Math Problem Statement

The expression p^2 + 4p finds the number of tiles in the pattern for figure number p. Move options to the blanks to explain why the expression represents the number of tiles in figure p.

Solution

The image you have uploaded shows a pattern of tiles where the first three figures represent increasing sizes of a cross-like shape. Each figure seems to be constructed with a center square and additional tiles extending from the sides. The given mathematical expression for the number of tiles in the pattern is p2+4pp^2 + 4p, where pp is the figure number. Below the pattern, there is a fill-in-the-blank explanation regarding how the formula corresponds to the pattern.

Here’s the explanation for the expression p2+4pp^2 + 4p in context:

  • The term p2p^2 represents the core square in the middle, which has dimensions p×pp \times p. For figure 1, p=1p = 1, so the center is a 1×11 \times 1 square. For figure 2, p=2p = 2, so it forms a 2×22 \times 2 square, and so on.

  • The term 4p4p accounts for the extra tiles that extend from each side of the square. There are 4 sides, and pp additional tiles are added to each side in figure pp.

Filling in the blanks:

  1. The pattern is made up of a pp by pp square plus an additional pp tiles added to each side.
  2. The total number of tiles is given by the number of tiles in the square, p2p^2, plus the number of additional tiles, 4p4p.

Would you like further clarification or more details on any part of this explanation?

Related Questions:

  1. How can we express the number of tiles for a general figure number pp in terms of algebraic expressions?
  2. What is the growth rate of the number of tiles as pp increases?
  3. How can you visually confirm the number of tiles added to each side for any figure?
  4. Can the expression p2+4pp^2 + 4p be factored? If yes, how would that help in understanding the pattern?
  5. How would the pattern evolve if more tiles were added beyond 4 per side?

Tip:

In pattern recognition, always look for how the shape evolves with each step, focusing on both the core structure and the additional elements added at each stage.

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Math Problem Analysis

Mathematical Concepts

Algebra
Pattern Recognition
Quadratic Expressions

Formulas

p^2 + 4p
Quadratic expression

Theorems

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Suitable Grade Level

Grades 7-9