1. Where in life is this useful?
a) Cooking: [pic]
b) Measurements (construction, remodeling, etc): [pic] c) Time: [pic]
d) Money: [pic]
2. Fractions with the same (“common”) denominators
Example: (without converting back & forth from mixed numbers):
[pic]
[pic]
3. Fractions with different denominators
In order to add (or subtract) fractions with different denominators (as a reminder, that’s the bottom number), you’ll need to convert them to have the same denominators. This is one place where we get to use the “least common multiple” that we talked about a while ago.
Let’s start with money, because we all do that conversion frequently, and without thinking about what we’re doing. If we add a quarter & a nickel, we know off the top of our head that we have 30 cents, or 30/100 of a dollar. But what is the math that we’re doing?
[pic]
First, we need to convert to a common denominator. For money, rather than worrying about the lowest common denominator, we automatically convert to hundredths. We do that by multiplying by one in the form of a fraction: [pic]. We can do this because multiplying a number by 1 does not change its value. So, we now have: [pic]. All we’ve done is converted the quarter to 25 cents and the nickel to 5 cents. From this point, we can add them: [pic]. We don’t give much thought to all these steps that we go through, but as soon as it is phrased as “adding fractions”, it seems to get much harder!
Now, let’s try a more abstract case. For no particular reason, we need to add 1/3 to 1/4. The LCM for 3 & 4 is 12 (if you get stuck finding the LCM, and don’t mind dealing with larger numbers, you can multiply the denominators and reduce your answer at the end). So: [pic]. With a little practice, you’ll be able to skip writing the...
...Definition: A Mixed Fraction is a
whole number and a fraction combined,
such as 1 3/4.
1 3/4
(one and threequarters)
To make it easy to add and subtract them, just convert to Improper Fractions first:
4/43/4
Quick Definition: An Improper fraction has a
top number larger than or equal to
the bottom number,
such as 7/4 or 4/3
(It is "topheavy")
7/4
(sevenfourths or sevenquarters)
Adding Mixed Fractions
I find this is the best way to add mixed fractions:
convert them to Improper Fractions
then add them (using Addition of Fractions)
then convert back to Mixed Fractions:
Example: What is 2 3/4 + 3 1/2 ?
Convert to Improper Fractions:
2 3/4 = 11/4
3 1/2 = 7/2
Common denominator of 4:
11/4 stays as 11/4
7/2 becomes 14/4
(by multiplying top and bottom by 2)
Now Add:
11/4 + 14/4 = 25/4
Convert back to Mixed Fractions:
25/4 = 6 1/4
When you get more experience you can do it faster like this:
Example: What is 3 5/8 + 1 3/4
Convert them to improper fractions:
3 5/8 = 29/8
1 3/4 = 7/4
Make same denominator: 7/4 becomes 14/8 (by multiplying top and bottom by 2)
And add:
29/8 + 14/8 = 43/8 = 5 3/8
Subtracting Mixed Fractions
Just follow the same method, but subtract instead of add:
Example: What is 15 3/4  8...
...Fraction (mathematics)
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, a fraction describes how many parts of a certain size there are, for example, onehalf, eightfifths, threequarters. A common, vulgar, or simple fraction (examples: \tfrac{1}{2} and 17/3) consists of an integer numerator, displayed above a line (or before a slash), and a nonzero integer denominator, displayed below (or after) that line. Numerators and denominators are also used in fractions that are not common, including compound fractions, complex fractions, and mixed numerals.
The numerator represents a number of equal parts, and the denominator, which cannot be zero, indicates how many of those parts make up a unit or a whole. For example, in the fraction 3/4, the numerator, 3, tells us that the fraction represents 3 equal parts, and the denominator, 4, tells us that 4 parts make up a whole. The picture to the right illustrates \tfrac{3}{4} or 3/4 of a cake.
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 such as the number 7 can be thought of as having an implied denominator of...
...Fractions
The problem here is to add and 
These two fractions do not have the same denominators (lower numbers), so we must first find a common denominator of the two fractions, before adding them together.
For the denominators here, the 8 and 14, a common denominator for both is 56.
With the common denominator, the
becomes a
and the
becomes a
So now our addition problem becomes this...
The problem here is to add and 
Since these two fractions have the same denominators (the numbersunder the fraction bar), we can add them together by simply adding the numerators (the 21 and 36 = 57), while keeping the same denominator (the 56).
Our answer here is:
The fraction is an improper fraction (the numerator is greater than the denominator).
While there is nothing incorrect about this, an improper fraction is typically
simplified further into a mixed number.
The whole number part of the mixed number is found by dividing the 57 by the 56.
In this case we get 1.
The fractional part of the mixed number is found by using the remainder of the division,
which in this case is 1 (57 divided by 56 is 1 remainder 1).
The final answer is: 
The problem here is to add and 
These two fractions do not have the same denominators (lower numbers), so we must first find a common denominator of the two...
...In order to teach students the concept of equivalence when working with fractions with unlike denominators or finding equivalent fractions, there are some skills that the students must already possess. These are as follows:
Students are able to both recognize and write fractions
Students understand the ‘breakdown’ of a fraction where the top is the numerator and the bottom is the denominator
Students must have some understanding of equivalence and what it means
Students must be able to both multiply and divide with relative ease
The concept of finding equivalent fractions could be introduced using manipulative. One of these manipulative that would be extremely useful would be the pies. The ‘pie kit’ could be made or purchased, but both would include a large amount of pies with each being cut into different sizes with each ‘slice’ having a specific value on it (i.e. 1/3, 1/4, 1/2, 1/16, etc.). Using this visual manipulative, students would able to see the equivalent of various fractions (i.e. 1/2 is equivalent to 2/4 and 3/3 is equivalent to 16/16).
The steps for finding equivalent fractions would begin by having an overhead (for a large class) or having the children at a large group table (for a smaller class). Either one of these ways would allow the teacher to utilize the pie manipulative to show the students equivalents. To begin the students could see...
...Lacsap’s Fractions
IB Math 20 Portfolio
By: Lorenzo Ravani
Lacsap’s Fractions Lacsap is backward for Pascal. If we use Pascal’s triangle we can identify patterns in Lacsap’s fractions. The goal of this portfolio is to ﬁnd an equation that describes the pattern presented in Lacsap’s fraction. This equation must determine the numerator and the denominator for every row possible.
Numerator
Elements of the Pascal’s triangle form multiple horizontal rows (n) and diagonal rows (r). The elements of the ﬁrst diagonal row (r = 1) are a linear function of the row number n. For every other row, each element is a parabolic function of n. Where r represents the element number and n represents the row number. The row numbers that represents the same sets of numbers as the numerators in Lacsap’s triangle, are the second row (r = 2) and the seventh row (r = 7). These rows are respectively the third element in the triangle, and equal to each other because the triangle is symmetrical. In this portfolio we will formulate an equation for only these two rows to ﬁnd Lacsap’s pattern. The equation for the numerator of the second and seventh row can be represented by the equation: (1/2)n * (n+1) = Nn (r) When n represents the row number. And Nn(r) represents the numerator Therefore the numerator of the sixth row is Nn(r) = (1/2)n * (n+1) Nn(r) = (1/2)6 * (6+1) Nn(r) = (3) * (7) Nn(r) = 21 Figure 2: Lacsap’s fractions....
...Partial Fractions
Introduction to Partial Fractions Given a rational function of the form p(x) q (x) where the degree of p(x) is less than the degree of q (x), the method of partial fractions seeks to break this rational function down into the sum of simpler rational functions. In particular, we are going to try to write the original function as a sum of rational functions where the degrees of the polynomials involved are as small as possible. The steps in the partial fractions method are as follows: 1. Make sure that the degree of the numerator is less than the degree of the denominator. (Below we will see what to do when that is not true.) 2. Factor the denominator as fully as possible. 3. Write the original rational function as a sum of fractions. Each of the terms in the sum gets one of the original factors as its denominator. The numerator of each new fraction is an arbitrary polynomial of degree one less than the degree of the denominator. 4. Use algebra to solve for the coefficients in the numerators of the new fractions. Here is a step by step tour through the method applied to the rational function 2x+4 x 4x5 Note that the degree of the numerator is one less than the degree of the denominator, as required in 1 above. Next, we factor the polynomial in the denominator.
2
1
2x+4 x 4x5
2
=
2x+4 (x  5)(x + 1)
We then attempt to write the...
...We can use the number line as a model to help us visualize adding and subtracting of signed integers. Just think of addition and subtraction as directions on the number line. There are also several rules and properties that define how to perform these basic operations.
To add integers having the same sign, keep the same sign and add the absolute value of each number.
To add integers with different signs, keep the sign of the number with the largest absolute value and subtract the smallest absolute value from the largest.
Subtract an integer by adding its opposite.
Watch out! The negative of a negative is the opposite positive number. That is, for real numbers,
(a) = +a
Here's how to add two positive integers:
4 + 7 = ?
If you start at positive four on the number line and move seven units to the right, you end up at positive eleven. Also, these integers have the same sign, so you can just keep the sign and add their absolute values, to get the same answer, positive eleven.
Here's how to add two negative integers:
4 + (8) = ?
If you start at negative four on the number line and move eight units to the left, you end up at negative twelve. Also, these integers have the same sign, so you can just keep the negative sign and add their absolute values, to get the same answer, negative twelve.
Here's how to add a positive integer to a negative integer:
3 + 6 = ?
If you start at negative three on the real number line...
...Mixed Fractions
(Also called "Mixed Numbers")
  A Mixed Fraction
is a
whole number
and a proper fraction
combined.
such as 1 3/4. 
1 3/4   
(one and threequarters)   
Examples
2 3/8  7 1/4  1 14/15  21 4/5 
See how each example is made up of a whole number and a proper fraction together? That is why it is called a "mixed" fraction (or mixed number).
Names
We can give names to every part of a mixed fraction:
Three Types of Fractions
There are three types of fraction:
Mixed Fractions or Improper Fractions
You can use either an improper fraction or a mixed fraction to show the same amount.
For example 1 3/4 = 7/4, as shown here:
1 3/4   7/4 
 =  
Converting Mixed Fractions to Improper Fractions
To convert a mixed fraction to an improper fraction, follow these steps:
 * Multiply the whole number part by the fraction's denominator. * Add that to the numerator * Write that result on top of the denominator. 
Example: Convert 3 2/5 to an improper fraction.
Multiply the whole number by the denominator:
3 × 5 = 15
Add the numerator to that:
15 + 2 = 17
Then write that down above the denominator, like this:
17 

5 
Converting Improper...
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