R4 · Lesson 5: Number Patterns & Sequences

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Why this matters for Grade 6: Grade 6 introduces expressions and equations (6.EE). Recognizing that "every output is the input plus 3" is the same idea as writing y = x + 3. This lesson bridges arithmetic patterns to the language of algebra you will use throughout Grade 6 and beyond.

Learn It

Sequences & Input/Output Rules

A sequence is a list of numbers that follows a rule. An arithmetic sequence adds (or subtracts) the same number each time. A geometric sequence multiplies (or divides) by the same number each time.

Example 1 — Arithmetic sequence (add 2):
2, 4, 6, 8, 10

Rule: add 2 each time. Next term: 8 + 2 = 10.

Example 2 — Arithmetic sequence (add 3):
3, 6, 9, 12

Rule: add 3 each time. Next term: 9 + 3 = 12.

Example 3 — Geometric sequence (multiply by 2):
1, 2, 4, 8, 16

Rule: multiply by 2 each time. Next term: 8 × 2 = 16.

Input/Output tables show a rule applied to a set of inputs:

Example 4 — Rule: y = x + 3

Input (x)	Output (y)
1	4
2	5
3	6
5	8

Each output = input + 3. When x = 5: y = 5 + 3 = 8.

Try It · Guided

Work These Together

1. What is the next term in the sequence: 3, 6, 9, 12, ___?

Hint

The rule is add 3 each time: 3, 6, 9, 12, 12 + 3 = 15.

2. What is the rule for this sequence: 1, 2, 4, 8, 16?

Hint

Check: 1 × 2 = 2 ✔, 2 × 2 = 4 ✔, 4 × 2 = 8 ✔, 8 × 2 = 16 ✔. The rule is multiply by 2.

Practice · On Your Own

You Try

3. What is the next term in the sequence: 2, 4, 6, 8, ___?

Hint

The rule is add 2 each time: 8 + 2 = 10.

4. The rule for an input/output table is y = x + 3. What is the output when the input is 7?

Hint

Replace x with 7: y = 7 + 3 = 10.

5. Which describes the rule for the sequence 5, 10, 15, 20, 25?

Hint

5 + 5 = 10 ✔, 10 + 5 = 15 ✔, 15 + 5 = 20 ✔. The rule is add 5 each time — these are the multiples of 5.

★

Exit Ticket

Show What You Know

6. The input/output rule is y = x + 3. What is the output when the input (x) is 9?

Hint

y = x + 3 = 9 + 3 = 12.

7. A sequence starts: 1, 2, 4, 8, 16. What is the next term?

Hint

The rule is multiply by 2 each time. 16 × 2 = 32.

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