Adding and Subtracting Fractions with Unlike Denominators

5.NF.A.1

In this lesson:

• Why unlike denominators can't be added directly
• Finding common denominators and converting fractions
• Adding, subtracting, and simplifying — including mixed numbers

Learning Objectives for This Lesson

By the end of this lesson, you will:

1. Explain why unlike denominators block direct addition
2. Find a common denominator using multiples
3. Convert fractions to equivalent common-denominator forms
4. Add and subtract fractions with unlike denominators
5. Simplify and check reasonableness

What You Already Know About Fractions

You can already add fractions with like denominators:

• Same-sized pieces → just count them
• But what if the pieces are different sizes?

"You ate 1/2 of a granola bar. Your friend ate 1/3. How much total?"

Different Denominators Mean Different-Sized Pieces

A half-piece and a third-piece are not the same size

Testing the Wrong Answer: Why Not Two-Fifths?

Someone says

• Quick check: Is more or less than ?
• — it's less than just the first fraction!
• We added something positive, so the answer must be more than
• Conclusion: Adding across doesn't work

Quick Check: Like or Unlike Denominators?

Can these be added directly? Why or why not?

Think about it — which pairs have same-sized pieces?

We Need Same-Sized Pieces First

The fix: re-cut both bars into same-sized pieces

• Halves and thirds → what size works for both?
• We need a number both 2 and 3 divide into evenly
• That number is called a common denominator

Re-Dividing Both Bars Creates Same-Sized Pieces

Re-cut into sixths → now we can count:

Finding Common Denominators with Multiples

For :

• Multiples of 2: 2, 4, 6, 8, 10, 12, ...
• Multiples of 3: 3, 6, 9, 12, 15, ...
• Least Common Denominator (LCD) = 6

Shortcut: Multiply the denominators: 2 × 3 = 6

Example: Find the LCD for Fourths and Thirds

For :

• Multiples of 4: 4, 8, 12, 16, 20, 24, ...
• Multiples of 3: 3, 6, 9, 12, 15, 18, ...
• LCD = 12

Shortcut check: 4 × 3 = 12 ✓ (matches the LCD this time)

Your Turn: Find Common Denominators

Find a common denominator for each pair:

1. and
2. and

Hint: Try listing multiples, then check the shortcut

Converting Fractions to a Common Denominator

To convert to sixths:

• Ask: How many sixths in one half?
• Multiply both parts by 3:

Same amount, just more, smaller pieces

Complete Procedure: One-Half Plus One-Third

• Find CD: 6
• Convert: ,
• Add:

The Standard's Example: Two-Thirds Plus Five-Fourths

Step 1: Common denominator:

Step 2: Convert: ,

Step 3: Add:

Subtraction Works the Same Way

Step 1: Common denominator: 12

Step 2: Convert: ,

Step 3: Subtract:

Check: and ✓

Your Turn: Complete the Conversion

• Common denominator: 15
• Convert : multiply by ___
• Convert : multiply by ___

Fill in the blanks, then check the next slide

Practice: Three Problems to Try

Solve each — find the common denominator, convert, and compute:

Pause and solve before checking answers

Answers to the Three Practice Problems

Now Let's Tackle Mixed Numbers

Problem:

Two strategies that both give the same answer:

• Strategy 1: Convert to improper fractions first
• Strategy 2: Add whole numbers and fractions separately

Strategy One: Use Improper Fractions

Convert:

LCD = 6:

Convert back:

Strategy Two: Add Parts Separately

Whole numbers:

Fractions:

Combine: ✓

Subtraction with Mixed Numbers: Use Improper Fractions

Convert:

LCD = 12:

Convert back:

Your Turn: Mixed Number Addition

Solve using either strategy:

Choose whichever strategy feels more natural to you

Simplifying: The Final Step Every Time

After computing, always check:

• Do the numerator and denominator share a common factor?
• If yes, divide both by that factor
• If the result is improper, convert to a mixed number

Three Simplifying Examples to Study Closely

Always Check: Does Your Answer Make Sense?

Reasonableness check — ask yourself:

• Added two fractions under 1? → Answer must be under 2
• Subtracted? → Answer must be less than the first fraction
• Is the answer close to your estimate?

Practice: Problems That Need Simplifying

Solve and simplify completely:

Remember: find CD → convert → compute → simplify → check

Answers to the Simplifying Practice Problems

3. (already simplest)

Key Takeaways and Common Mistakes

✓ Different denominators → find a common denominator first
✓ Convert both fractions, then add or subtract numerators
✓ Always simplify and check reasonableness

Never add denominators — they name piece size
Convert both fractions before combining
Simplify every time — look for shared factors

Coming Up Next in Fraction Operations

Next lesson: Solving word problems with fraction addition and subtraction (5.NF.A.2)

The complete procedure:

1. Find a common denominator
2. Convert both fractions
3. Add or subtract numerators
4. Simplify if possible
5. Check: is the answer reasonable?