Using Arrays to Multiply 2 Digit Numbers in Grade 3

Learn how to use arrays as a visual tool to understand and solve 2-digit multiplication problems effectively.

Array Fundamentals for 2-Digit Numbers
1 Core Concepts
Key Definitions:

Array: A rectangular arrangement of objects in equal rows and equal columns.

2-Digit Multiplication: Multiplication involving numbers with tens and ones places (e.g., 12, 23, 45).

Partial Products: Breaking down multiplication into smaller parts (tens × tens, tens × ones, etc.).

Place Value: Understanding that digits have different values based on their position (tens, ones).

Expanded Form: Writing numbers to show place value (e.g., 23 = 20 + 3).

Distributive Property: Breaking apart numbers to make multiplication easier.

2-Digit Array Structure:

For 2-digit multiplication, we break numbers into tens and ones:

\(23 \times 14 = (20 + 3) \times (10 + 4)\)

This creates 4 partial products: \(20 \times 10\), \(20 \times 4\), \(3 \times 10\), \(3 \times 4\)

Place Value: Break numbers into tens and ones for easier arrays
Distributive Property: Use to break complex problems into simpler ones
Partial Products: Add all partial products to get the final answer
2 Array Components for 2-Digit Numbers
(Tens + Ones) × (Tens + Ones) = Total
2-Digit Array Formula
Tens
×10
First digit
Ones
×1
Second digit
Partial
Multiply Parts
Smaller arrays
Total
Add All
Final answer
Tip 1: Always break 2-digit numbers into tens and ones first.
Tip 2: Draw separate arrays for each partial product.
Tip 3: Add all partial products to get the final answer.
Tip 4: Use different colors for different partial products.
Building Arrays for 2-Digit Numbers
(20 + 3) × (10 + 4) = 20×10 + 20×4 + 3×10 + 3×4
Distributive Property
2-Digit Array Construction Steps:

Step 1: Break both numbers into tens and ones (e.g., 23 = 20 + 3, 14 = 10 + 4)

Step 2: Create a grid showing the partial products

Step 3: Draw arrays for each partial product

Step 4: Calculate each partial product

Step 5: Add all partial products together

Step 6: Write the final multiplication equation

2-Digit Array Process:
  1. Decompose Numbers: Break each 2-digit number into tens and ones
  2. Create Grid: Set up a 2×2 grid for the partial products
  3. Calculate Partial Products: Multiply each combination of tens and ones
  4. Draw Visual Arrays: Represent each partial product with smaller arrays
  5. Sum Results: Add all partial products to get the final answer
  6. Verify: Check that your answer matches the expected result
23×14
(20+3)×(10+4)
4 partial products
Partial 1
20×10=200
Tens × Tens
Partial 2
20×4=80
Tens × Ones
Partial 3
3×10=30
Ones × Tens
2-Digit Array Rules:

• Always decompose numbers into tens and ones first

• Each partial product represents a smaller, more manageable multiplication

• The sum of all partial products equals the final result

• Draw arrays for each partial product to visualize the multiplication

• Double-check that your final sum matches the expected multiplication result

2-Digit Array Examples
3 Basic 2-Digit Arrays
Example 1: 12×13 Array

10×10 = 100

Using Arrays to Multiply 2 Digit Numbers | 3rd Grade Math | Chapter 1-Digit Multiplication

Using Arrays to Multiply 2 Digit Numbers in Grade 3

Learn how to use arrays as a visual tool to understand and solve 2-digit multiplication problems effectively.

2-Digit Array Fundamentals
1 Core Concepts
Key Definitions:

2-Digit Array: A rectangular arrangement of objects to represent multiplication of a 2-digit number by a 1-digit number.

Expanded Form: Breaking down a 2-digit number into tens and ones (e.g., 23 = 20 + 3).

Partial Products: Products found by multiplying the 1-digit number by the tens and ones separately.

Regrouping: Carrying over when products exceed 9 in any place value.

Area Model: Using rectangles to represent multiplication, where length × width = area.

Place Value: Understanding the value of digits based on their position (tens, ones).

2-Digit Array Structure:

For 2-digit by 1-digit multiplication, break down the 2-digit number:

\(23 \times 4 = (20 + 3) \times 4 = (20 \times 4) + (3 \times 4) = 80 + 12 = 92\)

Distributive Property: \(a \times (b + c) = (a \times b) + (a \times c)\)
Place Value Focus: Separate tens and ones for easier computation
Visual Connection: Links multiplication to area concepts
2 Array Components for 2-Digit Numbers
(Tens + Ones) × Single Digit = Partial Products Sum
2-Digit Array Formula
Tens
× Digit
First Partial Product
Ones
× Digit
Second Partial Product
Sum
Add Products
Final Answer
Place Value
Tens/Ones
Value Position
Tip 1: Always separate the 2-digit number into tens and ones first.
Tip 2: Multiply the tens and ones separately by the single digit.
Tip 3: Add the partial products to get the final answer.
Tip 4: Check your answer by counting all objects in the array.
Building 2-Digit Arrays
(Tens × Digit) + (Ones × Digit) = Total
2-Digit Array Calculation
2-Digit Array Construction Steps:

Step 1: Read the multiplication problem (e.g., 24×3)

Step 2: Break the 2-digit number into tens and ones (24 = 20 + 4)

Step 3: Draw a large array showing the tens portion (20×3)

Step 4: Draw a smaller array showing the ones portion (4×3)

Step 5: Count objects in each array separately

Step 6: Add the partial products to find the total

2-Digit Array Process:
  1. Separate the Number: Break the 2-digit number into tens and ones
  2. Create Two Arrays: One for tens and one for ones
  3. Calculate Tens Array: Multiply tens by the single digit
  4. Calculate Ones Array: Multiply ones by the single digit
  5. Combine Results: Add the partial products
  6. Verify Answer: Check by counting all objects in both arrays
23×4 Array
(20×4)+(3×4)
80+12=92
35×2 Array
(30×2)+(5×2)
60+10=70
41×3 Array
(40×3)+(1×3)
120+3=123
18×5 Array
(10×5)+(8×5)
50+40=90
2-Digit Array Rules:

• Always break the 2-digit number into tens and ones first

• Create separate arrays for tens and ones portions

• Count objects systematically in each array

• Add partial products to get the final result

• The total count should match the multiplication result

2-Digit Array Examples
3 Basic 2-Digit Arrays
Example 1: 23×4 Array
TENS PART: 20×4
Tens: 20×4 = 80
ONES PART: 3×4
Ones: 3×4 = 12

Total: 80 + 12 = 92

Equation: 23 × 4 = 92

Construction Steps:
  1. Break 23 into 20 + 3
  2. Create array for 20×4 (20 rows of 4 = 80)
  3. Create array for 3×4 (3 rows of 4 = 12)
  4. Add partial products: 80 + 12 = 92
4 Advanced 2-Digit Arrays
Example 2: 34×2 Array
TENS PART: 30×2
Using Arrays to Multiply 2 Digit Numbers | 3rd Grade Math | Chapter 1-Digit Multiplication

Using Arrays to Multiply 2 Digit Numbers in Grade 3

Learn how to use arrays as a visual tool to understand and solve 2-digit multiplication problems effectively.

Array Fundamentals for 2-Digit Numbers
1 Core Concepts
Key Definitions:

Array: A rectangular arrangement of objects in equal rows and equal columns.

2-Digit Multiplication: Multiplication involving numbers with two digits (10-99).

Partial Products: Breaking down 2-digit multiplication into smaller parts.

Place Value: Understanding tens and ones in 2-digit numbers.

Expanded Form: Writing numbers as the sum of their place values (e.g., 23 = 20 + 3).

Area Model: Using rectangles to represent multiplication visually.

2-Digit Array Structure:

For 2-digit multiplication, break down the problem using place value:

\(23 \times 4 = (20 + 3) \times 4 = (20 \times 4) + (3 \times 4) = 80 + 12 = 92\)

Place Value Focus: Separate tens and ones for easier calculation
Distributive Property: \(a \times (b + c) = (a \times b) + (a \times c)\)
Visual Connection: Arrays help understand the distributive property
2 Array Components
(Tens + Ones) × Single Digit = (Tens × Single) + (Ones × Single)
2-Digit Array Formula
Decompose
Separate Tens/Ones
Break down 2-digit number
Multiply
Each Part
Multiply each part separately
Combine
Add Results
Sum partial products
Verify
Check Answer
Confirm final result
Tip 1: Always decompose the 2-digit number into tens and ones.
Tip 2: Draw separate arrays for each part of the decomposition.
Tip 3: Add the partial products to get the final answer.
Tip 4: Check your work by counting all objects in the combined array.
Building 2-Digit Arrays
Decompose → Multiply Parts → Combine → Verify
2-Digit Array Process
2-Digit Array Construction Steps:

Step 1: Identify the 2-digit number and decompose it into tens and ones

Step 2: Identify the single digit multiplier

Step 3: Create separate arrays for the tens and ones parts

Step 4: Calculate the partial products

Step 5: Add the partial products to get the final result

Step 6: Verify by counting all objects in the combined array

2-Digit Array Process:
  1. Decompose: Break the 2-digit number into tens and ones
  2. Draw Tens Array: Create an array for the tens part
  3. Draw Ones Array: Create an array for the ones part
  4. Calculate Partial Products: Count objects in each array
  5. Combine Results: Add the partial products
  6. Verify: Check by counting the total objects
23×4
(20×4)+(3×4)
80+12=92
34×2
(30×2)+(4×2)
60+8=68
45×3
(40×3)+(5×3)
120+15=135
52×6
(50×6)+(2×6)
300+12=312
2-Digit Array Rules:

• Always decompose the 2-digit number into tens and ones

• Create separate arrays for each part of the decomposition

• Count systematically to avoid missing or double-counting objects

• Add partial products to get the final result

• Verify your answer by counting all objects in the combined array

2-Digit Array Examples
3 Basic 2-Digit Arrays
Example 1: 23×4 Array