senses: writing the facts, seeing the facts, hearing the facts spoken, speaking the facts (or singing the facts?). The more senses can be brought into the memorization process, the better.

It is never too late, even for adults, as the human brain is amazingly elastic. If you know what kind of learner you are (visual, audial, tactile) then use that to your advantage. The nice thing about memorization, especially for adults is that it expands the use you have of your mind. So make a point to learn and master the multiplication tables. You will be glad you did.


When Should I Take the SAT or ACT?

act-or-satWhen Should I Take the SAT or ACT?

By: Tyler Mathena

The SAT and ACT are daunting for many students. The scores on these two tests (or just one) can determine what colleges you can attend, what scholarships you can apply for, and whether or not you get into the honors program. All of these have the potential to impact your life, for better or worse, so it is best to be well-informed and prepared.

Everyone knows the magnitude of these tests, and a Google search for test taking tips returns over 18 million results. Most people know how, why, and where to take the test, but it is rarely discussed WHEN students should take it.

The reading and grammar concepts on both tests are mostly covered before high school, it is just a matter of recalling all of the rules and vocabulary. The math section, on the other hand, contains information that students learn in high school classes. After taking Geometry and Algebra, you should have learned most of the content, but do not be discouraged if there are questions you can not figure out! Some of these questions require a different approach or way of thinking, which is often where the strategy comes into play.

Even if you have not done all of the math courses to get a perfect score on the math section of the tests, you should still take it. I would recommend taking both tests at least once, starting towards the end of your sophomore year. This way, there is plenty of time to improve if you do not get the score you want the first time. After getting your score back, look over what your weak spots are and do whatever it takes to strengthen those areas. If you take the test again after studying from your previous results and do not improve, it may be wise to start working with a tutor once or twice a week to get to your preferred score.

For most college applications, you should take the ACT and SAT for the last time in September or October of your senior year. Most colleges have an option to submit your application early so that you know if you are accepted or not sooner. Early decision also automatically makes you eligible for certain scholarships at some colleges. The deadlines typically fall in mid-October to early November.

The SAT and ACT are important. These tests can determine what colleges you get into and how much you pay. Don’t make the mistake of waiting too long to take them. The best time to take the SAT or ACT is now. You can always retake it, as lower scores do not affect you at all, so get ahead of the game; you won’t regret it.