TACHS: Classification: match each shape to its shading
Learn to find the shape-to-shading pairing that every example keeps, ignore the changing counts and positions, and check every symbol in a choice.
How do you find the rule when the number and position of the symbols keep changing?
Short answer
The rule is a pairing: one shape is always dark and the other is always unfilled. State the pairing, then check every symbol in each choice. One symbol with the wrong shading rules the choice out.
Know first: Names of common shapes, Telling dark from unfilled
Step 1 of 5
The shading goes with the shape
In these questions, each example contains a few small symbols of two different shapes. The number of symbols changes, and they move around, but the shading always goes with the shape. One shape is always dark (solid) and the other is always unfilled (an outline only).
Counts and positions change on purpose, so they are noise. The rule has to hold for every single symbol in the figure.
| Feature | Across the examples | Use it? |
|---|---|---|
| How many symbols | changes | No |
| Where they sit | changes | No |
| Which shape is dark | stays the same | Yes |
| Which shape is unfilled | stays the same | Yes |
Step 2 of 5
Check every symbol
Worked example: Triangles and hexagons
Example 1: two dark triangles and one unfilled hexagon.
Example 2: one dark triangle and three unfilled hexagons.
Example 3: three dark triangles and one unfilled hexagon.
Each choice has four symbols. Which belongs with the examples?
- ATwo dark triangles, one unfilled hexagon, one dark hexagon
- BThree dark triangles, one unfilled triangle
- CTwo dark triangles, two unfilled hexagons
- DOne unfilled triangle, three unfilled hexagons
- ETwo dark hexagons, two dark triangles
- 1
State the rule
Every triangle is dark and every hexagon is unfilled, in all three examples. Rule: triangles dark, hexagons unfilled.
- 2
Check the choices symbol by symbol
Two dark triangles, one unfilled hexagon, one dark hexagon: the dark hexagon breaks the rule. Three dark triangles and an unfilled triangle: the unfilled triangle breaks it. One unfilled triangle and three unfilled hexagons: the triangle should be dark. Two dark hexagons and two dark triangles: the hexagons should be unfilled.
- 3
Confirm the answer
Two dark triangles and two unfilled hexagons. Every symbol fits.
- 4
See why the other choices are there
The first choice gets three of four symbols right and can look just like Example 1. Each wrong choice flips the shading of at least one symbol.
Answer
Two dark triangles and two unfilled hexagons.
Find the wrong step
The examples show dark triangles and unfilled hexagons. A student checks a choice with one dark triangle, two unfilled hexagons and one dark hexagon.
- 1
Confirms the rule: triangles dark, hexagons unfilled.
- 2
Checks the triangle. It is dark.
- 3
Sees two unfilled hexagons, decides the hexagons are fine, and accepts the choice.
What went wrong
Two hexagons that pass do not clear the third. That one is dark, which breaks the rule.
The fix
Check every symbol on its own. The dark hexagon rules this choice out.
Step 3 of 5
Find and apply the pairing
Check yourself · Question 1
Example 1: one dark diamond and two unfilled pentagons. Example 2: three dark diamonds and one unfilled pentagon, with a dark diamond on the far right. Example 3: two dark diamonds and two unfilled pentagons. Which rule fits all three?
Answer: C
The counts and positions change, but every diamond is dark and every pentagon is unfilled in all three examples.
Check yourself · Question 2
Example 1: one dark pentagon and one unfilled diamond. Example 2: two dark pentagons and two unfilled diamonds. Example 3: three dark pentagons and one unfilled diamond. Each choice has four symbols. Which belongs with the examples?
Answer: B
The rule is pentagons dark, diamonds unfilled. Only this choice keeps that pairing for all four symbols.
Trap answer: One dark pentagon, two unfilled diamonds, one dark diamond
Three out of four
- Why it looks right
- The first three symbols you check all fit, so it is easy to stop there.
- Why it's wrong
- The rule must hold for every symbol. The dark diamond breaks it.
- The right answer
- One dark pentagon, three unfilled diamonds.
Common mistake
I built my rule from the counts in the first example.
- Why it's tempting
- The first example is the first thing you study, and counts are easy to notice.
- Do this instead
- Counts change on purpose. Look for what stays the same in all three examples: which shape is dark and which is unfilled.
Step 4 of 5
What to remember
Remember
- The rule pairs each shape with a shading: one shape always dark, the other always unfilled.
- Ignore how many symbols there are and where they sit.
- Check every symbol in a choice. One mismatch rules it out.
Step 5 of 5
Practice on a real question
Use what you just learned on this question, then check the explanation.
Try another one
In your own words
In the practice question, which shape was dark and which was unfilled, and which symbol ruled out the closest wrong choice?