In previous units, students learned that the dilation of a line segment is longer or shorter in the ratio given by the scale factor. In the last lesson, students dilated a rectangle by a factor of \(k\) and observed that the area was not multiplied by \(k\). In this lesson, students analyze the result of scaling on area. This concept will be essential to creating a volume formula for pyramids later in the unit. Additionally, students will encounter the graph representing the equation \(y=\sqrt{x}\) in a geometric context when, in upcoming lessons, they analyze the relationship between scaled areas and factors of dilation. This will build on work students did in grade 8 evaluating square roots of small perfect squares and determining that \(\sqrt2\) is irrational.
Knowing the relationship between scale factor and area is a basis for understanding how measurement and dimension are related in geometry. As students observe a pattern in the area of a figure after several different dilations, they are looking for regularity in repeated reasoning (MP8).
Technology isn’t required for this lesson, but there are opportunities for students to choose to use appropriate technology to solve problems. We recommend making technology available.
Lesson overview
 4.1 Warmup: Squares and Roots (5 minutes)

4.2 Activity: Scaling Up a Rectangle (20 minutes)
 Includes "Are you Ready for More?" extension problem
 4.3 Activity: What About Other Shapes? (10 minutes)
 Lesson Synthesis
 4.4 Cooldown: Finding an Area (5 minutes)
Learning goals:
 Comprehend that when figures are dilated by a scale factor of \(k\), their areas are multiplied by \(k^2\).
Learning goals (student facing):
 Let’s see how the area of shapes changes when we dilate them.
Learning targets (student facing):
 I know that when figures are dilated by a scale factor of \(k\), their areas are multiplied by \(k^2\).
Standards:
 This lesson builds on the standards: CCSS.8.EE.A.2MS.8.EE.2CCSS.HSGSRT.A.1.bMS.GSRT.1bMO.8.EEI.A.2aMO.8.EEI.A.2bMO.G.SRT.A.1
 This lesson builds towards the standards: CCSS.HSFIF.C.7.bMS.FIF.7bCCSS.HSGGMDCCSS.HSGGMD.A.1MS.GGMD.1MO.A1.IF.C.7MO.G.GMD.A.1
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