One of the first things you notice in algebraic expressions (besides the sometimes haphazard mix of operations) are numbers that appear with a smaller number above them (like this 54). These smaller numbers are called exponents and, in this
post, I'll give a basic rundown of what they represent and a few basic rules that you will need to follow when dealing with them.
So, you're probably thinking, what do exponents represent anyway. In short, it's a special way of writing a special form of multiplication. I know it sounds hard to grasp, so I'll give you an example:
- Let's look a 3*3. Of course we know it as 9, but in dealing with the order of operations writing a number multiplied by itself may be combersome if you already have several parentheses in the expression. so the way that 3*3 would be written is 32
as your multiplying 3 by a second 3.
But what if you want to represent 4*4*4 or need to multiply 10 5's? Simply count up...
Four years ago, I came up with this math trick. Take a look at it, and at the end I'll show you why it works!
Let's play a game. I’m going to let you make up a math problem, and I will be able to tell you the answer from here. I can’t see what you’re doing, I’m not even in the same room as you, but I will still be able to tell you the correct answer.
Trust me. I’m a professional. Ready?
Okay. First, pick a number. It can be any number you wish, large or small. Now add 5 to that number. Got it? Okay, now double your new number (multiply by 2). Alright, now subtract 4 from the double.
Next, divide your new number by 2. Now, finally, subtract your original number from this new quotient. Got it? Okay. Here comes the cool part. Ready?
The answer is 3. Nifty, huh? What’s that? How’d I do it? Oh, magic.
Okay, okay, it’s not magic. The answer will always be 3, no matter what number you pick. Let’s illustrate this by...
In math you learn new terminologies and many significant things pop up. Guys, do you ever dream about analytical calculus? No? Well, why not!
As a high school student you learned algebra and pre-calculus and those are great, but you can really figure that there is more to math than just that. I assume you were dazed and confused. That's okay. Perhaps though you enjoyed your subjects. That is
There, you must try to learn analysis, because it is the most-funnest part of mathematics! Do you think I'm wrong? Well, begin with a subject like real analysis. During your study of analysis, you learn about continuity, metrics, and integration. I would
like to know more about metrics.
The weird thing is that math is everywhere. Sorry, but I like math because of this fact.
It takes a real scholar to learn math. Got me wrong? Gals sometimes support the most advanced mathematical conclusions. You can make their notions...
Suppose I place you at one end of a long, empty room. Your task is to get to the door at the other end of the room. Simple, right? But what if I told you that this simple task is actually mathematically impossible?
Think about it – in order to traverse the whole room, you first have to get to the halfway point, right? You'll have to travel one-half of the way there. And before you can get to that halfway point, you have to travel one-quarter of the way there (halfway
to the halfway point). And before you can get to the one-quarter point, you have to travel one-eighth of the way there (halfway to the quarter-way point). Since you have to go half of each distance before you can go the full distance, you'll never actually
get anywhere. The task requires an infinite number of steps, and you can never complete an infinite number of steps since there will always be another one. Furthermore, in order to even start your journey you would need to travel a specific distance, and...
Start this year off strong with good organizational and note taking skills. Make sure you understand the material and are not just taking notes aimlessly. Try to take in what your teacher is saying and don't be afraid to ask questions!! If you start taking
the initiative to learn and understand now, college will be a much more pleasant experience for you. Trust me!
Stay organized and plan your homework and study schedule!
Study with friends!
READ YOUR TEXTBOOK! :)
Remember, homework isn't busy work and a chance to copy down your notes, it is part of the learning process. This is especially important with math, as it builds on itself and understanding the basics will make the other subjects easier!
Have a fantastic and fun year!
There's a famous (and probably apocryphal) story about the mathematician Carl Friedrich Gauss that goes something like this:
Gauss was 9 years old, and sitting in his math class. He was a genius even at this young age, and as such was incredibly bored in his class and would always goof off and get into trouble. One day his teacher wanted to punish him for goofing off, and told him
that if he was so smart, why didn't he go sit in the corner and add up all the integers from 1 to 100? Gauss went and sat in the corner, but didn't pick up his pencil. The teacher confronted him, saying “Carl! Why aren't you working? I suppose you've figured
it out already, have you?” Gauss responded with “Yes – it's 5,050.” The teacher didn't believe him and spent the next ten minutes or so adding everything up by hand, only to find that Gauss was right!
So how did Gauss find the answer so fast? What did he see that his teacher didn't? The answer is simple, really – it's all about...
Normally, an equation has a single solution when it contains only one undefined variable. For example, take the equation 3x + 7 = 19.
3x + 7 = 19 [original equation]
3x = 12 [subtracted 7 from both sides]
x = 4 [divided both sides by 3]
This is one case of a larger trend in algebra. As I've already said, you can solve an equation for one answer when it contains a single variable. However, this is derived from the larger rule that you can solve a set of equations where there are as many
distinct equations as there are variables. These are called simultaneous equations, and occur any time that two equations are both true over a certain domain. In the more practical sense, this is what you should do if an exam asks you to solve for a value
and gives you two different equations to use.
To solve simultaneous equations, we can use three strategies...
Back when I was still in middle school, I was sitting at my kitchen table during a family gathering, and my uncle posed the following puzzle for me to solve: A vendor is selling apples for 10 cents apiece, oranges for 5 cents apiece, and peanuts two for
a penny. Someone comes along and buys exactly 100 items for exactly one dollar. How many apples, oranges and peanuts did that person buy?
I took out a sheet of paper and a pencil and came up with the answer in a couple of minutes. This astonished my uncle because, it turns out, he had posed this problem to two adults, including a geometry teacher, and they couldn't solve it in less than
a half hour. I had a bit of a reputation for mathematical cleverness, and he had posed this problem to stump me and test the extent of my cleverness. Decades later I still remember exactly how I solved it, probably because it was a boost to my ego to learn
that I was apparently smarter than a geometry teacher!
If ever a single academic subject has been under attack, it is algebra. Students repeatedly ask me, "Why is this important?" or "When am I ever going to use this?" and even the dreaded "Why should I care?" Recently parents are echoing the thoughts and
in several states alternatives to algebra under the umbrella term "trade math" are being added to curriculum so that students can opt out of upper level maths.
On one hand, I cannot blame my student's frustration and reluctance to give Algebra the time it deserves. I remember when I was a student initially introduced to Algebra. There is a basic, primal fear against seeing numbers and letters in the same equation
that is difficult to overcome and accept. I struggled with algebra, detested it even, but luckily I had many great teachers who helped me all the way into Calculus. Returning to Algebra as an adult I have found several methods which have helped me communicate
The majority of the students that I have often have the same problem -- they aren't grasping the information fast enough or they aren't really able to follow the lessons a teacher gives.
Sometimes, teachers aren't adaptive to every learning style for each student in their classroom. However, know that each student has the capability to learn math on their own. It is just necessary to have key characteristics to make it successful.
Every math student should have:
open communication between themselves and their teacher (inside and outside the classroom)
Always try to study outside of your home or dorm room. In our minds, those are places that we relax at and it can be difficult to turn your mind off from the distractions to study. Public libraries,
universities, coffee shops, and bookstores are the way to go...
This week's Math Journey builds on the material in
The Function Machine. If you have not yet read that journey, I suggest you do so now.
In The Function Machine we discussed why graphing a function is possible at all on a conceptual level – essentially, since every x value of a function has a corresponding y value, we can plot those corresponding values as an ordered
pair on a coordinate plane. Plot enough pairs and a pattern begins to emerge; we join the points into a continuous line as an indication that there are actually an infinite number of pairs when you account for all real numbers as possible x values.
But plotting point after point is a tedious and time-consuming process. Wouldn't it be great if there was a quick way to tell what the graph was going to look like, and to be able to sketch it after plotting just a few carefully-chosen points?
Well, there is! Mathematicians look for an assortment of clues that help to determine the shape of a...
Nailing an 800 on the math portion of the SAT can be a tricky feat, even if you are steadfastly familiar with all of the requisite formulas and rules. A difficult problem can overwhelm even the most prepared individual come test day. Time constraints,
test surroundings, and the overall weight of the exam can unnerve the most grounded students.
So what do you do when panic strikes and your mind draws a blank? How do you re-center yourself and charge forward with ferocity and confidence? What you do is this: write everything down from the problem. This is the most important part of the problem solving
process. As you peruse the question, write down the pertinent data and establish relationships by setting up equations. This exercise will help you see solutions that were previously difficult to decipher.
As you work on practice tests and sample problems, you must work diligently to form a solid habit of writing down important bits of information as you plow through...
Willpower is unique to humanity. It is the keystone characteristic that is directly responsible for our technological advancement over the last several hundred thousand years. Willpower can be defined as the capacity to restrain our impulses and resist
temptation in order to maximize our long-term success. It is the expulsion of energy to fight off innate survival based urges to exponentially increase future advantages and benefits. It is the driving force behind all civilizations, and it is what prods humankind
forward to learn and grow.
When we turn down a bite of cheesecake, step away from a mind numbing reality sitcom, or push off a nap to get some work done, the credit goes to willpower. It is this ghost like aura of control and discipline that we rely on to extend our existence and maximize
our accomplishments. When we watch highly successful individuals exercise routinely, read voraciously, and work tirelessly, we are impressed with their ability to resist...
MP3 players example
The profit in millions of dollars for an MP3 player can be done with the polynomial P=-4x^3 + 12X^2 + 16X. X represents the number of MP3 players produced annually. The company currently produces 3 million MP3 players and makes a profit of $ 48,000,000.
To figure out the least amount of MP3 players the company can produce and still make the same profit we need to solve for P.
Set P=48 This represents the total profit. 48=-4X^3 + 12X^2 + 16X
Step 2 subtract the 48 from 48 and from the end of the equation, like this 48=-4X^3 + 12X^2 + 16X - 48
After several months of carrying some pretty heavy textbooks around with me, I recently decided to switch to a Kindle Fire and start using electronic textbooks. Although there are times when a good old-fashioned book really cannot be replaced, I'm very
pleased with the weight of my tutoring bag now, and my students seem to be enjoying the switch as well.
I'm able to download textbooks for free in some cases ("Boundless" publishing), and I also have several different dictionaries and other reference books a tap away! Any other books I might find helpful for my students? Just a few clicks away. This also
frees up my paper textbooks to loan to my students in-between sessions.
Using a Kindle gives me the added benefit of being able to load educational applications to use for practice and reinforcement. Since we are in the 'computer testing' age, this also gives my students some extra practice in preparing for computerized exams.
I'm sure you'll...
When a young person takes their first higher math course, Algebra I, their brains are developing a different set of skills than arithmetic, and they are faced with abstract concepts. Most students are intellectually ready for this transition in the 8th
or 9th grade. Many students have to be patient, I as it sometimes takes a few months for the "aha" moment when it all clicks. I find that their are many ways to teach algebraic concepts that allow students to grasp the particular skill they are working on.
I use methods such as real-world examples, I foldable study guides, reviewing or introducing basic concepts such as inverses and identities, using colors, using hands-on materials like algebra tiles, and more. I often have students use highlighters or make
their own problems. After 28 years of teaching, I know which misconceptions are made most often, so I also what NOT to do and why. Research has repeatedly shown that success in Algebra II is the best predictor...
Come with me on a journey of division.
I have here a bag of M&Ms, which you and I and two of your friends want to share equally. I'm going to pour the bag out on the table and split it into four equal piles. For this example, “one bag” is our whole, and the best number to represent that whole would
be the number of M&Ms in the bag. Let's say there were 32. If I split those 32 M&Ms into four equal piles and asked you how many were in one pile, you could certainly just count them. But a quicker way would be to take that 32 and divide it by the number of
piles I'd made, which in this case is 4. You'd probably write that as:
32 ÷ 4 = 8
So there are 8 candies in each pile.
Seems easy enough with a large number of M&Ms, right? But what if there were less candies – what if our “whole” was less than the entire bag? Well, for a while we'd be okay – if there were 16, for example, we'd do the same thing and come up with piles of 4
"I can't do this. Why do I need to know this anyway? Can't we just use the computer to do this?"
We have all heard this from someone; ourselves, our spouses, friends and all too often we hear this from our children.
We have seen math as a difficult subject for our generation and now we are seeing math become even more "frustrating", "boring", and "intimidating" for many of our children. We have tried collaboration, individual tutoring and even extra home work as a
means toward improvement. But many of our efforts are met with failure, anger and even tears. What is the key to overcoming the math "Mount Everest"? While there is no band aid for healing math confusion, there are tips and strategies that are fundamental
in changing your child's view of math and developing "number sense".
Math Must Make Sense
The most important thing is to remember...
One of my Calculus students had an interesting Related Rates problem that I had to go home and think about for a while in order to figure out. The problem was set up as such:
A 25 inch piece of rope needs to be cut into 2 pieces to form a square and a circle. How should the rope be cut so that the combined surface area of the circle and square is as small as possible?
Here's what we'll need to do:
1. We will have to form equations that relate the length of the perimeter and circumference to the combined surface area.
2. We will then differentiate to create an equation with the derivative of the surface area with respect to lengths of rope.
3. Wherever this derivative equals 0 there will be a maxima or minima, and so we will set the derivative = to 0 and determine which critical points are minima...
Let's start off by defining some rules.
1) please no one post answers directly on this post, rather send all answers to me via a message. Comments will be deleted and the person will be disqualified from all future contests.
2) The first 5 people to respond correctly to this post will receive a free 1 hour tutoring session via the online platform in any subject that I am approved in. I will respond back to your message explaining the correct answer, how to get that answer,
whether you were correct, and, of course, the details in setting up your free session with me
3) If you are trying, but stumped...message me for a hint
4) Have fun!!
Now for the brainteaser!
Chicken McNuggets can be purchased in quantities of 6, 9, and 20 pieces. What is the largest amount of McNuggets that can NOT be purchased, using these quantites?
Happy Holidays everybody! I look...