Unit 3: Fractions
Fractions allow us to perform calculations with numbers that are not whole. For example, imagine you baked a pie. If your family eats half the pie for dessert one day, you could use fractions to determine that half a pie remains. If you eat another slice, you could use fractions yet again to see how much is left. We use fractions every day to calculate sale prices, measurements, money, and many other situations. In this unit, we explore how to add, subtract, multiply, divide, and reduce fractions.
Completing this unit should take you approximately 11 hours.
Upon successful completion of this unit, you will be able to:
- identify parts of a fraction;
- recognize fractions in lowest terms;
- recognize equivalent fractions;
- identify improper fractions and mixed numbers;
- convert between mixed numbers and improper fractions;
- determine the least common denominator between fractions;
- use equivalent fractions to add and subtract fractions and mixed numbers with like and unlike denominators;
- solve multiplication and division problems with fractions and mixed numbers; and
- solve real-world math problems involving fractions.
3.1: Identifying Parts of Fractions
Before we can perform calculations with fractions, we need to be able to identify the parts of a number written as a fraction.
Read the "Parts of a Fraction" section to identify the different parts of a number written as a fraction.
Read this section up to example 4.1 to see how fractions are used to describe real world scenarios, like the pieces of a pizza. Try examples 4.1 and 4.2 to translate between the language of fractions and diagrams.
3.2 Equivalent Fractions
Equivalent fractions are fractions written differently but with the same value. Imagine a pizza cut into four slices.
Each slice is 1/4 of the pizza. If you eat half the pizza, you are eating 1/2 the pizza. You can also say that you ate two slices, or 2/4 of the pizza. The fractions 1/2 and 2/4 are equivalent fractions because they have the same value.
Watch this video that uses number lines to show how different fractions are equivalent, which means to have the same value.
3.3: Proper and Improper Fractions
We can write fractions as proper fractions, where the numerator is less than the denominator. We can also write fractions as improper fractions, where the numerator is greater than the denominator. You should be able to identify these different types of fractions.
Read up to the "Positive Mixed Numbers" section to see some examples of proper and improper fractions using a number line.
3.4: Mixed Numbers and Improper Fractions
Mixed numbers are a different way to write improper fractions. In a mixed number, we write the whole number with the remainder as a fraction. For example, the improper fraction 3/2 would be written as 1 ½. This means that 3/2 is 1 + ½.
Read from the "Model Improper Fractions and Mixed Numbers" section up to the "Convert Between Improper Fractions and Mixed Numbers" section. Do examples 4.5–4.8 and check your answers. If you need more practice, do Try It exercises 4.9–4.16.
3.5: Converting Between Improper Fractions and Mixed Numbers
We often need to convert between fractional numbers written as improper fractions and mixed numbers.
Watch these two videos for examples of how to perform these types of calculations.
After you watch the videos, try this ungraded assessment. If you are unsure on any of the problems, use the hints for help.
3.6: Fractions in Lowest Terms
Before we can complete calculations involving fractions, we must be able to manipulate fractions to to change their denominators. As you will see, it is often easiest to have fractions with the same denominator when performing calculations like addition or multiplication.
One way we can manipulate fractions is by writing them in their lowest terms. A fraction in lowest terms has the lowest possible denominator. We can write the fraction 4/8 in lowest terms as ½.
Watch this video for another explanation of how to reduce fractions to lowest terms.
After you watch the videos, complete this quiz and check your answers. You can try problem sets 2 and 3 if you feel you need more practice.
3.7: Finding Common Denominators
To be able to perform calculations with fractions, we must first write fractions with the same denominator. This is called finding the common denominator. To do this, you will need to know how to write fractions in their lowest terms and find common multiples, both of which which we learned earlier.
Watch this video. Pause the video after the second example and try to determine the common denominator yourself. Then watch the rest of the video to see the solution.
3.8: Adding and Subtracting Fractions with Like Denominators
When we add or subtract fractions, we have to consider two different cases: when the fractions have the same denominator, and when the fractions have different denominators. First, we are going to focus on adding and subtracting fractions with the same denominator.
Read this section, which gives an overview of how to add and subtract fractions with the same denominator. Do examples 4.53–4.57 and 4.59–4.62 and check your answers. To see everyday examples of adding fractions, do the everyday math problems 312 and 313 at the end of the section.
3.9: Adding and Subtracting Fractions with Unlike Denominators
When we have fractions with different denominators, we need to take an extra step before we can add or subtract them. First, we must find the common denominator and rewrite the fractions using that common denominator.
Read this section for an overview of adding and subtracting fractions with different denominators. Do examples 4.67–4.72 and check your answers.
Watch this video for more examples of adding and subtracting fractions with like and unlike denominators.
3.10: Adding and Subtracting Mixed Numbers
There are two different ways to add and subtract mixed numbers. One way is to treat the whole numbers and fractions separately. The other approach is to convert mixed numbers into improper fractions and then use the addition and subtraction methods learned in the last two sections.
Read the "Add Mixed Numbers" section, which explains the two methods for adding mixed numbers. Do examples 4.83 and 4.84 using the method of adding the whole numbers and fractions separately. Then, do example 4.85 using the method of converting mixed numbers to improper fractions.
Then, read the "Subtract Mixed Numbers with a Common Denominator" section, which describes both methods for subtracting mixed numbers with common denominators. Do example 4.90 using the whole numbers and fraction method, and do example 4.91 using the improper fraction method.
Complete this assessment for more practice. If you need help, use the videos or hints.
3.11: Applications of Adding and Subtracting Fractions
Adding and subtracting fractions and mixed numbers comes up in a variety of real-world applications: for example, cooking, and making change from money.
- Watch these two videos to see real-world applications to fractions and mixed numbers.
3.12: Multiplying Fractions and Mixed Numbers
We now are ready to learn how to multiply fractions and mixed numbers.
Watch these two videos for examples of how to multiply fractions and mixed numbers. You will usually need to convert mixed numbers to improper fractions before multiplying.
After you watch the videos, complete these quizzes and check your answers. If you need more practice, do problem sets 2 and 3.
Here are some additional practice problems. If you need more practice, do problem sets 2 and 3.
3.13: Dividing Fractions and Mixed Numbers
The last important skill we need for fractions and mixed numbers is knowing how to divide them. Dividing fractions and mixed numbers follows similar rules to multiplying.
Watch these two videos to learn how do these calculations and see a few examples.
Watch these two videos to learn how to divide mixed numbers and see a few examples.
After you watch the videos, take this quiz and check your answers. If you need more practice, do problem sets 2 and 3.
3.14: Applications of Multiplying and Dividing Fractions
Multiplying and dividing fractions appear in a variety of real-world applications.
Watch these two videos for examples of multiplication and division problems.
After you watch the videos, complete these assessments.