Lesson 10

In this lesson, students apply dilations to polygons on a square grid, both with and without coordinates. The grid offers a way of measuring distances between points, especially points that lie at the intersection of grid lines. If point \(Q\) is three grid squares to the right and two grid squares up from \(P\) then the dilation with center \(P\) of \(Q\) with scale factor 4 can be found by counting grid squares: it will be twelve grid squares to the right of \(P\) and eight grid squares up from \(P\). The coordinate grid gives a more concise way to describe this dilation. If the center \(P\) is \((0,0)\) then \(Q\) has coordinates \((3,2)\). The image of \(Q\) after this dilation is \((12,8)\).

Students continue to find dilations of polygons, providing additional evidence that dilations map line segments to line segments and hence polygons to polygons. The scale factor of the dilation determines the factor by which the length of those segments increases or decreases. Using coordinates to describe points in the plane helps students develop language for precisely communicating figures in the plane and their images under dilations (MP6) and strategically use coordinates to perform and describe dilations (MP7). Student practice this type of language using 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.

Two activities in this lesson call for pre-printed slips cut from copies of a blackline master. Organizing these copies to hand out before the start of class, possibly by using plastic bags and paperclips, will help the lesson run smoothly.