Lecture Notes: Visual Representation of Factoring Patterns

Topic: Exercise 5.1.2 - Related Area Investigation

Objective

• Use areas of squares and rectangles to visually represent a factoring pattern.

Key Concepts

• Difference of Squares: A mathematical principle shown through geometric manipulation.

Visual Representation

1. Initial 5x5 Grid

   • Area: 25 (calculated as 5 x 5 or by counting squares).

2. Reconstruction of the Grid

   • Move four highlighted squares to form a new rectangle.
   • New Rectangle: 4 rows and 6 columns
     • Area: 24 (calculated as 4 x 6).

3. Comparison of Areas

   • Original Blue Square: 25
   • Yellow Rectangle: 24
   • Difference: 1 square left out after transformation.
   • Alternative area calculation by subtracting the leftover square (1 x 1) from the original blue square.

Mathematical Explanation

• Blue Rectangle:

  • Originally 5 rows, reduced to 4 by removing one square.
  • Added a column, resulting in dimensions 4 (5 - 1) by 6 (5 + 1).

• Factoring Pattern:

  • The visual demonstration confirms: ( (5 - 1) \times (5 + 1) = 5^2 - 1^2 )
  • Reduced to 24.

Generalization

• Variables:

  • Let ( a ) represent the length of the original square.
  • ( b ) represent the length of leftover square.

• Expression for New Rectangle:

  • Length: ( (a + b) )
  • Width: ( (a - b) )
  • Area of the rectangle: ( (a + b) \times (a - b) = a^2 - b^2 )

Conclusion

• The exercise visually and mathematically demonstrates the difference of squares concept through geometric manipulation of squares and rectangles.