SUPPORT MATERIALS FOR THE GEOMETRIC TRANSFORMATION WORKOUTS

PART 1: Evidence of Understanding

There is no single piece of evidence that can “prove” understanding. Rather, one builds a case for understanding via a preponderance of evidence.

The GTWs are a tool for

  1. providing experience and feedback for students to develop a robust understanding
  2. providing opportunities to make visible a variety of types of evidence of student understanding

General categories of evidence of understanding (not in any specific order)

Verbal: / “This one just slides over”, “move this one unit left and one unit down”
Visual: / ,
Dynamic: / Student moves an overhead transparency or uses computer software or smart board to indicate a translation
Written: / The flag is moved over and points the same way.
Gestural: / Student gestures with her hand to indicate a translation
Identification: / This one is a translation, (b) is the answer
Application: / Student uses a translation to solve a problem

The facilitator’s role is not just to elicit explanations, but to look for and highlight evidence of understanding within these explanations.

All evidence is not equally robust.

Given a list of four choices to identify the translation:

Teacher: Who got (b)?

Student: I did!

This is initial evidence (identification), but not necessarily robust. The facilitator can build on this:

Teacher: Explain why (b) is correct.

Student: It does this [moves hand horizontally]

This is adds some gestural evidence. Keep building a case:

Teacher: How does that represent a translation?

Student: It means slide it over but not change the way it is pointing.

This adds some verbal evidence. Can the student’s definition be refined?

Teacher: What do you mean by “slide over”?

Student: I mean everything is moved the same amount, the same distance.

More verbal evidence. The student refines the vague expression “slide over” to the more precise “everything is moved the same distance”. Does the student really understand that translations move every point in the plane? Not necessarily but this is still valuable evidence supporting the claim.

Teacher: OK, now what do mean by “not change the way its pointing?”.

Student: I don’t know. It just, you know, looks the same way.

So in a couple minutes of exchange the student has demonstrated some evidence of understanding. He has correctly identified, gestured, and verbalized evidence of understanding translations. He has also provided some evidence of the possible boundaries of his understanding, namely making precise the notion of a translation preserving directionality or the notion that a translation requires moving all points the same distance and same direction.

PART 2: Guide to the Geometric Transformation Workouts

These are designed to be “workouts” in the sense that they are not meant to introduce new concepts, but rather allow for practice and refinement around the ideas, definitions and language of geometric transformations.

These workouts are designed for students to work individually on them for a short time and then have a teacher led discussion with the class. The questions are often multiple choice or simple drawings. They are generally straightforward questions and are not meant to be lengthy explorations. The following format is suggested:

Students work individually for 5 minutes on the GTW

Teacher leads whole class discussion for 5-10 minutes

Here is an example discussion around the first problem on GTW#1:
Teacher: lets discuss the first one. Which one shows two figures equivalent by a translation?
Students: d!
Teacher: Raise your hand if you think d is correct.
Teacher: ok, can someone say why?
Student 1: cuz translations means to slide it over
Teacher: isn't (a) also a "slide over"
Student 1: no, its pointing the wrong direction
Student 2: yeah, you gotta go in the same direction
Teacher: say more what you mean by that
Student 2: you can't turn it. In (a) the flag is turned a different direction
Student 3: when you slide you have to keep the directions the same
Teacher: What about (c)? Aren't the flags in the same direction, or "orientation"?
Student 1: yeah so, OK (c) and (d) are both translations.
Student 2: no, i think you need the same size too, (c) is shrunk
Teacher: ok, so you are saying a translation slides a figure, but keeps its direction or orientation the same, and keeps its size the same?
Students: yeah
Teacher: OK, so lets look at number 2...

The key is to allow the students to use language and definitions around the transformations. Note that even though in the example the students all selected (d) which was the correct answer, the teacher led a brief discussion to help refine the language. So, as often is the case, getting the right answer is not the primary goal. The primary goal is to develop robust definitions and language around geometric transformations. A secondary goal is to provide experiences working with these transformations emphasizing the visual and dynamic aspects of the transformations.

The Field Guide to Geometric Transformations (laminated 2-sided matrix) provides definitions and language as a reference.

Most of the GTWs incorporate a grid. This design is intentional, as grids provide a helpful support when working with transformations. However, all of the GTWs can also be solved without a grid. It may be helpful to encourage students to solve the problems using a variety of tools, such as a grid, tracing/patty paper, and a ruler. Then consider how the choice of tool impacts their solution and their thinking.

Overview of the sequence of workouts:

GTW#1translations

GTW#2rotations

GTW#3dilations

GTW#4dilations

GTW#5reflections

GTW#6rotations

GTW#7combinations

GTW#8combinations

GTW#9combinations

Key Points/Answer Key:

GTW#1

  1. definition of translation, [answer: d]
  2. definition of translation, [answer: c]
  3. identify the translation and apply it. Note that it will be common for some students to see the translation as “down one and right one”, yielding d as an answer. This is a good time to discuss corresponding parts. The translation sends each point left two and down four. [answer: e]

GTW#2

  1. definition of rotation, [answer: b]
  2. definition of rotation. [answer: a (the center of rotation is one unit right and one unit down from the rightmost vertex of triangle a)] This is one is more difficult since the center of rotation is not given. Ruling out (d) as a translation and (b) as a reflection can be useful, but make sure to note that for some figures it is possible that there are multiple transformations that can work. For example:

Here are two pairs that are equivalent by a reflection in the blue line. There is no translation that can take the left hand flag to the right hand flag (directions are not preserved). However, there is a translation that takes one lollipop to the other.

  1. Identify the rotation and apply it. [answer: d] Note: the rotation that takes arrow 1 to 2 is centered at the grid point in the middle of the three arrows. The rotation is 1/3 of a complete revolution (or 120 degrees). But looking at the corresponding parts makes this easier. Square X is about one unit in the direction of the arrow from the point of arrow 1, so the rotation of square X will be one unit in the direction of arrow 2 from the point of arrow 2.

GTW#3

  1. Definition of dilation, with focus on the center of dilation [answer: c] Note that the scale factor is 3, hence the distance from a to c is 3 times the distance from b to c.
  2. Draw a dilated image.
  3. Draw a dilated image. Answer:

Note the dilated images of the points are indicated by primes (i.e. A goes to A’). Also note that the following is incorrect. It shows “half” a trapezoid, but it is NOT a dilation centered at X.

GTW#4

  1. Definition of dilation, combination of transformations [Answer: b] Note that “nontrivial” means that the transformation is not the identity transformation that takes every point to itself. A dilation of scale factor 1 is a trivial dilation. A translation that moves points 0 units is a trivial translation.
  2. Definition of dilation. [Answer: d]
  3. Draw a dilated image. Answer:

GTW#5

  1. Definition of reflection. [Answer: a]
  2. Applying reflections [Answer: reflect in line D then line A(or A than D)] Note: the order does not matter if the lines are perpendicular, otherwise order DOES matter. Try reflecting in line A then B, then B then A.
  3. Apply reflections [Answer: a]

GTW#6

  1. Definition of rotation, focus on center of rotation [Answer: c]
  2. Identify and apply a rotation. Answer:
  3. There are many (in fact infinite) combinations that will work. One is translate to the right 4 units then rotate a quarter turn (counterclockwise) around the leftmost point of triangle B.

GTW#7

  1. The answers are NO, YES, YES, MAYBE. Because the pac-man has symmetry, the pair is equivalent by a half-turn rotation center at the midpoint between the two pac-men. The MAYBE is because a dilation centered at the midpoint between the two pac-men with scale factor -1 will work. We are tending to restrict our scale factors to positive numbers, but technically negative scale factors can be used.
  2. Apply reflections. Answer:
  3. Apply rotations. Answers may vary, here is one rotating a quarter turn (90 degrees) around each vertex.

GTW#8

  1. Identify and apply translation. [Answer: b] The translation is right 3 units and up 1 unit.
  2. Answer: NO, NO, NO, NO. Note: this is an example of similar rectangles that are NOT equivalent by any single transformation. A combination of a dilation with scale factor 2 and a rotation and a translation will work.
  3. This is about similarity (defined as a combination of translation, rotation, reflection, and dilation). There are an infinite number of combinations that will work in this example. One is rotate a quarter turn counterclockwise centered at the upper point of triangle A. Then translate down 3 and right 5 (this should put the image of triangle A tucked into the right angle inside of triangle B. Now dilate with scale factor 2 centered at the right angled vertex of triangle B.

GTW#9

  1. Definition of dilation. [Answer: a]
  2. Apply rotations. Answer: a circle centered at C going through point P.
  3. Apply dilations. Answer a line (or ray if we restrict to positive scale factors) through points C and P. Note: this is one of the “big ideas”, namely, the set of dilations of a point form a line. This is at the heart of the relationship between similarity and linearity. Recall the dynamic diagram from the powerpoint slides:


PART 3: Evidence Eliciting Questions for Geometric Transformations:

Generic Questions:

  1. Can you say more?
  1. Can you draw a picture?
  1. Can you show me with your hands?
  1. Can you show me using an overhead transparency?
  1. Can you show me using patty paper?
  1. What is the definition?
  1. Can you label the diagram?
  1. Why does this not work?
  1. Why does this not satisfy the definition?
  1. Can you show it another way?

Translations:

  1. Does a translation move every point the same distance?
  1. Does a translation move every point the same direction?
  1. Can you create a transformation that moves every point the same distance, but is NOT a translation?
  1. Can you create a transformation that moves every point the same direction, but is NOT a translation?
  1. How many different translations move point A to point B (for some pair of fixed points A and B)
  1. If two lines are equivalent by a translation, must they be parallel?
  1. How would you show that a transformation was NOT a translation?
  1. How would you show that two figures were not equivalent by a translation?
  1. If two translations are combined, is the result a translation?

Rotations:

  1. Does a rotation move every point the same distance?
  1. Does a rotation move every point the same direction?
  1. Where is the center of rotation?
  1. How do you find the center of rotation?
  1. If you know center of a rotation, how would you find the angle of rotation?
  1. How many different rotations move point A to point B (for some pair of fixed points A and B)?
  1. If two lines are equivalent by a rotation, must they be parallel?
  1. How would you show that a transformation was NOT a rotation?
  1. How would you show that two figures were not equivalent by a rotation?
  1. If two rotations are combined, is the result a rotation?
  1. What points are not moved by a rotation?

Reflections:

  1. Does a reflection move every point the same distance?
  1. Does a reflection move every point the same direction?
  1. Where is the line of reflection?
  1. How do you find the line of reflection?
  1. If you know line of reflection, how would you find the image of a point by the reflection?
  1. How many different reflections move point A to point B (for some pair of fixed points A and B)?
  1. If two lines are equivalent by a reflection, must they be parallel?
  1. How would you show that a transformation was NOT a reflection?
  1. How would you show that two figures were not equivalent by a reflection?
  1. If two reflections are combined, is the result a reflection?
  1. What points are not moved by a reflection?

Dilations:

1. Does a dilation move every point the same distance?

  1. Does a dilation move every point the same direction?
  1. Where is the center of a dilation?
  1. How do you find the center of a dilation?
  1. If you know center of a dilation and the scale factor how would you find the image of a point by the dilation?
  1. If you know the scale factor of a dilation, but not the center of dilation could you find the image of a point by the dilation?
  1. If you know the center of a dilation, but not the scale factor could you find the image of a point by the dilation?
  1. How many different dilations move point A to point B (for some pair of fixed points A and B)?
  1. If two lines are equivalent by a dilation, must they be parallel?
  1. How would you show that a transformation was NOT a dilation?
  1. How would you show that two figures were not equivalent by a dilation?
  1. If two dilations are combined, is the result a dilation?
  1. What points are not moved by a dilation?
  1. How is the scale factor of a dilation related to the ratio of lengths of a line segment and its image by the dilation?

General questions about geometric transformations:

  1. What points are moved? How far? What direction(s)?
  1. What points stay fixed?
  1. What characterizes a transformation? What are the essential ingredients to uniquely describe a specific transformation?
  1. How do you show something is not a particular transformation?
  1. How do transformations behave in combination? Does the order of combination matter?
  1. Each transformation has an inverse, i.e. a transformation that “ undoes” it. For example the translation “move one unit left” has the inverse translation “move one unit right”. What are the inverses of other transformations?
  1. If two figures are equivalent by a transformation, is it unique? Could there be other transformations that create the equivalent?
  1. Transformations that preserve length are called isometries. Does an isometry preserve area? Does a transformation that preserves area need to be an isometry?

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