Advanced Multiples Pattern Exploration

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Mathematical patterns and geometric shapes

📐 Part 1: Geometric Pattern Analysis

1. Study this triangular number pattern where each figure shows dots arranged in triangular formations:

Figure 1: 1 dot | Figure 2: 3 dots | Figure 3: 6 dots | Figure 4: 10 dots

Complete this table showing the relationship between figure number and total dots:

2. A hexagonal pattern grows where Figure 1 has 7 dots (1 centre + 6 around), Figure 2 has 19 dots, Figure 3 has 37 dots. Create a detailed table showing the pattern for the first 6 figures, including the number of dots added each time.

3. In a square spiral pattern, the perimeter lengths follow this sequence: 8, 16, 24, 32, 40... Analyse this pattern and create a table showing figure number (n) versus perimeter length for figures 1-8.

🔍 Part 2: Multiplication Pattern Descriptions

4. For the triangular number pattern from Question 1, explain why the pattern works using multiplication concepts. Avoid simply saying "add 2, then add 3, then add 4..." Instead, describe the multiplicative relationship within each figure.

5. The sequence 5, 20, 45, 80, 125... represents the areas of expanding rectangles. Describe this pattern in terms of multiplication, explaining the relationship between the position number and the formula that generates each term.

6. A staircase pattern has steps where the number of blocks follows: 4, 12, 24, 40, 60... Explain why this pattern works by describing the multiplicative structure within each step of the staircase.

🧮 Part 3: Algebraic Rule Development

7. Study this input-output table:

Input (n): 1, 2, 3, 4, 5
Output: 6, 14, 24, 36, 50

a) Determine the algebraic rule that describes the relationship between input (n) and output:

b) Justify your rule by showing how it works for at least three examples from the table:

8. Create a generalisation: If a pattern follows the rule "output = n² + 2n + 3", explain what this tells us about how the pattern grows and predict the 10th and 15th terms. Show your working.

9. Challenge: The pattern 2, 8, 18, 32, 50... can be described by the rule 2n². Explain why this rule works by connecting it to a real-world geometric situation (such as areas or arrangements of objects). Then use your rule to find the 20th term.