How to Dilate a Figure on a Coordinate Plane Using the Rule (kx, ky)

📚 Understanding Dilation on the Coordinate Plane

Dilation is a transformation that changes the size of a figure. It either enlarges (stretches) or reduces (shrinks) the figure. The key is the scale factor, often represented by 'k'. When you dilate a figure on the coordinate plane using the rule $(kx, ky)$, you're multiplying the x and y coordinates of each point on the figure by the scale factor 'k'.

🎯 Objectives

• 🔍 Identify the scale factor in a dilation.
• 💡 Apply the rule $(kx, ky)$ to dilate a figure.
• 📝 Graph the original figure and its dilated image.
• 📐 Understand how the scale factor affects the size of the image.

🧮 Materials

• 📈 Coordinate plane graph paper
• ✏️ Pencils
• 📏 Rulers
• 🖍️ Colored pencils (optional, for distinguishing the original and dilated figures)

☀️ Warm-up (5 mins)

Before diving into dilation, let's quickly review plotting points on the coordinate plane.

• 📍 Plot the following points: A(2, 3), B(-1, 4), C(-2, -2), and D(3, -1).
• ➕ Briefly discuss what happens to the coordinates when reflecting over the x-axis or y-axis (a concept related to transformations).

👨‍🏫 Main Instruction

1. Defining Dilation and Scale Factor
   • 🔑 Explain that dilation changes the size but not the shape of a figure.
   • 🔢 The scale factor, $k$, determines whether the figure gets larger or smaller.
     • If $k > 1$, the figure enlarges.
     • If $0 < k < 1$, the figure reduces.
     • If $k = 1$, there is no change in size.
2. Applying the Rule (kx, ky)
   • ✍️ Show how to apply the rule $(kx, ky)$ to each coordinate point.
     • Example: If point A(2, 3) is dilated by a scale factor of 2, the new point A' becomes (2*2, 2*3) = A'(4, 6).
   • Visualize this with several examples on the coordinate plane.
3. Graphing the Dilated Image
   • 📊 Graph both the original figure and the dilated image on the same coordinate plane. Use different colors to distinguish them.
   • 📏 Measure the sides of both figures. Note how the lengths are scaled by the factor $k$.

📝 Practice Quiz

Dilation is a key concept in geometry. See if you can answer the questions below.

1. A triangle has vertices A(1, 1), B(2, 3), and C(4, 1). Dilate the triangle by a scale factor of 3. What are the new coordinates?
2. A square has vertices D(-2, 2), E(2, 2), F(2, -2), and G(-2, -2). Dilate the square by a scale factor of 0.5. What are the new coordinates?
3. A line segment has endpoints H(0, 4) and I(4, 0). Dilate the line segment by a scale factor of 2.5. What are the new coordinates?
4. A rectangle has vertices J(-1, -1), K(3, -1), L(3, -3), and M(-1, -3). Dilate the rectangle by a scale factor of 1.5. What are the new coordinates?
5. A pentagon has vertices N(0, 0), O(1, 2), P(3, 2), Q(4, 0), and R(2, -2). Dilate the pentagon by a scale factor of 0.75. What are the new coordinates?
6. A rhombus has vertices S(-2, 0), T(0, 2), U(2, 0), and V(0, -2). Dilate the rhombus by a scale factor of 4. What are the new coordinates?
7. A trapezoid has vertices W(-3, 1), X(-1, 3), Y(3, 3), and Z(5, 1). Dilate the trapezoid by a scale factor of 0.25. What are the new coordinates?

✅ Assessment

• ❓ Ask students to dilate various figures using different scale factors.
• ✍️ Have them explain in their own words how the scale factor affects the image.
• 🤝 Review common mistakes and reinforce the concept.