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Searching for Patterns — Module Hub & Visuals
Module 1. Build a sequence with sliders and watch it appear three ways at once — as a table, as points on a graph, and as a rule you can read aloud.
Lesson Slides & PDFs
Student-facing lesson slide PDFs are grouped by topic below for review, catch-up, and family support.
📖 Student Edition for this module (PDF) ✍ Skills Practice — Student Edition (PDF)
Graphic Organizers & Activities
Foldable notes and activities for each lesson, with the one to start with listed first. Sign in with your alias and PIN to open them.
Start Here — What Module 1 Is About
What this module is for, and what you will be able to do by the end of it.
Every sequence is a function with a special domain: the counting numbers \(\{1, 2, 3, \dots\}\). The first term is where the pattern starts; the step — a constant difference for arithmetic or a constant ratio for geometric — is the rule that drives it forward. Move the sliders below and watch the table, the graph, and the algebra stay in perfect agreement.
Key Vocabulary
The precise words for what you just investigated. The standards this module carries are in the next section.
An ordered list of numbers (terms). As a function its domain is the counting numbers \(\{1, 2, 3, \dots\}\) and each term \(a_n\) is an output.
The constant amount added from one term to the next in an arithmetic sequence: \(a_n - a_{n-1} = d\).
The constant multiplier from one term to the next in a geometric sequence: \(a_n \div a_{n-1} = r\).
Defines a term from the one before it, plus a starting value — e.g. \(a_1 = 3,\ a_n = a_{n-1} + 2\).
A closed-form formula for any term directly from \(n\) — e.g. \(a_n = 3 + 2(n-1)\). No earlier terms needed.
Domain, first/last term, and behavior (increasing, decreasing, growth, decay) — the structure you read off any function.
Standards in This Module
The Texas Essential Knowledge and Skills this module is built to teach. RDY marks a STAAR readiness standard.
TEKS & Function Key-Features
This lab targets the Sequences strand of Module 1, where scholars first meet recursive and explicit rules and connect them to the families of functions they'll graph all year.
A.12A simple recursive sequences · A.12C arithmetic sequences · A.12D geometric sequences. Identifying domain, range, and behavior connects to A.2A and A.3C.
Investigate the Pattern
What the lab is showing you, and four experiments worth running before you touch the vocabulary.
What you're seeing
- Three views, one pattern. The table, the graph, and the rule are the same sequence shown three different ways — the four blueprints of every function (here, three of them).
- Arithmetic = constant difference. Each term adds the same amount \(d\). On the graph the points land on a straight line — that's why \(d\) behaves like a slope.
- Geometric = constant ratio. Each term multiplies by the same number \(r\). The points curve — the seed of exponential growth and decay (Module 4).
- Recursive vs. explicit. The recursive rule tells you how to get the next term from the last; the explicit rule jumps straight to any term \(a_n\) without listing the others.
Try this
- Set arithmetic with \(a_1 = 3\) and \(d = 2\). Read the explicit rule, then predict \(a_{10}\) before dragging the term slider to check.
- Make \(d\) negative. What happens to the line — and to the "Behavior" readout? When does a sequence decrease?
- Switch to geometric with \(r = 2\). Compare the shape to the line. Now set \(r\) between 0 and 1 — that's decay.
- Find a setting where arithmetic and geometric look almost the same for the first two terms but split apart by term 6. Why does the ratio eventually win?
Pattern & Sequence Explorer
Pick a sequence type, then drag the three sliders. The highlighted step shows the constant difference or ratio at work.
Want to See This in 3D?
Optional, and always second. The lab above is where you measure and calculate; this is the same idea as an everyday object you can turn over and inspect. Open it when the lab already makes sense — or skip it entirely.
Ready for the full course map? Head back to Algebra I, read the Syllabus, or check the Pacing Guide to see where Module 1 sits in the year.
Module and sequence structure follow the TEA Bluebonnet Learning — Secondary Mathematics open curriculum (Edition 1, adapted from Carnegie Learning), licensed CC BY-NC 4.0. Classroom use is non-commercial.