Linda Griffith, Ph.D. and Belinda Robertson, M. S. E. University of Central Arkansas Conway, Arkansas
[email protected] [email protected] Lesson Title: Transformations! What is the Location? Subject: Mathematics – transformational geometry Grade: Grade 8 or High school geometry Common Core State Standards 8G.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. G-Co 2: Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g. translation versus horizontal stretch.. C-Co 3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. D-Co 4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. C-CO 5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
Description of Lesson: Time: four 50 minute class periods (more depending on the lever of the class). Materials: (For each student) Transformation handout, glue stick, miras,straw (coffee stirrer works best), paddy paper or tracing paper cut to the size of the coordinate grid, 10 copies of each of the following polygons in 4 different colors- rectangle ABCD, isosceles triangle EFG, trapezoid HIJK, isosceles right triangle LMN (for the class) tape, scissors, ruler, overhead transparencies of the handout and the triangles (optional). Preplanning: Prepare handout, send home figures to be cut out by the students, and gather other materials. Engage: Warm-up: Ask “What is a transformation?” Discuss the term and relate this term to things the students may be familiar with “Flips, Slides and Turns.” Define “Transformation.” Explore: Have the student glue down one of the ABCD rectangles and note this as the original or pre-image of the figure. Have one group glue theirs down in the 1st quadrant (at the same location), the second group glue theirs down in the 2 nd quadrant, the third group glue down in the 3rd quadrant, the fourth group in the 4th quadrant, the 5th group glue theirs down on the y-axis (where part of their figure is in the 1st and 2nd or the 3rd and 2
4th quadrants) and the 6th group glue theirs down where the figure is on either side of the x-axis. Have each group note their original locations. Note that the activity will be less difficult if the students make sure that the vertices of the figures are on coordinates that have whole numbers. They will continue to use this location for the original in each transformation. Have the students record the coordinates of the vertices in their Transformation handout.
Reflections: Have the students use miras to reflect the rectangle across the x-axis. When the students find the reflection, they need to glue a copy of the rectangle down and write down where the corresponding vertices are on the coordinate grid (A’, B’, C’ and D’). Ask several students to state an original coordinate and its corresponding reflection point. Make a list on the board for the class to see. Ask the class if they can see a pattern? If you give them a coordinate can they tell you the reflection coordinate? Finally ask then to give you a rule: For any point (a,b) the reflection of it over the xaxis is (a, -b). Read as “a opposite of b”. Have the students use miras to reflect the rectangle across the y-axis. When the students find the reflection, they need to glue a copy of the rectangle down and write down where the corresponding vertices are on the coordinate grid (A’’, B’’, C’’ and D’’). Ask several students to state an original coordinate and its corresponding reflection point. Make a list on the board for the class to see. Ask the class if they can see a pattern? If you give them a coordinate can they tell you the reflection coordinate? Finally ask then to give you a rule: For any point (a,b) the reflection of it over the yaxis is (-a, b). Read as “opposite of a, b”. Have the students use miras to reflect the rectangle across the line y = x. When the students find the reflection, they need to glue a copy of the rectangle down and write down where the corresponding vertices are on the coordinate grid (A’’’, B’’’, C’’’ and D’’’). Ask several students to state an original coordinate and its corresponding reflection point. Make a list on the board for the class to see. Ask the class if they can see a pattern? If you give them a coordinate can they tell you the reflection coordinate? Finally ask then to give you a rule: For any point (a,b) the reflection of it over the line y =x is (b, a).
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Translations: Using the same location of the original as they had in the reflection lesson, have the students glue down another rectangle and record its vertices’ coordinates. Have the students place a copy of the rectangle on top of the original and instruct them to slide the copy two units to the right. 1. glue the copy down in its new position (this is the image), 2. record the coordinate of the new figure. Indicate this is a horizontal translation, 3. Discuss and answer the questions that pertain to the horizontal translation. Make a list of students’ original vertices and the images’ vertices. 4. Ask the students, “If I give you a set of coordinates for a point, could you determine the coordinates of the image after the translation?” Give them a point and see if they can give you the coordinates of the image. (a, b) becomes (a+2, b) 5. Now ask them if you give them another point and a different horizontal translation (3 units to the left) could they give you the location of the image. (a, b) becomes (a+c, b) where c is the number of units moved and + indicate to the right or (a,b) becomes (a-c, b) where c is the number of units and – indicate to the left. Or (a,b) becomes (a+c, b) where c is the number of units moved horizontally. Do this same activity for the vertical translation. Fro example, choose 3 units down. (a,b) becomes (a,b+d) where d is the vertically translation Diagonal translation: State: “It is difficult to tell the length of a translation when we go diagonally across the grid. I want you to follow these directions.” 1. I want you to make a point at (2, -3) these are the combinations of the previous two translations. 2. Connect the origin to (2,-3) with a line segment. 3. Put one end of the coffee stirrer at the origin (0, 0). 4. Cross the original rectangle with the stirrer (keep the other end on the origin). 5. Put a copy of the rectangle over the stirrer and the original rectangle. 6. Tape the copy to the stirrer. 7. Slide the end of the stirrer along the line segment until it gets to the point (2,-3). 8. Tape the image to its new location and record the vertices coordinates. Collect original vertices and the corresponding image coordinates of the translation. (a, b) becomes (a+2, b+-3) In general: for a diagonal translation, (a,b) becomes (a+c, b+d) wher c is the horizontal move and d is the vertical move. 5
Rotation: Paddy paper and a brass brad 900 rotation counter clockwise / 2700 clockwise rotation Place a piece of paddy paper over the grid so that it covers the origin and the original rectangle ABCD. Remember to place your figures on the grid at the same location that you did on both the reflection and the translation. You should have the same original vertices points. 1. With your brass brad carefully punch through the paddy paper and the grid at the origin. 2. Tape down a copy of the rectangle over the original on the paddy paper. Make a mark on the paddy paper that matches up with the positive x-axis. This will be used to determine when rotations of 900 been completed. 3. Instruct the students to rotate the figure on the paddy paper 900 counter clockwise. They need to line up the line they made on the paddy paper to the positive y-axis. 4. Note where the vertices of the figure are now and glue down a copy of the figure on the coordinate grid. (They will have to carefully lift the paddy paper in order to glue down the copy of the figure.) 5. Have the students name the corresponding vertices of the new position of the rectangle A’B’C’D’. 6. Now have the students move the figure on the paddy paper so that it is back on the original figure and rotate the paddy paper 2700 clockwise (line up the line on the paddy paper with the positive y-axis. Note that the rectangle is in the same position as the 900 rotation. So a 900 rotation counter clockwise gives the same result as a 2700 clockwise rotation. 7. Record the coordinates of the image of the rectangle. 8. Ask several groups to report the coordinates of the vertices of their original positions and the corresponding vertices of their images after the rotation. Record these so the class can see. For the 900 counter clockwise and the 2700 clockwise rotation the students should say the opposite value of the y-coordinate becomes the x-coordinate and the value of the x-coordinate becomes the y-coordinate. (a, b) becomes (-b, a) 1800 rotation counter clockwise / 1800 clockwise rotation 1. Have the students move the paddy paper figure back on top of the original position. 2. Rotate the figure 1800 counter clockwise. 3. Follow directions 4-8 in the previous rotation but using the degrees for this rotation. The students should note that the x- and y-coordinates are now opposites of the original coordinates. (a, b) becomes (-a, -b). 6
2700 counter clockwise /900 clockwise rotation Complete the directions for the previous rotations but for 270 cc and 90 c. The result should be (a, b) becomes (-b, -a). Complete the questions in the handout ands discuss the outcome (patterns). Ask, “What do you think will happen if we used another figure?” Instruct the students to choose another figure, make a prediction of what the new coordinates will be, and perform the transformations.
Dilations: ruler, tape Enlargement Have the students tape a rectangle on the coordinate grid and record the coordinates of the vertices in their charts. Use a straight edge to connect the vertices to the rectangle to the origin. Have the students extend these segments so that the segments are 2 or 3 times as long as the original. Have the students connect the end of the line segments. They will have formed a new figure that is similar to the original. Have them label the vertices with corresponding letters. Have the students compare the length of the corresponding sides and record the scale factor of the dilation. Reduction Have the students draw rectangle ABCD on the coordinate grid and record the coordinates in their chart. Using a straight edge, have the students connect the vertices to the origin. Have the student find the midpoint of each line segment that connects the vertices to the origin and make a dot. The students should connect these dots in order. They will have created a rectangle that is similar to the original but the corresponding sides will be half the length of the original. Have them record the scale factor of the dilation. All sheets in the students’ books do not have to be completed. Some students will need more practice than others. Use your judgment when assigning the other figures. Supply some of your own figures is you want. If students can not remember what transformation results in what action, encourage them to choose a figure and complete the transformations that are remaining in their books. Explain: 7
Students will complete the handout and describe what happens to the coordinate of a figure after each transformation. Elaborate: The students will be able to give the coordinate for each transformation when given the original coordinates and the amount and direction of transformation without actually performing the transformation using a grid. Student Participation: Students will model the transformation for the class, discuss the outcomes, and make predictions about the coordinates based on their explorations. Evaluate: I have added an evaluation at the end of the students’ book. You may not want to put his in the students’ books. Instead you may want to give this to them after you have completed the unit or give parts of this after practicing each transformation.
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Combinations of Transformations: Students will explore various combinations of transformations. Students will start with a figure (preimage) on the plane in a position of their choice. They will perform and record a series of transformations on the figure that creates a new figure (image). You may want to limit the number of transformations to 2 or 3 until the students can identify the series correctly. Prepare several transformations with both preimage and image on the coordinate plane for the students to make predictions and have them check out their predictions using copies of the figure or georeflectors.
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Translation (Slide) Rotation (Turn) Reflection (Flip) Dilations (Stretch and Shrink) Name _____________________________ Class _________
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Figure
Rectangle
Original Coordinates (pre-image) A:
Reflection x-axis Coordinates (image) A’:
B:
B’:
C:
C’:
D:
D’:
(X, Y)
(X’, Y’)
Reflection over the x-axis What are the distances of the corresponding vertices of the figure (pre-image) and its image from the x-axis? A _______ A’ _______
B _______ B’ ________
C _______ C’ ______
D _______ D’ ________
How to the corresponding vertices relate (What is the rule?)
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Figure
Rectangle
Original Coordinates (pre-image) A:
Reflection y-axis Coordinates (image) A’:
B:
B’:
C:
C’:
D:
D’:
(X, Y)
(X’, Y’)
Reflection over the y-axis What are the distances of the corresponding parts of the figure and its image from the y-axis? A _______ A’ _______ B _______ B’ ________ C _______ C’ ______
D _______ D’ ________
How to the corresponding parts relate (What is the rule?)
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Figure
Rectangle
Original Coordinate (pre-image) A:
Reflection Line y= x Coordinates (image) A’:
B:
B’:
C:
C’:
D:
D’:
(X, Y)
(X’, Y’)
Reflection over the line y= x How do the distances from the line y = x of the corresponding parts of the figure and its reflections relate?
How do the corresponding parts relate (What is the rule?)?
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Figure
Rectangle
Original Coordinates (pre-image) A:
Reflection Line y= -x Coordinates (image) A’:
B:
B’:
C:
C’:
D:
D’:
(X, Y)
(X’, Y’)
Reflection over the line y= -x How do the distances from the line y = - x of the corresponding parts of the figure and its reflections relate?
How do the corresponding parts relate (What is the rule?)?
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Figure
Rectangle
Direction of translation
Original Coordinates (pre-image)
Horizontal Translation (image)
A:
A’:
B:
B’:
C:
C’:
D:
D’:
(X, Y)
(X’, Y’)
Distance of translation
Mathematical expression of translation
How do the coordinates of the original relate to the coordinates of the translations (What is the rule?)?
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Figure
Rectangle
Direction of translation
Original Coordinates Vertical Translation (pre-image) (image) A: A’: B:
B’:
C:
C’:
D:
D’:
(X, Y)
(X’, Y’)
Distance of translation
Mathematical expression of translation
How do the coordinates of the original relate to the coordinates of the translations (What is the rule?)?
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figure
Rectangle
Original Coordinates Diagonal Translation (pre-image) (image) A: A’: B:
B’:
C:
C’:
D:
D’:
(X, Y)
(X’, Y’)
Direction of translation
Distance of translation
Mathematical expression of translation
How do the coordinates of the original relate to the coordinates of the translations (What is the rule?)?
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Figure
Original Coordinates
Rotation Coordinates
(pre-image)
900cc/2700c
(image)
Rectangle
A:
A’:
B:
B’:
C:
C’:
D:
D’:
(X,Y)
(X’, Y’)
How do the corresponding parts of a 900 counter clockwise /2700clockwise rotation relate (what is the rule?)?
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Figure
Original Coordinates
Rotation Coordinates
(pre-image)
1800cc/1800c
(image)
Rectangle
A:
A’:
B:
B’:
C:
C’:
D:
D’:
(X,Y)
(X’, Y’)
How do the corresponding parts of an 1800 counter clockwise /1800clockwise rotation relate (what is the rule?)?
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Figure
Original Coordinates
Rotation Coordinates
(pre-image)
900cc/2700c
(image)
Rectangle
A:
A’:
B:
B’:
C:
C’:
D:
D’:
(X,Y)
(X’, Y’)
How do the corresponding parts of a 2700 counter clockwise /900clockwise rotation relate (what is the rule?)?
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Figure
Original Coordinates
Dilation (enlargement)
(pre-image)
Rectangle
A:
Scale factor ( (image) A’:
B:
B’:
C:
C’:
D:
D’:
(X,Y)
(X’, Y’)
)
How do the coordinates of the original figure (pre-image) relate to the dilation and its scale factor? (What is the rule?)
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Figure
Original Coordinates (pre-image)
Rectangle
Dilation (reduction)
A:
Scale factor ( (image) A’:
B:
B’:
C:
C’:
D:
D’:
(X,Y)
(X’, Y’)
)
How do the coordinates of the original figure (pre-image) relate to the dilation (image) and its scale factor? (What is the rule?)
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Figure
Original Coordinates (pre-image)
Reflection x-axis Coordinates (image)
(X’, Y’)
(X, Y)
Reflection over the x-axis What are the distances of the corresponding vertices of the figure (pre-image) and its image from the x-axis? __ _______
__’ _______
__ _______ __’ ______
__ _______
__’ ________
__ _______ __’ ________
How to the corresponding vertices relate (What is the rule?)
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Figure
Original Coordinates (pre-image)
Reflection y-axis Coordinates (image)
(X’, Y’)
(X, Y)
Reflection over the y-axis What are the distances of the corresponding parts of the figure and its image from the y-axis? __ _______ __’ _______ __ _______ __’ ________ __ _______ __ ’ ______
__ _______
__’ ________
How to the corresponding parts relate (What is the rule?)
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Figure
Original Coordinate (pre-image)
Reflection Line y= x Coordinates (image)
(X, Y)
(X’, Y’)
Reflection over the line y= x How do the distances from the line y = x of the corresponding parts of the figure and its reflections relate?
How do the corresponding parts relate (What is the rule?)?
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Figure
Original Coordinates (pre-image)
Reflection Line y= -x Coordinates (image)
(X’, Y’)
(X, Y)
Reflection over the line y= -x How do the distances from the line y = - x of the corresponding parts of the figure and its reflections relate?
How do the corresponding parts relate (What is the rule?)?
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Figure
Original Coordinates (pre-image)
(X’, Y’)
(X, Y)
Direction of translation
Horizontal Translation (image)
Distance of translation
Mathematical expression of translation
How do the coordinates of the original relate to the coordinates of the translations (What is the rule?)?
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Figure
Original Coordinates (pre-image)
Vertical Translation (image)
(X, Y)
(X’, Y’)
Direction of translation
Distance of translation
Mathematical expression of translation
How do the coordinates of the original relate to the coordinates of the translations (What is the rule?)?
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figure
Original Coordinates Diagonal Translation (pre-image) (image)
:
:
(X, Y)
(X’, Y’)
Direction of translation
Distance of translation
Mathematical expression of translation
How do the coordinates of the original relate to the coordinates of the translations (What is the rule?)?
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Figure
Original Coordinates
Rotation Coordinates
(pre-image)
900cc/2700c
(image)
(X’, Y’)
(X,Y)
How do the corresponding parts of a 900 counter clockwise /2700clockwise rotation relate (what is the rule?)?
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Figure
Original Coordinates
Rotation Coordinates
(pre-image)
1800cc/1800c
(image)
(X’, Y’)
(X,Y)
How do the corresponding parts of an 1800 counter clockwise /1800clockwise rotation relate (what is the rule?)?
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Figure
Original Coordinates
Rotation Coordinates
(pre-image)
900cc/2700c
(image)
(X’, Y’)
(X,Y)
How do the corresponding parts of a 2700 counter clockwise /900clockwise rotation relate (what is the rule?)?
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Figure
Original Coordinates
Dilation (enlargement)
(pre-image) Scale factor ( (image)
)
(X’, Y’)
(X,Y)
How do the coordinates of the original figure (pre-image) relate to the dilation and its scale factor? (What is the rule?)
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Figure
Original Coordinates (pre-image)
Dilation (reduction) Scale factor ( (image)
)
(X’, Y’)
(X,Y)
How do the coordinates of the original figure (pre-image) relate to the dilation (image) and its scale factor? (What is the rule?)
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Choose a figure (preimage) and a location for the figure on the coordinate grid. Record the locations of the vertices (or key points) of the figure. Perform and record a combination of 2 to 3 transformation to create an image. Record the corresponding points for the image. Are there other combinations that get the same image from the same preimage? If so record these. Cover up your list of transformations and see if your partner can perform the needed transformations to result in the same image.
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Choose a figure (preimage) and a location for the figure on the coordinate grid. Record the locations of the vertices (or key points) of the figure. Perform and record a combination of 3 to 4 transformation to create an image. Record the corresponding points for the image. Are there other combinations that get the same image from the same preimage? If so record these. Cover up your list of transformations and see if your partner can perform the needed transformations to result in the same image.
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Choose a figure (preimage) and a location for the figure on the coordinate grid. Record the locations of the vertices (or key points) of the figure. Perform and record a combination of transformation to create an image. Record the corresponding points for the image. Are there other combinations that get the same image from the same preimage? If so record these. Cover up your list of transformations and see if your partner can perform the needed transformations to result in the same image.
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Name ______________________________________ The coordinate of WXYZ are listed below. Without performing the actual transformations, state the coordinates for the images’ vertices.
Figure
Original Coord. (pre-image)
Horizontal Translation 4 units down
Vertical Translation 2 units left
Translation of 2 units up and 4 units right Coordinates
Rectangle ,,,Figure
W: (4, 5)
Coordinates W’:
Coordinates W’’:
W”’:
X: (8, 5)
X’:
X’’:
X”’:
Y: (8, 2)
Y’:
Y’’:
Y’”:
Z: (4, 2)
Z’:
Z’’:
Z”’:
Original Rotation Coord. 900 (pre-image) Counterclockwise
Rotation 180 Clockwise
Rotation 90 Clockwise
0
0
Rectangle
W: (4, 5)
Coordinates W””:
Coordinates W:
W:
X: (8, 5)
X””:
X5:
X6:
Y: (8, 2)
Y””:
Y5:
Y6:
Z: (4, 2)
Z””:
Z5:
Z6:
5
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Coordinates 6
Figure
Original Coord. (pre-image)
Reflection x-axis
Reflection y-axis
Reflection
Reflection
y=x
y= -x
Rectangle
W: (4, 5)
Coordinates W 7:
Coordinates W8:
Coordinates W9:
Coordinates W10:
X: (8, 5)
X7:
X8:
X9:
X10:
Y: (8, 2)
Y7:
Y8:
Y9:
Y10:
Z: (4, 2)
Z7:
Z8:
Z9:
Z10:
Figure
Original Coord. (pre-image)
Rectangle
Dilation
Dilation
Scale factor
Scale factor
3
¼
W: (4, 5)
Coordinates W11:
Coordinates W12:
X: (8, 5)
X11:
X12:
Y: (8, 2)
Y11:
Y12:
Z: (4, 2)
Z11:
Z12:
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