Using the image below, have students “I see, I think, I wonder” about the dilation shown. Have a discussion using these phrases to start students thinking about what happens to a figure during a dilation.

From Introduce/Engage handout (download from Materials section above)

Teacher Notes:

Possible questions to ask include:

Possible questions to ask include:

What are some characteristics that you noticed about dilations? [Possible student responses: extended line segments from center of dilation through each vertex on the figure, except for A (0,0), which will remain in the same place.

What happens to the figure that you dilate? What changes? [Possible student responses: when you are doubling, you are copying the segment one more time in addition to the length of the original segment.]

What does it mean to be 2 times larger? [Possible student responses: draw one additional segment, not two because that would be dilating by three.]

What is happening to the length of sides of the figure with each dilation? What is preserved? [Possible student responses: Dilations create similar figures. Orientation is preserved, but the size is not.

Multiple means of representation:
Students can use patty paper to trace the pre-image and informally compare the angle measure and side length to show similarity.

Explore

In pairs or groups have students look at their Dilations Activity Sheet (download from Materials section). In order to complete this activity students will need patty paper and a straight edge. Allow students to engage in the first problem of the activity. Make sure students are extending line AB along the y — axis and AC along the x — axis. A common misconception students may have when dilating the figure by a scale factor of two is drawing additional segments, which actually dilates the figure by a factor of three.

It is important for students to realize that when you are doubling, you are copying the segment one more time in addition to the length of the original segment.

After students have finished the first problem, bring the class together and discuss the construction of the new figure and what observations students made when constructing the dilation. Highlight student thinking that leads to general rules for dilations.

After finishing the initial discussion have students complete the second and third questions on the worksheet. Monitor student progress to catch misconceptions.

Synthesize

After students have completed the activity, share dilations and chart observations. Highlight student answers that lead to general rules for dilations.

Students may notice the coordinates will increase by the factor of dilation. This will remain true when the center of dilation is the origin. When discussing problem two, make sure students are drawing the correct rays along the diagonal of the figure.

Teacher Notes:

Possible rules: 1. [Student answers may vary. Sample student response: Scale factors greater than 1 made the shapes bigger. Scale factors less than one made the shape smaller.]
2. The shape of the figure and the measure of the angles are preserved.
3. With each dilation (stretch) or compression (shrink), I noticed that the original coordinates were multiplied by the scale factor to get the new coordinates.

Extension

Have students calculate the area of each figure in Activity 1 (the original triangle and both dilated triangles). Have them compare the side lengths of each triangle to the area of each triangle and make a conjecture about the relationship between the side lengths and area for each of the triangles.

Students should notice that the side lengths double when dilating by 2, but the area of the triangle will quadruple.

Have students follow this same structure for activity 2 and 3 by comparing side lengths and areas of dilated or compressed figures.

Exit Slip for the day: Dilate the figure below by a scale factor of 3. Describe the transformation in as much detail as possible with center of origin (0,0).

Figure from optional Dilations Assessment sheet (download from Materials section or links above)

Teacher Reflection

How were you able to gather and elicit evidence of student thinking and understanding from exploring dilations? How could this evidence help during your instruction for developing a general rule for dilations?

How did you connect the student-generated strategies and methods to the more formal procedure of rules for dilations?

What scaffolding or differentiation was needed in response to student thinking? How did you extend learning for students?