Lesson 5

Students have seen many examples of scale factors that are less than and greater than 1, but the goal of this warm up is to have students focus explicitly on how the size of the scale factor impacts the dilation of a figure. Several dilations of one figure, with the same center but different scale factors, are displayed and students are asked to make observations about the shapes.

If using the digital activity, students will be able to scale a triangle by sliding the scale factor button. Students will informally make conjectures about the relationship between the original and scaled triangles. For example, the triangles are similar, the angles do not change, etc. 

If students try to determine the actual scale factors between dilations, let them know that they only need to make an overall observation about the sizes of the dilations compared to Triangle D.

Important observations include

• The smaller triangles come from scale factors less than one while the bigger triangles correspond to scale factors larger than one
• The original triangle, itself, can be seen as coming from a scale factor of one
• All of the triangles have the same angles and orientation in the plane

A few other points worth discussing include

• Any of these triangles is a dilation of any other with center \(P\).
• The lines from \(P\) contain “corresponding vertices” of the triangles (in the sense that these points map to one another via the dilations taking one triangle to another).

This info gap activity gives students an opportunity to determine and request the information needed for a dilation, and to realize that using coordinates greatly simplifies talking about specific points. In order to perform a dilation, students will need to know the center of dilation (which can be communicated using the coordinate grid), the coordinates of the polygon that they are dilating (also communicated using the coordinate grid), and the scale factor. With this information, they can find the dilation as in previous activities.

The info gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for the information they need to solve the problem. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).

Here is the text of one of the cards for reference and planning:

Listen for how students request (and supply) information about the center of dilation and the location of the polygon that is being dilated. The coordinate grid helps name and communicate the location of points, which is essential in this activity. In addition to the location of points, listen for how students use “center of dilation” and “scale factor” in order to communicate this essential information.

Tell students they will continue to practice describing and drawing dilations using coordinates. Explain the Info Gap and consider demonstrating the protocol if students are unfamiliar with it. Arrange students in groups of 2. Provide access to geometry toolkits. In each group, distribute a problem card to one student and a data card to the other student. After you review their work on the first problem, give them the cards for the second problem and instruct them to switch roles.

After students have completed their work, share the correct answers and ask students to discuss the process of solving the problems. Some guiding questions: 

• “Other than the answer, what information would have been nice to have?”
• “How did using coordinates help in talking about the problem?”
• “If this same problem had a figure on a grid without coordinates, how would you talk about the points?”
• “What if there had been no grid at all? Would you still have been able to request or provide the needed information to perform the transformation?”

Highlight that coordinates allow us to unambiguously provide the location of a polygon’s vertices. In addition, reinforce the idea that in order to perform a dilation, we need to know the scale factor and the center of dilation. The coordinate grid again provides an efficient means to communicate the center of dilation.