Introduction to Fractions ½ ⅔ 1⅝

Introduction You often encounter situations in which whole numbers cannot give the exact information. Sometimes you need to talk about parts of wholes. Fractions, decimals, and percents are all ways of expressing quantities that are not whole numbers. Ex: 2 3 of the class did well on the quiz. 71% of the Earth’s surface is water The piece of wood is 0.25 in. thick

What is a fraction? A fraction is a quantity that indicates a part of a whole, a ratio of two quantities, or a division. Fractions are written in the form 𝑎 𝑏 , where a and b are whole numbers. The number above the bar is the numerator, and the number below the bar is the denominator. 𝑛𝑢𝑚𝑒𝑟𝑎𝑡𝑜𝑟 𝑑𝑒𝑛𝑜𝑚𝑖𝑛𝑎𝑡𝑜𝑟

Simplifying Fractions A fraction is simplified, or written in lowest terms, when the numerator and denominator are relatively prime. This means that you cannot divide them by any common factor besides 1. To simplify a fraction, divide the numerator and denominator by their greatest common factor. Are the following fractions in lowest terms? If not, simplify them. 3 6 5 8 4 12 9 30 15 100

Fraction Chart What are some fractions that line up on the chart? Why does that happen?

Equivalent Fractions The places on the fraction chart that line up are called equivalent fractions. Equivalent fractions are fractions that are equal in value , but have different numerators and denominators. Equivalent fractions simplify to become the same fraction.

How can you write a fraction equivalent to another fraction? You can make equivalent fractions by multiplying or dividing both the numerator and denominator of a fraction by the same amount. If you choose to divide, only divide so that the numerator and denominator would still be whole numbers. Let’s write some fractions that are equivalent to 4 20

Comparing Fractions to Benchmarks When you solve problems involving fractions, it is useful to estimate the size of each fraction. One way is to compare each fraction to benchmarks, or reference points. 0, 1 2 , and 1are all benchmarks. First, decide which two benchmarks each fraction falls between, then decide which benchmark each fraction is closest to.

Comparing Fractions to Benchmarks (cont.) Let’s decide which benchmark (0, ½, or 1) each fraction below is closest to. 2 6 5 9 10 12

Comparing Fractions There are a few ways to compare fractions. One way is to rename the fractions as equivalent fractions with the same denominator. Let’s try to do that with 4/5 and 7/8. This same method can be used to put fractions in order from least to greatest or greatest to least. Let’s put 2/5, 3/10, and 1/3 in order from least to greatest.

Comparing Fractions (cont.) Another way to compare fractions is to use cross products, as shown below. Let’s try with 4/9 and 3/7.

Finding Fractions Between Fractions Sometimes it’s easy to find fractions between fractions, like 1/5 and 4/5. With other fractions, like 5/8 and 6/8, it requires more work to find some of the fractions in between them. A good method to do this is to rename the fractions as equivalent fractions with larger numerators and denominators. Don’t be afraid to make your denominator too big. The greater the denominator is, the more fractions you’ll be able to find in between! Let’s practice.

Naming Fractions Greater than 1 Between every pair of whole numbers are many other numbers that can be labeled with fractions. A number such as 2 ½ is called a mixed number because it has a whole number part plus a fraction part.

Naming Fractions Greater than 1 (cont.) Mixed numbers can be rewritten as fractions with no whole number part. The numerator ends up being greater than the denominator, but it is still a fraction. When the numerator of a fraction is greater than the denominator, the fraction is called an improper fraction. Let’s change 2 ½ to an improper fraction.

Changing Improper Fractions to Mixed Numbers or Whole Numbers Now that we’ve practiced changing mixed numbers to improper fractions, let’s change some improper fractions to mixed numbers or whole numbers. You can do this by dividing the numerator by the denominator. 15/6 7/2 20/8 9/3