Overview
This lesson focused on understanding and performing dilations (resizing) of geometric figures on a square grid, using both estimation and coordinate methods, and applying these concepts to various shapes and transformations.
Estimating Scale Factor and Understanding Dilations
- A dilation moves points away from a center by a scale factor (S), enlarging or reducing the figure.
- Estimate scale factor by comparing distances from the center before and after dilation.
- A scale factor greater than 1 enlarges the figure; less than 1 reduces it.
- Dilation with a scale factor of 2 doubles the distance from the center to each point.
Performing Dilations on a Square Grid
- To dilate a point, count units from the center, then multiply each distance by the scale factor.
- For scale factor 2: repeat the movement from the center to each vertex to double the distance.
- For scale factor ½: halve the distance from the center to each point.
- Dilation keeps points aligned on straight lines (rays) from the center.
Using Coordinates for Dilations
- On a coordinate grid, use (x, y) distances from the center to each point.
- Multiply both the x and y differences by the scale factor to find the image's position.
- The new coordinates are calculated as:
- (center_x + scale factor × (point_x – center_x), center_y + scale factor × (point_y – center_y))
Card Sort Activity & Examples
- Practice dilations with various centers and scale factors, applying the above coordinate rule.
- Shapes expand or contract but stay on the same rays from the center.
- Check correctness by ensuring all images align with original-to-center rays.
Dilating Circles and Special Shapes
- When the center is the origin and the shape is a circle, multiply the radius by the scale factor.
- The origin remains fixed under dilation.
Homework Concepts & Practice
- Dilated triangle images have the same angles but sides are proportional, not necessarily congruent.
- Practice dilating quadrilaterals and other shapes with different centers and scale factors.
- Use supplementary and vertical angle rules for related angle questions.
- Describe combined transformations such as rotation plus translation.
Key Terms & Definitions
- Dilation — resizing a figure from a center point by a scale factor.
- Scale Factor (S) — the multiplier determining how much the figure enlarges or shrinks.
- Center of Dilation — the fixed point from which all points move.
- Ray — a straight line from the center through a point, guiding its dilation.
- Congruent — identical in form and size; dilated figures with S ≠1 are similar, not congruent.
- Supplementary Angles — two angles that sum to 180°.
Action Items / Next Steps
- Complete tonight’s homework on dilations and coordinate calculations.
- Practice dilating shapes with different scale factors and centers on graph paper.
- Review key dilation formula and check homework answers for alignment with rays from the center.