melt-institute- web viewuse transformations to assess whether two polygons are congruent....

Transformations

Mathematical Goals: Teachers will be able to State a definition for a geometric transformation. Use transformations to assess whether two polygons are congruent. Recognize and apply properties of reflections, rotations, and translations.

Pedagogical Goals: Teachers will be able to Anticipate difficulties students may have when learning geometric transformations. Consider particular aspects of polygons and transformations when selecting appropriate

examples for students. Critique and modify tasks that build on students’ understanding of geometric

transformations.

Technological Goals: Teachers will be able to use a technological tool to Use a DGE to apply reflections, rotations, and translations to points and polygons. Use the dragging feature to identify fixed points. Use measuring tools to examine lengths of segments, measures of angles, and distances

between points. Create animated sketches using transformations.

Mathematical Practices: Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure.

Length of session: 180 minutes (2 – 90 minute sessions)

Materials needed: Computer with Geometer’s Sketchpad, MysteryTransformations.gsp file, Transformations Participant Handout

Overview: In this session participants will utilize GSP to solve explore properties of geometric transformations. They will use this information to generate definitions of each transformation. They will then consider mystery transformations and discuss how using a DGE might impact students’ thinking when learning about transformations.

Estimated # of Minutes

Activity

20 minutes Introduction to Geometer’s Sketchpad Briefly introduce participants to Geometer’s Sketchpad (GSP). Show them

the menu bar at the top and the tool bar (typically found on one side). Allow participants time to explore what types of things they can do in Geometer’s Sketchpad (GSP). Encourage them to try creating points, lines, segments, and circles. See if they can determine how to measure, drag, or transform objects. The goal here is to let them get comfortable with where things are located in GSP. They should be actively engaged and talking

Adapted from: Hollebrands, K. F., & Lee, H. S. (2012). Geometric transformations. In Preparing to teach mathematics with technology: An integrated approach to geometry (63-96). Dubuque: Kendall Hunt.

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about what they are doing.5 minutes Introduction

In this session, we will focus on geometric transformations. When you hear this word, what do you think of? Participants will likely say reflections, rotations, translations, and dilations.

One place we can think informally about geometric transformations is in descriptions of dance. Consider the end of the clip: http://www.youtube.com/watch?v=tyZeGOsR9IA

What geometric transformations might you use to describe the dancers? Participants may say they could use a translation to line up the dancers or reflections for if the dancers are facing one another. When the dancers turn, they might say a rotation.

Through the PTMT wikispace, can also show video of animated dancer in GSP. http://tumteresources.wikispaces.com/file/view/Dancing+stick+figure.avi/338311984/Dancing stick figure.avi

55 minutes Translations If we wanted to model a chorus line of dancers like this one using GSP,

how might we do that? We could use translations.

Using GSP, create a stick figure to represent a single dancer. Now, because we want to have many dancers in a row, all moving synchronously, we are going to use a translation to create several other stick figures. There are two ways we are going to translate a geometric object using GSP: by polar coordinates or by a marked vector.

First, let’s consider translating by polar coordinates. Highlight your entire stick figure and from the transform menu, select “Translate”. Then, click on the “Polar” option. You will need to specify a distance and an angle. Enter 5cm for the distance and 0 degrees for the angle.

You should now see two stick figures. Note that the stick figure on the right is the image of the stick figure on the left (we call the one on the left the pre-image).

Questions to consider:1. What is true about the size and shape of the two stick figures? After

a translation the size, shape, and also the orientation of the two stick figures are the same.

2. Use the Arrow tool to drag a point on the leg of the original stick


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http://tumteresources.wikispaces.com/file/view/Dancing+stick+figure.avi/338311984/Dancing%20stick%20figure.avi

http://tumteresources.wikispaces.com/file/view/Dancing+stick+figure.avi/338311984/Dancing%20stick%20figure.avi

http://www.youtube.com/watch?v=tyZeGOsR9IA

figure (pre-image). Try dragging a point on the arm of this stick figure. What else changes when you perform this dragging? Why? As a point on the arm or on the leg of the original stick figure is manipulated, the length/direction of the arm/leg changes dynamically. These changes are also immediately seen in new stick figure, as the new stick figure (the image) is dependent on the original (pre-image).

3. Measure the distance between the endpoint of the left leg of the original stick figure and the corresponding point on the left leg of the new stick figure. Measure several other distances for pre-image/image pairs of points. What do you notice? Explain. The distance between the corresponding points is 5 cm, the value we entered for “fixed distance” previously. All of the other distances between points on the pre-image and the corresponding points on the image should be the same.

4. Translate your stick figure to create an entire chorus line of dancers that are equidistance apart. Describe how you chose to do this. One possible way is to begin by translating the original stick figure to create the new stick figure, as previously described. Then, this time, translate the new stick figure as if it was the original stick figure to get another “new” stick figure. Continue this process until you have your chorus line. A second option is to create a chorus line using only the original stick figure. To do this, allow your first translation to be a set distance, like 5 cm as previously described. Then, change the “fixed distance” for your next translation to be twice that of the first, say 10 cm. Repeat this process until you have a chorus line.

5. A student in a high school class asks what the 5 and 0 represent (the values we typed into the polar coordinate dialog box to perform the translation). How do you respond? Five, which is typed for the “fixed distance” value in the software, defines the magnitude of the translation. In other words, it determines the distance the object is translated. On the other hand, 0, which is typed for the “fixed angle” value in the software, defines the direction of the translation. For example, if we use 45 for the “fixed angle” value, the direction of the translation by the “fixed distance” value will be 45 degrees in the positive direction (counter-clockwise). If this value is 0, then the direction of translation will be 0 degrees in the positive direction (or parallel to the x-axis).

Open a new sketch and create a new stick figure. Construct a line segment and label it AB. Now we will consider how to translate an object by a vector using GSP. In order to translate by the vector AB, we first have to mark AB as our vector. Select points A and B, in that order, and then go to Transform – Mark Vector. Note that the direction the black animation goes is the direction your translation will take.


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Now, select your entire stick figure and go to Transform – Translate. This time, select by marked vector. Now you should have two stick figures.

Questions to consider:6. Translate the second stick figure to have three stick figures on the

screen. What do you notice about the size and shape of the three stick figures? All three stick figures should be the same.

7. How is the location of the stick figures related to the translation vector? The location of the stick figures is directly related to the translation vector because this vector determines the magnitude and direction of the translation. Thus, the new stick figures should be translated the distance of the vector and in the same direction as the vector.

8. Drag the line segment representing the translation vector so that the tail corresponds with a point on the original stick figure. Where is the head of the vector located? Why? Once you drag the tail to with a point on the original stick figure, the head of the vector is now located on the corresponding point of the new stick figure. This happens because we change the magnitude and direction of the vector, thus the image has to move to match this change.

9. Use the segment tool to create a segment that joins a point from the original stick figure to its corresponding image point. What do you notice about this segment and the vector? The lengths of the segment and the vector are the same and the segment and vector are parallel to each other.

10. Measure several distances of pre-image/image points as you did on your previous sketch. How do these distances relate to one another? How do they relate to the vector AB? The distance between any two corresponding points should be the same and this distance should be equivalent to the length of the vector.

11. What do you think will happen when you translate a line segment? What about a line? When you translate a line segment, a new line segment with the same length will be formed in the direction of the vector that is used to translate it. Also, this new line segment will be parallel to or coincident with the original line segment. Similarly, if you translate a line, in every case, the new line will be parallel to or coincident with the new line.

12. What do you think will happen when you translate an angle? The measure and direction of the new angle will be as same as the original angle.

13. What do you think will happen when you translate two lines that are parallel to each other? When you translate two lines that are parallel to each other, you get another pair of parallel lines that are parallel to the original lines.

14. You have had the opportunity to perform several different translations. Based on your experiences, describe a translation. Be sure to explain what stays the same and what changes. A translation


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maps each point in the domain to an image point as specified by a vector. Distance and angle measures are preserved under a translation, so it is an isometry.

15. Students often have difficulty reasoning about vectors. Because representations of vectors and rays are very similar, students confuse these two objects. Describe how you could assist students in understanding the differences between vectors and rays. One key difference that is important to point out to students is how a vector and ray differ in relation to length. A vector has two end points that provide a magnitude, so for this reason vectors are measurable. However, a ray has only one end point and extends indefinitely in one direction. Although representations of vectors and rays are, similar they are not the same (see below). When naming a vector we label the endpoints of the vector, but when naming a ray we label the endpoint of the ray and a point on the ray. This distinction might help students in understanding the differences in these two mathematical concepts.

16. How does this introduction of translations in a dynamic environment, using dancing and stick figures, compare with how you teach translations? What are the benefits and drawbacks of this approach? Generally, translation is introduced to students either using dot paper or using grid paper. For that reason, students have to follow a correct procedure and take the translation vector into consideration in order to translate a shape correctly. On the other hand, in a DGE students can translate a shape automatically and can observe the preserved and non-preserved properties of the shape. The prominent benefit of using a DGE is to be able to manipulate the shape dynamically and immediately observe the outcome. However, in a static geometry learning environment, students do not have this advantage. It is important to remember, however, that challenges can arise when using a DGE. For example, in GSP, a translation vector is represented by a line segment, thus students might have a difficult time determining which is the head and which is the tail of the translation vector.

40 minutes Reflections Although line dances are easily described using translations, partner dances

are different because the dancer and his or her partner are performing moves that mirror each other.


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Questions to consider:1. Explain how you could use the picture above to explain to students

that a translation would not be the appropriate transformation to use to describe the positions of the dancers. You can remind students of the properties that are preserved through translation by emphasizing that the orientation of the shapes does not change after translation. Then, you can analyze the picture in detail and point out: the male dancer put up his right hand and his left foot and looked over his left shoulder, while the female dancer put up her left hand and her right foot and looked over her right shoulder. In order to label this picture a translation, for example, you can stress that the male dancer must have put up his left hand and right foot and looked over his right shoulder to match what the female dancer did. Only then, would their orientations match and could be considered a translation. Since this doesn’t happen, a translation is not the appropriate transformation to use to describe the positions of the dancers.

In a new sketch, create a stick figure and a line or line segment named AB. Select AB and choose “Mark Mirror” from the transform menu. (Or just

double click AB.) Now, select your entire stick figure and select “Reflect” from the transform

menu. You should now see two stick figures. Questions to consider:

2. What do you notice about the size and shape of the two stick figures? The size and shape of the two stick figures are the same.

3. Drag a point on the leg of the original stick figure. What happens? Why? Drag other points of the pre-image or image of the stick figure. As a point on the leg of the original stick figure is manipulated, these changes are mirrored in the image for the corresponding point. In other words, as the leg of the pre-image changes, the corresponding leg of the new stick figure also changes. Moving any point on either figure immediately results in


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the same changes for the corresponding point of the other figure since these figures are dependent on each other through the reflection.

4. Predict what will happen if you drag an endpoint of the mirror line segment (point A for example). Drag point A. What happens? Why? The original stick figure will remains in the same place since it is independent from the line of reflection. On the other hand, the position of the new stick figure changes as the line of reflection changes since it is dependent on this line. In either case, the corresponding points of the figures should be equidistant from the line of reflection. For example, the point on the foot of the left leg of the pre-image is the same distance from the line of reflection as its corresponding point (the foot on the right leg of the image).

5. Predict what will happen if you drag point B. Drag point B. What happens? Why? If you drag point B, the same things happen as described in the previous answer because the image is dependent on the line of reflection.

6. Create a segment joining the hands of the two stick figures as shown below. What do you notice about this segment and the line of reflection? The line of reflection is the perpendicular bisector of the segment created that joins the hands (corresponding preimage/image points) of the two stick figures.

7. Based on your interactions with the sketch, provide a definition of a reflection. A reflection maps each point in the domain to a image point such that the distance between the preimage point and the line of reflection is the same as the distance between the image point and the line of reflection. A figure that is reflected across a line or line segment maintains its original size and shape but now mirrors the figure in orientation (orientation is reversed). As points on either figure are manipulated, the corresponding points of the other figure change in exactly the same way to keep the figures symmetric.

8. Describe at least three different properties of reflections. The size and shape of a figure/object do not change after a reflection, but the orientation does. The image will mirror the pre-image; in other words, if we “folded” our paper along the line of reflection, the two figures would match up exactly. Also, each point on the pre-image and its corresponding point on the image are equidistant from the line of reflection. Finally, any segment joining two points of the figures will be perpendicular to the line of reflection.

9. In order for students to develop an appropriate definition of reflection from this activity, it is imperative they focus in on the important properties of reflection. How can a teacher focus students on the relevant properties and assist students in understanding which features of the diagram are not important to consider? A teacher can ask students to investigate properties within the sketch


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by first allowing them to explore what happens when certain aspects are manipulated. Then, to hone in on the less apparent specifics, such as that each corresponding point is equidistant from the line of reflection, the teacher can specifically ask students to measure some of these distances.

10. A fixed point for a transformation is when the point is mapped to itself. Did you encounter any fixed points while you were dragging points around in the sketch? Describe fixed points in terms of input and output. Yes. A fixed point happens when the input and output are the same. In terms of the sketch, this occurs when you can drag a point so that its preimage and image coincide.

11. Let’s consider fixed points from an algebraic context. Describe a linear function for which there are infinite fixed points. Describe one or more linear functions for which there are zero fixed points. Describe one or more linear functions that have exactly one fixed point.f(x) = x has an infinite number of fixed pointsf(x) = x + 1 and f(x) = x – 6 both have zero fixed pointsf(x) = 5 and f(x) = 3x – 4 both have exactly one fixed point

12. Sometimes young students are taught to think about reflections as flips. What properties of reflection are highlighted by thinking about reflections as flips? What properties of reflection are not made as explicit when considering reflections as flips? When considering reflections as flips, it is easy for students to see the symmetrical aspects of reflection and that reflection preserves size and shape but changes the orientation. On the other hand, the idea that corresponding points are equidistant from the line of reflection is not as apparent.

30 minutes Rotations In keeping with the theme of dancing, how might we represent rotations?

We could think of someone doing a turn or twirl. In three dimensions, the dancers performing a spin are rotating their bodies

about a vertical axis of rotation that passes through the center of their bodies. So, to think about this from two dimensions, consider an aerial view of synchronized swimmers where the center of rotation is where their feet meet as in the picture below.


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Open a new sketch and create a stick figure to model one of the swimmers in the picture.

Create a point, not on your stick figure, and label it C. Mark point C as the center of rotation by selecting it and choosing “Mark

Center” from the transform menu (or double clicking it). Select the entire stick figure and choose “Rotate” from the transform menu. Enter the angle (in degrees) with which you wish to rotate the stick figure.

Let’s say 60 degrees. After selecting rotate, you should see a new stick figure.

Questions to consider:1. Consider the pre-image and image stick figures. Create a

description of the relationship between the two stick figures using a synchronized swimming scenario. After the transformation, the pre-image stick figure remains in its original location, while the new stick figure is located 60 degrees counterclockwise about point C from the original figure. This means that every angle created by selecting a point on the pre-image stick figure, point C, and then the corresponding point on the new stick figure equals 60 degrees, the “fixed angle” value we used in the software. In terms of synchronized swimming, this means that the two simmers are 60 degrees apart from the center of their rotation, typically the center of the circle created by their bodies.

2. What do you notice about the size and shape of the two stick figures? The size and shape of the two stick figures are the same.

3. Drag point C and describe what happens to the two stick figures. Explain. If you drag point C, then original stick figure is unchanged. However, the new stick figure moves to keep the angle of rotation a consistent 60 degrees.

4. Drag a point on one of the stick figures and describe what happens to the other stick figure. Does it matter which stick figure you drag? Explain. If you drag a point on either of the stick figures, the corresponding point changes in exactly the same way. Because the new stick figure is dependent on the original stick figure through point C, any manipulation of a point will change the corresponding


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point’s coordinates.5. If you constructed a circle with a center at point C that also passes

through a point on the original stick figure, what other point will the circle pass through? Use your DGE to test your hypothesis. The circle will also pass through the corresponding point on the new stick figure.

6. Describe at least three different properties of rotations. Rotations preserve both the size and shape of an object. However, the position of the object changes about the center of rotation. The angle created between any two corresponding points from the pre-image and image and the center of rotation is equal to the angle of rotation. This is true for any pair of corresponding points chosen.

7. Are there any fixed points under a rotation? There is one fixed point under a rotation: the center of rotation.

8. To perform a rotation, most DGEs require that you input a particular angle measure. Describe how this design feature of the technology may influence student thinking about rotations. Because you must manually input an angle measure, if the teacher always provides specific angles of rotations, the students may not realize that other angles of rotation are possible. It is important to let them experiment trying all types of angle measures before honing in on the most useful ones. But, by allowing students to measure the various angles created by pairs of corresponding points and the center of rotation, the idea that we rotated by a certain angle measure may become more apparent. They can also drag the rotation point and observe that the angle measures of the rotation remain fixed.

9. Are there particular angles of rotation that would be more or less helpful to use with students? Explain. Students generally are more familiar some angle values, such as 30, 45, 60, 90, 120, and 180 degrees, thus it is important that they investigate these angle measures at some point. These particular angle measures are useful when showing the full rotation of an object back onto itself because they all are factors of 360. For example, if I rotate an object 12 times by an angle of 30 degrees, the last rotation should place the object back onto itself. Some less helpful, yet still worth exploring, angle measures are random measures, such as 22 degrees, 79.4 degrees, etc. and negative measures. The random measures will not always allow you to completely rotate a figure back onto itself and the negative measures should be examined to discuss differences in how the figure rotates.

30 minutes Mystery Transformations Questions to consider:

1. Open the file MysteryTransformations.gsp. In this sketch, there are 4 tabs. On the first, there are seven pairs of pre-image/image points. On the remaining tabs, there are shapes. Drag the points to


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determine what transformation was used to create each image point. If a translation was used, then also describe the translation vector. If a rotation was applied, identify the center of rotation and measure the angle of rotation. If a reflection was used, then indicate where the line of reflection should be located.First tab: (Points)

o Translation: Points A and B; points G and Ko Rotation: Points E and F; Points H and Lo Reflection: Points C and D; Points I and M; Points J and No The translation vector of points A and B is AB; similarly, the

translation vector of G and K is GK.o The center of rotation will lie on the perpendicular bisector

line segment between the two points. Then, dragging the points together will identify the center of rotation; where the points coincide is the center of rotation. For points E and F, this happens to be the midpoint of EF. Thus, the angle of rotation for this set of points is 180 degrees. Finding the center of rotation for points H and L is a little trickier. First, construct HL and its perpendicular bisector. We know the center of rotation must lie on the perpendicular bisector, so to determine where, let the midpoint of HL be the center of a circle and construct the circle so that H and L are on the circle. Now, the center of rotation must be one of the points where the circle intersects the perpendicular bisector. By dragging H and L until they coincide, we can determine which is the center of rotation. Now, we can find the angle of rotation, which is 90 degrees.

o The line of reflection can be found by constructing the perpendicular bisector of the line segment joining the two points.

Second tab: (Triangles) This is a rotation.Third tab: (Quadrilaterals) This is a reflection.Fourth tab: (Pentagons) This is a translation.

2. Describe your strategies for identifying each of the mystery transformations. If the distance between the two points is always the same while dragging, then this transformation is a translation. If the points always move in opposite directions while dragging, this kind of transformation is a reflection. If the points move in the same direction while dragging, then this transformation is a rotation.

3. Which transformation was the most difficult to determine? Why? Typically, the most difficult transformations to determine are either rotations or reflections. The key to distinguishing between these two is to pay attention to the direction the points move.


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TransformationsParticipant Handout

TranslationsQuestions to consider:

Questions to consider:1. What is true about the size and shape of the two stick figures?2. Use the Arrow tool to drag a point on the leg of the original stick figure (pre-image). Try

dragging a point on the arm of this stick figure. What else changes when you perform this dragging? Why?

3. Measure the distance between the endpoint of the left leg of the original stick figure and the corresponding point on the left leg of the new stick figure. Measure several other distances for pre-image/image pairs of points. What do you notice? Explain.

4. Translate your stick figure to create an entire chorus line of dancers that are equidistance apart. Describe how you chose to do this.

5. A student in a high school class asks what the 5 and 0 represent (the values we typed into the polar coordinate dialog box to perform the translation). How do you respond?

*New sketch – translate by marked vector

Questions to consider:1. Translate the second stick figure to have three stick figures on the screen. What do you

notice about the size and shape of the three stick figures?2. How is the location of the stick figures related to the translation vector?3. Drag the line segment representing the translation vector so that the tail corresponds with

a point on the original stick figure. Where is the head of the vector located? Why?4. Use the segment tool to create a segment that joins a point from the original stick figure

to its corresponding image point. What do you notice about this segment and the vector?5. Measure several distances of pre-image/image points as you did on your previous sketch.

How do these distances relate to one another? How do they relate to the vector AB?6. What do you think will happen when you translate a line segment? What about a line?7. What do you think will happen when you translate an angle? 8. What do you think will happen when you translate two lines that are parallel to each

other?9. You have had the opportunity to perform several different translations. Based on your

experiences, describe a translation. Be sure to explain what stays the same and what


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changes.10. Students often have difficulty reasoning about vectors. Because representations of vectors

and rays are very similar, students confuse these two objects. Describe how you could assist students in understanding the differences between vectors and rays.

11. How does this introduction of translations in a dynamic environment, using dancing and stick figures, compare with how you teach translations? What are the benefits and drawbacks of this approach?

Reflections

Questions to consider:1. Explain how you could use the picture above to explain to students that a translation

would not be the appropriate transformation to use to describe the positions of the dancers.

2. What do you notice about the size and shape of the two stick figures?3. Drag a point on the leg of the original stick figure. What happens? Why? Drag other

points of the pre-image or image of the stick figure.4. Predict what will happen if you drag an endpoint of the mirror line segment (point A for

example). Drag point A. What happens? Why?5. Predict what will happen if you drag point B. Drag point B. What happens? Why?6. Create a segment joining the hands of the two stick figures as shown below. What do you

notice about this segment and the line of reflection?7. Based on your interactions with the sketch, provide a definition of a reflection.8. Describe at least three different properties of reflections.9. In order for students to develop an appropriate definition of reflection from this activity,

it is imperative they focus in on the important properties of reflection. How can a teacher focus students on the relevant properties and assist students in understanding which features of the diagram are not important to consider?

10. A fixed point for a transformation is when the point is mapped to itself. Did you encounter any fixed points while you were dragging points around in the sketch? Describe fixed points in terms of input and output.

11. Let’s consider fixed points from an algebraic context. Describe a linear function for which there are infinite fixed points. Describe one or more linear functions for which there are zero fixed points. Describe one or more linear functions that have exactly one


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fixed point.12. Sometimes young students are taught to think about reflections as flips. What properties

of reflection are highlighted by thinking about reflections as flips? What properties of reflection are not made as explicit when considering reflections as flips?

Rotations

Questions to consider:1. Consider the pre-image and image stick figures. Create a description of the relationship

between the two stick figures using a synchronized swimming scenario.2. What do you notice about the size and shape of the two stick figures?3. Drag point C and describe what happens to the two stick figures. Explain.4. Drag a point on one of the stick figures and describe what happens to the other stick

figure. Does it matter which stick figure you drag? Explain.5. If you constructed a circle with a center at point C that also passes through a point on the

original stick figure, what other point will the circle pass through? Use your DGE to test your hypothesis.

6. Describe at least three different properties of rotations.7. Are there any fixed points under a rotation?8. To perform a rotation, most DGEs require that you input a particular angle measure.

Describe how this design feature of the technology may influence student thinking about rotations.

9. Are there particular angles of rotation that would be more or less helpful to use with students? Explain.

Mystery TransformationsQuestions to consider:

1. Open the file MysteryTransformations.gsp. In this sketch, there are 4 tabs. On the first, there are seven pairs of pre-image/image points. On the remaining tabs, there are shapes. Drag the points to determine what transformation was used to create each image point. If a translation was used, then also describe the translation vector. If a rotation was applied, identify the center of rotation and measure the angle of rotation. If a reflection was used, then indicate where the line of reflection should be located.

2. Describe your strategies for identifying each of the mystery transformations.


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3. Which transformation was the most difficult to determine? Why?


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melt-institute- web viewuse transformations to assess whether two polygons are congruent....

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