There is some confusion among students about the distinction between "undefined" and "indeterminate". This article will give a couple examples of both, along with some explanation.
First, here's a generic example from algebra, where we solve an equation by finding the value of the variable.
2x + 4 = 12
The solution to this equation is the value of x that will make the equation true. We can simplify by subtracting 4 from both sides:
2x = 8
Then, divide both sides by 2:
x = 4
Placing 4 back into the original equation in place of x:
2(4) + 4 = 12
8 + 4 = 12
12 = 12
With that illustration in mind, let's look at an expression that is said to be undefined:
Rather than just saying "we can't divide by 0", let's see what happens when we turn this into an equation and try...
Settle in, folks, today's a long one.
The Function Machine, we learned that functions can be depicted as curves graphed on a coordinate plane. In
What Does the Function Look Like?, we learned how to tell the general shape of a function's graph based on characteristics of its equation, and vice versa. Today, we'll be focusing on linear equations (meaning any equation that graphs into a straight line).
The defining characteristic of a linear equation is that the highest power of x in the equation is x to the first. This denotes that for every y value, there is exactly one corresponding x value. Of course, there is always exactly one corresponding y value
for every x, but this is one of those “square is a rectangle; rectangle is not necessarily a square” moments. We know there's exactly one y for every x because we choose our x's independently and the y's are dependent on them. There can't be more than one
y for any given x; you've only got one output...
Here are 48 of my favorite math words in 12 groups of 4. Each group has words in it that can be thought of at the same time or are a tool for doing math.
What are your favorite math words? If you aren't sure, search for "mathematical words" and pick a few.
The primary student baseline communication skills a student should learn from their tutor are the following:
Precise use of vocabulary
Express complete thoughts
Interpreting and following instructions
These baseline communication skills are common in academia, particularly Mathematics. Any behaviors, thoughts, attitudes, philosophies, etc. that hinders these baseline communication skills presents learning hindrances for the students and tutors.
Let me know your thoughts.
Purpose: This series shares tips on how to identify, manage, and overcome Mathematics Negative Self Talk (NST). We cannot avoid NST totally because the NST about Math skills in general is a widely accepted habit.
So what is Mathematics NST anyway? Mathematics NST is when we speak in our minds or to others about an inability to learn, do, and/or understand Mathematics in general. Focus here is what we cannot do or have never done in Mathematics. For example, "I hate
Math." "I can't do Math!" "This is too complicated!" " I could never do Math!" "My parents aren't good at Math either." "What can we use Algebra for anyway?" "The teacher is confusing me." The NST phrases list is endless, but also popular in today’s
Downside of NST: NST in Math is simply a bad habit of thinking and attitude. This habit limits learning...
Reading Formulas can make or break how a student comprehends the formula when alone - outside the presence of the teacher, instructor, tutor, or parent.
Formula For Perimeter of Rectangle: P = 2l + 2w
How To Read: The Perimeter of a Rectangle is equal to two (2) times the Length of the longer side of the rectangle (L) plus two (2) times the Width of the shorter side of the rectangle (W).
When is reading formulas like this necessary? At three particular moments, reading this formula in this manner can be effective.
When students are initially learning what the formula means
When student are learning what it means when they should already know (remediation).
When students want to remind themselves (basics learning study skill habit)
Remember, Formulas at their introduction are complete statements or thoughts. Students cannot and will not recall complete thoughts or statements...
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