HCPSS Curriculum Framework
Common Core Geometry, Unit 1
Transformations, Similarity, and Congruence
Overview (Big Ideas):
Transformations and the Coordinate Plane
Students are expected to use coordinates to prove simple geometric theorems algebraically.
Building on their work with the Pythagorean theorem in 8th grade to find distances, students use
a rectangular coordinate system to verify geometric relationships, including properties of special
triangles and quadrilaterals and slopes of parallel and perpendicular lines. For example, students
will prove or disprove that a figure defined by four given points in the coordinate plane is a
rectangle.
Students will also experiment with transformations in the plane. In Grade 8, students had
experience with transformations: translations, reflections, rotations, and dilations. Comparisons
of transformations will provide the foundation for understanding similarity and congruence.
Additionally, students will prove theorems, using a variety of formats, and solve problems about
triangles, quadrilaterals, and other polygons. Students will be exposed to multiple ways of
writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and
using diagrams without words. They also apply reasoning to complete geometric constructions
and explain why they work. Students should be encouraged to focus on the validity of the
underlying reasoning while exploring a variety of formats for expressing that reasoning.
Similarity
Students apply their earliest experience with dilations and proportional reasoning to build a
formal understanding of similarity. The only transformations that do not absolutely preserve
congruence are dilations. Focus will be on teh power of dilations to effect size change in
geometric figures. Dilations preserve angle measure while enlarging or reducing lengths
proportionally. Students will be able to determine if two figures are similar, essentially a dilation
of each other, by comparing their angles for congruence and their sides for
proportionality. Students will discover that the other transformations (reflections, rotations,
shifts, translations) produce a similarity in a 1:1 ratio and thus are a special subset of similarity
transformations, namely transformations that produce congruent figures. (A metaphor for this
would be that of a square being a specifically defined rectangle in that way that a congruence is a
specifically defined similarity).
Congruence
Students will build on their prior experience with rigid motions, including translations,
reflections and rotations, in order to develop notions about what it means for two objects to be
congruent. Rigid motions are at the foundation of the definition of congruence, and students
should recognize that they preserve distance and angle. Students will continue to develop their
understanding that congruence is a special case of similarity with a 1:1 ratio. Students should
experiment with rigid motions in the coordinate plane in order to carry given polygons onto
themselves, in order to verify major properties of parallelograms, and in order to determine the
congruence of given polygons. Students should be encouraged to use a variety of tools, such as
protractors, compasses, MIRA boards, patty paper, graph paper, and geometry software in order
to transform given figures to prove congruence. In addition, students should use a variety of
methods for proof, both formal and informal. Students will apply their understanding of
congruence to copy segments and angles using a variety of tools and construction methods.
Enduring Understandings:
1. A diagram is a sophisticated mathematical device for thinking and communicating. A
diagram is a built geometric artifact, with both a history- a narrative of successive
construction- and a purpose. A diagram is not a picture. It needs to be interpreted:
learning how to read a diagram can be like learning a new language.
2. Empirical verification is an important part of the process of proving, but it can never, by
itself, constitute a proof. Geometry uses a wide variety of kinds of proofs.
3. The processes of proving include a variety of activities, such as developing conjectures,
considering the general case, exploring with examples, looking for structural similarities
across cases, and searching for counterexamples.
4. Making sense of others’ arguments and determining their validity are proof-related
activities.
5. A proof can have many different valid representational forms, including narrative,
picture, diagram, two-column presentation, or algebraic form.
6. Underlying any geometric theorem is an invariance- something that does not change
while something else does.
7. Similar geometric figures have angles that are congruent and segments that are
proportional in length.
8. Similar geometric figures can be created by transformations. All transformations create
similar geometric figures. Dilations, in particular, create figures that are similar, but may
not be congruent.
9. Congruence is also similarity. It is just a more specifically defined similarity where the
ratio of lengths is 1:1.
Sources:
Ellis, A. B., Bieda, K., & Knuth, E. (2012). Developing Essential Understanding of Proof and
Proving. Reston, VA: The National Council of Teachers of Mathematics, Inc.
Pimm, D., Sinclair, N., & Skelin, M. (2012). Developing Essential Understanding of Geometry.
Reston,
VA: The National Council of Teachers of Mathematics, Inc.
Essential Questions:
Transformations and the Coordinate Plane
● How is visualization essential to the study of geometry?
● How does the concept of rigid motion connect to the concept of congruence?
● How does geometry explain or describe the structure of our world?
● How do geometric constructions enhance understanding of the geometric properties of
objects?
● How can reasoning be used to establish or refute conjectures?
● What are the characteristics of a valid argument?
● What is the role of deductive or inductive reasoning in validating a conjecture?
● What facts need to be verified in order to establish that two figures are congruent?
● How can the properties of geometric figures be verified using a coordinate plane?
● What is the connection between parallel and perpendicular lines in coordinate geometry
as it relates to systems of equations studied in Algebra?
● What is the connection between the distance formula and the Pythagorean theorem?
● How can coordinate geometry be used to prove simple theorem algebraically?
Similarity
● How can transformations be used to explain similarity?
● How is similarity of geometric figures applied and verified?
● What is the relationship between transformations that produce congruent figures and
transformations that produce similar figures?
Congruence
● Which transformations preserve distance and angle?
● How can a figure be transformed to carry on to itself?
● What tools can be used to perform given transformations?
● What are the properties of various parallelograms and how can they be verified using
rigid motions?
● Can transformations be applied in any sequence in order to carry a shape onto itself?
● How do congruence and similarity relate? How are they similar? How are they
different?
Common Misconceptions:
Transformations and the Coordinate Plane
● When proving geometric theorems, students struggle with finding the most efficient
process.
●
When proving geometric theorems, students may have gaps in their reasoning and miss
key steps.
Similarity
● Students do not understand the relationship between congruence and similarity. All
figures that are congruent are in fact similar or a size with a scale factor of 1.
● Students have difficulty with transformations and should be encouraged to use tools
strategically.
● Students struggle with the use of certain geometric tools.
Congruence
● Students have difficulty with transformations and should be encouraged to use tools
strategically.
● Students confuse congruence with equality. It should be stressed that congruence refers
to figures, while equality refers to values and distances.
● Students struggle with the use of certain geometric tools.