ZingPath: Operations with Fractions

An equivalent fraction is a fraction that represents the same portion of a model as another fraction.

For example, and are equivalent, even though these two fractions don't look alike, because they represent the same portion of a model.

The models below represent these fractions:

This fraction is because 3 out of the 6 portions are shaded.

This fraction is because 1 out of the 2 portions is shaded. However, these fractions are equivalent because even though they are broken into different amounts they still represent the same part of the whole rectangle.

Models can sometimes be used to determine equivalency.

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To determine whether the fraction models above are equivalent, ask whether they have equal parts shaded or represented. Because the top fraction model is not shaded in completely, we can conclude that they are not equivalent.

In order to determine whether to fractions are equivalent, you can rename the fractions by finding the least common multiple/denominator.

For example, to determine whether is equivalent to , first you need to have a common denominator. The denominators in this problem are 2 and 4. If you wrote out all the multiples of 2 and 4 and found the lowest one they have in common, this would be the least common multiple.

Notice that in the multiples above the lowest number they have in common is 4, therefore, 4 is common denominator. This is also known as the least common denominator.

After finding the least common denominator, the fractions must be renamed.

In the problem above, the common denominator is 4, therefore only the fraction that doesn't have a 4 as a denominator needs to be renamed, which is .

In order to rename , you have to multiple both the numerator and denominator by the same number to form an equivalent fraction. In this case that number would be 2.

Once fractions have been renamed, or have a common denominator, equivalency can be determined.

To determine whether is equivalent to , first we rename the fractions; according to the information above we use and . Once the two fractions have a common denominator, then we only need to look at the value of the numerators. Since 2 is less than 4, we can conclude that these two fractions are not equivalent.

If presented with an improper fraction, the same steps apply in order to determine equivalency.

For example, and .

To find out if these two improper fractions are equivalent you can draw a model to represent each fraction and determine whether they represent the same value or you can find the common denominator of the two denominators (4 and 6) and rename the fractions to see if they are equivalent.

After listing all the multiples to find the common denominators, you will discover that the least common denominator of 4 and 6 would be 12. Use 12 as your new denominator to rename the fractions as follows:

Because and don't represent the same value, you can conclude that these fractions are not equivalent.

The following key vocabulary terms will be used throughout this Activity Object:

• common denominators - denominators that are the same number, usually found by finding the least common multiple of previous denominators
• denominator - the number written below the line of a fraction; represents the total number of equal parts in the whole
• equivalent - having the same value; equal
• equivalent fractions - fractions that represent the same value, but don't necessarily have the same numbers
• fraction - a number of the form , that represents part of a whole.
• improper fractions - a fraction where the numerator is a larger number than the denominator
• integer - the numbers made from the set of whole numbers and their opposites ....1, 0, 1, ...
• least common multiple (LCM) - the smallest multiple two or more numbers have in common; the LCM can be determined by listing the multiples in order for each number or by using prime factorization
• numerator - the number written above the line of a fraction; represents the number of equal parts being considered or counted