Unit Map for Geometry
2015 – 2016
Felisa Rincon de Gautier Institute for Law
and Public Policy
Math Department
Mr. Balkaran
Topic A: Scale Drawings
Timeframe: 5-6 days
Number of lessons in this unit: 5
Learning Outcomes
Common Core Learning Standards addressed in this Unit:
G-SRT.A.1
 Verify experimentally the properties of dilations given by a center and a scale factor:
a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
G-SRT.B.4
 Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle. Theorems include: a line parallel to one side of a
triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
Standards for Mathematical Practices addressed in this Unit:
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively
3. Construct viable arguments and critique the reasoning of others
4. Model with mathematics.
5. Use appropriate tools strategically
6. Attend to precision.
Enduring Understandings for this Unit:
Students begin with a review of scale drawings in
Lesson 1, followed by two lessons on how to
systematically create scale drawings. The study of
scale drawings, specifically the way they are
constructed under the Ratio and Parallel Methods,
gives us the language to examine dilations. The
comparison of why both construction methods
result in the same image leads to two theorems:
the Triangle Side Splitter Theorem and the
Dilation Theorem. Note that while dilations are
Essential Questions for this Unit:
 What are properties of a well-scaled drawing of a figure?
 What is the term for the constant of proportionality by which all lengths are scaled in a
well-scaled drawing?
 If we were to take any of the scale drawings in our examples and place them in a
different location or rotate them on our paper, would it change the fact that the drawing is
still a scale drawing?
 What are the key properties of a scale drawing relative to its original figure?
 How can it be confirmed that what is drawn by the Ratio Method is in fact a scale
drawing?
 How are dilations and scale drawings related?
defined in Lesson 2, it is the Dilation Theorem in
Lesson 5 that begins to tell us how dilations
behave (G-SRT.A.1, G-SRT.A.4).
Content of this Unit:
- Scale Drawings
- Scale Factors
- Ratio Method
- Parallel Method
Skills of this Unit:
- Explain the properties of scale drawings and are able to create them.
- Create scale drawings of polygonal figures by the Ratio Method.
- Create scale drawings of polygonal figures by the Parallel Method.
- Explain why angles are preserved in scale drawings created by the Parallel Method using the
theorem on parallel lines cut by a transversal.
- Justify the Dilation Theorem.
- Apply the Dilation Theorem to show that the scale drawings constructed using the Ratio and
Parallel Methods have a scale factor that is the same as the scale factor for the dilation.
Key Vocabulary & Language of this Unit:
Scale drawing, Scale factor, Dilation, Rigid
Motions Composition, Pythagorean Theorem,
Slope.
Resources used in this Unit:
- JMAP.org Resources for each Standard
- TI Smartview
- Regents Reference table
- Big Ideas Math Geometry by Ron Larson
- Engaged NY Geometry
- Regentsprep.org
Assessments
Formative Assessments:
- Mini dry erase boards
- Daily exit slips
- Pair, group, and class discussions
- Homework assignments
- Quizzes
- Results and observations from daily learning activities and teaching
strategies
- Questioning
- Warm up collection and review
- Peer- and self-assessments
- Math Sprint
Summative Assessments:
- Class Gallery Walk
- Unit Exam
- Weekly Quiz
Instructional Pathway
Learning Activities & Teaching Strategies Used in This Unit
CC = compare/contrast topic
DT = discussion topics
E = extension topic/problems
EA = error analysis: teacher models a common mistake and students determine where the mistake was made
GW = gallery walk
OEQ = open-ended question: question that has many possible correct answers
MR = multiple representations (e.g. verbal, algebraic, numeric, symbolic, graphical)
MS = math sprint: quick timed assessment for building fluency and reviewing
PA = peer assessment with whiteboard and during gallery walk
RWC = real world connection
T = use of technology
TPS = think-pair-share
WB = whiteboard: students do/correct their work on mini whiteboards and share with a partner, group, or class
WP = writing prompt: for collection (i.e. exit slip) or journal
Standards
Aim
Lesson Content
-
G-SRT.A.1
G-SRT.B.4
1. What are scale
drawings and how
are we able to create
them?
Lesson 1 reviews the properties of a
scale drawing before studying
relationship between dilations and scale
drawings in Lessons 2 and 3. Students’
focus on scaling triangles using
construction tools and skills learned in
Module 1. The lesson begins by
exploring how to scale images using
common electronics. After students
work on scaling triangles given various
pieces of initial information, we tie it all
together by showing how triangle
scaling can be used in programming a
phone to a scale a complex image.
Grouping Structures
(I) = individual
(P) = with a partner
(G) = in a student group
(C) = whole class
Activities & Strategies
-
-
MS/TPS opener: A common feature on cell
phones. Then have students make analogy
with bicycle and determine which image
below appears to be a well- scaled image of
the original?
MR: Have students use compass to perform
dilation of a given triangle.
E: Have students model off example on
created a scaled image of a given image.
Students then discuss what are the
similarities and differences between the
original image and the scaled image.
-
G-SRT.A.1
G-SRT.B.4
2. How do we create
scale drawings of
polygonal figures by
the Ratio Method?
-
G-SRT.A.1
G-SRT.B.4
3. How do we make
scale drawings using
the parallel method?
-
G-SRT.A.1
G-SRT.B.4
4. How does the Ratio
Method compare to
the Parallel Method?
Lesson 2 introduces students to a
systematic way of creating a scale
drawing: the Ratio Method, which relies
on dilations. Students dilate the vertices
of the provided figure and verify that
the resulting image is in fact a scale
drawing of the original. It is important
to note that we approach the Ratio
Method as a method that strictly dilates
the vertices. After some practice with
the Ratio Method, students dilate a few
other points of the polygonal figure and
notice that they lie on the scale drawing.
Lesson 3, students learn the Parallel
Method as yet another way of creating
scale drawings. The lesson focuses on
constructing parallel lines with the use
of setsquares, although parallel lines can
also be constructed using a compass and
straightedge; setsquares will reduce the
time needed for the construction.
Straightedges, compasses, and
setsquares are needed for this lesson.
Lesson 4 challenges students to
understand the reasoning that connects
the Ratio and Parallel Methods that
have been used in the last two lessons
for producing a scale drawing. The first
key idea is the two triangles with the
same base that have vertices on a line
parallel to the base are equal in area.
The second key idea is that two
triangles with different bases, but equal
altitudes will have a ratio of areas that is
-
-
-
-
MS opener: Have students use prior
knowledge to describe what a dilation is? In
addition, have students provide an example.
OEQ: Students create a Frayer Model on
dilations.
Scaffolding: Figures can be made as simple
or as complex as desired- a triangle will
involve fewer segments to keep track of than
a figure.
MS opener: Dani dilated a triangle from
center O. She says that she completed the
drawing using parallel lines. How could she
have done this? Explain.
Scaffolding: Steps should be modeled for all
students, but depending on student needs,
these images may need to be made larger or
teachers may need to model the steps more
explicitly.
MS: Have students apply their knowledge of
triangles to calculate the areas of two triangles
and compare them.
equal to the ratio of their bases.
-
G-SRT.A.1
G-SRT.B.4
5. What is the Dilation
Theorems?
In Grade 8 students learned about the
Fundamental Theorem of Similarity
(FTS), which contains the concepts that
are in the Dilation Theorem presented in
the lesson. We call it the Dilation
Theorem at this point in the module
because students have not yet entered
into the formal study of similarity.
-
Students will participate in a Quick Write.
Write how a figure is transformed under a
dilation with a scale factor r=1, r > 1, and 0
< r < 1.
Unit Exam
Differentiation strategies used in this unit & modifications embedded within this unit to provide access for all learners
-
Use of projected calculator emulator (TI Smartview).
When developing list of vocabulary words for the unit, provide students with the opportunity to draw graphic representations/examples of the
words and add them to the word wall.
Provide multiple representations of solutions and solution sets throughout unit.
Choice provided to students in writing prompts.
Use of open-ended questions provides students at varying levels with entry points.
Students are provided with independent think time prior to answering questions in any grouping setting.
Real world connections help to build relevance.
Development of Academic & Personal Behaviors and 21st Century Skills
-
Students maintain portfolios and contribute work from this unit to it; students are provided with opportunities to revise their work, after
receiving feedback from peers and teacher (including the unit performance task).
Through the use of strategic grouping, driven by formative assessment data, students interact with pairs, groups, and the whole class on a
routinely changing basis; tasks and classroom activities are developed to promote independence (e.g. the use of “Ask Three Before Me” and
related strategies), effective collaboration, and leadership (group leaders are rotated).
-
Students monitor their own progress on daily and weekly progress trackers; differentiated support is provided to students based on the skills
they still need help with.
Instructional Shifts
Instructional Shift: Focus
Where in this unit is there evidence of focusing
deeply on the concepts that are prioritized in
the standards?
Instructional Shift: Coherence
How does this unit build upon knowledge of
prior years, and how does it support future
coursework?
Instructional Shift: Rigor
Where is there evidence of rigor in this unit?
The sequence of lessons in this unit provide
students with the opportunity to develop an in
depth understanding of similarity. Diagrams
are developed throughout the unit to help
students associate scale factoring and dilation to
rigid motions.
These lessons build upon the skills developed in
7th grade and 8th grade pertaining to dilating
images. This unit prepares students for
upcoming units that involve students to draw
out facts from given information and diagrams
to reach a conclusion.
Several rigorous activities are planned for this
unit, including multiple opportunities to answer
higher order thinking questions (including
open-ended questions), engage in discussions
with peers, work through extension activities,
and analyze errors. Students compare and
contrast different mathematical concepts and
engage in peer assessment in multiple
situations.