Understanding Dilation in Geometry

Objective:

Students will be able to understand and apply the concept of dilation, including identifying the center, determining the scale factor, and recognizing similarities between dilated figures and the original figure.

Assessment:

Students will be assessed through a worksheet that includes various figures for dilation. They will need to identify the center of dilation, calculate the scale factor, and determine the dilated image's dimensions based on a given scale factor.

Key Points:

• Dilation: An enlargement or reduction of a figure that preserves shape but not size. All dilations are similar to the original figure.

• Center and Scale Factor: Dilations have a center and a scale factor. The center is the point of reference for the dilation, and the scale factor (k) tells us how much the figure stretches or shrinks.

  • Formula for Dilation:

  • If the center of dilation is at point O and the scale factor is k:

  • The coordinates of the dilated point A' are given by:

    • A' = k * OA

  • The distances between points are all multiplied by the scale factor k.

Opening:

• Introduction of the topic with real-world examples of dilation (e.g., resizing images on a computer)

• Engage students with a teaser question: "How can we change the size of a shape without changing its shape?"

Introduction to New Material:

• Explain definition of dilation and give examples

• Introduce the concept of a center of dilation and its significance

• Provide examples of calculating scale factor in dilations

• Common misconception to anticipate: Confusing scale factor with the actual measurements of the dilated figure

Guided Practice:

• Demonstrate how to find the center of dilation and calculate the scale factor with guided examples

• Scaffold questioning from basic to complex, allowing students to practice identifying the key components of dilation

• Monitor student performance by circulating the classroom and providing feedback as needed

Independent Practice:

• Assign a worksheet with figures for dilation where students need to identify the center, calculate the scale factor, and determine the dimensions of the dilated figure

• Encourage students to work independently and ask for assistance if needed

Closing:

• Review key concepts of dilation by asking students to share one thing they learned about dilations today

Extension Activity:

• Challenge early finishers to create their own set of dilated figures using a given scale factor and center of dilation. They can then swap with a partner to solve each other's dilation problems.

Homework:

• For homework, students should practice dilating different figures at home and come prepared to discuss their findings in the next class.

Standards Addressed:

• CCSS.MATH.CONTENT.HSG.CO.A.3: Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

• CCSS.MATH.CONTENT.HSG.CO.A.4: Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, and parallel lines.