Mathematical Induction

by Andy Jiang | Apr 18, 2017 | Math

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

by Andy Jiang | Mar 28, 2017 | Math

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

by Kira Pyronneau | Mar 28, 2017 | Math

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.

by Chuck Summers | Jan 2, 2017 | Math

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.

Should Students Memorize Multiplication Facts?

by Chuck Summers | Oct 6, 2016 | Math

Should 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.