Equivalent Fractions Comparing Fractions
Part of the Math for Young Minds curriculum — designed for neurodivergent students, grounded in real-world examples.
📋 Session plan (for teachers)
Session 5 — Equivalent fractions & comparing fractions
Grade 4 · Math for Young Minds Total time: ~22 minutes Common Core: 4.NF.A.1, 4.NF.A.2 Today's idea: Two fractions can look different and still be the same amount — and that helps us compare them.
What students will be able to do
By the end of this session, the student can:
- Generate equivalent fractions by multiplying the top and bottom by the same number.
- Compare two fractions by rewriting them with a common denominator.
- Place fractions on a number line in order.
Materials
- Pre-cut fraction strips per pair (halves, thirds, fourths, sixths, eighths, twelfths)
- Worksheet (one per student)
- Pencils
Substitution: If you don't have pre-cut strips, draw 6 same-length bars on the board and divide them into 2, 3, 4, 6, 8, and 12 equal parts. Strips of paper torn to the same length also work.
New words
| Word | Meaning we use in class |
|---|---|
| common denominator | A denominator that works for two or more fractions — letting us compare them. |
Heads-up — common confusions
- Students often say the fraction with the bigger numerator is bigger. Watch for this with
3/4vs5/8. Slow down and line up the strips. - Some will add the same number to top and bottom. The rule is multiply top and bottom by the same number. Name this out loud.
- Remind them:
1/2 = 2/4 = 3/6 = 4/8 = 6/12. Same amount, different names.
Plan
1 · Hello & today's idea — 2 min
"Today we're going to find out how one fraction can have many names — and how that helps us figure out which fraction is bigger."
Hold up a 1/2 strip and a 2/4 strip side by side.
Ask: "These have different numbers on them. But look — same length. How can that be?"
Take 1–2 quick answers. Don't correct. Just listen.
2 · Hands-on explore — 6 min
Hand each pair their fraction strips.
Prompt: "Line up your 1/2 strip. Now find every strip that matches its length exactly. Write down each one."
Let them work. Listen for:
- Are they laying strips flat and lining up the edges?
- Do they notice
1/2 = 2/4 = 3/6 = 4/8 = 6/12?
After ~3 minutes, pause everyone:
"What did you find? How many names did 1/2 have today?"
Take 2–3 responses. Then try one more:
"Now line up
2/3. What strips match it?"
You're listening for 4/6, 8/12.
3 · Connect to the math — 4 min
Now name what just happened.
"When two fractions take up the same amount, we call them equivalent. And there's a shortcut — we don't have to lay out strips every time."
Write on the board:
2 ×2 4
─ ────────► ─
3 ×2 6
"Multiply the top AND the bottom by the same number. You get a new name for the same fraction."
Show two more:
2 ×3 6 2 ×4 8
─ ──── ── ─ ──── ──
3 9 3 12
"Careful — we multiply top and bottom. We do not add."
Now the comparing idea:
"If two fractions have the same bottom number — the same denominator — we can just look at the tops. A common denominator is a bottom that works for both."
4 · Practice with support — 8 min
Pass out the worksheet.
Do problem 1 together on the board:
"Write 4 fractions equivalent to
2/3."
Work through it out loud: multiply top and bottom by 2, then 3, then 4, then 5.
Answers: 4/6, 6/9, 8/12, 10/15.
Now let students try on their own. Circulate.
Problem 2 (solo): Compare 3/4 and 5/8. Which is bigger?
→ Rewrite 3/4 as 6/8. So 6/8 > 5/8. 3/4 is bigger.
Problem 3 (solo): Compare 2/3 and 3/4. Find a common denominator.
→ 8/12 vs 9/12. 3/4 is bigger.
Problem 4 (stretch): Order from smallest to biggest: 1/2, 2/3, 3/4, 5/6.
→ Rewrite all with denominator 12: 6/12, 8/12, 9/12, 10/12. Order: 1/2, 2/3, 3/4, 5/6.
If a student is stuck, invite them to pull out the fraction strips again before doing the math.
5 · What we did + Try at home — 2 min
"Today you learned that one fraction can have many names. Multiply top and bottom by the same number to find them. And when two fractions share a denominator, comparing is easy."
Wave the take-home:
"Tonight, find something you can split into halves, thirds, or fourths — a candy bar, a folded paper, a pizza, a measuring cup. Compare two slices. Which is bigger?"
Observation rubric — what to notice in this session
Use this during the session, not as a test. One observation per student is plenty.
| Where the student is | What you'd see |
|---|---|
| Developing | Relies on the fraction strips to compare. May still try to add to top and bottom, or compare numerators alone. |
| Using | Multiplies top and bottom to make equivalent fractions. Finds a common denominator to compare two fractions. |
| Extending | Picks an efficient common denominator (like 12 for thirds and fourths). Orders three or more fractions confidently. |
No fail state. "Developing" today is "using" next week.
What's next (Session 6)
Building on this, Session 6 — Adding & subtracting fractions puts fractions together (and takes them apart). When the bottoms are the same, it's easier than you'd think — and now you already know how to make the bottoms match.
✏️ Worksheet (for students)
Math for Young Minds · Grade 4
Session 5 — Equivalent fractions & comparing fractions
[ Hello ] → [ Explore ] → [ Connect ] → [ Practice ← we are here ] → [ Try at home ]
Today's big idea
The same amount can have many names.
1/2 = 2/4 = 3/6 = 4/8 = …
To make an equivalent fraction, multiply the top AND the bottom by the same number. (Adding doesn't work — only multiplying!)
Example we did together
Look at the fraction strips:
┌─────────────────────────────┐
1 │ 1 │
├──────────────┬──────────────┤
1/2│ 1/2 │ 1/2 │
├──────┬───────┼──────┬───────┤
1/4│ 1/4 │ 1/4 │ 1/4 │ 1/4 │
├──┬───┼──┬────┼───┬──┼───┬───┤
1/8│1/8│1/8│1/8│1/8│1/8│1/8│1/8│1/8
└──┴───┴──┴────┴───┴──┴───┴───┘
We see: 1/2 = 2/4 = 4/8. Same amount — different names!
To get 2/4 from 1/2: (1 × 2) / (2 × 2) = 2/4 ✓
Problem 1 — together
Write 4 fractions equivalent to 2/3.
Multiply top and bottom by 2, then 3, then 4, then 5:
2 × __ 2 × __ 2 × __ 2 × __
────── = ────── = ────── = ──────
3 × __ 3 × __ 3 × __ 3 × __
= ____ = ____ = ____ = ____
Use this space to draw fraction strips and check:
┌─────────────────────────────────────────────────────────┐
│ │
│ │
│ │
│ │
│ │
│ │
│ │
│ │
└─────────────────────────────────────────────────────────┘
Problem 2 — on your own
Compare 3/4 and 5/8. Which is bigger?
Hint: rewrite 3/4 with denominator 8.
3 × __ __
────── = ────
4 × __ 8
Now compare with 5/8. Circle the bigger fraction: 3/4 5/8
Draw fraction strips to show your thinking:
┌─────────────────────────────────────────────────────────┐
│ │
│ │
│ │
│ │
│ │
│ │
│ │
│ │
└─────────────────────────────────────────────────────────┘
Answer sentence: / > /
Problem 3 — on your own
Compare 2/3 and 3/4. Find a common denominator first.
A denominator that works for both 3 and 4 is: ____
Rewrite both fractions:
2 × __ __ 3 × __ __
────── = ──── ────── = ────
3 × __ __ 4 × __ __
Circle the bigger fraction: 2/3 3/4
Draw it here:
┌─────────────────────────────────────────────────────────┐
│ │
│ │
│ │
│ │
│ │
│ │
│ │
│ │
└─────────────────────────────────────────────────────────┘
Problem 4 — stretch
Order from smallest to biggest: 1/2, 2/3, 3/4, 5/6
A common denominator that works for 2, 3, 4, and 6 is 12. Rewrite each:
1/2 = ____/12 2/3 = ____/12
3/4 = ____/12 5/6 = ____/12
Now place them on the number line:
0 1
├───┬───┬───┬───┬───┬───┬───┬───┬───┬───┬───┬───┤
0/12 1 2 3 4 5 6 7 8 9 10 11 12/12
Write them in order, smallest to biggest:
____ < ____ < ____ < ____
Hint: bigger numerator doesn't always mean bigger fraction! You have to check the bottom too.
Today's words
| Word | What it means |
|---|---|
| common denominator | A denominator that works for two or more fractions — letting us compare them |
Try at home tonight (1 minute)
Find something you can split into halves, thirds, or fourths. Compare two slices — which is bigger?
- A candy bar split unevenly between you and a friend
- A piece of paper folded into 6, and another folded into 4
- A pizza cut into 8 slices vs. one cut into 6 slices
- A measuring cup filled to 1/2 vs. one filled to 2/3
- Two glasses of juice — one half full, one three-quarters full
Write the two fractions and circle the bigger one:
____/____ ____/____
Tell a grown-up which slice you'd pick — and why!
Next time (Session 6): Adding & subtracting fractions — when the bottoms match, it's easier than you'd think.
🏠 Family guide (for parents)
Math for Young Minds · Grade 4 · Session 5
A note for grown-ups: today we worked with equivalent fractions
What your child did today
In class today, we explored equivalent fractions — different-looking fractions that are actually the same amount.
The big idea: 1/2 is the same as 2/4, which is the same as 3/6, which is the same as 4/8. Same slice of the world, different names.
We used fraction strips — halves, thirds, fourths, sixths, eighths, twelfths — and lined them up to see which pieces matched. Then we used that idea to compare two fractions, like 3/4 and 5/8, by rewriting them so the bottoms match.
Why this matters
Fractions are where a lot of older math lives — measuring, dividing, ratios, algebra later on. The trick of "rename it so the bottoms match" is one your child will use for years. We're not racing through it. Understanding first. Speed comes later, on its own.
🏠 Try this tonight (1 minute)
Find something at home you can split into halves, thirds, or fourths. Compare two pieces — which is bigger, and how do you know?
Easy starters around the house:
| Thing | Try this |
|---|---|
| A candy bar split unevenly | Which piece is bigger — 1/3 or 1/4? |
| Paper folded into 6 vs into 4 | Which fold makes bigger pieces? |
| Pizza cut into 8 vs 6 slices | Which slice is bigger? |
| Measuring cup | Fill to 1/2, then to 2/3 — which is more? |
A short script:
- "If we split this into 4 pieces vs 3 pieces — which pieces are bigger?"
- "How do you know?"
- "Can you show me with your hands?"
More pieces means smaller pieces. That surprises a lot of kids — and it's worth pausing on.
Words your child is learning
- Common denominator — a bottom number that works for two fractions, so we can compare them fairly
If your child says…
"This is easy." Great. Ask them: "Is 3/4 bigger or smaller than 5/6? How do you know?" That one makes you stop and think.
"This is hard." Also great. Pull out a piece of paper and fold it. Fold one into halves, another into fourths. Let them see and touch the pieces. The fraction strips from class do the same job. We're not in a rush.
"I don't want to." That's okay. Skip the worksheet feeling. Just ask one question at dinner — "is your slice bigger than mine?" — and let it go. One minute is plenty.
What's next
In our next session, your child will start adding and subtracting fractions. When the bottoms are the same, it's easier than you'd think — and we'll build slowly from there.
Thanks for taking a minute tonight. These small kitchen-table moments are where math lives.
— Math for Young Minds
🔑 Cheat sheet (visual)
🍰 Same size, different slices
Picture 1 — One half = many names
┌───────────────────────────┐
│█████████████ │ 1/2
├─────────────┬─────────────┤
│██████ │ │ 2/4
├──────┬──────┼──────┬──────┤
│████ │ │ │ │ 3/6
├───┬──┴─┬────┼───┬──┴─┬────┤
│██ │ │ │ │ │ │ 4/8
└───┴────┴────┴───┴────┴────┘
Same shaded amount → 1/2 = 2/4 = 3/6 = 4/8 ✨
Picture 2 — Make equivalents (×, not +)
×2 ×3 ×4
┌──┐ ┌──┐ ┌──┐
2 │ │ 4 │ │ 6 │ │ 8
─ ─ ─ ─
3 │ │ 6 │ │ 9 │ │ 12
└──┘ └──┘ └──┘
×2 ×3 ×4
2 4 6 8 10
─ = ─ = ─ = ── = ──
3 6 9 12 15
How to read the sign
multiply top AND bottom
by the SAME number
│
2 ×3 │ 6
─ ─────► ─
3 ×3 9
│
│
✅ multiply ❌ don't add
Picture 3 — Compare 3/4 vs 5/8
┌───────┬───────┬───────┬───────┐
│███████│███████│███████│ │ 3/4
└───┬───┴───┬───┴───┬───┴───┬───┘
│ │ │ │
┌───┴─┬─────┼───┬───┴─┬─────┴─┐
│█████│█████│███│█████│ │ 5/8
└─────┴─────┴───┴─────┴───────┘
3 6 6 5
─ = ─ so ─ > ─
4 8 8 8
Picture 4 — Common denominator
common denominator: a bottom that works for BOTH fractions — so we can compare.
Compare 2/3 vs 3/4
2 ×4 8 3 ×3 9
─ ─────► ── ─ ─────► ──
3 ×4 12 4 ×3 12
8 9
── < ──
12 12
→ 3/4 is bigger
When are two fractions equal?
| ✅ Equivalent when... | ❌ NOT equivalent when... |
|---|---|
2/3 → ×2/×2 → 4/6 |
2/3 → +1/+1 → 3/4 |
| same shaded area | different shaded area |
| multiplied top & bottom by same number | added the same number to top & bottom |
Number line check
0 1
├───────────────┼───────────────┼───────────────┤
1/2
0 1
├───────┼───────┼───────┼───────┼───────┼───────┤
1/6 2/6 3/6 4/6 5/6
↑
same spot as 1/2
Try this in your head
Order from smallest to biggest:
1 2 3 5
─ ─ ─ ─
2 3 4 6
➤ Use a common denominator of 12:
Answer:
6/12, 8/12, 9/12, 10/12→ 1/2, 2/3, 3/4, 5/6