Adding and Subtracting Fractions (with Mixed Numbers)

This chapter supports future elementary teachers in helping students add and subtract fractions with
like and unlike denominators, including mixed numbers. It emphasizes visual models, common
denominators, and estimation for reasonableness.

1. Fractions with Like Denominators

Rule: When denominators match, add/subtract the numerators and keep the denominator.

Add: [latex]\frac{a}{d} + \frac{b}{d} = \frac{a+b}{d}[/latex]

Subtract: [latex]\frac{a}{d} - \frac{b}{d} = \frac{a-b}{d}[/latex]

Example (Add):

[latex]\displaystyle \frac{3}{8} + \frac{1}{8} = \frac{4}{8} = \frac{1}{2}[/latex]

Example (Subtract):

[latex]\displaystyle \frac{7}{10} - \frac{3}{10} = \frac{4}{10} = \frac{2}{5}[/latex]

Teaching note: Use fraction strips or area models to show “same-sized parts.”

2. Fractions with Unlike Denominators

Key idea: Create a common denominator (usually the LCM of the denominators), then add/subtract.

1. Find a common denominator [latex]D[/latex] (often [latex]\mathrm{LCM}(d_1, d_2)[/latex]).
2. Convert each fraction to an equivalent fraction with denominator [latex]D[/latex].
3. Add or subtract the numerators; keep denominator [latex]D[/latex].
4. Simplify if possible.

Example (Add):

[latex]\displaystyle \frac{2}{3} + \frac{5}{12} = \frac{8}{12} + \frac{5}{12} = \frac{13}{12} = 1\frac{1}{12}[/latex]

Example (Subtract):

[latex]\displaystyle \frac{5}{6} - \frac{1}{4} = \frac{10}{12} - \frac{3}{12} = \frac{7}{12}[/latex]

Visual model: Show both fractions on a number line partitioned into [latex]D[/latex] equal parts to
make the addition/subtraction visible.

3. Mixed Numbers

Two strategies:

• Convert to improper fractions, operate, then convert back to mixed number if desired.
• Add/subtract whole parts and fraction parts separately (renaming/borrowing when needed).

3.1 Adding Mixed Numbers

Example (Convert):

[latex]\displaystyle 2\frac{1}{3} + 1\frac{3}{4} = \frac{7}{3} + \frac{7}{4} = \frac{28}{12} + \frac{21}{12} = \frac{49}{12} = 4\frac{1}{12}[/latex]

Example (Add parts):

Whole parts: [latex]2 + 1 = 3[/latex]. Fraction parts: [latex]\frac{1}{3} + \frac{3}{4} = \frac{4}{12} + \frac{9}{12} = \frac{13}{12} = 1\frac{1}{12}[/latex].
Total: [latex]3 + 1\frac{1}{12} = 4\frac{1}{12}[/latex].

3.2 Subtracting Mixed Numbers

Example (Borrow/Rename):

[latex]\displaystyle 5\frac{1}{6} - 2\frac{3}{4}[/latex]

Common denominator for sixths and fourths is [latex]12[/latex]. Rename [latex]5\frac{1}{6} = 4 + \left(1 + \frac{1}{6}\right) = 4 + \frac{7}{6} = 4\frac{7}{6}[/latex].
Convert to twelfths: [latex]4\frac{7}{6} = 4 + \frac{14}{12}[/latex], and [latex]2\frac{3}{4} = 2 + \frac{9}{12}[/latex].
Subtract: whole parts [latex]4-2=2[/latex]; fraction parts [latex]\frac{14}{12} - \frac{9}{12} = \frac{5}{12}[/latex].
Result: [latex]2\frac{5}{12}[/latex].

4. Estimation and Reasonableness

• Round fractions to friendly benchmarks (e.g., [latex]\frac{1}{2}[/latex], [latex]\frac{1}{4}[/latex], [latex]\frac{3}{4}[/latex]) to predict results.
• Check that sums/differences make sense (e.g., adding two numbers less than 1 should yield less than 2).

5. Common Misconceptions

• Adding denominators: [latex]\frac{1}{4} + \frac{1}{4} \neq \frac{2}{8}[/latex] (should be [latex]\frac{2}{4} = \frac{1}{2}[/latex]).
• Skipping the common denominator step: Students try [latex]\frac{2}{3} + \frac{1}{5} = \frac{3}{8}[/latex] (incorrect).
• Not simplifying: Omitting reduction (e.g., leaving [latex]\frac{4}{10}[/latex] instead of [latex]\frac{2}{5}[/latex]).
• Whole–fraction mixing errors: Losing track of the whole-number part in mixed numbers.

6. Practice Problems

1. [latex]\frac{3}{5} + \frac{1}{10}[/latex]
2. [latex]\frac{7}{12} - \frac{1}{8}[/latex]
3. [latex]\frac{5}{6} + \frac{2}{9}[/latex]
4. [latex]2\frac{3}{8} + 1\frac{1}{6}[/latex]
5. [latex]4\frac{5}{12} - 1\frac{2}{3}[/latex]
6. [latex]\frac{11}{18} - \frac{5}{12}[/latex]

7. Teaching Tips for Future Educators

• Use multiple representations: area models, number lines, and fraction strips.
• Stress unit size: Common denominators create same-sized parts—make this explicit.
• Language routines: Have students say, “I’m renaming to twelfths,” not “I’m cross-multiplying.”
• Fluency with LCMs: Build quick strategies (e.g., testing multiples) to find common denominators.
• Estimate first: Benchmarking to [latex]\frac{1}{2}[/latex] and [latex]1[/latex] catches many errors early.