Apply pattern recognition and rule-based reasoning to unfamiliar problems - TACHS Ability

SECTION 1

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.

~3000 BCE • Ancient Number Patterns

Babylonian scribes noticed patterns in numbers and created some of the first multiplication tables. They used repeating patterns to predict floods along the Tigris and Euphrates rivers.

~300 BCE • Euclid's Logical Rules

The Greek mathematician Euclid wrote a book called Elements. He showed that you can prove things by following strict rules, one step at a time.

1202 • Fibonacci's Famous Sequence

Leonardo Fibonacci described a number sequence (1, 1, 2, 3, 5, 8, 13…) where each number is the sum of the two before it. This pattern appears in flowers, shells, and galaxies.

1950s • Computers Learn Rules

Early computer scientists taught machines to follow rule-based reasoning. Computers could solve puzzles by checking many possible paths, one rule at a time.

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.

SECTION 2

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

Look at every detail. Compare items in a sequence side by side. Ask: What changes? What stays the same?

Find the Pattern

A pattern is anything that repeats or changes in a predictable way. It could be a number going up by 3, a shape rotating, or colors alternating.

State the Rule

Turn the pattern into an "if-then" rule. Example: "Each number is the previous number plus 5." This is your rule.

Apply the Rule

Use your rule to predict the next item, fill in a blank, or choose the correct answer. Then check: does the rule work for every item, not just one?

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KEY TAKEAWAY

Think of pattern recognition like being a detective. A detective collects clues (observe), figures out what the clues have in common (find the pattern), writes a theory (state the rule), and then tests that theory (apply the rule). If the theory doesn't fit all the clues, the detective tries a new one!

SECTION 3

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.

The top row shows the four-step method. The bottom row shows the number sequence with each jump of +4 highlighted. The green circle is the answer we found by applying the rule.

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.

SECTION 4

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.

ARITHMETIC (ADDING/SUBTRACTING) PATTERN

next number = previous number + d

Here, d is the common difference — the amount you add (or subtract) each time. Example: 5, 8, 11, 14 → d = 3.

GEOMETRIC (MULTIPLYING/DIVIDING) PATTERN

next number = previous number × r

Here, r is the common ratio — the number you multiply by each time. Example: 2, 6, 18, 54 → r = 3.

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.

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TIP: Look for Multiple Rules

Some tricky problems have more than one rule happening at the same time. For example, a shape might rotate 90° AND change shading at each step. Always check if more than one thing is changing.

SECTION 5

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.

This chart shows five common types of pattern and reasoning problems. Number sequences and shape patterns are the most common on the TACHS.

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

Quick-reference strategies for each problem type

SECTION 6

Worked Example: Cracking an Unfamiliar Problem

Let's walk through a problem step by step. Imagine you see this on a test:

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PROBLEM

Look at the sequence: 4, 7, 12, 19, 28, ? What number comes next? (A) 35 (B) 37 (C) 39 (D) 41

Step-by-Step Solution

1
Step 1 — Observe the Sequence

Write out the numbers: 4, 7, 12, 19, 28. At first glance, this doesn't look like a simple "+3" or "×2" pattern. Don't panic — just move to step 2.
2
Step 2 — Find the Differences

Subtract each number from the one after it. 7 − 4 = 3, 12 − 7 = 5, 19 − 12 = 7, 28 − 19 = 9. So the differences are: 3, 5, 7, 9. Those differences are going up by 2 each time!

Differences: 3, 5, 7, 9 (increasing by 2)
3
Step 3 — State the Rule

The rule is: "Each difference is 2 more than the previous difference." The next difference should be 9 + 2 = 11.

Next difference = 11
4
Step 4 — Apply the Rule

Add 11 to the last number: 28 + 11 = 39.

Answer: (C) 39
5
Step 5 — Check Your Work

Look back: differences were 3, 5, 7, 9, 11. Each goes up by 2. ✓ The pattern holds for every step, so we can be confident the answer is correct.

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KEY TAKEAWAY

When the differences between numbers aren't the same, take it one level deeper — look at the differences of the differences. It's like peeling layers off an onion until you find the simple pattern hiding inside.

SECTION 7

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

Strategy comparison for pattern problems

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KEY TAKEAWAY

Think of your strategies like tools in a toolbox. A hammer is great for nails, but terrible for screws. If the first strategy you try doesn't work, don't force it — just pick a different tool!

SECTION 8

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)

From TACHS reasoning to high school math

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.

SECTION 9

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.

Question 1: CONCEPTUAL

What is the main difference between an arithmetic pattern and a geometric pattern?

Show answer

In an arithmetic pattern, you add (or subtract) the same number each time. In a geometric pattern, you multiply (or divide) by the same number each time. For example, 2, 5, 8, 11 is arithmetic because each term adds 3. The sequence 3, 6, 12, 24 is geometric because each term multiplies by 2. Knowing which type you're dealing with tells you which strategy to use.

Question 2: BASIC CALCULATION

What is the next number in the sequence? 10, 16, 22, 28, ?

Show answer

First, find the differences: 16 − 10 = 6, 22 − 16 = 6, 28 − 22 = 6. The common difference is 6 (an arithmetic pattern). Apply the rule: 28 + 6 = 34. The answer is 34. You can double-check by verifying that every jump is exactly 6.

Question 3: INTERMEDIATE

Look at this sequence: 1, 4, 9, 16, 25, ?. Find the next number. (Hint: if the differences aren't equal, look at the second differences.)

Show answer

The differences are 3, 5, 7, 9. The second differences are all 2. So the next first-difference should be 9 + 2 = 11. Therefore the next number is 25 + 11 = 36. You may also notice that these are perfect squares: 1², 2², 3², 4², 5², 6². Either approach gives you 36.

Question 4: APPLIED

A password uses this rule: "Start with the letter A. Move forward in the alphabet by 2, then by 3, then by 4, and so on." The first few letters are A, C, F, J. What is the sixth letter in the sequence?

Show answer

Letter positions: A = 1, C = 3, F = 6, J = 10. The jumps are +2, +3, +4. Continuing: the next jump is +5, so 10 + 5 = 15 → the letter O. The jump after that is +6, so 15 + 6 = 21 → the letter U. The sixth letter is U. This problem uses increasing differences just like the number problems, but with letters instead.

Question 5: CRITICAL THINKING

A grid has three rows and three columns. In each row, the shapes follow a rule. Row 1: filled circle, empty square, filled triangle. Row 2: empty circle, filled square, empty triangle. Row 3: filled circle, empty square, ?. What goes in the missing cell, and what two rules did you use?

Show answer

There are two rules working at the same time. Rule 1 (columns — shape): column 1 is always a circle, column 2 is always a square, column 3 is always a triangle. Rule 2 (rows — shading pattern): shading alternates filled, empty, filled across each row. Row 3 follows filled, empty, so the third cell must be filled. Putting both rules together, the answer is a filled triangle. On the TACHS, always check for a rule across rows AND a separate rule down columns.

SUMMARY

Lesson Summary

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!