Fractions

Lesson

In our chapter What's a Unit Fraction?, we discovered that to identify a shaded fraction, or what fraction something is of another, we need to know:

- How many pieces the original shape is cut up into. This number goes on the bottom of the fraction. And,
- How many parts do we have, or are shaded. This number goes on the top of the fraction.

We also learned that a unit fraction is a fraction with a $1$1 in the numerator.

Let's now look at some formal mathematical names for parts of the fraction, and types of fractions.

The **number on the bottom** of the fraction is called the DENOMINATOR. The denominator tells you how many pieces the original shape is cut up into.

The **number on the top** of the fraction is called the NUMERATOR. The numerator tells you how many parts we are interested in.

We also already know that a unit fraction is a fraction with a $1$1 in the numerator.

A proper fraction is a fraction whose *numerator is less than the denominator*. The top is less than the bottom. This means its total size is less than $1$1. These are all proper fractions, $\frac{1}{2}$12, $\frac{3}{8}$38, $\frac{11}{25}$1125 and $\frac{158}{236}$158236.

An improper fraction is a fraction whose *numerator is greater than the denominator*. The bottom is less than the top. This means its size is more than $1$1. These are all improper fractions $\frac{16}{7}$167, $\frac{23}{10}$2310 and $\frac{9}{8}$98.

A mixed number is a fraction that* contains both whole and fractional parts*. It also has a size larger than $1$1. These are all mixed numbers: $1\frac{4}{7}$147, $16\frac{1}{2}$1612 and $3\frac{7}{8}$378.

We need to convert between mixed numbers and improper fractions.

These two fractions have the same value:

$\frac{4}{3}$43 and $1\frac{1}{3}$113, they are just written in two different forms.

$\frac{4}{3}$43 is $4$4 *thirds*, we know that $3$3 *thirds *is one whole, and this $1$1 gets written out the front of our mixed fraction. After we take the $3$3 *thirds *from the $4$4 *thirds *we have $1$1 *third *remaining, and this last piece we write next to the whole number.

Some more conversions:

State the numerator in the fraction $\frac{1}{6}$16.

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State the denominator in the fraction $\frac{6}{2}$62.

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Express $\frac{17}{4}$174 as a mixed number.

Express $3\frac{6}{7}$367 as an improper fraction.

Apply simple linear proportions, including ordering fractions