Solved Exercises on Division Using Rectangular Arrays in Grade 3

Master division using rectangular arrays: visual models, row-column partitioning, and real-world applications through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Basic Array Division
Exercise 1
Use an array to solve 12 ÷ 3. Draw the array and show how it represents the division.
Definition:

Division Using Arrays: A method where a rectangular arrangement of objects is divided into equal rows or columns to visualize and solve division problems.

Array Division Method:
  1. Draw the total number of items: Create a rectangle with the dividend number of items
  2. Identify the divisor: Determine how many items should be in each row or column
  3. Partition the array: Group items into equal rows or columns
  4. Count the groups: The number of rows or columns is the quotient
  5. Verify: Multiply to check that the answer is correct
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4 rows of 3
12 ÷ 3 = 4
Total items
12
Items per group
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Number of groups
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Step 1: Create the array

Draw 12 items arranged in rows and columns

Step 2: Group by the divisor

Group items into sets of 3 (the divisor)

Step 3: Count the groups

Count how many groups of 3 we have (4 groups)

Step 4: State the answer

12 ÷ 3 = 4

12 ÷ 3 = 4
Final answer:

12 ÷ 3 = 4 because when 12 items are arranged in an array and grouped by 3s, there are 4 groups.

Applied rules:

• Array Partitioning: Divide the array into equal groups

• Visual Counting: Count the number of equal groups formed

• Verification: Multiply the quotient by the divisor to check

2 Row-Based Division
Exercise 2
Create an array for 15 ÷ 5 by arranging items in rows. Show how many rows you get.
Definition:

Row-Based Division: Arranging items in rows of equal length to visualize division, where the divisor determines the number of items per row.

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Total items
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Items per row
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Number of rows
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Step 1: Arrange items in rows of 5

Each row has exactly 5 items

Step 2: Count the total number of rows

There are 3 complete rows

Step 3: Relate to division

15 total items ÷ 5 items per row = 3 rows

Step 4: Verify the answer

3 rows × 5 items per row = 15 items total ✓

15 ÷ 5 = 3 rows
Final answer:

15 ÷ 5 = 3 because when 15 items are arranged in rows of 5, there are 3 rows.

Applied rules:

• Row Formation: Items are arranged in horizontal groups

• Equal Distribution: Each row contains the same number of items

• Counting Principle: The number of rows is the quotient

3 Column-Based Division
Exercise 3
Use an array to solve 18 ÷ 6 by arranging items in columns. Show how many columns you get.
Definition:

Column-Based Division: Arranging items in columns of equal height to visualize division, where the divisor determines the number of items per column.

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Total items
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Items per column
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Number of columns
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Step 1: Arrange items in columns of 6

Each column has exactly 6 items

Step 2: Count the total number of columns

There are 3 complete columns

Step 3: Relate to division

18 total items ÷ 6 items per column = 3 columns

Step 4: Verify the answer

3 columns × 6 items per column = 18 items total ✓

18 ÷ 6 = 3 columns
Final answer:

18 ÷ 6 = 3 because when 18 items are arranged in columns of 6, there are 3 columns.

Applied rules:

• Column Formation: Items are arranged in vertical groups

• Equal Height: Each column contains the same number of items

• Vertical Counting: The number of columns is the quotient

Division Using Arrays Overview
Total Items ÷ Items per Group = Number of Groups
Array Division Formula
Method 1
Rows
Horizontal groups
Method 2
Columns
Vertical groups
Method 3
Equal Groups
Same size groups
Method 4
Verification
Multiply to check
Key definitions:

Array: A rectangular arrangement of objects in rows and columns

Row: A horizontal line of objects in an array

Column: A vertical line of objects in an array

Division: The operation of splitting a number into equal groups

Quotient: The result of a division problem

Array Division Strategies:
  1. Identify the dividend: The total number of items to arrange
  2. Identify the divisor: The number of items per group (row or column)
  3. Draw the array: Create the rectangular arrangement
  4. Count groups: Count the number of equal rows or columns
  5. State the answer: The number of groups is the quotient
Key Principle: Arrays provide a visual representation of equal groups in division.
Visual Clarity: Arrays make it easy to see how items are grouped equally.
Tip 1: Start by deciding whether to make rows or columns based on the divisor.
Tip 2: Draw arrays neatly to easily count rows or columns.
Tip 3: Always verify your answer by multiplying the quotient by the divisor.
Solution: Exercises 4 to 5
4 Complex Array Problem
Exercise 4
A farmer plants 24 carrots in an array with 4 carrots in each row. How many rows does he have? Show your work using an array.
Definition:

Real-World Array Problem: Applying array division to practical scenarios involving equal distribution of objects in rows or columns.

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Total carrots
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Carrots per row
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Number of rows
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Step 1: Identify the total number of items

The farmer has 24 carrots to plant

Step 2: Identify the number of items per group

Each row will have 4 carrots

Step 3: Arrange in an array

Create rows with 4 carrots each

Step 4: Count the number of rows

Count how many rows of 4 were formed (6 rows)

Step 5: Verify the answer

6 rows × 4 carrots per row = 24 carrots total ✓

24 ÷ 4 = 6 rows
Final answer:

The farmer has 6 rows because 24 ÷ 4 = 6

Applied rules:

• Real-World Application: Division solves practical distribution problems

• Array Modeling: Visual representation helps understand the problem

• Verification: Multiply back to confirm the division is correct

5 Mixed Array Challenge
Exercise 5
Show two different ways to divide 20 using arrays: (a) 20 ÷ 4 and (b) 20 ÷ 5. Draw both arrays and explain the relationship.
Definition:

Multiple Representations: The same total number can be divided in different ways by changing the number of items per group, resulting in different numbers of groups.

20 ÷ 4 = 5

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5 rows of 4 items each

20 ÷ 5 = 4

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4 rows of 5 items each

First equation
20 ÷ 4 = 5
Second equation
20 ÷ 5 = 4
Relationship
Multiplication fact family
Step 1: Solve 20 ÷ 4

20 items in groups of 4 = 5 groups

Step 2: Solve 20 ÷ 5

20 items in groups of 5 = 4 groups

Step 3: Identify the pattern

Same total, different group sizes, different number of groups

Step 4: Connect to multiplication

4 × 5 = 20 and 5 × 4 = 20

20 ÷ 4 = 5 and 20 ÷ 5 = 4
Final answer:

20 ÷ 4 = 5 (5 rows of 4) and 20 ÷ 5 = 4 (4 rows of 5), showing how the same total can be grouped differently.

Applied rules:

• Multiple Solutions: Same dividend, different divisors yield different quotients

• Fact Family Connection: Division facts relate to multiplication facts

• Inverse Relationship: Division and multiplication are inverse operations

Advanced Array Division Concepts
If a × b = c, then c ÷ a = b and c ÷ b = a
Multiplication-Division Relationship
Key definitions:

Factor: A number that divides another number evenly

Product: The result of multiplying two or more numbers

Dividend: The number being divided

Divisor: The number that divides the dividend

Quotient: The result of division

Advanced Array Strategies:
  1. Rotate the array: Switch between rows and columns to find easier divisions
  2. Factor pairs: Use multiplication facts to help with division
  3. Partial arrays: Break large arrays into smaller, manageable sections
  4. Verification: Always check by multiplying the quotient by the divisor
  5. Connection building: Link arrays to real-world rectangular arrangements
Key Pattern: The same total can be arranged in different array configurations.
Rectangular Models: Arrays connect to area and geometry concepts.
Tip 1: Arrays help visualize the connection between multiplication and division.
Tip 2: Practice switching between rows and columns for the same total.
Tip 3: Use arrays to understand remainders in division.
Advanced rules:

• Array Flexibility: Same total, different arrangements (e.g., 12 = 3×4 or 2×6)

• Dimension Relationship: Number of rows × number of columns = total items

• Inverse Operations: Multiplication and division undo each other

• Commutative Connection: Array rotation reflects multiplication commutativity

Example
Division
Array Representation
Simple
12 ÷ 3 = 4
4 rows of 3
Medium
18 ÷ 6 = 3
3 rows of 6
Advanced
24 ÷ 8 = 3
3 rows of 8

Questions & Answers

Question: Why do we use arrays to learn division? Can't we just divide normally?

Answer: Arrays are like a picture of division! Here's why they're helpful:

  • Visual Understanding: You can SEE how items are grouped equally
  • Concrete Connection: Shows the actual process of dividing into equal groups
  • Foundation Building: Prepares you for more complex division later
  • Pattern Recognition: Helps you see relationships between multiplication and division
  • Error Prevention: Easier to catch mistakes when you can see the groups

Arrays help you UNDERSTAND division, not just memorize it!

Question: How can I help my child practice division using arrays at home?

Answer: Here are some practical activities to try:

  • Use objects: Arrange toys, coins, or snacks in rows and columns
  • Draw arrays: Sketch dots or squares in grids to represent problems
  • Cooking: Arrange food items in arrays (cookies on baking sheets)
  • Art projects: Create arrays with stickers, beads, or drawings
  • Games: "I spy an array" around the house (tiles, windows, etc.)
  • Grid paper: Use graph paper to create neat arrays

The key is making it hands-on and visual!

Question: What happens if I can't make equal rows or columns? Like with 13 items?

Answer: Great question! When you can't make equal groups, you get a remainder:

Example with 13 ÷ 4:

  • You can make 3 full rows of 4: 3 × 4 = 12
  • You have 1 item left over: 13 - 12 = 1
  • So: 13 ÷ 4 = 3 remainder 1

The leftover items are called the "remainder." In arrays, they might sit outside the main rectangular formation.

This introduces you to division with remainders!