standard_description: Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).
problem_activity: Optional activity
The goal of this activity is for students to start to develop a systematic, point-by-point sequence of transformations that will work to take
any
pair of congruent polygons onto one another. This is especially important because when transformations are used in triangle congruence proofs in a subsequent lesson, students will be justifying how they know that a certain transformation will take one triangle onto another. Finding a single transformation that takes each vertex of one triangle onto each corresponding vertex of another cannot be generalized because it is specific to one pair of congruent triangles. In a proof, students must define sequences of rigid transformations that take a given triangle to any triangle with congruent corresponding parts, and justify why that sequence is guaranteed to take each vertex to its corresponding vertex.
Monitor for students who:
Find a single, separate transformation that takes
\(A\)
to
\(A’\)
and
\(B\)
to
\(B’\)
, so that when
\(B\)
goes to
\(B’\)
,
\(A\)
no longer remains on
\(A’\)
.
Look for a single transformation that takes quadrilateral
\(ABCD\)
onto quadrilateral
\(A’B’C’D’\)
all at once.
Line up one or two points and then get stuck.
Systematically line up pairs of corresponding points, then use transformations with fixed points to line up edges. You might find students who use translation, then rotation, then reflection, as well as students who use reflections across perpendicular bisectors and then edges.
Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).
Digital
Launch
Engagement: Develop Effort and Persistence.
Connect a new concept to one with which students have experienced success. For example, remind students about the sequence of translations and rotations that take
\(ABC\)
to
\(DEF\)
in the previous activity. Ask students how they can use the same idea to describe the sequence of translations and rotations that take parallelogram
\(ABCD\)
to
\(A’B’C’D’\)
.
Supports accessibility for: Social-emotional skills; Conceptual processing
Student Facing
For each question, describe a sequence of translations, rotations, and reflections that will take parallelogram
\(ABCD\)
to parallelogram
\(A'B'C'D'\)
.
1.
2.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Engagement: Develop Effort and Persistence.
Connect a new concept to one with which students have experienced success. For example, remind students about the sequence of translations and rotations that take
\(ABC\)
to
\(DEF\)
in the previous activity. Ask students how they can use the same idea to describe the sequence of translations and rotations that take parallelogram
\(ABCD\)
to
\(A’B’C’D’\)
.
Supports accessibility for: Social-emotional skills; Conceptual processing
Student Facing
For each question, describe a sequence of translations, rotations, and reflections that will take parallelogram
\(ABCD\)
to parallelogram
\(A'B'C'D'\)
.
[IMG]
[IMG]
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
In this unit, we have been focusing on rigid transformations in two dimensions. By thinking carefully about precise definitions, we can extend many of these ideas into three dimensions. How could you define rotations, reflections, and translations in three dimensions?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
Select previously identified students to share in this order:
Look for a single transformation that moves all the points at once.
Line up one or two points and then get stuck.
Systematically line up pairs of corresponding points, then use transformations with fixed points to line up edges.
For each strategy shared, focus on what question that strategy helps to answer or what is unique and valuable about that strategy.
If a student who shares looked for a single transformation, focus on the elegance of finding a single rigid motion, as well as inviting the student to share what was hard about their strategy.
If a student who shares got stuck, emphasize the strategic choice to try to line up just one or two points at a time, and why that might be easier than finding one or two more complicated transformations.
If a student who shares systematically lined up one point at a time using transformations with fixed points to ensure that previously lined-up points did not move, emphasize why this strategy could be used for lots of different starting conditions.
Speaking: MLR 7 Compare and Connect
. Use this routine when students present their sequence of translations, rotations, and reflections that will take parallelogram
\(ABCD\)
to parallelogram
\(A’B’C’D’\)
. Ask students to consider what is the same and what is different about the sequences of transformations. Draw students’ attention to the way that transformations are described precisely (that is, order, numbers, words). These exchanges strengthen students’ mathematical language use and reasoning about transformations.
Design Principle(s): Maximize meta-awareness


aligns: no