📘 Figure it Out – Questions & Answers
Q1. Count the number of sides in each shape in the sequence of Regular Polygons. Which number sequence do you get? What about the number of corners in each shape? Do you get the same number sequence? Can you explain why?
✅ Answer:
- Number of sides: 3, 4, 5, 6, 7, 8, 9, 10, …
- Number of corners (vertices): 3, 4, 5, 6, 7, 8, 9, 10, …
✔ Yes, both sequences are the same because:
👉 In any polygon, the number of sides = number of corners (vertices).
Q2.Count the number of lines in each shape in the sequence of Complete Graphs. Which number sequence do you get? Can you explain why?
✅ Answer:
Number of lines (edges):
- K₂ → 1
- K₃ → 3
- K₄ → 6
- K₅ → 10
- K₆ → 15
👉 Sequence: 1, 3, 6, 10, 15, …
✔ Explanation:
Each new point joins with all previous points, so lines increase in a triangular number pattern.
Q3. How many little squares are there in each shape of the sequence of Stacked Squares? Which number sequence does this give? Can you explain why?
✅ Answer:
Number of small squares:
👉 1, 4, 9, 16, 25, …
✔ Sequence: Square numbers (n²)
✔ Explanation:
Each figure forms an n × n grid, so total squares = n².
Q4. How many little triangles are there in each shape of the sequence of Stacked Triangles?
Which number sequence does this give? Can you explain why?
✅ Answer:
Number of triangles:
👉 1, 3, 6, 10, 15, …
✔ Sequence: Triangular numbers
✔ Explanation:
Each row adds one more triangle than the previous row.
Q5. To get from one shape to the next shape in the Koch Snowflake sequence, one replaces each line segment by a “spiked bump”.
How many total line segments are there in each shape?
What is the corresponding number sequence?
✅ Answer:
Number of line segments:
👉 3, 12, 48, 192, …
✔ Sequence: 3 × (4ⁿ)
✔ Explanation:
Each line segment is replaced by 4 smaller segments, so total segments multiply by 4 each step.
🔥 Final Summary
Polygons → counting numbers
Graphs & triangles → triangular numbers
Squares → square numbers
Snowflake → powers of 4
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