Equivalent Fractions: Why They Work

Today's Big Ideas:

• Visual models show why fractions are equivalent
• Multiplying by preserves value
• Generating families of equivalent fractions

Your Learning Objectives for Today

By the end of this lesson, you can:

1. Explain why subdividing doesn't change the amount
2. Demonstrate why multiplying top and bottom works
3. Justify why multiplying by equals 1
4. Verify equivalence with models and number lines
5. Generate equivalent fractions by choosing

You Already Know Some Equivalent Fractions

From Grade 3, you learned:

But why is this true?

• The numbers look completely different
• How can equal ?
• What's really happening here?

Today we'll answer this question!

Seeing Equivalence in an Area Model

• Started with (2 of 3 columns shaded)
• Split each column into 4 equal rows: now
• The shaded area did not change!

Subdivision Does Not Change the Amount

Key Insight:
When you split every piece into the same number of smaller pieces:

• The number of parts changes (3 → 12)
• The size of each part changes (bigger → smaller)
• The total shaded amount stays the same

This is why:

Number Line: Same Point, Different Names

• Top line divided into thirds: point at
• Bottom line divided into twelfths: point at
• The point did not move — same location, different names

Another Example:

Area model: Split each half into 3 equal pieces

• → 2 halves, 1 shaded
• → 6 sixths, 3 shaded
• Same shaded area

Number line: Same point at two different partition levels

• Halfway between 0 and 1 on both number lines

The pattern holds!

Turn and Talk: Explain the Model

Question: Why is the shaded area the same with 8 pieces instead of 2?

Talk to your partner:

• What stayed the same? What changed?
• Why didn't the amount change?

Listen for: same region, just counted with smaller pieces

Your Turn: Predict the Equivalent Fraction

Given: An area model of (3 of 4 columns shaded)

Task: Split each column into 3 equal rows.

• Total small rectangles? ____
• Shaded rectangles? ____
• Equivalent fraction?

Answer: 12 total, 9 shaded —

Connecting the Visual to the Symbols

What we did visually:

• Split each of the 3 columns into 4 rows
• Shaded parts:
• Total parts:

What we wrote symbolically:

Why Multiplying by Works

Key insight: (all parts shaded = one whole)

So:

The value doesn't change because we multiplied by 1!

The General Principle for Any Fraction

For any fraction and any whole number :

Because:

This works for every fraction and every nonzero value of .

Explain to Your Neighbor Why It Works

Question: Why does multiplying numerator and denominator by the same number keep the fraction equivalent?

Listen for:

• "We multiplied by , which equals 1"
• "We subdivided each piece equally, so the amount stayed the same"

Both answers are valid!

Worked Example:

Choose :

Verify with area model:

• Start with 5 columns, 3 shaded
• Split each column into 3 rows
• Result: 15 small rectangles, 9 shaded ✓

Both methods confirm:

Generating a Family of Equivalent Fractions

Start with . Choose different values of :

• :
• :
• :

Infinitely many equivalent fractions exist!

Recognizing Whether Two Fractions Are Equivalent

Question: Are and equivalent?

Strategy: Find a common multiplier

• Does ? Yes,
• Does ? Yes, ✓

Conclusion: . They are equivalent.

Your Turn: Generate Three Equivalents

Given:

Task: Choose three values of and generate three equivalents.

• :
• :
• :

Compare with a partner — did you pick the same ?

Show Me on Your Whiteboard

Write on your whiteboard:

One equivalent fraction for

Teacher circulates to check student work. Look for correct application of multiplying both numerator and denominator by the same n.

Common correct answers: 10/12, 15/18, 20/24, 25/30, etc.

Common Mistakes and How to Avoid Them

• Multiplying only one part: — multiply both by
• Adding instead of multiplying: — only works
• ✓ Different-looking fractions can name the same amount
• ✓ The rule works for every fraction, not just familiar ones

Key Takeaways from Today's Lesson

• Subdivision preserves amount — more pieces, same total
• Multiply by — that's why it works
• Infinitely many equivalents exist for every fraction
• Visual and symbolic reasoning connect — models match the algebra

What's Next in Fraction Learning?

Today: Explained why

Next lesson: Comparing fractions with different denominators

• Use equivalent fractions to build common denominators
• Compare and using and

This is the foundation for all future fraction work!