Lesson 3

In this warm-up, students analyze the results of the transformation rule \((x,y)\rightarrow (3x,3y)\). They connect the geometric definition of a dilation to this coordinate rule that produces a dilation.

The goal of the discussion is to connect the geometric definition of a dilation to the coordinate rule \((x,y) \rightarrow (3x,3y)\). Here are some questions for discussion:

• “Why does it make sense that the ratios of the legs of the triangles were both \(1:3\)?” (This is because we multiplied each coordinate by 3, and the legs are just the vertical and horizontal distances to the points.)
• “Look at the definition of a dilation on your reference chart. How does it match what’s happening the coordinate rule \((x,y) \rightarrow (3x,3y)\)?” (For scale factor \(k\), a dilation takes a point \(k\) times farther away from the center of dilation. Here, the rule \((x,y) \rightarrow (3x,3y)\) took the image of \(B\) 3 times farther from the origin than the original.)

In this activity students match graphs to rules in coordinate transformation notation. Then they analyze both the rules and images to decide which represent similarity transformations and which represent rigid transformations.

Students will use a variety of strategies to explain which figures are similar or congruent (counting, Pythagorean Theorem, recognizing right angles, properties of isosceles triangles, and trigonometry). Monitor for students who use these different methods.

Some students may not be sure how to work with the rule \((x,y)\rightarrow (y,\text-x)\). Ask these students, “What are the coordinates of point \(A\)?” (\((3,\text-4)\).) “Which of those is the \(x\)-coordinate?” (3.) “In the transformation rule, where does the \(x\) land, and what happens to it?” (The \(x\)-coordinate lands in the \(y\) spot, and its sign is opposite.) Another option is to suggest students write out \(x=3\) and \(y=\text-4\), then substitute each value into the transformation rule.

If students suggest that figures are congruent or similar simply because they look that way, tell them that they need to provide more backing for their answer. What are some ways we can verify that 2 figures are congruent or similar? Remind students that for triangles, they’ve learned some shortcuts that they can use here.

Select previously identified students to share their reasoning. Ask them how they calculated side lengths or angle measures. The key point to emphasize is that similar figures have congruent angles and proportional sides, while congruent figures have congruent angles and sides. Students know some shortcuts for triangles, but for the rectangles there is no shortcut.

In this activity students work backwards to figure out a transformation rule. They are given the opportunity to come up with their own method of organizing their work. Monitor for students who have methods of clearly documenting information they are given as well as students who record their attempts at a rule. Once students write a rule, they analyze the figures to decide if the transformation takes shapes to congruent shapes.

Give students 2–3 minutes of quiet work time. If most students haven’t come up with a method to organize their information at that point, invite a student who has made a table to share.

If students struggle to write a rule, ask them to start by writing out the pattern they see in words. For example, they may write, “The \(x\)-coordinate stays the same and the \(y\)-coordinate doubles.” Then ask how they could put those words into coordinate transformation notation.

The goal is to use the language of distance and angle preserving moves to describe the 2 transformations. Here are some questions for discussion:

• “Look at the corresponding side lengths and angles in the 2 pairs of triangles. How do they compare?” (In triangles \(DEF\) and \(D'E'F'\), all sets of corresponding sides and all sets of corresponding angles are congruent. In triangles \(ABC\) and \(A'B'C'\), neither the corresponding sides nor the corresponding angles are congruent.)
• “What would a transformation look like if it kept the angles the same but not the side lengths?” (It could be a dilation. The corresponding sides' lengths would need to be proportional for this to be true.)
• “Is it possible for a transformation to keep side lengths the same but not keep the angles the same?” (This isn’t possible in a triangle. For a square, it could be transformed into a rhombus. However, none of our standard transformations (translation, reflection, rotation, or dilation) would accomplish this.)