Mathematics Grade 8 – Unit 1 (SAMPLE)

Transcription

Mathematics
Grade 8 – Unit 1 (SAMPLE)
Possible time frame:
20 days
In this unit students will explore translations, reflections, and rotations. This is the first time students have been introduced to transformations. This unit should begin by
transforming points to help develop the properties of each transformation. Once students have developed an understanding of each transformation and some of their
properties, begin verifying the properties through experiments. Through experiments students should solidify their understanding of each transformation and fully develop the
properties of each. Design experiments with the initial and transformed geometric item along with a particular transformation (or series of transformations) that was
performed. Students will then determine if the proposed transformation (or series of transformations) actually works and possibly alter the proposed transformation (or series
of transformations) to one that does carry the initial item onto the transformed one. Students will then apply their understandings of the transformations to two-dimensional
figures. The criteria for figure congruence will be established through rigid motion and applied to help students create a sequence of transformations that would carry one figure
onto another. You may consider beginning transformations of figures on a blank grid and then extend to the coordinate plane. Students will begin to formalize the effects of
translations, reflections, and rotations in terms of coordinates.
Major Cluster Standards
Standards Clarification
Understand congruence and similarity using physical models, transparencies, or geometry software.
8.G.A.1 The skill of transforming geometric items as
8.G.A.1 Verify experimentally the properties of rotations, reflections, and translations:
well as the properties of transforming these items will
a. Lines are taken to lines, and line segments to line segments of the same length.
extend to develop and establish the criteria for figure
b. Angles are taken to angles of the same measure.
similarity in Unit 3. Standard 8.G.A.1 does not include
c. Parallel lines are taken to parallel lines.
the transformation of figures.
Unit 1: Transformations and Congruence
Understand congruence and similarity using physical models, transparencies, or geometry software.
8.G.A.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.
8.G.A.2 This same skill is used in Unit 3 (8.G.A.4) with
the addition of dilations.
8.G.A.3 Students are still responsible for knowing the
properties of translations, reflections, and rotations in
Unit 3. Dilations are not included in Unit 1. They are
addressed in Unit 3.
8.G.A.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using
coordinates.
Focus Standards of Mathematical Practice
MP.3 Construct viable arguments and
critique the reasoning of others.
As students investigate transformations, they attend to precision (MP.6) as they use appropriate terminology to describe and verify
MP.5 Use appropriate tools
the properties of the various transformations. They also select and use tools such as geometry software, coordinate planes, and
strategically.
tracing strategically (MP.5). Students construct viable arguments and critique the reasoning of others (MP.3) as they describe the
effect of transformations. As students investigate those effects, they attend to structure (MP.7) by recognizing the common attributes
MP.6 Attend to precision.
and properties generated by the transformations.
MP.7 Look for and make use of
structure.
Review the Grade 8 sample year-long scope and sequence associated with this unit plan.
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Mathematics
What will students know and be able to do by the end of this unit?
Students will demonstrate an understanding of the unit focus and meet the expectations of the Common Core State Standards on the unit assessments.
Standards
Unit Assessment
Objectives and
Formative Tasks
The major clusters for this
unit include:
Students will demonstrate
mastery of the content
through assessment items
and tasks requiring:
Objectives and tasks
aligned to the CCSS prepare
students to meet the
expectations of the unit
assessments.
•
•
Understand congruence
and similarity using
physical models,
transparencies, or
geometry software.
Understand congruence
and similarity using
physical models,
transparencies, or
geometry software.
•
•
•
•
Conceptual
Understanding
Procedural Skill and
Fluency
Application
Math Practices
Concepts and Skills
Each objective is broken
down into the key concepts
and skills students should
learn in order to master
objectives.
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Mathematics
Sample End-of-unit Assessment Items:
1)
Tammy applies the translation (x – 3, y + 2) to a line segment. What do we know is true about
the resulting line segment?
2) A 130˚ angle is rotated 180 degrees clockwise. What is the measure of the angle image?
A. 180˚
B. 130˚
C. 50˚
D. 310˚
3) Maria plots the points X, Y, and Z to create ∆𝑋𝑌𝑍. Translate ∆𝑋𝑌𝑍 right 5 units and down 2
units. Label 𝑋 → 𝑋 ′ , 𝑌 → 𝑌 ′ , 𝑎𝑛𝑑 𝑍 → 𝑍 ′ .
Y
X
Z
4) Joseph graphs a set of parallel line segments below. Describe the final image of the line
segments after a reflection over the y-axis and a translation(𝑥, 𝑦) → (𝑥 + 2, 𝑦 − 1).
A.
B.
C.
D.
E.
The lines intersect at (2, -1).
The lines intersect at (-2, 1).
The lines do not intersect and are the same distance apart as the original set of lines.
The lines do not intersect and are further apart than the original set of lines.
The lines do not intersect and are closer together than the original set of lines.
5) Brooke transforms a right angle using rotations, reflections, and translations. Is it possible that
Brooke’s final image is an obtuse angle? Explain.
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6)
What happens to a vertical line if you rotate it 90 degrees clockwise?
A. The image line is 90 times as long.
B. The image is a vertical line.
C. The image is a line 90 inches long.
D. The image is a horizontal line.
7)
Which best describes the transformation from original to image, and tells whether the two
figures are similar or congruent?
A.
B.
C.
D.
8)
Translation, similar
Rotation, congruent
Reflection, similar
Reflection, congruent
Plot the points T(0, 0), U(2, 2), and V(5, 0) to create ∆𝑇𝑈𝑉, then rotate the triangle 90˚
counterclockwise about the origin.
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9)
A pentagon has a perimeter of 40 cm. What is the perimeter of the pentagon after a translation
of (𝑥, 𝑦) → (𝑥 − 7, 𝑦 + 5)and a rotation of 90˚clockwise about the origin is applied? How do
you know?
10) Triangle ABC has the vertices A(2,0), B(4, 2), and C(3, 4). Name the ordered pair of C’ after a
reflection across the x-axis.
11) Suppose the streets in a neighborhood follow a coordinate grid. A car is at a stop sign at the
location (6, 3). If the car must travel north 3 blocks and west 2 blocks to reach its destination,
what is the ordered pair representing the destination?
A.
(8, 6)
B. (4, 6)
C. (9, 1)
D. (9, 5)
12) Give a sequence of translations, reflections, and/or reflections in which ∆𝐴𝐵𝐶 → ∆𝐸𝐷𝐶 .
13) In number 12 above, ∆𝐴𝐵𝐶 is mapped onto ∆𝐸𝐷𝐶. As a result, what do we know is true about
the relationship between ∆𝐴𝐵𝐶 and ∆𝐸𝐷𝐶?
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14) Give a sequence of translations, reflections, or rotations that will transform figure A to figure B.
A
B
15) Select all transformations from the list below that could transform the original to the image in
the coordinate grid below?
Original
Image
A.
B.
C.
D.
E.
F.
G.
Rotate 180˚ about the origin.
Rotate 90˚ clockwise about the origin, translate down 6 units
Rotate 90˚ counterclockwise about the origin, translate down 6 units
Rotate 90˚counterclockwise about the origin, translate down 1 unit and right 6 units
Rotate 90˚counterclockwise about the origin, translate down 6 units and left 1 unit
Rotate 90˚ counterclockwise about the origin, translate down 1 unit, reflect over the y-axis
Rotate 90˚ clockwise about the origin, translate down 1 unit, reflect over the y-axis
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Sample End of Unit Assessment Task:
Complete all parts. You may use a ruler and/or protractor to complete the task.
X
Y
a.
Rotate
Z
XYZ 90˚counter clockwise about the origin. Draw and label the image,
X’Y’Z’.
b. Name the ordered pairs for: X’ ____________
Y’ ____________
Z’ ____________
c. Reflect
X’Y’Z’ over the y-axis. Draw and label
X”Y”Z”.
d. What is the m XYZ? _______________ What is the m X”Y”Z”?_______________
How do the measures of these angles compare?
Explain why this is reasonable for an angle that is rotated and reflected.
e.
Sarah thinks that she can produce the X”Y”Z” from XYZ with one transformation, a 180˚
rotation about the origin. Is she correct? Explain why or why not.
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Triangles ABC and PQR are shown below in the coordinate plane.
a. Show that ∆𝐴𝐵𝐶 is congruent to ∆𝑃𝑄𝑅 with a reflection followed by a translation.
b.
If you reverse the order of your reflection and translation in part (a) does it still map ∆𝐴𝐵𝐶 to
∆𝑃𝑄𝑅?
c. Find a second way different from your work in parts (a) or (b), to map ∆𝐴𝐵𝐶 to ∆𝑃𝑄𝑅 using
translations, rotations, and/or reflections.
(Source: http://www.illustrativemathematics.org/illustrations/1232 )
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Mathematics
Sample End-of-unit Assessment Item Responses:
1)
Standard 8.G.A.1a
The resulting image is a line segment of the same length.
2)
Standard 8.G.A.1b
Answer choice B. 130˚.
3) Standard 8.G.A.1a and 8.G.A.3
Y’
X’
Z’
4) Standard 8.G.A.1c and 8.G.A.3
Answer choice C. The lines do not intersect and are the same distance apart as the original set of
lines.
5) Standard 8.G.A.1b
No, Brooke’s final image will be a right angle because rotations, reflections, and translations are
congruency transformations and produce congruent angles.
6) Standard 8.G.A.1a
Answer choice D
7) Standard 8.G.A.2
Answer choice D. Reflection, congruent.
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Mathematics
8)
Standard 8.G.A.3
9) Standard 8.G.A.2
The perimeter of the hexagon after the transformations is 40 cm because a translation and a
rotation produce images congruent to the original. Since the hexagon image will be congruent, the
perimeter of the image will be the same as the perimeter of the original.
10) Standard 8.G.A.3
C’ (3, -4)
Answer choice B. (4, 6).
All of the following are correct:
• 180˚rotation about point C
• Reflection of ∆𝐴𝐵𝐶 over y-axis, then a reflection over 𝐶𝐷.
• Reflection of ∆𝐴𝐵𝐶 over 𝐶𝐵, then a reflection over the y-axis.
∆𝐴𝐵𝐶 ≅ 𝐸𝐷𝐶
Rotate 90˚ clockwise, reflect over the x-axis, and translate down 1 unit.
Answer choices:
B. Rotate 90˚ clockwise about the origin, translate down 6 units
D. Rotate 90˚counterclockwise about the origin, translate down 1 unit and left 6 units
F. Rotate 90˚ counterclockwise about the origin, translate down 1 unit, reflect over the y-axis
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Mathematics
Sample End of Unit Assessment Task Response:
Complete all parts. You may use a ruler and/or protractor to complete the task.
X
Y
Z
X’
X”
Z’
Z”
Y’
a.
Rotate
point)
Y”
XYZ 90˚counter clockwise about the origin. Draw and label the image,
b. Name the ordered pairs for: X’ __(-4, -2)______
Y’ __(-1, -5)______
Z’ __(-1, -3)______
c. Reflect
X’Y’Z’ over the y-axis. Draw and label
X’Y’Z’. (1
(1point for all correct)
X”Y”Z”. (1 point)
d. What is the m XYZ? _____45˚_______ What is the m X”Y”Z”?______ 45˚ ____
(1 point for both)
How do the measures of these angles compare? The angles are congruent. (1 pt.)
Explain why this is reasonable for an angle that is rotated and reflected. Rotations and
reflections produce congruent transformations, so the angles should be congruent after
transformation. (2 points)
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Mathematics
e.
Sarah thinks that she can produce the X”Y”Z” from XYZ with one transformation, a 180˚
rotation about the origin. Is she correct? Explain why or why not.
No, the coordinates for X”Y”Z” are X”(4, -2), Y”(1, -5), and Z”(1, -3), but the coordinates
of the angle if it is rotated 180˚ are X(2, -4), Y(5, -1), and Z (3, -1). (2 points)
Or
In order to produce 180˚ rotation, two reflections are necessary.
Or
Any other student responses that are accurate and demonstrate strong mathematical
reasoning.
a. Show that ∆𝑨𝑩𝑪 is congruent to ∆𝑷𝑸𝑹 with a reflection followed by a translation.
Reflect across the y-axis, then translate down 6 units.
There are other possible
combinations of
reflections and
translations, but this is
the one the students will
most likely choose.
However, other solutions
will need to be checked
for accuracy.
b.
If you reverse the order of your reflection and translation in part (a) does it still map ∆𝑨𝑩𝑪 to
∆𝑷𝑸𝑹?
This answer is dependent on the answer in part (a). Using the reflection and translation
chosen above, the answer would be yes. ∆𝐴𝐵𝐶 translated down 6 units and then
reflected over the y-axis, would result in ∆𝑃𝑄𝑅. See below.
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Mathematics
c.
Find a second way different from your work in part (a), to map ∆𝑨𝑩𝑪 to ∆𝑷𝑸𝑹 using
translations, rotations, and/or reflections.
There are several possible answers. Below are a few samples.
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Mathematics
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Possible Pacing and Sequence of Standards
Content and Practice Standards
Understand congruence and similarity using
physical models, transparencies, or geometry
software.
8.G.A.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.
8.G.A.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.
Possible Pacing and Sequence
Days 1 – 5
Objectives:
Students will understand and perform the rigid motions of translations, reflections, and rotations of
points and simple figures.
Students will determine the coordinates of figures transformed on the coordinate plane using the rigid
motions of translations, reflections, and rotations.
Students will apply their understandings of translations, reflections, and rotations on simple figures by
predicting the coordinates of the new image given the coordinates of the original image.
Concepts and Skills:
• Perform a translation on a blank grid and a coordinate grid.
• Perform a reflection on a blank grid and a coordinate grid.
• Perform a rotation about a designated point on a blank grid and on a coordinate grid.
Sample Tasks:
Have students perform a translation, reflection, and rotation with their names.
8.G.A.3 Describe the effect of dilations, translations,
a. Have students write their initials on a blank grid. Then have them translate the initials 8 units to
rotations, and reflections on two-dimensional figures
the right and 5 units down. (You could have different translation directions written on notecards
using coordinates.
and give every student a different translation. Afterwards they could share with a classmate (MP
3)).
b. Have students fold a blank grid vertically. Have them write their entire first name and reflect it
Possible Connections to Standards for
across the vertical line.
Mathematical Practices
c. Have students write their first and last initial in block letters on a coordinate grid. Instruct
students to pick a point to be the point of rotation. Then have them rotate the initials 90˚
MP.3 Construct viable arguments and critique the
clockwise and then 180˚counterclockwise.
reasoning of others. Students should have the
opportunity to share a series of transformations to
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Mathematics
prove or disprove that two figures are congruent.
Classmates should be instructed to give feedback.
MP.5 Use appropriate tools strategically.
Students will need to use ruler and protractors as
necessary to compare and measure lines and angles.
MP.6 Attend to precision.
The rigid motion of the transformations requires
that students be very precise in the transformations
and the naming of ordered pairs.
Days 6 – 9
Objectives:
Students will verify experimentally that lines are taken to lines of the same length, angles are taken to
angles of the same measure, and parallel lines are taken to parallel lines.
• Verify that translations, reflections, and rotations produce images of the same length by
measuring and comparing lengths of original line segments to images of the line segment.
• Verify that translations, reflections, and rotations produce images of angles with the same angle
measure by measuring and comparing angle measures of original angles to images of the angles.
• Verify that translations, reflections, and rotations produce images of parallel lines the same
distance apart as the original lines by measuring and comparing distances between parallel line
segments to images of the parallel lines.
MP.7 Look for and make use of structure.
At some point in this unit, students need to list the xand y- coordinates of a figure before and after a
Sample Tasks:
reflection and describe the patterns seen. The same
can be done for translations and rotations.
1. Measure and record the length the line segment.
2. Reflect the line segment over the y-axis. Measure and record the length the image.
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3. Rotate the line segment 180˚ clockwise about the origin. Measure and record the length the
image.
4. Translate the line segment to the left 5 units and down 3 units. Measure and record the length
the image.
5. What can you conclude about line segments that are reflected, rotated, and/or translated?
1. Measure and record the measure of the angle.
2. Reflect the angle over the x-axis. Measure and record the angle measure of the image.
3. Rotate the angle 90˚ clockwise about the origin. Measure and record the angle measure of the
image.
4. Translate the line segment to the right 3 units and down 4 units. Measure and record the angle
measure of the image.
5. What can you conclude about angles that are reflected, rotated, and/or translated?
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1. Measure and record the distance between the parallel lines.
2. Reflect the parallel lines over the y-axis. Measure and record the distance between the parallel
lines.
3. Rotate the parallel lines 90˚ counterclockwise about the origin. Measure and record the distance
between the parallel lines.
4. Translate the parallel lines to the right 2 units and up 1 unit. Measure and record the distance
between the parallel lines.
5. What can you conclude about parallel lines that are reflected, rotated, and/or translated?
Days 10 – 14
Objectives:
Students will perform series of transformations to explain why two figures are congruent.
• Explain how transformations can be used to prove that two figures are congruent.
• Perform a series of transformations (reflections, rotations, and/or translations) to prove or disprove
that two figures are congruent.
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Mathematics
Sample Task:
1. See Illustrative Mathematics for a sample tasks which could be used during instruction.
2.
Have students play Shape Mods
http://www.mathplayground.com/ShapeMods/ShapeMods.html an online transformation game.
Days 15 - 17
Objective:
Students will study a real-world example of the use of transformations and apply their understandings of
transformations to create their own game.
Application Task Description:
Students are asked to create a game using transformations (rotations, reflections, and translations) and
series of transformations. Students must use their understandings of each transformation and the
properties of congruence in its creation. This is a real-world extension of the study of congruent
transformations.
Days 18 - 20: End of Unit Assessment
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Application Task:
Source for picture: http://en.wikipedia.org/wiki/Ms._Pac-Man
Above is the game screen for Ms. Pac-man. Reflections, translations, and rotations were used to place
the “walls” between which Ms. Pan-Man can travel. Take a minute to locate reflection, translations and
rotations of the “walls” in the screen above.
Design a game of your own below. Create your own game and game characters. Be sure your game
meets the following requirements.
• Identify the goal of your game. The goal may be for your game character to eat all of the dots, as
in Ms. Pac-man, or you may have a different goal altogether.
• Use at least 2 of the 3 figures below. You may also use other figures you create. Draw any
figures you create below.
• Use at least 3 different figures in the game.
• Each different figure used must be a different color. When each figure is transformed, the new
image must be the same color as the original figure.
• The interior walls of your game must be the figures you have created or chosen and the images
of those figures which have been reflected, translated, and rotated.
• There must be at least one use of each type of transformation.
• There must be at least 2 series of transformations. (You will likely have many more.)
• Exterior walls and other objects may be drawn, if needed, to complete your game.
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Mathematics
Use the grid below to draw any additional figures you plan to use in your game that will be transformed
using reflections, rotations, and/or translations. Be sure to use a different color for each different figure.
Create your game using the grid on the next page.
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Use the blank grid below to create your game.
Complete parts A through H on the following page.
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Mathematics
After you have created your game, complete the following about your game.
A. Identify a reflection you used in the game. Explain how you know this is a reflection.
B. Identify a translation you used in the game. Explain how you know this is a translation.
C. Identify a rotation you used in the game. Explain how you know this is a rotation.
D. Identify and list the steps of two transformations that involved a series of reflections,
translations, and/or rotations.
Next, exchange games with your partner. Do not show them your answers to parts A through D above.
Using your partner’s game, complete the following. After you identify the transformations, discuss your
findings with your partner.
E. Identify a reflection used in your partner’s game. Explain how you know this is a
reflection.
F. Identify a translation used in your partner’s game. Explain how you know this is a
translation.
G. Identify a rotation used in your partner’s game. Explain how you know this is a rotation.
H. Identify and list the steps of one transformation that involves a series of reflections,
Now, play your partner’s game. ☺
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Mathematics
Application Task Exemplar Response:
This is a sample response. This is an open ended task and student will create very different games that
meet the criteria.
Pac-Man finds Smiley
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Mathematics
c
a
Z
c’
a’
e’ Y
b
d
e
b’
d’
After you have created your game, complete the following about your game.
A. Identify a reflection you used in the game. Explain how you know this is a reflection.
𝑎 → 𝑎′ 𝑟𝑒𝑓𝑙𝑒𝑐𝑡𝑒𝑑 𝑜𝑣𝑒𝑟 𝑡ℎ𝑒 𝑑𝑎𝑠ℎ𝑒𝑑 𝑙𝑖𝑛𝑒.
Check student’s explanation.
B. Identify a translation you used in the game. Explain how you know this is a translation.
𝑏 → 𝑏 ′ 𝑇𝑟𝑎𝑛𝑠𝑙𝑎𝑡𝑒𝑑 𝑑𝑜𝑤𝑛 𝑜𝑛𝑒, 𝑟𝑖𝑔ℎ𝑡 𝑜𝑛𝑒.
C. Identify a rotation you used in the game. Explain how you know this is a rotation.
𝑐 → 𝑐 ′ 𝑅𝑜𝑡𝑎𝑡𝑒𝑑 90° 𝑐𝑜𝑢𝑛𝑡𝑒𝑟𝑐𝑙𝑜𝑐𝑘𝑤𝑖𝑠𝑒 𝑎𝑏𝑜𝑢𝑡 𝑝𝑜𝑖𝑛𝑡 𝑍.
D. Identify and list the steps of two transformations that involved a series of reflections,
𝑑 → 𝑑′ 𝑅𝑒𝑓𝑙𝑒𝑐𝑡𝑒𝑑 𝑜𝑣𝑒𝑟 𝑑𝑜𝑡𝑡𝑒𝑑 𝑙𝑖𝑛𝑒, 𝑡ℎ𝑒𝑛 𝑡𝑟𝑎𝑛𝑠𝑙𝑎𝑡𝑒𝑑 𝑑𝑜𝑤𝑛 3 𝑎𝑛𝑑 𝑟𝑖𝑔ℎ𝑡 6.
𝑒 → 𝑒 ′ 𝑅𝑜𝑡𝑎𝑡𝑒𝑑 90° 𝑐𝑙𝑜𝑐𝑘𝑤𝑖𝑠𝑒 𝑎𝑏𝑜𝑢𝑡 𝑝𝑜𝑖𝑛𝑡 𝑌, 𝑡ℎ𝑒𝑛 𝑡𝑟𝑎𝑛𝑠𝑙𝑎𝑡𝑒𝑑 𝑟𝑖𝑔ℎ𝑡 𝑜𝑛𝑒.
Next, exchange games with your partner. Do not show them your answers to parts A through D above.
Using your partner’s game, complete the following. After you identify the transformations, discuss your
findings with your partner. Solutions for this section will vary based on student observations.
E. Identify a reflection used in your partner’s game. Explain how you know this is a
reflection.
F. Identify a translation used in your partner’s game. Explain how you know this is a
translation.
G. Identify a rotation used in your partner’s game. Explain how you know this is a rotation.
H. Identify and list the steps of one transformation that involves a series of reflections,
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Mathematics Grade 8 – Unit 1 (SAMPLE)

Transcription

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