4.1 Visualize fractions  (Page 5/11)

Find equivalent fractions

We used fraction tiles to show that there are many fractions equivalent to $\frac{1}{2}.$ For example, $\frac{2}{4},\frac{3}{6},$ and $\frac{4}{8}$ are all equivalent to $\frac{1}{2}.$ When we lined up the fraction tiles, it took four of the $\frac{1}{8}$ tiles to make the same length as a $\frac{1}{2}$ tile. This showed that $\frac{4}{8}=\frac{1}{2}.$ See [link] .

We can show this with pizzas, too. [link] (a) shows a single pizza, cut into two equal pieces with $\frac{1}{2}$ shaded. [link] (b) shows a second pizza of the same size, cut into eight pieces with $\frac{4}{8}$ shaded.

This is another way to show that $\frac{1}{2}$ is equivalent to $\frac{4}{8}.$

How can we use mathematics to change $\frac{1}{2}$ into $\frac{4}{8}?$ How could you take a pizza that is cut into two pieces and cut it into eight pieces? You could cut each of the two larger pieces into four smaller pieces! The whole pizza would then be cut into eight pieces instead of just two. Mathematically, what we’ve described could be written as:

These models lead to the Equivalent Fractions Property, which states that if we multiply the numerator and denominator of a fraction by the same number, the value of the fraction does not change.

When working with fractions, it is often necessary to express the same fraction in different forms. To find equivalent forms of a fraction, we can use the Equivalent Fractions Property. For example, consider the fraction one-half.

So, we say that $\frac{1}{2},\frac{2}{4},\frac{3}{6},$ and $\frac{10}{20}$ are equivalent fractions.

Locate fractions and mixed numbers on the number line

Now we are ready to plot fractions on a number line. This will help us visualize fractions and understand their values.

Let us locate $\frac{1}{5},\frac{4}{5},3,3\frac{1}{3},\frac{7}{4},\frac{9}{2},5,$ and $\frac{8}{3}$ on the number line.

We will start with the whole numbers $3$ and $5$ because they are the easiest to plot.

The proper fractions listed are $\frac{1}{5}$ and $\frac{4}{5}.$ We know proper fractions have values less than one, so $\frac{1}{5}$ and $\frac{4}{5}$ are located between the whole numbers $0$ and $1.$ The denominators are both $5,$ so we need to divide the segment of the number line between $0$ and $1$ into five equal parts. We can do this by drawing four equally spaced marks on the number line, which we can then label as $\frac{1}{5},\frac{2}{5},\frac{3}{5},$ and $\frac{4}{5}.$