📋 Session plan (for teachers)

Session 6 — Equivalent fractions

Grade 3 · Math for Young Minds Total time: ~22 minutes Common Core: 3.NF.A.3 Today's idea: The same amount can be written as different fractions. We call those equivalent.

What students will be able to do

By the end of this session, the student can:

• See that the same amount can be written as different fractions (like 1/2 = 2/4 = 3/6 = 4/8).
• Use fraction strips to spot equivalent pairs.
• Understand that multiplying the top and bottom by the same number gives an equivalent fraction.

Materials

• Pre-cut paper strips of equal length (5–6 per pair)
• Colored markers or crayons
• Worksheet (one per student)
• Pencils

Substitution: If you don't have pre-cut strips, cut any paper into long, equal rectangles. Sticky notes lined up in a row also work. The strips just need to all be the same length to start.

New words

Heads-up — common confusions

• Many kids think 1/2 and 2/4 are different just because the numbers look different. Fold the strips and lay them side by side — let their eyes do the convincing.
• Some will try to find equivalents by adding the same number to top and bottom. That's the wrong move. The rule is multiply top and bottom by the same number.
• A student may find one equivalent and stop. Push them to see the whole pattern: 1/2 = 2/4 = 3/6 = 4/8...

Plan

1 · Hello & today's idea — 2 min

"Today we're going to find out something a little surprising — the same amount of a strip can have more than one name."

Hold up one paper strip.

Ask: "If I fold this in half and color one part, what fraction did I color?"

Wait for "one half" or "1/2". Write 1/2 on the board.

"Keep that in your head. We're going to find other ways to write that same amount."

2 · Hands-on explore — 7 min

Hand each pair:

• 5–6 paper strips of equal length
• Markers or crayons

Problem 1 — together:

"Fold one strip into halves. Fold another into fourths. Color in 1/2 of the first strip and 2/4 of the second. Then lay them on top of each other."

Walk around. Listen for:

• Are the strips lined up at the same edge?
• Are they noticing the colored parts are the same length?

After ~3 minutes, pause everyone:

"What did you notice when you put them side by side?"

You're listening for "they're the same" or "they cover the same amount".

Write on the board:

1/2  =  2/4

"Same length. Different fraction names. We say these are equivalent."

3 · Connect to the math — 4 min

Now show the pattern.

Draw on the board:

1/2  =  2/4  =  3/6  =  4/8

"Look at the top numbers: 1, 2, 3, 4. Look at the bottom numbers: 2, 4, 6, 8. The bottom is always double the top."

Point at 1/2 → 2/4:

"To go from 1/2 to 2/4, we multiplied the top by 2 and the bottom by 2. Same number on top and bottom."

Warn them:

"We do NOT add the same number to top and bottom. That breaks it. We multiply."

4 · Practice with support — 7 min

Pass out the worksheet.

Problem 2 — solo: Use strips folded into thirds and sixths. Color 1/3 of one and 2/6 of the other. Are they the same?

→ Yes — 1/3 = 2/6. Let them fold, color, and compare. Circulate.

Problem 3 — solo: Write three different fractions that equal 1/2.

→ Look for 2/4, 3/6, 4/8 (or 5/10, etc.). If a student is stuck, point back to the pattern on the board.

Problem 4 — stretch: If 1/2 = 2/4, what did we do to the top and bottom? What if I multiply 1/2 by 5/5 — what do I get?

→ 5/10. Multiplied top and bottom by the same number.

If a student is stuck on the stretch, invite them to draw or fold a strip into tenths first.

5 · What we did + Try at home — 2 min

"Today you learned that the same amount can have different fraction names. We call them equivalent. And we get them by multiplying the top and bottom by the same number."

Wave the family guide:

"Tonight, look at a measuring cup at home. Find the 1/2 line and the 2/4 line. Are they the same? What about 1/4 and 2/8?"

Observation rubric — what to notice in this session

Use this during the session, not as a test. One observation per student is plenty.

No fail state. "Developing" today is "using" next week.

What's next (Session 7)

We leave fractions for a bit. Session 7 — Area and perimeter moves to shapes — how much space is inside a shape (area) vs how far around it (perimeter).

✏️ Worksheet (for students)

Math for Young Minds · Grade 3

Session 6 — Equivalent fractions

[ Hello ]  →  [ Explore ]  →  [ Connect ]  →  [ Practice ← we are here ]  →  [ Try at home ]

Today's big idea

The same amount can be written as different fractions.

When two fractions cover the same length, they are equivalent.

So 1/2 = 2/4 = 3/6 = 4/8. Same amount, different names.

Example we did together

  ┌───────────────┬───────────────┐
  │ ▓▓▓▓▓▓▓▓▓▓▓▓▓ │               │   1/2
  └───────────────┴───────────────┘

  ┌───────┬───────┬───────┬───────┐
  │ ▓▓▓▓▓ │ ▓▓▓▓▓ │       │       │   2/4
  └───────┴───────┴───────┴───────┘

Same length covered! So 1/2 = 2/4.

Problem 1 — together

Take two strips of equal length.

• Fold the first into halves. Color 1/2.
• Fold the second into fourths. Color 2/4.

Tape or draw your strips here:

┌─────────────────────────────────────────────────────────┐
│                                                         │
│                                                         │
│                                                         │
│                                                         │
│                                                         │
│                                                         │
│                                                         │
│                                                         │
└─────────────────────────────────────────────────────────┘

What do you notice?

1/2 _____ 2/4 (write = or ≠)

Problem 2 — on your own

Take two more strips.

• Fold one into thirds. Color 1/3.
• Fold the other into sixths. Color 2/6.

Draw your strips here:

┌─────────────────────────────────────────────────────────┐
│                                                         │
│                                                         │
│                                                         │
│                                                         │
│                                                         │
│                                                         │
│                                                         │
│                                                         │
└─────────────────────────────────────────────────────────┘

Are they the same length? ____

Fill in: 1/3 = ____ / ____

Problem 3 — on your own

Write three different fractions that equal 1/2.

Use strips or drawings to check.

┌─────────────────────────────────────────────────────────┐
│                                                         │
│                                                         │
│                                                         │
│                                                         │
│                                                         │
│                                                         │
│                                                         │
│                                                         │
└─────────────────────────────────────────────────────────┘

  1/2  =  ____/____  =  ____/____  =  ____/____

Careful: do not add the same number to top and bottom. That doesn't work!

Problem 4 — stretch

Look at this pair:

        1            2
       ───   =      ───
        2            4

• What did we do to the top? Multiply by ____
• What did we do to the bottom? Multiply by ____

Now try this one. Multiply 1/2 by 5/5:

   1     5         ____
  ─── × ───   =   ──────
   2     5         ____

Top times top, bottom times bottom. The fraction looks different — but it's the same amount!

Today's words

Try at home tonight (1 minute)

Find a measuring cup in the kitchen. Look at the lines on the side.

• Is 1/2 at the same line as 2/4?
• Is 1/4 at the same line as 2/8?

Other things to try:

• Find a recipe that uses fractions — spot any equivalents
• Fold a piece of paper to compare 1/3 and 2/6
• Split a chocolate bar with someone — see how many ways you can do it
• Fold a strip into eighths and find a fraction equal to 1/2

Show a grown-up what you found tomorrow.

🏠 Family guide (for parents)

Math for Young Minds · Grade 3 · Session 6

A note for grown-ups: today we explored equivalent fractions

What your child did today

In class today, we worked with equivalent fractions using folded paper strips.

The big idea: the same amount can be written in more than one way. For example, 1/2, 2/4, 3/6, and 4/8 all cover the same length on a strip.

Your child folded strips into halves, fourths, thirds, sixths, and eighths. They colored parts and lined the strips up side by side. They saw with their own eyes that 1/2 = 2/4 and 1/3 = 2/6.

At the end, we started noticing a pattern: when you multiply the top and bottom by the same number, you get an equivalent fraction.

Why this matters

Fractions trip a lot of people up because the numbers look different even when the amount is the same. Seeing 1/2 and 2/4 as the same length — not just two strange labels — is what makes the rest of fractions click later. We're not drilling rules. We're building the picture first, so the rules feel obvious when they come. Understanding before speed.

🏠 Try this tonight (1 minute)

Pull out a measuring cup from the kitchen. Look at the lines on the side together.

Ask:

• "Can you find the 1/2 line?"
• "Is there a 2/4 line? Is it in the same place?"
• "What about 1/4 and 2/8?"

Other quick options if no measuring cup:

If your child says the two amounts are the same, ask how they know. That's the gold.

Words your child is learning

• Equivalent — equal in value; a different way of writing the same number

If your child says…

"This is easy." Great. Ask them to write three different fractions that all equal 1/2. Then ask if they can keep going forever. (They can.)

"This is hard." Also great. Get out a piece of paper and fold it in half, then in half again. Color in half. Count the fourths that got colored. The paper does the explaining. No rush.

"I don't want to." Fair. Skip the worksheet feeling and just ask one question while you're cooking: "Is 1/2 cup the same as 2/4 cup?" One question, one minute. That's enough for tonight.

A small heads-up

A common slip is trying to make equivalent fractions by adding the same number to the top and bottom. That doesn't work — it's multiplying both by the same number. If you see that mistake, no worries. Just pull out a folded strip and compare. The picture will sort it out.

What's next

In our next session, we leave fractions for a bit and move to shapes — area and perimeter. How much space is inside, versus how far around. Different question, same careful pace.

Thanks for taking a minute tonight. These small kitchen-table moments are where math lives.

— Math for Young Minds

🔑 Cheat sheet (visual)

🔑 Same amount, different names

The big idea

   1/2   =   2/4   =   3/6   =   4/8

   ░░░░░░░░░░░░░░░░░░░░
   ██████████░░░░░░░░░░   ← 1/2

   ░░░░░░░░░░░░░░░░░░░░
   █████░████░░░░░░░░░░   ← 2/4   (same length!)

Equivalent = equal in value, written a different way. ✨

Picture 1 — Fold a strip in halves and fourths

   halves:   |██████████|░░░░░░░░░░|        →  1/2 shaded
              \________ 1 of 2 ______/

   fourths:  |█████|█████|░░░░░|░░░░░|      →  2/4 shaded
              \____ 2 of 4 ____/

➤ Same length covered. 1/2 = 2/4 ✅

Picture 2 — Thirds and sixths

   thirds:   |██████|░░░░░░|░░░░░░|         →  1/3
              \_ 1 of 3 _/

   sixths:   |███|███|░░░|░░░|░░░|░░░|      →  2/6
              \_ 2 of 6 _/

➤ Same length. 1/3 = 2/6 ✅

Picture 3 — A whole family equals 1/2

   |█████|░░░░░|                  1/2
   |██|██|░░|░░|                  2/4
   |█|█|█|░|░|░|                  3/6
   |▌|▌|▌|▌|░|░|░|░|              4/8

All cover the same half of the strip.

How to make an equivalent fraction

            × 2                        × 5
        ┌────────┐                 ┌────────┐
        │        ▼                 │        ▼
        1   =   2                  1   =   5
        ─       ─                  ─       ──
        2   =   4                  2       10
        │        ▲                 │        ▲
        └────────┘                 └────────┘
            × 2                        × 5

Multiply top and bottom by the same number.

When are two fractions equivalent?

Try this in your head

        1            ?
        ─    =      ──
        2           10

➤ Multiply top and bottom by ____ ?

Answer: × 5 → 5/10