Comparing Fractions With Different Denominators

Lesson 2 of 2: Fraction Equivalence and Ordering

In this lesson:

• Three strategies for comparing any two fractions
• Why the whole has to match before you compare

Your Learning Objectives for Today

By the end of this lesson, you can:

1. Compare fractions using a common denominator
2. Compare fractions using a common numerator
3. Use the benchmark to compare quickly
4. Explain why comparisons need the same whole
5. Record results with , , or and justify them

Which Granola Bar Piece Is Bigger?

• You get of a bar; your friend gets
• The numbers don't obviously say who has more
• Make your prediction before we check

Quick Recall on Equivalent Fractions

Before we start, a quick check:

• Name a fraction equivalent to
• Why is multiplying top and bottom by the same number allowed?

Why the Old Trick Doesn't Work

To compare and , you might try:

• Compare numerators: vs. ? That only works with matching denominators
• Compare denominators: vs. ? That only works with matching numerators
• Neither number alone decides it — the pieces are different sizes

Rewrite Both With a Common Denominator

The fix: rewrite both fractions with the same denominator.

• Equivalent fractions (4.NF.A.1) let us do this without changing the value
• For thirds and fourths, 12 works for both
• Once the pieces match, just compare the numerators

Rewriting Two Thirds and Three Fourths

• and
• Same bars, same shading — just cut into 12 equal pieces now
• , so

Confirming It on a Number Line

• and both plotted on the same line from 0 to 1
• sits farther to the right, so it is the bigger fraction

A Faster Second Comparison Example

Compare and . Common denominator: 24.

• and
• , so

Predict: Is Bigger Than ?

Which is true?

• (a)
• (b)
• (c)

Pick one before the next slide — no computing, just a gut guess.

Reveal: These Two Are Actually Equal

Common denominator for 4 and 8 is 8.

• (multiply top and bottom by 2)
• — same fraction, same amount
• Equal is a real answer — it connects right back to 4.NF.A.1

Predict: Is Bigger Than ?

Someone says: "8 is bigger than 4, so ." Are they right?

• (a) Yes, bigger denominator means bigger fraction
• (b) No, something else is going on

Predict before we check.

Reveal: Smaller Pieces, Not Bigger

• Rewrite as (multiply top and bottom by 2)
• Now compare: , so
• Rule: a bigger denominator means smaller pieces, not a bigger fraction

Watch Out: Cross-Multiplying in Disguise

Seen the "cross-multiply" shortcut for vs. ?

• 2 × 7 = 14 and 3 × 5 = 15 — really just the numerators of and
• It works, but always be ready to explain why

Quick Check: Two Fifths or Three Fourths

Compare and using a common denominator.

Try it on paper — what denominator will you pick, and why?

One Shortcut Down, Two to Go

Common denominators make the pieces match.

• Next: a shortcut that matches the number of pieces instead
• Same idea, different lever to pull

Common Numerators: Compare Piece Size

• and — the same 3 pieces are shaded in both
• Fifths are bigger pieces than sevenths (fewer total parts)
• So

When Numerators Aren't Equal Yet

• and
• Now both fractions have numerator 6
• Ninths are bigger than tenths, so
• That means

An Efficient Case for Common Numerators

• — matches the 8 in
• Fourteenths are bigger than fifteenths, so
• That means

Quick Check: Which Is Bigger?

Which is bigger, or ? Why — without computing anything?

A Shortcut That Skips Rewriting

Before you find a common denominator or numerator... is more or less than ?

• This one question can settle a comparison instantly
• No rewriting needed — just compare to a landmark

The Doubling Test for One Half

• Double the numerator, then compare it to the denominator
• : 2 × 3 = 6 < 8, so
• : 2 × 5 = 10 > 9, so
• Below vs. above settles it:

Testing a Second Benchmark Example

• : 2 × 5 = 10 < 12, so
• : 2 × 4 = 8 > 7, so
• Different sides of , so

Predict: Does the Benchmark Decide This?

Compare and using the doubling test.

• Test both fractions against first
• Will one shortcut answer settle it this time?

Try it before the next slide.

When the Benchmark Isn't Enough

• and both test greater than
• The benchmark narrows it down but can't finish here
• Switch to common denominators:

Benchmarks of Zero and One

• A fraction near 0 has a tiny numerator compared to its denominator (like )
• A fraction near 1 has a numerator close to its denominator (like )
• — no computation needed

Who Really Ate More Pizza?

• Maya eats of her pizza
• Jordan eats of his pizza
• We know — so Maya ate more... right?

The Pizzas Weren't the Same Size

• Maya's pizza: a small 8-inch personal pizza
• Jordan's pizza: a large 18-inch family pizza
• of the big pizza is actually more food

Comparisons Need the Same Whole

• is still true — but only for the same-size whole
• Area models: both rectangles or circles must be the same size
• Number lines: both fractions go on the same 0-to-1 line

Three Tools, One Choice Each Time

• vs : same denominator, compare numerators
• vs : benchmark settles it
• vs : common denominators,
• Every tool matches something before comparing

Your Turn: Four Comparisons, No Hints

• vs
• vs
• vs
• vs

Pick a strategy for each. Record , , or , and justify with a model or a sentence.

Four Things That Trip People Up

• Bigger denominator does not mean bigger fraction — check piece size
• Cross-multiplying works, but you must be able to explain why
• Different-looking fractions can still be exactly equal
• only holds when both pieces come from the same whole

What You Now Know About Fractions

✓ Common denominators, common numerators, and benchmarks are three ways to compare
✓ Every strategy works by matching something — piece size, piece count, or a landmark
✓ A comparison only makes sense when both fractions share the same whole

Next Up: Adding Fractions Together

You now know how to find a common denominator. Next lesson uses that same move to add fractions with unlike denominators instead of just comparing them.