Why Patterns and Rules Matter
Humans have always looked for patterns in the world around them. Thousands of years ago, farmers noticed that seasons repeat in a cycle. They used that pattern to decide when to plant crops. This is one of the earliest examples of pattern recognition (noticing something that repeats or follows a rule).
Over time, thinkers began writing down the rules behind patterns. This idea of following step-by-step rules is called rule-based reasoning (using a set of rules, or "if-then" statements, to figure out an answer). Let's look at how this skill developed over history.
Today, tests like the TACHS check whether you can spot patterns and follow rules even when a problem looks brand-new. The good news? You can get better at this skill with practice.
Core Principles of Pattern Recognition & Rule-Based Reasoning
Before you can solve unfamiliar problems, you need to understand four key ideas. Think of these as your reasoning toolkit.
Observe Carefully
Find the Pattern
State the Rule
Apply the Rule
Seeing Patterns in Action
The diagram below shows a simple number pattern and the four-step process you use to crack it. Follow the arrows from left to right.
Notice how each step builds on the one before it. You can't state a rule until you find the pattern, and you can't find the pattern until you observe carefully. On the TACHS, the problems may use numbers, shapes, or pictures, but the four-step process is always the same.
How Patterns and Rules Work
Number Patterns
Many TACHS questions give you a sequence of numbers. Your job is to find the rule that connects them. Here are two common types.
Shape and Picture Patterns
Some problems use shapes instead of numbers. In these problems, look for changes in size, rotation (turning), shading (filled vs. empty), and position (where the shape sits). These are the four most common changes tested on reasoning sections.
If-Then Rules
An if-then rule tells you what to do when a certain condition is true. For example: "If the shape is a triangle, shade it. If it is a circle, leave it empty." When you see a new shape in the sequence, you check the condition and follow the rule.
Types of Pattern Problems You'll See
The TACHS Ability section can throw many kinds of pattern and reasoning problems at you. The diagram below groups them into categories so you know what to expect.
| Problem Type | What to Look For | Strategy |
|---|---|---|
| Number Sequence | Differences or ratios between numbers | Subtract each pair; look for a constant or growing difference |
| Shape Pattern | Rotation, reflection, shading, size changes | Track one feature at a time across the sequence |
| Matrix / Grid | Rules across rows AND down columns | Check each row for a rule, then check each column |
| Analogy | A relationship between the first pair | Name the relationship in words, then apply it to the new pair |
| Odd-One-Out | A shared rule that most items follow | Find what makes one item different from the rest |
Worked Example: Cracking an Unfamiliar Problem
Let's walk through a problem step by step. Imagine you see this on a test:
Strengths and Pitfalls of Each Strategy
Not every strategy works for every problem. Knowing when a strategy is strong — and when it might trick you — will save time on test day.
| Strategy | Strengths | Watch Out For… |
|---|---|---|
| Find the difference | Quick and easy for add/subtract patterns | Doesn't work when the operation is multiply or divide |
| Find the ratio | Great for doubling/tripling patterns | May give non-whole-number ratios, which are harder to spot |
| Track one feature at a time | Keeps you organized on shape problems | You might miss a feature if you stop tracking too early |
| Test the answer choices | Works when you're stuck — try each answer and see which fits | Slower; use this as a backup, not your first move |
| Look at second differences | Unlocks harder patterns where differences change | You need at least four numbers to confirm the pattern |
Connecting to Advanced Reasoning
The skills you learn here don't just help on the TACHS. They connect to bigger ideas you'll see in high school and beyond. The table below shows how today's lesson links to more advanced topics.
| What You Learn Now | Where It Leads |
|---|---|
| Finding arithmetic patterns (add/subtract) | Arithmetic sequences and series in Algebra 2 |
| Finding geometric patterns (multiply/divide) | Geometric sequences, exponential growth, compound interest |
| Stating if-then rules | Logic and proofs in Geometry; computer programming |
| Looking at second differences | Quadratic functions (y = ax² + bx + c) in Algebra 1 and 2 |
| Tracking multiple features in shapes | Geometric transformations (rotation, reflection, translation) |
Every time you solve a pattern problem, you are building the same thinking muscles used by scientists, coders, and engineers. Pattern recognition is one of the most important skills in the world.
Practice Problems
Try each problem on your own before reading the answer. Use the four-step process: Observe → Find the Pattern → State the Rule → Apply.
Lesson Summary
Pattern recognition means noticing what repeats or changes in a predictable way. Rule-based reasoning means turning that pattern into a clear rule (like an if-then statement) and applying it. On the TACHS, you'll meet number sequences, shape patterns, matrix grids, analogies, and odd-one-out problems.
Always follow the four-step process: Observe every detail, Find the pattern by looking at differences or changes, State the rule in your own words, and Apply the rule to find the answer. If the first differences aren't equal, check the second differences. For shape problems, track one feature at a time. With practice, these skills become second nature!