Growing Patterns and Number Sequences - First Grade Mathematics

Not all patterns simply repeat the same items over and over; some patterns love to grow bigger and bigger, while others shrink smaller and smaller! In mathematics, a pattern that changes by a predictable mathematical rule on every step is called a growing pattern or a shrinking pattern. Think of a baby plant growing one new leaf each week, or a countdown clock ticking backward toward blastoff. Growing and shrinking patterns are your first exciting steps into algebra! In this chapter, you will master identifying growth rules, completing number sequences, and building geometric staircases.

What Is a Growing Pattern?

In a growing pattern, each step is larger than the step before it, increasing by a regular amount.

The Growing Cube Staircase (+1 Rule):
Step 1:  [#]                          (1 Cube)
Step 2:  [#][#]                       (2 Cubes - Grew by 1!)
Step 3:  [#][#][#]                    (3 Cubes - Grew by 1!)
Step 4:  [#][#][#][#]                 (4 Cubes - Grew by 1!)
Step 5:  [#][#][#][#][#]              (5 Cubes - Grew by 1!)

Rule: "ADD 1 EACH TIME" (+1)

Finding the Growth Rule

To figure out what is happening in a growing pattern, compare two neighboring steps: Look at Step 1 and Step 2: "How many were added?" Look at Step 2 and Step 3: "Were the same amount added again?" If the same quantity is added every time, you have discovered the pattern rule!

Growing by 2s, 5s, and 10s

Growing patterns can grow by any number! When we write them with numerals, they become number sequences.

Growing Number Sequences:
Pattern A:  2, 4, 6, 8, 10, 12...      Rule: "Add 2" (+2)
Pattern B:  5, 10, 15, 20, 25, 30...   Rule: "Add 5" (+5)
Pattern C:  10, 20, 30, 40, 50, 60...  Rule: "Add 10" (+10)
Pattern D:  1, 4, 7, 10, 13, 16...     Rule: "Add 3" (+3)

Predicting Future Steps

Once you know the rule for Pattern D is "Add 3," you can predict the next number after 16 simply by adding 3: 16 + 3 = 19! The next number is 19.

What Is a Shrinking Pattern?

A shrinking pattern is the reverse: each step decreases by a predictable amount!

A Shrinking Number Sequence (-2 Rule):
Start at 20:
20 -> 18 -> 16 -> 14 -> 12 -> 10 -> 8...
Rule: "SUBTRACT 2 EACH TIME" (-2)

The Countdown Rule

Whenever numbers decrease in order, look for a subtraction rule. Sequence: 15, 12, 9, 6, 3... Compare 15 and 12: it decreased by 3. Compare 12 and 9: it decreased by 3 again. Rule: "Subtract 3 each time" (-3)!

Practice Exercises

Exercise 1: Find the rule and the next two numbers in this growing sequence: 3, 6, 9, 12, , .

Exercise 2: Look at this pattern of dots: Step 1: * * Step 2: * * * * Step 3: * * * * * * How many dots are added at each step? How many dots will be in Step 4?

Exercise 3: Find the rule and the next two numbers in this shrinking sequence: 50, 40, 30, 20, , .

Exercise 4: Explain the difference between a repeating pattern (like AB) and a growing pattern.

Solutions and Explanations

Solution to Exercise 1: The rule is "Add 3" (+3). The next two numbers are 15 (12 + 3) and 18 (15 + 3).

Solution to Exercise 2: Exactly 2 dots are added at each step (the rule is +2). Step 4 will have 8 dots (6 + 2 = 8).

Solution to Exercise 3: The rule is "Subtract 10" (-10). The next two numbers are 10 (20 - 10) and 0 (10 - 10).

Solution to Exercise 4: In a repeating pattern, the exact same items repeat in a circular cycle without changing size (like red, blue, red, blue). In a growing pattern, each step changes and increases in quantity according to an addition rule (like 1, 2, 3, 4).