Formatting Blackline Masters
|Word/Phrase |+ |( |– |Definition/Rule |Example |
|transformation | | | | | |
|pre-image | | | | | |
|image | | | | | |
|rigid transformation | | | | | |
|(rigid motion) | | | | | |
|non-rigid transformation | | | | | |
|(non-rigid motion) | | | | | |
|orientation | | | | | |
|isometry | | | | | |
|reflection | | | | | |
|line of reflection | | | | | |
|translation | | | | | |
|rotation | | | | | |
|center of rotation | | | | | |
|degree of rotation | | | | | |
|clockwise | | | | | |
|counterclockwise | | | | | |
|dilation | | | | | |
|center of dilation | | | | | |
|scale factor | | | | | |
|similarity transformation | | | | | |
|composite transformation | | | | | |
|glide reflection | | | | | |
Procedure:
1. Examine the list of words/phrases in the first column.
2. Put a + next to each word/phrase you know well and for which you can write an accurate example and definition. Your definition and example must relate to this unit of study.
3. Place a ( next to any words/phrases for which you can write either a definition or an example, but not both.
4. Put a – next to words/phrases that are new to you.
This chart will be used throughout the unit. As your understanding of the concepts listed changes, you will revise the chart. By the end of the unit, you should have all plus signs. Because you will be revising this chart, write in pencil.
|Word/Phrase |+ |( |– |Definition/Rule |Example |
|transformation | | | |A correspondence between two sets of points | |
| | | | |such that each point in the pre-image has a | |
| | | | |unique image and that each point in the image | |
| | | | |has exactly one pre-image; a change in size, | |
| | | | |orientation, or position of a figure in space. | |
|pre-image | | | |The original object that is to be transformed. | |
|image | | | |The “copy” of the object that has been | |
| | | | |transformed. | |
|rigid transformation | | | |A transformation that preserves measurements of| |
|(rigid motion) | | | |segments and angles; also called an isometry | |
| | | | |(see below). | |
|non-rigid transformation | | | |A transformation that does not preserve | |
|(non-rigid motion) | | | |measures of segments and angles; the shape of | |
| | | | |the pre-image may not be preserved either. | |
|orientation | | | |The location (position and angle) of an object | |
| | | | |in space in relation to a set of reference | |
| | | | |axes. | |
|isometry | | | |A transformation that preserves measurements | |
| | | | |and more specifically distances between points;| |
| | | | |a transformation that preserves distances is | |
| | | | |also bound to preserve angle measures; a | |
| | | | |congruence transformation. | |
|reflection | | | |A transformation in which each point in the | |
| | | | |pre-image has an image that is the same | |
| | | | |distance from the line of reflection (see | |
| | | | |below). For a point on the line of reflection, | |
| | | | |the image is itself; aka “flip.” | |
|line of reflection | | | |The perpendicular bisector of the segment | |
| | | | |joining each point (pre-image) and its image. | |
|translation | | | |A transformation which moves an object a fixed | |
| | | | |distance in a fixed direction; a composite of | |
| | | | |two reflections over parallel lines; aka slide.| |
|rotation | | | |A transformation that turns a figure about a | |
| | | | |fixed point called the center of rotation; a | |
| | | | |composite of two reflections over intersecting | |
| | | | |lines; aka “turn.” | |
|center of rotation | | | |A fixed point about which a figure is rotated; | |
| | | | |the point where two intersecting lines of | |
| | | | |rotation meet?? | |
|angle (degree) of rotation | | | |Rays drawn from the center of rotation to a | |
| | | | |point on the pre-image and its image form the | |
| | | | |angle of rotation (measured in degrees). | |
|clockwise | | | |Rotation of an object to the right indicated by| |
| | | | |a negative angle of rotation. | |
|counterclockwise | | | |Rotation of an object to the left indicated by | |
| | | | |a positive angle of rotation. | |
|dilation | | | |A transformation that produces an image that is| |
| | | | |the same shape as the pre-image but is a | |
| | | | |different size; a stretch or shrink of the | |
| | | | |pre-image. | |
|center of dilation | | | |A fixed point in the plane about which all | |
| | | | |points are expanded (stretched) or contracted | |
| | | | |(shrunk). | |
|scale factor | | | |The ratio by which an object is enlarged or | |
| | | | |reduced; if greater than 1 the image is an | |
| | | | |enlargement; if between 0 and 1 the dilation is| |
| | | | |a reduction; if the scale factor equals 1, the | |
| | | | |figures are congruent. | |
|similarity transformation | | | |A transformation that is the composite of | |
| | | | |dilations and/or reflections; a non-rigid | |
| | | | |transformation; the shape of the pre-image is | |
| | | | |preserved but the size is changed. | |
|composite transformation | | | |The result of two or more successive | |
| | | | |transformations. | |
|glide reflection | | | |A type of composite transformation where a | |
| | | | |figure is reflected then translated. | |
Procedure:
1. Examine the list of words/phrases in the first column.
2. Put a + next to each word/phrase you know well and for which you can write an accurate example and definition. Your definition and example must relate to this unit of study.
3. Place a ( next to any words/phrases for which you can write either a definition or an example, but not both.
4. Put a – next to words/phrases that are new to you.
This chart will be used throughout the unit. As your understanding of the concepts listed changes, you will revise the chart. By the end of the unit, you should have all plus signs. Because you will be revising this chart, write in pencil.
Trace the following polygons on the patty paper or tracing paper given to you.
What Are Transformations?
When learning about transformations, one might first look at the parts of the word. Transformation can be separated into the prefix trans- and the word formation. The prefix trans- means “changing thoroughly” and formation means “the act of giving or taking form, shape, or existence.” Taken together, a transformation can be described as “the act of changing a form or shape.” Specifically, in geometry, a transformation may change the position, orientation, or size of a figure in the plane.
Look at the transformations below. The figure drawn with the solid lines is called the pre-image, or the original object that is being transformed. The figure drawn with dashed lines is called the image, or the “copy” of the object that has been transformed.
pre-image image
image pre-image
Figure 1
Figure 2
The most basic transformation is a reflection. A reflection can be easily described as a “flip”, however, that is not the most accurate definition. A reflection is a transformation in which each point in the pre-image has an image that is the same distance from the line of reflection. The line of reflection is the perpendicular bisector of the segment joining each point on the pre-image with its corresponding point on the image. Look at the example below. The image is labeled A’B’C’D’ (read A prime, B prime, C prime, D prime—the use of the apostrophe on the letter is universally accepted to show that a figure is the image of a transformation).
Figure 3
Line m is the line of reflection in the figure and P is the point where line m intersects the segment joining A and A’. Using a ruler, measure the distance from A to A’. Now, measure the distance from A to P and the distance from P to A’. You should notice that AP and PA’ are equal. Use a protractor to measure the angles formed at P. You should see that the angles all measure 90 degrees. How do those measurements relate to the definition of the line of reflection given earlier?
Another basic transformation is a translation, or “slide.” When a translation is performed, the pre-image is moved a fixed distance in a fixed direction. The directions for performing a translation could state to move the pre-image 5 inches to the right in which each point on the pre-image is moved 5 inches to the right of the location to form the image. Translations can also be thought of as the composition of two reflections over parallel lines. Look at the example below.
pre-image image
Figure 4
Notice there are two reflections. Lines m and n are parallel. The resulting image has the same orientation as the pre-image, but has been moved to the right by 10 cm. A question to think about: does the distance of the translation have any relationship to the distance between the parallel lines?
A third transformation is called a rotation. This transformation may also be referred to as a turn. A rotation is a transformation that turns a figure about a fixed point, called the center of rotation, through a fixed angle of rotation (measured in degrees). The path the figure follows during the rotation would form a circle around the center of rotation if the figure were rotated 360 degrees. A rotation can be performed with any degree measure and can be considered a clockwise rotation or a counterclockwise rotation. A clockwise rotation will turn a figure to the right around the center of rotation while a counterclockwise rotation will turn a figure to the left around the center of rotation. All positive degree measures are assumed to indicate a counterclockwise rotation, while all negative degree measures are assumed to indicate a clockwise rotation.
Figure 5
In Figure 5 above, X is the center of rotation. The angle of rotation is formed by drawing a segment from one point on the pre-image to the center of rotation then drawing the required angle using the center of rotation (X) as the vertex. The angle used in this figure is 90° clockwise, or -90°. Question to think about: What would happen to the image if the center of rotation was moved but the angle of rotation remained the same?
A rotation can also be defined as a composite of two or more reflections over intersecting lines. Consider the example below.
Figure 6
The intersection of lines m and n becomes the center of rotation. These lines happen to be perpendicular. Notice how the image has been rotated in a counterclockwise direction around the center (point of intersection). How could you determine what the angle of rotation is for this diagram? Go back to the definition of rotation discussed earlier for some ideas.
The three transformations discussed so far have one thing in common. If you look at all of the images and compare them to their corresponding pre-images, you will notice that the measures of the segments and angles have not changed (go ahead—measure them if you wish!). Since the images have the same shape and are the same size as the pre-images, they are congruent. Each of these transformations is called an isometry. An isometry is a transformation that preserves measurements of segments and angles and therefore produces an image congruent to its pre-image. A transformation that is an isometry is also sometimes referred to as a rigid motion or rigid transformation.
In geometry, there is one more important transformation. A dilation is a transformation that produces an image that is the same shape as the pre-image but is a different size. Sometimes they are referred to as a stretch or shrink (also called an enlargement or reduction). Each dilation is focused at the center of dilation, or a fixed point about which all points are enlarged or reduced. How much the figure is enlarged or reduced depends upon the scale factor, the ratio by which an object is enlarged or reduced. If the scale factor is greater than 1, the image is an enlargement of the pre-image. If the scale factor is between 0 and 1, the image is a reduction of the pre-image.
Figure 7
In Figure 7 above, the center of dilation is A. The measure of segment AB’ is 2 times the measure of segment AB. Therefore, the image A’B’C’D’E’ is an enlargement of the pre-image ABCDE, and the scale factor is 2. Notice, the measures of the corresponding segments are not equal, however the measures of the corresponding angles are (you can verify this by using your protractor). Therefore, dilation is not an isometry. Dilation is a non-rigid transformation, or a non-rigid motion. Because the corresponding angles have the same measure and the corresponding sides are proportional, these figures are similar which means dilation is a similarity transformation.
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D
C
A
B
A’
B’
C’
D’
m
P
m
n
X
m
n
pre-image
image
B’
B
C’
C
A
E’
E
D’
D
................
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