Steps to Dilate a Figure on a Coordinate Plane (Origin-Centered)

📚 Understanding Dilation: A Teacher's Guide

This lesson plan provides a structured approach to teaching students how to dilate figures on a coordinate plane, specifically focusing on origin-centered dilations.

🎯 Learning Objectives

• 🧭 Students will be able to identify the center of dilation and the scale factor.
• 📐 Students will be able to apply the scale factor to the coordinates of a figure.
• 📈 Students will be able to accurately plot the dilated image on a coordinate plane.
• ✍️ Students will be able to describe the relationship between the original figure and its dilated image.

🧰 Materials

• 🌐 Graph paper
• ✏️ Pencils
• 📏 Rulers
• 🖍️ Colored pencils (optional, for distinguishing between original and dilated figures)
• 💻 Access to graphing software (optional)
• 📄 Worksheet with practice problems

Warm-up Activity (5 minutes)

• 🧠 Coordinate Plane Review: Briefly review the coordinate plane, focusing on identifying coordinates of points.
• ❓ What is Scaling?: Ask students what they understand about the concept of scaling or resizing something (e.g., making a picture bigger on a phone).

Main Instruction

1. ✏️ Introduction to Dilation

   • 🔍 Define dilation as a transformation that changes the size of a figure.
   • 📍 Explain that the center of dilation is a fixed point from which the figure is enlarged or reduced. In this lesson, we focus on the origin (0,0) as the center.
   • 🔢 Define the scale factor ($k$) as the ratio of the lengths of the corresponding sides of the image and the pre-image.

2. 📐 Dilation Rule

   • 📝 Explain that when dilating a figure with respect to the origin, each coordinate $(x, y)$ of the original figure is multiplied by the scale factor $k$ to obtain the corresponding coordinate $(kx, ky)$ of the dilated image.
   • 📌 Therefore, the rule for dilation centered at the origin is: $(x, y) \rightarrow (kx, ky)$.

3. 📈 Step-by-Step Example

   • ✍️ Provide a step-by-step example of dilating a triangle with vertices A(1, 1), B(2, 1), and C(1, 2) by a scale factor of 2.
   • 📍 Step 1: Identify the coordinates of the pre-image (A(1, 1), B(2, 1), C(1, 2)).
   • 🔢 Step 2: Multiply each coordinate by the scale factor 2: A'(2, 2), B'(4, 2), C'(2, 4).
   • 🌐 Step 3: Plot the dilated image A'B'C' on the coordinate plane.

4. 💭 Discussion

   • 💡 Discuss what happens when the scale factor is greater than 1 (enlargement) and when it is between 0 and 1 (reduction).
   • 🤔 Ask students to predict the effect of different scale factors on various figures.

✅ Assessment

Use these questions to gauge student understanding.

• ❓Dilate the point (3, -2) by a scale factor of 4. What are the new coordinates?
• ❓A triangle has vertices (0, 0), (2, 0) and (0, 2). What are the coordinates of the vertices after a dilation by a scale factor of 1.5?
• ❓A square has vertices (1, 1), (1, 2), (2, 2) and (2, 1). What are the coordinates of the vertices after a dilation by a scale factor of 0.5?