CLASS 6 PATTERNS IN MATHS
Figure it out Page 12
Question 1:
*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 in the sequence of Regular Polygons? Do you get the same number sequence? Can you explain why this happens?*
Solution:
- *Regular polygons* are shapes where all sides and angles are equal. Examples include the triangle, square, pentagon, hexagon, etc.
The sequence of the number of sides for regular polygons is:
- *Triangle*: 3 sides
- *Square*: 4 sides
- *Pentagon*: 5 sides
- *Hexagon*: 6 sides
- *Heptagon*: 7 sides
- And so on.
So, the number sequence is *3, 4, 5, 6, 7, 8,...*.
*Number of corners (vertices)*:
- The number of corners in a regular polygon is the same as the number of sides.
Thus, the number sequence for the corners is also *3, 4, 5, 6, 7, 8,...*.
*Explanation:*
- For any regular polygon, the number of sides is always equal to the number of corners (or vertices) because each side forms one angle, which corresponds to one corner. Therefore, both the sequences are the same.
Question 2:
*Count the number of lines in each shape in the sequence of Complete Graphs. Which number sequence do you get? Can you explain why?*
Solution:
- A *complete graph* is a graph where every pair of vertices is connected by a unique line (or edge).
The number of lines (edges) in a complete graph with n vertices can be calculated using the formula:
\(\frac{n(n-1)}{2}\)
Let's calculate for the first few graphs:
- *Complete graph with 2 vertices (K2)*: \(\frac{2(2-1)}{2} = 1\) line
- *Complete graph with 3 vertices (K3)*: \(\frac{3(3-1)}{2} = 3\) lines
- *Complete graph with 4 vertices (K4)*: \(\frac{4(4-1)}{2} = 6\) lines
- *Complete graph with 5 vertices (K5)*: \(\frac{5(5-1)}{2} = 10\) lines
So, the number sequence for the lines in complete graphs is *1, 3, 6, 10, 15,...*.
*Explanation:*
- The sequence formed is the sequence of triangular numbers because each time you add a new vertex, it connects to all the previous vertices, creating additional lines equal to the number of previous vertices.
Question 3:
*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?*
Solution:
- The sequence of stacked squares typically follows the pattern where each shape in the sequence is made by stacking squares.
- If the first shape has 1 square, the second shape will have 1 + 2 squares, the third shape will have 1 + 2 + 3 squares, and so on.
This results in the sequence:
- 1 (just 1 square)
- 1 + 2 = 3 squares
- 1 + 2 + 3 = 6 squares
- 1 + 2 + 3 + 4 = 10 squares
Thus, the number sequence is *1, 3, 6, 10, 15,...* (which is the sequence of triangular numbers).
*Explanation:*
- The sequence of numbers represents the sum of the first n natural numbers, where n is the number of stacked layers. The formula for the `n`th triangular number is given by \(\frac{n(n+1)}{2}\).
Question 4:
*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?*
Solution:
- Similar to the stacked squares, if we look at stacked triangles, the first shape has 1 triangle, the second shape has 3 triangles (1 + 2), the third shape has 6 triangles (1 + 2 + 3), and so forth.
This results in the sequence:
- 1 (1 triangle)
- 1 + 2 = 3 triangles
- 1 + 2 + 3 = 6 triangles
- 1 + 2 + 3 + 4 = 10 triangles
So, the number sequence is also *1, 3, 6, 10, 15,...* which again corresponds to the sequence of triangular numbers.
*Explanation:*
- Just like with the squares, the sequence for the triangles also follows the triangular number pattern, as each layer adds an additional number of triangles equal to the layer's number.
Question 5:
*To get from one shape to the next shape in the Koch Snowflake sequence, one replaces each line segment ‘—’ by a ‘speed bump’. As one does this more and more times, the changes become tinier and tinier with very, very small line segments. How many total line segments are there in each shape of the Koch Snowflake? What is the corresponding number sequence?*
Solution:
- The Koch Snowflake is formed by iteratively replacing each line segment with four new segments in the shape of a "bump."
- The sequence of the number of line segments starts with the initial triangle having 3 segments.
The sequence is as follows:
- After 1 iteration: Each of the 3 segments is replaced by 4 new segments, resulting in \(3 \times 4 = 12\) segments.
- After 2 iterations: Each of the 12 segments is replaced by 4 new segments, resulting in \(12 \times 4 = 48\) segments.
- After 3 iterations: Each of the 48 segments is replaced by 4 new segments, resulting in \(48 \times 4 = 192\) segments.
This gives us the sequence:
- 3 (initial triangle)
- 12 (after 1st iteration)
- 48 (after 2nd iteration)
- 192 (after 3rd iteration)
- And so forth...
The sequence can be generalized as *3, 12, 48, 192,...*
Explanation:
- The number of line segments increases by a factor of 4 at each iteration because each line segment is replaced by 4 new ones. Hence, the sequence is generated by multiplying by 4 at each step after the initial 3 segments.
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