How to Explain Patterns and Problem Solving to Third Graders

Patterns are the heartbeat of mathematics.

Every mathematical concept—from counting to algebra to calculus—is built on recognizing and using patterns. Third grade is when students begin to see patterns systematically and use them to solve increasingly complex problems.

What Are Patterns?

A pattern is anything that repeats or follows a rule.

Repeating Patterns

The same sequence over and over:

• 🔴🔵🔴🔵🔴🔵... (red, blue, red, blue...)
• A, B, C, A, B, C, A, B, C...
• 1, 2, 1, 2, 1, 2...

The question: What comes next? What's in position 10?

Growing Patterns

Each step grows according to a rule:

• 2, 4, 6, 8, 10... (add 2 each time)
• 1, 4, 7, 10, 13... (add 3 each time)
• 1, 2, 4, 8, 16... (double each time)

The question: What's the rule? What comes next? What's the 20th term?

Patterns in Tables

The multiplication table is full of patterns:

• The 5s column ends in 0 or 5
• The 9s digits always add to 9 (18: 1+8=9; 27: 2+7=9)
• The 2s are all even numbers
• Diagonals show interesting relationships

Discovering these patterns makes math memorable.

Teaching Pattern Recognition

Step 1: Identify the Pattern

What's happening? Describe it in words.

Pattern: 3, 6, 9, 12, 15, ...
Description: "We're adding 3 each time" or "These are the multiples of 3"

Step 2: Express the Rule

Can you write a rule that generates the pattern?

Rule: Start at 3, add 3 each time.
Or: The pattern shows 3 × 1, 3 × 2, 3 × 3, 3 × 4, 3 × 5...

Step 3: Extend the Pattern

What comes next?

15 + 3 = 18
18 + 3 = 21
...

Step 4: Find Specific Terms

What's the 10th number in the pattern?

If the pattern is 3 × (position), then the 10th term is 3 × 10 = 30.

Types of Number Patterns

Addition Patterns

• Add same number: 4, 7, 10, 13, 16... (+3)
• Add increasing amounts: 2, 3, 5, 8, 12... (+1, +2, +3, +4)

Multiplication Patterns

• Double: 1, 2, 4, 8, 16, 32...
• Triple: 2, 6, 18, 54...
• Times tables: 7, 14, 21, 28, 35...

Mixed Patterns

• Alternating operations: 2, 6, 4, 8, 6, 10... (+4, -2, +4, -2)
• Two rules: 1, 2, 4, 5, 7, 8, 10... (+1, +2, +1, +2)

Input-Output Tables

Third graders work with function tables:

In	Out
1	4
2	5
3	6
4	?

Finding the rule: What happens to each input?

• 1 → 4 (add 3)
• 2 → 5 (add 3)
• 3 → 6 (add 3)

Rule: Add 3
Missing output: 4 + 3 = 7

More Complex Tables

In	Out
2	6
3	9
4	12
5	?

Rule: Multiply by 3
Missing output: 5 × 3 = 15

Problem-Solving Strategies

Third grade emphasizes systematic problem solving.

Strategy 1: Read Carefully

Many errors come from misreading. Teach:

• Read the whole problem first
• Read it again, underlining important information
• Identify: What do I know? What do I need to find?

Strategy 2: Visualize

Draw a picture or diagram:

• Circles for groups
• Number lines for sequences
• Boxes for unknown values

Strategy 3: Break It Down

Multi-step problems become manageable when broken into parts.

Problem: "Maria has 24 stickers. She gives 6 to Jake and then shares the rest equally among 3 friends. How many does each friend get?"

Step 1: Stickers after giving to Jake: 24 - 6 = 18
Step 2: Divide among 3 friends: 18 ÷ 3 = 6

Each friend gets 6 stickers.

Strategy 4: Work Backwards

Sometimes you know the ending and need to find the beginning.

Problem: "After spending $5 and then earning $8, Marcus has $15. How much did he start with?"

Work backwards:

• Before earning $8: $15 - $8 = $7
• Before spending $5: $7 + $5 = $12

Marcus started with $12.

Strategy 5: Guess and Check

Make an educated guess, check if it works, adjust.

Problem: "Two numbers add to 14 and one is 4 more than the other. What are the numbers?"

Guess 1: 7 and 7? No, they're equal.
Guess 2: 5 and 9? 5 + 9 = 14 ✓ Is 9 four more than 5? 5 + 4 = 9 ✓

The numbers are 5 and 9.

Strategy 6: Look for Patterns

Sometimes patterns reveal the answer.

Problem: "A snail climbs 3 feet up a wall each day but slides back 1 foot each night. If the wall is 10 feet tall, how many days to reach the top?"

Pattern:

• Day 1: Up 3, back 1 = net 2 feet (at 2 ft)
• Day 2: Up 3, back 1 = net 2 feet (at 4 ft)
• Day 3: net 2 feet (at 6 ft)
• Day 4: net 2 feet (at 8 ft)
• Day 5: Up 3 = reaches 11 ft (past the top!)

The snail reaches the top on Day 5.

Multi-Step Word Problems

The Common Core specifically emphasizes two-step problems in third grade.

Identifying Multi-Step Problems

Single step: "There are 24 cookies shared among 6 friends. How many does each get?"

Multi-step: "There are 24 cookies. 4 are chocolate. The rest are shared equally among 4 friends. How many does each friend get?"

Step 1: 24 - 4 = 20 non-chocolate cookies
Step 2: 20 ÷ 4 = 5 cookies each

The Two-Step Process

1. Solve the first part: What's the first thing you need to find?
2. Use that answer: How does that result help you finish?

Common Multi-Step Types

Total then share:
"Buy 3 packs of 8 pencils, share among 4 students."
Step 1: 3 × 8 = 24 total
Step 2: 24 ÷ 4 = 6 each

Combine then compare:
"Tom has 15 cards, Maria has 22. How many more does Maria have than both combined?"
Wait—this doesn't make sense! Good problems require careful reading.

Add then subtract:
"Start with 45 books, receive 12 more, then give away 18. How many left?"
Step 1: 45 + 12 = 57
Step 2: 57 - 18 = 39

Making Sense of Answers

Always ask: Does this answer make sense?

Problem: "25 students are in class. 8 are absent. How many are present?"
Wrong answer: 33
Check: Wait—there are only 25 students total. 33 doesn't make sense!
Correct: 25 - 8 = 17

Teaching reasonableness catches calculation errors and builds number sense.

Common Mistakes (And How to Fix Them)

Mistake 1: Using the Wrong Operation

Error: Multiplying when the problem requires division.

Fix: Act it out. Draw a picture. Ask: "Is the answer going to be bigger or smaller?"

Mistake 2: Stopping Too Soon

Error: Solving only the first step of a multi-step problem.

Fix: Re-read the question. "Did I actually answer what was asked?"

Mistake 3: Ignoring Extra Information

Error: Using all numbers mentioned, even irrelevant ones.

Fix: Practice identifying what's needed. Cross out irrelevant information.

Mistake 4: Not Checking Reasonableness

Error: Accepting any answer without thinking about whether it makes sense.

Fix: Always ask: "Could this be right? Is it too big? Too small?"

Building Pattern and Problem-Solving Skills

Daily Pattern Hunts

Look for patterns everywhere:

• House numbers on a street
• Tile floors
• Music rhythms
• Nature (petals, leaves)

Question Games

"I'm thinking of a rule. You give me a number, I'll give you back a number."

• You: 2 → I say: 5
• You: 3 → I say: 6
• You: 10 → I say: 13
• What's my rule? (Add 3)

Story Problems

Create problems from daily life:

• Sharing snacks
• Calculating game scores
• Planning purchases

Logic Puzzles

Age-appropriate puzzles build reasoning:

• Sudoku (kid versions)
• Logic grid puzzles
• "Who lives where?" problems

The Bottom Line

Patterns and problem solving are where mathematics becomes thinking.

Calculation is a tool. The real power is in:

• Seeing relationships
• Recognizing structure
• Breaking complex problems into steps
• Checking that answers make sense

When your third grader can find the pattern in 5, 8, 11, 14... AND solve a two-step word problem by breaking it down... AND check whether their answer is reasonable—they're developing the mathematical thinking that will serve them for life.

That's the goal. That's what third grade is building.