For Topics 8 and 9, the students should know: Fractions are a part of a whole. The bottom number in the fraction is called the denominator. The top number is called the numerator. Equivalent fractions are equal to one another. Models can be used to represent equivalent fractions. For example: Benchmark Fractions are known fractions commonly used for estimating with fractions. Examples of Benchmark Fractions are 1/4, 1/2, and 3/4. For example, we know that 5/8 must be a little more than 1/2 since we know that 4/8 is equivalent, or equal to 1/2.
Number lines can also be used to represent equivalent fractions and show benchmark relationships: Multiplication or division can be used to create equivalent fractions. As long as you follow the rule that whatever is done to the denominator (bottom number) must also be done to the numerator (top number). In other words, in the example below, 1/2 is equivalent to 2/4 because 1 X 2 = 2 and 2 X 2 =4. Both the numerator and denominator were multiplied by 2 to create the equivalent fraction 2/4. In the examples below, you can see how equivalent fractions are created using multiplication. Division can also be used to create equivalent fractions, as in the last example. But remember, whatever is done to the numerator (top) must be done to the denominator (bottom) to make the fractions equivalent.
Comparing fractions means determining which is greater or less. There are several ways to compare fractions: 1. Of course, you could always draw a picture to create a model to see which is greater. 2. When the denominators are the same, you just compare the numerators. The greater numerator is the greater fraction. You can create equivalent fractions in order to do this. 3/4 > 1/4 3. If the numerators are the same, it is the opposite. The greater the denominator, the smaller the number. This is because the denominator tells how many parts, or pieces, there are in the whole. The larger the denominator, the more the whole needed to be divided up, so the pieces will be smaller. 3/6 > 3/12 4. The butterfly method can be used to compare fractions. Cross multiply and put the product on the top of each numerator. Compare the answers. When adding or subtracting fractions the denominators must be the same. Being able to create equivalent fractions makes this easier.
A fraction that has a numerator less than its denominator is called a proper fraction. 3/4 is a proper fraction. A fraction that has a numerator that is greater than its denominator is called an improper fraction. 7/4 is an improper fraction. A number that consists of a whole number AND a fraction is called a mixed number. Mixed numbers can be turned into improper fractions by drawing a picture, or following this formula: Denominator X Whole number + Numerator = Improper Fraction Improper fractions can be turned into mixed numbers through simple division. How many times the denominator can go into the numerator will be the whole number. The remainder then goes on top of the same denominator.
Composing means putting together. When we compose fraction pieces, we put them together. For example: 1/4 + 1/4 + 1/4 = 3/4 Composing is necessary when adding fractions. Decomposing means breaking apart. When we decompose fractions or mixed numbers we break them apart into smaller pieces. For example 3/4 = 1/4 + 1/4 + 1/4 Decomposing is necessary when subtracting mixed numbers. For example: 3 1/4 2 3/4 = If we decompose 3 1/4 into 2 + 4/4 + 1/4 = 2 5/4 it will make subtracting simple. Here is another example - Remember: When adding or subtracting fractions, the denominators must be the same!