Question

1. grid with figure a and b 2. grid with figure a and b 3. grid with figure a and b 4. is there a mathematical relationship between a scale factor from figure a to figure b and its reverse scale factor from figure b to figure a? describe how you figured out the relationship so that a 5th grader can understand how you did it.

Explanation:

Step1: Recall Scale Factor Basics

A scale factor from \( A \) to \( B \) (\( k_{A\to B} \)) means we multiply \( A \)'s dimensions by \( k \) to get \( B \)'s. For reverse ( \( B \) to \( A \), \( k_{B\to A} \) ), we do the opposite.

Step2: Use an Example (Like Grid Figures)

Take Figure 3: Let’s say \( A \) is a rectangle with length 9 (grid units) and \( B \) has length 3.

• Scale factor \( A \to B \): \( k_{A\to B}=\frac{\text{Length of } B}{\text{Length of } A}=\frac{3}{9}=\frac{1}{3} \).
• Scale factor \( B \to A \): \( k_{B\to A}=\frac{\text{Length of } A}{\text{Length of } B}=\frac{9}{3}=3 \).

Step3: Observe the Relationship

Notice \( \frac{1}{3} \) and \( 3 \) are reciprocals (since \( \frac{1}{3}\times3 = 1 \)). Test with another figure (e.g., Figure 1: If \( A \)’s square is 3x3, \( B \)’s is 1x1. \( k_{A\to B}=\frac{1}{3} \), \( k_{B\to A}=3 \) – same reciprocal pattern).

Answer:

The scale factor from \( A \) to \( B \) and from \( B \) to \( A \) are reciprocals (multiplicative inverses). To find this, pick a side length from \( A \) and \( B \) (using grid squares), divide \( B \)’s length by \( A \)’s for \( A\to B \) scale factor, and \( A \)’s by \( B \)’s for \( B\to A \). These two numbers multiply to 1 (like \( \frac{1}{3} \) and \( 3 \) or \( \frac{1}{2} \) and \( 2 \)), so they’re reciprocals.