Question 7 of 19intermediate💡 UnderstandShort Answer3 marks

Can you find a similar pictorial explanation for why adding counting numbers up and down, i.e., 1, 1 + 2 + 1, 1 + 2 + 3 + 2 + 1, ..., gives square numbers?

Correct Answer

Yes, a pictorial explanation can be made by arranging dots in a square grid.

Example 1: 1+2+3+2+1=91 + 2 + 3 + 2 + 1 = 9 This forms a 3×33 \times 3 square where:

  • The middle row has 3 dots
  • The rows above and below have 2 dots each
  • The top and bottom rows have 1 dot each

Example 2: 1+2+3+4+3+2+1=161 + 2 + 3 + 4 + 3 + 2 + 1 = 16 This forms a 4×44 \times 4 square.

Conclusion: The pattern works because when we count up from 1 to nn and back down to 1, we get n2n^2. Visually, this creates a square shape with the longest row (nn dots) in the middle, and symmetrically decreasing rows above and below. The total number of dots equals n×n=n2n \times n = n^2, which is a perfect square.

Exercise: Figure it Out (Section 4) | Q: 1 | (Page Page 8)
For More Understanding

Explanation

The chapter shows in section 1.4 that adding counting numbers up and down gives square numbers: 1=1, 1+2+1=4, 1+2+3+2+1=9, 1+2+3+4+3+2+1=16, and so on. The pictorial explanation involves arranging dots in a square pattern where the middle row has the maximum count and rows decrease symmetrically above and below, forming a perfect square shape.

Solution Steps

  1. Arrange dots in a square grid to represent the sum.

  2. Show that 1+2+3+2+1=91 + 2 + 3 + 2 + 1 = 9 forms a 3×33 \times 3 square.

  3. Show that 1+2+3+4+3+2+1=161 + 2 + 3 + 4 + 3 + 2 + 1 = 16 forms a 4×44 \times 4 square.

  4. Observe that counting up from 1 to nn and back down to 1 results in n2n^2.

  5. Conclude that the total dots equal n×n=n2n \times n = n^2, a perfect square.

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