Understand Division as Missing Factor - 3rd Grade Math

1. What is $32\div 8$? Think: what times 8 equals 32?
   Answer: 4 Since 4 times 8 equals 32, dividing 32 by 8 gives 4. Choice A, 32, is the dividend itself, not the quotient. Choice C, 5, is close to the answer but does not match the fact family. Choice D, 256, comes from multiplying 32 by 8 instead of dividing.
2. What is $35 \div 7$?
   Answer: 5 5 is correct because 7 times 5 equals 35, so 35 divided by 7 is 5. 4 is incorrect because 7 times 4 equals 28, not 35. 42 is incorrect because 7 times 6 equals 42, not 35. 7 is incorrect because it repeats the divisor instead of the quotient.
3. Jessica has 63 marbles. She puts them into bags with 9 marbles in each bag. To find how many bags she needs, she could solve $63 \div 9$ or solve which missing factor problem?
   Answer: $9 \times ? = 63$ The correct missing-factor problem is D) 9 x ? = 63, since dividing 63 by 9 asks what number times 9 equals 63. Choice A uses the wrong total (72 instead of 63). Choice B incorrectly swaps which number should be the product. Choice C shows multiplication rather than a missing-factor equation.
4. What number times 7 equals 42? Use this fact to find $42 \div 7$.
   Answer: 6 Since 6 times 7 equals 42, dividing 42 by 7 gives 6, matching choice D. Choice A reverses the operation. Choices B and C are close numbers but do not satisfy the multiplication fact.
5. What is $72 \div 9$?
   Answer: 8 8 is correct because $9 \times 8 = 72$, so $72 \div 9 = 8$. 81 is incorrect; it's the product of $9 \times 9$, not related to dividing 72 by 9. 7 is incorrect because $9 \times 7 = 63$, not 72. 9 is incorrect; it's the divisor itself, not the quotient.
6. Emma has $42$ stickers in $6$ equal groups; solve $6 \times ?=42$.
   Answer: 7 This question tests understanding division as an unknown-factor problem (CCSS.3.OA.6), specifically recognizing that division can be solved by finding the missing factor in a multiplication equation. Division and multiplication are inverse operations—they undo each other. When you see a division problem like $42 \div 6$ (stickers per group), you can think of it as a multiplication question: 'What number times 6 equals 42?' or '6 times what number equals 42?' This is the same as solving the equation $? \times 6=42$ or $6 \times ?=42$. If you know your multiplication facts, you can use them to divide: Since $7 \times 6=42$, then $42 \div 6=7$. The missing factor (7) is the quotient. Fact families show this relationship: $7 \times 6=42$, $6 \times 7=42$, $42 \div 7=6$, $42 \div 6=7$ are all related. In this problem, we need to find how many stickers per group with 42 stickers in 6 groups, solving $6 \times ?=42$. Using the missing factor approach: We know $6 \times 7=42$ from multiplication facts, so $42 \div 6=7$. Choice B is correct because $7 \times 6=42$, so 7 is the missing factor that makes 42 when multiplied by 6, which means $42 \div 6=7$ stickers per group. This demonstrates understanding that division finds the unknown factor in multiplication. Choice C is incorrect because it provides 36, which might come from $6 \times 6=36$ or using a wrong fact. This error occurs when students make calculation errors or confuse the numbers. To help students understand division as missing factor: Explicitly teach the connection—'$42 \div 6$ means: what times 6 equals 42?' Practice fact families: if $7 \times 6=42$, then $42 \div 7=6$ (division finds the other factor). Use arrays: '6 rows of how many equals 42 total? $6 \times ?=42$' Model thinking aloud: 'I need to find $56 \div 7$. I think: 7 times what equals 56? I know $7 \times 8=56$, so $56 \div 7=8$.' Have students write both equations (division and missing factor multiplication) side by side. Check division answers by multiplying (if $42 \div 6=7$, check: does $7 \times 6=42$? Yes!). This reinforces the inverse relationship. Watch for students who can multiply but struggle with division—show them they already know division by knowing multiplication facts.
7. What is $48 \div 6$?
   Answer: 8 Dividing 48 by 6 gives 8, since 6 times 8 equals 48, matching choice D. Choices A and B are close numbers but do not satisfy the division. Choice C comes from multiplying instead of dividing.
8. Sarah arranges 36 books into equal rows with 9 books in each row. She writes the equation $9 \times n = 36$ to find the number of rows. What does $n$ represent in her equation?
   Answer: The number of rows of books she made In the equation $9 \times n = 36$, the 9 represents books per row, 36 is the total books, so $n$ must be the number of rows. Choice B is 36, choice C is 9, and choice D would be remainder in division with a remainder.
9. What is $56\div 8$? Think: what times 8 equals 56?
   Answer: 7 The correct answer is A) 7, since 7x8=56. Choice B (64) is the result of 8x8, not 56 divided by 8. Choice C (8) confuses the divisor with the quotient. Choice D (5) is too small; 5x8=40, not 56.
10. What is $32\div 8$? Think: what times $8$ equals $32$?
    Answer: 4 Since 8 x 4 = 32, the answer to 32 divided by 8 is 4, making Choice C correct. Choice B (8) repeats the known factor instead of solving for the unknown one. Choice A (5) is close to the correct value but does not satisfy the equation. Choice D (24) comes from subtracting 8 from 32 instead of dividing.
11. Use the fact family $6\times 7=42$ to find $42\div 7$.
    Answer: 6 This question tests understanding division as an unknown-factor problem (CCSS.3.OA.6), specifically recognizing that division can be solved by finding the missing factor in a multiplication equation. Division and multiplication are inverse operations—they undo each other. When you see a division problem like 42÷7, you can think of it as a multiplication question: 'What number times 7 equals 42?' or '7 times what number equals 42?' This is the same as solving the equation ?×7=42 or 7×?=42. If you know your multiplication facts, you can use them to divide: Since 6×7=42, then 42÷7=6. The missing factor (6) is the quotient. Fact families show this relationship: 6×7=42, 7×6=42, 42÷6=7, 42÷7=6 are all related. In this problem, we need to use the fact family 6×7=42 to find 42÷7. Using the missing factor approach: We know 7×6=42 from the fact family, so 42÷7=6. Choice C is correct because the fact family shows 7×6=42, so 6 is the missing factor that makes 42 when multiplied by 7, which means 42÷7=6. This demonstrates understanding that division finds the unknown factor in multiplication. Choice B is incorrect because it provides 49, which might come from multiplying 7×7=49 instead of using the given fact. This error occurs when students confuse operations or use wrong numbers. To help students understand division as missing factor: Explicitly teach the connection—'42÷7 means: what times 7 equals 42?' Practice fact families: if 7×6=42, then 42÷7=6 (division finds the other factor). Use arrays: '7 rows of how many equals 42 total? 7×?=42' Model thinking aloud: 'I need to find 56÷7. I think: 7 times what equals 56? I know 7×8=56, so 56÷7=8.' Have students write both equations (division and missing factor multiplication) side by side. Check division answers by multiplying (if 42÷7=6, check: does 6×7=42? Yes!). This reinforces the inverse relationship. Watch for students who can multiply but struggle with division—show them they already know division by knowing multiplication facts.
12. Alex knows that $72 \div 8 = 9$. He wants to check his answer by writing a multiplication equation. Which equation should he write to verify his division is correct?
    Answer: $8 \times 9 = 72$ When you solve a division problem, you can always check your work using multiplication because division and multiplication are opposite operations. Think of division as asking "what number times the divisor gives me the dividend?" In Alex's problem, $72 \div 8 = 9$ means "72 divided by 8 equals 9." To verify this is correct, you need to check if 8 times 9 really does equal 72. If it does, then your division is right! Choice C gives us $8 \times 9 = 72$, which is exactly what we need. When we multiply the divisor (8) by the quotient (9), we get back to our original dividend (72). This confirms that $72 \div 8 = 9$ is correct. Let's see why the other choices don't work. Choice A shows $9 \times 7 = 63$, but this uses 7 instead of 8 and gives us 63 instead of our original number 72. Choice B gives us $72 \times 8 = 576$, which multiplies our dividend by the divisor instead of multiplying the divisor by the quotient - this creates a much larger number that doesn't help us check our work. Choice D shows $8 + 9 = 17$, but addition won't help us verify division since these operations aren't related. Remember this checking strategy: to verify any division problem, multiply the divisor by the quotient. If you get back to your original dividend, your division is correct!
13. Carlos wants to find $56 \div 7$. He thinks: "What number times 7 gives me 56?" Which equation shows his thinking?
    Answer: $7 \times ? = 56$ Carlos is using division as a missing factor problem. $56 \div 7$ means "what times 7 equals 56?" which is $7 \times ? = 56$. Choice B multiplies instead of finding a missing factor, choice C subtracts, and choice D uses addition instead of multiplication.
14. What number makes this equation true? $9 \times ? = 72$
    Answer: 8 9 x 8 = 72, so the missing number is 8, making Choice B correct. Choice A (7) and Choice D (9) are close numbers that don't satisfy the equation. Choice C (63) comes from multiplying 9 x 7 instead of finding the correct factor.
15. What is $35 \div 7$?
    Answer: 5 5 is correct because 7 times 5 equals 35, so 35 divided by 7 is 5. 42 is incorrect because 7 times 6 equals 42, not 35. 7 is incorrect because it repeats the divisor instead of the quotient. 4 is incorrect because 7 times 4 equals 28, not 35.
16. Use the fact family with 6, 7, and 42 to find $42 \div 6$.
    Answer: 7 Since 6 times 7 equals 42, dividing 42 by 6 gives 7. Choice A, 8, is not part of this fact family. Choice C, 6, is the divisor itself, not the quotient. Choice D, 36, comes from subtracting 6 from 42 instead of dividing.
17. Emma arranges 45 stickers in equal groups. She knows each group has 5 stickers. To find the number of groups, she writes $5 \times g = 45$. What is the value of $g$?
    Answer: $g = 9$ groups When you see a multiplication equation with an unknown variable like $5 \times g = 45$, you're looking at a division problem in disguise. Emma has 45 total stickers arranged in groups of 5 stickers each, and she needs to find how many groups she made. To solve $5 \times g = 45$, think about what number times 5 equals 45. You can work backwards by dividing: $g = 45 ÷ 5 = 9$. You can check this by multiplying: $5 \times 9 = 45$ ✓. So Emma has 9 groups of stickers. Looking at the wrong answers: Choice A suggests $g = 45$, but that would mean $5 \times 45 = 225$ stickers, which is way too many. Choice B says $g = 5$, which would give us $5 \times 5 = 25$ stickers total—not enough. Choice D proposes $g = 50$, leading to $5 \times 50 = 250$ stickers, again far too many. The correct answer is C: $g = 9$ groups. Remember this strategy: when you have multiplication equations like "something times what equals a number," you can always find the missing piece by dividing the total by the known factor. So if you see $a \times ? = b$, then $? = b ÷ a$. This connection between multiplication and division will help you solve similar word problems quickly.
18. Which multiplication equation helps solve $48 \div 6$ as a missing factor?
    Answer: $6 \times 8 = 48$ The multiplication fact that helps solve 48 divided by 6 as a missing factor is 6 times 8 equals 48, so A is correct. Choice B (8 times 6 equals 40) is an incorrect equation, since 8 times 6 actually equals 48, not 40. Choice C (6 times 6 equals 48) is false, since 6 times 6 equals 36, not 48. Choice D restates the division problem instead of the related multiplication fact.
19. An array has 32 dots in 8 rows; what is $32 \div 8$?
    Answer: 4 Dividing 32 dots into 8 equal rows means each row has 32 divided by 8, which is 4 dots. Choice A, 8, is the number of rows, not the answer to the division. Choice B, 3, is too small to be correct. Choice C, 40, comes from adding 32 and 8 instead of dividing.
20. Solve $24 \div 6$ using a missing factor: $6 \times \ ? = 24$.
    Answer: 4 Since $6 \times 4 = 24$, the missing factor is 4, so $24 \div 6 = 4$. Choice A (18) comes from subtracting 6 from 24 instead of finding the missing factor. Choice B (3) is off by one from the correct answer. Choice C (6) repeats the known factor instead of solving for the missing one.

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