Multiply Multi-Digit Whole Numbers - 5th Grade Math

A teacher buys $24$ boxes of markers. Each box has $36$ markers. Use the standard algorithm to find the product $36 \times 24$ . Which value is the correct product?
Answer: The product is 864. Multi-digit multiplication uses place value to break down numbers into their ones, tens, and higher components for accurate computation. Partial products are created by multiplying the first number by each digit of the second number, accounting for the digit's place value, such as 36 × 4 and 36 × 20 for 36 × 24. In the standard algorithm, digits are aligned by writing the partial product for the tens place shifted one position to the left, often adding a zero at the end. This connects to the area model, where the rectangle is divided into sections like 30 × 20, 30 × 4, 6 × 20, and 6 × 4, summing to the total product of 864. A common misconception is forgetting to multiply by the tens value, leading to incorrect additions like just 36 × 2 + 36 × 4 = 144 instead of using 20. The algorithm works because it systematically combines these place-value-based products. This ensures reliable results for real-world problems, like finding total markers in 24 boxes of 36 each.
A student solves $624 \times 15$ using the standard algorithm and place value. Which claim about the multiplication is incorrect?
Answer: You can line up the numbers in any way because place value does not matter in multiplication. Multi-digit multiplication uses place value to expand numbers like 15 into 1 ten and 5 ones, facilitating the breakdown of the operation. Partial products include multiplying 624 by 5 for the ones and by 10 for the tens, then adding them to find the total. Aligning digits involves placing the numbers right-aligned by place value, with the tens partial product shifted left to represent its higher value. This connects to reasoning with the area model, where the product is the sum of areas from 624 × 10 and 624 × 5 sections. A misconception is thinking place value doesn't affect alignment, but it does, as misalignment ignores the true value of digits and causes errors. The algorithm works by systematically applying place value to each multiplication step, ensuring comprehensive coverage. It generalizes effectively because it builds on the foundational idea that numbers are sums of their place value parts, making complex multiplications manageable.
Lisa is calculating $456 \times 89$ using the standard algorithm. She correctly finds that $456 \times 9 = 4104$ and $456 \times 80 = 36480$ . However, when she adds these partial products, she gets $76584$ . What error did Lisa make?
Answer: She made an error when adding the two partial products together Lisa's partial products are correct: $456 \times 9 = 4104$ and $456 \times 80 = 36480$ . However, when adding $4104 + 36480$ , the correct sum is $40584$ , not $76584$ . Lisa made an addition error. The correct answer to $456 \times 89$ is $40584$ . Choice A and B are incorrect because both partial products are calculated correctly. Choice D is wrong because $76584$ is not the correct final answer.
Compute $67\times 24$ using the standard algorithm. Be sure to align digits by place value when adding partial products. What is the product?
Answer: 1,608 Multi-digit multiplication uses place value to break down and multiply numbers by their ones, tens, and other components. Partial products are the outcomes of multiplying by each digit with its place value, like 4 ones and 2 tens. Aligning digits requires positioning partial products so their place values match when adding. This aligns with the area model, representing multiplication as summed areas of place value rectangles. One misconception is forgetting to align by place, leading to incorrect column additions. The algorithm functions by methodically incorporating each place's contribution. It works universally by maintaining the integrity of the base-ten system.
A student multiplies $504\times 27$ using partial products based on place value. The student writes $504\times 7=3,528$ and $504\times 20=10,080$ and then combines the partial products. Which final product is correct?
Answer: 13,608 Multi-digit multiplication uses place value to create partial products like multiplying by 7 ones and 20 for the tens. Partial products break the task into smaller, place-specific multiplications that are then added. Aligning digits ensures the twenties product starts in the tens place for correct summation. This relates to the area model by representing sections for hundreds times tens, tens times tens, etc. A misconception is adding without place adjustments, undervaluing the partials. The algorithm works via the distributive property on place values. It generalizes to larger numbers by maintaining consistent place handling.
Compute $58\times 42$ using the standard algorithm. What is the product?
Answer: 2,436 Multi-digit multiplication uses place value to decompose numbers and multiply their components separately before combining. Partial products involve multiplying the entire top number by each digit of the bottom number, respecting their places like ones or tens. Aligning digits means placing the tens partial product starting from the tens column to reflect its value. This relates to the area model, where each partial product corresponds to an area of a rectangle segmented by place values. A misconception is aligning all partial products in the ones place, which disregards the tens multiplier. The algorithm works by ensuring each partial product is scaled by its place value factor. Overall, it provides a reliable way to compute products by honoring numerical structure.
Compute $63 \times 25$ using the standard algorithm. Which value is the correct product?
Answer: The product is 1575. Multi-digit multiplication uses place value to handle digits in ones, tens, and beyond systematically. Partial products come from multiplying by each digit with its place value, like 63 × 5 and 63 × 20 for 63 × 25. Aligning involves placing the tens partial product shifted left, often with a trailing zero for clarity. This ties to the area model, dividing into rectangles such as 60 × 20, 60 × 5, 3 × 20, and 3 × 5, totaling 1575. A misconception is omitting the zero in the tens partial product, which might lead to misadding and wrong totals like 126 + 315 = 441 instead. The algorithm works because it builds the product layer by layer using place values. It ensures consistent results for various multi-digit scenarios.
Compute $65 \times 48$ using the standard algorithm. Which value is the correct product (and would match combining $65\times8$ and $65\times40$ )?
Answer: The product is 3120. Multi-digit multiplication uses place value to handle multi-digit numbers by breaking them down. Partial products are the results of multiplying by each digit's value, such as 65 × 8 and 65 × 40 for 65 × 48. Alignment involves shifting the higher place partial product to the left before summing. This ties to the area model, summing sub-areas like 60 × 40, 60 × 8, 5 × 40, and 5 × 8 to 3120. A misconception is forgetting the ×10 for tens, leading to adding 520 + 260 = 780 wrongly. The algorithm works by integrating place values seamlessly. This ensures accuracy in diverse multiplication tasks.
A student is multiplying $219\times 63$ using the standard algorithm. The student correctly computed $219\times 3=657$ for the ones digit. What should the student do for the 6 in 63 and how should it be aligned?
Answer: Compute $219\times 60=13,140$ and write it starting in the tens place. Multi-digit multiplication uses place value to treat digits like 6 in 63 as 60 for the tens partial product. Partial products are computed for each place, such as ones and tens, with appropriate scaling. Aligning digits involves starting the tens product in the tens column, often by shifting or adding a zero. This connects to the area model, dividing into grids for each place value multiplication. A misconception is skipping place value shifts, which collapses the product's magnitude. The algorithm works by distributing over place-expanded forms. It generalizes to ensure accuracy in multiplying any multi-digit combinations.
Sarah needs to find $275 \times 48$ . She decides to use the fact that $48 = 50 - 2$ to make the calculation easier. If she calculates $275 \times 50 = 13750$ , what should she do next to find the correct answer?
Answer: Subtract $275 \times 2 = 550$ to get $13750 - 550 = 13200$ Using the distributive property, $275 \times 48 = 275 \times (50 - 2) = 275 \times 50 - 275 \times 2$ . Since Sarah calculated $275 \times 50 = 13750$ , she needs to subtract $275 \times 2 = 550$ to get $13750 - 550 = 13200$ . Choice A adds instead of subtracts. Choice C and D don't use the distributive property correctly and would give incorrect results.
A student multiplies $312\times 48$ using the standard algorithm. Which final product is correct?
Answer: 14,976 Multi-digit multiplication uses place value to handle digits in their correct positional weights during computation. Partial products are formed by multiplying by each digit of the multiplier, incorporating its place value. Aligning digits requires positioning each partial product according to the place value, shifting left for tens and higher. This ties into the area model, where the total area is the sum of sub-areas defined by place value breakdowns. A common misconception is treating all digits equally without place adjustments, resulting in misaligned sums. The algorithm works because it breaks down the problem into distributive property applications. It generalizes by scaling to more digits while preserving place value accuracy.
A student uses an area model to represent $46\times 35$ by breaking the numbers into tens and ones: $46=40+6$ and $35=30+5$ . Which total is the correct product?
Answer: 1,610 Multi-digit multiplication uses place value to break numbers into tens and ones for an area model approach. Partial products are the four sub-products from combining tens and ones of each number, then summed. Aligning digits in the model ensures each product is placed in its correct place value section. The area model directly represents the reasoning by showing the total as combined areas of a rectangle. A misconception is thinking the model only works for two-digit numbers, but it extends to more digits. The algorithm works by expanding both numbers and multiplying all combinations. It generalizes to verify standard algorithm results through visual place value confirmation.
A school is ordering new textbooks. They need $147$ books for each of their $23$ classrooms. Each book costs $28. What is the total amount the school will spend on textbooks?
Answer: $94668 for the total textbook expenditure This requires two multiplication steps. First, find total books needed: $147 \times 23 = 3381$ books. Then multiply by cost per book: $3381 \times 28 = 94668$ . The total cost is $94668. Choice A represents $147 + 23 + 28$ . Choice B represents $147 \times 28$ (one classroom's cost). Choice D represents only the total number of books without multiplying by the cost per book.
A student tries to multiply $76 \times 52$ using the standard algorithm. Their work shows partial products of $76 \times 2 = 152$ and $76 \times 5 = 380$ , then they add $152 + 380 = 532$ . Which statement best describes the mistake using place value reasoning?
Answer: The student forgot that the 5 in 52 means 5 tens, so the second partial product should be $76 \times 50$ , not $76 \times 5$ . Multi-digit multiplication uses place value to expand numbers like 52 into 5 tens and 2 ones, which helps identify errors in computation. Partial products should be 76 × 2 = 152 for ones and 76 × 50 for tens, but the student mistakenly used 76 × 5 instead. Aligning digits requires right-alignment by place value, with the tens partial product shifted to reflect the extra factor of 10. This connects to the area model, splitting the multiplication into areas for 76 × 50 and 76 × 2, summed together. A common misconception is ignoring the tens place multiplier, treating 5 as just 5 instead of 50, leading to a smaller product. The algorithm works by breaking down numbers via place values and multiplying distributively. It generalizes well because it consistently applies these principles to yield correct results across various multi-digit problems.
A warehouse ships $126$ packages per day. Each package contains $37$ items. If the warehouse operates for $15$ days, which calculation will give the total number of items shipped?
Answer: $126 \times 37 \times 15 = 69930$ total items shipped To find total items, multiply packages per day by items per package by number of days: $126 \times 37 \times 15 = 69930$ items. First $126 \times 37 = 4662$ , then $4662 \times 15 = 69930$ . Choice A adds all values instead of multiplying. Choice B calculates items for one day then adds days instead of multiplying. Choice C adds packages per day to the product of items per package and days, which doesn't represent the situation correctly.
Roberto multiplies $234 \times 67$ and gets $15678$ . To check his work, he decides to use the fact that $67 = 70 - 3$ . Which expression should Roberto calculate to verify his answer?
Answer: $234 \times 70 - 234 \times 3$ and compare to $15678$ Using the distributive property, $234 \times 67 = 234 \times (70 - 3) = 234 \times 70 - 234 \times 3$ . Roberto should calculate $234 \times 70 - 234 \times 3 = 16380 - 702 = 15678$ to verify. Choice A adds instead of subtracts. Choice C multiplies the partial products instead of using the distributive property. Choice D uses addition throughout instead of multiplication.
A student multiplies $52\times 19$ and writes these partial products: - $52\times 9=468$ - $52\times 10=520$ Then the student adds them. Which value is the correct product of $52\times 19$ (showing the partial products were combined correctly using place value)?
Answer: 988 Multi-digit multiplication uses place value to handle numbers by their positional values, like interpreting 1 in 19 as 10. Partial products are calculated for each place, such as multiplying by 9 ones and 1 ten separately. Aligning digits means ensuring the ten's product is shifted left to reflect its value. This connects to the area model, summing areas of rectangles defined by place value breakdowns. One misconception is treating the tens digit as just its face value without adjustment, causing misalignment. The algorithm works by accurately aggregating these adjusted products. It provides a universal method for multiplication by preserving numerical positions.
A student is solving $39\times 47$ with the standard algorithm. They correctly find $39\times 7=273$ . For the tens digit, they write $39\times 4=156$ and add it without shifting left. Which option shows the correct product of $39\times 47$ when place value is used correctly?
Answer: 1,833 Multi-digit multiplication uses place value to interpret digits correctly, such as a 4 in the tens place meaning 40. Partial products come from multiplying by each digit separately, like by 7 ones and then by 4 tens, creating intermediate results. Aligning digits involves shifting the tens partial product left to account for the place value multiplier. This ties into the area model, where the total area is the sum of sub-areas representing products of place value pairs. One misconception is adding partial products without shifting, which treats tens as ones and underestimates the product. The algorithm works by correctly scaling and summing these components. It ensures precision across various number sizes by respecting place value rules.
Use the standard algorithm for $84\times 36$ . Which step is correct for the tens partial product?
Answer: Multiply $84\times 3=252$ and write 252 starting in the tens place. Multi-digit multiplication uses place value to expand numbers, treating digits as multiples of powers of ten. Partial products are formed by multiplying the whole number by each digit's value, such as 3 tens meaning a shift in placement. Aligning digits requires positioning the tens partial product to start in the tens column for accurate addition. This process mirrors the area model, with sections corresponding to ones-by-ones, ones-by-tens, and so on. A misconception is writing the tens product without shifting, which ignores the place value and leads to errors. The algorithm succeeds through systematic decomposition and recombination. It generalizes effectively because it builds on the foundational structure of our number system.
An area model for $34 \times 26$ is split into $(30 + 4)$ and $(20 + 6)$ . The partial products should be $30\times20$ , $30\times6$ , $4\times20$ , and $4\times6$ . Which value is the correct total product (showing the partial products are combined correctly)?
Answer: The product is 884. Multi-digit multiplication uses place value to expand numbers into ones and tens for systematic calculation. Partial products arise from multiplying by each decomposed part, like 30 × 20, 30 × 6, 4 × 20, and 4 × 6 for 34 × 26. Digits are aligned by positioning partial products based on their place values when adding. This directly connects to the area model, where the total area sums the partial areas to 884. A misconception is misadding partials without proper combination, potentially yielding 600 + 180 + 80 + 24 incorrectly if not summed fully. The algorithm works because it mirrors the distributive property over place values. It provides a reliable framework for multi-digit problems.

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