Division as repeated subtraction

3.OA.A.23.OA.A.3 · take · grade 3

Archetype: Division as the Inverse of Multiplication · step in a 4-type progression

▶ Practice — 11 problems

Do the division using the same method as in the example.

Example: When you add two division facts, the dividends add together and the quotients add together.

$6 \div 3 = 2$
$9 \div 3 = 3$
$\overline{15 \div 3 = 5}$

Using the same method, find the number for each box below.

$12 \div 3 = \square$
$\square \div 3 = \square$
$\overline{27 \div 3 = \square}$

Show solution

Understand

The example shows that when you stack two division facts that share the same divisor, the dividends add and the quotients add (6 div 3 = 2 and 9 div 3 = 3 combine to 15 div 3 = 5). Using that same adding rule, we must fill the boxes so that 12 div 3 plus another fact equals 27 div 3.

Givens

Example: 6 div 3 = 2, 9 div 3 = 3, and adding gives 15 div 3 = 5.
The rule: with the same divisor, add the dividends and add the quotients.
The new stack is 12 div 3 = box, then box div 3 = box, summing to 27 div 3 = box.

Unknowns

The quotient of 12 div 3.
The middle dividend and its quotient.
The quotient of 27 div 3.

Constraints

All three facts use divisor 3.
The first two dividends must add to 27, and the first two quotients must add to the bottom quotient.

Plan

#5 Look for a Pattern · also uses: #11 Work Backwards

The example reveals a pattern: dividends add and quotients add. We follow that pattern, and we work backwards from the known total 27 to find the missing middle dividend.

Execute

#5 Look for a Pattern 3.OA.A.2

Divide 12 by 3. Since 3 times 4 is 12, the quotient is 4. So 12 div 3 = 4.

12 \div 3 = 4

Grade 3 division: 12 split into groups of 3 makes 4 groups.

#11 Work Backwards 3.NBT.A.2

The pattern says the top two dividends add to the bottom dividend: 12 plus the middle dividend equals 27. So the middle dividend is 27 minus 12 = 15.

27 - 12 = 15

Grade 3 subtraction: since the dividends must add to 27, the missing one is 27 take away 12.

#5 Look for a Pattern 3.OA.A.2

Divide 15 by 3: 3 times 5 is 15, so 15 div 3 = 5.

15 \div 3 = 5

Grade 3 division: 15 split into groups of 3 makes 5 groups.

#5 Look for a Pattern 3.OA.A.3

Divide 27 by 3 to get 9. This matches the pattern because the top two quotients add: 4 plus 5 equals 9.

27 \div 3 = 9, \quad 4 + 5 = 9

Grade 3: the quotients add the same way the dividends do, confirming 9 is right.

Answer: 12 div 3 = 4; 15 div 3 = 5; 27 div 3 = 9

Review

Dividends: 12 + 15 = 27 (correct total). Quotients: 4 + 5 = 9, and 27 div 3 is indeed 9. Both the dividend sum and the quotient sum line up with the example's rule.

Skip-count by 3 to fill the boxes: 3, 6, 9, 12 (4 steps for 12), continue to 15 (5 steps), and 27 (9 steps), giving the same quotients 4, 5, and 9.

Standards · min grade 3

3.OA.A.2 Interpret whole-number quotients of whole numbers — Dividing 12 and 15 by 3 to get the quotients.
3.NBT.A.2 Fluently add and subtract within 1000 — Finding the missing middle dividend as 27 minus 12.
3.OA.A.3 Solve multiplication and division word problems within 100 — Confirming the quotients add to the bottom quotient.

💡 When the divisor stays the same, dividends add and quotients add, so the boxes are 4, 15, 5, and 9!