standard_description: Extend polynomial identities to the complex numbers. For example, rewrite x^(2) + 4 as (x + 2i)(x – 2i).
problem_activity: Activity
In this activity, students apply their understanding of the properties of rigid transformations to
\(180^\circ\)
rotations of a line about a point on the line in order to establish the vertical angle theorem. Students have likely already used this theorem in grade 7, but this lesson informally demonstrates why the theorem is true. The demonstration of the vertical angle theorem exploits the structure of parallel lines and properties of both 180 degree rotations (studied in the previous lesson) and rigid transformations. This lesson is a good example of MP7, investigating the structure of different mathematical objects.
Students begin the activity by rotating a line with marked points
\(180^\circ\)
about a point on the line. Unlike the whole-class example discussed in the launch, this line contains marked points other than the center of rotation. Then students rotate an angle
\(180^\circ\)
about a point on the line to draw conclusions about lengths and angles. Finally, students are asked to consider the intersection of two lines, the angles formed, and how the measurements of those angles can be deduced using a
\(180^\circ\)
rotation about the intersection of the lines, which is the vertical angle theorem.
While students are working, encourage the use of tracing paper to show the transformations directly over the original image in order to help students keep track of what is happening with lines in each
\(180^\circ\)
rotation.
Launch
Rotations require the students to think about rotating an entire figure. It would be good to remind students about this before the start of this activity. This might help students see what is happening in the first question better.
Before students read the activity, draw a line
\(\ell\)
with a marked point
\(D\)
for all to see. Ask students to picture what the figure rotated
\(180^\circ\)
around point
\(D\)
looks like. After a minute of quiet think time, invite students share what they think the transformed figure would look like. Make sure all students agree that
\(\ell’\)
looks “the same” as the original. If no students bring it up in their explanations, ask for suggestions of features that would make it possible to quickly tell the difference between the
\(\ell’\)
and
\(\ell\)
, such as another point or if the line were different colors on each side of point
\(D\)
.
Provide access to tracing paper.
Engagement: Internalize Self Regulation.
Chunk this task into more manageable parts to differentiate the degree of difficulty or complexity. For example, to get students started, display only the first problem and diagram. Once students have successfully completed the three parts, introduce the remaining problems, one at a time. Invite students to think aloud about the relationships between the angles after the 180 degree rotation.
Supports accessibility for: Organization; Attention
Student Facing
The diagram shows a line with points labeled
\(A\)
,
\(C\)
,
\(D\)
, and
\(B\)
.
On the diagram, draw the image of the line and points
\(A\)
,
\(C\)
, and
\(B\)
after the line has been rotated 180 degrees around point
\(D\)
.
Label the images of the points
\(A’\)
,
\(B’\)
, and
\(C’\)
.
What is the order of all seven points? Explain or show your reasoning.
[IMG]
The diagram shows a line with points
\(A\)
and
\(C\)
on the line and a segment
\(AD\)
where
\(D\)
is not on the line.
Rotate the figure 180 degrees about point
\(C\)
. Label the image of
\(A\)
as
\(A’\)
and the image of
\(D\)
as
\(D’\)
.
What do you know about the relationship between angle
\(CAD\)
and angle
\(CA’D’\)
? Explain or show your reasoning.
[IMG]
The diagram shows two lines
\(\ell\)
and
\(m\)
that intersect at a point
\(O\)
with point
\(A\)
on
\(\ell\)
and point
\(D\)
on
\(m\)
.
Rotate the figure 180 degrees around
\(O\)
. Label the image of
\(A\)
as
\(A’\)
and the image of
\(D\)
as
\(D’\)
.
What do you know about the relationship between the angles in the figure? Explain or show your reasoning.
[IMG]
Student Response
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Anticipated Misconceptions
In the second question, students may not understand that rotating the figure includes both segment
\(CA\)
and segment
\(AD\)
since they have been working with rotating one segment at a time. Explain to these students that the figure refers to both of the segments. Encourage them to use tracing paper to help them visualize the rotation.
Activity Synthesis
The focus of the discussion should start with the relationships students find between the lengths of segments and angle measures and then move to the final problem, which establishes the vertical angle theorem as understood through rigid transformations. Questions to connect the discussion include:
"What relationships between lengths did we find after performing transformations?" (They are the same.)
"What relationships between angle measures did we find after performing transformations?" (They are the same.)
"What does this transformation informally prove?" (Vertical angles are congruent.)
If time permits, consider discussing how the vertical angle theorem was approached in grade 7, namely by looking for pairs of supplementary angles. Pairs of vertical angles have the same measure because they are both supplementary to the
same
angle. The argument using 180 degree rotations is different because no reference needs to be made to the supplementary angle. The 180 degree rotation shows that both pairs of vertical angles have the same measure directly by mapping them to each other!
Representing, Speaking: MLR7 Compare and Connect.
Use this routine when students share what they noticed about the relationships between the angle measures. Ask students to consider what changes and what stays the same when rigid transformations are applied to lines and segments. Draw students’ attention to the associations between the rigid transformation, lengths of segments, and angle measures. These exchanges strengthen students’ mathematical language use and reasoning based on rigid transformations of lines and will lead to the informal argument of the vertical angle theorem.
Design Principle(s): Maximize meta-awareness


aligns: no