Divide multi-digit whole numbers with a standard algorithm

Learning goal: Use a standard long-division algorithm accurately and explain how each quotient digit corresponds to a place-value amount. Estimate, carry out the calculation and check with multiplication. Fluency requires practice across varied problems, not memorizing one completed layout.

Divide 7,452 by 23. An estimate helps: 23 × 300 = 6,900 and 23 × 400 = 9,200, so the quotient lies between three hundred and four hundred. A result near thirty or three thousand would be unreasonable. The first quotient digit therefore belongs in the hundreds place.

Consider the first two dividend digits, 74, as seventy-four hundreds. Three groups of twenty-three hundreds make sixty-nine hundreds. Write 3 in the quotient's hundreds place and subtract 69 from 74, leaving five hundreds. Bring down the next dividend digit, 5 tens. The remaining five hundreds become fifty tens; together with five tens, there are fifty-five tens to divide.

Twenty-three fits into fifty-five twice, so write 2 in the quotient's tens place. Two groups of twenty-three tens make forty-six tens. Subtracting leaves nine tens. Bring down the final digit, 2 ones, to obtain ninety-two ones. Twenty-three fits into ninety-two four times, so write 4 in the ones place and subtract ninety-two, leaving zero.

The quotient is 324. Its digits represent 300 + 20 + 4 groups. The corresponding products are 6,900 + 460 + 92 = 7,452. The standard layout compresses those place-value products, while the expanded check explains why the steps work. Bringing down a digit combines the remainder from one place with the next lower place; it is not an unexplained movement of ink.

A zero quotient digit must be recorded when no group fits at a particular place. For 9,648 ÷ 24, dividing ninety-six hundreds by 24 gives four hundreds in the quotient because 24 × 400 = 9,600. Four tens remain before the final eight ones are considered. Twenty-four does not fit into four tens even once at the tens place, so the tens quotient digit is zero. Combining those four tens with eight ones gives forty-eight, allowing two ones. The quotient is 402, not 42.

When a trial quotient digit gives a product larger than the current amount, reduce it. When a remainder is at least the divisor, the digit is too small. For example, during a division by 23, a remainder of 25 means another group of 23 could have been included at that place. A valid final whole-number remainder is zero or greater and smaller than the divisor.

Not every division ends with zero remainder. 7,457 ÷ 23 gives 324 remainder 5, because 23 × 324 + 5 = 7,457. It can also be written 324 + 5/23. The remainder's practical meaning depends on the context. If 7,457 objects are packed in full boxes of 23, there are 324 full boxes and five loose objects. To hold every object in boxes of capacity 23 requires 325 boxes, with the last partly filled.

Keep the dividend and divisor in the intended order. “Objects per box” is a group size, so total objects divided by that size counts boxes. Dividing 23 by 7,457 would describe a different comparison. Write the equation and units before calculating when a word problem is involved.

Check the full relationship: divisor × quotient + remainder = dividend. An estimate detects a misplaced digit; the exact inverse check detects arithmetic errors. Neither should be replaced by the statement that an answer looks familiar.

Worked place-value check: 12,096 ÷ 36 = 336. The first step divides 120 hundreds by 36: 36 × 300 = 10,800 fits and 36 × 400 = 14,400 does not, so the first quotient digit is 3 in the hundreds place. The quotient digits are three hundreds, three tens and six ones; writing 36 or 3,360 would misplace them. Check 36 × (300 + 30 + 6) = 10,800 + 1,080 + 216 = 12,096.

Worked remainder interpretation: For 7,457 objects in boxes holding 23, a learner reports 324 boxes can hold everything. The inverse check leaves five objects unpacked. The calculation 324 remainder 5 is correct, but the capacity question requires one additional box.

Guided algorithm: Work through 7,452 ÷ 23 in standard long-division form. At each bring-down step, name the units being regrouped. Compare the written products with the expanded hundreds, tens and ones products in the diagram.

Independent performance: Solve several multi-digit divisions, including a zero quotient digit, an exact quotient and a nonzero remainder. Include new divisions such as 6,552 ÷ 26, 9,315 ÷ 45 and 4,000 ÷ 37, which were not worked in the lesson. Estimate first, use a standard algorithm without a copied model and verify each result by multiplication and addition. Explain two different practical interpretations of one remainder. An observer records accuracy, efficient use of the procedure and whether corrections are meaningfully checked.