G.CO.A.2 Lesson Seeds: Defining Similarity and Congruence
Experiment with transformations in the plane.
G.CO.A.2 Represent transformations in the plane using, e.g., transparencies and
geometry software; describe transformations as functions that take points in the plane as
inputs and give other points as outputs. Compare transformations that preserve distance
and angle to those that do not (e.g., translation versus horizontal stretch).
Opening Question:
Construct two figures for each of the following scenarios.
a) figures that are different from each other
b) figures that are almost the same as each other
c) figures that are exactly the same as each other
Allow students to work in groups or partners with multiple tools in order to
construct each set of figures.
Discussion Questions:
These questions are intended to formatively assess prior knowledge and to begin
student discussion of similarity versus congruence. Students should be encouraged to
answer in their own words and to critique each other’s assessments. If students are
unsure about an answer, have them revisit their tools from the opening question in
order to make a more informed decision.
1. How do you know your figures were the same? If they are not exactly the
same, how are they different? How did you interpret “almost the same”? Can
you construct figures that are closer to being the same than your first pair
but are still slightly different?
2. What do you think are the qualifications that make two figures exactly the
same?
3. Can two figures be the same type of figure but be different from each other in
multiple ways?
4. What makes two figures look like each other?
5. Can two figures look like each other but be different sizes? Can two figures
be the same size but look like different figures? Which of these scenarios
creates figures that look more alike?
6. How do angle measurements affect sameness? How do side lengths affect
sameness? Which one has more of an effect on sameness?
7. What would it take for two figures to be proportional to each other?
Other Activities:
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Have students construct two figures that are proportional. Instruct them to
measure all sides and angles in order to make conjectures about proportional
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figures. Have students do the same with two identical figures, and discuss
the 1:1 ratio of the second set of figures.
Have students complete the graphic organizer for Similar and Congruent
figures by writing information they know about each term as well as
sketching examples.
Have students construct multiple similar figures using appropriate tools,
once they have developed a better understanding about similarity. Discuss
proportionality using their figures. Have them do the same with congruent
figures. Discuss which is more difficult to create as well as which has more
options. Make the connection that congruence is one specific example of
similarity that has a 1:1 ratio.
Have students discuss transformations in relationship to similarity and
congruence. Which transformations preserve congruence? Which create
similar figures?