Home›Lesson Plans›Mathematics›Functions · Grade 9

Arithmetic & Geometric Sequences

Ninth graders learn to recognize and extend arithmetic sequences (adding a common difference) and geometric sequences (multiplying by a common ratio), and to use formulas to find any term.

Grade 9Sequences55 minutes1 class periodGradual ReleaseExplicit teaching4 StandardsCommon Core
Start the Lesson
Lesson at a Glance

Everything you need before the bell rings

Learning Objectives

Students will be able to…

  • ✓Identify an arithmetic sequence.
  • ✓Identify a geometric sequence.
  • ✓Find the common difference/ratio.
  • ✓Find any term.
Essential Question

2, 5, 8, 11… versus 2, 6, 18, 54… — both are patterns, but they grow very differently. What’s the rule behind each, and how do you find the 100th term?

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Lesson Phases
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Vocabulary Terms
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Standards Aligned
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Interactive Task
Sequence Explorer · Interactive

The Sequence Explorer

Project this and tap each idea. Have students learn the difference between adding (arithmetic) and multiplying (geometric) sequences!

🔢 Explore sequences — tap eachTry it

The Lesson · Gradual Release (I Do · We Do · You Do)

55 minutes, five moves

Tap any phase to open the teacher moves and student actions.

1

Launch — Two Kinds of Growth

5 min

2,5,8,11 vs. 2,6,18,54 — how do these patterns differ?

👩‍🏫 Teacher Moves

  • Show both sequences.
  • Ask about the rule.
  • Set the goal: sequences.

🎒 Student Actions

  • See both.
  • Find the rules.
  • Predict the idea.
2

I Do — Add vs. Multiply

10 min

Teacher models.

👩‍🏫 Teacher Moves

  • Arithmetic adds a common difference.
  • Geometric multiplies by a ratio.
  • Show the formulas.

🎒 Student Actions

  • Learn arithmetic.
  • Learn geometric.
  • See formulas.
3

We Do — Explore Together

13 min

Class uses the Sequence Explorer.

👩‍🏫 Teacher Moves

  • Tap each idea.
  • Find d or r.
  • Extend the sequence.

🎒 Student Actions

  • Tap each.
  • Find d/r.
  • Extend it.
4

You Do — On Your Own

12 min

Students work sequences.

👩‍🏫 Teacher Moves

  • Classify sequences.
  • Find d or r.
  • Find a term with the formula.

🎒 Student Actions

  • Classify them.
  • Find d/r.
  • Find a term.
5

Close — Close

5 min

One sequence.

👩‍🏫 Teacher Moves

  • Give a sequence.
  • Find the next term.
  • Hand out the exit ticket.

🎒 Student Actions

  • Find the term.
  • Explain it.
  • Complete the exit ticket.
Standards Alignment

Built to the standards you report on

Aligned to the Common Core State Standards for Mathematics (High School Algebra).

CCSS
F-BF.2

Write arithmetic and geometric sequences recursively and explicitly.

CCSS
F-LE.2

Construct linear and exponential functions from a sequence.

CCSS
F-IF.3

Recognize that sequences are functions.

CCSS
F-BF.1

Write a function that describes a relationship.

Differentiation

One lesson, every learner

Multilingual Learners

ELL / EMERGING READERS
  • Sequence-type example cards.
  • Sentence frame: “This is arithmetic/geometric because ___.”
  • Extend with counters.

Support & Access

IEP / 504
  • Start with arithmetic.
  • Find the rule by pattern.
  • Extend by one term.

Stretch & Extend

GIFTED / EARLY FINISHERS
  • Write explicit formulas.
  • Find a term far out (a₅₀).
  • Compare growth rates.
Materials

What to gather

  • 📽️Projector / board
  • 🧮Calculators
  • 💻The Sequence Explorer
  • 📄Sequence charts
  • ✏️Pencils
  • 🎫Exit-ticket slips
Vocabulary

Key terms — hover for a quick definition

sequencean ordered list of numbersterma number in a sequencearithmetic sequenceadds a common differencegeometric sequencemultiplies by a common ratiocommon difference (d)what you add each timecommon ratio (r)what you multiply by each timeexplicit formulafinds any term directlyrecursivedefined using the previous term
Evaluate

Exit Ticket

Preview the three formative checks. Tap “Sample answer” to see what mastery looks like — hide them before you print for students.

QUESTION 1
Is 3, 7, 11, 15 arithmetic or geometric? What is d or r?
Arithmetic, with a common difference of 4.
QUESTION 2
Is 2, 4, 8, 16 arithmetic or geometric? What is d or r?
Geometric, with a common ratio of 2.
QUESTION 3
What is the next term in 5, 10, 20, 40?
80 (geometric, times 2).

Going deeper? The 50th term.

Have students use the explicit formula aₙ = a₁ + (n−1)d to find a far-out term without listing them all. A printable sequences sheet is in the Math library.

Study · Flashcards

Study the key terms

Tap a card to flip it, then rate whether you knew it. Built from this lesson’s vocabulary.

🃏 Arithmetic & Geometric SequencesFlip
Card 1
Term
Tap to flip →
Meaning
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Practice · Quiz

Check your understanding

A quick self-check with instant feedback, drawn from this lesson’s key terms.

📝 Arithmetic & Geometric SequencesQuiz
Score: 0
1 / 6
Question 1
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Practice · Worksheet

Printable worksheet

A print-and-go review sheet with a built-in answer key. Tap “Show answer key” to reveal answers, or print the clean version for students.

🖨️ Arithmetic & Geometric SequencesPrint
Name: ________________________
Date: ____________

Part A · Write the word that matches each meaning

Word bank: arithmetic sequence, common difference (d), common ratio (r), explicit formula, geometric sequence, recursive, sequence, term
  1. what you multiply by each time
  2. adds a common difference
  3. an ordered list of numbers
  4. multiplies by a common ratio
  5. defined using the previous term
  6. a number in a sequence
  7. what you add each time
  8. finds any term directly

Part B · Show what you learned

  1. Is 3, 7, 11, 15 arithmetic or geometric? What is d or r?
  2. Is 2, 4, 8, 16 arithmetic or geometric? What is d or r?
  3. What is the next term in 5, 10, 20, 40?
Answer key — Part A: 1) common ratio (r) · 2) arithmetic sequence · 3) sequence · 4) geometric sequence · 5) recursive · 6) term · 7) common difference (d) · 8) explicit formula
Part B: 1) Arithmetic, with a common difference of 4. 2) Geometric, with a common ratio of 2. 3) 80 (geometric, times 2).