Kid Einsteins · Class 6 · Division Facts · Grades 3–4 · NYC Math Class
Division Facts & the Inverse of Multiplication — 25 Practice Questions with Theory & Hidden Answers (Kid Einsteins Class 6)
A complete class-ready lesson on division facts taught as the inverse of multiplication: fact families, sharing vs. grouping models, division by 2–10, an intro to remainders, and 25 practice questions with click-to-reveal step-by-step answers. Built for the SOMATH Kid Einsteins program (grades 3–4) on the Upper West Side of Manhattan.
This is Class 6 of the SOMATH Kid Einsteins multiplication & division arc — the class where students realize that they already know their division facts. If they can say 6 × 7 = 42, they already know that 42 ÷ 7 = 6 and 42 ÷ 6 = 7. Division is not a new set of facts to memorize — it’s the same facts, asked backwards.
Every question below has a hidden button that reveals the answer and the reasoning, so a student can practice honestly and then check their thinking. Written by the same team that teaches Kid Einsteins at SOMATH, a math-focused school on the Upper West Side of NYC run by cofounder Marcelo Ambrozio (Northwestern) and cofounder Vivianne Wright (Harvard).
- Warm-up: read the fact-family theory aloud and write one on the board (10 min).
- Chant division facts by 2, 5, and 10 out loud — the “easy” ones (5 min).
- Students attempt the 25 questions with the answers hidden (25 min).
- Reveal answers as a group and re-teach any that missed 3+ students (10 min).
What’s in this lesson
- Division is the inverse of multiplication
- Fact families — three numbers, four facts
- Two pictures of division: sharing and grouping
- The easy division facts: ÷2, ÷3, ÷4, ÷5, ÷10
- The tricky division facts: ÷6, ÷7, ÷8, ÷9
- When it doesn’t divide evenly — remainders
- Special cases: ÷1, ÷itself, and why you can’t divide by 0
- 25 practice questions
- 5 hard word problems for grade 5 (Q26–30)
- Answer key summary
- About SOMATH & Kid Einsteins
1. Division is the inverse of multiplication
Undoing what multiplication did
Multiplication and division are opposite operations. They undo each other. If you multiply and then divide by the same number, you get back where you started.
Example: 6 × 7 = 42. To undo the ×7, we divide by 7: 42 ÷ 7 = 6. To undo the ×6, we divide by 6: 42 ÷ 6 = 7.
This means: every multiplication fact you already know is also a division fact. Students who’ve memorized their times tables through 10 already know 100 multiplication facts — and 100 division facts — whether they realize it yet or not. Class 6 is the class where we make them realize it.
The three ways to write the same relationship:
- Multiplication: 6 × 7 = 42
- Division (undo the 7): 42 ÷ 7 = 6
- Division (undo the 6): 42 ÷ 6 = 7
2. Fact families — three numbers, four facts
Three numbers that share four facts
A fact family is a set of three numbers that form two multiplication facts and two division facts. The three numbers are always the two factors and their product.
5 × 4 = 20
20 ÷ 4 = 5
20 ÷ 5 = 4
7 × 6 = 42
42 ÷ 6 = 7
42 ÷ 7 = 6
Notice: the biggest number is always the answer when you multiply, and always the starting number (dividend) when you divide. If a student learns their multiplication facts as fact families instead of as separate facts, the division facts come along for free.
Fact family with a repeated factor. A square number like 25 or 49 has a smaller fact family — two facts instead of four:
25 ÷ 5 = 5
3. Two pictures of division: sharing and grouping
Sharing (partitive) vs. grouping (measurement)
Multiplication has one picture (an array or a rectangle). Division has two pictures, and both matter for word problems.
Sharing (partitive). “20 stickers, split fairly between 4 kids. How many stickers per kid?” We know how many groups (4) and we’re looking for how many go in each group.
20 dots split into 4 equal groups → 5 per group. 20 ÷ 4 = 5 (per kid).
Grouping (measurement). “20 stickers, packed into bags of 4. How many bags?” We know how many go in each group (4), we’re looking for how many groups.
20 dots packed into groups of 4 → 5 groups. 20 ÷ 4 = 5 (bags).
Both are 20 ÷ 4 = 5. The answer is the same (5), but the “5” means different things: 5 per kid vs. 5 bags. The word problem tells you which picture to draw.
4. The easy division facts: ÷2, ÷3, ÷4, ÷5, ÷10
The ones that click first
Division by 2, 5, and 10 usually clicks first because the multiplication tables for these numbers are the easiest to memorize.
| Rule | Example | Why |
|---|---|---|
| ÷2 = half | 18 ÷ 2 = 9 | Half of 18 |
| ÷5 | 35 ÷ 5 = 7 | Count by 5s: 5, 10, 15, 20, 25, 30, 35 → 7 counts |
| ÷10 | 80 ÷ 10 = 8 | Drop the zero: 80 → 8 |
| ÷4 = half twice | 32 ÷ 4 = 8 | 32 → 16 → 8 |
| ÷3 | 27 ÷ 3 = 9 | Skip-count by 3s until you hit 27 (9 counts) |
÷4 shortcut: to divide by 4, halve twice. This works because 4 = 2 × 2, so dividing by 4 undoes multiplying by 2 twice.
5. The tricky division facts: ÷6, ÷7, ÷8, ÷9
Fall back on the multiplication fact
The tricky division facts — ÷6, ÷7, ÷8, ÷9 — are the ones students get stuck on. The trick: don’t try to divide, ask a multiplication question instead.
Instead of asking “What is 56 ÷ 7?” ask yourself “7 times what equals 56?”
7 × 7 = 49 (too small). 7 × 8 = 56 (yes). So 56 ÷ 7 = 8.
This is the whole reason we drilled ×7, ×8, ×9 with area models in Class 3. Every multiplication fact you memorized then, you now cash in for a division fact.
Shortcuts you already know from multiplication:
- ÷8 — if you learned ×8 as “double three times,” then ÷8 is halve three times. Example: 56 ÷ 8 → 56 → 28 → 14 → 7.
- ÷9 — the 9-times digit trick still helps. The digits of every multiple of 9 add to 9 (or a multiple of 9). If you see 63, add: 6 + 3 = 9, so 63 is a multiple of 9. Then ask “9 × what = 63?” = 7.
- ÷6 — break 6 into 2 × 3. Divide by 2 (halve), then divide by 3. Example: 48 ÷ 6 → 48 ÷ 2 = 24, then 24 ÷ 3 = 8.
6. When it doesn’t divide evenly — remainders
What’s left over
Sometimes a number doesn’t split evenly. Example: 23 ÷ 5. We know 5 × 4 = 20 and 5 × 5 = 25. 20 fits, 25 is too big. So 23 ÷ 5 is 4 with something left over. That leftover is called the remainder.
23 ÷ 5 = 4 R 3
Read: “23 divided by 5 is 4 remainder 3.” It means: 5 goes into 23 four times (using up 20), and 3 is left over.
The check. A division-with-remainder answer is right if:
quotient × divisor + remainder = dividend
For our example: 4 × 5 + 3 = 20 + 3 = 23. ✅
Rule to remember: the remainder must always be less than the divisor. If the remainder is bigger than the divisor, another whole group fits — the quotient is too small.
7. Special cases: ÷1, ÷itself, and why you can’t divide by 0
Three rules every 3rd grader should own
- Any number ÷ 1 = itself. 42 ÷ 1 = 42. (One group of 42.)
- Any number ÷ itself = 1. 42 ÷ 42 = 1. (One group of 42 fits into 42.)
- 0 ÷ any number = 0. 0 ÷ 42 = 0. (Zero stickers split between 42 kids → nobody gets any.)
- Any number ÷ 0 = undefined. 42 ÷ 0 = ???. We’re asking “0 × what = 42?” No such number exists — anything × 0 is 0, never 42. So division by 0 has no answer.
Rule of thumb: dividing by 0 is a rule for life — it’s undefined, forever. If a student sees ÷0 on a homework problem, that problem is broken.
25 practice questions
Read the question. Try the problem in your head or on scratch paper. Then click the button to reveal the answer and the reasoning.
Question 1
Write the whole fact family for the numbers 3, 8, and 24. (Two multiplication facts and two division facts.)
Answer:
3 × 8 = 24
8 × 3 = 24
24 ÷ 3 = 8
24 ÷ 8 = 3
The biggest number (24) is the answer when you multiply, and it’s the starting number (dividend) when you divide.
Question 2
If 6 × 7 = 42, what is 42 ÷ 6?
Answer: 7
Division undoes multiplication. If 6 × 7 = 42, then dividing 42 by 6 “undoes” the ×6 and gets us back to 7. Every multiplication fact gives you two division facts.
Question 3
Solve: 36 ÷ 4
Answer: 9
Ask “4 × what = 36?” → 4 × 9 = 36. So 36 ÷ 4 = 9.
Or use the ÷4 shortcut — halve twice: 36 → 18 → 9. Same answer.
Question 4
Solve: 45 ÷ 5
Answer: 9
Ask “5 × what = 45?” → 5 × 9 = 45. Or skip-count by 5s: 5, 10, 15, 20, 25, 30, 35, 40, 45 — that’s 9 fives.
Question 5
Solve: 80 ÷ 10
Answer: 8
Dividing by 10 is the easy one — just drop the zero. 80 ÷ 10 = 8.
The full reasoning: 10 × 8 = 80, so 80 ÷ 10 = 8.
Question 6
Solve: 63 ÷ 9
Answer: 7
Check the digit trick first: 6 + 3 = 9, so 63 is a multiple of 9. Then ask “9 × what = 63?” → 9 × 7 = 63. So 63 ÷ 9 = 7.
Question 7
Solve: 56 ÷ 8
Answer: 7
Ask “8 × what = 56?” → 8 × 7 = 56.
Or halve three times: 56 → 28 → 14 → 7. Same answer, because ÷8 = halve, halve, halve.
Question 8
Solve: 48 ÷ 6
Answer: 8
Break 6 into 2 × 3: 48 ÷ 2 = 24, then 24 ÷ 3 = 8.
Or ask directly “6 × what = 48?” → 6 × 8 = 48. So 48 ÷ 6 = 8.
Question 9
Solve: 72 ÷ 8
Answer: 9
Ask “8 × what = 72?” → 8 × 9 = 72. So 72 ÷ 8 = 9.
Question 10
Solve: 54 ÷ 6
Answer: 9
Ask “6 × what = 54?” → 6 × 9 = 54.
Or break 6 into 2 × 3: 54 ÷ 2 = 27, 27 ÷ 3 = 9. Same answer.
Question 11
Fill in the missing number: ? ÷ 7 = 6
Answer: 42
If ÷7 gives 6, the original number is 7 × 6 = 42. Multiplication undoes the division: 42 ÷ 7 = 6.
Question 12
Fill in the missing number: 45 ÷ ? = 9
Answer: 5
Ask “what times 9 equals 45?” → 5 × 9 = 45. So we divided by 5. Check: 45 ÷ 5 = 9. ✅
Question 13
Solve: 100 ÷ 10
Answer: 10
10 × 10 = 100, so 100 ÷ 10 = 10. (Or drop the zero: 100 → 10.)
Question 14
Solve: 49 ÷ 7
Answer: 7
This is a square-number fact family: 7 × 7 = 49, so 49 ÷ 7 = 7. Only one division fact for a square number.
Question 15
Solve: 64 ÷ 8
Answer: 8
Another square: 8 × 8 = 64, so 64 ÷ 8 = 8.
Or halve three times: 64 → 32 → 16 → 8. Same answer.
Question 16
Solve: 23 ÷ 5 (with remainder)
Answer: 4 R 3
5 × 4 = 20 (fits, 3 left). 5 × 5 = 25 (too big). So 23 ÷ 5 = 4 R 3.
Check: 4 × 5 + 3 = 20 + 3 = 23. ✅
The remainder (3) is less than the divisor (5), which is what we want.
Question 17
Solve: 50 ÷ 7 (with remainder)
Answer: 7 R 1
7 × 7 = 49 (fits, 1 left). 7 × 8 = 56 (too big). So 50 ÷ 7 = 7 R 1.
Check: 7 × 7 + 1 = 49 + 1 = 50. ✅
Question 18
Solve: 37 ÷ 4 (with remainder)
Answer: 9 R 1
4 × 9 = 36 (fits, 1 left). 4 × 10 = 40 (too big). So 37 ÷ 4 = 9 R 1.
Check: 9 × 4 + 1 = 36 + 1 = 37. ✅
Question 19
Solve: 17 ÷ 1
Answer: 17
Any number divided by 1 is itself. One group of 17 has 17 things in it.
Question 20
Solve: 17 ÷ 17
Answer: 1
Any number divided by itself is 1. One group of 17 fits into 17.
Question 21
Solve: 0 ÷ 12
Answer: 0
Zero split between 12 friends — nobody gets any. Zero divided by anything (except zero itself) is 0.
Question 22
Solve: 12 ÷ 0
Answer: undefined
Dividing by zero has no answer. We’d be asking “zero times what equals 12?” — but anything times 0 is 0, never 12. So you can never divide by 0. This is a rule for life.
Question 23
Word problem (sharing): Ms. Kim has 28 pencils to share fairly among 4 students. How many pencils does each student get?
Answer: 7 pencils per student
This is a sharing problem: we know the number of groups (4 students) and we’re looking for how many pencils per group.
28 ÷ 4 = 7. Each student gets 7 pencils.
Check: 4 × 7 = 28. ✅
Question 24
Word problem (grouping): A bakery packages 54 cupcakes into boxes of 6. How many boxes do they fill?
Answer: 9 boxes
This is a grouping problem: we know how many go in each group (6) and we’re looking for how many groups.
54 ÷ 6 = 9. They fill 9 boxes.
Check: 9 × 6 = 54. ✅
Notice both Q23 and Q24 are division, but the “5” and “9” mean different things — one is per student, the other is number of boxes.
Question 25
Challenge (remainder in a word problem): A teacher has 29 markers to share fairly among 4 students. How many markers does each student get, and how many are left over for the teacher?
Answer: 7 markers per student, 1 left over
4 × 7 = 28 (fits — every student gets 7). 4 × 8 = 32 (too many). So each student gets 7, and 29 − 28 = 1 marker is left over.
As a division statement: 29 ÷ 4 = 7 R 1.
Check: 7 × 4 + 1 = 28 + 1 = 29. ✅
In a real classroom, the leftover marker often goes back in the teacher’s drawer — that’s what a remainder looks like in real life.
5 hard word problems for grade 5
Multi-step problems that use division as the inverse of multiplication, long division with remainders, average as total ÷ count, division as a fraction, and unit-rate reasoning. For students who have finished Class 6 and are ready to stretch.
Question 26
Multi-step (division then multiplication): A shipping company charges the same amount for every box. Ms. Alvarez paid $84 for 6 boxes. At that same rate, how much would she pay for 12 boxes?
Answer: $168
Step 1 (division): find the cost of one box. 84 ÷ 6 = 14, so one box costs $14.
Step 2 (multiplication): 12 boxes at $14 each. 12 × 14 = 168.
So 12 boxes cost $168.
Shortcut for this one: 12 boxes is exactly double 6 boxes, so the price is also double: 2 × $84 = $168. Both methods agree — noticing the shortcut is a grade-5 habit worth building.
Why the unit-rate method always works: when a rate is constant (same price per box), we can always divide to find the unit rate, then multiply by any new quantity we want. This is the same idea as “division and multiplication undo each other,” used forward: divide to unpack the rate, multiply to repack.
Check: 168 ÷ 12 = 14. ✅
Question 27
Long division with interpreted remainder: A school is taking 245 students on a field trip. Each bus holds 32 students. What is the smallest number of buses the school must rent so that every student gets a seat?
Answer: 8 buses
245 ÷ 32 = 7 R 21.
Check: 7 × 32 + 21 = 224 + 21 = 245. ✅
The pure division answer says 7 buses hold 224 students — but that leaves 21 students without a seat. Those 21 students still need a bus, so we must round up to 8 buses.
Interpreting the remainder is the whole trick in grade 5. The word problem tells you what to do with it:
- Round up when leftovers still need to be handled (buses, boxes to pack all items, trips).
- Drop it (round down) when leftovers can’t form a complete unit (how many full teams of 6 from 25 kids? → 4 teams, 1 kid sits out).
- Keep it as the answer when the question asks “how many left over?”
Question 28
Average (total ÷ count): Over 5 days, Malik read a total of 340 pages. On days 1–4 he read 62, 71, 58, and 79 pages. What is Malik’s average pages per day, and how many pages did he read on day 5?
Answer: average = 68 pages/day; day 5 = 70 pages
Average: average = total ÷ number of days = 340 ÷ 5 = 68 pages per day.
Day 5: add days 1–4: 62 + 71 + 58 + 79 = 270. Then day 5 = 340 − 270 = 70 pages.
The core idea: an average is just the total shared equally among the count. It answers the sharing (partitive) question: “If Malik had read the same number every day, how many would that be?” Division is the tool that unpacks a total into equal shares — the inverse of multiplication (5 × 68 = 340).
Check: 62 + 71 + 58 + 79 + 70 = 340. ✅ And 340 ÷ 5 = 68. ✅
Question 29
Division as a fraction (mixed number quotient): Four friends want to share 7 pizzas equally, with no leftovers. Write the amount each friend gets as a fraction and as a mixed number.
Answer: 7/4 of a pizza = 1¾ pizzas per friend
A big grade-5 idea: a ÷ b is the same as the fraction a/b.
Here, 7 pizzas shared among 4 friends is 7 ÷ 4 = 7/4.
To turn 7/4 into a mixed number, ask “how many whole 4/4s fit in 7/4?” → 4/4 = 1 whole, so 7/4 = 4/4 + 3/4 = 1 3/4.
Check both ways:
• With division: 7 ÷ 4 = 1 R 3, which is 1 whole and 3 out of 4 more → 1 3/4. ✅
• Backwards (multiplication): 4 × 1 3/4 = 4 × 7/4 = 28/4 = 7 pizzas. ✅
Why this matters: in grade 5, division stops being “whole answer + remainder” and starts being an exact fractional answer. There is no leftover pizza — every slice gets shared. The remainder from long division (the 3) becomes the numerator of the fractional part (3/4).
Question 30
Unit-rate challenge (multi-step): A factory packs 1,248 candles into boxes of 24 candles per box. The boxes are then loaded onto trucks that hold 8 boxes per truck. How many full trucks can the factory ship, and how many boxes are left over after those full trucks leave?
Answer: 6 full trucks; 4 boxes left over
Step 1 — how many boxes total? 1,248 ÷ 24 = 52 boxes.
Check: 24 × 52 = 24 × 50 + 24 × 2 = 1,200 + 48 = 1,248. ✅
Step 2 — how many full trucks? 52 ÷ 8 = 6 R 4.
Check: 8 × 6 + 4 = 48 + 4 = 52. ✅
So the factory ships 6 full trucks (carrying 48 boxes = 1,152 candles) and has 4 boxes (96 candles) left waiting for the next truck.
Why we drop the remainder this time: the question asked for full trucks — 4 boxes is not a full load, so it doesn’t count. Compare with Q27, where we rounded up because every student still needed a seat. Same math, opposite interpretation. Grade 5 is where students learn to read the sentence and choose.
Answer key summary
| Q# | Answer | Q# | Answer | Q# | Answer |
|---|---|---|---|---|---|
| 1 | fact family | 11 | 42 | 21 | 0 |
| 2 | 7 | 12 | 5 | 22 | undefined |
| 3 | 9 | 13 | 10 | 23 | 7 per student |
| 4 | 9 | 14 | 7 | 24 | 9 boxes |
| 5 | 8 | 15 | 8 | 25 | 7 each, 1 left |
| 6 | 7 | 16 | 4 R 3 | 26 | $168 |
| 7 | 7 | 17 | 7 R 1 | 27 | 8 buses |
| 8 | 8 | 18 | 9 R 1 | 28 | avg 68; day 5 = 70 |
| 9 | 9 | 19 | 17 | 29 | 7/4 = 1¾ |
| 10 | 9 | 20 | 1 | 30 | 6 trucks, 4 left |
About SOMATH & Kid Einsteins
SOMATH — School of Math is a math-focused school on the Upper West Side of Manhattan for students in grades 1–12. The Kid Einsteins program (grades 3–4) drills multiplication and division fluency, fractions and decimals, area and perimeter, ratios, and structured word problems in small groups of 6–8 students. This is Class 6 of the Kid Einsteins arc.
Classes are taught by cofounder Marcelo Ambrozio (Northwestern-trained, 15+ years teaching math in NYC) and the SOMATH team.
Location: 226 W 79th St, 1st Floor, New York, NY 10024 (Upper West Side)
Phone: (646) 668-6151
Email: hello@schoolofmath.us
Book a free evaluation for your child — we’ll assess where they are and place them in the right small group.
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