# Lesson 10

Dilations on a Square Grid

### Lesson Narrative

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.

### Learning Goals

Teacher Facing

• Create a dilation of a polygon on a square grid given a scale factor and center of dilation.
• Identify the image of a figure on a coordinate grid given a scale factor and center of dilation.
• Identify what information is needed to dilate a polygon on a coordinate grid. Ask questions to elicit that information.

### Student Facing

Let’s dilate figures on a square grid.

### Required Preparation

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.

### Student Facing

• I can apply dilations to figures on a square grid.
• I can apply dilations to polygons on a rectangular grid if I know the coordinates of the vertices and of the center of dilation.
• If I know the angle measures and side lengths of a polygon, I know the angles measures and side lengths of the polygon if I apply a dilation with a certain scale factor.