Question

reverse it!
figure b is a scaled copy of figure a. for each set of figures, calculate the scale factor:

• from figure a to figure b.
• from figure b to figure a.

1. image of figure 1
2. image of figure 2
3. image of figure 3

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:

For the scale factor calculations (taking problem 1 as an example, assuming grid units):

Step 1: Determine side lengths of A and B

Let's assume a side of a square in the grid is 1 unit. For figure A (the square part), suppose its side length is 3 units. For figure B (the smaller square), its side length is 1 unit.

Step 2: Scale factor from A to B

Scale factor \( k_{A
ightarrow B} = \frac{\text{Length of B}}{\text{Length of A}} = \frac{1}{3} \)

Step 3: Scale factor from B to A

Scale factor \( k_{B
ightarrow A} = \frac{\text{Length of A}}{\text{Length of B}} = \frac{3}{1} = 3 \)

Relationship between the scale factors:

If the scale factor from A to B is \( \frac{a}{b} \) (where \( a \) is the length of B and \( b \) is the length of A), then the scale factor from B to A is \( \frac{b}{a} \). They are reciprocals of each other. For example, if A to B is \( \frac{1}{3} \), B to A is \( 3 \) (since \( \frac{1}{3} \times 3 = 1 \)). To check, pick a side length of A (say 3 units) and B (1 unit). Scaling A by \( \frac{1}{3} \) gives B ( \( 3 \times \frac{1}{3} = 1 \) ). Scaling B by 3 gives A ( \( 1 \times 3 = 3 \) ). So one is the reciprocal of the other.

Answer:

The scale factor from \( A \) to \( B \) and from \( B \) to \( A \) are reciprocals (their product is 1). For example, if \( A \) to \( B \) is \( \frac{1}{3} \), \( B \) to \( A \) is \( 3 \) (since \( \frac{1}{3} \times 3 = 1 \)).