Michael is making a geometric pattern using pins to form triangles.
The first 3 shapes in the pattern are shown below.
- Draw the next shape in the pattern. (1 mark)
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- Complete the table of values using the pattern. (2 marks)
Number of triangles \((t)\) \(\ \ 1\ \ \) \(\ \ 2\ \ \) \(\ \ 3\ \ \) \(\ \ 4\ \ \) Number of pins \((p)\) \(\ \ 3\ \ \) - Write the rule connecting the number of pins \((p)\) to the number of triangles \((t)\). (2 marks)
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- How many pins will be needed to make \(25\) triangles? (2 marks)
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Show Answers Only
a. \(\text{Shape number }4\)
b. \(\text{Table of values}\)
| \(\text{Number of triangles }(t)\) | \(\ \ 1\ \ \) | \(\ \ 2\ \ \) | \(\ \ 3\ \ \) | \(\ \ 4\ \ \) |
| \(\text{Number of pins }(p)\) | \(\ \ 3\ \ \) | \(\ \ 5\ \ \) | \(\ \ 7\ \ \) | \(\ \ 9\ \ \) |
c. \(p=2\times t+1\)
d. \(51\)
Show Worked Solution
a. \(\text{Shape number }4\)
b. \(\text{Table of values}\)
| \(\text{Number of triangles }(t)\) | \(\ \ 1\ \ \) | \(\ \ 2\ \ \) | \(\ \ 3\ \ \) | \(\ \ 4\ \ \) |
| \(\text{Number of pins }(p)\) | \(\ \ 3\ \ \) | \(\ \ 5\ \ \) | \(\ \ 7\ \ \) | \(\ \ 9\ \ \) |
c. \(\text{Rule: The number of pins}=2\times \text{(the number of triangles)}+1\)
\(\therefore\ \text{Rule: }\ p=2\times t+1\)
d. \(\text{Find the value of }p\ \text{when }t=25\)
\(p=2\times t+1=2\times 25+1=51\)