Home›Lesson Plans›Mathematics›Algebra II · Grade 11

Sequences & Series: Patterns in Numbers

Students explore two fundamental number patterns — arithmetic sequences (adding a common difference) and geometric sequences (multiplying by a common ratio) — and learn to identify, extend, and describe each with a rule.

Grade 11Sequences50 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 arithmetic sequences.
  • ✓Identify geometric sequences.
  • ✓Find a common difference or ratio.
  • ✓Extend a sequence.
Essential Question

2, 4, 8, 16… or 2, 4, 6, 8… — both look like simple patterns, but they grow in completely different ways. How do you tell them apart and predict the next term?

0
Lesson Phases
0
Vocabulary Terms
0
Standards Aligned
0
Interactive Task
Arithmetic or Geometric? · Interactive

Arithmetic or Geometric?

Project this and sort each pattern or clue into arithmetic (add a difference) or geometric (multiply by a ratio).

⚖️ Sort each — arithmetic or geometric?Try it
➕ Arithmetic (add)
✖️ Geometric (multiply)
Tap one, then tap a bucket.
The Lesson · Gradual Release (I Do · We Do · You Do)

50 minutes, five moves

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

1

Launch — Two Kinds of Growth

5 min

2, 4, 6, 8 vs. 2, 4, 8, 16 — what’s the difference?

👩‍🏫 Teacher Moves

  • Show both sequences.
  • Ask how each grows.
  • Set the goal: sequence types.

🎒 Student Actions

  • Compare them.
  • Find the pattern.
  • Predict the rule.
2

I Do — Difference vs. Ratio

10 min

Teacher models.

👩‍🏫 Teacher Moves

  • Arithmetic: add a common difference.
  • Geometric: multiply by a common ratio.
  • Extend each.

🎒 Student Actions

  • Learn arithmetic.
  • Learn geometric.
  • Extend one.
3

We Do — Sort Together

13 min

Class uses Arithmetic or Geometric?

👩‍🏫 Teacher Moves

  • Sort each pattern.
  • Find the difference or ratio.
  • Predict the next term.

🎒 Student Actions

  • Sort them.
  • Find d or r.
  • Predict next.
4

You Do — On Your Own

12 min

Students analyze sequences.

👩‍🏫 Teacher Moves

  • Classify sequences.
  • Find d or r.
  • Write a rule and extend.

🎒 Student Actions

  • Classify them.
  • Find d or r.
  • Extend them.
5

Close — Close

5 min

One sequence.

👩‍🏫 Teacher Moves

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

🎒 Student Actions

  • Classify it.
  • Find the next term.
  • 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 II).

CCSS
F-BF.2

Write arithmetic and geometric sequences recursively and explicitly.

CCSS
F-LE.2

Construct linear and exponential functions, including from sequences.

CCSS
F-IF.3

Recognize that sequences are functions defined on the integers.

CCSS
A-SSE.4

Derive and use the formula for a geometric series.

Differentiation

One lesson, every learner

Multilingual Learners

ELL / EMERGING READERS
  • Sequence pattern cards.
  • Sentence frame: “This sequence ___ each time.”
  • Highlight the operation.

Support & Access

IEP / 504
  • Start with clear add/multiply patterns.
  • Provide the first few terms.
  • Find d or r together.

Stretch & Extend

GIFTED / EARLY FINISHERS
  • Write explicit formulas.
  • Find the sum of a series.
  • Solve a real growth problem.
Materials

What to gather

  • 📽️Projector / board
  • 📓Math notebooks
  • 💻Arithmetic or Geometric?
  • 🧮Calculators
  • ✏️Pencils
  • 🎫Exit-ticket slips
Vocabulary

Key terms — hover for a quick definition

sequencean ordered list of numberstermone number in a sequencearithmetic sequenceadds a common differencegeometric sequencemultiplies by a common ratiocommon differencethe number added each time (d)common ratiothe factor multiplied each time (r)seriesthe sum of a sequence’s termsexplicit formulaa rule for the nth 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
What is the difference between an arithmetic and a geometric sequence?
An arithmetic sequence adds a common difference each term; a geometric sequence multiplies by a common ratio.
QUESTION 2
Find the next term: 5, 10, 20, 40, ___.
80 (geometric, multiply by 2).
QUESTION 3
What is the common difference in the sequence 7, 11, 15, 19?
4 (each term adds 4).

Going deeper? Sum a series.

Have students use formulas to find the sum of the first n terms of an arithmetic or geometric series. A printable sequences-and-series 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.

🃏 Sequences & SeriesFlip
Card 1
Term
Tap to flip →
Meaning
0

Nice work!

Practice · Quiz

Check your understanding

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

📝 Sequences & SeriesQuiz
Score: 0
1 / 6
Question 1
0%

Nice work!

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.

🖨️ Sequences & SeriesPrint
Name: ________________________
Date: ____________

Part A · Write the word that matches each meaning

Word bank: arithmetic sequence, common difference, common ratio, explicit formula, geometric sequence, sequence, series, term
  1. adds a common difference
  2. the factor multiplied each time (r)
  3. an ordered list of numbers
  4. the sum of a sequence’s terms
  5. one number in a sequence
  6. a rule for the nth term
  7. the number added each time (d)
  8. multiplies by a common ratio

Part B · Show what you learned

  1. What is the difference between an arithmetic and a geometric sequence?
  2. Find the next term: 5, 10, 20, 40, ___.
  3. What is the common difference in the sequence 7, 11, 15, 19?
Answer key — Part A: 1) arithmetic sequence · 2) common ratio · 3) sequence · 4) series · 5) term · 6) explicit formula · 7) common difference · 8) geometric sequence
Part B: 1) An arithmetic sequence adds a common difference each term; a geometric sequence multiplies by a common ratio. 2) 80 (geometric, multiply by 2). 3) 4 (each term adds 4).