Math is Beautiful – Adding Fractions Is A Piece Of Cake!

“Adding fractions is a piece of cake!” Have you ever heard that statement from someone who has learned how to handle fractions? Do you feel the same way?

Since my math tutoring methodology is a little different than that of most teachers, my students have learned to respond to the phrase: “Learning fractions is a piece of PIZZA!”

I have used my PIZZA example to help students understand what a fraction is, what “like” and “unlike” fractions are, how to add or subtract all types of fractions, how fractions are related to percentage, and much, much more.

This is how to use a pizza to help one understand fractions.

Drawing The “Whole” Pizza.

Draw 3 circles of equal size to represent 3 pizzas.  Label the pizzas “A”, “B and “C”.

Then draw a straight line through the center of each pizza to indicate that the pizzas are cut in half.

Label the corresponding halves of each pizza as “A”, “B”, and “C” so that you can refer to these same halves later in this discussion.  Note that the portion of the whole pizza represented by “A”, “B”, and “C” will always be one half of the pizza..

Understanding The Definition Of A Fraction

“A fraction consists of two numbers, called the Numerator and the Denominator, which are separated by a straight line”.

When written in sentence form the fraction would look like this:  “N/D” where “N” represents the Numerator and “D” represents the Denominator.

The “D” number tells us how many pieces the “whole” has been cut into.  In our case, since the “whole” pizzas were cut into 2 pieces, the “D” part of our fractions would be 2.  (I like to refer to the denominator as a “size” like when we refer to a shoe size because using this terminology tends to simplify the addition process.)

The “N” number tells us how many pieces the “whole” is being considered or identified.  In our case the lettered half of each pizza would 1.

The diagrams show that each of the three “whole” pizzas are made up of two halves.  Therefore, the fraction that would represent the pieces of each pizza would be shown as 1/2, and, in my terminology, would be called “1 piece size 2″.

FIGURE 1

Defining “Equivalent” Fractions.

“Equivalent fractions are fractions that represent the same portion of the whole”.

Since the pieces “A”, “B” and “C” are the same size, the fractions representing these pieces are equal to each other and are called “equivalent fractions”.

We can show their equality by the following expression:

1/2  =  1/2  =  1/2

Defining “Like Fractions”.

“Like fractions” are fractions that have the same denominator, regardless of their numerator.

Since the fractions representing the pieces A, B and C have the same denominator, 2, they are called “like fractions”.

Also, since their numerators are also the same, they meet the definition of “equivalent” fractions.

Defining “Unlike Fractions”.

“Unlike fractions” are fractions that have different denominators, regardless of their numerator.

Since the denominators are different, the fractions represent different size pieces of the whole.

The denominator defines how many pieces the whole has been cut into.  The more pieces into which the whole has been cut, the smaller the size of each piece.

To illustrate what is meant by “unlike fractions”, modify the diagrams that were used earlier to illustrate “like fractions”.

In Figure 2 (below), make no changes to pizza “A”

Draw one additional line through the center of pizza “B” so as to show 4 pieces of the same size rather than just 2.

Draw two additional lines the center of pizza “C” so as to show 8 pieces of the same size rather than just 2.

The visual and fractional representation of each piece would then look like the following illustration:

FIGURE 2

Note that all of the “A” pieces are the same size.  In my terminology they are “size 2″.

That also hold true for all of the “B” pieces and all of the “C” pieces.  All of the “B” pieces would be size 4 and all of the “C” pieces would be size 8. However, none of the pieces in any one pizza is the same size of any of the pieces in any of the other pizzas since some pieces are “size 2″, some are “size 4″ and some are “size 8″.

You can see that the size 8 pieces are smaller than the size 4 pieces, and the size 4 pieces are smaller than the size 2 pieces.  By definition:

“If the denominators for the fractional representation of the pieces are different, their fractional representations are referred to as “unlike fractions”.

Adding unlike fractions is not allowed in math.  To see why the addition of unlike fractions is not allowed, consider the following example:

Assume that you had a collection of coins.  Some of the coins were pennies,         some were nickels, and some were dimes.  If you were asked how many    pennies you had, you could add all of the pennies together and come up with a sum of pennies.

You could do the same thing for the nickels and the dimes and come up with       a sum for each.

However, you could not add all of the coins together and call the sum as     being the sum of pennies, nor the sum of nickels, nor the sum of dimes.

This example demonstrates why we have a rule in math that states:

“You can only add like things”.

This rule applies to every math function, regardless of whether we are dealing with numbers, or fractions, or letters.

Adding Unlike Fractions

Since the rule says that we can only add like things, how could we add 1 piece from pizza “A” plus 1 piece from pizza “B” plus 1 piece from pizza “C”?  The expression would look like this:

1/2 + 1/4 + 1/8 = how much?

To solve this expression, the pieces in each pizza have to be the same size.  Since we are dealing with a pizza, making the pieces all the same size is easy, and doing so should enable you to “see” what is meant by generating “equivalent” fractions.

Look at Figure 3 and compare it to Figures 1 and 2.  Note that in Figure 3, pizza “A” is cut into 8 pieces, as was pizza “B”.

FIGURE 3

If we compare the number of pieces that are in the same space that we had called piece “A” in Figure 2, we see that we now have 4 pieces rather than just 1.  Thus, instead of having 1 piece size 2, we now have 4 pieces size 8.

If we count the number of pieces that are in the same space that we had called piece “B” in Figure 2, we see that we now have 2 pieces rather than just 1.  Thus, instead of having 1 piece size 4, we now have 2 pieces size 8.

Now that all of the pieces are the same size, we can add the pieces together.

Let us now compare the original addition expression to an equivalent addition expression.

We were asked to add 1 piece size 2 to 1 piece size 4 and to 1 piece size 8.

1/2 + 1/4 + 1/8  =  ?

But, with our trusty pizza knife, we changed 1 piece size 2 to 4 pieces size 8, and changed the 1 piece size 4 into 2 pieces size 8.  Now the expression would look like this:

4/8 + 2/8 +  1/8  = ?

Why is the fraction 1/2 equivalent to the fraction 4/8?  The answer is because the two fractions represent exactly the same portion of the whole pizza.

The same reasoning applies in being able to say that the fraction 1/4 is equivalent to the fraction 2/8.

Now let me show you why I like to refer to fractions as “pieces size …..”.  Using my terminology the addition problem could be stated as:

How many pieces size 8 would we have if we added 4 pieces size 8 plus 2   pieces size 8 plus 1 piece size 8?

The answer would be: 4 pieces + 2 pieces + 1 piece = 7 pieces size 8  or  7/8.

The rule for adding like fractions is:

“Add the numerators together to find the new numerator, and write their sum over a common denominator”.

The rule is pretty simple to understand, but too many people tend to add all the numerators together to find the new numerator, but also add all the denominators together to find what they think is the “common denominator”.  In order to help people apply the addition rule properly, I show them how to visualize the fractions as “number of pieces size …” because the denominator of the added fractions does not change.

How Do We Subtract Fractions?

One of the things that makes math beautiful is that functions that are related follow the same rules.

Subtraction is related to addition because the subtraction function is the direct opposite of the addition function.  As such, since we can add only like things, we can subtract only like things.

If our problem is to find the result of subtracting 1/8 from 1/2, the expression would look like this:

1/2 – 1/8 =?   or, converting to equivalent fractions,  4/8 – 1/8 = ?

Using my terminology the question can be stated as “4 pieces size 8 minus 1 piece size 8 will equal how many pieces size 8?

The answer is 4 pieces – 1 piece  = 3 pieces   or 3 pieces size 8  or  3/8.

How Do We Generate Equivalent Fractions Without A Pizza?

Let us examine what two equivalent fractions look like.  Using a pizza we have shown that the fraction 1/2 is equivalent to the fraction 4/8.

Now ask yourself, what did we do to the numerator to change it from a 1 to a 4?  What did we do to the denominator to change it from a 2 to an 8?

We multiplied the numerator and denominator by the same number.

Since Math Is Beautiful, if multiplying the numerator and denominator by the same number generated equivalent fractions for our pizza problem, then by multiplying any fraction in any problem by the same number, we can generate equivalent fractions.

The number that we choose to multiply numerator and denominator by depends on the fractions that we are trying to add together.  In our pizza example the numbers were “nice” numbers because or problem was to add 1/2 to 1/8.

We needed equivalent fractions with the common denominator size 8. We could change the denominator 2 into the denominator 8 by simply multiplying the number 2 by 4 to get 8.  Piece of cake! (Or should I have said piece of pizza?)

But what do we have to do if we are trying to add fractions that have “ugly” numbers?  For example, how would we add 2/3 + 5/8?

There is no whole number that I can multiply the number 3 by to get the number 8.  However, because math is beautiful, we can change both fractions into equivalent fractions that will have the same denominator.

One of the simplest procedures is to multiply the numerator and denominator of the first fraction by the denominator of the second fraction.  Then multiply the numerator and denominator of the second fraction by the denominator of the first fraction.  In our example the procedure would look like this:

2/3 becomes 2×8 / 3×8   or 16/24     and 5/8 becomes 5×3 / 8×3  or 15/24

Since the two fractions were changed into equivalent fractions size 24, we can add the pieces together to get:

16 pieces size 24 plus 15 pieces size 24 equals 31 pieces size 24

or

2/3 + 5/8 = 16/24 + 15/24 = 31/24

What if we were required to generate a fractional answer with a denominator 3 or 8?

That would mean that we would have to convert 5/8 to a fraction with a denominator of 3 or convert 2/3 to a fraction with the denominator of 8.  Would the same rule for converting the fractions to like fractions still work?

The answer is yes!!! Again, the beauty of math is that:

if a rule works for one number, it will work for all numbers.

The resulting like fractions may not be “pretty”, but they will be equivalent.

How can we do this?  I will tell you how in another presentation in which I discuss how to multiply and divide any fractions.

This completes my presentation of “Math is Beautiful – Adding Fractions Is A Piece Of Cake!”.

Summary Of This Presentation

A fraction consists of two numbers separated by a straight line.

The number to the right (or below) the line is called the “numerator” and tells us how many pieces of a “whole” is cut into.

The number to the left (or above) the line is called the “denominator” and tells us how many pieces of the “whole” that we have.

Fractions that have the same denominator are called “like” fractions.

Fractions that have different denominators are called “unlike” fractions.

Fractions that represent the same portion of the whole are called “equivalent” fractions.

Any fraction can be converted into its equivalent fraction if its numerator and denominator are multiplied by the same thing.

You can add (or subtract) only “like” things.

To add or subtract “unlike” fractions, the fractions must be converted into equivalent “like” fractions prior to addition.

How do you add or subtract any fraction?  The same way that you would eat a pizza …

one bite at a time.

Warning:  If you review all of my articles you may be tempted to adopt my favorite phrase:

Math is Beautiful

I would appreciate receiving any comments or feedback that may help me improve this article.

Please send the comments to me at genefantasia@cox.net or math-tutor@cox.net

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