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Teaching unit · Grade 5 (ages 10 to 11)

Long division

Sharing by place value, the Divide-Multiply-Subtract-Bring down cycle, remainders, and a first two-digit divisor

About four to five lessons of 45 to 60 minutes

Start here · hook

How do you share a big prize so nobody feels cheated?

Your team wins a jar of 96 tokens at the fair, and there are 4 of you. Nobody wants to count them out one at a time into four piles like little kids. There is a faster, fair way to split any pile, however big, and it works even when the number is awkward and something is left over.

That method is long division. It looks like a tall staircase of numbers, but underneath it is just careful sharing: deal out the big place values first, swap what is left over down to the next place, and keep going. Master it here and you can fairly split 96, or 852, or a bill among friends, without ever losing track.

Learning objective

What students will be able to do

Students will divide up to three-digit numbers by a one-digit divisor using the standard long-division algorithm, understand each step as place-value sharing, interpret and record a remainder, and extend the method to a friendly two-digit divisor.

Success criteria
  • I can share a number by dealing out the tens first, then the ones.
  • I can carry out the Divide, Multiply, Subtract, Bring down cycle in order.
  • I can line up each digit of the quotient over the right place.
  • I can write and explain a remainder, and check it is smaller than the divisor.
  • I can divide by a friendly two-digit number such as 12.
Curriculum anchor

Standards this unit teaches

  • 4.NBT.B.6Common Core (US)
    Divide by a one-digit number

    Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and the relationship between multiplication and division, and illustrate and explain the calculation with equations, rectangular arrays or area models.

  • 5.NBT.B.6Common Core (US)
    Divide by a two-digit number

    Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and the relationship between multiplication and division, and illustrate and explain the calculation with equations, rectangular arrays or area models.

  • AC9M5N09Australian Curriculum v9 (ACARA)
    Model practical problems (Year 5)

    Use mathematical modelling to solve practical problems involving additive and multiplicative situations, including money, by choosing operations and efficient calculation strategies and checking that the result is reasonable.

  • AC9M6N06Australian Curriculum v9 (ACARA)
    Division and interpreting remainders (Year 6)

    Solve problems that require division, using efficient mental and written strategies, and interpret any remainder according to the context, expressing the result as a whole number, a fraction or a decimal. Australia introduces the formal recorded division method here, so this US Grade 5 unit reaches toward the Year 6 descriptor.

Before you start

Prior knowledge

This unit builds on skills students should already have met. Revisit any that are shaky first.

Key vocabulary

Words to teach and display

Dividend
the number being shared out, the total you start with
Divisor
the number you are dividing by, how many groups or how big each group is
Quotient
the answer to a division, how much each group gets
Remainder
what is left over when the sharing does not come out even
Regroup
swap a leftover from one place value down to the next, such as 1 ten becoming 10 ones
Long division
a written method that shares a number one place-value column at a time
Teaching sequence

Teach it: concrete, pictorial, abstract

The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.

1. Division is fair sharing

Concrete

Start with the meaning, not the method. Put out 96 counters and share them equally among 4 cups. When the sharing is done, each cup holds 24. Long division is only a faster, written way to do exactly this: split the whole into equal groups without dealing one at a time.

Name the parts as you go. The 96 you start with is the dividend, the 4 groups is the divisor, and the 24 in each cup is the quotient. Long division finds the quotient quickly by sharing whole place-value columns at once instead of single counters.

9624friend 124friend 224friend 324friend 4
96 shared equally into 4 groups is 24 in each group. That 24 is the quotient.
Check for understanding, ask
  • In 96 divided by 4, which number is the dividend and which is the divisor?
  • If we shared 96 into 3 groups instead, would each group get more or fewer?

2. Share the tens first, then the ones

Pictorial

The trick that makes long division fast is to share by place value. For 96 divided by 4, do not deal ones. Deal the tens first. 96 is 9 tens and 6 ones. Share the 9 tens among 4: each group gets 2 tens (that is 8 tens used), and 1 ten is left over.

That leftover ten cannot stay a ten when you are sharing into 4, so regroup it: 1 ten becomes 10 ones, and with the 6 ones already there that makes 16 ones. Share 16 ones among 4: each group gets 4. So each group has 2 tens and 4 ones, which is 24.

Splitting 96 into a part that shares evenly (80) and the part that carries down (16) is the whole idea of the written method. The picture below shows the two chunks that each divide by 4 cleanly: 80 gives 20 each and 16 gives 4 each, for 24.

96808 tens, shares to 20 each161 ten + 6 ones = 16, shares to 4 each
96 = 80 + 16. Share each chunk by 4: 20 each plus 4 each is 24 each.
Check for understanding, ask
  • Why can the leftover ten not stay a ten when we share into 4?
  • What does the 1 leftover ten become when we regroup it?

3. The written method: Divide, Multiply, Subtract, Bring down

Abstract

Now record that sharing as the standard algorithm. It repeats one four-step cycle for each digit, working left to right: Divide, Multiply, Subtract, Bring down. Some classes remember the order with a phrase, but the meaning is the place-value sharing from step 2.

Set out 96 divided by 4 with 4 outside and 96 inside. Cycle one, on the tens digit 9: Divide (4 into 9 is 2), Multiply (2 times 4 is 8), Subtract (9 minus 8 is 1), Bring down (the 6, making 16). Cycle two, on the 16: Divide (4 into 16 is 4), Multiply (4 times 4 is 16), Subtract (16 minus 16 is 0), nothing left to bring down. The answer sits on top: 24.

Worked example

Work out 852 divided by 6 using long division.

  1. Cycle 1, hundreds digit 8: 6 into 8 is 1, write 1 above the 8. Multiply 1 times 6 is 6. Subtract 8 minus 6 is 2. Bring down the 5 to make 25.
  2. Cycle 2, the 25: 6 into 25 is 4, write 4 above the 5. Multiply 4 times 6 is 24. Subtract 25 minus 24 is 1. Bring down the 2 to make 12.
  3. Cycle 3, the 12: 6 into 12 is 2, write 2 above the 2. Multiply 2 times 6 is 12. Subtract 12 minus 12 is 0. Nothing left to bring down.
  4. Check by multiplying back: 142 times 6 is 852.

Answer: 852 divided by 6 is 142.

Check for understanding, ask
  • What are the four steps of the cycle, in order?
  • After you subtract, what do you always do next while digits remain?
  • How can you check a division answer using multiplication?

4. When it does not come out even: remainders

Abstract

Not every share is exact. Divide 74 by 5. Share the 7 tens: 5 into 7 is 1 ten (5 used), 2 tens left. Regroup to make 24 ones, and 5 into 24 is 4 (20 used), with 4 ones left that cannot be shared into 5 groups. That leftover 4 is the remainder.

Write it as 74 divided by 5 is 14 remainder 4. The golden rule: a remainder must always be smaller than the divisor. If your leftover is 5 or more when dividing by 5, the last quotient digit was too small, so go back and make it bigger.

The bar below shows the fair share: five equal groups of 14, and 4 tokens that will not split evenly into the 5 groups.

Worked example

Work out 74 divided by 5.

  1. Tens: 5 into 7 is 1, write 1. Multiply 1 times 5 is 5. Subtract 7 minus 5 is 2. Bring down the 4 to make 24.
  2. Ones: 5 into 24 is 4, write 4. Multiply 4 times 5 is 20. Subtract 24 minus 20 is 4.
  3. No digits left to bring down, and 4 is smaller than 5, so 4 is the remainder.
7414141414144left over
Five equal groups of 14 use 70, and 4 are left over. So 74 divided by 5 is 14 remainder 4.

Answer: 74 divided by 5 is 14 remainder 4.

Check for understanding, ask
  • Why must a remainder always be smaller than the divisor?
  • If dividing by 6 leaves you with 8, what went wrong?

5. A first two-digit divisor

Abstract

The method does not change when the divisor has two digits, you just lean on the times table of the divisor. Try 156 divided by 12. The 1 alone is smaller than 12, so start with 15: 12 into 15 is 1, and carry on from there.

It helps to think in chunks that 12 divides cleanly. 156 is 120 plus 36. 120 divided by 12 is 10, and 36 divided by 12 is 3, so the answer is 13. The written algorithm reaches the same 13, one digit at a time.

Worked example

Work out 156 divided by 12.

  1. Look at 15 (the 1 alone is too small): 12 into 15 is 1, write 1 above the 5. Multiply 1 times 12 is 12. Subtract 15 minus 12 is 3. Bring down the 6 to make 36.
  2. Now 36: 12 into 36 is 3, write 3 above the 6. Multiply 3 times 12 is 36. Subtract 36 minus 36 is 0.
  3. Check: 13 times 12 is 156.
156120120 divided by 12 = 103636 divided by 12 = 3
156 = 120 + 36. Divide each chunk by 12: 10 plus 3 is 13.

Answer: 156 divided by 12 is 13.

Check for understanding, ask
  • Why do we start with 15 and not with the 1 in 156?
  • Break 288 into two friendly chunks that each divide by 12.
Watch for

Common misconceptions and how to address them

MisconceptionYou always divide the bigger digit by the smaller one, so 4 into 3 becomes 3 into 4.

Why it happens: Students remember that division makes things smaller and flip any awkward step to keep the big number on top.

How to address it: Keep the divisor fixed on the outside. If it will not go (4 into 3), the quotient digit is 0 for that place and you bring down the next digit. The order of a division is fixed by the problem, it is not chosen to be convenient.

MisconceptionIf a place does not divide, just skip it and carry on.

Why it happens: Students only write a quotient digit when the step feels tidy, so they drop the 0 and the answer loses a place value.

How to address it: Every digit you bring down earns a digit on top, even a 0. In 618 divided by 6, the tens step is 6 into 1, which is 0, so a 0 must be written: the answer is 103, not 13.

618600600 divided by 6 = 1001818 divided by 6 = 3
618 = 600 + 18. The tens place gives nothing, so the quotient is 103, with a 0 in the tens place.

MisconceptionThe remainder can be any size, so 74 divided by 5 is 13 remainder 9.

Why it happens: Students stop dividing one step early and dump everything left into the remainder.

How to address it: A remainder must be smaller than the divisor. A leftover of 9 when dividing by 5 still holds another group of 5, so share it: the quotient was one too small. The correct answer is 14 remainder 4.

MisconceptionIt does not matter which column the quotient digit sits above.

Why it happens: Students treat the answer as a loose string of digits rather than a place-value number.

How to address it: Each quotient digit lines up above the last digit you brought down. Keeping the columns straight, on grid paper if needed, is what turns 24 into twenty-four rather than a jumble.

MisconceptionThe remainder is just an extra bit that means nothing.

Why it happens: Students record remainder 4 as a leftover symbol without asking what it stands for.

How to address it: The remainder is real leftover stuff, and the context decides what to do with it. Sharing 74 cookies into 5 boxes leaves 4 cookies over; needing 5 cookies per party bag from 74 means you can fill 14 bags with 4 to spare.

Do it together

Guided practice (with answers)

  1. 1. Work out 84 divided by 4.

    Answer: 21. Tens: 4 into 8 is 2. Ones: 4 into 4 is 1. So 21.

  2. 2. Work out 96 divided by 3.

    Answer: 32. Tens: 3 into 9 is 3. Ones: 3 into 6 is 2. So 32.

  3. 3. Work out 90 divided by 4.

    Answer: 22 remainder 2. Four 22s make 88, and 2 are left, which is smaller than 4.

  4. 4. Work out 738 divided by 6.

    Answer: 123. Hundreds: 6 into 7 is 1 remainder 1, bring down to 13; 6 into 13 is 2 remainder 1, bring down to 18; 6 into 18 is 3.

  5. 5. Work out 156 divided by 12.

    Answer: 13. Start with 15: 12 into 15 is 1 remainder 3, bring down to 36; 12 into 36 is 3.

  6. 6. A shop packs 125 pencils into boxes of 10. How many full boxes, and how many pencils are left over?

    Answer: 12 full boxes with 5 pencils left over, because 125 divided by 10 is 12 remainder 5.

On their own

Independent practice worksheets

Set the matching ChalkBee division worksheets for independent work. The answer keys are computed in code, so they are never wrong. Start with exact one-digit-divisor divisions, then bring in remainders, then word problems where the remainder must be read in context.

Grade 5 division worksheet (Set A)long division by one-digit divisorsGrade 5 division worksheet (Set B)more division practice, including remaindersGrade 4 division worksheet (Set A)short division facts, the fluency long division rests onGrade 5 word problems worksheet (Set A)problems where the remainder must be interpretedAll division worksheetsthe full set, six sets per grade
Reach every student

Differentiation

Support
  • Keep sharing counters into cups alongside the written steps so each line of the algorithm has a concrete meaning.
  • Give a multiplication facts strip for the divisor (the 4 times table for divide-by-4) so the Divide step is a lookup, not a struggle.
  • Use grid or squared paper so the place-value columns stay lined up.
  • Start with dividends whose every place divides exactly (like 84 divided by 4) before introducing regrouping and remainders.
Extension
  • Divide four-digit dividends, and dividends where a middle place gives a 0 in the quotient.
  • Interpret remainders in context, deciding when to round the quotient up, down, or express the remainder as a fraction.
  • Divide by other friendly two-digit divisors (11, 15, 20, 25) using chunking to check the algorithm.
  • Bridge to Year 6 (AC9M6N06) by writing a remainder as a fraction of the divisor, such as 74 divided by 5 is 14 and 4 fifths.
Check it stuck

Assessment: exit ticket

A three-question exit ticket for the last five minutes, sampling an exact division, one with a remainder, and a word problem.

  1. 1. Work out 78 divided by 3.

    Answer: 26. Tens: 3 into 7 is 2 remainder 1, bring down to 18; 3 into 18 is 6. Check 26 times 3 is 78.

  2. 2. Work out 85 divided by 4.

    Answer: 21 remainder 1. Four 21s make 84, and 1 is left over.

  3. 3. Four friends share a $92 restaurant bill equally. How much does each pay?

    Answer: $23 each, because 92 divided by 4 is 23.

For the teacher

Teacher notes and timings

  • Rough timing across four to five lessons: Lesson 1 sharing and place-value sharing (sections 1 to 2), Lesson 2 the written cycle (section 3), Lesson 3 remainders (section 4), Lesson 4 a two-digit divisor (section 5), Lesson 5 mixed practice and the exit ticket.
  • This unit assumes fluent times tables and short division. If the Divide step is slow, the whole algorithm feels hard, so revisit multiplication facts first.
  • Language to keep saying: share the tens first, regroup the leftover, bring down the next digit, the remainder is smaller than the divisor. These pre-empt most of the misconceptions.
  • Insist that every brought-down digit earns a digit on top, including a 0, and that the columns stay straight. Most long-division errors are place-value slips, not arithmetic slips.
  • Curriculum note and a US and AU divergence: the US teaches the recorded long-division algorithm across Grade 4 (one-digit divisors, 4.NBT.B.6) and Grade 5 (two-digit divisors, 5.NBT.B.6). ACARA does not mandate the formal algorithm at Year 5; it emphasises efficient strategies and modelling (AC9M5N09) and introduces recorded division with interpreted remainders at Year 6 (AC9M6N06). So this unit maps to Australian Years 5 and 6.
  • Present mode and print both work: use the Print button for a clean teacher copy or a student handout, and project the page to teach straight from the diagrams.
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