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Pattern Sleuth Academy

A comprehensive fourth-grade unit exploring number and shape patterns, uncovering unstated mathematical features, and informally explaining patterns under standard 4.OA.C.5.

Addition Step Rules

Students generate addition and subtraction sequences, extend terms using given rules, and identify underlying features such as alternating odd and even numbers.

Multiplication Factor Patterns

Students explore multiplication and division patterns, identifying non-explicit features including repeating last digits, constant parity, and rapid growth.

Shape Growth Sequences

Students generate and extend repeating and growing geometric shape sequences, identifying structural features such as symmetry, perimeter growth, and alternating components.

Feature Sleuth Analysis

Students analyze complex and multi-step patterns, formulate hypotheses about hidden features, and write informal mathematical explanations for why patterns continue.

Pattern Sleuth Academy

A comprehensive fourth-grade unit exploring number and shape patterns, uncovering unstated mathematical features, and informally explaining patterns under standard 4.OA.C.5.

Addition Patterns Worksheet

1

Addition Patterns Worksheet

Standard 4.OA.C.5 • Discover the Rule & Hidden Features

Name:

Date:

Directions: Find the rule, extend the sequence by 2 terms, and identify the hidden feature. 20 Questions

1. Pattern: 1, 4, 7, 10, ,

Rule:

Feature: Parity: [ alternates / stays odd ]

2. Pattern: 2, 6, 10, 14, ,

Rule:

Feature: All terms are: [ even / odd ]

3. Pattern: 5, 10, 15, 20, ,

Rule:

Feature: Last digits alternate between:

4. Pattern: 3, 9, 15, 21, ,

Rule:

Feature: All numbers in sequence are: [ odd / even ]

5. Pattern: 4, 14, 24, 34, ,

Rule:

Feature: Every term ends with the digit:

6. Pattern: 40, 37, 34, 31, ,

Rule:

Feature: Parity: [ alternates / stays even ]

7. Pattern: 7, 9, 11, 13, ,

Rule:

Feature: All numbers remain:

8. Pattern: 12, 20, 28, 36, ,

Rule:

Feature: All terms are multiples of 2 and:

9. Pattern: 80, 75, 70, 65, ,

Rule:

Feature: Ones digits alternate between 0 and:

10. Pattern: 1, 8, 15, 22, ,

Rule:

Feature: Parity: [ alternates / stays odd ]

11. Pattern: 6, 12, 18, 24, ,

Rule:

Feature: Every term is an number.

12. Pattern: 2, 11, 20, 29, ,

Rule:

Feature: Parity: [ alternates / stays even ]

13. Pattern: 50, 46, 42, 38, ,

Rule:

Feature: All numbers remain: [ even / odd ]

14. Pattern: 8, 18, 28, 38, ,

Rule:

Feature: The ones digit is always:

15. Pattern: 100, 89, 78, 67, ,

Rule:

Feature: Parity: [ alternates / stays even ]

16. Pattern: 13, 16, 19, 22, ,

Rule:

Feature: Parity alternates between:

17. Pattern: 9, 18, 27, 36, ,

Rule:

Feature: The sum of digits in each term equals:

18. Pattern: 21, 19, 17, 15, ,

Rule:

Feature: All numbers remain:

19. Pattern: 25, 50, 75, 100, ,

Rule:

Feature: Last 2 digits repeat in cycle of: [ 2 / 4 ]

20. Pattern: 0, 12, 24, 36, ,

Rule:

Feature: All terms are divisible by 3 and:

Pattern Detective Rule: Check the difference between consecutive numbers to deduce the rule! Page 1 of 1

Addition Rules Homework Sheet

H1

Addition Rules Homework Sheet

Standard 4.OA.C.5 • Deduce the Rule, Discover Features & Explain

Name:

Date:

Directions: Examine each number pattern. (a) Determine and write the rule, (b) calculate Terms 5 and 6, (c) identify an apparent feature not in the rule, and (d) informally explain why it continues.

Question 1 • Given Sequence Pattern Investigation

Term 11

Term 24

Term 37

Term 410

Term 5

Term 6

1. What is the rule?

2. Identify Apparent Feature: What do you notice about whether the terms are odd or even?

3. Informal Explanation: Why will this alternating pattern continue forever?

Question 2 • Given Sequence Pattern Investigation

Term 12

Term 26

Term 310

Term 414

Term 5

Term 6

1. What is the rule?

2. Identify Apparent Feature: Notice the parity. Are there any odd numbers in this sequence?

3. Informal Explanation: Why can an odd number never appear based on this rule?

Question 3 • Given Sequence Pattern Investigation

Term 15

Term 215

Term 325

Term 435

Term 5

Term 6

1. What is the rule?

2. Identify Apparent Feature: Examine the ones place of each number in the sequence.

3. Informal Explanation: Why does this operation never change the digit in the ones place?

Question 4 • Given Sequence Pattern Investigation

Term 160

Term 255

Term 350

Term 445

Term 5

Term 6

1. What is the rule?

2. Identify Apparent Feature: What repeats in the last digit and parity of the numbers?

3. Informal Explanation: Why do the units digits alternate between 0 and 5?

Pattern Detective Tip: Test your rule with the first two pairs before writing it down! Homework Page 1 of 4

Multiplication Patterns Worksheet

2

Multiplication Patterns Worksheet

Standard 4.OA.C.5 • Discover the Rule & Factor Properties

Name:

Date:

Directions: Find the rule, extend the sequence by 2 terms, and identify the hidden feature. 20 Questions

1. Pattern: 2, 4, 8, 16, ,

Rule:

Feature: Every term is an [ even / odd ] number.

2. Pattern: 1, 3, 9, 27, ,

Rule:

Feature: Multiplying odd by odd results in:

3. Pattern: 3, 6, 12, 24, ,

Rule:

Feature: After term 1, all terms are: [ even / odd ]

4. Pattern: 5, 10, 20, 40, ,

Rule:

Feature: After 5, all terms end in digit:

5. Pattern: 1, 4, 16, 64, ,

Rule:

Feature: Last digits cycle between 4 and:

6. Pattern: 1, 10, 100, 1000, ,

Rule:

Feature: Each step adds one more: [ zero / tens ]

7. Pattern: 2, 10, 50, 250, ,

Rule:

Feature: Every term after 2 ends in:

8. Pattern: 64, 32, 16, 8, ,

Rule:

Feature: All terms remain: [ even / odd ] until 1.

9. Pattern: 7, 14, 28, 56, ,

Rule:

Feature: After term 1, all terms are:

10. Pattern: 4, 12, 36, 108, ,

Rule:

Feature: All terms are divisible by:

11. Pattern: 3, 9, 27, 81, ,

Rule:

Feature: Units digit repeats: [ 3, 9, 7, 1 ]

12. Pattern: 5, 25, 125, 625, ,

Rule:

Feature: Every term ends in the digit:

13. Pattern: 6, 12, 24, 48, ,

Rule:

Feature: All terms are multiples of 6 and:

14. Pattern: 81, 27, 9, 3, ,

Rule:

Feature: All numbers in sequence are:

15. Pattern: 10, 30, 90, 270, ,

Rule:

Feature: The ones digit is always:

16. Pattern: 1, 6, 36, 216, ,

Rule:

Feature: Starting with 6, every term ends in:

17. Pattern: 8, 16, 32, 64, ,

Rule:

Feature: Last digits cycle: [ 8, 6, 2, 4 ]

18. Pattern: 100, 50, 25, ,

Rule:

Feature: Parity switches from even to:

19. Pattern: 2, 20, 200, 2000, ,

Rule:

Feature: The leading digit is always:

20. Pattern: 9, 18, 36, 72, ,

Multiplication Rules Homework Sheet

H2

Multiplication Rules Homework Sheet

Standard 4.OA.C.5 • Deduce Multiplication Rules & Explain Features

Name:

Date:

Directions: Examine each number pattern. (a) Determine and write the rule, (b) calculate Terms 5 and 6, (c) identify an apparent feature not in the rule, and (d) informally explain why it continues.

Question 1 • Given Sequence Pattern Investigation

Term 12

Term 26

Term 318

Term 454

Term 5

Term 6

1. What is the rule?

2. Identify Apparent Feature: What parity (even or odd) do all terms in this sequence share?

3. Informal Explanation: Why does multiplying an even number by 3 always result in an even number?

Question 2 • Given Sequence Pattern Investigation

Term 15

Term 225

Term 3125

Term 4625

Term 5

Term 6

1. What is the rule?

2. Identify Apparent Feature: What do you notice about the digit in the ones place of every term?

3. Informal Explanation: Why will any product of a number ending in 5 multiplied by 5 always end in 5?

Question 3 • Given Sequence Pattern Investigation

Term 11

Term 22

Term 34

Term 48

Term 516

Term 632

1. What is the rule?

2. Identify Apparent Feature: List the ones digits after 1: 2, 4, 8, 6, 2... What repeating cycle appears?

3. Informal Explanation: Why does doubling 6 give 12 (ends in 2) and restart the same digit pattern?

Question 4 • Given Sequence Pattern Investigation

Term 13

Term 212

Term 348

Term 4192

Term 5

Term 6

1. What is the rule?

2. Identify Apparent Feature: Starting with 12, what common factor do all numbers share?

3. Informal Explanation: Why must every future term in this sequence remain a multiple of 12?

Pattern Detective Tip: Divide any term by the previous term to find the multiplication factor! Homework Page 2 of 4

Shape Sequences Worksheet

3

Shape Sequences Worksheet

Standard 4.OA.C.5 • Discover Shape Rules & Geometric Features

Name:

Date:

Directions: Find the rule, complete the next 2 shapes or values, and identify the hidden feature. 20 Questions

1. Pattern: ▲, ■, ▲, ■, ▲,

Rule:

Feature: Even positions are always: [ squares / triangles ]

2. Pattern: ●, ●, ◆, ●, ●, ◆,

Rule:

Feature: Diamonds appear at multiples of:

3. Pattern: △(3), □(4), ⬠(5), ⬡(6), ,

Rule:

Feature: Side counts alternate: [ odd / even ]

4. Pattern: ★, ✪, ★, ✪, ★, ✪,

Rule:

Feature: Term 15 will be a: [ solid ★ / circled ✪ ]

5. Pattern: 1 dot, 3 dots, 5 dots, 7 dots, ,

Rule:

Feature: Every figure contains an number of dots.

6. Pattern: ↑, →, ↓, ←, ↑,

Rule:

Feature: Rotation cycle repeats every turns.

7. Grid dots: 1, 4, 9, 16, ,

Rule:

Feature: Square counts alternate: [ odd / even ]

8. Pattern: ▲, ▼, ▲, ▼, ▲,

Rule:

Feature: Points down when position is: [ even / odd ]

9. Triangle blocks: 1, 3, 6, 10, ,

Rule:

Feature: Amount added each step increases by:

10. Pattern: ⬟(5), ⬡(6), ⬟(5), ⬡(6),

Rule:

Feature: Odd positions always have sides.

11. Pattern: ◼, ◻, ◻, ◼, ◻, ◻,

Rule:

Feature: Black squares appear at step 1, 4, 7 (add )

12. Ring dots: 4, 8, 12, 16, ,

Rule:

Feature: All dot counts are multiples of:

13. Pattern: ◐, ◑, ◐, ◑, ◐,

Rule:

Feature: Shaded side alternates: [ left / right ]

14. Square Perimeters: 4, 8, 12, 16, ,

Rule:

Feature: Perimeters are all: [ even / odd ] numbers.

15. Pattern: ✚, ✖, ✚, ✖, ✚,

Rule:

Feature: Each step rotates the cross by: degrees.

16. T-blocks: 4, 7, 10, 13, ,

Rule:

Feature: Total block count alternates:

17. Pattern: ●, ■, ▲, ●, ■, ▲,

Rule:

Feature: Triangles only occur at multiples of:

18. Hex tiles: 6, 12, 18, 24, ,

Rule:

Feature: Every tile count is divisible by 2, 3, and:

19. Dots: 1, 2, 4, 8, ,

Shape Growth Homework Sheet

H3

Shape Growth Homework Sheet

Standard 4.OA.C.5 • Deduce Shape Rules, Perimeters & Explain Features

Name:

Date:

Directions: Examine each geometric pattern. (a) Determine and write the rule of growth, (b) calculate Terms 5 and 6, (c) identify an apparent feature not in the rule, and (d) informally explain why it continues.

Question 1 • Growing Block Tower Pattern Investigation

Fig 11 block

Fig 23 blocks

Fig 35 blocks

Fig 47 blocks

Fig 5

Fig 6

1. What is the rule of growth?

2. Identify Apparent Feature: Examine the total block count. Are any figures even?

3. Informal Explanation: Why does starting with 1 and continually adding 2 always produce odd numbers?

Question 2 • Square Tile Perimeters (Side 1, 2, 3...) Pattern Investigation

Side 1P = 4

Side 2P = 8

Side 3P = 12

Side 4P = 16

Side 5

Side 6

1. What is the rule for finding perimeter?

2. Identify Apparent Feature: What number do all perimeters divide by evenly, and what is their parity?

3. Informal Explanation: Why can a square with whole-number sides never have an odd perimeter?

Question 3 • Rotating Arrow Sequence Pattern Investigation

Step 1↑ Up

Step 2→ Right

Step 3↓ Down

Step 4← Left

Step 5

Step 6

1. What is the rotation rule?

2. Identify Apparent Feature: Which position numbers (odd or even) contain horizontal arrows (→ or ←)?

3. Informal Explanation: Why does rotating by a quarter-turn alternate between vertical and horizontal?

Question 4 • Triangular Dot Clusters Pattern Investigation

Fig 13 dots

Fig 26 dots

Fig 39 dots

Fig 412 dots

Fig 5

Fig 6

1. What is the dot rule?

2. Identify Apparent Feature: Notice the parity of total dots. What alternating pattern appears?

3. Informal Explanation: Why does adding 3 dots each step cause the total dot parity to alternate?

Pattern Detective Tip: Count the sides, dots, or rows of each figure to find the rule of change! Homework Page 3 of 4

Feature Sleuth Worksheet

4

Feature Sleuth Worksheet

Standard 4.OA.C.5 • Discover Complex Rules & Unstated Features

Name:

Date:

Directions: Find the rule, extend each sequence by 2 terms, and identify the hidden feature. 20 Questions

1. Pattern: 1, 3, 7, 15, ,

Rule:

Feature: Every single term is: [ odd / even ]

2. Pattern: 4, 9, 7, 12, 10, ,

Rule:

Feature: Every 2 steps increases by:

3. Pattern: 1▲, 2■, 3▲, 4■, 5▲, ,

Rule:

Feature: Squares always have an [ even / odd ] number.

4. Pattern: 2, 7, 22, 67, ,

Rule:

Feature: Parity: [ alternates / stays even ]

5. Pattern: 6, 10, 14, 18, 22, ,

Rule:

Feature: Last digits cycle through: [ 6, 0, 4, 8, 2 ]

6. Pattern: ○, ◐, ●, ◑, ○,

Rule:

Feature: Shading cycle length is: steps.

7. Pattern: 50, 44, 38, 32, ,

Rule:

Feature: All terms remain: [ even / odd ] numbers.

8. Pattern: 1, 4, 10, 22, ,

Rule:

Feature: After term 1, all numbers are:

9. Pattern: 1, 6, 11, 16, 21, ,

Rule:

Feature: The ones digit alternates between 1 and:

10. Pattern: 2, 4, 8, 16, 32, ,

Rule:

Feature: Last digits cycle: [ 2, 4, 8, 6 ]

11. Pattern: ⬟, ⬟, ⬡, ⬟, ⬟, ⬡,

Rule:

Feature: 6-sided hexagons only appear at multiples of:

12. Pattern: 100, 93, 86, 79, ,

Rule:

Feature: Parity: [ alternates / stays even ]

13. Pattern: 3, 5, 9, 17, ,

Rule:

Feature: Every term is an number.

14. Pattern: ★, ★★, ★★★★, ★★★★★★★★, ,

Rule:

Feature: Each term is double the previous: [ count / size ]

15. Pattern: 15, 30, 45, 60, ,

Rule:

Feature: Ones digits alternate between 5 and:

16. Pattern: ▲, ▲▲, ▲, ▲▲, ▲,

Rule:

Feature: Triangle count parity alternates: [ odd / even ]

17. Pattern: 3, 15, 27, 39, ,

Rule:

Feature: All numbers remain:

18. Pattern: 1, 5, 25, 125, ,

Rule:

Feature: Every term after 1 ends in digit:

19. Dots: 1, 4, 9, 16, ,

Rule:

Feature: Differences added are consecutive odd: [ 3, 5, 7, 9 ]

Secret Features Homework Sheet

H4

Secret Features Homework Sheet

Standard 4.OA.C.5 • Deduce Complex Rules & Mathematical Proofs

Name:

Date:

Directions: Examine each number pattern. (a) Determine and write the rule, (b) calculate Terms 5 and 6, (c) identify an apparent feature not in the rule, and (d) informally explain why it continues.

Question 1 • Given Sequence Pattern Investigation

Term 11

Term 23

Term 37

Term 415

Term 5

Term 6

1. What is the two-step rule?

2. Identify Apparent Feature: Look at the parity of each term. Can an even number ever appear?

3. Informal Explanation: Why does doubling any whole number and adding 1 always result in an odd number?

Question 2 • Given Sequence Pattern Investigation

Term 19

Term 218

Term 327

Term 436

Term 5

Term 6

1. What is the rule?

2. Identify Apparent Feature: Add the individual digits of each term (e.g., 1+8, 2+7). What is always the sum?

3. Informal Explanation: Adding 9 is like adding 10 and subtracting 1 (+10, -1). Why does this keep the digit sum constant?

Question 3 • Given Sequence Pattern Investigation

Term 12

Term 26

Term 35

Term 49

Term 5

Term 6

1. What is the alternating rule?

2. Identify Apparent Feature: Compare Term 1 (2), Term 3 (5), and Term 5. What net change occurs every two steps?

3. Informal Explanation: Why do the odd-positioned terms (1st, 3rd, 5th...) alternate between even and odd?

Question 4 • Triangular Stack Blocks Pattern Investigation

Fig 11 block

Fig 23 blocks

Fig 36 blocks

Fig 410 blocks

Fig 5

Fig 6

1. What is the rule of addition for each new figure?

2. Identify Apparent Feature: Label each figure's block count as odd or even. What 4-step repeating cycle appears?

3. Informal Explanation: Why do the parities repeat in pairs of two odd numbers followed by two even numbers?

Pattern Detective Tip: Congratulations! You have mastered discovering rules and explaining secret features! Homework Page 4 of 4

Rule:

Feature: Digits in each term sum to:

Pattern Detective Rule: Check what factor is multiplied by each term to uncover the rule! Page 1 of 1

Rule:

Feature: After term 1, all dot counts are: [ even / odd ]

20. Pattern: ↗, ↘, ↗, ↘, ↗,

Rule:

Feature: 20th arrow will point: [ up-right ↗ / down-right ↘ ]

Pattern Detective Rule: Count changes in sides, positions, or dots to find the rule! Page 1 of 1

20. Pattern: 0, 9, 18, 27, 36, ,

Rule:

Feature: Sum of digits in non-zero multiples of 9 is:

Master Detective Rule: Hidden features always have a mathematical reason behind them! Page 1 of 1

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