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

Learning Objectives

By the end of this lesson, you will be able to:

1. Explain using a visual model why subdividing changes the number of parts but not the amount
2. Demonstrate that multiplying numerator and denominator by the same number produces an equivalent fraction
3. Justify why multiplying by is the same as multiplying by 1
4. Verify that two fractions are equivalent using area models and number lines
5. Generate equivalent fractions by choosing different values of

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: Area Model

What happened?

• Started with (2 out of 3 columns shaded)
• Split each column into 4 equal rows
• Now we have (8 out of 12 small rectangles shaded)
• 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

On the number line:

• Top line divided into thirds: point at
• Bottom line divided into twelfths: point at
• The point did not move!

and name the same location.

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

Question:
Why is the shaded area the same even though we have 8 pieces instead of 2?

Talk to your partner:

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

Listen for: "We split the parts but didn't add or remove shading" or "Same region, just counted differently"

Your Turn: Predict the Equivalent Fraction

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

Task: Draw lines to split each column into 3 equal rows. Then:

• How many small rectangles total? ____
• How many small rectangles shaded? ____
• What is the equivalent fraction?

Expected answer: 12 total, 9 shaded, so

Connecting Visual to Symbolic

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 and any whole number :

Because:

This works for every fraction and every nonzero value of .

Explain to Your Neighbor

Question:
Why can we multiply the numerator and denominator by the same number and get an equivalent fraction?

Possible answers to listen for:

• "Because we're multiplying by which equals 1, and multiplying by 1 doesn't change the value"
• "Because we're subdividing each piece equally, so the amount stays the same"

Both visual and symbolic reasoning 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 Fraction Family

Start with:

Choose different values of :

• :
• :
• :
• :

There are infinitely many equivalent fractions!

Recognizing Equivalent Fractions

Question: Are and equivalent?

Strategy: Look for a common multiplier

• Does ? Yes, if
• Does ? Yes, if
• Same works for both! ✓

Conclusion:

They are equivalent.

Your Turn: Generate Three Equivalents

Given:

Task: Choose three different values of and generate three equivalent fractions.

Example choices:

• :
• :
• :

Then compare with a partner-did you choose the same values?

Expected answers: 4/10, 6/15, 8/20 or other valid equivalents

Show Me: Whiteboard Check

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

Error 1: Multiplying only numerator or only denominator

• or
• ✓ Both must be multiplied by the same

Error 2: Adding the same number instead of multiplying

• (added 3 to both)
• ✓ Must multiply, not add (only works)

Error 3: Thinking different numbers can't be equal

• ✓ Different fractions can name the same amount

Error 4: Thinking equivalence only works for "special" fractions

• ✓ Works for ALL fractions, ANY nonzero

Key Takeaways

1. Subdivision preserves amount

• Splitting pieces equally changes the count, not the total

2. Multiplying by is why it works

• The identity property of multiplication explains equivalence

3. Infinitely many equivalents

• Every fraction has infinitely many equivalent forms

4. Visual and symbolic reasoning connect

• Area models and number lines show what the algebra describes

What's Next?

Today: Explained why

Next lesson: Comparing fractions with different denominators

• Standard 4.NF.A.2
• Use equivalent fractions to create common denominators
• Compare and by converting to and

The skill you learned today is the foundation for all future fraction work!