MATHEMATICS GRADE 8
HOUSTON ISD PLANNING GUIDE
4TH SIX-WEEKS
Planning Guide User Information
Unit 11: Geometric Transformations and
Dimensional Changes
Part 1: Geometric Transformations on the
Coordinate Plane
Time Allocations
Unit
Part 1
4 lessons (90-minutes each)
or
8 lessons (45-minutes each)
2 lessons (90-minutes each)
or
4 lessons (45-minutes each)
Unit Overview
Geometric Transformations and Dimensional Changes – Students investigate geometric transformations and their
relationships, and the effects of dimensional changes on area and perimeter of two-dimensional geometric figures.
Part 1: Geometric Transformations on the Coordinate Plane starts here.
TEKS/SEs (district clarifications/elaborations in italics)
Ⓡ MATH.8.6A Generate similar figures using dilations including enlargements and reductions, describe the relationship
between the pre-image and the image using scale factor and magnitude, and apply scale factors in problem solving
situations.
Ⓢ MATH.8.6B Graph dilations, reflections, and translations on a coordinate plane and describe the relationships
between the pre-image and the image.
Ⓢ MATH.8.7B Use geometric concepts (including symmetry, similarity, congruence, and transformations) and
properties of two- and three-dimensional figures to solve problems in fields such as art and architecture.
Ⓡ MATH.8.9B Use proportional relationships in similar two-dimensional figures or similar three-dimensional figures to
find missing measurements.
MATH.8.14A Identify and apply mathematics to everyday experiences, to activities in and outside of school, with
other disciplines, and with other mathematical topics.
 English Language Proficiency Standards
ELPS C.1a Use prior knowledge
and experiences to understand
meanings in English.
ELPS C.1g Demonstrate an
increasing ability to distinguish
between formal and informal
English and an increasing
knowledge of when to use each one
commensurate with grade-level
learning expectations.
ELPS C.3b Expand and internalize
initial English vocabulary by learning
and using high-frequency English
words necessary for identifying and
describing people, places, and
objects, by retelling simple stories
and basic information represented
or supported by pictures, and by
learning and using routine language
needed for classroom
communication.
ELPS C.4g Demonstrate
comprehension of increasingly
complex English by participating
in shared reading, retelling or
summarizing material, responding
to questions, and taking notes
commensurate with content area
and grade level needs.
College and Career Readiness Standards
CCRS 3.A1 Identify and represent
the features of plane and space
figures.
CCRS 3.B1. Identify and apply
transformations to figures.
CCRS 3.B2 Identify the symmetries
of a plane figure.
CCRS 3.B3 Use congruence
transformations and dilations to
investigate congruence, similarity,
and symmetries of plane figures.
CCRS 10.A2 Connect
mathematics to the study of other
disciplines.
CCRS 10.B1 Use multiple
representations to demonstrate
links between mathematical and
real world situations.
geometric figures
transformation
Key Concepts
change
Academic Vocabulary
graph
rule
 - English Language Proficiency Standards (ELPS)
- Literacy Leads the Way Best Practices
- Aligned to Upcoming State Readiness Standard
- State Process Standard Ⓡ - State Readiness Standard Ⓢ - State Supporting Standard Ⓣ - TAKS Tested Objective (only 11th grade)
© Houston ISD Curriculum
2012 – 2013
Page 1 of 8
MATHEMATICS GRADE 8
HOUSTON ISD PLANNING GUIDE
4TH SIX-WEEKS
Content-Specific Vocabulary
image
magnitude
pre-image
similar
Essential Understandings / Guiding Questions
Congruent figures and similar figures can undergo size and/or position changes while maintaining proportional
relationships.
1. What is the difference between congruent and similar figures?
2. How are the corresponding parts of similar figures related?
3. How are dilations different from translations and reflections?
Geometric figures and their transformations can be represented on a coordinate plane.
1. What happens to a figure if it is translated? reflected?
2. How is the image of a transformed geometric figure related to its pre-image?
3. How can geometric figures and their transformations be represented on a coordinate plane?
Assessment Connections
Performance Expectation – Students will generate congruent and similar figures using transformations, graph those
figures on a coordinate plane and analyze the relationships between the image and pre-image of those
transformations.
Journal Writing – students create their own example of one type of transformation (dilation, reflection, or translation)
and describe the relationship between the image and the pre-image.
Formative Assessment – Student Council President – utilizes the table feature of graphing calculators to assist
students in their investigation of dilations.
®
SpringBoard Course 1 – Embedded Assessment #2: “Graphic Geometry” – #1– 7
STAAR Sample Item – Item #7 (MATH.8.6A) and Item #8 (MATH.8.7B)
Texas English Language Proficiency Assessment System (TELPAS): End-of-year assessment in listening,
speaking, reading, and writing for all students coded as LEP (ELL) and for students who are LEP but have parental
denials for Language Support Programming (coded WH). For the Writing TELPAS, teachers provide five writing
samples – one narrative about a past event, two academic (from science, social studies, or mathematics), and two
others.
Instructional Considerations
Information in this section is provided to assist the teacher with the background knowledge needed to plan instruction
that facilitates students to internalize the Key Concepts and Essential Understandings for this unit. It is recommended
that teachers thoroughly read this section before implementing the strategies and activities in the Instructional
Strategies section.
Prerequisites and/or Background Knowledge for Students
Students have worked with translations and reflections since fourth grade and graphed them on a coordinate plane in
seventh grade. (Ⓡ MATH.7.7B, Ⓢ MATH.8.6B)
Background Knowledge for Teacher
Critical Content
Explore and use reflections on a coordinate plane;
Explore and use translations on a coordinate plane; and
Explore and use dilations on a coordinate plane.
Introduction
MATH.8.6B is the students’ first experience with generating the algebraic rules involved in transforming figures on the
coordinate plane in all four quadrants. The use of “movement arrows” on the plane showing the transformation from
the pre-image to the image will help students visualize and generalize the rules involved in moving between the
corresponding vertices of the figures.
 - English Language Proficiency Standards (ELPS)
- Literacy Leads the Way Best Practices
- Aligned to Upcoming State Readiness Standard
- State Process Standard Ⓡ - State Readiness Standard Ⓢ - State Supporting Standard Ⓣ - TAKS Tested Objective (only 11th grade)
© Houston ISD Curriculum
2012 – 2013
Page 2 of 8
MATHEMATICS GRADE 8
HOUSTON ISD PLANNING GUIDE
4TH SIX-WEEKS
Instructional Considerations
Instructional Accommodations for Diverse Learners
 MATH.8.7B relates transformations to experiences that students have in the real world. This helps to make the
concepts more concrete. For example, relate a translation to a dance step, a reflection to an image in a mirror, and a
dilation to the use of a copy machine that enlarges and reduces the size of an original. C.1a
Connections to other areas of mathematics:
Proportionality is an underlying “big idea” that runs throughout middle school mathematics. Dilations and similar figures
are a natural way to link geometric concepts to previously learned skills in proportionality such as setting up proportions
and solving for the “unknown” in a proportional relationship. Applying these skills in a school-based situation such as
creating signs for a student council election allows students to relate their prior knowledge to something that will be
perceived as non-threatening and engaging and to enhance that knowledge with the use of graphing calculators.
Instructional Strategies / Activities
The strategies and activities in this section are designed to assist the teacher to provide learning experiences to
ensure that all learners achieve mastery of the TEKS SEs for this unit. It is recommended that the strategies and
activities in this section be taught in the order in which they appear.
Identifying Similarities and Differences
KWL (Turn The Light On)
The use of a KWL chart before beginning transformational geometry will help the teacher decide what kind of review
activities are necessary. Concrete activities such as the use of a geoboard helps students experience
transformations before they move to the use of grid paper or graphing calculators.
 Playing The Transformation Game is a fun, engaging, and non-threatening way for students to review and
demonstrate their knowledge of the basic concepts of transformations. It should be noted that the vocabulary used in
this game is informal (slide, flip, turn) and the students should be encouraged to “translate” this informal language
into the more formal vocabulary appropriate for this grade level (translation, reflection, and rotation) – see Resources.
(MATH.8.6A, MATH.8.6B) C.1g
Summarizing and Note-Taking
Two-Column Notes (Pen/cil To Paper)
After students finish playing the game above, they should summarize their definitions and findings using two-column
notes.
Identifying Similarities and Differences
When translating figures, students should compare and contrast the coordinates of the original figures and the
coordinates of the transformed figures to establish patterns and generate rules (Activity: Translating Shapes – see
Resources). (MATH.8.6B)
Instructional Accommodations for Diverse Learners
Nonlinguistic Representations
 Multiple representations of the transformation operations in translations should be discussed; for example, verbal
descriptions, pictorial representations depicting both the pre-image and the image on a coordinate plane, and
pictorial representations showing the pre-image and a “movement arrow” on the coordinate plane, an algebraic rule,
and a t-chart (Activity: Translations and Congruence – see Resources). Students should be able to verbalize their
®
understandings of the connections between the various representations. (SpringBoard Mathematics with Meaning:
Course 1, Activity 5.7 “Symmetry and Transformations”) C.3b
For example, a triangle with vertices L(−4, 4), M(−1, 4), and N(−2, 1) is translated to an
image with vertices L (−1, 0), M (2, 0), and N (1, −3). The students should analyze the
relationship between the pre-image and the image and generalize to a verbal description
of the rule and then to a symbolic representation of the rule. They might notice that these
Vertices can be found by moving 3 units left and 4 units down or by adding 3 to the
x-coordinates and −4 to the y-coordinates.
 - English Language Proficiency Standards (ELPS)
- Literacy Leads the Way Best Practices
- Aligned to Upcoming State Readiness Standard
- State Process Standard Ⓡ - State Readiness Standard Ⓢ - State Supporting Standard Ⓣ - TAKS Tested Objective (only 11th grade)
© Houston ISD Curriculum
2012 – 2013
Page 3 of 8
MATHEMATICS GRADE 8
HOUSTON ISD PLANNING GUIDE
4TH SIX-WEEKS
Instructional Strategies / Activities
Symbolically this is represented:
Original (Pre-image)  Add (3, −4)  Image becomes:
L(−4, 4)  (−4 + 3, 4 + −4)  L (−1, 0)
The use of a line of reflection and its role in generating reflections is an important concept for student understanding.
Begin by using the x- and y-axis as lines of reflection. Students need to understand that the pre-image and image are
equidistant from the line of reflection and that there are resulting relationships between the corresponding vertices of
the figures (Activity: Reflections, Concepts in Motion: Reflections – see Resources).
For example, the coordinates of the vertices of a triangle are X(2,3),Y(3, 1), and Z(−2, 2).
After reflection over the x-axis, the coordinates of its image are X (2,−3), Y (3, −1), and
Z (−2, −2).
Have the students examine the relationship between the coordinates of each figure and
generalize to a rule.
Reflection Rule:
opposites
same
X (2, 3)
Y (3, 1)
Z (−2, 2)
X (2, −3)
Y (3, −1)
Z (−2, −2)
For additional strategies to assist diverse learners, access Recommendations for Accommodating Special Needs
Students: Math 8, Cycle 4, Unit 11, Part 1.
Dilations
In Formative Assessment – Student Council President, students use the TABLE feature of graphing calculators to
explore the results of dilations in a familiar situation. Scaffolding questions such as those below will help guide
students in their investigation.
o What makes two figures similar?
o How does an enlargement or reduction affect the shape of a figure?
o How does it affect the size of a figure?
o What is a scale factor?
o How is the scale factor of an enlargement determined? A reduction?
o How does the new figure compare with the original in an enlargement? In a reduction? (MATH.8.9B,
MATH.8.14A)
Students need to extend their previous experiences with dilations into the coordinate plane by graphing and using
algebraic rules to generate ordered pairs for enlarged or reduced figures. They should be able to express magnitude
such as twice as large, half as large, etc. (Activities: Dilations, Dilations and Similarity – see Resources).
Identifying Similarities and Differences
 Students should identify similarities and differences between the resulting figures in translations, reflections, and
dilations. The orientation of the figures, the scale factor, and the fact that the resulting figures are either congruent
(translations and reflections) or similar (dilations) should be observed and discussed (Activity 8.6A Enlargements and
Activity 8.6B Transformations in the Mathematics Toolkit – see Resources). C.4g
 - English Language Proficiency Standards (ELPS)
- Literacy Leads the Way Best Practices
- Aligned to Upcoming State Readiness Standard
- State Process Standard Ⓡ - State Readiness Standard Ⓢ - State Supporting Standard Ⓣ - TAKS Tested Objective (only 11th grade)
© Houston ISD Curriculum
2012 – 2013
Page 4 of 8
MATHEMATICS GRADE 8
HOUSTON ISD PLANNING GUIDE
4TH SIX-WEEKS
Instructional Strategies / Activities
Extensions for Pre-AP
To meet the instructional needs of Pre-AP students, provide challenging problem-solving scenarios such as the one
below:
o Gary wanted two copies of his favorite 6-inch by 4-inch photo. One copy will be an enlargement for his girlfriend,
and the other will be a reduction for his mother. First, he had the photo enlarged by a scale factor of 1.5. Then
he had the enlargement reduced by a scale factor of 0.75. What are the dimensions of the final reduced photo?
Justify your answer. (MATH.8.6A, MATH.8.7B)
Resources
Adopted Instructional Materials
®
SpringBoard Mathematics with
Meaning: Middle School 1
5.7 “Symmetry and
Transformations”
Glencoe, Texas Mathematics, Course
3:
TWE/SE, 6-6 & 6-7, pp. 330 – 339
Ch. 6 Resource Masters, pp. 37 –
42
Geometry Lab: Tessellations, pp.
340 – 341
Teaching Math with Manipulatives,
pp. 60, 63 – 65
Supporting Resources
The Transformation Game
Translating Shapes
Translations and Congruence
Reflections
Dilations
Dilations and Similarity
Activity 8.6A Enlargements
Activity 8.6B Transformations
Recommendations for
Accommodating Special Needs
Students: Math 8, Cycle 4, Unit 11,
Part 1
Online Resources
Concepts in Motion: Reflections
Mathematics Toolkit – UT Dana
Center
The following websites include
information and activities that
highlight connections between
mathematics and the fine arts.
What are Tessellations?
Tessellations
Tessellation Town
Tessellate!
 - English Language Proficiency Standards (ELPS)
- Literacy Leads the Way Best Practices
- Aligned to Upcoming State Readiness Standard
- State Process Standard Ⓡ - State Readiness Standard Ⓢ - State Supporting Standard Ⓣ - TAKS Tested Objective (only 11th grade)
© Houston ISD Curriculum
2012 – 2013
Page 5 of 8
MATHEMATICS GRADE 8
HOUSTON ISD PLANNING GUIDE
4TH SIX-WEEKS
Planning Guide User Information
Unit 11: Geometric Transformations and
Dimensional Changes
Part 2: Dimensional Changes in Area and Perimeter
of Similar Figures
Time Allocations
Unit
Part 2
4 lessons (90-minutes each)
or
8 lessons (45-minutes each)
2 lessons (90-minutes each)
or
4 lessons (45-minutes each)
Unit Overview
Geometric Transformations and Dimensional Changes – Students investigate geometric transformations and their
relationships, and the effects of dimensional changes on area and perimeter of two-dimensional geometric figures.
Part 2: Dimensional Changes in Area and Perimeter of Similar Figures starts here.
TEKS/SEs (district clarifications/elaborations in italics)
Ⓢ MATH.8.10A Using concrete or pictorial models as well as verbal or algebraic descriptions, describe the resulting
effects on perimeter and area when dimensions of a figure are changed proportionally.
 English Language Proficiency Standards
ELPS C.1h Develop and expand repertoire of learning strategies such as reasoning inductively or deductively,
looking for patterns in language, and analyzing sayings and expressions commensurate with grade-level learning
expectations.
College and Career Readiness Standards
CCRS 3.A1 Identify and represent
the features of plane and space
figures.
CCRS 3.B1. Identify and apply
transformations to figures.
CCRS 3.B2 Identify the symmetries
of a plane figure.
CCRS 3.B3 Use congruence
transformations and dilations to
investigate congruence, similarity,
and symmetries of plane figures.
CCRS 4.C1 Find the perimeter and
area of two-dimensional figures.
CCRS 4.C3 Determine indirect
measurements of figures using
scale drawings, similar figures,
the Pythagorean theorem, and
basic trigonometry.
Key Concepts
change
geometric figures
Academic Vocabulary
model
proportional
Content-Specific Vocabulary
area
dimensional change
perimeter
Essential Understandings / Guiding Questions
Two-dimensional geometric figures exhibit proportional relationships when they experience dimensional changes in
area and perimeter.
1. How is the perimeter of a dilated figure related to the perimeter of its pre-image?
2. How is the area of a dilated figure related to the area of its pre-image?
Assessment Connections
Performance Expectation – Students will generate congruent and similar figures using transformations, graph those
figures on a coordinate plane, and analyze the relationships between the image and pre-image of those
transformations.
Expository Writing – Students explain what happens to the area and perimeter of a rectangle when the dimensions
are doubled, and when the dimensions are cut in half.
Formative Assessment – By the Sea – involves creating reflections and dilations on the coordinate plane and
changes in perimeter and area in the resulting figures.
STAAR Sample Item – Item #11 (MATH.8.10A)
 - English Language Proficiency Standards (ELPS)
- Literacy Leads the Way Best Practices
- Aligned to Upcoming State Readiness Standard
- State Process Standard Ⓡ - State Readiness Standard Ⓢ - State Supporting Standard Ⓣ - TAKS Tested Objective (only 11th grade)
© Houston ISD Curriculum
2012 – 2013
Page 6 of 8
MATHEMATICS GRADE 8
HOUSTON ISD PLANNING GUIDE
4TH SIX-WEEKS
Assessment Connections
Texas English Language Proficiency Assessment System (TELPAS): End-of-year assessment in listening,
speaking, reading, and writing for all students coded as LEP (ELL) and for students who are LEP but have parental
denials for Language Support Programming (coded WH). For the Writing TELPAS, teachers provide five writing
samples – one narrative about a past event, two academic (from science, social studies, or mathematics), and two
others.
Instructional Considerations
Information in this section is provided to assist the teacher with the background knowledge needed to plan instruction
that facilitates students to internalize the Key Concepts and Essential Understandings for this unit. It is recommended
that teachers thoroughly read this section before implementing the strategies and activities in the Instructional
Strategies section.
Prerequisites and/or Background Knowledge for Students
Seventh-grade students determined missing measurements in similar figures. (Ⓡ MATH.7.6D)
Background Knowledge for Teacher
Critical Content
Analyze dimensional changes in the area and perimeter of similar figures.
Instructional Accommodations for Diverse Learners
Before beginning these investigations, it may be necessary to review the concrete applications of perimeter and area to
help students visualize the concepts involved in dimensional changes. For example, build a rectangle with sides 2 units
by 3 units and then build an enlarged figure of 4 units by 6 units. The students can easily make a connection between
the doubled length and widths and the resulting perimeter and area changes. This concrete experience could serve as
a warm-up activity for the measurement experiences referenced in Instructional Strategies.
Perimeter and Area
Students should discover that the perimeters of similar figures have a ratio that equals the scale factor and areas that
equal the square of the scale factor.
Instructional Strategies / Activities
The strategies and activities in this section are designed to assist the teacher to provide learning experiences to
ensure that all learners achieve mastery of the TEKS SEs for this unit. It is recommended that the strategies and
activities in this section be taught in the order in which they appear.
Cues, Questions, and Advance Organizers
Think-Pair-Share (Let’s Talk)
Students work with a partner to discuss and model using centimeter grid paper:
o keeping the perimeter constant and changing the dimensions of a figure;
o keeping the area constant and changing the dimensions of a figure; and
o changing the dimensions of a figure proportionally and describe the resulting effects on both the perimeter and
®
the area of the figure. (SpringBoard Mathematics with Meaning: Course 2, Activity 4.5 “Changing Dimensions”)
The following activities investigate proportional changes in perimeter and area – see Resources:
o Perimeter and Area Changes
o Proportional Changes – Area and Perimeter
o Activity 8.10A Changing Dimensions in the Mathematics Toolkit.
 - English Language Proficiency Standards (ELPS)
- Literacy Leads the Way Best Practices
- Aligned to Upcoming State Readiness Standard
- State Process Standard Ⓡ - State Readiness Standard Ⓢ - State Supporting Standard Ⓣ - TAKS Tested Objective (only 11th grade)
© Houston ISD Curriculum
2012 – 2013
Page 7 of 8
MATHEMATICS GRADE 8
HOUSTON ISD PLANNING GUIDE
4TH SIX-WEEKS
Instructional Strategies / Activities
Instructional Accommodations for Diverse Learners
Cooperative Learning
 In small groups, give students a set of figures with proportionally changed lengths and widths. The students
measure each figure and find its area and perimeter.
They should then create wall charts with the figures pasted down and columns filled in with the data for the length,
width, area, and perimeter.
The charts should be examined for patterns that describe the relationships observed in the measurements and those
relationships should be recorded in verbal or algebraic form.
For example, doubling the length and width of a rectangle yields a perimeter that is double the original perimeter, but
the area is four times as large as the original. C.1h
For additional strategies to assist diverse learners, access Recommendations for Accommodating Special Needs
Students: Math 8, Cycle 4, Unit 11, Part 2.
In Formative Assessment – By the Sea, students plot geometric figures on the coordinate plane and create dilated
images of each. They compare and contrast the changes from pre-image to image including changes in perimeter and
area in the resulting figures. They also investigate reflections of the original figures
Extensions for Pre-AP
To meet the instructional needs of Pre-AP students, provide challenging problem-solving scenarios such as the one
below:
o Polygon ABCD is similar to polygon FGHI. Each side of polygon ABCD is 3 ¼ times longer than the
corresponding side of polygon FGHI. Find the perimeter and area of polygon ABCD.
Resources
Adopted Instructional Materials
®
SpringBoard Mathematics with
Meaning: Middle School 2
4.5 “Changing Dimensions”
Glencoe, Texas Mathematics, Course
3:
TWE/SE, 4-5, pp. 205 – 210
Chapter 4 Resource Masters, p. 36
Supporting Resources
Perimeter and Area Changes
Proportional Changes – Area and
Perimeter
Activity 8.10A Changing Dimensions
Recommendations for
Accommodating Special Needs
Students: Math 8, Cycle 4, Unit 11,
Part 2
Online Resources
Mathematics Toolkit – UT Dana
Center
 - English Language Proficiency Standards (ELPS)
- Literacy Leads the Way Best Practices
- Aligned to Upcoming State Readiness Standard
- State Process Standard Ⓡ - State Readiness Standard Ⓢ - State Supporting Standard Ⓣ - TAKS Tested Objective (only 11th grade)
© Houston ISD Curriculum
2012 – 2013
Page 8 of 8