Rationale
Due to both pedagogical time constraints and the emphasis (testing-wise) on the manipulation and understanding of the coordinates when it comes to dilation, the following three activities will not be used to transition dilations into similarity or provide students with proportional reasoning in regards to similarity. Those activities are valuable, but are not contained here within. Rather, the following activities aim to give students mastery over a range of interpretations of CCGPS standard MCC9-12.G.SRT1:
MCC9‐12.G.SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor:
a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
This standard gives rise to EOCT (now the EOG) questions that require students to map coordinates of a pre-image given any center of dilation & scale factor, identify a center of dilation given a pre-image and image, identify lines that are parallel due to dilations, and more. It is an implicitly loaded standard and thus aptly suited by a three-day unit, in which three major characteristics.
The first lesson, “How Big?” reintroduces students to dilations, transformations they first learned in Coordinate Algebra. The technology, Geogebra, gives students an easy way to view continuous strings of dilations based on any center, as well as the impact of the dilation on lines and segment lengths. This activity, more so than the second, quite perfectly aligns with the standard, and it is with technology that I believe the standard was meant to be taught with in the first place. Experimentation of dilations by hand would not be fruitful at all, as far as I see it, and Geogebra is a free software that allows for sliders and can compete with any other dynamic geometry software (DGS).
The second activity is not an application of dilations, but rather (as previously mentioned) addresses dilating in the coordinate plane. Students develop the formal rule for dilating coordinates from the origin, and make use of a clever repositioning of the coordinate plane and auxiliary triangles to develop a similar, complicated rule for dilating from a point not at the origin. Though this guided exploration might could be carried out with paddy paper, it would require much more work and the auxiliary triangles would not be in place to scaffold students’ understandings. In my opinion, the auxiliary triangles are what make this sketch, and they would be very difficult to incorporate with any more primitive form of technology.
The third day is a short exploration of the notion of proportionality. I know from personal experience that students obsess over integer scale factors but have serious trouble grasping the proportionality of side lengths in dilated figures. It is very surprising, but the relationship between multiplication and division is often lost on students. Thus the last activity, though deceptively simple, is hard to get “correct.” The prompts are accordingly short. The goal of this activity is to get students to pay attention to the ratios of the side lengths – not scale factors or centers of dilation. Though they can use their prior knowledge of these two to try and make the sketch “work” – actually, it is encouraged – they must do so intentionally all in order to reach the same result (equality of ratios).
The three activities address the three major facets of the transformation that is a “dilation” and build in a logical manner: from dilation as “changing the size of something” to analyzing exactly how and where something can be dilated to analyzing a subtle result of the fickle dilation.
MCC9‐12.G.SRT.1 Verify experimentally the properties of dilations given by a center and a scale factor:
a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
This standard gives rise to EOCT (now the EOG) questions that require students to map coordinates of a pre-image given any center of dilation & scale factor, identify a center of dilation given a pre-image and image, identify lines that are parallel due to dilations, and more. It is an implicitly loaded standard and thus aptly suited by a three-day unit, in which three major characteristics.
The first lesson, “How Big?” reintroduces students to dilations, transformations they first learned in Coordinate Algebra. The technology, Geogebra, gives students an easy way to view continuous strings of dilations based on any center, as well as the impact of the dilation on lines and segment lengths. This activity, more so than the second, quite perfectly aligns with the standard, and it is with technology that I believe the standard was meant to be taught with in the first place. Experimentation of dilations by hand would not be fruitful at all, as far as I see it, and Geogebra is a free software that allows for sliders and can compete with any other dynamic geometry software (DGS).
The second activity is not an application of dilations, but rather (as previously mentioned) addresses dilating in the coordinate plane. Students develop the formal rule for dilating coordinates from the origin, and make use of a clever repositioning of the coordinate plane and auxiliary triangles to develop a similar, complicated rule for dilating from a point not at the origin. Though this guided exploration might could be carried out with paddy paper, it would require much more work and the auxiliary triangles would not be in place to scaffold students’ understandings. In my opinion, the auxiliary triangles are what make this sketch, and they would be very difficult to incorporate with any more primitive form of technology.
The third day is a short exploration of the notion of proportionality. I know from personal experience that students obsess over integer scale factors but have serious trouble grasping the proportionality of side lengths in dilated figures. It is very surprising, but the relationship between multiplication and division is often lost on students. Thus the last activity, though deceptively simple, is hard to get “correct.” The prompts are accordingly short. The goal of this activity is to get students to pay attention to the ratios of the side lengths – not scale factors or centers of dilation. Though they can use their prior knowledge of these two to try and make the sketch “work” – actually, it is encouraged – they must do so intentionally all in order to reach the same result (equality of ratios).
The three activities address the three major facets of the transformation that is a “dilation” and build in a logical manner: from dilation as “changing the size of something” to analyzing exactly how and where something can be dilated to analyzing a subtle result of the fickle dilation.