Identify how the first figures are alike in certain ways

People have been sorting and grouping shapes for thousands of years. Ancient builders needed to know which stones fit together. Scientists needed to organize plants and animals into groups. The idea of classification (putting things into groups based on shared traits) is one of the most important thinking skills you can develop.

On the TACHS exam, figure classification questions test how well you can spot patterns. You are given three figures that share something in common. Your job is to figure out what they share and then pick a fourth figure that belongs with them.

The big question these problems ask is: "What do these figures have in common, and which answer choice shares that same trait?" Let's learn exactly how to answer that.

Before you can find what figures share, you need to know what features to look for. Think of each figure like a person at a costume party. You can describe each person by their hat, their shoes, or their shirt color. Figures can be described by their own set of features.

Let's look at a diagram that shows how three figures can share a feature even though they look different at first glance. In the example below, all three figures are different shapes, but they all share the same shading. That shading is the common trait.

Notice that the shapes are all different — circle, square, and triangle. The sizes are slightly different too. The only feature they all share is the striped fill pattern. That is how you know shading is the rule, not shape or size.

Figure classification is not about math formulas. It is about following a clear thinking process. Here is a checklist you can use every time you face one of these questions.

The Feature Checklist

1. Step 1 — Count sides. Do all three figures have the same number of sides? If yes, shape might be the rule.
2. Step 2 — Check shading. Are they all solid, all outline, or all striped? Shading is tested very often.
3. Step 3 — Compare size. Are they all the same size, all large, or all small?
4. Step 4 — Look at orientation. Are they all pointing the same direction or rotated the same way?
5. Step 5 — Count inner parts. Does each figure contain a smaller shape inside it? If so, the number of inner shapes might be the rule.
6. Step 6 — Check for combined rules. Sometimes two features matter at the same time (for example, all are small AND all are shaded).

By eliminating features that change across the three figures, you narrow down the one (or two) features that stay the same. That is your rule.

The diagram below organizes the features you should check into categories. Use it as a mental map when you take the test.

Let's walk through a full problem together. Imagine you see three given figures: a large solid circle, a large solid square, and a large solid triangle. The answer choices are: (A) a small solid circle, (B) a large outline hexagon, (C) a large solid pentagon, and (D) a large striped rectangle.

Some TACHS figure classification questions go beyond simple features. Here are a few advanced patterns you might see on the test.

When you run into a harder question and none of the basic features seem to match, try thinking about broader categories. Instead of asking "Are they all triangles?" ask "Do they all have straight edges?" or "Do they all have an even number of sides?" These bigger-picture questions can unlock the rule.

Try these five problems on your own. For each one, identify the shared rule among the three given figures first, then pick the answer choice that follows the same rule.