Multiplying and Dividing Multi-Digit Numbers Fluency Practice

Master multi-digit multiplication and division through these 5 detailed exercises with step-by-step solutions, rules, and visual learning aids.

Solution: Exercises 1 to 3
1 2-Digit Multiplication
Exercise 1
Calculate: \(24 \times 13\)
Definition:

Multi-digit multiplication: Method of multiplying numbers with multiple digits using partial products

Standard algorithm method:
  1. Multiply the ones digit of the bottom number by the top number
  2. Multiply the tens digit of the bottom number by the top number (shift one place left)
  3. Add the partial products together
Expression
\(24 \times 13\)
Partial Products
\(72 + 240\)
Result
\(312\)
24
×
13
×3
72
← 24×3
×10
240
← 24×10
Total
312
Step 1: Multiply ones digit (3) by top number (24)

\(3 \times 24 = 72\)

Step 2: Multiply tens digit (1) by top number (24)

\(1 \times 24 = 24\), but since it's in the tens place: \(24 \times 10 = 240\)

Step 3: Add partial products

\(72 + 240 = 312\)

\(24 \times 13 = 312\)
Final answer:

\(24 \times 13 = 312\)

Applied rules:

• Place Value: Each digit has a specific place value (ones, tens, hundreds)

• Distributive Property: \(24 \times 13 = 24 \times (10 + 3) = (24 \times 10) + (24 \times 3)\)

• Regrouping: Carry over when products exceed 9 in any place value

Key Concept: The standard algorithm breaks down multiplication into smaller, manageable parts using the distributive property.
2 2-Digit Division
Exercise 2
Calculate: \(84 \div 4\)
Definition:

Long division: Method of dividing multi-digit numbers by repeatedly subtracting multiples of the divisor

Expression
\(84 \div 4\)
Division
\(21\)
Step 1: How many times does 4 go into 8? → 2 times
Step 2: \(2 \times 4 = 8\), subtract from 8 → 0
Step 3: Bring down the 4
Step 4: How many times does 4 go into 4? → 1 time
Step 5: \(1 \times 4 = 4\), subtract from 4 → 0
Step 1: Divide tens place

How many groups of 4 can we make from 8 tens? Answer: 2 groups

Step 2: Subtract and bring down

\(8 - (2 \times 4) = 0\), bring down the 4 to make 4 ones

Step 3: Divide ones place

How many groups of 4 can we make from 4 ones? Answer: 1 group

Step 4: Check remainder

\(4 - (1 \times 4) = 0\), so there's no remainder

\(84 \div 4 = 21\)
Final answer:

\(84 \div 4 = 21\)

Applied rules:

• Division Algorithm: Dividend = Divisor × Quotient + Remainder

• Place Value: Work from highest place value to lowest

• Estimation: Use multiplication facts to determine how many times divisor fits

Key Concept: Division is the inverse operation of multiplication. Always check: \(21 \times 4 = 84\)
3 3-Digit Multiplication
Exercise 3
Calculate: \(123 \times 45\)
Definition:

3-digit multiplication: Extends the standard algorithm to numbers with three digits

Expression
\(123 \times 45\)
Partial Products
\(615 + 4920\)
Result
\(5535\)
123
×
45
×5
615
← 123×5
×40
4920
← 123×40
Total
5535
Step 1: Multiply ones digit (5) by 123

\(5 \times 123 = 615\)

Step 2: Multiply tens digit (4) by 123

\(4 \times 123 = 492\), but since it's in the tens place: \(492 \times 10 = 4920\)

Step 3: Add partial products

\(615 + 4920 = 5535\)

Step 4: Verify with estimation

\(120 \times 50 = 6000\), our answer 5535 is close to 6000 ✓

\(123 \times 45 = 5535\)
Final answer:

\(123 \times 45 = 5535\)

Applied rules:

• Distributive Property: \(123 \times 45 = 123 \times (40 + 5) = (123 \times 40) + (123 \times 5)\)

• Place Value: Each digit's position determines its multiplier

• Regrouping: Carry over when products exceed 9 in any place value

Key Concept: For each digit in the bottom number, multiply by the top number and shift the result to the correct place value.
Multiplication & Division Rules and Methods
\(a \times b = c\)
Multiplication
Multiplication
\(24 \times 13 = 312\)
Standard Algorithm
Division
\(84 \div 4 = 21\)
Long Division
Check
\(21 \times 4 = 84\)
Verification
Key definitions:

Multiplication: Repeated addition of equal groups

Division: Splitting a number into equal parts

Partial Products: Products of each digit multiplied by the other number

Place Value: The value of a digit based on its position

Multiplication methodology:
  1. Align numbers: Place values lined up correctly
  2. Multiply each digit: Start with ones, move to higher place values
  3. Shift products: Each new product moves one place to the left
  4. Add partial products: Sum all partial products
Division methodology:
  1. Divide: How many times divisor goes into dividend
  2. Multiply: Divisor by quotient digit
  3. Subtract: Result from dividend
  4. Bring down: Next digit
  5. Repeat: Until no more digits
Tip 1: Estimate first! Round numbers to check if answer makes sense.
Tip 2: Always check division: Quotient × Divisor = Dividend.
Tip 3: In multiplication, each digit in bottom number creates a new row.
Tip 4: Count decimal places in multiplication to place decimal in answer.
Common errors: Misaligning place values, forgetting to carry over, miscalculating partial products.
Fluency tips: Practice basic multiplication facts, use estimation, check answers.
Essential formulas:

• Distributive Property: \(a(b + c) = ab + ac\)

• Division Algorithm: Dividend = Divisor × Quotient + Remainder

• Multiplication by Powers of 10: Add zeros for each power of 10

• Zero Property: Any number × 0 = 0

Solution: Exercises 4 to 5
4 3-Digit Division
Exercise 4
Calculate: \(432 \div 12\)
Definition:

Multi-digit division: Long division with 3-digit dividends and 2-digit divisors

Expression
\(432 \div 12\)
Division
\(36\)
Step 1: How many times does 12 go into 43? → 3 times (12×3=36)
Step 2: \(43 - 36 = 7\), bring down the 2 to make 72
Step 3: How many times does 12 go into 72? → 6 times (12×6=72)
Step 4: \(72 - 72 = 0\), no remainder
Step 1: Divide hundreds and tens place

Look at first two digits: 43. How many groups of 12 fit in 43? Answer: 3 groups (\(12 \times 3 = 36\))

Step 2: Subtract and bring down next digit

\(43 - 36 = 7\), bring down the 2 to make 72

Step 3: Divide remaining part

How many groups of 12 fit in 72? Answer: 6 groups (\(12 \times 6 = 72\))

Step 4: Check remainder

\(72 - 72 = 0\), so there's no remainder

\(432 \div 12 = 36\)
Final answer:

\(432 \div 12 = 36\)

Verification: \(36 \times 12 = 432\) ✓
Applied rules:

• Long Division Steps: Divide, Multiply, Subtract, Bring Down, Repeat

• Estimation: Use known facts to estimate quotients

• Verification: Always check by multiplying quotient by divisor

Key Concept: When the divisor doesn't fit into the first digit(s), consider more digits until it fits.
5 Word Problem
Exercise 5
A school has 24 classrooms. Each classroom has 28 desks. How many desks are there in total?
Definition:

Word problems: Real-world applications requiring mathematical operations

Problem
24 classrooms × 28 desks each
Calculation
\(24 \times 28\)
Result
\(672\)
24
×
28
×8
192
← 24×8
×20
480
← 24×20
Total
672
Step 1: Identify operation needed

We need total desks, so multiply number of classrooms by desks per classroom

Step 2: Set up multiplication

\(24 \times 28\)

Step 3: Multiply ones digit (8) by 24

\(8 \times 24 = 192\)

Step 4: Multiply tens digit (2) by 24

\(2 \times 24 = 48\), but since it's in the tens place: \(48 \times 10 = 480\)

Step 5: Add partial products

\(192 + 480 = 672\)

Step 6: State the answer in context

There are 672 desks in total at the school

There are 672 desks in total.
Final answer:

There are 672 desks in total at the school.

Verification: \(24 \times 28 = 672\) ✓
Applied rules:

• Problem Solving: Identify what is being asked and relevant numbers

• Operations: Determine which mathematical operation to use

• Context: Always state the answer in the context of the problem

Key Concept: Word problems require translation from words to mathematical expressions, then solving using appropriate operations.
Multiplication & Division Laws, Methods, Rules & Definitions
\(a \times b = c\)
Multiplication
Key definitions:

Factor: Numbers that are multiplied together

Product: The result of multiplication

Dividend: Number being divided

Divisor: Number doing the dividing

Quotient: Result of division

Remainder: Amount left over after division

Complete multiplication methodology:
  1. Set up: Write numbers vertically with place values aligned
  2. Start from right: Multiply bottom number's rightmost digit by top number
  3. Carry over: If product > 9, write ones digit, carry tens to next column
  4. Move left: Repeat for next digit in bottom number, shift result one place left
  5. Add results: Sum all partial products
Complete division methodology:
  1. Divide: Look at first digit(s) of dividend, divide by divisor
  2. Multiply: Multiply divisor by quotient digit
  3. Subtract: Subtract result from current dividend part
  4. Bring down: Bring down next digit from dividend
  5. Repeat: Continue until all digits processed
Tip 1: Use area models to visualize multiplication: 24×13 = (20+4)×(10+3).
Tip 2: Check division by multiplying quotient by divisor and adding remainder.
Tip 3: Estimate first to catch calculation errors.
Tip 4: Remember: multiplication and division are inverse operations.
Common errors: Misaligning columns, forgetting to carry, incorrect place value shifts.
Exam preparation: Practice basic facts, show work clearly, verify answers.
Formulas to know by heart:

• Distributive Property: \(a(b + c) = ab + ac\)

• Division Algorithm: Dividend = Divisor × Quotient + Remainder

• Commutative Property: \(a \times b = b \times a\)

• Associative Property: \((a \times b) \times c = a \times (b \times c)\)

• Identity Properties: \(a \times 1 = a\), \(a \div 1 = a\)

Fluency Tips:
  • Practice daily with timed drills
  • Memorize multiplication tables up to 12×12
  • Use mental math strategies like breaking apart numbers
  • Always estimate to check reasonableness
  • Review place value concepts regularly

Questions & Answers

Question: I get confused when multiplying larger numbers like 247 × 36. How do I keep track of where to put each digit?

Answer: Great question! The key is understanding place value. Here's a systematic approach:

  • When multiplying by the ones digit (6 in 36), write the result normally
  • When multiplying by the tens digit (3 in 36), shift the result one place to the left (add a zero at the end)
  • When multiplying by the hundreds digit, shift two places left (add two zeros)

For 247 × 36:

  • First: 247 × 6 = 1482 (write in ones, tens, hundreds, thousands)
  • Second: 247 × 3 = 741, but since 3 is in tens place: 7410 (shifted one place left)
  • Third: Add 1482 + 7410 = 8892

Think of it as: 247 × 36 = 247 × (30 + 6) = (247 × 30) + (247 × 6)

Question: When doing long division, how do I know when to bring down the next digit?

Answer: The long division process follows a consistent pattern: Divide, Multiply, Subtract, Bring Down, Repeat.

  • Divide: How many times does the divisor go into the current part of the dividend?
  • Multiply: Multiply the divisor by your answer
  • Subtract: Subtract your result from the current part
  • Bring Down: This happens AFTER subtraction. Bring down the next digit from the dividend
  • Repeat: Continue the cycle with your new number

Example: 432 ÷ 12

  • Divide: 12 into 43 = 3, Multiply: 3×12=36, Subtract: 43-36=7
  • Bring Down: the 2 to make 72
  • Divide: 12 into 72 = 6, Multiply: 6×12=72, Subtract: 72-72=0
  • No more digits to bring down, so we're done!

Question: My child struggles with multiplication fluency. How can I help them practice at home?

Answer: Building multiplication fluency takes consistent practice with engaging methods:

  • Daily practice: Just 10-15 minutes daily is better than longer sessions once a week
  • Games and apps: Use multiplication games, flashcards, or apps like XtraMath
  • Real-world connections: Practice with items around the house (eggs in cartons, tiles, etc.)
  • Patterns and tricks: Teach shortcuts like doubling for ×2, or adding a zero for ×10
  • Progress tracking: Keep a chart to celebrate improvements

Focus on building conceptual understanding first (what multiplication means), then work on speed. Fluency comes with understanding!

Also encourage estimation skills - if they're calculating 24×13, they should know the answer is close to 20×10=200.