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Combining Fractional Quantities

Adding and subtracting fractions is about combining or comparing quantities. The key idea is that the pieces must be the same size before we can add or subtract them directly. This is why common denominators are so important. A denominator tells us the size of the pieces, so fractions with the same denominator are already written in terms of the same-sized parts.

For example, 2/7 + 3/7​ can be added directly because both fractions are made of sevenths. We are combining 2 sevenths and 3 sevenths, giving 5 sevenths. But with 1/2 + 1/3, the pieces are different sizes. Before we can combine them, we need to rename the fractions using a common denominator so that both fractions are written with the same-sized pieces.

In this topic, we also look at regrouping with fractions and mixed numbers. Sometimes a sum creates more than one whole, such as 5/6 + 4/6 = 9/6 = 1 3/6. Other times, subtraction requires regrouping, such as rewriting 3 1/4 as 2 5/4​ so that a fractional amount can be subtracted. These ideas are closely connected to whole-number regrouping, but students often need visual models to see why the steps make sense.

As future teachers, it is important to help students understand that adding and subtracting fractions is not just about following a rule. It is about making sure the pieces being combined or compared are the same size, then reasoning about the quantity.

 

Student Learning Goals

By the end of this topic, students should be able to:

  • Explain why fractions need common denominators before they can be added or subtracted.

  • Add and subtract fractions with common denominators.

  • Find common denominators for fractions with unlike denominators.

  • Use visual models to represent fraction addition and subtraction.

  • Add and subtract mixed numbers.

  • Explain regrouping with fractions and mixed numbers.

  • Connect fraction operations to the meaning of the numerator, denominator, and whole.

Key Vocabulary

  • Common Denominator - A shared denominator that allows fractions to be written with the same-sized pieces.

  • Unlike Denominators - Fractions with different denominators.

  • Equivalent Fractions - Fractions that have different names but represent the same amount.

  • Mixed Number - A number made of a whole number and a fraction, such as 2 1/3.

  • Improper Fraction - A fraction where the numerator is greater than or equal to the denominator.

  • Regrouping - Rewriting a number in an equivalent form to make addition or subtraction possible.

  • Fraction Sum - The result of adding fractions.

  • Fraction Difference - The result of subtracting fractions.

Strategies

  • Add with common denominators - 2/9 + 5/9 - ​Add the number of ninths: 7/9.

  • Subtract with common denominators - 7/8 - 3/8 - Subtract the number of eighths: 4/8 = 1/2.

  • Add with unlike denominators - 1/2 + 1/4 - Rename 1/2​ as 2/4​, then add 2/4 + 1/4 = 3/4.

  • Subtract with unlike denominators - 5/6 - 1/3 - ​Rename 1/3​ as 2/6​, then subtract 5/6 - 2/6 = 3/6 = 1/2.

  • Add mixed numbers - 2 3/5 + 1 4/5 - Add wholes and fractions: 3 7/5 = 4 2/5.

  • Subtract mixed numbers with regrouping - 4 1/3 - 2 2/3 - ​Regroup 4 1/3 as 3 4/3, then subtract.

Teacher Connection

Fraction addition and subtraction are often difficult because students may try to apply whole-number rules to fractions. For example, a student may add 1/2 + 1/3​ and get 2/5​ because they add across the top and across the bottom. This usually shows that the student is not thinking about the size of the pieces.

A helpful teacher response is to return to meaning:

  • Are these pieces the same size?

  • What does the denominator tell us?

  • What would we need to do before combining these pieces?

  • Can we rename one or both fractions so the pieces match?

  • How could we show this with a model?

For mixed numbers, it is also helpful to connect regrouping to whole-number subtraction. Just as 1 ten can be renamed as 10 ones, 1 whole can be renamed as 4/4, 5/5, 6/6​, and so on, depending on the denominator.

Quick Reflection Question

Why is 1/2 + 1/3​ not equal to 2/5​? What model or explanation could you use to help a student understand this?