3A · Chapter 4 · Lesson 5
Division with Remainders
Describe a remainder as what remains after forming full equal groups.
Today's challenge
43 oranges are packed into bags holding 5 each. How many FULL bags and how many oranges left? Try your own method, then compare it with a model.
What you'll learn
- Describe a remainder as what remains after forming full equal groups.
- Explain the connection between a diagram and a number sentence.
- Solve a fresh application without step-by-step hints.
Before this lesson
Equal-group division.
Prerequisite competency: 3A-C04-L03
Entry · 2 questions
Warm-up: prerequisite check
Try on paper before continuing. If the questions are hard, review the prerequisite concepts.
- E1
40 counters are shared equally among 4 children. How many does each receive?
Show answer and reasoning
Answer: 10
4 equal groups contain 40 counters: 40÷4=10; verify 10×4=40.
- E2
65 counters are shared equally among 5 children. How many does each receive?
Show answer and reasoning
Answer: 13
5 equal groups contain 65 counters: 65÷5=13; verify 13×5=65.
Learn the concept
Words to know
- remainder
- items left after making complete equal groups
- divisor
- the number of groups or the size of each group
Describe a remainder as what remains after forming full equal groups.
Count how many groups of the divisor fit into the dividend. Anything less than one full group is the remainder. Check dividend = divisor × quotient + remainder.
Why this works
Sometimes a total does not fill equal groups exactly. Forty-three counters make 7 complete groups of 6 with 1 left, because 43 = 7 × 6 + 1. The remainder must be smaller than the group size. If every child must ride in a van, a partly filled van may still be needed.
Check the meaning
If the story asks for containers to hold EVERY item, a partly filled container may increase the required count by one.
Concrete → Pictorial → Abstract
- Build: Use counters, grid paper, or labeled bundles to represent the story or quantity.
- Draw: Sketch an array, place-value exchange, or equal-unit bar; label what is known.
- Calculate: Write an equation matching the representation and solve.
- Verify: Use an inverse operation or a second strategy to check.
For advanced students: if you can already explain this concept, start with the independent or challenge set.
Worked examples and immediate checks
Explain each step aloud or in writing. Work the fresh question before revealing its solution.
Worked example 1
Reason and verify
43 oranges are packed into bags holding 5 each. How many FULL bags and how many oranges left?
- Count how many groups of the divisor fit into the dividend. Anything less than one full group is the remainder. Check dividend = divisor × quotient + remainder.
- 43=5×8+3; since 3<5, 8 bags are full and 3 oranges remain.
- If the story asks for containers to hold EVERY item, a partly filled container may increase the required count by one.
Result: 8 bags, 3 left
Understanding check 1 — your turn
37 oranges are packed into bags holding 2 each. How many FULL bags and how many oranges left?
Reveal reasoning
Answer: 18 bags, 1 left — 37=2×18+1; since 1<2, 18 bags are full and 1 oranges remain.
Worked example 2
Reason and verify
95 oranges are packed into bags holding 6 each. How many FULL bags and how many oranges left?
- Count how many groups of the divisor fit into the dividend. Anything less than one full group is the remainder. Check dividend = divisor × quotient + remainder.
- 95=6×15+5; since 5<6, 15 bags are full and 5 oranges remain.
- If the story asks for containers to hold EVERY item, a partly filled container may increase the required count by one.
Result: 15 bags, 5 left
Understanding check 2 — your turn
76 oranges are packed into bags holding 3 each. How many FULL bags and how many oranges left?
Reveal reasoning
Answer: 25 bags, 1 left — 76=3×25+1; since 1<3, 25 bags are full and 1 oranges remain.
Worked example 3
Reason and verify
162 oranges are packed into bags holding 7 each. How many FULL bags and how many oranges left?
- Count how many groups of the divisor fit into the dividend. Anything less than one full group is the remainder. Check dividend = divisor × quotient + remainder.
- 162=7×23+1; since 1<7, 23 bags are full and 1 oranges remain.
- If the story asks for containers to hold EVERY item, a partly filled container may increase the required count by one.
Result: 23 bags, 1 left
Understanding check 3 — your turn
130 oranges are packed into bags holding 4 each. How many FULL bags and how many oranges left?
Reveal reasoning
Answer: 32 bags, 2 left — 130=4×32+2; since 2<4, 32 bags are full and 2 oranges remain.
Guided · 4 questions
Guided practice
Practice with the model visible. Explain the units at each step.
- G1
206 oranges are packed into bags holding 6 each. How many FULL bags and how many oranges left?
Show answer and reasoning
Answer: 34 bags, 2 left
206=6×34+2; since 2<6, 34 bags are full and 2 oranges remain.
- G2
288 oranges are packed into bags holding 7 each. How many FULL bags and how many oranges left?
Show answer and reasoning
Answer: 41 bags, 1 left
288=7×41+1; since 1<7, 41 bags are full and 1 oranges remain.
- G3
93 oranges are packed into bags holding 8 each. How many FULL bags and how many oranges left?
Show answer and reasoning
Answer: 11 bags, 5 left
93=8×11+5; since 5<8, 11 bags are full and 5 oranges remain.
- G4
167 oranges are packed into bags holding 9 each. How many FULL bags and how many oranges left?
Show answer and reasoning
Answer: 18 bags, 5 left
167=9×18+5; since 5<9, 18 bags are full and 5 oranges remain.
Independent · 8 questions
Independent practice
Work without relying on the model or checking answers prematurely.
- I1
43 oranges are packed into bags holding 6 each. How many FULL bags and how many oranges left?
Show answer and reasoning
Answer: 7 bags, 1 left
43=6×7+1; since 1<6, 7 bags are full and 1 oranges remain.
- I2
107 oranges are packed into bags holding 7 each. How many FULL bags and how many oranges left?
Show answer and reasoning
Answer: 15 bags, 2 left
107=7×15+2; since 2<7, 15 bags are full and 2 oranges remain.
- I3
177 oranges are packed into bags holding 8 each. How many FULL bags and how many oranges left?
Show answer and reasoning
Answer: 22 bags, 1 left
177=8×22+1; since 1<8, 22 bags are full and 1 oranges remain.
- I4
268 oranges are packed into bags holding 9 each. How many FULL bags and how many oranges left?
Show answer and reasoning
Answer: 29 bags, 7 left
268=9×29+7; since 7<9, 29 bags are full and 7 oranges remain.
- I5
73 oranges are packed into bags holding 2 each. How many FULL bags and how many oranges left?
Show answer and reasoning
Answer: 36 bags, 1 left
73=2×36+1; since 1<2, 36 bags are full and 1 oranges remain.
- I6
19 oranges are packed into bags holding 3 each. How many FULL bags and how many oranges left?
Show answer and reasoning
Answer: 6 bags, 1 left
19=3×6+1; since 1<3, 6 bags are full and 1 oranges remain.
- I7
53 oranges are packed into bags holding 4 each. How many FULL bags and how many oranges left?
Show answer and reasoning
Answer: 13 bags, 1 left
53=4×13+1; since 1<4, 13 bags are full and 1 oranges remain.
- I8
103 oranges are packed into bags holding 5 each. How many FULL bags and how many oranges left?
Show answer and reasoning
Answer: 20 bags, 3 left
103=5×20+3; since 3<5, 20 bags are full and 3 oranges remain.
Challenge · 2 questions
Challenge practice
Solve original application and explanation problems. Draw and defend a useful model.
- C1
Thirty-five students need vans holding at most eight students each. How many vans are necessary?
Show answer and reasoning
Answer: 5 vans
Four vans hold only 32. Since three students remain, a fifth van is required, even though 35÷8 is 4 remainder 3.
- C2
Fifty-three children need teams of at most 8. If every child must join a team, how many teams are necessary and how many on the last team?
Show answer and reasoning
Answer: 7 teams; 5 on last
Six full teams hold 48, leaving 5 children. A seventh smaller team is necessary.
Exit · 3 questions
Exit check
Solve independently; revisit the skill if an important explanation is missing.
- E1
178 oranges are packed into bags holding 6 each. How many FULL bags and how many oranges left?
Show answer and reasoning
Answer: 29 bags, 4 left
178=6×29+4; since 4<6, 29 bags are full and 4 oranges remain.
- E2
258 oranges are packed into bags holding 7 each. How many FULL bags and how many oranges left?
Show answer and reasoning
Answer: 36 bags, 6 left
258=7×36+6; since 6<7, 36 bags are full and 6 oranges remain.
- E3
50 oranges are packed into bags holding 8 each. How many FULL bags and how many oranges left?
Show answer and reasoning
Answer: 6 bags, 2 left
50=8×6+2; since 2<8, 6 bags are full and 2 oranges remain.