You will have noticed that 36 is both a triangular number and a square number! That is, 36 dots can be arranged perfectly both in a triangle and in a square. Make pictures in your notebook illustrating this! This shows that the same number can be represented differently, and play different roles, depending on the context. Try representing some other numbers pictorially in different ways!
36 as a Triangular Number: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36 This shows 36 is the 8th triangular number (8 dots on each side).
36 as a Square Number: = 36 This shows 36 is the 6th square number (6² = 36).
Other Numbers with Multiple Representations:
1:
- 1st triangular number
- 1st square number
1225:
- 49th triangular number
- 35th square number
4:
- square
- 2 + 2 (sum of two even numbers)
10:
- Triangular number: 1 + 2 + 3 + 4
- 2 rows of 5 dots
Conclusion: Numbers have multiple representations depending on the mathematical context.
Explanation
The chapter highlights that 36 is both a triangular number (8th triangular number: 1+2+3+4+5+6+7+8=36) and a square number (6²=36). This illustrates that numbers can play different roles in different contexts. Students are encouraged to explore other numbers that have multiple representations.
Solution Steps
Represent 36 as a triangular number: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36 (8th triangular number)
Represent 36 as a square number: = 36 (6th square number)
Identify other numbers with multiple representations: 1, 1225, 4, 10
Conclude that numbers have multiple representations depending on context