1. Exploring Transformations
Core Mathematics Partnership
Building Mathematical Knowledge and
High-Leverage Instruction for Student Success
8:00 – 10:15
July 28, 2016

2. Classification of Shapes Spatial Visualization
Essential Understandings of Geometry
Classification of Shapes
Features or properties of geometric shapes can be analyzed and described to define and refine classification schemes with growing precision.
Spatial Visualization
Spatial relationships and spatial structuring involves developing, attending to, and learning how to work with imagery, as well as to specify locations.
Geometry is the branch of mathematics that addresses spatial sense and geometric reasoning.
Transformations
Transformation involves working with geometric phenomena in ways that build on spatial intuition by explaining what does and does not change when moving and altering the objects and the space that they occupy.

3. Learning Intention
We are learning Common Core expectations for rigid motions and congruence.
We will be successful when we can
Describe and recognize the different types of rigid motion (isometry) in a plane;
Understand the Common Core definition of Congruence, and use that definition to show that two shapes or figures are, or are not, congruent.

4. Motion Commotion
Cut out the “Motion Commotion” strip, and the solution strip, from the Motion Commotion handout.
Cut along all the dotted lines on the “Motion Commotion” strip, so that the top half of the strip has foldable flaps.
Choose one of the figures at the bottom of the handout, and cut it out.

5. Motion Commotion
Trace your figure onto the left-hand box on the “Motion Commotion” strip.
Perform a (secret!) transformation on your figure, and draw the result on the second box.
Your transformation should be either a quarter or half turn, or a reflection across a horizontal or vertical line.
Fold down the flap to cover the result of your transformation, then describe the transformation on the flap.

6. Motion Commotion
Repeat for the remaining boxes, applying each transformation to the result of the previous one.

7. Motion Commotion Trade “Motion Commotion” strips with a partner.
Predict the results of your partner’s transformations, and record your predictions on your solution strip.
What was challenging about this process?

8. What is a “Rigid Motion”?
Come to a consensus in your table groups as to the definition of “rigid motion”. Be prepared to report out.
Provide a representation of your definition.

9. What is a “Rigid Motion”?
Definition
A rigid motion (in a plane) is a motion of the points in the plane that preserves distances: if point P moves to P’ and point Q moves to Q’, then the distance between P’ and Q’ is the same as the distance between P and Q.
Note that a rigid motion moves every point in the plane, not just the points of some shape or figure.
A rigid motion is often called an isometry.

10. What is a “Rigid Motion”?
Definition
A rigid motion, or isometry, (in a plane) is a motion of the points in the plane that preserves distances: if point P moves to P’ and point Q moves to Q’, then the distance between P’ and Q’ is the same as the distance between P and Q.
Why is it important for students to have a good understanding of isometries?
What types of isometry do you know?

11. Name that Isometry
Translation

12. Name that Isometry
Rotation

13. Name that Isometry
Reflection

14. Types of Rigid Motion (Isometries)
With your table group, come to consensus on a definition (be as precise as you can!) of each of the following types of isometry:
A translation
A rotation
A reflection
What information do you have to give to specify one of these isometries completely?
Facilitators will probably have to explain: For example, if I was talking about a translation, and I wanted you to know exactly which translation I was talking about, what information would I have to give you?

15. Name that Isometry
Glide Reflection

16. Types of Plane Isometry
Any isometry (in a plane) is one of the following four types:
A translation
A rotation
A reflection
A glide reflection
The CCSSM refer to translations, rotations, and reflections as basic rigid motions. (The standards never refer to glide reflections.)

17. What is Congruence?
At your tables, come to consensus as to what it means to say that two shapes or figures are congruent.
Provide a representation of your understanding.

18. CCSSM Definition of Congruence
Two geometric figures are congruent if there is a sequence of (one or more) basic rigid motions which takes one of them onto the other.

19. CCSSM Congruence Standard
Understand congruence and similarity using physical models, transparencies, or geometry software.
8.G.1 Verify experimentally the properties of rotations, reflections, and translations:
a Lines are taken to lines, and line segments to line segments of the same length.
b Angles are taken to angles of the same measure.
c Parallel lines are taken to parallel lines.
Where in our work this summer have we seen ideas related to this standard?

20. CCSSM Congruence Standard
Understand congruence and similarity using physical models, transparencies, or geometry software.
8.G.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
Where in our work this summer have we seen ideas related to this standard?

21. Is Elizabeth Correct?
Elizabeth took the parallelogram below and performed some flips, slides, and turns with it. When she finished, she claimed she had a rectangle. Is it possible that her claim was correct? Why or why not?

22. Make a conjecture: What will happen if you reflectB l A m n
Make a conjecture:
What will happen if you reflect
ABC over n and then over m?

23. Make a conjecture: What will happen if you reflectB l A m n
Students will do it on the patty paper and then give a description of the outcome including how it compared to their conjecture.
The result is a translation, through a distance equal to twice the distance between the lines, and in the direction perpendicular to the lines and pointing from the first line to the second.
1 translation = 2 reflections
- What would have happened if you had done the two reflections in the opposite order?
Make a conjecture:
What will happen if you reflect
ABC over n and then over m?

24. Make a conjecture: What will happen if you reflectB l A m n
Make a conjecture:
What will happen if you reflect
ABC over n and then over l?

25. Make a conjecture: What will happen if you reflectB A l m n
Students will do it on the patty paper and then give a description of the outcome including how it compared to their conjecture.
The result is a rotation. The point of rotation is the intersection of the perpendicular lines. Therefore the corresponding points of the figures are equidistant from the intersection of the perpendicular lines (the point of rotation).
1 rotation = 2 reflections
Make a conjecture:
What will happen if you reflect
ABC over n and then over l?

26. Make a conjecture: What will happen if you reflectB l A n
Make a conjecture:
What will happen if you reflect
ABC over n and then over l?

27. Make a conjecture: What will happen if you reflectB A l n
Students will do it on the patty paper and then give a description of the outcome including how it compared to their conjecture.
Possible conjectures: A rotation? Why. A translation? Why. A reflection? Why.
Why can’t it be a glide reflection? (only 2 reflections. A glide reflection must be 3: 2 for the glide + 1 for the reflection)
Make a conjecture:
What will happen if you reflect
ABC over n and then over l?

28. Are there other possibilities?
Think about our previous discoveries.
Students will do it on the dot paper and then give a description of the outcome including how it compared to their conjecture.
- The distance between the parallel lines is fixed, however they can appear anywhere between the two original figures.
Given the translation, find two parallel lines such that successive reflections in the lines result in the given translation.
Are there other possibilities?

29. What Do These Examples Show?
ANY isometry (rigid motion) in the plane is in fact a composition of, at most, 3 reflections.
Reflections are in some sense the most basic type of isometry.

30. Learning Intention
We are learning Common Core expectations for rigid motions and congruence.
We will be successful when we can
Describe and recognize the different types of rigid motion (isometry) in a plane;
Understand the Common Core definition of Congruence, and use that definition to show that two shapes or figures are, or are not, congruent.

31. Core Mathematics Partnership Project
Disclaimer
Core Mathematics Partnership Project
University of Wisconsin-Milwaukee,
This material was developed for the Core Mathematics Partnership project through the University of Wisconsin-Milwaukee, Center for Mathematics and Science Education Research (CMSER). This material may be used by schools to support learning of teachers and staff provided appropriate attribution and acknowledgement of its source. Other use of this work without prior written permission is prohibited—including reproduction, modification, distribution, or re-publication and use by non-profit organizations and commercial vendors.
This project was supported through a grant from the Wisconsin ESEA Title II, Part B, Mathematics and Science Partnerships.