Solved Exercises on Using Pattern-Finding Skills to Copy and Complete Tables in Grade 4

Master pattern-finding skills to copy and complete tables: number patterns, geometric patterns, arithmetic sequences, and pattern completion through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Arithmetic Pattern Table
Exercise 1
Copy and complete the table:
Input Output
2 7
3 9
4 11
5 __
6 __
7 __
Definition:

Input-Output table: A table showing the relationship between input and output values following a specific rule

Pattern finding method:
  1. Find the relationship between input and output
  2. Identify the pattern rule
  3. Apply the rule to find missing outputs
Input
2, 3, 4, 5, 6, 7
Rule
Multiply by 2, add 3
Output
7, 9, 11, 13, 15, 17
Step 1: Analyze the given input-output pairs

Input 2 → Output 7 (2 × 2 + 3 = 7)

Input 3 → Output 9 (3 × 2 + 3 = 9)

Input 4 → Output 11 (4 × 2 + 3 = 11)

Step 2: Identify the pattern rule

Multiply the input by 2, then add 3

Pattern rule: Input × 2 + 3 = Output

Step 3: Apply the rule to find missing outputs

Input 5: 5 × 2 + 3 = 10 + 3 = 13

Input 6: 6 × 2 + 3 = 12 + 3 = 15

Input 7: 7 × 2 + 3 = 14 + 3 = 17

Step 4: Verify the pattern

All inputs follow the rule: Input × 2 + 3 = Output ✓

Input Output
2 7
3 9
4 11
5 13
6 15
7 17
Final answer:

Outputs: 13, 15, 17

Applied rules:

• Input-Output relationship: Each input produces an output following a specific rule

• Pattern recognition: Look for consistent mathematical operations

• Rule application: Apply the identified rule to all inputs

2 Geometric Pattern Table
Exercise 2
Copy and complete the table:
Step Shape Pattern
1 🟦
2 🟦 🔵
3 🟦 🔵 🔴
4 __
5 __
6 __
Definition:

Geometric pattern table: A table showing the relationship between step number and geometric pattern formation

Step
1, 2, 3, 4, 5, 6
Rule
Add next color
Pattern
🟦, 🟦 🔵, 🟦 🔵 🔴, ...
Step 1: Analyze the given pattern sequence

Step 1: 🟦 (Blue Square)

Step 2: 🟦 🔵 (Blue Square + Blue Circle)

Step 3: 🟦 🔵 🔴 (Blue Square + Blue Circle + Red Triangle)

Step 2: Identify the pattern rule

At each step, add the next shape/color to the sequence

Pattern: Blue Square, Blue Circle, Red Triangle, Blue Square, Blue Circle, Red Triangle...

Step 3: Apply the rule to find missing patterns

Step 4: Add Yellow Star to existing pattern

Step 5: Add Green Diamond to existing pattern

Step 6: Add Orange Pentagon to existing pattern

Step 4: Verify the pattern

Each step adds one new shape to the sequence ✓

Step Shape Pattern
1 🟦
2 🟦 🔵
3 🟦 🔵 🔴
4 🟦 🔵 🔴 🟨
5 🟦 🔵 🔴 🟨 🟢
6 🟦 🔵 🔴 🟨 🟢 🟠
Final answer:

Step 4: 🟦 🔵 🔴 🟨, Step 5: 🟦 🔵 🔴 🟨 🟢, Step 6: 🟦 🔵 🔴 🟨 🟢 🟠

Applied rules:

• Geometric pattern: Identify the sequence of shapes/colors

• Sequential addition: Each step adds one new element

• Repeating cycle: Recognize if shapes follow a cycle

3 Multiplication Pattern Table
Exercise 3
Copy and complete the table:
Number Triple
2 6
4 12
6 18
8 __
10 __
12 __
Definition:

Multiplication pattern table: A table showing the relationship between a number and its multiple

Number
2, 4, 6, 8, 10, 12
Rule
Multiply by 3
Triple
6, 12, 18, 24, 30, 36
Step 1: Analyze the given number-triple pairs

2 → 6 (2 × 3 = 6)

4 → 12 (4 × 3 = 12)

6 → 18 (6 × 3 = 18)

Step 2: Identify the pattern rule

Multiply the number by 3 to get its triple

Pattern rule: Number × 3 = Triple

Step 3: Apply the rule to find missing triples

8 × 3 = 24

10 × 3 = 30

12 × 3 = 36

Step 4: Verify the pattern

All numbers follow the rule: Number × 3 = Triple ✓

Number Triple
2 6
4 12
6 18
8 24
10 30
12 36
Final answer:

Tripled values: 24, 30, 36

Applied rules:

• Multiplication pattern: Each number is multiplied by a constant factor

• Factor identification: Divide output by input to find the multiplier

• Consistency: Apply the same operation throughout

Rules and methods, laws,...
\(Output = Rule(Input)\)
Input-Output Relationship
Arithmetic Pattern
Add Constant
Input × 2 + 3 = Output
Geometric Pattern
Sequential Addition
Step 1: 🟦, Step 2: 🟦 🔵, ...
Multiplication Pattern
Multiply by Constant
Number × 3 = Triple
Key definitions:

Pattern: A regular, repeated sequence following a specific rule

Input-Output table: A table showing the relationship between input and output values

Pattern rule: The mathematical operation that transforms input to output

Complete methodology:
  1. Analyze the table: Look at the given input-output pairs carefully
  2. Find the relationship: Check if adding, subtracting, multiplying, or dividing gives the output
  3. Test the rule: Apply the rule to multiple pairs to confirm
  4. Apply the rule: Use the rule to find missing outputs
  5. Verify: Check that the completed table makes sense
Tip 1: Always check at least 2 or 3 input-output pairs to confirm the pattern rule.
Tip 2: For number patterns, try addition/subtraction first, then multiplication/division.
Tip 3: For geometric patterns, look for repeating cycles or sequential additions.
Tip 4: Write down the operations you notice to visualize the pattern rule.

Common errors: Misidentifying the pattern rule, applying inconsistent operations, skipping verification steps.
Pattern recognition: Practice with various sequence types, pay attention to small differences, verify results.
Formulas to know by heart:

• Input-Output rule: Output = f(Input) where f is the pattern rule

• Arithmetic pattern: Output = Input × multiplier + constant

• Geometric pattern: Look for shape/color cycles

• Pattern verification: Always check that your rule works for multiple pairs

Solution: Exercises 4 to 5
4 Mixed Pattern Table
Exercise 4
Copy and complete the table:
Position Value
1 1
2 4
3 9
4 __
5 __
6 __
Definition:

Square number pattern: Each value is the square of its position (n²)

Position
1, 2, 3, 4, 5, 6
Rule
Square the position
Value
1, 4, 9, 16, 25, 36
Step 1: Analyze the given position-value pairs

Position 1 → Value 1 (1² = 1)

Position 2 → Value 4 (2² = 4)

Position 3 → Value 9 (3² = 9)

Step 2: Identify the pattern rule

Take the square of the position number

Pattern rule: Position² = Value

Step 3: Apply the rule to find missing values

Position 4: 4² = 16

Position 5: 5² = 25

Position 6: 6² = 36

Step 4: Verify the pattern

All positions follow the rule: Position² = Value ✓

Position Value
1 1
2 4
3 9
4 16
5 25
6 36
Final answer:

Values: 16, 25, 36

Applied rules:

• Square number pattern: Recognize perfect squares

• Position analysis: Consider the position of each term

• Verification: Ensure the identified rule works for all known terms

5 Complex Pattern Table
Exercise 5
Copy and complete the table:
Input Output
1 3
2 6
3 11
4 __
5 __
6 __
Definition:

Complex pattern: A combination of operations following a specific rule

Input
1, 2, 3, 4, 5, 6
Rule
Square + 2
Output
3, 6, 11, 18, 27, 38
Step 1: Analyze the given input-output pairs

Input 1 → Output 3 (1² + 2 = 3)

Input 2 → Output 6 (2² + 2 = 6)

Input 3 → Output 11 (3² + 2 = 11)

Step 2: Identify the pattern rule

Square the input and add 2

Pattern rule: Input² + 2 = Output

Step 3: Apply the rule to find missing outputs

Input 4: 4² + 2 = 16 + 2 = 18

Input 5: 5² + 2 = 25 + 2 = 27

Input 6: 6² + 2 = 36 + 2 = 38

Step 4: Verify the pattern

All inputs follow the rule: Input² + 2 = Output ✓

Input Output
1 3
2 6
3 11
4 18
5 27
6 38
Final answer:

Outputs: 18, 27, 38

Applied rules:

• Complex pattern: May involve multiple operations

• Pattern analysis: Look for combinations of operations

• Verification: Ensure the identified rule works for all known terms

Key Concepts: Using Pattern-Finding Skills to Complete Tables
\(Output = Rule(Input)\)
Input-Output Relationship
Key definitions:

Pattern: A predictable sequence following a specific rule

Input-Output table: A table showing the relationship between input and output values

Pattern rule: The mathematical operation that transforms input to output

Function: A relationship where each input has exactly one output

Complete methodology:
  1. Observe: Carefully examine the given input-output pairs
  2. Compare: Look for relationships between inputs and outputs
  3. Test: Apply potential rules to multiple pairs
  4. Confirm: Verify the rule works consistently
  5. Apply: Use the rule to find missing outputs
  6. Validate: Check the final table makes sense
Tip 1: For number patterns, first try addition/subtraction (arithmetic), then multiplication/division (geometric).
Tip 2: When dealing with perfect squares, consider if terms are 1², 2², 3², etc.
Tip 3: For geometric patterns, look for repeating cycles or sequential additions.
Tip 4: Always verify your pattern rule with at least two consecutive pairs.

Tip 5: Write out the pattern rule explicitly to avoid mistakes.
Common errors: Identifying the wrong pattern type, miscalculating differences/ratios, not verifying the pattern.
Pattern recognition: Practice with various sequence types, pay attention to small differences, verify results.
Problem-solving strategies: Work systematically, check your work, consider multiple approaches.
Formulas to know by heart:

• Arithmetic pattern: Output = Input × multiplier + constant

• Geometric pattern: Output = Input × common ratio

• Perfect squares: 1²=1, 2²=4, 3²=9, 4²=16, 5²=25, 6²=36, 7²=49, 8²=64, 9²=81, 10²=100

• Pattern verification: Always check that your rule works for multiple pairs

Questions & Answers

Question: I have trouble figuring out what the pattern rule is in a table. How can I identify it?

Answer: Great question! Here's how to identify the pattern rule in a table:

  1. Look for simple operations first: Try adding or subtracting the same number
  2. Check for multiplication: See if the output is a multiple of the input
  3. Try combinations: Maybe multiply first, then add a number
  4. Test with all pairs: Make sure your rule works for every given input-output pair

Example: If Input 2 → Output 7, try 2 × 2 + 3 = 7, then check if this works for other pairs!

Question: How do I handle tables with geometric patterns (shapes, colors)? I don't understand the rules.

Answer: Geometric pattern tables follow the same principles as number patterns! Here's how to approach them:

  1. Identify the sequence: Look for the order of shapes, colors, or other attributes
  2. Count the cycle length: How many different elements make up one complete repetition?
  3. Continue the pattern: Repeat the same sequence of shapes or attributes

Example: If Step 1 is 🟦, Step 2 is 🟦 🔵, Step 3 is 🟦 🔵 🔴, the pattern is adding one new shape each time!

Question: What should I do when a table has a complex pattern that uses two different operations? I got confused on a problem like Input→Output: 1→3, 2→6, 3→11...

Answer: Some complex patterns combine operations! Let's analyze your example: 1→3, 2→6, 3→11...

Looking closely:

  • 1 → 3: Could be 1² + 2 = 1 + 2 = 3
  • 2 → 6: Could be 2² + 2 = 4 + 2 = 6
  • 3 → 11: Could be 3² + 2 = 9 + 2 = 11

The pattern is: Square the input, then add 2

When facing complex patterns:

  1. Check if it's arithmetic or geometric first
  2. If not, try combining operations (square, cube, multiply, add)
  3. Write out the operations to see the pattern clearly
  4. Always verify your rule works for all given pairs
Complete Summary: Use Your Pattern-Finding Skills to Complete Tables
Key Definitions:

Pattern: A regular, repeated sequence that follows a specific rule. Patterns help us predict what comes next.

Input-Output Table: A table showing the relationship between input values and their corresponding output values.

Pattern Rule: The mathematical operation(s) that transform input values to output values.

Function: A relationship where each input has exactly one output.

Perfect Square: A number that can be expressed as n² where n is a whole number (like 1, 4, 9, 16, 25...).

Core Rules and Principles:

• Consistency: The same rule must apply to every input-output pair.

• Verification: Always check that your pattern rule works for multiple pairs.

• Recognition: Look for arithmetic (add/subtract) before geometric (multiply/divide).

• Complexity: Some patterns may combine multiple operations.

Step-by-Step Method:
  1. Observe: Carefully look at the first few input-output pairs in the table.
  2. Compare: Find the relationship between inputs and outputs.
  3. Test: Apply your suspected rule to multiple pairs to confirm.
  4. Apply: Use the confirmed rule to find missing outputs.
  5. Verify: Check that your completed table follows the pattern.
Simple Examples

Arithmetic: Input × 2 + 1 = Output (1→3, 2→5, 3→7...)

Geometric: Input × 3 = Output (1→3, 2→6, 3→9...)

Shape Pattern: Step 1: 🟦, Step 2: 🟦 🔵, Step 3: 🟦 🔵 🔴...

Advanced Examples

Perfect Squares: Input² = Output (1→1, 2→4, 3→9, 4→16...)

Combination: Input² + Input = Output (1→2, 2→6, 3→12...)

Cycling Shapes: Step 1: 🟦, Step 2: 🔵, Step 3: 🔴, Step 4: 🟦, ...

Tips: Always check at least 2 consecutive pairs to confirm your pattern rule. Write out the operations to visualize the pattern.
Common Pitfalls: Don't assume the first pair you see represents the pattern. Always verify with multiple pairs.
Memorization Tips: Remember perfect squares up to 10² = 100. For geometric patterns, divide consecutive terms to find the ratio.
Key Takeaways:

• Tables always follow a consistent rule that applies to every input-output pair.

• Arithmetic patterns involve addition or subtraction of a constant.

• Geometric patterns involve multiplication or division by a constant.

• Complex patterns may combine multiple operations.

• Always verify your answer by applying the rule to check if it makes sense.