📋 Session plan (for teachers)

Session 5 — Multiplying fractions

Grade 5 · Math for Young Minds Total time: ~22 minutes Common Core: 5.NF.B.4 Today's idea: Multiplying by a fraction means scaling — taking a fraction's worth of something.

What students will be able to do

By the end of this session, the student can:

• Multiply a fraction by a whole number (like 4 × 2/3).
• Multiply two fractions (like 2/3 × 3/4) using an area picture.
• Explain why multiplying by a fraction less than 1 makes the answer smaller.

Materials

• Graph paper
• Colored pencils (at least 2 colors)
• Worksheet (one per student)
• Pencil

Substitution: No graph paper? Draw rectangles on plain paper. Any two colors of pencil, marker, or crayon will work — even pen and pencil for "shaded" vs "double-shaded".

New words

Heads-up — common confusions

• Students will be surprised that 2/3 × 3/4 is smaller than both 2/3 and 3/4. Name this out loud — it's the big idea.
• Some will multiply only the tops, or only the bottoms. Remind them: tops × tops, bottoms × bottoms.
• Some will reach for common denominators. Gently: "That's for adding. For multiplying, we don't need them."

Plan

1 · Hello & today's idea — 2 min

"Today we multiply fractions. Grown-ups do this every time they cook half a recipe or measure part of a board."

Write on the board: "Half of a half is…?"

Take a few guesses. Don't correct yet.

"By the end of today, you'll know — and you'll see why it's smaller, not bigger."

2 · Hands-on explore — 6 min

Pass out graph paper and colored pencils.

Prompt 1 — whole number times fraction:

"Draw 4 rectangles. Shade 2/3 of each one. How much did you shade in total?"

Let students work. Walk around. Listen for: "It's 2/3 four times" or "8 thirds."

Prompt 2 — fraction times fraction:

"Now draw one rectangle. Split it into 4 columns. Shade 3 of them in one color — that's 3/4."

"Now split the same rectangle into 3 rows. With a second color, shade 2 of the rows — that's 2/3 of the 3/4."

"Where the two colors overlap — that's the answer. Count those little boxes."

Pause everyone:

"How many tiny boxes are in the whole rectangle? How many are double-shaded?"

You're listening for 12 total, 6 double-shaded.

3 · Connect to the math — 4 min

Name what just happened.

"When we scale by a fraction, we take a fraction's worth of something. 2/3 × 3/4 means '2/3 of 3/4'."

Write on the board:

   2     3     2 × 3     6     1
   — ×   — =  ———————  = ── = ──
   3     4     3 × 4    12     2

"Multiply the tops. Multiply the bottoms. That's it. No common denominators."

Now circle the big idea:

"Look — 6/12 is smaller than 2/3, and smaller than 3/4. Multiplying by a fraction less than 1 always makes things smaller. You're taking a part of a part."

4 · Practice with support — 7 min

Pass out the worksheet.

Problem 1 (together): Compute 4 × 2/3 using a picture.

Draw 4 rectangles on the board. Shade 2/3 of each. Count thirds together: 8/3, which is 2 and 2/3.

Problem 2 (solo): 2/3 × 3/4. They already did the picture — now write the math. Answer: 6/12 = 1/2.

Problem 3 (solo): 1/2 × 1/2. Encourage the area picture: split a square in half one way, then in half the other way. Answer: 1/4.

Problem 4 (stretch): "A recipe needs 3/4 cup of sugar. You're making half a batch. How much sugar?"

If stuck, ask: "What does 'half a batch' mean we multiply by?" Answer: 1/2 × 3/4 = 3/8 cup.

Circulate. If a student forgets to multiply both tops and bottoms, point at the picture, not the rule.

5 · What we did + Try at home — 2 min

"Today you learned to scale by a fraction. Multiply tops, multiply bottoms. And a fraction of a fraction is smaller than either one."

Take-home:

"Tonight, try a real recipe — or pretend to cook. Halve a cookie recipe. Double a pancake recipe. Cut a salad in thirds. Multiply each fraction in the ingredients list."

Observation rubric — what to notice in this session

Use this during the session. One observation per student is plenty.

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

What's next (Session 6)

Building on this, Session 6 — Dividing fractions (intro) flips the surprise: when we divide by a fraction, the answer sometimes gets bigger than what we started with. We'll use pictures again to see why.

✏️ Worksheet (for students)

Math for Young Minds · Grade 5

Session 5 — Multiplying fractions

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

Today's big idea

To multiply fractions, multiply the tops, then multiply the bottoms. That's it.

When you scale by a fraction less than 1, the answer gets smaller. Weird but true!

Example we did together

Here's an area model for 1/2 × 1/3:

        ← 1/3 →
      ┌─────┬─────┬─────┐
  1/2 │ ▓▓▓ │     │     │
      ├─────┼─────┼─────┤
      │     │     │     │
      └─────┴─────┴─────┘

  The shaded piece is 1 out of 6 little boxes.

  1/2  ×  1/3  =  1/6

Top × top = 1. Bottom × bottom = 6. Done.

Problem 1 — together

Compute 4 × 2/3 using a picture.

Draw 4 rectangles, each split into thirds. Shade 2/3 of each one.

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

Count all the shaded thirds:

  4  ×  2/3  =  ____  =  ____
              (improper)  (mixed)

Problem 2 — on your own

Compute 2/3 × 3/4 with an area model.

Draw a rectangle. Split it into 3 rows and 4 columns. Shade 2 rows going one way and 3 columns going the other way. The overlap is your answer.

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

  2/3  ×  3/4  =  ____ / ____  =  ____

👀 Notice: the answer is smaller than 2/3 AND smaller than 3/4!

Problem 3 — on your own

Compute 1/2 × 1/2.

Draw a square. Split it in half one way, then in half the other way. Shade the overlap.

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

  1/2  ×  1/2  =  ____

Half of a half is a quarter. That's why scaling by 1/2 shrinks things.

Problem 4 — stretch

A recipe needs 3/4 cup of sugar. You're making half a batch.

How much sugar do you need?

        ← 3/4 cup →
      ┌─────┬─────┬─────┬─────┐
  1/2 │     │     │     │     │
      ├─────┼─────┼─────┼─────┤
  1/2 │     │     │     │     │
      └─────┴─────┴─────┴─────┘

Shade half of the 3/4. How many little boxes did you shade? How many little boxes are in one whole cup?

  1/2  ×  3/4  =  ____ / ____  =  ____  cup

Reminder: no common denominators needed. Just top × top, bottom × bottom.

Today's words

Try at home tonight

Find a real recipe in the kitchen (or pretend!). Pick one and multiply each fraction in the ingredient list:

• Halve a cookie recipe (multiply every amount by 1/2)
• Double a pancake recipe (multiply every amount by 2)
• Cut a salad in thirds (multiply every amount by 1/3)
• Make 1 and 1/2 batches of something fun

Write one ingredient's new amount here:

  ____  ×  ____  =  ____  of  ____________

Show a grown-up. Maybe even cook it.

Next time: Session 6 — Dividing fractions. Plot twist: dividing by a fraction can make the answer bigger. 🤯

🏠 Family guide (for parents)

Math for Young Minds · Grade 5 · Session 5

A note for grown-ups: today we multiplied fractions

What your child did today

In class today, we worked on multiplying fractions — the kind of math grown-ups actually use in kitchens and home projects.

The big idea: multiplying by a fraction means scaling something to a fraction's worth of its size.

We drew rectangles on graph paper, split them into rows and columns, and shaded the parts. That picture showed us that 2/3 × 3/4 is 6/12, which is 1/2. We also noticed something surprising: when you multiply by a fraction less than 1, the answer gets smaller, not bigger.

Why this matters

This is the math behind doubling a recipe, halving a paint mix, or scaling a budget. We're building the picture first — the rectangle, the rows, the columns — so the rule (multiply tops, multiply bottoms) feels like it makes sense, not magic. No timed tests. Understanding before speed.

🏠 Try this tonight (1 minute)

Pick a real recipe — or pretend. Halve it or double it together. For each fraction in the recipe, multiply.

Try one of these:

Short script:

• "The recipe needs 3/4 cup of sugar. We're making half. How much?"
• "Multiply the tops. Multiply the bottoms."
• "Is the answer smaller or bigger than what we started with? Why?"

One fraction is enough. You don't need to finish the recipe.

Words your child is learning

• Scale by a fraction — make something a fraction's worth of its original size. Halving a recipe is scaling by 1/2.

If your child says…

"This is easy." Good. Ask them why 2/3 × 3/4 ends up smaller than either piece you started with. That's the surprise worth sitting with.

"This is hard." Also good. Grab graph paper. Draw a rectangle, split it into rows and columns, shade the parts. The picture does the heavy lifting. The rule comes after the picture makes sense.

"I don't want to." Fair. Skip the worksheet feel. Open the fridge, find one container, and ask: "If we used half of this, how much would that be?" One question. Done. Math lives in those small moments.

A note on common mix-ups

You don't need common denominators to multiply fractions — that's only for adding. For multiplying, just multiply the tops, then multiply the bottoms. That's it.

What's next

In our next session, we'll start dividing fractions. There's another surprise waiting: dividing by a fraction sometimes makes the answer bigger than what you started with. We'll see why.

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

— Math for Young Minds

🔑 Cheat sheet (visual)

🔑 Multiplying fractions = scale by a fraction

Picture 1 — Whole × fraction: 4 × 2/3

  ┌───┬───┐  ┌───┬───┐  ┌───┬───┐  ┌───┬───┐
  │ ▓ │ ▓ │  │ ▓ │ ▓ │  │ ▓ │ ▓ │  │ ▓ │ ▓ │
  └───┴───┘  └───┴───┘  └───┴───┘  └───┴───┘
   2 thirds   2 thirds   2 thirds   2 thirds

   4  groups  of  2/3   =   8/3   =   2 2/3

Count the shaded pieces: 8 thirds. ✨

Picture 2 — Fraction × fraction: 2/3 × 3/4

Take 3/4 of a square. Then take 2/3 of that.

        ← 3/4 wide →
   ┌───┬───┬───┲━━━┓
   │ ▓ │ ▓ │ ▓ ┃   ┃  ↑
   ├───┼───┼───╂───┨  2/3
   │ ▓ │ ▓ │ ▓ ┃   ┃  tall
   ┝━━━┿━━━┿━━━╋━━━┩  ↓
   │   │   │   │   │
   └───┴───┴───┴───┘

   shaded: 6 little boxes
   total : 12 little boxes

     2     3       2 × 3      6      1
    ─── × ───  =  ───────  =  ──  =  ─
     3     4       3 × 4      12     2

😲 Smaller than both 2/3 and 3/4!

How to read the sign

       top × top
                     ┌──── multiply the tops
                     │
        2     3      ↓     2 × 3       6
       ─── × ───  =  ───────────  =  ────
        3     4      3 × 4       12
                     ↑
                     └──── multiply the bottoms
                          (no common denominator needed!)

Say it: "two-thirds of three-fourths." The word of = ×.

When does the answer shrink?

Multiplying by less than 1 → you get a part of the other number.

You try — 1/2 × 1/2

   ┌───┬───┐         ┌───┬───┐
   │ ▓ │ ▓ │   →     │ ▓ │   │
   ├───┼───┤         ├───┼───┤
   │   │   │         │   │   │
   └───┴───┘         └───┴───┘
   half shaded      half of that half

➤ 1/2 × 1/2 = ___

Answer: 1/4

🍰 Stretch — half a batch

Recipe needs 3/4 cup sugar. Make 1/2 batch.

    1     3       1 × 3       3
   ─── × ───  =  ───────  =  ───  cup
    2     4       2 × 4       8

➤ 3/8 cup of sugar 🥄

🏠 Take home: halve a cookie recipe · double pancakes · cut a salad in thirds. Multiply every fraction!

🔮 Next time: dividing by a fraction — sometimes the answer gets bigger!