Complete the tables and describe the rules for the given geometric patterns.

Here are the solutions to the geometric patterns worksheet.

Top Section: Geometric Patterns (Triangles)

1.1 Complete the table below.

Diagram number	1	2	3	4	5	6	7
Number of matches	3	5	7	9	$\boxed{11}$	$\boxed{13}$	$\boxed{15}$

1.2 How do you calculate the next number of matches in the pattern?
To calculate the next number of matches, you add 2 to the previous number of matches.

1.3 Identify the rule.
The rule is $2n + 1$, where $n$ represents the diagram number.

Middle Section: Geometric Patterns (Squares formed by matches)

2.1 Draw the next diagram of the pattern (Diagram 4).
Diagram 1 has 1 square (4 matches).
Diagram 2 has 2 squares in a row (7 matches).
Diagram 3 has 3 squares in a row (10 matches).
Diagram 4 will have 4 squares in a row. It will look like Diagram 3 with an additional square attached to the right, sharing one vertical matchstick.

2.2 Complete the following table.
The pattern in the table is adding 3 to the number of matches for each subsequent diagram.

Diagram number	1	2	3	4	5	6	9
Number of matches	6	9	12	$\boxed{15}$	$\boxed{18}$	$\boxed{21}$	$\boxed{30}$

2.3 Write the rule of the match pattern in your own words.
The rule is to start with 6 matches for the first diagram and add 3 matches for each subsequent diagram. Algebraically, the rule is $T_n = 3n + 3$, where $n$ is the diagram number.

Bottom Section: Geometric Patterns (Squares and Dots)

1.1 Draw the next diagram in this geometric pattern (Diagram 4).
Diagram 1 has 1 square.
Diagram 2 has 2 squares in a row.
Diagram 3 has 3 squares in a row.
Diagram 4 will have 4 squares in a row. It will look like Diagram 3 with an additional square attached to the right.

1.2 Count the number of squares used in each diagram and complete the table.

Diagram number	1	2	3	4	5	7	9	10
Number of squares	1	2	3	$\boxed{4}$	$\boxed{5}$	$\boxed{7}$	$\boxed{9}$	$\boxed{10}$

1.3 Identify the rule.
The rule is that the number of squares is equal to the diagram number. Algebraically, $S = n$, where $S$ is the number of squares and $n$ is the diagram number.

2.1 Draw the next two diagrams (Diagram 4 and Diagram 5).
This pattern represents triangular numbers.
Diagram 1: 1 dot
Diagram 2: 3 dots (arranged as a triangle)
Diagram 3: 6 dots (arranged as a triangle)
Diagram 4 will have 10 dots, arranged as a triangle with 4 dots on the bottom row.
Diagram 5 will have 15 dots, arranged as a triangle with 5 dots on the bottom row.

2.2 Complete the table below.
The pattern follows the triangular number sequence, $T_n = \frac{n(n+1)}{2}$.

Diagram number	1	2	3	4	5	7	10	11
Number of dots	1	3	6	$\boxed{10}$	$\boxed{15}$	$\boxed{28}$	$\boxed{55}$	$\boxed{66}$

2.3 Identify the rule.
The rule is $T_n = \frac{n(n+1)}{2}$, where $T_n$ is the number of dots and $n$ is the diagram number.

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