2-C Numbers and Operations: Fractions

Proportional relationships

A ratio is a comparison of two numbers using a fraction, a colon, or the word "to". Rates are ratios with different units, while ratios have the same units. Rates are often expressed as unit rates and are read using the word "per" instead of "to." It is common to reference a unit rate in everyday language.

Rates and Ratios

Verbal statement

Four dogs for every three cats

Fraction

4 dogs/3 cats

Colon

4 dogs: 3 cats OR 4:3

Unit Rate

A unit rate is a rate with a denominator of 1. Examples of unit rates include 60 miles per hour (60 miles/1 hour ), $3 per box ( $3/1 box), or 22 students per teacher ( 22 students/1 teacher ). Any rate can be converted to a unit rate by dividing the numerator of the fraction by the denominator. When two ratios are equivalent, they can be set equal to one another to form a proportion

Working with rational fractions

Students learn unit fractions using fractional pieces of circles or bars and also by breaking up a number line into fractional pieces. From the number line, students learn the placement of fractions and their size relative to one another and to 0 and 1. The table that follows gives examples of some, not all, unit fractions.

Number representation

The same number can be written several ways. Be prepared to know how to write a number in different forms or recognize equivalent forms of the same number. Also, be prepared to compare numbers written in different forms. Fractions, decimals, and percents are all interchangeable and are acceptable in any of the three forms. Each of these is a number out of 100, so base-10 blocks create an effective visual when first learning percents and decimals.

Form

Base-10 blocks

Example

When interchanging fractions, decimals, and percents, we think of this out of 100. To represent this pictorially, we can use a base-10 block with 100 squares. The image to the right has 30 squares shaded or 30 out of 100, which is 30 100

Form

Fraction

Example

The fraction 30 100 may also be written as 3 10. Decimal To convert a fraction to a decimal, always divide the numerator (top number) by the denominator (bottom number); numerator ÷ denominator. 30 ÷ 100 = 0.30 or 0.3

Form

Percent

Example

To convert a decimal to a percent, move the decimal point 2 place values to the right. This works because a fraction and its decimal are out of 1, and a percent is out of 100. Moving the decimal point two places to the right is the same as multiplying by 100. Example: To convert 0.30 to a percent, move the decimal 2 places to the right: 0.30 = 30% This is the same as 0.30 × 100 = 30. Moving the decimal is the shortcut; make sure you always know the math behind the shortcuts.