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What is $3 \div 3$?

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Explanation: Any number divided by itself is 1: $3 \div 3 = 1$.

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Division

Division Practice Quiz with a Step-by-Step Interactive Lesson

Use the quiz at the top of the page to practice division. If you want a refresher on division facts, remainders, and long division steps, click Start lesson to open a step-by-step guide.

How this division practice works

1. Take the quiz: answer the questions at the top of the page to practice division facts, mental math, and basic division.
2. Open the lesson (optional): review division strategies with examples, quick checks, and reminders for remainders and estimation.
3. Retry: return to the quiz and apply the strategy right away to improve speed, accuracy, and confidence.

What you’ll learn in the division lesson

Meaning, models & vocabulary

Division as sharing equally (partitive division)
Division as grouping (quotative division)
Dividend, divisor, quotient, and remainder

Division & multiplication relationship

Division as the inverse of multiplication
Fact families: $a\times b=c$ connects to $c\div a=b$ and $c\div b=a$

Division facts & mental strategies

Patterns for ÷1, ÷2, ÷5, ÷10 and powers of ten
Halving and doubling to simplify
Use multiplication to check: $q\times d + r = n$

Remainders & long division

Remainders explained: $n = d\times q + r$ with $0 \le r < d$
Estimate, divide, multiply, subtract, bring down (repeat)
Interpreting remainders in word problems (round, leftover, or fractional answer)

Back to the quiz

When you’re ready, return to the quiz at the top of the page and continue practicing division until it feels automatic.

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Division
Lesson

Step-by-step guide

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Division Lesson

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Lesson Overview

Lesson overview

Purpose: Build a clear understanding of division and learn reliable strategies for division facts, remainders, and long division.

Success criteria

Explain division as sharing equally (partitive) and as grouping (quotative).
Use the inverse relationship between multiplication and division to solve problems.
Use fact families to connect $a\times b=c$ with $c\div a=b$ and $c\div b=a$.
Understand and interpret remainders using $n = d\times q + r$.
Use efficient strategies for division facts (÷1, ÷2, ÷5, ÷10, halving, and checking with multiplication).
Divide larger numbers using estimate → divide → multiply → subtract → bring down (long division steps).

Key vocabulary

Dividend: the number being divided (in $a\div b$, $a$ is the dividend).
Divisor: the number you divide by (in $a\div b$, $b$ is the divisor).
Quotient: the result of division.
Remainder: what is left over when a division does not divide evenly.

Quick pre-check

Pre-check 1: Which expression means “share 12 among 3 equal groups”?

$12\times 3$ $12\div 3$ $12+3$ $12-3$

Hint: Division is used when you split into equal groups or ask how many groups fit.

Pre-check 2: Compute $12\div 3$.

Hint: If $3\times 4=12$, then $12\div 3=4$.

Sharing & Grouping

Division as sharing and as grouping

Learning goal: Recognize two meanings of division and choose the correct model for a word problem.

Key idea

Division answers “How many in each group?” or “How many groups?” depending on the situation:

Sharing equally (partitive): Split $n$ items into $d$ equal groups. How many in each group? That is $n\div d$.
Grouping (quotative): Make groups of size $d$. How many groups fit into $n$? That is also $n\div d$.

Worked example

Example: $15\div 3$

Sharing: 15 stickers shared into 3 equal groups gives 5 in each group.
Grouping: How many groups of 3 fit in 15? There are 5 groups.
So, $15\div 3 = 5$.

Try it

Try it 1: “24 cookies are shared equally among 6 friends.” What operation matches?

$24\div 6$ $24\times 6$ $24+6$ $24-6$

Hint: Sharing equally means divide the total by the number of groups.

Try it 2: Compute $49\div 7$.

Hint: Use a related multiplication fact: $7\times ? = 49$.

Summary

Division can mean sharing equally or counting how many groups fit.
Both models use the same notation $n\div d$.

Inverse Relationship

Division and multiplication are inverses

Learning goal: Use multiplication to solve division problems and check answers quickly.

Key idea

Division “undoes” multiplication. If $a\times b=c$, then: $\,c\div a=b$ and $\,c\div b=a$. This relationship builds division facts from multiplication facts.

Worked example

Example: $56\div 8$

Think: $8\times ? = 56$. Since $8\times 7 = 56$, the quotient is 7.
Check: $7\times 8 = 56$. Correct.

Try it

Try it 1: Compute $48\div 8$.

Hint: Find the number that makes $8\times ? = 48$.

Try it 2: If $9\times 4 = 36$, which division sentence is true?

$36\div 4 = 9$ $36\div 9 = 5$ Both $36\div 4 = 9$ and $36\div 9 = 4$ $36\div 4 = 13$

Hint: Fact families connect one multiplication fact to two division facts.

Summary

Use multiplication facts to find division quotients faster.
Always check with $q\times d$ to confirm.

Facts & Strategies

Efficient strategies for division facts

Learning goal: Use patterns and mental strategies to divide quickly and accurately.

Key patterns

÷1: the quotient is the same number
÷10: move the decimal one place left (for base-10 numbers)
÷5: divide by 10, then double
÷2: halve
÷4: halve twice
÷8: halve three times
Check: $q\times d = n$ (or $q\times d + r = n$ with remainders)

Worked example

Example: $72\div 9$

Think: $9\times ? = 72$. Since $9\times 8 = 72$, the quotient is $8$.
Check: $8\times 9 = 72$.

Try it

Try it 1: Compute $81\div 9$.

Hint: Find $?$ so that $9\times ? = 81$.

Try it 2: Compute $120\div 8$ using halving.

Hint: ÷8 means halve three times: $120\to 60\to 30\to 15$.

Summary

Use patterns first (÷1, ÷2, ÷5, ÷10, powers of ten).
Use halving for ÷4 and ÷8, and multiplication to check results.

Remainders

Remainders and what they mean

Learning goal: Understand remainders and represent division with the equation $n = d\times q + r$.

Key idea

When a number does not divide evenly, we can write: $\,n = d\times q + r$, where $0 \le r < d$.
This shows the quotient $q$ and the remainder $r$.

Worked example

Example: $29\div 6$

$6\times 4 = 24$, and $29-24 = 5$.
So $29\div 6 = 4$ remainder $5$.
Check: $6\times 4 + 5 = 29$.

Try it

Try it 1: What is the quotient in $31\div 7$? (quotient only)

Hint: Find the biggest multiple of 7 that is ≤ 31.

Try it 2: What is the remainder in $31\div 7$? (remainder only)

Hint: If $7\times 4 = 28$, then $31-28 = 3$.

Worked solution

Find the largest multiple of 7 that fits in 31: $7\times 4=28$.
Subtract: $31-28=3$.
So $31\div 7 = 4$ remainder $3$, and $31 = 7\times 4 + 3$.

Summary

Remainders satisfy $0 \le r < d$.
Use $d\times q + r$ to check the division result.

Long Division

Long division step-by-step

Learning goal: Divide multi-digit numbers using estimate and repeated subtraction of multiples (the long division algorithm).

Key steps

Estimate: How many times does the divisor fit?
Multiply: Multiply the divisor by the estimate.
Subtract: Find what remains.
Bring down: Bring down the next digit and repeat.
Check: divisor × quotient + remainder = dividend.

Worked example

Example: $84\div 6$

Estimate: $6$ goes into $8$ one time. Multiply: $1\times 6=6$. Subtract: $8-6=2$.
Bring down 4 → 24. Estimate: $6$ goes into $24$ four times. Multiply: $4\times 6=24$. Subtract: $24-24=0$.
So $84\div 6 = 14$.

Try it

Try it: Compute $96\div 8$.

Hint: $8$ goes into $9$ once (remainder 1), bring down 6 to make 16, then $16\div 8=2$. So the quotient is 12.

Worked solution

$96\div 8$: $8$ goes into $9$ one time → remainder $1$. Bring down 6 → 16.
$16\div 8 = 2$ remainder $0$.
So $96\div 8 = 12$. Check: $12\times 8 = 96$.

Summary

Long division repeats estimate → multiply → subtract → bring down.
Checking with multiplication prevents common mistakes.

Divide First

Order of operations: divide first

Learning goal: Evaluate expressions that contain division and addition/subtraction.

Key idea

When an expression contains $+$ or $−$ and $ \div $, do division first (then add or subtract).

Worked example

Example: $18 - 12\div 3$

Step 1: Divide: $12\div 3=4$.
Step 2: Subtract: $18-4=14$.
So, $18 - 12\div 3 = 14$.

Try it

Try it 1: Compute $5 + 18\div 3$.

Hint: Divide $18\div 3$ first, then add 5.

Try it 2: Compute $20 - 28\div 7$.

Hint: Divide $28\div 7$ first.

Summary

In mixed expressions, do division before addition/subtraction.
Work step-by-step to avoid common order-of-operations errors.

Applications & Real Life

Why division matters

Learning goal: Connect division to equal sharing, rates, averages, fractions, and everyday decision-making.

Where you use division

Sharing equally: splitting snacks, prizes, or money.
Rates: distance ÷ time, cost ÷ quantity (unit price).
Averages: total ÷ number of items.
Fractions and decimals: division creates fractional results.
Geometry: find side length from area ÷ width (rearranging formulas).

Worked example: unit price

Example: A pack costs $12 for 3 items.

Unit price = total cost ÷ number of items = $12\div 3 = 4$.
Answer: Each item costs $4.

Try it

Try it 1: A 30 km trip takes 6 hours. What is the average speed (km/h)?

Hint: Average speed = distance ÷ time.

Final recap

Division models sharing equally and grouping.
Division and multiplication are inverses: use facts to solve faster.
Remainders fit the form $n = d\times q + r$.
Long division repeats estimate → multiply → subtract → bring down.
Division appears everywhere: rates, averages, unit prices, and fractions.

Next step: Close this lesson and try your quiz again. If you miss a question, reopen the book and review the page that matches the skill.