In previous lessons, students learned what a dilation is and practice dilating points and figures on a circular grid, on a square grid, on a coordinate grid, and with no grid. In this lesson, they work on a coordinate grid and use the coordinates to communicate precisely the information needed to perform a dilation. Students use the info gap structure. The student with the problem card needs to dilate a polygon on the coordinate grid. In order to do so, they need to request the coordinates of the polygon’s vertices and the center of dilation as well as the scale factor. After obtaining all of this information from the partner with the data card, the student performs the dilation. The focus here is on deciding what information is needed and communicating clearly to request the information and explain why it is needed.
One important use of coordinates in geometry is to facilitate precise and concise communication about the location of points (MP6). This allows students to indicate where the center of the dilation is and also to communicate the vertices of the polygon that is dilated.
- Describe (orally) a figure on a coordinate grid and its image under a dilation, using coordinates to refer to points.
- Describe (orally) several dilations of one figure with the same center but different scale factors.
- Identify what information is needed to dilate a polygon on a coordinate grid. Ask questions to elicit that information.
Let’s look at dilations in the coordinate plane.
Print and cut out copies of the blackline master for the Info Gap activity (one set of two problem cards and two data cards per pair of students).
- I can apply dilations to polygons on a rectangular grid if I know the coordinates of the vertices and of the center of dilation.
center of a dilation
The center of a dilation is a fixed point on a plane. It is the starting point from which we measure distances in a dilation.
In this diagram, point \(P\) is the center of the dilation.
A dilation is a transformation in which each point on a figure moves along a line and changes its distance from a fixed point. The fixed point is the center of the dilation. All of the original distances are multiplied by the same scale factor.
For example, triangle \(DEF\) is a dilation of triangle \(ABC\). The center of dilation is \(O\) and the scale factor is 3.
This means that every point of triangle \(DEF\) is 3 times as far from \(O\) as every corresponding point of triangle \(ABC\).
To create a scaled copy, we multiply all the lengths in the original figure by the same number. This number is called the scale factor.
In this example, the scale factor is 1.5, because \(4 \boldcdot (1.5) = 6\), \(5 \boldcdot (1.5)=7.5\), and \(6 \boldcdot (1.5)=9\).
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