Download Transformations, Coordinate Geometry

NYCDOE Magnet Program District 25 & 28
JHS185 Magnet School
Grade 8
Essential Question: How does transformational geometry allow us to analyze and
predict population changes?
Suggested Time Frame: 5 weeks
Magnet Theme: Social Justice
Stage 1- Desired Results
Standards-Based Learning Goals:
8.G.1 Identify pairs of vertical angles as congruent
8.G.2 Identify pairs of supplementary and complementary angles
8.G.3 Calculate the missing angle in a supplementary or complementary pair
8.G.4 Determine angle pair relationships when given two parallel lines cut by a transversal
8.G.5 Calculate the missing angle measurements when given two parallel lines cut by a transversal
8.G.6 Calculate the missing angle measurement when given two intersecting lines and an angle.
8.G.7 Describe and identify transformations in the plane, using proper function notation (rotations, reflections,
translations, and dilations.)
8.G.8. Draw the image of a figure under rotations of 90 and 180 degrees
8.G.9. Draw the image of a figure under a reflection over a given line
8.G.10. Draw the image of a figure under a translation
8.G.11 Draw the image of a figure under a dilation
8.G.12 Identify the properties preserved and not preserved under a reflection, rotation, translation, and dilation
Concepts
Big Ideas for this Unit:
Magnet School Theme:
 Change/continuity
 Social Justice
 Balance/Relationships
Relevant/Connected Big Idea:
 Interdependence/Connections
 Angle relationships are crucial in how we plan and design
 Patterns
cities.
 Regions in population density maps can be represented as
geometric figures
o Translation, Reflection , Rotation can be related to
movements in the population (e.g. relocation)
o Dilation can be related to population growth or
shrinkage
Enduring Understandings:
Students will understand that:
 Changes in life and the world can be represented
as translations, rotations, reflections or dilations
 Reflections of geometric forms can be considered
as creating balance
 Transformations of a geometric figure mimic
population transformation (relocation,
gentrification)
 Changes in demographics can be studied as
transformations
Content (nouns)
Students will know…
 Angle relationships
o Vertical angles
o Interior angles
o Exterior angles
o Alternate interior angles
o Alternate exterior angles
o Corresponding angles
o Complementary angles
o Supplementary angles
o Adjacent angles
 Parallel lines
 Perpendicular lines
 Transversal
Overarching Essential Question(s):
 How does transformational geometry allow us to analyze,
predict, and prepare for population changes?
Content and Skills
Skills (verbs)
Students will be able to…
 Identify parallel lines and distinguish them from
intersecting lines
 Identify a transversal
 Identify and label pairs of angles in angle relationships
 Distinguish and explain the difference between one angle
relationship and the other.
 Determine which pairs of angles formed by parallel lines
and a transversal are congruent and which are
supplementary
 Determine the measure of an angle when its complement
or supplement is given.
 Construct an equation to determine the measure of angles
in angle relationships when the measures are given as
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Congruent
Transformations
Rotations
Reflections
Translations
Dilations
Properties preserved
Properties not preserved
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algebraic expressions.
Describe transformations in the plane
Graph and identify transformations in the plane using
proper notation
Draw the image of a figure under rotations of 90°
clockwise and counterclockwise using proper notation
Draw the image of a figure under rotations of 180° using
proper notation
Draw the image of a figure under translations
Draw the image of a figure under reflections of x-axis, yaxis, and line x=y.
Draw the image of a figure under dilations.
Draw the image of a figure under rotation, reflection,
translations, or dilations after being rotated, reflected,
translated or dilated.
Identify the properties preserved and not preserved under
a reflection, rotation, translations, and dilation.
Stage 2- Summative Assessment Evidence
If students understand, know and are able to do the items in Stage 1, they should be able to show their
understanding by completing an authentic task found in the world beyond the classroom.
G- (goal)
The students will create a land-use proposal for the fictional town of Greenville. They will use transformational and
analytical geometry to create a proposal that deals with emergency evacuation situations and population growth or
shrinkage.
R- (role)
The students are urban planners. They are working in groups that are competing to have their proposal approved by the
Greenville Office of City Planning.
A- (audience)
The audience is the Greenville Office of City Planning. The proposals from each group of urban planners will be
reviewed and voted on.
S- (situation)
Greenville is a newly founded town. The town has a small population, but expects population growth in the coming
years. The town government wants to have a development plan in place to deal with population growth. They want a
map that shows the town’s boundaries after doubling in size.
Greenville also wants a map that shows what the town could do if they had to evacuate and relocate to another area.
This map should show the main roads, which form a set of parallel lines and transversal. The urban planners will decide
where the roads go. Then, they will use tools to measure only one angle. They will use analytical geometry skills to
determine the angle measurement formed by the intersecting roads.
The urban planners must write a letter to accompany their proposal. The letter should describe how they came up with
the population growth map, relocation maps, and how they decided to put in the roads.
P- (purpose and product)
 Each map below must include coordinates of all vertices. Coordinates must also be included for each
transformation. The transformation that is applied in each step must be identified and explained (i.e. what is the
transformation and how do figures move or change under the transformation)
 Population growth map (dilation)
o For scale factor of 2.
 Population shrinkage map (dilation)
o For scale factor of 0.5.
 Hurricane relocation maps
o Relocation (translation with rule of x+3,y-4)
o Relocation and change (translation with same rule, and rotation of 90-degrees clockwise)
o Relocation and change (translation with same rule, and rotation of 90-degrees counterclockwise)
o Relocation and change (translation with same rule, and rotation of 180-degrees)
o Relocation and change (translation with same rule, and reflection across x-axis)
o Relocation and change (translation with same rule, and reflection across y-axis)
 Town description letter for relocated town (roads and angle relationships)
o Urban planners place North, South, and Main Street on the relocated-town map, indicating the angles
formed by the roads.
o In a letter, they describe their vision for the relocated town.
 Accompanying each map with a graph drawn on a graph paper.
S- (standards for performance)
Transform each map and label them correctly.
Correctly identify and describe the different transformations.
Draw the map with roads, and correctly determine and label the angle measurements.
Correctly identify the angle relationships.
Culminating Project
You are urban planners. You are working in groups that competing to have your proposal approved the Greenville Office
of City Planning. You will use transformational and analytical geometry to create a proposal that deals with emergency
evacuation situations, population growth or shrinkage, and various economic developments.
Greenville is a newly founded town. The town has a small population, but expects population growth in the coming years.
The town government wants to have a development plan in place to deal with population growth. They want a map that
shows the town’s boundaries, if the town were to grow double, triple, etc. in size.
At its present size, they are also requiring a map that shows what the town could do if they had to evacuate and relocate to
another area. This is because the town is currently located in an area frequently struck by hurricanes. The town
government wants a relocation plan that show where the town would be located if they had to evacuate. Also to be
included in the plan is a proposal for how the town could be changed.
The council requires a few details in the map. North Avenue and South Avenue are two roads in Greenville. They are
parallel to each other. Main Street runs diagonal to North and Main Avenues. The proposal map must include
measurements for the angles formed by these roads. The urban planners will use this information in a descriptive letter
that will accompany their proposal.
The proposals from each group of urban planners will be reviewed by the Review Board of the Greenville Office of City
planning, as well as the Greenville City Council.
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Each map below must include coordinates of all vertices. Coordinates must also be included for each transformation.
The transformation that is applied in each step must be identified and explained (i.e. what is the transformation and
how do figures move or change under the transformation)
Population growth map (dilation)
o For scale factor of 2.
Population shrinkage map (dilation)
o For scale factor of 0.5.
Hurricane relocation maps
o Relocation (translation with rule of x+3,y-4)
o Relocation and change (translation with same rule, and rotation of 90-degrees clockwise)
o Relocation and change (translation with same rule, and rotation of 90-degrees counterclockwise)
o Relocation and change (translation with same rule, and rotation of 180-degrees)
o Relocation and change (translation with same rule, and reflection across x-axis)
o Relocation and change (translation with same rule, and reflection across y-axis)
Town description letter for relocated town (roads and angle relationships)
o Urban planners place North, South, and Main Street on the relocated-town map, indicating the angles formed
by the roads.
o In a letter, they describe their vision for the relocated town.
Each map with a graph drawn on a graph paper and tape them on to a presentation cardboard.
Rubric for Culminating Project
Unit Essential Question:
4
3
2
Complete
Good solid
response with
response with Explanation
Explanation
detailed
clear
unclear.
explanation.
explanation.
Use of
Visuals
Mechanics
1
Misses key
points.
Correct maps
or sketch with
Incorrect maps
Correct maps
No maps or
appropriate
with incorrect
or sketch.
sketch.
labels.
labeling
No math
errors.
No major math
May be some Major math
errors or
serious math
errors or
serious flaws
errors or flaws serious flaws
in reasoning.
in reasoning. in reasoning.
Shows
Shows
Shows
complete lack
Demonstrated complete
Shows some
substantial
of
Knowledge understanding.
understanding.
understanding.
understanding.
Does not meet
Hardly meets
Goes beyond
Meets
the
Requirements
the
requirements. requirements.
requirements
requirements.
.
Mini-Unit
Title
Line and
angle
relationships
Translation in
a coordinate
plane
Big Ideas of
the miniunit/
Concept
Statement
Interdepende
nce/Connect
ions
Fairness
Change/cont
inuity
Balance/relat
ionships
Pattern
Knowledge
Important
content to
know about
mini-unit
(nouns)
Vertical angles
Interior angles
Exterior
angles
Alternate
interior angles
Alternate
exterior angles
Correspondin
g angles
Complementa
ry angles
Supplementary
angles
Adjacent
angles
Parallel lines
Perpendicular
lines
Transversal
Congruent
Transformatio
ns
Translations
Dilations
Properties
preserved
Properties not
preserved
Skills
What should students be able to do?
(verbs)
Identify parallel lines and distinguish
them from intersecting lines
Identify a transversal
Identify and label pairs of angles in angle
relationships
Connection
to Magnet
Theme
Possible Topical
Essential/ Focus
Questions
Mini-Unit
Assessment
How can you
describe your
vision for a
relocated town
(roads and
angle
relationships)?
If measure of angle D
is 53° and angle D and
angle E are
complementary, what
is measure of angle E?
Given one angle,
find the other angle
if they are
complement to each
other.
Angles PQR and STU
are supplementary. If
measure angle PQR =
x-15 and measure angle
STU = x-65, find the
measure of each angle.
Given one angle,
find the other angle
if they are
supplement to each
other.
Distinguish and explain the difference
between one angle relationship and the
other.
Determine which pairs of angles formed
by parallel lines and a transversal are
congruent and which are supplementary
Identify all congruent
angles in
vertical angles.
Determine the measure of an angle when
its complement or supplement is given.
Identify all congruent
angles when two
parallel lines cut by a
transversal line
Construct an equation to determine the
measure of angles in angle relationships
when the measures are given as algebraic
expressions.
Describe transformations in the plane
Graph and identify transformations in
the plane using proper notation
Draw the image of a figure under
translations
Identify the properties preserved and not
preserved under a reflection, rotation,
How does
transformation
al geometry
allow us to
analyze and
predict
population
changes?
The vertices of triangle
MNP are M(4,-2),
N(0,2), and P(5,2).
Graph the triangle and
the image of triangle
MNP after a
translation 3 units left
and 6 units up.
Given sets of
coordinates, students
will translate them
according to the
given rules (some
units to the left or
right and some units
up or down).
Benchmarks,
Scaffolding
Towards
Culminating
Project
Draw maps
roads (maps)
and identify
angle
relationships
for relocated
town
Draw
hurricane
relocation
maps in which
translation is
involved
translations, and dilation.
Reflection in a
coordinate
plane
Change/cont
inuity
Balance/relat
ionships
Rotation in a
coordinate
plane
Change/cont
inuity
Balance/relat
ionships
Dilation in a
coordinate
plane
Change/cont
inuity
Balance/relat
ionships
Transformatio
ns
Reflections
Properties
preserved
Properties not
preserved
Draw the image of a figure under
reflections of x-axis, y-axis, and line x=y.
Identify the properties preserved and not
preserved under a reflection, rotation,
translations, and dilation.
Transformatio
ns
Rotations
Properties
preserved
Properties not
preserved
Draw the image of a figure under rotations
of 90° clockwise and counterclockwise
using proper notation
Transformatio
ns
Dilations
Properties
preserved
Properties not
preserved
Draw the image of a figure under dilations.
Draw the image of a figure under rotation,
reflection, translations, or dilations after
being rotated, reflected, translated or
dilated.
Draw the image of a figure under rotations
of 180° using proper notation
Identify the properties preserved and not
preserved under a reflection, rotation,
translations, and dilation.
Identify the properties preserved and not
preserved under a reflection, rotation,
translations, and dilation.
How does
transformation
al geometry
allow us to
analyze and
predict
population
changes?
The vertices of a figure
are A(-2,3), B(0,5),
C(3,1), and D(3,3).
Graph the figure and
the image of the figure
after a reflection over
the y-axis
Given sets of
coordinates, students
will reflect them
according to the
given rules (across xaxis, y-axis, or line of
y=x).
Draw
hurricane
relocation
maps in which
reflection
across x-axis
and y-axis are
involved
How does
transformation
al geometry
allow us to
analyze and
predict
population
changes?
A figure has vertices
A(-4,5), B(-2,4), C(1,2), D(-3,1), and E(5,3). Graph the figure
and the image of the
figure after a rotation
of 180°
Given sets of
coordinates, students
will rotate them 90°
clockwise, 90°
counterclockwise, or
180°.
How does
transformation
al geometry
allow us to
analyze and
predict
population
changes?
A trapezoid has
vertices A(-1,2), B(0,3),
C(4,1), and D(2,-1).
Graph the figure and
the image of the figure
after a dilation of a
scale factor of 2.
Given sets of
coordinates, students
will dilate them with
a given scale factor).
Draw
hurricane
relocation
maps in which
rotation of
90°
counterclock
wise, 90°
clockwise and
180° are
involved
Draw
population
growth and
population
shrinkage
maps
WHERE is the student going and what is expected
HOOK with needed skills to experience and explore
Opportunity to REVISE and RETHINK their understanding
Allow students to EVALUATE work and implications
TAILOR work to student needs
Be ORGANIZED to maximize engagement
Session1
Session2
Session3
Session4
Session5
Content Focus:
Adjacent & vertical angles.
Content Focus:
Identify complementary &
supplementary angles.
Content Focus:
Calculate the missing angle
in a supplementary or
complementary pair.
Content Focus:
Continue  Calculate the
missing angle in a
supplementary or
complementary pair.
Content Focus:
Determine line and angle
relationships in which two
parallel lines cut by a
transversal
Hook:
In parallelogram SRQT,
angles R and Q are
supplementary. If measure
of angle R = (3x+22)° and
measure of angle Q = (5x2)°, what is the measure of
angle Q?
Hook:
The measure of the
supplement of an angle is
15° less than four times the
measure of the complement.
Find the measure of the
angle.
Hook:
Two streets are parallel to
each other and cut by a
railroad track diagonally.
Identify angles which are
congruent.
Daily Assessment:
Glencoe workbook p. 201203.
Daily Assessment:
Pre-Algebra p. 495-496 #69, 22-25
Hook:
Beth used a pizza cutter to
mate 2 intersection cuts and
divide a pizza into 4 slices.
She expressed the angle of
one slice as (4x + 20)° and
the one opposite as (3x +
35)°. What was the measure
of each angle?
Daily Assessment:
Glencoe workbook p. 195197
Hook:
If a-short-hand points to 1,
a-long-hand points to 7, and
a-third-hand points to 9on a
clock, what kind of angles
do they formed? What are
their measures?
Daily Assessment:
Given several sets of right
angles and straight angles
that are each formed by two
angles, students are going to
determine the missing
angles.
Weekly Assessment:
Multiple Choice and Extended Response questions testing the sessions’ content.
What have the students produced that scaffolds towards the units culminating assessment?
The students have drawn a relocated-town map, indicating the angles formed by the roads.
Daily Assessment:
Have students draw two
parallel lines and a
transversal. Ask them to
name all of the congruent
angles.
WHERE is the student going and what is expected
HOOK with needed skills to experience and explore
Opportunity to REVISE and RETHINK their understanding
Allow students to EVALUATE work and implications
TAILOR work to student needs
Be ORGANIZED to maximize engagement
Session6
Session7
Session8
Session9
Session10
Content Focus:
Calculate missing angles
when given two parallel
lines cut by a transversal.
Content Focus:
Calculate missing angles
(algebraically) when given two
parallel lines cut by a
transversal.
Content Focus:
Continue  calculate
missing angles
(algebraically) when given
two parallel lines cut by a
transversal.
Content Focus:
Describe and identify
transformations
Content Focus:
Translation
Hook:
A road crosses two parallel
railroad tracks. Given an
angle, find the missing
angles.
Daily Assessment:
Have students draw two
parallel lines and a
transversal. Give them one
of the angle measures and
have them fill in the
measures of the other
angles.
Hook:
To measure the angle between a
sloped ceiling and a wall, a
carpenter uses a plumb line (a
string with a weight attached).
Given one angle algebraically,
determine the relationship and
find the missing angles.
Hook:
A road crosses two parallel
railroad tracks. Given two
angles as expressions, find
the missing angles.
Daily Assessment:
Pre-Algebra p. 496 #29-33
Hook:
Michael saw snowflakes
falling down and he
wondered what they have to
do with transformations?
Daily Assessment:
Glencoe workbook p. 221222
Daily Assessment:
Line m and l are parallel lines
and t is a transversal. Find the
value of x if measure of angle 2
= 2x +3 and measure of angle 4
= 4x – 7. (Draw the figure and
place angle 2 and 4 so that they
form vertical angle
relationships).
Weekly Assessment:
Multiple Choice and Extended Response questions testing the sessions’ content.
What have the students produced that scaffolds towards the units culminating assessment?
The students have drawn a relocated map with translation with rule of x+3, y-4.
Hook:
Translation can be related to
movements in the
population such as
relocation due to hurricane
Daily Assessment:
Glencoe p. 238-240.
WHERE is the student going and what is expected
HOOK with needed skills to experience and explore
Opportunity to REVISE and RETHINK their understanding
Allow students to EVALUATE work and implications
TAILOR work to student needs
Be ORGANIZED to maximize engagement
Session11
Session12
Session13
Session14
Session15
Content Focus:
Reflection.
Content Focus:
Rotation 90° clockwise
Content Focus:
Rotation 180°.
Content Focus:
Dilation
Hook:
Which transformation exists
when you look into a
mirror?
Hook:
Rotation can be related to
movements in the blades of
a ceiling fan, a dartboard or
a ferris wheel
Content Focus:
Rotation 90°
counterclockwise
Hook:
Given a-downward-facing
picture, students have to
rotate it so that it faces
upward.
Daily Assessment:
Pre-Algebra Lesson 10-3 #14 or
Glencoe workbook p. 231233.
Daily Assessment:
Given a set of coordinates of
a figure, students will draw
and write the coordinates of
its image after a rotation of
90° clockwise.
Hook:
Population growth (dilation
with a scale factor greater
than 1)
Population shrinkage
(dilation with a scale factor
smaller than 1)
Hook:
Given a landscape
orientation picture, students
have to rotate it into a
portrait orientation so that
they can upload it into an
online profile.
Daily Assessment:
Given a set of coordinates of
a figure, students will draw
and write the coordinates of
its image after a rotation of
90° counterclockwise.
Daily Assessment:
Given a set of coordinates of
a figure, students will draw
and write the coordinates of
its image after a rotation of
180°.
Weekly Assessment:
Multiple Choice and Extended Response questions testing the sessions’ content.
What have the students produced that scaffolds towards the units culminating assessment?
The students have drawn almost all of the required relocated maps.
Daily Assessment:
Given a set of coordinates of
a figure and scale factors,
students will draw and write
the coordinates of their
images.
WHERE is the student going and what is expected
HOOK with needed skills to experience and explore
Opportunity to REVISE and RETHINK their understanding
Allow students to EVALUATE work and implications
TAILOR work to student needs
Be ORGANIZED to maximize engagement
Session16
Session17
Session18
Session19
Session20
Content Focus:
The properties preserved
and not preserved
Content Focus:
Content Focus:
Content Focus:
Content Focus:
Hook:
Hook:
Hook:
Hook:
Daily Assessment:
Daily Assessment:
Daily Assessment:
Hook:
For the relocation proposal
to be useful, students have
to understand the properties
preserved and not
preserved under each
transformation
Daily Assessment:
Daily Assessment:
Glencoe workbook p. 248250.
Weekly Assessment:
Multiple Choice and Extended Response questions testing the whole unit’s content.
What have the students produced that scaffolds towards the units culminating assessment?
The students are completing their project.
Unit Resources
Books:
Pre-Algebra, Glencoe (McGraw Hill), 2005
Glencoe – New York Review Series (Debra L. Harley & Erica J. Shatz)
Websites:
Teacher Materials:
Other: