Problem Solving Strategies with Examples | Arithmetic Patterns

Strategy Examples 1 to 3

1 Pattern Recognition Strategy

Exercise 1

Find the missing number in this pattern: 4, 8, 12, 16, __, 24
What is the rule for this pattern?

Definition:

Arithmetic Sequence: A pattern where each number increases by the same amount

Strategy: Find the Difference

Look at the first few numbers in the sequence
Find the difference between consecutive numbers
Apply the same difference to find the missing number

→

Step 1: Examine the sequence

We have: 4, 8, 12, 16, __, 24

Step 2: Calculate the difference

8 - 4 = 4, 12 - 8 = 4, 16 - 12 = 4

The rule is: add 4 each time!

Step 3: Apply the rule

16 + 4 = 20, and 20 + 4 = 24 ✓

The missing number is 20

Final answer:

The sequence is: 4, 8, 12, 16, 20, 24

The rule is: add 4 each time

Applied rules:

• Pattern Recognition: Find the consistent change between numbers

• Consistent Addition: Same number added each time

• Verification: Check that the pattern continues correctly

2 Growing Pattern Strategy

Exercise 2

Mia has 3 stickers on Monday, 6 on Tuesday, 9 on Wednesday.
If this pattern continues, how many stickers will she have on Friday?

Definition:

Growing Pattern: Numbers that increase following a rule each time period

Mon
3

→

Tue
6

→

Wed
9

→

Thu
12

→

Fri
15

Step 1: Organize the given information

Monday: 3 stickers, Tuesday: 6 stickers, Wednesday: 9 stickers

Step 2: Determine the pattern

6 - 3 = 3, 9 - 6 = 3

She gains 3 stickers each day!

Step 3: Continue the pattern

Thursday: 9 + 3 = 12 stickers

Friday: 12 + 3 = 15 stickers

Mia will have 15 stickers on Friday

Final answer:

Mia will have 15 stickers on Friday

Applied rules:

• Daily Increase: Same number added each day

• Sequential Counting: Continue the pattern day by day

• Verification: Check that the pattern holds consistently

3 Repeating Pattern Strategy

Exercise 3

The pattern is: Apple, Banana, Orange, Apple, Banana, Orange, ...
What fruit comes next in the 16th position?

Definition:

Repeating Pattern: Items that repeat in the same order

🍎

1st

🍌

2nd

🍊

3rd

🍎

4th

🍌

5th

🍊

6th

🍎

7th

🍌

8th

🍊

9th

🍎

10th

🍌

11th

🍊

12th

🍎

13th

🍌

14th

🍊

15th

16th

Step 1: Identify the repeating unit

The pattern repeats every 3 fruits: Apple, Banana, Orange

Step 2: Use division to find position

To find the 16th fruit: 16 ÷ 3 = 5 remainder 1

This means after 5 full cycles, we're at the 1st position in the next cycle

Step 3: Identify the fruit at that position

The 1st position in the pattern is Apple (🍎)

The 16th fruit is an Apple

Final answer:

The fruit in the 16th position is an Apple

Applied rules:

• Repeating Unit: Identify what part repeats

• Division Method: Use division to find position in cycle

• Modulo Operation: Remainder tells us the position in the repeating unit

Problem Solving Strategies

Strategy 1: Find the Difference

When to use: When numbers increase or decrease by the same amount

How to use: Subtract consecutive numbers to find the pattern

Example: 5, 10, 15, 20 → 10-5=5, 15-10=5, 20-15=5 → Rule: add 5

Strategy 2: Identify Repeating Units

When to use: When shapes, colors, or objects repeat in order

How to use: Find the smallest group that repeats

Example: Red, Blue, Green, Red, Blue, Green → Unit: Red, Blue, Green

Strategy 3: Use Division for Positions

When to use: To find what comes at a specific position in a repeating pattern

How to use: Divide position number by unit length, use remainder

Example: ABABAB... position 7: 7÷2=3r1 → Position 1 in unit → A

Strategy 4: Visual Representation

When to use: When struggling to see the pattern

How to use: Draw the pattern or make a table

Example: Drawing circles to represent number patterns

🔍

Look Carefully

Examine the sequence to identify how numbers change

➕

Calculate Differences

Find how much each number increases or decreases

🔄

Spot Cycles

Look for repeating units in shape or number patterns

✅

Verify Solution

Check that your answer fits the established pattern

Detailed Summary & Examples

Core Concepts & Definitions

Arithmetic Pattern

A sequence where each number changes by the same amount. Example: 2, 4, 6, 8 (+2 each time)

Repeating Pattern

A sequence that repeats the same group of items. Example: Red, Blue, Green, Red, Blue, Green

Pattern Rule

The instruction that tells how to continue the pattern. Example: "Add 5" or "Circle, Square, Triangle"

Position Finding

Using division to determine what item appears at a specific position in a repeating pattern

Step-by-Step Problem Solving Methods

Method 1: Find Missing Numbers in Arithmetic Patterns

Look at the numbers before and after the blank
Calculate the difference between consecutive known numbers
Apply the same difference to find the missing number
Verify by checking the entire sequence

Example: 3, 6, 9, __, 15 → 6-3=3, 9-6=3 → Missing: 9+3=12

Method 2: Extend Growing Patterns

Identify the starting number(s)
Determine how much is added each time
Continue adding the same amount
Verify by checking the pattern consistency

Example: 10, 15, 20, ... → Start: 10, Add: 5 → Next: 25, 30, 35

Method 3: Find Position in Repeating Patterns

Identify the repeating unit (how many items repeat)
Divide the target position by the unit length
Use the remainder to find the position in the unit
Match the remainder to the corresponding item

Example: ABCABCABC... position 11: 11÷3=3r2 → Item 2 in ABC → B

Multiple Examples: Simple to Advanced

Simple Example

Pattern: 2, 4, 6, __, 10

Difference: 4-2=2, 6-4=2

Missing: 6+2=8

Answer: 8

Intermediate Example

Pattern: Triangle, Circle, Square, Triangle, __, Square

Unit: Triangle, Circle, Square

Position 5: 5÷3=1r2

Answer: Circle

Advanced Example

Pattern: 5, 10, 15, 20, ... (find 25th term)

Rule: Start at 5, add 5 each time

Formula: 5 + (position-1)×5

Answer: 5 + 24×5 = 125

Complex Example

Pattern: XYZXYZXYZ... (find 47th letter)

Unit: XYZ (length 3)

47÷3=15r2

2nd in unit is Y

Answer: Y

Key Memorization Tips

• Pattern Memory: "Same difference means addition pattern"

• Position Rule: "Divide position by unit length, remainder shows where in cycle"

• Verification: "Always check your answer by continuing the pattern"

• Visual Aid: "Draw circles around the repeating unit"

• Count Aloud: "Say the pattern out loud to hear the rhythm"

Important Rules to Remember:

• Same Difference: In arithmetic sequences, subtract to find the common difference

• Repeating Unit: For shape patterns, identify the smallest repeating group

• Position Finding: Use division to find where in a cycle a position falls

• Verification: Always check that your answer follows the pattern

• Consistency: The same rule must apply to every part of the pattern

Tip 1: Write out the pattern clearly to see the relationship between numbers.

Tip 2: Use visual aids like drawing the repeating shapes to understand cycles.

Tip 3: Practice with small numbers first before moving to larger ones.

Tip 4: Always verify your answer by continuing the pattern to see if it makes sense.

Problem Solving Strategies with Examples for Arithmetic Patterns in Grade 3