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6.NS.B.2Common CoreMathThe Number SystemGrade 6

6.NS.B.2: Dividing Multi-Digit Numbers with the Standard Algorithm

In plain English: 6.NS.B.2 is the Common Core grade 6 math standard that asks students to divide multi-digit whole numbers quickly and accurately with the standard algorithm, often called long division. Students estimate each quotient digit, handle zeros and remainders, check answers with multiplication, and decide what a remainder means in a word problem.

Fluently divide multi-digit numbers using the standard algorithm.

Common Core State Standards for Mathematics · Domain: The Number System (NS) · Cluster: Compute fluently with multi-digit numbers and find common factors and multiples.
Also written as 6.NS.2 · Official standard

01

Lesson Plan

60-70 min

Overview

Students practice dividing multi-digit whole numbers with the standard algorithm, the written method often called long division. In 8,976 ÷ 34 = 264, the dividend is 8,976 (the number being divided), the divisor is 34 (the number you divide by), and the quotient is 264 (the answer). When the divisor does not go in evenly, the amount left over is the remainder, and it is always less than the divisor.

The standard asks for fluency: dividing accurately, in a reasonable time, and knowing why each step works. Students repeat four steps (divide, multiply, subtract, bring down), estimate each quotient digit by rounding the divisor, and fix a digit that is too big or too small. They learn why some quotients contain a zero, check every answer with divisor × quotient + remainder = dividend, and decide in word problems whether to round up, drop the remainder, or use the remainder as the answer. All numbers are whole numbers, with 2-digit and 3-digit divisors, as in grade 6. Dividing decimals is the next standard, 6.NS.B.3.

Learning Objectives

By the end of this lesson, students will be able to:

  • Divide multi-digit whole numbers by 2-digit and 3-digit divisors with the standard algorithm
  • Estimate each quotient digit by rounding the divisor, and adjust a digit that is too big or too small
  • Place quotient digits in the correct place, including zeros
  • Check a quotient with multiplication: divisor × quotient + remainder = dividend
  • Decide what a remainder means in a word problem

Prior Knowledge Required

Students should already be comfortable with:

  • Rounding whole numbers to any place 4.NBT.A.3
  • Dividing up to 4-digit numbers by 1-digit numbers with place value strategies 4.NBT.B.6
  • Interpreting remainders in word problems 4.OA.A.3
  • Multiplying multi-digit whole numbers with the standard algorithm 5.NBT.B.5
  • Dividing up to 4-digit numbers by 2-digit numbers with strategies such as area models 5.NBT.B.6

Lesson Procedure

60-70 minutes of class time across 5 phases.

  1. Warm-Up10 minutes

    Show the prompt. Students write an answer on mini whiteboards without dividing on paper, then share how they decided.

    Warm-Up Prompt

    "A school orders 4,512 pencils in boxes of 48. Is the number of boxes closer to 9, 90 or 900? How do you know without dividing?"

    The answer is about 90, because 48 × 100 = 4,800, which is a little more than 4,512. Some students will say 9 or 900 because they count digits instead of thinking about place value. The exact answer is 94 boxes, and 48 × 94 = 4,512. Tell students that today's method gives the exact answer, and an estimate like this one tells them whether the answer is sensible.

  2. Direct Instruction20-25 minutes

    Part 1: The four steps. Work 8,976 ÷ 34 with the class, as in Diagram 1. Look at the dividend from the left until the digits make a number at least as big as the divisor: 8 is too small, so start with 89. Then repeat these steps:

    1. Divide: estimate how many times the divisor fits. Round 34 to 30: 89 ÷ 30 is about 2. Write 2 above the 9, because 89 means 89 hundreds.
    2. Multiply: 2 × 34 = 68. Write it under 89.
    3. Subtract: 89 - 68 = 21. The difference must be less than the divisor. If it is not, the quotient digit is too small.
    4. Bring down the next digit of the dividend (7) to make 217, and divide again: 217 ÷ 34 is about 6, and 6 × 34 = 204. Then 217 - 204 = 13, bring down 6, and 136 ÷ 34 = 4 exactly. The quotient is 264.

    Part 2: What the digits mean. Diagram 2 shows the same division as a bar drawn to scale. The 2 in the hundreds place means 200 groups of 34 (6,800), the 6 means 60 groups (2,040), and the 4 means 4 groups (136). Together, 200 + 60 + 4 = 264 groups. This is the same idea as the area models students used in grade 5, written in a shorter form.

    • 2-digit divisor, no remainder

      Divide 8,976 by 34 (Diagram 1).

      Equation: 89 ÷ 34 → 2, 217 ÷ 34 → 6, 136 ÷ 34 → 4, so 8,976 ÷ 34 = 264. Check: 34 × 264 = 8,976

    • A zero in the quotient

      Divide 17,591 by 29.

      Equation: 175 ÷ 29 → 6 (174), remainder 1; bring down 9: 19 is less than 29, so write 0; bring down 1: 191 ÷ 29 → 6 (174), remainder 17. So 17,591 ÷ 29 = 606 R17. Check: 29 × 606 + 17 = 17,591

    • Adjusting an estimated digit

      Divide 1,862 by 38.

      Equation: Round 38 to 40: 186 ÷ 40 → 4, 4 × 38 = 152, remainder 34. Bring down 2: 342 ÷ 40 → 8, but 342 - 8 × 38 = 38, which is not less than 38, so the digit is 9: 9 × 38 = 342. So 1,862 ÷ 38 = 49

    • 3-digit divisor

      Divide 45,678 by 123.

      Equation: Round 123 to 120: 456 ÷ 123 → 3 (369), remainder 87; 877 ÷ 123 → 7 (861), remainder 16; 168 ÷ 123 → 1, remainder 45. So 45,678 ÷ 123 = 371 R45. Check: 123 × 371 + 45 = 45,678

    • Word problem: rounding up

      915 students are going on a field trip. Each bus holds 52 students. How many buses are needed?

      Equation: 915 ÷ 52 = 17 R31. Seventeen buses carry 884 students, and 31 students still need a seat, so the school needs 18 buses

    Part 3: Checking and remainders. Every division can be checked with multiplication: divisor × quotient + remainder = dividend. In a word problem, the remainder can mean three different things. Sometimes you round up (the extra students still need a bus). Sometimes you drop the remainder (only full boxes can be sold). Sometimes the remainder is the answer (how much is left over). Students can also write a remainder as a fraction of the divisor when the story allows equal parts, such as miles per day.

  3. Guided Practice15 minutes

    Pairs solve four problems on whiteboards. One partner says each step out loud (divide, multiply, subtract, bring down) while the other writes; they switch roles for each problem. Pairs check each answer with multiplication before they look at the key.

    Guided practice problems and answers
    ProblemAnswer
    1. 6,384 ÷ 56114 (56 × 114 = 6,384)
    2. 9,207 ÷ 31297 (31 × 297 = 9,207)
    3. 20,735 ÷ 47441 R8 (47 × 441 + 8 = 20,735)
    4. A factory packs 1,000 pencils in boxes of 24. How many full boxes, and how many pencils are left?1,000 ÷ 24 = 41 R16: 41 full boxes and 16 pencils left

    Listen for students who write a remainder that is bigger than the divisor, and ask them what the remainder says about the digit they chose. In Problem 2, the second step is 300 ÷ 31. Rounding 31 to 30 suggests 10, but a quotient digit is never more than 9: 9 × 31 = 279 fits, with 21 left.

  4. Independent Practice10-15 minutes

    Students solve five problems on their own and check each one with multiplication. (1) 7,452 ÷ 18 (414). (2) 12,006 ÷ 58 (207). (3) 39,000 ÷ 125 (312). (4) 5,000 ÷ 64 (78 R8). (5) A school's volunteers log 2,340 minutes of community service in one month. How many hours is that? (2,340 ÷ 60 = 39 hours.) Students who finish early time themselves on Problem 1 again and compare.

  5. Closure5 minutes

    Exit ticket: (1) Divide 3,588 ÷ 26 and check your answer. (138; 26 × 138 = 3,588.) (2) Sam says 4,230 ÷ 21 = 21 R9. Find the mistake and fix it. (After 42 ÷ 21 = 2, bringing down 3 gives 3, which is less than 21, so Sam must write 0 in the tens place. The answer is 201 R9, and 21 × 201 + 9 = 4,230.)

Differentiation Strategies

For Struggling Students

  • Give a multiples list for the divisor (1 × 34 through 9 × 34) so students choose each digit by comparing, not guessing
  • Use grid paper so every digit sits in its own column and zeros are not skipped
  • Start with 2-digit divisors and quotients without zeros, then add problems with a zero in the quotient

For Advanced Students

  • Ask for a division by a 2-digit number whose quotient has two zeros, and have a partner solve it
  • Ask students to explain, with place value, why a remainder must be less than the divisor
  • Give a 6-digit dividend and a 3-digit divisor, such as 506,184 ÷ 318, and ask for an estimate first and the exact quotient second

Assessment Guidance

What to Look For

Check that students start with the right partial dividend and write each quotient digit above the last digit they used. Watch for skipped zeros, remainders greater than or equal to the divisor, and subtraction slips. Ask students to estimate before dividing and to compare the estimate with the answer. Every answer should have a multiplication check. For word problems, listen for a reason when a student rounds up, drops the remainder, or gives the remainder as the answer.

02

Classroom Activities

3 Activities

1

Error Analysis Cards

15 minPairs

Pairs get 6 cards. Each card shows a finished division by an imaginary student. Pairs check each card with multiplication, decide whether it is right, and rewrite any wrong card with the correct steps and a note that names the mistake.

Error Cards (6 cards)

  • Card A: 7,812 ÷ 36 = 217
  • Card B: 6,090 ÷ 29 = 21
  • Card C: 4,875 ÷ 39 = 125
  • Card D: 3,250 ÷ 45 = 72 R10
  • Card E: 9,523 ÷ 42 = 225 R73
  • Card F: 8,142 ÷ 23 = 397 R11 (the card shows 81 - 69 = 22 in the first step)

Answer Key

  • Cards A, C and D are correct
  • Card B drops the final 0: the answer is 210
  • Card E has a remainder bigger than the divisor, so the last digit is too small: the answer is 226 R31
  • Card F has a subtraction error (81 - 69 = 12): the answer is 354

Discussion Questions

  • How many cards are correct? (Three: A, C and D)
  • Which card could you reject without multiplying at all? (Card E, because 73 is more than 42)
  • How does the multiplication check show the error on Card F? (23 × 397 + 11 = 9,142, not 8,142)

Modification for Distance Learning

Post the cards as images on a shared slide deck, one per slide. Pairs type the check and a one-sentence note under each card, then compare notes with another pair in a breakout room.

2

Long Division Relay

20 minGroups of 4

Each group gets 4 problem sheets. Each student owns one step of the algorithm (divide, multiply, subtract, bring down), does that step, and passes the sheet to the next student until the problem is finished. Groups race to finish all 4 sheets with correct checks.

Relay Problems (4 sheets)

  • Sheet 1: 5,688 ÷ 24 (237)
  • Sheet 2: 30,015 ÷ 87 (345)
  • Sheet 3: 14,448 ÷ 172 (84)
  • Sheet 4: 61,250 ÷ 245 (250)

Procedure

  • Before starting, the group writes an estimate at the top of each sheet (for Sheet 1, 6,000 ÷ 25 = 240)
  • Pass the sheet after each step; the next student may say "stop" and fix an error before doing a step
  • When a sheet is finished, the whole group checks it with multiplication and compares it with the estimate
  • Rotate the step roles for each new sheet

Discussion Questions

  • Which sheet has a 0 in its quotient? (Only Sheet 4)
  • On Sheet 3, why does the quotient have only 2 digits? (144 is less than 172, so the first digit goes above the 4 in the tens place)
  • Which step caused the most "stop" calls in your group?

Challenge Variation

Each group writes a relay sheet of its own with a 5-digit dividend and a 3-digit divisor, makes sure the quotient has a zero in it, and trades with another group.

3

What Does the Remainder Mean?

15 minPairs

Pairs solve 4 story cards with the standard algorithm. The quotients all have remainders, and each story needs a different answer: round up, drop the remainder, give the remainder, or write the remainder as a fraction.

Story Cards (4 cards)

  • Card 1: A bakery packs 1,450 cookies in boxes of 36. How many full boxes can it sell? (40 R10, so 40 full boxes)
  • Card 2: 638 people ride vans that hold 15 people each. How many vans are needed? (42 R8, so 43 vans)
  • Card 3: A 1,000-foot roll of twine is cut into 48-foot pieces. How many feet are left over? (20 R40, so 40 feet)
  • Card 4: A family drives 1,125 miles in 12 days, the same distance each day. How far do they drive each day? (93 R9, so 93 9/12 = 93 3/4 miles)

Procedure

  • Divide and check with multiplication
  • Circle the quotient, the remainder, or both, depending on what the story asks
  • Write one sentence that answers the question with a unit

Discussion Questions

  • Which card is answered by rounding up? (Only Card 2)
  • In which card is the remainder itself the answer? (Card 3)
  • Why can Card 4 use a fraction, but Card 2 cannot?

03

Diagrams & Visual Aids

2 diagrams

Diagram 1: The Standard Algorithm, Step by Step

The standard algorithm for 8,976 ÷ 34 2 6 4 34 8 9 7 6 6 8 - 2 1 7 2 0 4 - 1 3 6 1 3 6 - 0 89 ÷ 34 → 2 (2 × 34 = 68), subtract: 21 the 2 is in the hundreds place: 200 groups of 34 217 ÷ 34 → 6 (6 × 34 = 204), subtract: 13 the 6 is in the tens place: 60 groups of 34 136 ÷ 34 → 4 (4 × 34 = 136), subtract: 0 the 4 is in the ones place: 4 groups of 34 Remainder 0, so 8,976 ÷ 34 = 264 Check: 34 × 264 = 8,976
8,976 ÷ 34 worked with the standard algorithm. Each quotient digit sits above the last digit of the number it divides: 89 hundreds, 217 tens and 136 ones. The notes show the divide, multiply and subtract steps, and what each digit means. The remainder is 0, and 34 × 264 = 8,976 checks the answer.

Diagram 2: What the Quotient Digits Mean

What the quotient digits mean: 8,976 split into groups of 34 Bar drawn to scale: each part is a number of groups of 34 200 × 34 = 6,800 60 × 34 = 2,040 4 × 34 = 136 0 6,800 8,976 200 + 60 + 4 = 264 groups of 34, with nothing left over
A bar drawn to scale for 8,976. It splits into 200 groups of 34 (6,800), 60 groups of 34 (2,040) and 4 groups of 34 (136). The quotient digits 2, 6 and 4 count these groups in the hundreds, tens and ones places.

04

Homework Assignment

~30 min

6.NS.B.2 Homework: Long Division Practice

Directions: Use the standard algorithm for every problem. Write an estimate first, show every step, and check each answer with divisor × quotient + remainder = dividend. Answer word problems in a sentence with a unit.

Part 1: Divide and Check (Problems 1-3)

  1. 8,265 ÷ 15
  2. 23,712 ÷ 38
  3. 50,418 ÷ 67

Part 2: Zeros and 3-Digit Divisors (Problems 4-5)

  1. Divide 27,880 by 136. Explain how you knew where to write the first digit of the quotient.
  2. Dani says 16,245 ÷ 54 = 3 R45. Estimate the quotient to show that Dani's answer is too small, find the mistake, and give the correct answer with a check.

Part 3: Word Problems (Problem 6)

  1. (a) A stadium section has 1,876 seats in rows of 28 seats. How many rows does it have? (b) Volunteers pack 2,500 flyers in stacks of 75. How many full stacks can they make, and how many flyers are left over?

Rubric

CriterionFull Credit (2 pts)Partial Credit (1 pt)No Credit (0 pts)
AccuracyAll quotients and remainders correctOne quotient or remainder wrongTwo or more wrong
Steps and Place ValueEvery step shown, digits in the right places, zeros includedSteps shown with one place value slipSteps missing
Estimates and ChecksEstimate and multiplication check for every problemSome estimates or checks missingNo estimates or checks
Word ProblemsRemainder used correctly, answer in a sentence with a unitCorrect division but remainder misused or unit missingWrong division

05

Quiz: 20 Questions

Interactive, with answers

Instructions

Use scrap paper for the standard algorithm and estimate before you divide. Your score updates after each answer, and Reset quiz clears the page for another try.

Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.

0 of 20 answered · 0 correct

  1. Question 1 of 20 · Multiple Choice

    What is 7,344 ÷ 36?

  2. Question 2 of 20 · Multiple Choice

    Which is the best estimate for 6,213 ÷ 29?

  3. Question 3 of 20 · Multiple Choice

    Which of these CANNOT be the remainder when a whole number is divided by 47?

  4. Question 4 of 20 · Multiple Choice

    What is 5,106 ÷ 23?

  5. Question 5 of 20 · Multiple Choice

    A student finds 3,917 ÷ 52 = 75 R17. Which expression should equal 3,917 if the answer is right?

  6. Question 6 of 20 · Multiple Choice

    A farm packs 4,380 eggs into cartons of 12. How many cartons does it fill?

  7. Question 7 of 20 · Multiple Choice

    A sports banquet expects 430 guests. Each table seats 12 people. How many tables are needed so every guest has a seat?

  8. Question 8 of 20 · Multiple Choice

    A pancake recipe uses 180 grams of flour per batch. A bag holds 2,500 grams. After making as many full batches as possible, how much flour is left?

  9. Question 9 of 20 · Multiple Choice

    What is 38,025 ÷ 125?

  10. Question 10 of 20 · Multiple Choice

    In 4,968 ÷ 54, above which digit of the dividend should the first digit of the quotient go?

  11. Question 11 of 20 · Multiple Choice

    In 2,730 ÷ 42, the first step is 273 ÷ 42. What digit goes in the quotient for this step?

  12. Question 12 of 20 · Multiple Choice

    A school spends $9,360 on 72 desks that all cost the same. What is the price of one desk?

  13. Question 13 of 20 · Multiple Choice

    Which division has a 0 in its quotient?

  14. Question 14 of 20 · Multiple Choice

    What is 91,512 ÷ 246?

  15. Question 15 of 20 · Short Answer

    Divide 9,648 by 72. Show your steps and a check.

  16. Question 16 of 20 · Short Answer

    Estimate 31,684 ÷ 89, then find the exact quotient and check it.

  17. Question 17 of 20 · Short Answer

    Divide 48,275 by 305. Give the quotient and the remainder, and check your answer.

  18. Question 18 of 20 · Short Answer

    Luis says 12,642 ÷ 21 = 62. Explain his mistake and give the correct quotient.

  19. Question 19 of 20 · Short Answer

    A zoo expects 1,230 visitors on a school day. Each tour group can have at most 35 visitors. How many tour groups are needed so every visitor is in a group?

  20. Question 20 of 20 · Short Answer

    A charity shares $2,475 equally among 18 families. How much money does each family get? Write the remainder as a fraction of a dollar.

0 of 20 answered · 0 correct

06

Frequently Asked Questions

10 Questions

What does 6.NS.B.2 mean?

6.NS.B.2 means students can divide multi-digit whole numbers accurately and efficiently with the standard algorithm, or long division. That includes 2-digit and 3-digit divisors, quotients with zeros, and remainders. "Fluently" means students divide correctly, in a reasonable time, and can explain why each step works.

What is the standard algorithm for division?

The standard algorithm is the written long division method. Students divide the first part of the dividend that is big enough, multiply the quotient digit by the divisor, subtract, bring down the next digit, and repeat until no digits are left. Each quotient digit goes above the last digit it used, which keeps the place values right.

Is 6.NS.B.2 a grade 5 or grade 6 standard?

It is a grade 6 standard. In grade 5 (5.NBT.B.6), students divide up to 4-digit numbers by 2-digit numbers with strategies such as area models and partial quotients. Grade 6 is the first grade where the standard algorithm is required and fluency is expected.

How do students estimate each quotient digit?

Round the divisor to the nearest ten or hundred and divide mentally. For 2,236 ÷ 43, round 43 to 40: 223 ÷ 40 is about 5, and 5 × 43 = 215 fits. If the product is too big, try one less. If the difference after subtracting is not less than the divisor, try one more.

Why is there sometimes a zero in the quotient?

A zero goes in the quotient when the number after bringing down a digit is still smaller than the divisor. For 9,690 ÷ 19, the steps give 96 ÷ 19 → 5 and 19 ÷ 19 → 1, and then 0 is less than 19, so the ones digit is 0: the quotient is 510. Skipping that zero gives 51, which is 10 times too small.

What do you do with a remainder in a word problem?

It depends on the question. Round up when everyone or everything must be included, such as buses for students. Drop the remainder when only full groups count, such as full boxes to sell. Give the remainder when the question asks what is left over. When the story allows equal parts, such as money or distance, the remainder can be written as a fraction of the divisor.

How can students check a division answer?

Multiply the divisor by the quotient and add the remainder. The result must equal the dividend. An estimate is a second check: if the estimate is about 300 and the answer is 35, a digit is missing or in the wrong place.

What are common mistakes in long division?

A common mistake is skipping a zero in the quotient. Others are writing the first digit in the wrong place, choosing a digit that leaves a remainder bigger than the divisor, subtraction slips, and forgetting to bring down the last digit. A multiplication check catches all of them, and grid paper helps students keep digits in columns.

Do students still need long division if they have calculators?

Yes. The algorithm builds number sense about place value and estimation, and students use it to check whether a calculator answer is reasonable. It is also the method students use in grade 7 to write fractions as decimals, and later to divide polynomials in high school.

How does 6.NS.B.2 connect to later math?

6.NS.B.2 leads straight into dividing decimals (6.NS.B.3), which uses the same steps. In grade 7, students use long division to turn fractions into decimals and see that the decimals end or repeat (7.NS.A.2). In Algebra II, polynomial long division follows the same divide, multiply, subtract and bring down pattern (HSA.APR.D.6).