Length of one Total # of # of
side of pattern triangular white
(in cm) Shapes used triang-
to make pattern ular
shapes
used
2 4 3
3 9 6
4 16 10
? ? ?
6 ? ?
? ? ?
? ? ?
20 ? ?
Number of gray triangular shapes used 1, 3,6,? ? ? ? ?
observation #1:
what happens when you square the numbers in the first column?
observation #2
in the third column:
second # - first # = ??
third # - second # = ??
observation #3
visualize billiard balls racked up
sum of balls after 1 row = 1
sum of balls after 2 rows = 3
sum of balls after 3 rows = 6
....
Ok Reiny i get that i have to square the 1st row to come up with the 2nd row, but i still don't get how i come up with the 3rd row?
2 4 3
3 9 6
4 16 10
? ? ?
6 ? ?
? ? ?
? ? ?
20 ? ?
Third column
6-3 = 3
10-6 = 4
then
15-5 = 5
21-15 = 6
28-21 = 7
so
3
6
10
15
21
28
etc
To find the missing values in the table, we need to identify the patterns or relationships between the given values. Let's analyze each row of the table individually.
First Row:
Length of one side of the pattern (in cm): 2
Total number of triangular shapes used to make the pattern: 4
Number of white triangular shapes used: 3
Second Row:
Length of one side of the pattern (in cm): 3
Total number of triangular shapes used to make the pattern: 9
Number of white triangular shapes used: 6
Third Row:
Length of one side of the pattern (in cm): 4
Total number of triangular shapes used to make the pattern: 16
Number of white triangular shapes used: 10
From these three rows, we can observe the following patterns:
1. The total number of triangular shapes used is equal to the square of the length of one side of the pattern. (total = side length squared)
2. The number of white triangular shapes used is less than the total number of triangular shapes used. It seems to increase by 2 with each increment in the length of one side of the pattern.
Based on these patterns, we can continue to fill in the missing values:
For the fourth row:
Length of one side of the pattern (in cm): 5
Total number of triangular shapes used to make the pattern: 25 (5^2)
Number of white triangular shapes used: 14 (previous value + 2)
For the fifth row:
Length of one side of the pattern (in cm): 6
Total number of triangular shapes used to make the pattern: 36 (6^2)
Number of white triangular shapes used: 18 (previous value + 2)
For the remaining rows, we can continue using the same patterns:
For the sixth row:
Length of one side of the pattern (in cm): 7
Total number of triangular shapes used to make the pattern: 49 (7^2)
Number of white triangular shapes used: 22 (previous value + 2)
For the seventh row:
Length of one side of the pattern (in cm): 8
Total number of triangular shapes used to make the pattern: 64 (8^2)
Number of white triangular shapes used: 26 (previous value + 2)
For the eighth row:
Length of one side of the pattern (in cm): 20
Total number of triangular shapes used to make the pattern: 400 (20^2)
Number of white triangular shapes used: 38 (previous value + 2)
Therefore, the missing values in the table would be as follows:
? 25 14
6 36 18
7 49 22
8 64 26
20 400 38
The sequence for the number of gray triangular shapes used would be:
1, 3, 6, 10, 14, 18, 22, 26