Question

1. 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: Define Scale Factor

Let’s say figure A has a side length of \( a \), and when we scale it to figure B, the side length becomes \( b \). The scale factor from A to B, let's call it \( k_{A
ightarrow B} \), is how many times we multiply \( a \) to get \( b \). So \( k_{A
ightarrow B}=\frac{b}{a} \) (like if A has a side of 2 and B has a side of 4, then \( k_{A
ightarrow B}=\frac{4}{2} = 2 \)).

Step2: Find Reverse Scale Factor

Now, the reverse scale factor is from B to A. So we want to find how many times we multiply \( b \) to get \( a \). Let's call this \( k_{B
ightarrow A} \). So \( k_{B
ightarrow A}=\frac{a}{b} \) (using the same example, \( k_{B
ightarrow A}=\frac{2}{4}=\frac{1}{2} \)).

Step3: Relationship Between Them

Notice that \( k_{A
ightarrow B}=\frac{b}{a} \) and \( k_{B
ightarrow A}=\frac{a}{b} \). If we multiply these two scale factors together: \( k_{A
ightarrow B}\times k_{B
ightarrow A}=\frac{b}{a}\times\frac{a}{b} = 1 \). So they are reciprocals of each other (because the product of a number and its reciprocal is 1). For example, if the scale factor from A to B is 2, the scale factor from B to A is \( \frac{1}{2} \), and \( 2\times\frac{1}{2}=1 \).

Answer:

Yes, there is a mathematical relationship. The scale factor from figure A to figure B and the reverse scale factor from figure B to figure A are reciprocals (their product is 1). To figure this out: 1. Pick a simple example (like A has a side of 2 and B has a side of 4). 2. Find the scale factor from A to B (4 ÷ 2 = 2). 3. Find the scale factor from B to A (2 ÷ 4 = 1/2). 4. Notice that 2 and 1/2 multiply to 1, so they are reciprocals.