Primary Mathematics/Fractions

Learning to use fractions (Visually)

Fractions or rational numbers are in essence the same as division, however we use them more often to express numbers less than one - for instance a half or a quarter. Fractions have a numerator (on the top) and a denominator (on the bottom). If a fraction is larger than 1 then the numerator will be larger.

Modern methods to teach fractions

Today's modern methods of teaching math and fractions are drastically different than how they were taught just 10 years ago. The difference between these methods is that the later method explores the visual evidence for certain ways of manipulating fractions and whereas the earlier approach simply used variables from the beginning. Tiles of different colors sorted into groups can be useful in representing fractions visually.

Origami and Fractions

There is perhaps little emphasis these days which shows the elaborate "visualization" that math requires. Students used to be taught merely by equation, but to my understanding, by teaching them such methods they tend to take the "cooking approach" to problems in that they have an inadequate sense of "visualizing" the concept the problem in their mind. What I want to do is emphasize the creative aspect of fractions while at the same time exploring the richness of why such problems are true.

Why I call this section Origami and math is because they are very much related to each other. The thing about Origami which is rich in math is that essentially folding a piece of paper proves that such a fraction exists!

In order to do this experiment, you need the following materials:

1. Square piece of paper
2. pencil

Steps to seeing some nice fractions

First, we have to say to ourselves, "This piece of paper is 1 piece of paper"

Now, we will explore fractions by seeing how "much" of the remaining paper we see as we fold. Every time we see the paper, we will write down the fraction of the paper on the front of it. By the time we get done, we'll have lots of fractions written on a piece of paper.

Step 1: Obtain square piece of paper. Write "1" on it.

Step 2: fold piece of paper in half. Write "1/2" on it.

Step 3: Fold again in half. write "1/4"

Step 4: Fold again in half. write "1/8"

Step 5: Unfold the piece of paper and write lines in where there are folds in the paper

Notice: if there is a 1 on top, then whatever number is on the bottom, it takes that many "pieces" to make a "whole". For example, 1/8 needs 8 pieces to become 1 whole.

The square model vs. the circle model

A square divided into fractions	A circle divided into fractions

The money approach: 1, 1/4, 1/2, 1/10, 1/20, 1/100

A very practical way to learn fractions is the use of money, as we use it everyday.

Questions:

1. How many quarters are in a dollar?
2. How many dimes are in a dollar?
3. How many nickels are in a dollar?
4. How many pennies are in a dollar?

As said from the previous section, if there is a one on top and some number on bottom, that means it needs that much pieces to make it a whole (a whole means 1 by the way)

For example, 1/10 is the value of a dime (10 cents). You need 10 dimes (10 x 10 = 1 dollar) to get one full dollar.

Whole number fractions

First of all, as always, instead of looking at complicated variable jargon, we will instead look at certain ways to "view" certain types of number. Just like art, you don't need to be an accomplished artist to draw, but rather you just need to know how to look at things better (in this case, numbers)

Since fractions have both a top (called a numerator. think "topinator") and a bottom (denominator, which "downominator" which is divided by a bar, we have to "adjust our thinking" so that we can recognize what our friendly fractions might look like.)

For example,

Q: 5 ÷ 5 reads "5 divided by 5". What does that look in Fraction form?

Well, since we are prospective mathematicians (and artists...) we will look at the magic of what the divided sign actually means:

o === o

(my divided sign) What it actually means is that dots tell you to "Make me a number!" (because Zeroes often become lonely...they want to have value in life...) and you see that bridge which devides them too says that, "In order to separate such ZEROES in life and making more zeroes, we say always divide your numbers from each other."

A: ${\frac {5}{5}}=1$, which also reads "5 divided by 5".

Since we know how to recognize some fractions, we have to "see" with our naked eyes what some fractions actually mean. The best preparation that you will have in our "artist" training is that whole numbers actually have "1" on the bottom. You may ask yourself, "How can that be? Is that even possible?"

Well Just for fun, we will go through this little exercise to show that there is indeed ones on the bottom when you have whole numbers.

Example: express "5" in a fraction form.

Look at our previous example about the placement of things. We now know what a denominator is and a numerator is. Since the whole number is the number of top, all we have to do is see what number is on bottom, i wonder what it is.

${\frac {5}{?}}$	In order to figure out what goes on bottom, we can figure it out many ways. Fractions are in ways actually like ratios. Such as if there were two ice creams for me and none for you, my ratio would be 2:1 and yours would be 0:1...Sad isn't it? Getting back to the point, we must think about why and how it becomes a whole number.
${\frac {(X)(X)(X)(X)(X)}{(X)}}$	The upper part represents five oranges. the lower represents 1.	$={\frac {5}{1}}$

So our lesson is, if there is any whole number, "1" will always go on the bottom. So if I say, "What's the number that goes on the bottom if the number is 1,2,3,4,5,6,7, or any other whole number?"

Of course, your answer should be "1", "1", "1", "1", "1", "1", "1" and always "1." I want you to remember that!

Multiplying fractions

In general, multiplying fractions involves a simple formula:

${\frac {A}{B}}*{\frac {C}{D}}={\frac {A*C}{B*D}}$

Where A, B, C, and D represent the numbers within the fractions. If this seems intimidating, there are methods used to tackle the multiplication more easily.

Traditional multiplication method

1. Know what the fractions are. Write it out.

2. If you wrote it out correctly, ignore the fractions part and look at it this way.

For example:

${\frac {5}{3}}*{\frac {5}{3}}={\frac {?}{?}}$

If you wrote it correctly, this is what it should look like. Also, it would be helpful if you wrote it equal to the mysteriously equal question marks, "?/?." They're there so you can see where to write your answers.

3. Using your finger or card, cover or hide the bottom (called the denominator) so that you can concentrate on multiplying the top. Focus solely on multipling the upper portion:

${\frac {5}{}}*{\frac {5}{}}={\frac {?}{}}$

5 x 5 = ? (what is 5 times 5? what is the value (in cents) of 5 nickels? why 25 of course!) So if 25 is equals ?, then simply "erase" the "?" and write "25." Its that simple!

Again, since you know whats on top, do the same on the bottom:

${\frac {}{3}}*{\frac {}{3}}={\frac {}{?}}$

Again, what is 3 x 3? Of course, it is 9. So, again, erase the "?" and write down "9"

Once you remove the finger or card, you are presented with the answer: ${\frac {5}{3}}*{\frac {5}{3}}={\frac {25}{9}}$

Multiplying Fractions with nice pictures

Another good way to prove that answers are actually correct is to use a diagram or tiles to prove that it is correct. So how we do that is first write out one of our fractions on one side and the other fraction on the other side. To see what i mean, here is an example:

Here is our little lesson. We will multiply 3/4 x 1/2 = ?.

1/4 1/4 1/4 1/4

So just imagine the figure to the right is one whole 4 story "chocobar" (i will remind you that one chocobar that has 4 stories still is one chcobar) You will see that each story represents a fraction. 1/4 means its needs 4 pieces to make a whole.

0000 __________________ 000000/ | OOOO Mr. Anaconda | "CHOBARS ARE YUMMY!" \ _______ | "I hope I eat a lot | / \ | of them!" | / \ | | / \ | | / \ / \-/ \-/

But the thing is, Mr. Anaconda only has enough energy to eat once a week. His mouth is 1/2-full with chocobar and the deepest he can eat is 3/4 a chocobar. How much of the chocobar has he eaten?

The fraction is set up like this:

${\frac {3}{4}}*{\frac {1}{2}}={\frac {?}{?}}$

Visually, it will look like this:

1/2	1/2
1/8 1/8 1/8 1/8 1/8 1/8 1/8 1/8		1/4
		1/4
		1/4
		1/4

So, simplified from our little illustration is our little chocobar broken up into nice convenient blocks so that Mr. Anaconda can eat it.

1/2	1/2
1/8 1/8 1/8 1/8 1/8 1/8 1/8 1/8		1/4
		1/4
		1/4
		1/4

From the image, you can count 3/8ths of the chocobar was eaten.

Dividing fractions

As mentioned earlier, a fraction is one number divided by another. Because of textual limitations, fractions here will be represented using numbers and slashes. For example, the fraction one half will be represented as 1/2. The fraction two thirds will be represented as 2/3.

Now, on to some specifics. You may already know how to multiply fractions. Remember all you have to do is multiply the two top numbers together and put them above your two bottom numbers multiplied together. If know how to multiply fractions, dividing fractions will also be easy, and only requires one extra step.

In order to divide two fractions, you invert the second fraction and multiply. This means that you switch around the numerator (top) and denominator (bottom) of the fraction before multiplication.

Let's look at examples using our two fractions. First, let's divide the fraction 1/2 by the fraction 1/2. Since we are dividing a number by itself, we should expect to get an answer of 1.

Here is our problem.

${\frac {1}{2}}\div {\frac {1}{2}}={\frac {1}{2}}*{\frac {2}{1}}={\frac {1*2}{2*1}}={\frac {2}{2}}=1$

Now let's try another example.

${\frac {1}{2}}\div {\frac {2}{3}}={\frac {1}{2}}*{\frac {3}{2}}={\frac {1*3}{2*2}}={\frac {3}{4}}$

Following these simple steps will let you solve any fraction division problem.

There is also an alternate way to divide fractions using an equivalent fraction approach. You obtain this by multiplying the numerator and denominator so that they're equivalent, cancelling out the denominator, and creating a fraction from the two remaining numerators.

${\frac {1}{2}}\div {\frac {2}{3}}={\frac {3}{6}}\div {\frac {4}{6}}={\frac {3}{4}}$

Here is another example:

${\frac {2}{5}}\div {\frac {3}{4}}={\frac {8}{20}}\div {\frac {15}{20}}={\frac {8}{15}}$

It works because the common denominator will always be ${\frac {n}{n}}=1$

Adding fractions

Fractions that have the same or "Common" denominator are called "Like" fractions.

1/3, 2/3

(one-third, two-thirds)

To add Like fractions together such as these:

${\frac {1}{3}}+{\frac {2}{3}}=?$

1. Add the numerators (the top numbers):

• 1 + 2 = 3 (one plus two equals three)

2. Use the common denominator (the bottom numbers):

• /3

All together it looks like this:

${\frac {1}{3}}+{\frac {2}{3}}={\frac {3}{3}}$

3. Simplify the answer as much as you can by dividing the denominator into the numerator.

• ${\frac {3}{3}}=1$

If the numerator is now larger than the denominator it might look like this:

4/5 + 3/5 = 7/5 (four-fifths plus three-fifths equals seven-fifths)

This is simplified by creating a mixed number:

• 5 goes into 7 one time, with 2 left over. Put the remainder (2) over the denominator (5).
• ${\frac {7}{5}}=1{\frac {2}{5}}$

To add or subtract fractions with different denominators, you must first convert them to equivalent fractions with common denominators.

Multiplying to get equivalent fractions

In some cases, you may need to add 1/2 to 1/3:

${\frac {1}{2}}+{\frac {1}{3}}=?$

You will first have to convert the fractions into like fractions. The easiest means of doing so is to multiply the top and bottom of the left fraction by the bottom on the right, and the top and bottom of the right fraction by the bottom on the left:

${\frac {1*3}{2*3}}+{\frac {1*2}{3*2}}=?$

${\frac {3}{6}}+{\frac {2}{6}}=?$

The addition can now take place as expected:

${\frac {3}{6}}+{\frac {2}{6}}={\frac {5}{6}}$