Divide by Two-Digit Divisors
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5th Grade Math › Divide by Two-Digit Divisors
A student says, “$1{,}248\div 26=48$ because $26\times 48=1{,}248$.” Another student says, “No, the quotient is $480$ because $26$ goes into $124$ about $4$ times, so add a zero.” Use place value reasoning and the multiplication relationship to decide. Which claim about $1{,}248\div 26$ is incorrect?
The claim that the quotient is 48 is incorrect.
Both claims are correct because division and multiplication are related.
The claim that the quotient is 480 is incorrect.
Neither claim can be checked with multiplication.
Explanation
Division with two-digit divisors uses place value to analyze claims, like evaluating 1,248 ÷ 26 by breaking down the numbers. Estimating the quotient, 1,248 is close to 1,300, and 1,300 ÷ 26 ≈ 50, pointing away from 480 toward 48. Checking with multiplication, 26 × 48 = 1,248, while 26 × 480 = 12,480, shows 480 is incorrect. This connects to error analysis strategies, using place value to identify mistakes in reasoning. A common misconception is appending zeros without considering place value, leading to inflated quotients like 480. Such reasoning helps distinguish correct from incorrect claims, ensuring accurate division. It generalizes to critical thinking in mathematical verification.
A garden club has $1{,}584$ seeds and wants to plant them in equal rows with $33$ seeds in each row. Estimation: $1{,}584$ is close to $1{,}650$, and $1{,}650\div 33=50$, so the quotient should be a little less than $50$. Use multiplication to check: $33\times\text{quotient}=1{,}584$. What is the quotient of $1{,}584\div 33$?
16
33
48
480
Explanation
Division with two-digit divisors applies place value for organization, such as in 1,584 ÷ 33. Estimation: 1,584 near 1,650, 1,650 ÷ 33 = 50, adjusting to 48. Verify by multiplying: 33 × 48 = 1,584, ensuring accuracy. This relates to the long division process, emphasizing place value at each digit. Common misconception is over-rounding without correction, but multiplication checks it. Place value reasoning guarantees complete and correct division. It promotes effective strategies for real-world problems.
A school orders $1{,}344$ stickers and packs them equally into bags of $21$ stickers each. Use an estimate to think about the size of the quotient (for example, $21 \times 60 = 1{,}260$) and then use multiplication to check your exact answer.
What is the quotient of $1{,}344 \div 21$?
64
84
168
640
Explanation
Division with two-digit divisors uses place value to handle the dividend by its digit positions, making large numbers more approachable. Estimating for 1,344 ÷ 21, 21 × 60 = 1,260 is less than 1,344, and 21 × 70 = 1,470 is too high, so the quotient is in the 60s. Multiplication checks the exact quotient by ensuring it times 21 equals 1,344, as in 64 × 21 = 1,344. This ties into the area model, where you visualize rectangles of 21 units wide to fit into 1,344. A common misconception is adding extra zeros to the quotient when dealing with even divisors, leading to answers like 640. Place value reasoning ensures each step accounts for the correct magnitude, aiding precision. This strategy generalizes to support accurate division by integrating estimation with multiplicative verification.
A class has $1{,}296$ centimeter cubes to build equal towers. Each tower uses $27$ cubes. Use place value reasoning (like $27 \times 40 = 1{,}080$ and then add more groups of $27$) and explain how multiplication can check the quotient.
What is the quotient of $1{,}296 \div 27$?
27
36
48
480
Explanation
Division with two-digit divisors uses place value to decompose dividends like 1,296 into easier parts for division. Estimating 1,296 ÷ 27, 27 × 40 = 1,080, then adding 27 × 8 = 216 reaches 1,296, suggesting 48. Multiplication checks by verifying 48 × 27 = 1,296. This ties to the partial quotients strategy, adding groups step-by-step. One misconception is inverting the numbers, like dividing 27 by 1,296. Reasoning with place value supports precise group counting in division. It generalizes by using multiplication to validate and refine estimates effectively.
A teacher has $864$ pencils to pack equally into boxes that hold $24$ pencils each. Use place value reasoning (for example, break $864$ into hundreds, tens, and ones) to decide the quotient. How can multiplication help check the result of $864 \div 24$?
What is the quotient?
18
36
72
360
Explanation
Division with two-digit divisors uses place value by breaking down the dividend into hundreds, tens, and ones to simplify the process. To estimate the quotient for 864 ÷ 24, note that 24 × 30 = 720, which is less than 864, and 24 × 40 = 960, which is too much, so the quotient is between 30 and 40. Using multiplication to check, multiply the proposed quotient by 24 and see if it equals 864, such as 36 × 24 = 864. This connects to the strategy of partial quotients, where you add groups like 30 groups of 24 (720) and then 6 more (144) to reach 864. A common misconception is forgetting to account for the remainder after the first partial quotient, leading to an underestimate. By applying place value reasoning, you ensure each part of the dividend is divided accurately, supporting precise calculations. This method generalizes to any division problem by reinforcing the inverse relationship between multiplication and division for verification.
A bakery has $1{,}248$ cookies to pack equally into trays. Each tray holds $26$ cookies.
A student breaks $1{,}248$ into $1{,}040 + 208$ and says, “Since $26 \times 40 = 1{,}040$ and $26 \times 8 = 208$, the quotient is $40 + 8$.” This uses place value and the idea that division can be explained using multiplication relationships.
Which strategy correctly divides the numbers?
Because $1{,}248$ is close to $1{,}300$, choose $50$ without checking by multiplication.
Divide $1{,}248$ by $2$ and then divide by $6$ because $26 = 2 + 6$.
Use $26 \times 40$ and $26 \times 8$ to make $1{,}248$, so the quotient is $48$.
Treat $26$ like $2$ and compute $1{,}248 \div 2 = 624$, so the quotient is $624$.
Explanation
Division with two-digit divisors uses place value to break down numbers like 1,248 into components for accurate strategies. Estimating 1,248 ÷ 26, 26 × 40 = 1,040 and 26 × 8 = 208 sum to 1,248, identifying the correct approach as adding partial quotients. Multiplication verifies strategies by ensuring the product matches the dividend. This connects to the distributive property in division. A misconception is adding divisor parts instead of multiplying factors. Place value reasoning helps select valid methods for division. It generalizes by fostering strategy evaluation through estimation and checking.
A band has $1{,}540$ flyers to hand out equally to $28$ students. Use estimation to think about the size of the quotient (for example, $28 \times 50 = 1{,}400$) and then use multiplication to check the exact answer.
What is the quotient of $1{,}540 \div 28$?
28
45
55
550
Explanation
Division with two-digit divisors uses place value by considering the positions in numbers like 1,540 to divide efficiently. For estimation in 1,540 ÷ 28, 28 × 50 = 1,400 is less, and adjusting to 28 × 55 = 1,540 fits exactly. Multiplication confirms the quotient by checking if it times 28 equals 1,540. This connects to the scaffold method, building the quotient incrementally. A misconception is scaling up the quotient unnecessarily, leading to answers like 550. Place value reasoning ensures accurate scaling and adjustment in division. It generalizes by linking estimation to multiplication for reliable results in various contexts.
A library has $1{,}176$ books to place equally on $28$ shelves. A student says, “I can check my division by multiplying the quotient by $28$ to get $1{,}176$.” Use place value reasoning (like $28 \times 40 = 1{,}120$ and then adjust) to find the quotient.
What is the quotient of $1{,}176 \div 28$?
14
42
84
420
Explanation
Division with two-digit divisors uses place value by decomposing the dividend, such as viewing 1,176 as 1,100 + 76, to facilitate easier division. For estimating 1,176 ÷ 28, calculate 28 × 40 = 1,120, which is close to 1,176, suggesting the quotient is around 40 and needs slight adjustment upward. Multiplication serves to check by multiplying the quotient back by 28 to confirm it equals 1,176, like 42 × 28 = 1,176. This connects to the partial products strategy, adding multiples such as 40 × 28 and 2 × 28 to reach the total. A misconception is mistaking the divisor for a single digit and ignoring the tens place, which can double the error in the quotient. Using place value reasoning promotes step-by-step accuracy in division tasks. It generalizes by building confidence in verifying answers through the multiplication-division connection.
A coach has $1{,}152$ water bottles to share equally among $32$ players. A student estimates $1{,}152 \approx 1{,}280$, and since $32 \times 40 = 1{,}280$, the quotient should be close to $40$. Use place value reasoning and the relationship between multiplication and division to find the exact quotient.
What is the quotient of $1{,}152 \div 32$?
32
36
48
360
Explanation
Division with two-digit divisors uses place value to divide larger numbers by considering their expanded form, like treating 1,152 as 1,000 + 152. Estimating the quotient for 1,152 ÷ 32 involves approximating 1,152 as 1,280, and since 32 × 40 = 1,280, the exact quotient should be near 40 but adjusted lower. Multiplication checks the result by verifying if the quotient times 32 equals 1,152, for example, 36 × 32 = 1,152. This relates to the long division algorithm, where you divide step-by-step into the hundreds and then the remaining tens and ones. One misconception is over-relying on estimates without exact calculation, which might lead to accepting 40 instead of adjusting to 36. Place value reasoning helps break down complex divisions into manageable parts, ensuring accuracy. Overall, this approach supports reliable division by linking estimation, calculation, and verification through multiplication.
A coach has $864$ water bottles to pack equally into boxes that each hold $24$ bottles. Using place value reasoning, you can think $24\times 30=720$ and $24\times 6=144$, and $720+144=864$. Which statement gives the correct quotient and shows how multiplication can check the division $864\div 24$?
The quotient is $34$ because $24\times 34=816$ and that is close enough to $864$.
The quotient is $360$ because $24$ goes into $864$ about $36$ times and then you add a zero.
The quotient is $36$ because $24\times 36=864$, so multiplication checks the division.
The quotient is $3{,}456$ because $864\times 24=3{,}456$.
Explanation
Division with two-digit divisors uses place value to break down the dividend into manageable parts, like hundreds and tens. Estimating the quotient involves finding how many times the divisor fits into larger place values, such as seeing that 24 goes into 720 thirty times since 24×30=720. Using multiplication to check means multiplying the estimated quotient by the divisor to verify if it equals the dividend, confirming 24×36=864. This connects to the partial quotients strategy, where you add quotients from each place value, like 30+6=36. A common misconception is adding an extra zero to the quotient, but that ignores the actual place value relationships. Reasoning with place value helps ensure the quotient is accurate by aligning the multiplication back to the original dividend. Overall, this approach builds confidence in division by linking it to familiar multiplication facts.