🔑 Key Concepts
- A fractional scale factor between \(0\) and \(1\) makes a figure smaller.
- The image moves closer to the center of dilation when \(0<k<1\).
- The center of dilation stays fixed and does not move.
- When the center is not the origin, use the center point in the dilation formula.
- The dilation rule is \( (x,y)\rightarrow \left(a+k(x-a),\,b+k(y-b)\right) \), where \((a,b)\) is the center of dilation.
✏️ Worked Examples
🧠 Math Vocabulary
- Dilation: a transformation that changes the size of a figure while keeping the same shape.
- Scale Factor: the value \(k\) that tells how much the figure is enlarged or reduced.
- Fractional Scale Factor: a scale factor between \(0\) and \(1\), such as \(\displaystyle \frac{1}{2}\) or \(\displaystyle \frac{2}{3}\), that makes a figure smaller.
- Center of Dilation: the fixed point from which all distances are measured during a dilation.
- Image: the new figure after the dilation, labeled with prime symbols such as \(A'\), \(B'\), and \(C'\).
- Preimage: the original figure before the dilation.
- Non-Rigid Transformation: a transformation that changes the size of a figure. Dilations are non-rigid when \(k\neq1\).
- Dilation Formula: the coordinate rule \( (x,y)\rightarrow \left(a+k(x-a),\,b+k(y-b)\right) \), where \((a,b)\) is the center of dilation.
💡 Main Idea
A fractional scale factor makes the image smaller and pulls each point closer to the center of dilation. When the center of dilation is not the origin, you can use the dilation formula to calculate each image point without relying only on the graph.
📚 What You Should Already Know
Students should know how to plot ordered pairs, subtract integers, multiply fractions by whole numbers, and understand that a scale factor less than \(1\) creates a reduction.
🚀 What Comes Next
After practicing fractional scale factors, students can compare enlargements and reductions, identify scale factors from graphs, and connect dilations to similar figures and proportional side lengths.
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