📐 Place Value

Uses column alignment for ones, tens, hundreds[web:222]

Essential for correct positioning of digits

➕ Partial Products

Multiplies digit by digit, creating intermediate results[web:229]

Added together for final answer

🔄 Carrying Over

When product exceeds 9, carry to next column[web:222]

Critical skill for accurate calculation

Set Up the Problem

Write the larger number on top and the smaller number below it. Align digits by place value (ones, tens, hundreds). Draw a line underneath for the answer.[web:222]

   234
×   56
------

Multiply by Ones Digit

Multiply 6 (ones) by each digit in 234, right to left. 6×4=24, 6×3=18, 6×2=12. Write result carrying when needed.[web:222][web:227]

   234
×   56
------
  1404

Multiply by Tens Digit

Put a 0 as placeholder. Multiply 5 (tens) by each digit in 234. 5×4=20, 5×3=15, 5×2=10. Write below, shifted left.[web:222][web:229]

   234
×   56
------
  1404
11700

Add Partial Products

Add all the partial products using column addition. 1404 + 11700 = 13104. This is your final answer![web:222]

   234
×   56
------
  1404
11700
------
13104

Formula 1: Distributive Property

Long multiplication is based on the distributive property:

\[a \times (b + c) = (a \times b) + (a \times c)\]

Example: \(234 \times 56 = 234 \times (50 + 6) = (234 \times 50) + (234 \times 6)\)

Formula 2: Place Value Expansion

Breaking numbers by place value:

\[(a \times 100 + b \times 10 + c) \times (d \times 10 + e)\]

Example: \((2 \times 100 + 3 \times 10 + 4) \times (5 \times 10 + 6)\)

Formula 3: Partial Products Sum

Final answer is sum of all partial products:[web:229]

\[\text{Result} = \sum_{i=0}^{m-1} \sum_{j=0}^{n-1} (a_i \times b_j \times 10^{i+j})\]

Where \(a_i\) and \(b_j\) are individual digits, \(i\) and \(j\) are position indices.

Formula 4: General Multiplication

For any two numbers:

\[A \times B = \sum_{i} (a_i \times 10^i) \times \sum_{j} (b_j \times 10^j)\]

Expands to all combinations of digit products with proper place value.

Formula 5: Carrying Rule

When digit product ≥ 10:

\[\text{Carry} = \lfloor \frac{\text{Product}}{10} \rfloor, \quad \text{Write} = \text{Product} \mod 10\]

Example: If 6×8=48, write 8, carry 4 to next column.

📊 Lattice Method (Grid Method)

Also called Italian method or Chinese method, this ancient technique uses a diagonal grid to organize partial products. Each cell contains a single-digit multiplication split into tens and ones.[web:228][web:231]

Steps:[web:228]

1. Draw a grid with rows and columns for each digit
2. Draw diagonal lines in each cell
3. Write multiplicands along top and right side
4. Multiply each pair, split product across diagonal
5. Add numbers along diagonals, carrying as needed
6. Read answer from left and bottom edges

📋 Box/Area Method

This method uses a rectangular grid where each cell represents partial products based on place value. It connects to the area model of multiplication and makes the distributive property visible.[web:227]

Example: 23 × 47

1. Prerequisites

Master times tables (especially 1-12), understand place value, know how to multiply single digits by multi-digit numbers, and be comfortable with addition.[web:226]

2. Use Concrete Materials

Start with base-ten blocks, place value charts, and area models before moving to the abstract algorithm. Visualization helps conceptual understanding.[web:236]

3. Break Into Steps

Don't teach all 16+ steps at once. Start with 2-digit by 1-digit, then gradually increase complexity. Build fluency at each level.[web:226]

4. Color Coding

Use different colors for ones, tens, hundreds to reinforce place value. Highlight carrying numbers to make them visible during practice.[web:236]

5. Common Errors

Watch for: forgetting placeholders (zeros), misaligning columns, incorrect carrying, forgetting to add carried numbers, and addition errors in final step.

6. Practice & Fluency

Regular practice with varied problem types builds automaticity. Use estimation to check reasonableness of answers and catch major errors quickly.

🌍 The lattice multiplication method was used in medieval Europe and ancient China, dating back over 1000 years![web:228]

📚 The word "multiply" comes from Latin "multiplicare" meaning "to increase manifold or many times"!

🧮 Before calculators, slide rules and multiplication tables were essential tools for scientists and engineers!