TACHS: Classification: link sides to dots
Learn to turn each example into a pair of numbers, sides and dots, find the fixed difference, and count the answer polygon's sides without slipping.
How do you find the rule that connects a polygon to the number of dots inside it?
Short answer
Count sides and dots in each example and find how many fewer dots there are than sides. Then count the answer polygon's sides carefully, corner by corner, and apply the same difference.
Know first: Polygon names, from triangle to octagon, Subtracting small numbers
Step 1 of 5
Sides and dots are tied by a fixed difference
In these questions, each example is a polygon with some dots inside it. The polygons differ and the dot counts differ, but the two numbers are tied by a fixed rule: dots equal sides, or there are 1, 2 or 3 fewer dots than sides.
The answer choices usually show the same polygon with different numbers of dots. So everything depends on counting that polygon's sides correctly.
| Shape | Sides |
|---|---|
| Triangle | 3 |
| Square or diamond | 4 |
| Pentagon | 5 |
| Hexagon | 6 |
| Heptagon | 7 |
| Octagon | 8 |
Small hexagons, heptagons and octagons look alike, so count the corners every time.
Step 2 of 5
Find the difference, then count
Worked example: Three fewer dots than sides
Example 1: a square with 1 dot.
Example 2: a hexagon with 3 dots.
Example 3: an octagon with 5 dots.
All five choices show the same seven-cornered polygon with different numbers of dots. Which belongs?
- A3 dots
- B4 dots
- C5 dots
- D6 dots
- E7 dots
- 1
Find the difference
4 − 1 = 3, 6 − 3 = 3, 8 − 5 = 3. Rule: 3 fewer dots than sides.
- 2
Count the answer polygon
Start at the top corner and move clockwise. You pass 7 corners before you return, so it has 7 sides.
- 3
Apply the rule
7 − 3 = 4 dots.
- 4
See why the other choices are there
3 dots comes from miscounting the polygon as a hexagon. 5 dots comes from miscounting it as an octagon. 6 dots uses a 1-fewer rule, and 7 dots uses dots equal sides. Neither fits the examples.
Answer
4 dots.
Find the wrong step
A student counts the sides of an octagon before applying a rule.
- 1
Picks a corner to start from.
- 2
Counts corners while glancing back and forth between the polygon and the choices, and arrives at 9.
What went wrong
Looking away loses your place, so a corner gets counted twice. An octagon has 8 corners.
- 3
Subtracts using the 9.
The fix
Mark the starting corner in your mind, count clockwise without looking away, and stop when you reach it again.
Step 3 of 5
Find and apply the rule
Check yourself · Question 1
Example 1: a pentagon with 5 dots. Example 2: a triangle with 3 dots. Example 3: an octagon with 8 dots. Which rule fits all three?
Answer: C
Each polygon has exactly as many dots as sides: 5 and 5, 3 and 3, 8 and 8.
Check yourself · Question 2
Example 1: a square with 3 dots. Example 2: a hexagon with 5 dots. Example 3: a triangle with 2 dots. All five choices show the same eight-cornered polygon. How many dots does the right choice have?
Answer: C
Each example has 1 fewer dot than sides: 4 − 3, 6 − 5, 3 − 2. The polygon has 8 sides, so 8 − 1 = 7 dots.
Trap answer: 6 dots
A one-side miscount
- Why it looks right
- The rule, 1 fewer, is right, so the answer feels checked.
- Why it's wrong
- The polygon has 8 sides. Counting 7 is a single slip, and the choices are built around it.
- The right answer
- 7 dots, since 8 − 1 = 7.
Common mistake
I checked the rule on one example and moved on.
- Why it's tempting
- A triangle with 2 dots fits "1 fewer," so the rule seems settled.
- Do this instead
- One example can fit several rules. A triangle with 2 dots also fits "always 2 dots." Confirm the difference on all three examples.
Step 4 of 5
What to remember
Remember
- Write each example as sides and dots, and find the difference that holds for all three.
- Count the answer polygon's sides from a fixed starting corner, going one way around.
- Wrong choices sit one side off and use rules that fit no example.
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, what difference between sides and dots did the examples share, and how many sides did the answer polygon have?