Fractions Made Easy

Fraction

A fraction (from Latin: fractus, "broken") represents a part of a whole or, 

more generally, any number of equal parts. 

When spoken in everyday English, we specify how many parts of a certain size there are, for example, one-half, five-eighths and three-quarters.

A common or "vulgar" fraction, such as 1/2, 5/8, 3/4, etc., consists of a numerator and adenominator—the numerator representing a number of equal parts and the denominator indicating how many of those parts make up a whole. 

An example is 3/4, in which the numerator, 3, tells us that the fraction represents 3 equal parts, and the denominator, 4, tells us that 4 parts equal a whole. The picture to the right illustrates 3/4 of a cake.

A cake with one quarter removed. The remaining three quarters are shown. Dotted lines indicate where the cake may be cut in order to divide it into equal parts. Each quarter of the cake is denoted by the fraction 1/4.

Fractional numbers can also be written without using explicit numerators or denominators, 

by using decimals, 

percent signs, or 

negative exponents (as in 0.01, 1%, and 10⁻² respectively, 

all of which are equivalent to 1/100).

 An integer (e.g. the number 7) has an implied denominator of one, meaning that the number can be expressed as a fraction like 7/1.

Other uses for fractions are to represent ratios and to represent division. 

Thus the fraction 3/4 is also used to represent the ratio 3:4 (the ratio of the part to the whole) and 

the division 3 ÷ 4 (three divided by four). 

In mathematics the set of all numbers which can be expressed as a fraction m/n, where m and n are integers and n is not zero, is called the set of rational numbers and is represented by the symbol Q. 

The word fraction is also used to describe continued fractions,algebraic fractions (quotients of algebraic expressions), and expressions that contain irrational numbers, such as √2/2  and π/4 

http://en.wikipedia.org/wiki/Fraction_(mathematics)

Forms of fractions

A common fraction (also known as a vulgar fraction or simple fraction) is a rational number written as an ordered pair of integers, called the numerator and denominator, separated by a line. 

Proper and improper common fractions

A common fraction is said to be a proper fraction if the absolute value of the numerator is less than the absolute value of the denominator—that is, if the absolute value of the entire fraction is less than 1. A vulgar fraction is said to be an improper fraction (U.S., British or Australian) or top-heavy fraction (British, occasionally North America) if the absolute value of the numerator is greater than or equal to the absolute value of the denominator (e.g. )

Mixed numbers

A mixed numeral (often called a mixed number, also called a mixed fraction) is the sum of a whole number and a proper fraction. This sum is implied without the use of any visible operator such as "+". For example, in referring to two entire cakes and three quarters of another cake, the whole and fractional parts of the number are written next to each other: .

This is not to be confused with the algebra rule of implied multiplication. When two algebraic expressions are written next to each other, the operation of multiplication is said to be "understood". In algebra,  for example is not a mixed number. Instead, multiplication is understood: .

Complex fractions

In a complex fraction, either the numerator, or the denominator, or both, is a fraction or a mixed number

Compound fractions

Fractions Made Easy

 All it takes is to look at it in terms of everyday objects, like pizza!

Table of Contents

1. The Basics
2. Numerators and Denominators
3. Kinds of Fractions
4. Type of Fractions Quiz
5. Converting between Mixed and Improper Forms
6. Comparing fractions
7. Equivalent Fractions
8. Comparing Fractions Quiz!
9. Least Common Denominators and Reciprocals
10. LCD and Reciprocals Quiz!
11. Adding and Subtracting Fractions
12. Multiplying and Dividing Fractions
13. A little fraction arithmetic quiz!
14. Lowest Terms
15. Lowest Terms Quiz
16. Fractions stuff on Amazon
17. Love 'em or hate 'em.
18. Other Fun Math Resources!
19. Comments, suggestions, outpourings of emotion

 http://www.squidoo.com/fractions-made-easy

 

 

Vector Spaces

 Vector Spaces

Vector spaces are the subject of linear algebra. Today, vector spaces are applied throughout mathematics, science and engineering.

 A vector space is nothing more than a collection of vectors that satisfies a set of axioms.

 vector space origin

http://www.math.montana.edu/frankw/ccp/multiworld/virtual/Vectors1/learn.htm 

Mathematics Contents 9 (City School)

Mathematics Contents  9 (City School)

1. Direct & Indirect Proportions                          

     D-2;  

     Ex. 2b, 2c, 2e, 2f

2. Algebraic Multiplication and Formulae

     D-2,

     Ex. 4f, 

     Q3

     4j, 4k

     Q 1 - 5

3. Indices & Standard Form

    D-3, 

    ????; 2e

4. Pythagorus Theorem

   D - 2

  Ex 6a; 6b

  Q 1 - 16

5. Trigonometical Ratios

  D-3

  Ex 6a; 6b, 6c; 6d; 6e; 6f; 6g

  Q 1 - 10

6. Volume & Surface

   D -2 

   Ex. 7b; 7c

7. Mensuration, Arc Length, Surface Area, ??? Radian

    D -3 

   Ex. 12a, 12b

8. Sets

  D-2

Ex. 10a; 10b, 10c, 10d

LIMITS

 

CENTRAL TO CALCULUS is the value of the slope of a line,

, but when

the terms become almost

0 0

.  To evaluate the slope, that rate of change,

under those vanishing conditions, requires the idea of a limit.  And central to the idea of a limit is the idea of a sequence of rational numbers.

A sequence of rational numbers

We encounter such a sequence in geometry when we determine a value for π, which is the ratio of the circumference of a circle to the diameter.  To do that, we inscribe in the circle a regular polygon. The ratio of the perimeter of the polygon to the diameter, which we can actually calculate, will be an approximation to π.  And as we increase the number of sides -- that is, if we consider a sequence of polygons:  60 sides, 61 sides, 62, 63, 64, and so on -- then the sequence of those ratios gets closer and closer to π.  Now, the circle is never equal to any polygon. But by considering a sufficiently large number of sides, the difference between the circle and that polygon -- the error -- will be less than a certain small number that we specify.  Less, say, than

0.00000000000000000000000000000001

That is the idea of a sequence approaching a limit, or a boundary.
By that process we can approximate the value of π, which is the limit
of the ratio of perimeters to diameter, as closely as we care to.

Problem 1.    The student surely can recognize the number that is the limit of this sequence of rational numbers.

3,  3.1,  3.14,  3.141,  3.1415,  3.14159,  3.141592, . . .

To see the answer, pass your mouse over the colored area.
To cover the answer again, click "Refresh" ("Reload").

π

We speak of a sequence as being infinite, which, in analogy with the sequence of natural numbers, is a brief way of saying that, no matter which term we name, there is a rule or a pattern that allows us to name one more.

See The mathematical existence of numbers.

The limit of a variable

Consider this sequence of values of a variable x:

1.9,  1.99,  1.999,  1.9999,  1.99999, . . .

Those values are getting closer and closer to 2 -- they are approaching 2 as their limit.  2 is the smallest number such that no matter which term of that sequence we name, it will be less than 2.

By "closer and closer" we mean the following.  Choose an extremely small positive number.  For example, "1 over the national debt"  Then we can name a term of that sequence such that the absolute value of the difference between it and 2  will be less than that small number -- and the same will be true of any subsequent term that we name.
(We say the absolute value because the terms are less than 2, and so the difference itself will be negative.)
When a variable x approaches a number l as a limit, we symbolize that as  x l.  Read:  "The values of x approach l as a limit," or simply, "x approaches l."  In the example above,  x 2.  "x approaches 2."
We also say that a sequence converges to a limit.  The sequence above converges to 2.

By a sequence in what follows, we mean an ordering of rational numbers according to a rule or an indicated pattern.  Here, for example, is a sequence that approaches 0:

0.1,  0.01,  0.001,  0.0001,  0.00001,  and so on.

Left-hand and right-hand limits

Now the sequence we chose were values less than 2.  Hence we say that x approaches 2 from the left.  We write

x 2−

But we can easily construct a sequence of values of x that converges to 2 from the right; that is, a sequence of values that are more than 2.
For example,

2.2,  2.1,  2.01,  2.001,  2.0001,  2.00001, .  .  .

In this case, we write  x 2+ .

But again, no matter what small number we specify, if we go far enough out in that sequence, the value of a difference |x − 2| will be less than that small number.  And so will all subsequent differences that we might name.

Again, when we say that the values of x "approach a limit l," that limit is never a value of x as it approaches l.  There is always a difference between the values of x and their limit.  The limit is the boundary beyond which no member of the sequence will pass.

We summarize this in the following definition.  But first, x is not the only variable.  y is a variable.  And y will be a function of x -- f(x) -- which is also a variable.  And when we come to the definition of the derivative, Δx or h will be the variable.  In the following, then, we will use the letter v to represent any variable.

DEFINITION 2.1.  The limit of a variable.  We say that a sequence of values of a variable v approaches a limit l (a number which is not a term in the sequence)  if, beginning with a certain term of the squence, the absolute value of  v − l  for that term and any subsequent term we might name  is less than an extremely small positive number that we specify.

When that condition is satisfied, we write  v l.

Thus if a sequence of values of a variable approaches a limit, there is always a difference between the limit and the terms of the sequence.  But the difference becomes less than the specified small number of the definition, which is the error we are willing to tolerate. Therefore, because we specify that small number, we say that we can come as close to that limit as we please.

If Δx, then, is the variable that approaches the limit 0 (as it does when we determine the derivative), then Δx is never equal to 0.

The limit of a function

We have defined the limit of a variable, but what we typically have is a function of a variable -- which is also a variable.  For example,

y = f(x) = x².

Now, a sequence of values of x will force a sequence of values of f(x). The question is:  If the values of x approaches a limit, will the corresponding values of f(x) also approach a limit?  If that is the case -- if f(x) approaches a limit L when x approaches a limit l -- then we write

"The limit of f(x) as x approaches l, is L."

In fact, let us see what happens to  f(x) = x²  as x 2−.  Suppose again that x assumes this sequence of values:

1.9,  1.99,  1.999,  1.9999,  1.99999, . . .

x² will then assume this sequence:

(1.9)²,  (1.99)²),  (1.999)²,  (1.9999)²,  (1.99999)², . . .

It is easy to see that x² approaches 2² = 4.

Again, this means that, beginning with a certain term of the x² sequence, the absolute values of the differences between the terms and 4  will be less than any extremely small positive number that we might specify.

Moreover, if we consider a sequence  x 2+:

2.2,  2.1,  2.01,  2.001,  2.0001,  2.00001, .  .  .

then x² becomes this sequence:

(2.2)²,   (2.1)²,   (2.01)² ,   (2.001)²,   (2.0001)²,  .  .  .

And that sequence also approaches 4.  Therefore 4 is the limit of x² whether x approaches 2 from the right or from the left.  And so we can drop the + or − signs and simply write:

To summarize:

DEFINITION 2.2.  A function "has a limit."  We say that a

function f(x) "has a limit" L as x approaches , if  for every sequence of values of x that approach as a limit (Definition 2.1) -- whether from the left or from the right -- the corresponding values of f(x) approach L as a limit (Definition 2.1).

If that is the case, then we write:

"The limit of f(x) as x approaches l  is  L."

In other words, for the limit of f(x) to exist as x approaches l , the left-hand and right-hand limits must be equal.

if and only if

When we say, then, that a function has a limit, we mean that Definition 2.2 has been satified. In practice, it is not necessary to actually produce the requirements of the definition. The theorems on limits imply them.

The most important limit -- the limit that differential calculus is about -- is called the derivative. All the other limits studied in Calculus I are logical fun and games, never to be heard from again.

Now here is an example of a function that does not approach a limit:

As x approaches 2 from the left,  f(x) approaches 1.  As x approaches 2 from the right,  f(x) approaches 3.  The left- and right-hand limits are not equal.  Therefore,  f(x) does not approach any limit as x approaches 2. Definition 2.2,

http://www.themathpage.com/acalc/limits.htm#sequence 

Four habits of highly effective math teaching

these basic habits or principles that can keep your math teaching on the right track.

 
Habit 1: Let It Make Sense
Habit 2: Remember the Goals
Habit 3: Know Your Tools
Habit 4: Living and Loving Math

Habit 1: Let It Make Sense

Let us strive to teach for understanding of mathematical concepts and procedures, the "why" something works, and not only the "how".

This understanding, as I'm sure you realize, doesn't always come immediately. It may take even several years to grasp a concept. For example, place value is something kids understand partially at first, and then that deepens over a few years.
This is why many math curricula use spiraling: they come back to a concept the next year, and the next. And this can be very good if not done excessively (like for 5-6 years is probably excessive).
However, spiraling also has its own pitfalls: if your child doesn't get a concept, don't blindly "trust" the spiraling and think, "Well, she gets it the next year when the book comes back around to it."
The next year's schoolbook won't necessarily present the concept at the same level - the presentation might be too difficult. If a child doesn't "get it", they might need a very basic instruction for the concept again.
The "how" something works is often called procedural understanding: the child knows how to work long division, or the procedure of fraction addition or fraction division, for example. It is often possible to learn the "how" mechanically without understanding why it works. Procedures learned this way are often forgotten very easily.
The relationship between the "how" and the "why" - or between procedures and concepts - is complex. One doesn't always come totally before the other, and it also varies from child to child. And, conceptual and procedural understanding actually help each other: conceptual knowledge (understanding the "why") is important for the development of procedural fluency, while fluent procedural knowledge supports the development of further understanding and learning.
Try alternating the instruction: teach how to add fractions, and let the student practice. Then explain why it works. Go back to some practice. Back and forth. Sooner or later it should 'stick' - but it might be next year instead of this one, or after 6 months instead of in this month.
As a rule of thumb, don't totally leave a topic until the student both knows "how", and understands the "why".
Tip: you can often test a student's understanding of a topic by asking HIM to produce an example, preferably with a picture or other illustration: "Tell me an example of multiplying fractions by whole numbers, and draw a picture." Whatever gets produced can tell the teacher a lot about what has been understood.

---

Habit 2: Remember the Goals

What are the goals of your math teaching? Are they...

• to finish the book by the end of school year
• make sure the kids pass the test ...?

Or do you have goals such as:

• My student can add, simplify, and multiply fractions
• My student can divide by 10, 100, and 1000.

These are all just "subgoals". But what is the ultimate goal of learning school mathematics?
Consider these goals:

• Students need to be able to navigate their lives in this ever-so-complex modern world.
  This involves dealing with taxes, loans, credit cards, purchases, budgeting, shopping. Our youngsters need to be able to handle money wisely. All that requires good understanding of parts, proportions, and percents.
• Another very important goal of mathematics education as a whole is to enable the students to understand information aroud us. In today's world, this includes quite a bit of scientific information. Being able to read through it and make sense of it requires knowing big and small numbers, statistics, probability, percents.
• And then one more. We need to prepare our students for further studies in math and science. Not everyone ultimately needs algebra, but many do, and teens don't always know what profession they might choose or end up with.
• I'd like to add one more broad goal of math education: teaching deductive reasoning. Of course geometry is a good example of this, but when taught properly, other areas of school math can be as well.
  
• Then one more goal that I personally feel fairly strongly about: let students see some beauty of mathematics and learn to like it, or at the very least, make sure they do not feel negatively about mathematics.
  

The more you can keep these big real goals in mind, the better you can connect your subgoals to them. And the more you can keep the goals and the subgoals in mind, the better teacher you will be.
For example, adding, simplifying, and multiplying fractions all connects with a broader goal of understanding parts or part and whole. It will soon lead to ratios, proportions, and percent. Also, all fraction operations are a needed basis for solving rational equations and doing the operations with rational expressions (during algebra studies).
Tying in with the goals, remember that the BOOK or CURRICULUM is just a tool to achieve the goals -- not a goal in itself. Don't ever be a slave to any math book.

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Habit 3: Know Your Tools

Math teacher's tools are quite numerous nowadays.
First of all of course comes a black or white board, or paper - something to write on, pencil, compass, protractor, ruler, eraser.
And the book you're using.
Then we also have computer software, animations and activities online, animated lessons and such.
There are workbooks, fun books, worktexts, online texts.
Then all the manipulatives, abacus, measuring cups, scales, algebra tiles, and so on.
And then there are games, games, games.
The choices are so numerous it's daunting. What's a teacher to do?
Well, you just have to get started somewhere, probably with the basics, and then add to your "toolbox" little by little as you have opportunity.
There is no need to try 'hog' it all at once. It's important to learn how to use any tool you might acquire. Quantity won't equal quality. Knowing a few "math tools" inside out is more beneficial than a mindless dashing to find the newest activity to spice up your math lessons.

Basic tools

1. The board and/or paper to write on. Essential. Easy to use.
2. The book or curriculum. Choosing a math curriculum is often difficult for homeschoolers. Check my curriculum pages for some help. Two things to keep in mind:
   

   i) Now matter what book you're using, YOU as the teacher have the control. Don't be a slave to the curriculum. You can skip pages, rearrange the order in which to teach the material, supplement it, and so on.
   
   ii) Don't despair if the book you're using doesn't seem to be the perfect choice for your student. You can quite likely sell it on homeschool swap boards, and buy some other one.

3. Manipulatives. I once saw a question asked by a homeschooling parent, on the lines, "What manipulatives must I use and when?" The person was under the impression that manipulatives are a 'must' thing.
   Manipulatives are definitely emphasized in these days. They are usually very good, but they're not the end goal of math education, and there is no need to go hog wild over them.
   Manipulatives are something the student manipulates with his hands to get a better grasp of something. But the goal is to learn to do math without them.
   Some very helpful manipulatives are
   
   • A 100-bead basic abacus
   • something to illustrate hundreds/tens/ones place value. I made my daughter ten-bags by putting marbles into little plastic bags.
   • some sort of fraction manipulatives. You can just make pie models out of cardboard, even.
   Often, drawing pictures can take place of manipulatives, especially after the first few grades and on.
   Check out also some virtual manipulatives.
4. Geometry and measuring tools. These are pretty essential, I'd say. For geometry however, dynamic software can these days replace compass and ruler and easily be far better.

The extras

These are, obviously, too many to even start listing.

• Some game or games are good for drilling basic facts. Games are nice for about any topic. Here's one that I played with playing cards with my dd; and now she seems to have learned the sums that add to 10. And here's a game that's worth 1000 worksheets. Of course the internet is full of online math games.
• I would definitely use some math software if teaching graphing, algebra, or calculus. Check MathProf for example, or Math Mechanixs. I've listed a few more here.
• If you're ready to add something new to your toolbox from the online world, try The Math Forum's MathTools - a library of technology tools, lessons, activities, and support materials. Check also my pages listing math games and activities online.

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Habit 4: Living and Loving Math

You are the teacher. You show the way - also with your attitudes, your way of life.
Do you use math often in your daily life? Is using mathematical reasoning, numbers, measurements, etc. a natural thing to you every day?
And then: do you like math? Love it? Are you happy to teach it? Enthusiastic?
Both of these tend to show up in how you teach, but especially so in a homeschooling enviroment, because at home you're teaching your kids a way of life, and if math is a natural part of it or not.
Math is not a drudgery, nor something just confined to math lessons.
Some ideas:

• Let it make sense. This alone can usually make math quite a difference and kids will stay interested.
• Read through some fun math books, such as Theoni Pappas books, or puzzle-type books. Get to know some interesting math topics besides just schoolbook arithmetic. And, there are lots of story books (math readers) to teach math concepts - see a lists by concepts here.
• Try including a bit about math history. This might work best in a homeschooling environment where there is no horrible rush to get through the thick book before the year is over. At LivingMath.net you can find a math history course designed for homeschoolers.
• When you use math in your daily life, explain how you're doing it, and include the children if possible. Figure it out together.

Miscellaneous Math Teaching tips

• The child needs to know the basic addition and multiplication facts very well, or she will have difficulties with fractions, decimals, etc.  These basic facts need to be known by heart.
  
  One of the best ways to start children with math is to have them skip-count up and down from a very young age.  Use a number line to show what the 'skips' or steps mean.  if your child can master the skip counting by twos, threes, fours, etc., she has learned a lot about addition and later on multiplication tables will be an easy fare!  See also this article How to drill multiplication tables.
• Use manipulatives and pictures in your teaching.  Almost all mathematical concepts can be illustrated with pictures, which can even take a place of concrete manipulatives.  For example, if you can condition your child to draw lots of pie pictures when studying fractions, he/she can learn to visualize fractions as 'pies'.  Then he/she won't make the addition mistake 1/2 + 1/4 = 2/6.  Also certain kind of software can take place of the manipulatives.
• In geometry have your child or children DRAW a lot.  See examples in the Math Mammoth Geometry ebook.
• When studying time, money, measuring, homeschoolers have an advantage since they can study those subjects in their natural settings.  Involve your child when you measure, count money, check the time.
• In middle school years, it's important to get familiar with functions, relations, and patterns - these develop algebraic thinking. Check this article about algebraic reasoning from MathCounts.org.
• If you need to know the whys and wherefores of some particular math topic deeper than the textbook tells you, check Dr. Math's archives.  The answers provided there are mathematically "sound doctrine", whereas math textbooks can contain all kinds of errors.

http://www.homeschoolmath.net/teaching/teaching.php

Difference Between Rate and Ratio

A ratio is a comparison of two numbers and can be written multiple ways (like 1/6 or 1:6). You typically do not use units, but if you do, they are often the same. If you have 4 oranges and your friend as 6 oranges, the ratio of your oranges to his is 4/6, which simplifies to 2/3 or 0.666.

A rate is typically distance per unit time, such as 20 miles/hour. This rate is speed, in which you travel 20 miles per 1 hour. For rates, your units are different and often distance and time. Another rate could be some other number per unit time, such as 100kb/sec. In this case, you could say "for every one second, my computer can download 100kb of data." In three seconds, your computer would download 300kb given this rate.

Rate pertains to fixed quantity between 2 things while a ratio is the relationship between lots of things. 

A unit rate can be written as 12 kms per hour or 10km/1hr; a unit ratio can be written in this manner 10:1 or is read as 10 is to 1. 

A rate usually pertains to a certain change while a ratio is the difference of something.

.

Rate and ratios are very important in explaining the equivalence from one and the other. A rate cannot be one if ratio does not exist.

You don’t even notice that these two are still being used in our day to day living like calculating bank interest, product cost and many more. Life has been made easier because of these two.

Read more: http://www.differencebetween.com/difference-between-rate-and-vs-ratio/#ixzz1diyYi5tz




The P/E ratio (price-to-earnings ratio)