This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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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 | 11 | 13 | 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 | 15 | 18 | 21 | 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 | 4 | 5 | 7 | 9 | 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 = (n(n+1))/(2). | Diagram number | 1 | 2 | 3 | 4 | 5 | 7 | 10 | 11 | | :------------- | :- | :- | :- | :- | :- | :- | :-- | :-- | | Number of dots | 1 | 3 | 6 | 10 | 15 | 28 | 55 | 66 | 2.3 Identify the rule. The rule is T_n = (n(n+1))/(2), where T_n is the number of dots and n is the diagram number. Last free one today — make