Unit 5, Lesson 3
Types of Transformations
Integrated Math 1
3.1
Warm-up
Instructional Routines
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NoneActivity Narrative
In this Warm-up, students analyze the results of the transformation rule . They connect the geometric definition of a dilation to this coordinate rule that produces a dilation.
Launch
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NonePoint was transformed using the coordinate rule .
- Add these auxiliary points and lines to create 2 right triangles: Label the origin . Plot points and . Draw segments and .
- How do triangles and compare? How do you know?
- What must be true about the ratio ?
Activity Synthesis
The goal of the discussion is to connect the geometric definition of a dilation to the coordinate rule . Here are some questions for discussion:
- “Why does it make sense that the ratios of the legs of the triangles were both ?” (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 in the coordinate rule ?” (For scale factor , a dilation takes a point times farther away from the center of dilation. Here, the rule took the image of 3 times farther from the origin than the original.)
Student Lesson Summary
Triangle has been transformed in two different ways:
- , resulting in triangle
- , resulting in triangle
Triangles A B C, D E F, and X Y C on coordinate plane. Horizontal x axis from negative 9 to 9. Vertical y axis from negative 12 to 6. Triangle A B C with vertices A at 2 comma negative 2, B at 3 comma negative 4 and C at 6 comma 0. Triangle D E F with vertices D at 2 comma 2, E at 4 comma 3 and F and 0 comma 6. Triangle X Y C with vertices X at 2 comma negative 6, Y at 3 comma negative 12, and C at 6 comma 0.
Let’s analyze the effects of the first transformation. If we calculate the lengths of all the sides, we find that segments and each measure units, and each measure 5 units, and and each measure units. The triangles are congruent by the Side-Side-Side Triangle Congruence Theorem. That is, this transformation leaves the lengths and angles in the triangle the same—it is a rigid transformation.
Not all transformations keep lengths or angles the same. Compare triangles and . Angle is larger than angle . All of the side lengths of are larger than their corresponding sides. The transformation stretches the points on the triangle 3 times farther away from the -axis. This is not a rigid transformation. It is also not a dilation since the corresponding angles are not congruent.