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Question 13 Marks

Try and redraw each sequence in Table 3 in your notebook. Can you draw the next shape in each sequence? Why or why not? After each sequence, describe in your own words what is the rule or pattern for forming the shapes in the sequence.

Answer

→ Regular Polygons - This sequence shows polygons that have equal sides and starts with a triangle having 3 sides and in each step in the sequence adds one more side to the previous shape.

Rule $n-$ number of sides, $n \in N$
$n+1=$ number of sides in the next shape
→ Complete Graphs - This sequence shows that each point is connected to every other point and starts with two points connected by a line, three points form a triangle, four points form a square and so on. In each step, the number of lines increases.

→ Stacked Squares - In this sequence, squares are stacked on each other to get the large square. In each steps of the sequence (starting with one square), the number of squares increases in a pattern.

→ Stacked Triangle In this sequence, triangles are stacke to form a large triangle. In each step of the sequence starting with one triangle add more number of triangles to form a large triangle.

→ Koch Snowflakes In this sequence, starts with the triangle every side of the snowflake is replaced with 4 new sides. Each of these sides is a third of the length of the side, it is replacing or in this sequence, starts with triangle and in next step the line segment is replaced by a 'speed bump'

. In each step, the chances become tinier and tinier with very very small line.

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Question 23 Marks

Can you recognize the pattern in each of the sequences in Table ?

Answer

(a) Regular Polygons: Triangle, quadrilateral, pentagon, hexagon, (the number of sides increases by 1 at a time). Hence it is a continuous number sequence starting from 3.

Here number of lines is as follows:
K₂ = 1
K₃ = 3
K₄ = 6
K₅ = 10
K₆ = 15
The series formed is 1, 3, 6, 10, 15,…….
Hence it is a triangular number sequence.

(c) Stacked Squares:
The number of small squares in each:

The series formed is 1, 4, 9, 16, 25,…….
Hence it is a presentation of squares numbers sequence.

(d) Stacked Triangles
The number of small triangles in each:

Hence it is also a square number sequence shown by triangles.

(e) Koch Snowflake
Number of sides in each becomes 4 times.

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