Solve novel reasoning questions using spatial and figural content - TACHS Ability

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Why Do We Study Spatial Reasoning?

Have you ever tried to figure out how a flat piece of cardboard folds into a box? Or noticed a repeating design on a tile floor and wondered what comes next? These are examples of spatial and figural reasoning — the ability to think about shapes, patterns, and how things fit together in space.

People have been interested in this kind of thinking for a very long time. Ancient builders, artists, and mathematicians all relied on spatial skills. Over the years, researchers studied how our brains handle shapes and patterns. They discovered that these skills help us solve problems in science, engineering, art, and everyday life.

~300 BC • Euclid's Elements

The Greek mathematician Euclid wrote a famous book about geometry. It taught people to reason about shapes, angles, and spatial relationships using logic.

1920s • First Spatial Tests

Psychologists began creating tests that used shapes and figures (instead of words) to measure reasoning ability. These tests worked across different languages.

1938 • Raven's Progressive Matrices

John C. Raven published a famous test using grids of shapes. Test-takers had to find the missing piece by spotting the pattern. This idea still influences tests like the TACHS.

Today • TACHS & Modern Assessments

The TACHS (Test for Admission into Catholic High Schools) uses spatial and figural questions to measure your reasoning skills. These questions test how well you think — not just what you have memorized.

The big question these tests try to answer is: Can you figure out a rule or pattern you have never seen before, just by looking at shapes and figures? That is exactly what you will learn to do in this lesson.

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Core Principles of Spatial & Figural Reasoning

Before we jump into practice, let's build a toolbox of thinking skills. Every spatial reasoning question is really asking you to do one (or more) of these things.

Pattern Recognition

Find the rule that connects a series of shapes. The rule might involve size, shading, rotation, or the number of sides.

Spatial Transformation

Imagine flipping, rotating, or sliding a shape in your mind. Predict what it looks like after the change.

Figure Analogies

Think "Shape A is to Shape B as Shape C is to ___?" You find how two shapes relate, then apply that same relationship to a new pair.

Odd One Out

Look at a group of figures and find the one that does NOT fit. The odd figure breaks a rule that all the others follow.

Matrix Completion

A grid of shapes is missing one piece. You study the rows and columns to discover the pattern, then pick the piece that completes the grid.

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KEY TAKEAWAY

Think of spatial reasoning like being a detective at a crime scene. Each shape or figure is a clue. Your job is to study the clues, figure out the hidden rule, and use it to predict the missing piece. You don't need to memorize facts — you just need to look carefully and think logically!

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Seeing Patterns in Action

Let's look at a visual example. The diagram below shows a common type of TACHS reasoning question: a figure analogy. Study the relationship between the first pair of shapes, then figure out what the missing shape should be.

In Pair 1, the plain square (Shape A) becomes a shaded square with an X (Shape B). You apply the same two changes — add shading and draw an X — to the triangle (Shape C) to find Shape D.

Notice the two-step process. First, you study Pair 1 and list every change: the shape got shading and an X inside. Then you apply those same changes to Shape C to predict Shape D. The shape itself (triangle vs. square) stays the same — only the modifications carry over.

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How Spatial Reasoning Questions Work

Spatial reasoning questions on the TACHS don't use formulas the way a math test does. Instead, they rely on a set of visual operations — things that can happen to a shape. Let's learn the most common operations you will need to recognize.

The Six Visual Operations

Operation	What It Means	Example
Rotation	The shape turns clockwise or counterclockwise by a certain angle (like 90° or 180°).	An arrow pointing up turns to point right (90° clockwise).
Reflection (Flip)	The shape flips over a line, like looking in a mirror.	A letter 'b' becomes 'd' when flipped horizontally.
Scaling	The shape gets bigger or smaller but keeps the same proportions.	A small circle becomes a large circle.
Shading / Color Change	Part or all of the shape changes from white to gray, black, or striped.	A white triangle becomes a striped triangle.
Addition / Removal	A new element is added inside or around the shape, or a piece is taken away.	A dot appears in the center of a square.
Shape Change	The number of sides increases or decreases following a pattern.	Triangle → square → pentagon (sides increase by 1).

Common visual operations tested on spatial reasoning questions

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Pro Tip

Most TACHS questions combine two or three of these operations at once. For example, a shape might rotate AND change shading at the same time. Always check for more than one change!

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KEY TAKEAWAY

Think of each shape like a character in a video game. The "operations" are power-ups or changes that happen to the character at each level. Your job is to figure out which power-ups were used so you can predict what the character looks like at the next level.

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Types of Spatial Questions You Will See

The TACHS Ability section can ask spatial questions in several formats. Knowing what to expect helps you work faster. Below is a diagram showing the four main types and a strategy for each.

Each card shows a question type, a brief description, and a quick strategy. On the TACHS, you may see any of these four formats. Practice identifying which type a question is before you start solving it.

When you sit down with a spatial question, the very first thing to do is identify the type. Is it asking you to continue a series? Find the odd one out? Complete an analogy or a grid? Once you know the type, you can use the right strategy from the table above.

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Worked Example: Solving a Series Completion

Let's walk through a full problem step by step. Imagine you see the following series of shapes:

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Question

A sequence shows: (1) a small white triangle pointing up, (2) a medium gray triangle pointing right, (3) a large black triangle pointing down. What is the fourth shape in the series?

Series Completion — Step by Step

1
Step 1 — Identify the Question Type

You see shapes in a row that change from one to the next. This is a series completion question. Your goal is to find what comes next.
2
Step 2 — Track Size Changes

Shape 1 is small, Shape 2 is medium, and Shape 3 is large. The size increases each step. If the pattern continues, Shape 4 should be extra-large.

Size pattern: small → medium → large → extra-large
3
Step 3 — Track Shading Changes

Shape 1 is white, Shape 2 is gray, Shape 3 is black. The shading gets darker each step. Since black is already the darkest, the pattern may cycle back. Shape 4 could be white again, or could use a new pattern like stripes. On the TACHS, look at the answer choices for the best match.

Shading pattern: white → gray → black → (cycles back to white)
4
Step 4 — Track Rotation Changes

Shape 1 points up, Shape 2 points right, Shape 3 points down. Each shape rotates 90° clockwise. So Shape 4 should point left.

Rotation pattern: up → right → down → left (90° clockwise each time)
5
Step 5 — Combine All Patterns

Put the three patterns together. Shape 4 is an extra-large, white (if cycling), left-pointing triangle. Look at the answer choices and pick the one that matches all three features.

Answer: Extra-large white triangle pointing left.

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STRATEGY SUMMARY

Break every problem into separate "channels" — size, shading, rotation, number of sides, and inner details. Track each channel one at a time, like adjusting the volume, bass, and treble on a stereo separately. Then combine them at the end for your answer.

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Strategies, Strengths & Common Mistakes

Spatial reasoning is a skill you can improve with practice. Let's look at some strategies that help and some common traps to avoid.

Strong Strategy ✅	Common Mistake ❌
Check all features (size, shape, shading, position) one at a time.	Only looking at one feature and ignoring the rest.
Look at EVERY answer choice before picking one.	Choosing the first answer that seems "close enough."
Use elimination: cross out answers that break even one rule.	Trying to find the right answer without eliminating wrong ones first.
In a matrix, check rows AND columns for patterns.	Only checking rows and forgetting columns (or vice versa).
Mentally rotate or flip the shape instead of guessing.	Confusing rotation with reflection. A 90° turn is NOT the same as a flip!

Strategies versus common mistakes on spatial reasoning questions

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KEY TAKEAWAY

Imagine you are ordering a pizza. If you only check the crust and ignore the toppings, you might end up with pineapple when you wanted pepperoni! In the same way, you have to check every feature of a shape — not just one — to make sure your answer is right.

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Connecting to More Advanced Reasoning

The spatial reasoning skills you build now will help you in many subjects later on. Let's see how the basics connect to more advanced thinking.

What You Learn Now	Where It Goes Next
Rotating shapes in your mind	Geometry proofs about rotations, reflections, and translations in high school math
Spotting visual patterns in a series	Recognizing number patterns, functions, and algebraic sequences
Figuring out how a flat net folds into a 3-D box	Surface area and volume calculations in geometry, and 3-D modeling in engineering
Finding the odd one out in a set of figures	Classifying data, identifying outliers in statistics, and logical reasoning in science
Completing a matrix grid	Solving systems of equations and understanding matrices in advanced algebra

How current spatial skills connect to future coursework

The TACHS tests these skills because they predict how well you will handle complex problem-solving in high school. Think of this test as a preview. The better you get at reasoning with shapes and figures now, the easier it will be to learn geometry, physics, and computer science later.

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Practice Problems

Try these five problems on your own. Each one is a little harder than the last. Read carefully, look at every feature, and use the strategies you learned.

Question 1: CONCEPTUAL

A figure analogy question shows: a large white circle is to a small black circle, as a large white square is to ___. What is the answer, and how do you know?

Show answer

The answer is a small black square. Looking at the first pair, two changes happen: the size goes from large to small, and the shading goes from white to black. You apply those exact same two changes to the large white square. The shape (square) stays the same, but it becomes small and black. Always track size, shading, and shape separately to find the right answer.

Question 2: BASIC CALCULATION

In a series of shapes, each shape rotates 90° clockwise from the previous one. The first shape is an arrow pointing up. What direction does the arrow point in the fourth shape?

Show answer

The arrow points left. Starting from up and rotating 90° clockwise each time: Shape 1 = up, Shape 2 = right (90°), Shape 3 = down (180°), Shape 4 = left (270°). After four 90° turns, you would return to the starting position (up), but we stop at the fourth shape. Count each rotation carefully — it helps to use your hand or a pencil to trace the direction.

Question 3: INTERMEDIATE

Five figures are shown: (A) a triangle with a dot inside, (B) a square with a dot inside, (C) a pentagon with a dot inside, (D) a hexagon with a star inside, (E) a heptagon (7 sides) with a dot inside. Which figure is the odd one out, and why?

Show answer

Figure D is the odd one out. The pattern across the other four figures is that the number of sides increases by one (3, 4, 5, 6, 7) and each figure has a dot inside. Figure D is a hexagon (6 sides), which fits the side-count pattern, but it has a star inside instead of a dot. That breaks the rule about the inner element. When looking for the odd one out, always check both the outer shape and the inner details.

Question 4: APPLIED

A 3×3 matrix grid is shown. Row 1: small circle, medium circle, large circle. Row 2: small triangle, medium triangle, large triangle. Row 3: small square, medium square, ___. What goes in the blank? Now look at the columns. Column 1: small circle, small triangle, small square. Does the column pattern confirm your answer?

Show answer

The blank should be a large square. In each row, the shape stays the same while the size increases from small to medium to large. Row 3 has squares, so the missing piece is a large square. Checking the columns: Column 3 has large circle, large triangle, and should have a large square — the column pattern (same size, different shapes increasing in side count) also confirms this. Whenever you solve a matrix, always verify your answer by checking both rows and columns.

Question 5: CRITICAL THINKING

A figure analogy shows: Shape A is a white circle with a horizontal line through the middle. Shape B is a gray circle with a vertical line through the middle. Shape C is a white diamond with two horizontal lines through it. What should Shape D look like? Explain every change you need to apply.

Show answer

Shape D should be a gray diamond with two vertical lines through it. Here is how to find the answer. First, compare Shape A to Shape B. There are two changes: (1) the shading changes from white to gray, and (2) the line rotates 90° (horizontal becomes vertical). Now apply both changes to Shape C. Change 1: The white diamond becomes a gray diamond. Change 2: The two horizontal lines rotate 90° and become two vertical lines. The shape itself (diamond) does not change, just like the circle stayed a circle in the first pair. Combining all changes gives a gray diamond with two vertical lines. This problem shows why tracking each feature separately is so important — if you miss the line rotation, you would pick the wrong answer.

SUMMARY

Lesson Summary

Summary

Spatial and figural reasoning is the ability to spot patterns and rules in shapes, figures, and visual arrangements. On the TACHS, you will see four main question types: series completion (what comes next?), figure analogies (A is to B as C is to ___), odd one out (which doesn't belong?), and matrix completion (fill in the missing cell in a grid).

The key to solving these questions is to break each figure into separate features: size, shape, shading, rotation, and inner details. Track each feature one at a time, then combine them for your final answer. Use elimination to cross out choices that break even one rule. Practice these strategies, and you will build confidence for the TACHS Ability section!