Adding and Subtracting Fractions

Introduction

Procedural fluency with addition and subtraction of fractions means performing these operations accurately, efficiently, and with understanding of common denominators, equivalent fractions, and simplification.
Pre-service teachers should be able to model these procedures clearly and anticipate common student misconceptions.

Adding Fractions with Like Denominators

When fractions have the same denominator, simply add the numerators and keep the denominator.

[latex]\dfrac{a}{d}+\dfrac{b}{d}=\dfrac{a+b}{d}[/latex]

Example 1
[latex]\dfrac{3}{8}+\dfrac{4}{8}=\dfrac{7}{8}[/latex]

Steps

1. Add numerators.
2. Keep the denominator.
3. Simplify if needed.

Adding Fractions with Unlike Denominators

To add fractions with different denominators, you must create equivalent fractions with a common denominator.

Example 2
[latex]\dfrac{2}{3}+\dfrac{1}{4}[/latex]

Step 1: Find the least common denominator (LCD).
LCD of 3 and 4 is 12.

Step 2: Rewrite fractions.
[latex]\dfrac{2}{3}=\dfrac{8}{12}, \quad \dfrac{1}{4}=\dfrac{3}{12}[/latex]

Step 3: Add numerators.
[latex]\dfrac{8}{12}+\dfrac{3}{12}=\dfrac{11}{12}[/latex]

Subtracting Fractions

Subtraction follows the same process, but you subtract numerators after finding a common denominator.

[latex]\dfrac{a}{b}-\dfrac{c}{d}=\dfrac{ad-bc}{bd}[/latex]

Example 3
[latex]\dfrac{5}{6}-\dfrac{1}{4}[/latex]

LCD = 12
[latex]\dfrac{5}{6}=\dfrac{10}{12}, \quad \dfrac{1}{4}=\dfrac{3}{12}[/latex]

Subtract:
[latex]\dfrac{10}{12}-\dfrac{3}{12}=\dfrac{7}{12}[/latex]

Adding and Subtracting Mixed Numbers

Rule: Convert to improper fractions, perform the operation, then convert back to a mixed number if needed.

Example 4 — Addition
[latex]1\dfrac{2}{3}+2\dfrac{1}{4}[/latex]

Convert to improper fractions:
[latex]\dfrac{5}{3}+\dfrac{9}{4}[/latex]

LCD = 12
[latex]\dfrac{20}{12}+\dfrac{27}{12}=\dfrac{47}{12}=3\dfrac{11}{12}[/latex]

Example 5 — Subtraction
[latex]3\dfrac{1}{2}-1\dfrac{2}{3}[/latex]

Convert to improper:
[latex]\dfrac{7}{2}-\dfrac{5}{3}[/latex]

LCD = 6
[latex]\dfrac{21}{6}-\dfrac{10}{6}=\dfrac{11}{6}=1\dfrac{5}{6}[/latex]

Word Problems with Fractions

Example
Jenny measured [latex]\dfrac{3}{4}[/latex] cup of flour and [latex]\dfrac{2}{3}[/latex] cup of flour. How many cups total?

LCD = 12
[latex]\dfrac{9}{12}+\dfrac{8}{12}=\dfrac{17}{12}=1\dfrac{5}{12}[/latex]
Jenny measured 1 and 5/12 cups of flour in total.

Simplifying Results

Always reduce fractions to lowest terms.

Example
[latex]\dfrac{9}{12}=\dfrac{3}{4}[/latex]

If an answer is improper, convert to a mixed number.

Example
[latex]\dfrac{17}{12}=1\dfrac{5}{12}[/latex]

Common Student Errors and Corrections

H5P Practice Activities

H5P 1: Fill-in-the-Blank — Add Fractions (Like Denominators)

Prompt: Solve and simplify.

H5P Placeholder:

Examples

1. [latex]\dfrac{3}{8}+\dfrac{4}{8}[/latex] → 7/8
2. [latex]\dfrac{1}{5}+\dfrac{2}{5}[/latex] → 3/5
3. [latex]\dfrac{5}{9}+\dfrac{3}{9}[/latex] → 8/9

H5P 2: Fill-in-the-Blank — Add Fractions (Unlike Denominators)

Prompt: Find the LCD, then add.

H5P Placeholder:

Examples

1. [latex]\dfrac{2}{3}+\dfrac{1}{4}[/latex] → 11/12
2. [latex]\dfrac{3}{5}+\dfrac{1}{2}[/latex] → 1 1/10
3. [latex]\dfrac{7}{8}+\dfrac{1}{6}[/latex] → 1 5/24

H5P 3: Fill-in-the-Blank — Subtract Fractions

Prompt: Subtract and simplify.

H5P Placeholder:

Examples

1. [latex]\dfrac{5}{6}-\dfrac{1}{4}[/latex] → 7/12
2. [latex]\dfrac{7}{8}-\dfrac{1}{3}[/latex] → 13/24
3. [latex]\dfrac{3}{4}-\dfrac{1}{2}[/latex] → 1/4

H5P 4: Multiple Choice — Mixed Numbers

Prompt: Choose the correct sum or difference.

H5P Placeholder:

Example:
[latex]1\dfrac{2}{3}+2\dfrac{1}{4}[/latex]
a) 3 5/12
b) 3 11/12 ✅
c) 4 1/12
d) 2 11/12

H5P 5: Word Problems — Fraction Contexts

Prompt: Solve and write your answer in lowest terms or as a mixed number.

H5P Placeholder:

1. A recipe uses [latex]\dfrac{3}{4}[/latex] cup of sugar and [latex]\dfrac{2}{3}[/latex] cup of flour. How much total? → 1 5/12
2. Mia walked [latex]\dfrac{5}{6}[/latex] mile before lunch and [latex]\dfrac{3}{4}[/latex] mile after. How far total? → 1 7/12
3. A board [latex]3\dfrac{1}{2}[/latex] feet long is cut by [latex]1\dfrac{3}{4}[/latex] feet. How long is left? → 1 3/4

Reflection for Pre-Service Teachers

• Why must denominators be equal before adding or subtracting?
• What visual or hands-on tools can help students understand the need for a common denominator?
• How can teachers help students check the reasonableness of their answers (e.g., through estimation)?
• What vocabulary (e.g., “simplify,” “LCD,” “regroup”) should be explicitly taught?

Summary

To build procedural fluency in adding and subtracting fractions, students must:

1. Find a common denominator.
2. Add or subtract numerators.
3. Simplify the result.
4. Convert improper fractions to mixed numbers if needed.

When taught systematically—with consistent modeling and visual support—these procedures empower learners to work confidently with fractions.