Dividing multidigit numbers and operating with decimals
The standard division algorithm with a multidigit divisor, and fluent add, subtract, multiply and divide with decimals
About four lessons of 45 to 60 minutes
A two-digit divisor, and a decimal point that will not sit still
By now you can divide by a single digit and add simple decimals. Grade 6 pushes both skills further: dividing by a two-digit number such as 24, and confidently adding, subtracting, multiplying and dividing decimals of any size, not just tenths and hundredths.
Both skills lean on habits you already trust. Long division with a bigger divisor is still Divide, Multiply, Subtract, Bring down, just leaning on bigger times-table facts. Decimal operations are still the same four operations you already know, with one extra job: keeping the decimal point exactly where place value says it belongs.
- 4,536 raffle tickets packed 24 to a box4,536 divided by 24 is 189 full boxes
- $12.45 plus $7.80line up the decimal points: $20.25
- $15.20 minus $6.75 change$8.45 change
- $1.20 per litre, 7.8 litres of fuel$9.36 divided by $1.20 per litre checks the pump reading: 7.8 litres
What students will be able to do
Students will fluently divide multidigit whole numbers, including by a two-digit divisor, using the standard long-division algorithm, and will fluently add, subtract, multiply, and divide multidigit decimals using the standard written algorithms, correctly placing the decimal point in every result.
- I can divide a multidigit number by a two-digit divisor using long division.
- I can add and subtract decimals by lining up the decimal points.
- I can multiply decimals and place the decimal point using the total number of decimal places.
- I can divide a decimal by a decimal by first making the divisor a whole number.
- I can check a decimal answer is reasonable by estimating first.
Standards this unit teaches
- 6.NS.B.2Common Core (US)Divide multidigit numbers
Divide multidigit numbers fluently using the standard algorithm.
- 6.NS.B.3Common Core (US)Operate with decimals fluently
Add, subtract, multiply, and divide multidigit decimals fluently using the standard algorithms.
- AC9M6N06Australian Curriculum v9 (ACARA)Division with remainders (Year 6)
Solve division problems with efficient strategies, interpreting remainders by context and expressing results as whole numbers, decimals or fractions.
- AC9M7N06Australian Curriculum v9 (ACARA)Four operations with rationals (Year 7)
Use the four operations with positive fractions, decimals and percentages, choosing efficient strategies to solve problems. Australia's descriptor for full fluency across all four decimal operations sits at Year 7, so the decimal half of this unit runs about a year ahead of the ACARA placement.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
- Grade 5 long division teaching unitthe Divide-Multiply-Subtract-Bring down cycle this unit extends to bigger divisors
- Grade 5 decimals: tenths and hundredths teaching unitreading decimal place value, essential for placing the decimal point correctly
- Grade 5 adding and subtracting decimals teaching unitlining up decimal points, extended here to multiplying and dividing
- How to teach long divisionthe written method this unit scales up to a two-digit divisor
Words to teach and display
- Standard algorithm
- the fixed, reliable written method for an operation, such as long division or column multiplication
- Divisor
- the number you are dividing by
- Decimal point
- the dot separating the whole-number part of a number from its fraction part
- Aligning decimal points
- lining up numbers so their decimal points sit in the same column, so digits of the same place value add or subtract correctly
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. Long division with a two-digit divisor
AbstractThe Divide, Multiply, Subtract, Bring down cycle does not change with a bigger divisor, but each Divide step now needs an estimate against a two-digit number instead of a single times table. For 4,536 divided by 24, look for how many 24s fit into each chunk of the dividend, checking with a quick multiplication.
Break the dividend into friendly chunks first as a sanity check: 4,536 is close to 4,800 (24 x 200) and 4,320 (24 x 180), so the answer should land somewhere near 190. That estimate protects against a wildly wrong quotient digit before the written algorithm even starts.
Work out 4,536 divided by 24 using long division.
- 45 divided by 24: 24 into 45 is 1 (24 x 1 = 24). Subtract: 45 - 24 = 21. Bring down the 3 to make 213.
- 213 divided by 24: 24 x 8 = 192, 24 x 9 = 216 (too big), so 24 into 213 is 8. Subtract: 213 - 192 = 21. Bring down the 6 to make 216.
- 216 divided by 24: 24 x 9 = 216 exactly. Subtract: 216 - 216 = 0.
- Check by multiplying back: 189 x 24 = 4,536.
Answer: 4,536 divided by 24 = 189.
- Why is checking 24 x 8 and 24 x 9 useful before writing the quotient digit for 213 divided by 24?
- How can multiplying the quotient back by the divisor check a long-division answer?
2. Adding and subtracting decimals: line up the point
AbstractDecimal addition and subtraction use the exact column method you already know for whole numbers, with one extra rule: line up the decimal points first, so every column holds digits of the same place value. Pad missing decimal places with zeros so every number has the same number of decimal digits before you start.
$12.45 + $7.80: line up the decimal points (12.45 and 7.80 both have two decimal places already). Add column by column, right to left, exactly like whole-number addition: 12.45 + 7.80 = 20.25.
Find $15.20 minus $6.75.
- Line up the decimal points: 15.20 and 6.75.
- Subtract column by column, right to left, regrouping where needed: 15.20 - 6.75.
- The hundredths column: 0 - 5 needs regrouping from the tenths. Working through the columns gives 8.45.
Answer: $15.20 - $6.75 = $8.45.
- Why must the decimal points be lined up before adding or subtracting?
- What do you do if one decimal has fewer digits after the point than the other?
3. Multiplying and dividing decimals
AbstractTo multiply decimals, ignore the decimal points at first and multiply the digits as whole numbers, then count the total decimal places in both factors and place the point that many places from the right in the answer. 3.2 x 1.5: multiply 32 x 15 = 480. Both factors together have 2 decimal places (1 from 3.2, 1 from 1.5), so the answer is 4.80, or 4.8.
To divide by a decimal, first turn the divisor into a whole number by multiplying both the divisor and the dividend by the same power of ten, then divide as usual. 9.36 divided by 1.2: multiply both by 10 to get 93.6 divided by 12, which divides cleanly to 7.8.
Find 6.72 divided by 1.4.
- Multiply both numbers by 10 to make the divisor a whole number: 67.2 divided by 14.
- 14 into 67: 14 x 4 = 56, remainder 11. Bring down the 2 to make 112.
- 14 into 112: 14 x 8 = 112 exactly.
- So 67.2 divided by 14 = 4.8.
Answer: 6.72 divided by 1.4 = 4.8.
- Why can you multiply both the divisor and the dividend by 10 without changing the answer?
- In 3.2 x 1.5, how many total decimal places do the two factors have, and how does that decide where the decimal point goes in the answer?
Common misconceptions and how to address them
MisconceptionWhen adding or subtracting decimals, line up the digits on the right (like whole numbers) instead of lining up the decimal points.
Why it happens: Whole-number addition lines up on the right (the ones place), so students carry that habit over without noticing decimals need the decimal point, not the right edge, as the anchor.
How to address it: Always line up the decimal points first; pad shorter decimals with trailing zeros so every number has the same number of decimal places before adding. 3.4 + 12.75 lined up correctly is 03.40 + 12.75, not 3.4 sitting under the wrong columns.
MisconceptionIn decimal multiplication, you place the decimal point by lining up the factors' decimal points, the same way you do for addition.
Why it happens: Students over-apply the 'line up the decimal points' rule from addition and subtraction to multiplication, where a completely different rule (counting total decimal places) applies.
How to address it: For multiplication, multiply the digits as whole numbers first, ignoring the decimal points, then count the total decimal places across both factors and place the point that many places from the right in the product. 3.2 x 1.5: 32 x 15 = 480, 2 total decimal places, so 4.80.
MisconceptionYou cannot divide by a decimal, since division needs a whole-number divisor.
Why it happens: Students have only practised dividing by whole numbers and assume a decimal divisor breaks the method.
How to address it: Multiply both the divisor and the dividend by the same power of ten to make the divisor a whole number first -- this does not change the answer, because multiplying both parts of a division by the same amount keeps the ratio the same. 9.36 divided by 1.2 becomes 93.6 divided by 12.
Guided practice (with answers)
1. Find 1,764 divided by 36.
Answer: 49. Check: 36 x 49 = 1,764.
2. Find 5.6 + 3.75.
Answer: 9.35, lining up the decimal points (5.60 + 3.75).
3. Find 10.4 - 2.65.
Answer: 7.75, lining up the decimal points (10.40 - 2.65).
4. Find 2.5 x 3.4.
Answer: 8.5. 25 x 34 = 850, with 2 total decimal places, giving 8.50.
5. Find 6.72 divided by 1.4.
Answer: 4.8. Multiply both by 10 to get 67.2 divided by 14 = 4.8.
Independent practice worksheets
Set the matching ChalkBee division and decimals worksheets for independent work. The answer keys are computed in code, so they are never wrong. Alternate between long division and decimal-operation sets so both fluency skills stay warm.
Differentiation
- Keep a two-digit-divisor times-table strip (such as the 24 times table up to 24 x 10) for the Divide step, so estimating is a lookup rather than mental multiplication under pressure.
- Use grid or squared paper for both long division and decimal column operations, so digits stay aligned in the correct place-value columns.
- Estimate every decimal answer first (round each number to the nearest whole) before computing exactly, to catch a misplaced decimal point immediately.
- Separate the two skills across different days at first (division one day, decimal operations another) before mixing them in the same practice set.
- Divide a four- or five-digit dividend by a two-digit divisor, including cases with a remainder expressed as a decimal.
- Solve multi-step problems that combine division and decimal operations, such as finding a per-item decimal cost from a multidigit total.
- Explore why multiplying both the divisor and dividend by the same power of ten never changes a division's answer, connecting to equivalent fractions.
- Estimate first, then compute exactly, for decimal problems with three or more decimal places.
Assessment: exit ticket
A three-question exit ticket for the last five minutes, sampling long division and two decimal operations.
1. Find 2,208 divided by 48.
Answer: 46. Check: 48 x 46 = 2,208.
2. Find 4.63 + 2.9.
Answer: 7.53, lining up the decimal points (4.63 + 2.90).
3. Find 3.6 x 2.25.
Answer: 8.1. 36 x 225 = 8,100, with 3 total decimal places, giving 8.100, which is 8.1.
Teacher notes and timings
- Rough timing across four lessons: Lesson 1 two-digit-divisor long division (section 1), Lesson 2 adding and subtracting decimals (section 2), Lesson 3 multiplying and dividing decimals (section 3), Lesson 4 mixed practice and the exit ticket.
- Language to keep saying: estimate first, line up the decimal points for adding and subtracting, count total decimal places for multiplying, make the divisor a whole number for dividing. These target the three main misconceptions directly.
- This unit assumes fluent one-digit-divisor long division and simple decimal addition from Grade 5; if either is shaky, revisit those units before pushing to a two-digit divisor and full decimal fluency.
- Curriculum note: ACARA v9's division-with-remainders descriptor sits at Year 6 (AC9M6N06), matching the division half of this unit closely. Full fluency across all four decimal operations is placed at Year 7 in Australia (AC9M7N06), so the decimal half of this unit runs about a year ahead of the ACARA placement.
- Present mode and print both work: use Present to walk through the two-digit-divisor cycle and a decimal multiplication live, then print for independent fluency practice.