Question

lesson 4: dilations on a square grid
learning goal: lets dilate figures on a square grid!
activity 1: estimating a scale factor
point c is the dilation of point b with center of dilation a and scale factor s.
estimate s (the scale factor). be prepared to explain your reasoning.

1. is the scale factor greater than 1? how do you know?
2. is the scale factor greater than 2? how do you know?
3. is the scale factor greater than 3? how do you know?
4. is the scale factor greater or less than 2.5? how do you know?

activity 2: dilations on a grid

Explanation:

1. Is the scale factor greater than 1?

In dilation, if the image (point C) is farther from the center of dilation (point A) than the original point (point B), the scale factor \(s=\frac{AC}{AB}\). Since \(AC>AB\), \(s > 1\).

2. Is the scale factor greater than 2?

Count the number of units (assuming a grid - like distance measurement conceptually). If \(AB\) is considered as 1 unit segment (in terms of distance from \(A\) to \(B\)) and \(AC\) is more than 2 times the length of \(AB\) (by visual estimation of the relative distances from \(A\)), \(s>2\).

3. Is the scale factor greater than 3?

If \(AB\) is taken as a reference length, and by visual comparison of \(AC\) and \(AB\) (assuming we can use a non - formal unit of distance based on the grid - like arrangement of points), \(AC<3AB\), so \(s < 3\).

4. Is the scale factor greater or less than 2.5?

By visual estimation of the ratio \(\frac{AC}{AB}\) (using the idea of relative distances from \(A\) to \(B\) and \(A\) to \(C\) as if on a grid), if we assume \(AB\) as a basic length unit, \(AC<2.5AB\), so \(s<2.5\).

Answer:

1. Yes. Because \(AC > AB\) and \(s=\frac{AC}{AB}\).
2. Yes. By visual comparison of \(AC\) and \(AB\) (assuming a relative distance measure).
3. No. By visual comparison of \(AC\) and \(AB\) (assuming a relative distance measure).
4. Less than 2.5. By visual comparison of \(AC\) and \(AB\) (assuming a relative distance measure).