Fractions and Decimals

Equivalent Fractions

We teach students to find equivalent fractions by using models, and then by multiplying or dividing by one whole. 

Whenever you multiply or divide a number by 1, the value of the number stays the same.  This is the same with fractions.  Whenever you multiply or divide a fraction by a form of one, the value of the fraction stays the same. 

For example, 1/2 x 2/2 = 2/4, so 1/2 and 2/4 are equivalent fractions.

Students learn how to compare fractions in many ways.  As stated above, you can only compare fractions of the same sized whole. 

They first learn two generalizations:

1. When comparing fractions with common denominators, you look to the numerator, and the fraction with the larger numerator is the larger fraction.  This is because you have more equal sized parts.  For example, 3/4 is greater than 2/4, because 3 pieces are more than 2 pieces.  See the first picture below for a visual example.

2. When comparing fractions with common numerators, you look to the denominator.  As the denominators get larger, the pieces get smaller, so the fraction with the smaller denominator is the larger fraction.  This is because the pieces are larger.  For example, 1/4 is larger than 1/6, because the fourths are larger pieces than the sixths, so one large piece is greater than one small piece.  See the second picture below for a visual example.

When comparing fractions with unlike denominators and unlike numerators, students learn the following three strategies: 

1. Compare to a Benchmark of One-Half - Think about the fractions in terms of a half.  If one fraction is less than a half and the other is greater than a half, then the fraction that is greater than a half is greater than the other fraction.  For example, 1/4 is less than 4/6, and I can prove this because 1/4 is less than 2/4 and 4/6 is greater than 3/6.  This strategy does not work when both fractions are less than or greater than a half.  See the third picture below for a visual example.

2. Compare using a Model - Students can draw two rectangular models stacked on top of each  other to compare fractions.  You want to make sure the wholes drawn are the same size, and the parts are drawn equally within each whole.  It is easiest for students to draw rectangles stacked on top of each other, rather than circles or rectangles / squares drawn side by side.  See the first picture below for a visual example.

3. Find Common Denominators and Compare - Once students have learned how to find equivalent fractions, we teach them how to compare after finding common denominators.  See the picture fourth picture below for a visual example.

Finally, we teach students to check their work with cross multiplication.  See the fifth picture below for a visual example.

Adding and Subtracting Fractions

Students learn that unit fractions are fractions with a one in the numerator, representing one part of the whole.  All wholes can be broken up into unit fractions.  For example 1/4 + 1/4 + 1/4 + 1/4 = 4/4 = 1

When adding and subtracting fractions, the denominators need to be the same. You add or subtract the numerators, while the denominator stays the same. 

To help students understand this, we teach the students to think of the numerators as adjectives and the denominators as nouns.  For example, two slices plus one slice equals three slices, just like two sixths plus one sixth equals three sixths. 

The denominator always stays the same!

Multiply Fractions by a Whole Number

Students learn how to multiply a fraction by a whole number in three different ways.  They first learn how to use a model, as pictured in the following two examples.  They then learn how to use repeated addition to solve, as pictured first below.  Finally, they learn how to solve using the formula, as pictured second below.

Students learn how to decompose fractions to convert them between mixed numbers and improper fractions. 

For example: 2 and 2/3 means 3/3 plus 3/3 plus 2/3, which equals 8/3

and

5/2 means 2/2 plus 2/2 plus 1/2, which equals 2 and 1/2. 

See the two pictures below for the formula to convert between mixed numbers and improper fractions.