# Lesson 3

Scaled Relationships

### Lesson Narrative

In previous lessons, students looked at the relationship between a figure and a scaled copy by finding the scale factor that relates the side lengths and by using tracing paper to compare the angles. This lesson takes both of these comparisons a step further.

• Students study corresponding distances between points that are not connected by segments, in both scaled and unscaled copies. They notice that when a figure is a scaled copy of another, corresponding distances that are not connected by a segment are also related by the same scale factor as corresponding sides.
• Students use protractors to test their observations about corresponding angles. They verify in several sets of examples that corresponding angles in a figure and its scaled copies are the same size.

Students use both insights—about angles and distances between points—to make a case for whether a figure is or is not a scaled copy of another (MP3). Practice with the use of protractors will help develop a sense for measurement accuracy, and how to draw conclusions from said measurements, when determining whether or not two angles are the same.

Students then deepen their understanding of scale factors by classifying scale factors by size (less than 1, exactly 1, or greater than 1) and notice how each class of factors affects the scaled copies (MP8). They see that the scale factor that takes an original figure to its copy and the one that takes the copy to the original are reciprocals (MP7). This means that the scaling process is reversible, and that if Figure B is a scaled copy of Figure A, then Figure A is also a scaled copy of Figure B.

One of the activities, Scaling a Puzzle, is optional. In Scaling a Puzzle, students scale the 6 pieces of a puzzle individually and then assemble them to make a scaled copy of the puzzle. The individual pieces are rectangular with line segments partitioning them into regions. Students need to think strategically about which measurements to take in order to scale the pieces accurately.

### Learning Goals

Teacher Facing

• Explain (orally and in writing) that corresponding angles in a figure and its scaled copies have the same measure.
• Identify (orally and in writing) corresponding distances or angles that can show that a figure is not a scaled copy of another.
• Recognize (orally and in writing) the relationship between a scale factor of a scaled copy to its original figure is the “reciprocal” of the scale factor of the original figure to its scaled copy.
• Recognize that corresponding distances in a figure and its scaled copy are related by the same scale factor as corresponding sides.

### Student Facing

Let’s find relationships between scaled copies.

### Required Preparation

Make sure students have access to their geometry toolkits, especially rulers and protractors.

### Student Facing

• I can describe the effect on a scaled copy when I use a scale factor that is greater than 1, less than 1, or equal to 1.
• I can explain how the scale factor that takes Figure A to its copy Figure B is related to the scale factor that takes Figure B to Figure A.

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