Lesson 1.1: Halves and Quarters

New Concept

This lesson introduces fractions as a way to represent a part of a whole. We'll explore benchmark fractions like one-half ($\frac{1}{2}$) and one-fourth ($\frac{1}{4}$), learning to identify, compare, and represent them in different ways.

What’s next

Next, you'll work through interactive examples and practice cards to build your skills in finding and comparing halves and quarters of various quantities.

Property

To find one-half of something, we divide it into 2 equal pieces and take 1 of them.

When representing a fraction from a group of objects, we use the format:

Fractions such as $\frac{6}{12}$ and $\frac{30}{60}$ are different ways to write $\frac{1}{2}$.

Examples

• To find $\frac{1}{2}$ of 18 apples, we divide the group into two equal parts: $18 \div 2 = 9$. So, one-half of 18 apples is 9 apples. The fraction can be written as $\frac{9}{18} = \frac{1}{2}$.

• A video is 80 minutes long. Halfway through is $\frac{1}{2}$ of the total time. We find this by dividing 80 by 2, which is 40 minutes. This can be represented as the fraction $\frac{40}{80}$.

Property

If we divide one whole into four parts, each part is one-fourth of the whole. One-fourth is also called one quarter. Taking one-half of half a whole gives you one-fourth, or $\frac{1}{4}$. Three-fourths, or $\frac{3}{4}$, represents three of those four equal parts.

Examples

• To find one-fourth of a 24-hour day, you divide 24 by 4, which gives 6 hours. So, $\frac{1}{4}$ of a day is 6 hours.

• A pizza is cut into 8 slices. Three-fourths of the pizza would be $\frac{3}{4}$ of 8 slices. Since $\frac{1}{4}$ is $8 \div 4 = 2$ slices, $\frac{3}{4}$ is $3 \times 2 = 6$ slices.

Property

The denominator of a fraction (the bottom) tells us how many pieces are in one whole. The numerator (the top) tells us how many pieces are in the part.

Although it has a different numerator and denominator, the fraction $\frac{15}{60}$ is the same as $\frac{1}{4}$, because 15 pieces are one-fourth of 60 pieces. Both fractions give the same portion of one whole.

Examples

• In the fraction $\frac{5}{8}$, the numerator is 5 and the denominator is 8. This represents 5 pieces out of a total of 8 equal pieces.

• If a class has 25 students and 12 are boys, the fraction of boys is $\frac{12}{25}$. Here, 12 is the numerator (the part) and 25 is the denominator (the whole).

Property

We can show fractions of a distance on a number line. To find a fractional part of a total length on a number line, first divide the total length by the denominator, and then multiply the result by the numerator to find the correct position.

Examples

• To mark $\frac{1}{4}$ of 20 on a number line, you first find the value: $20 \div 4 = 5$. You would place a mark at the number 5 on a line from 0 to 20.

• Let's find $\frac{2}{5}$ of 10 on a number line. First, $\frac{1}{5}$ of 10 is $10 \div 5 = 2$. Then, $\frac{2}{5}$ of 10 is $2 \times 2 = 4$. The fraction is located at the 4-unit mark.

Property

The fractions $\frac{1}{2}$, $\frac{1}{4}$, and $\frac{3}{4}$ are called benchmark fractions because we have a good intuitive feel for their size. We can estimate other fractions by comparing them to the benchmarks. To compare a fraction to a benchmark, write the benchmark fraction with the same denominator.

Examples

• Is the fraction $\frac{7}{20}$ closer to $\frac{1}{4}$ or $\frac{1}{2}$? With a denominator of 20, $\frac{1}{4} = \frac{5}{20}$ and $\frac{1}{2} = \frac{10}{20}$. Since 7 is closer to 5 than to 10, $\frac{7}{20}$ is closer to $\frac{1}{4}$.

• A team won 14 out of 30 games. Is this fraction, $\frac{14}{30}$, closest to $\frac{1}{4}$, $\frac{1}{2}$, or $\frac{3}{4}$? We know $\frac{1}{2}$ of 30 is 15. Since 14 is very close to 15, the fraction is closest to $\frac{1}{2}$.