How to Explain Multi-Digit Division to Fifth Graders

"How many times does 23 go into 87?"

This question is at the heart of why fifth-grade division feels so hard. Single-digit division is one thing—but two-digit divisors require estimation, trial and error, and genuine number sense.

Let's break down how to teach this challenging skill.

Why Multi-Digit Division Matters

Long division with multi-digit divisors is one of the most complex algorithms students learn in elementary school. It matters because:

• It builds estimation and number sense
• It prepares students for decimal division
• It develops perseverance and systematic thinking
• It's essential for middle school math and beyond

Understanding What Division Means

Before tackling the algorithm, ensure students understand what division asks:

"How many groups of 23 are in 851?"

Or equivalently:

"If I share 851 things equally among 23 people, how many does each person get?"

A Concrete Example

"You have 851 stickers to put in albums. Each album holds 23 stickers. How many albums do you need?"

851 ÷ 23 = ?

This real-world context helps students see that division answers meaningful questions.

The Standard Algorithm: Step by Step

Let's solve 851 ÷ 23.

The Setup

        ___
    23 ) 851

Step 1: Divide

"Does 23 go into 8?" No (23 > 8)
"Does 23 go into 85?" Yes! How many times?

Estimate: 23 is close to 20. How many 20s in 85? About 4.
Check: 23 × 4 = 92. Too big!
Try: 23 × 3 = 69. That works!

         3
        ___
    23 ) 851

Step 2: Multiply

3 × 23 = 69

         3
        ___
    23 ) 851
         69

Step 3: Subtract

85 - 69 = 16

         3
        ___
    23 ) 851
         69
         --
         16

Step 4: Bring Down

Bring down the 1 to make 161.

         3
        ___
    23 ) 851
         69
         --
         161

Step 5: Repeat

"How many 23s in 161?"

Estimate: How many 20s in 161? About 8.
Check: 23 × 8 = 184. Too big!
Try: 23 × 7 = 161. Perfect!

         37
        ___
    23 ) 851
         69
         --
         161
         161
         ---
           0

Answer: 851 ÷ 23 = 37

The Check

37 × 23 = 851 ✓

The Memory Aid: "Divide, Multiply, Subtract, Bring Down"

Some students use the phrase:

• Does
• McDonald's
• Sell
• Burgers?

Or: Dad, Mom, Sister, Brother

But understanding beats memorization. Each step has a purpose:

• Divide: How many groups?
• Multiply: That many groups gives us...
• Subtract: What's left over?
• Bring Down: Add the next piece to what's left

The Partial Quotients Method

This method makes the reasoning more visible:

Solve 851 ÷ 23:

    23 ) 851
       - 230    (23 × 10 = 230)    → 10
         ---
         621
       - 230    (23 × 10 = 230)    → 10
         ---
         391
       - 230    (23 × 10 = 230)    → 10
         ---
         161
       - 161    (23 × 7 = 161)     → 7
         ---
           0                       ----
                                    37

This method shows: "I took out 10 groups, then 10 more, then 10 more, then 7 more. Total: 37 groups."

It's less efficient but much clearer for understanding.

Estimation: The Secret Weapon

Why Estimation Matters

When dividing by 23, students must repeatedly answer "how many 23s in ___?" Good estimation makes this manageable.

Estimation Strategies

Round the divisor:

• 23 → 20 (easy to work with)
• "How many 20s in 85?" About 4. Try 4 or 3.

Round both numbers:

• 851 ÷ 23 ≈ 850 ÷ 25 = 34 (estimate)
• Or: 800 ÷ 20 = 40 (rough estimate)

Use multiplication facts:

• "I know 23 × 10 = 230 and 23 × 5 = 115"
• Use these as anchors

Estimation Before Calculating

"About how much is 7,452 ÷ 36?"

Think: 7,200 ÷ 36 = 200 (since 36 × 200 = 7,200)

So the answer is around 200. If a student gets 20 or 2,000, the estimate catches the error.

Handling Remainders

When Division Doesn't Come Out Even

Solve 875 ÷ 23:

         38 R1
        ____
    23 ) 875
         69
         --
         185
         184
         ---
           1

Answer: 875 ÷ 23 = 38 R1

Check: 38 × 23 + 1 = 874 + 1 = 875 ✓

Interpreting Remainders

The remainder's meaning depends on context:

"875 stickers in albums of 23"
→ 38 full albums, with 1 sticker left over

"875 students in buses holding 23"
→ Need 39 buses (38 full + 1 more for the remaining student)

"875 dollars shared among 23 people"
→ $38 each, with $1 left (or keep dividing for cents)

Visual Models

The Array/Area Model

Think of 851 ÷ 23 as: "What's the width if the area is 851 and the length is 23?"

         23
    +----------+
    |          |
    |   851    |  ? = 37
    |          |
    +----------+

The Number Line

Jump back from 851 in groups of 23:

851 → 828 → 805 → 782 → ... → 23 → 0

Count the jumps = 37

Hands-On Activities

Division with Play Money

Use play money to solve 851 ÷ 23:

• Start with 8 hundreds, 5 tens, 1 one
• Group into piles of $23
• Trade as needed
• Count the groups

The Estimation Game

Flash division problems. First person to estimate within 10% wins:

• 456 ÷ 12 ≈ ?
• 892 ÷ 28 ≈ ?
• 1,547 ÷ 42 ≈ ?

Create Division Stories

Write a word problem for 624 ÷ 24. Discuss what the remainder means in each story.

Error Analysis Challenge

Find the mistake:

         25
        ____
    34 ) 918
         68
         --
         238
         170
         ---
          68   ← Something's wrong!

(Error: 34 × 2 = 68, not 34 × 5 = 170 for the tens)

Common Mistakes and How to Fix Them

Mistake 1: Poor Estimation

Problem: Student tries 23 × 9 = 207 for "how many 23s in 85?"

Fix: Build estimation skills. "Is 85 more or less than 200? Less. So we need fewer than 9 groups."

Mistake 2: Forgetting to Bring Down

Wrong:

         3
        ___
    23 ) 851
         69
         --
         16   ← Stopped here!

Fix: Use the DMSB checklist. After subtracting, always ask "Is there another digit to bring down?"

Mistake 3: Quotient Digit in Wrong Place

Wrong:

        3 7
        ___
    23 ) 851   ← 3 should be over the 5, not the 8

Fix: The first quotient digit goes directly above the last digit you're dividing into. "23 into 85" → the answer goes over the 5.

Mistake 4: Subtraction Errors

Fix: Have students check each subtraction. Or use addition: "69 + ? = 85"

Mistake 5: Not Checking the Answer

Fix: Make checking mandatory. Quotient × Divisor + Remainder = Dividend. Every time.

Building Fluency

Start with Friendly Divisors

• Dividing by 10, 20, 30... (just place value)
• Dividing by 11, 12, 25 (familiar facts)
• Then move to "harder" two-digit divisors

Use Related Facts

If you know 23 × 4 = 92, then:

• 920 ÷ 23 = 40
• 92 ÷ 23 = 4
• 230 × 4 = 920

Practice in Context

Real problems build engagement:

• "A factory makes 5,376 widgets. They pack 24 per box. How many boxes?"
• "The trip is 1,825 miles. We drive about 65 miles per hour. About how many hours?"

Practice Ideas for Home

The Checking Habit

Give your child completed division problems. Their job: check each one and find any errors.

Mental Math Warm-Ups

• "What's 240 ÷ 20?"
• "What's 360 ÷ 12?"
• "About how many 25s are in 500?"

Real-Life Division

• Plan a party: "We have 150 pieces of candy for 24 guests. How many each?"
• Road trips: "285 miles left, traveling 60 mph. About how many hours?"

The Reasonableness Test

After solving, ask: "Does this answer make sense?"

• 851 ÷ 23 = 37. Is that reasonable?
• Well, 23 × 30 = 690 and 23 × 40 = 920. So 37 is between 30 and 40. ✓

Connecting to Future Concepts

Decimal Division

When dividing 8.51 ÷ 2.3, the same algorithm applies:

• Move decimals to make the divisor whole
• Divide normally
• Place the decimal point

Fractions and Division

851 ÷ 23 = 851/23

Understanding division as a fraction opens up simplifying, equivalent fractions, and more.

Algebra

Dividing polynomials uses a very similar process to long division!

The Bottom Line

Multi-digit division is genuinely hard. It requires:

• Strong multiplication facts
• Good estimation skills
• Careful organization
• Patience and persistence

But here's the thing: every struggle with "how many 23s in 161?" is building number sense. Every estimation is strengthening mathematical thinking.

The goal isn't robotic execution of an algorithm—it's developing the kind of flexible numerical reasoning that serves students for life.

So when your fifth grader groans at a long division problem, remind them: this is brain-building work. And with practice, it does get easier.