standard_description: Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
problem_activity: Activity
The purpose of this task is to use rigid transformations to describe an important picture that students have seen in grade 6 when they developed the formula for the area of a triangle. They first found the area of a parallelogram to be
\(\text{base} \boldcdot \text{height}\)
and then, to find
\(\frac{1}{2}\text{base} \boldcdot \text{height}\)
for the area of a triangle, they “composed” two copies of a triangle to make a parallelogram. The language “compose” is a grade 6 appropriate way of talking about a
\(180^\circ\)
rotation. The focus of this activity is on developing this precise language to describe a familiar geometric situation.
Students need to remember and use an important property of 180 degree rotations, namely that the image of a line after a 180 degree rotation is parallel to that line. This is what allows them to conclude that the shape they have built is a parallelogram (MP7).
Launch
Arrange students in groups of 2. Provide access to geometry toolkits. A few minutes of quiet work time, followed by sharing with a partner and a whole-class discussion.
Action and Expression: Internalize Executive Functions.
To support development of organizational skills, check in with students within the first 2–3 minutes of work time. Look for students who struggle with visualizing the 180 degree rotation using center
\(M\)
. Consider pausing for a brief discussion to invite 1–2 pairs of students to demonstrate and explain how to do the rotation.
Supports accessibility for: Memory; Organization
Student Facing
Here is triangle
\(ABC\)
.
Draw midpoint
\(M\)
of side
\(AC\)
.
Rotate triangle
\(ABC\)
180 degrees using center
\(M\)
to form triangle
\(CDA\)
. Draw and label this triangle.
What kind of quadrilateral is
\(ABCD\)
? Explain how you know.
[IMG]
Student Response
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Student Facing
Are you ready for more?
In the activity, we made a parallelogram by taking a triangle and its image under a 180-degree rotation around the midpoint of a side. This picture helps you justify a well-known formula for the area of a triangle. What is the formula and how does the figure help justify it?
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Students may struggle to see the
\(180^\circ\)
rotation using center
\(M\)
. This may be because they do not understand that
\(M\)
is the center of rotation or because they struggle with visualizing a
\(180^\circ\)
rotation. Offer these students patty paper, a transparency, or the rotation overlay from earlier in this unit to help them see the rotated triangle.
Activity Synthesis
Begin the discussion by asking, "What happens to points
\(A\)
and
\(C\)
under the rotation?" (They end up at
\(C\)
and
\(A\)
, respectively.) This type of rotation and analysis will happen several times in upcoming lessons.
Next ask, "How do you know that the lines containing opposite sides of
\(ABCD\)
are parallel?" (They are taken to one another by a 180 degree rotation.) As seen in the previous lesson, the image of a
\(180^\circ\)
rotation of a line
\(\ell\)
is parallel to
\(\ell\)
. Students also saw that when
\(180^\circ\)
rotations were applied to a
pair
of parallel lines it resulting in a (sometimes) new pair of parallel lines which are also parallel to the original lines. The logic here is the same, except that only one line is being rotated
\(180^\circ\)
rather than a pair of lines. This does not need to be mentioned unless it is brought up by students.
Finally, ask students "How is the area of parallelogram
\(ABCD\)
is related to the area of triangle
\(ABC\)
?" (The area of the parallelogram
\(ABCD\)
is twice the area of triangle
\(ABC\)
because it is made up of
\(ABC\)
and
\(CDA\)
which has the same area as
\(ABC\)
.) Later in this unit, area of shapes and their images under rigid transformations will be studied further.
Writing, Speaking: Math Language Routine 1 Stronger and Clearer Each Time.
This is the first time Math Language Routine 1 is suggested as a support in this course. In this routine, students are given a thought-provoking question or prompt and are asked to create a first draft response. Students meet with 2–3 partners to share and refine their response through conversation. While meeting, listeners ask questions such as, “What did you mean by . . .?” and “Can you say that another way?” Finally, students write a second draft of their response reflecting ideas from partners and improvements on their initial ideas. The purpose of this routine is to provide a structured and interactive opportunity for students to revise and refine their ideas through verbal and written means.
Design Principle(s): Optimize output (for justification)
How It Happens:
Use this routine to provide students a structured opportunity to refine their justification for the question asking “What kind of quadrilateral is
\(ABCD\)
? Explain how you know.” Give students 2–3 minutes to individually create first draft responses in writing.
Invite students to meet with 2–3 other partners for feedback.
Instruct the speaker to begin by sharing their ideas without looking at their written draft, if possible. Listeners should press for details and clarity.
Provide students with these prompts for feedback that will help individuals strengthen their ideas and clarify their language: “What do you mean when you say...?”, “Can you describe that another way?”, “How do you know the lines are parallel?”, and “What happens to lines under rotations?” Be sure to have the partners switch roles. Allow 1–2 minutes to discuss.
Signal for students to move on to their next partner and repeat this structured meeting.
Close the partner conversations and invite students to revise and refine their writing in a second draft. Students can borrow ideas and language from each partner to strengthen the final product.
Provide these sentence frames to help students organize their thoughts in a clear, precise way: “Quadrilateral
\(ABCD\)
is a ____ because.…” and “Another way to verify this is.…”.
Here is an example of a second draft:
“
\(ABCD\)
is a parallelogram, I know this because a 180-degree rotation creates new lines that are parallel to the original lines. In this figure, the 180-degree rotation takes line
\(AB\)
to line
\(CD\)
and line
\(BC\)
to line
\(AD\)
. I checked this by copying the triangle onto patty paper and rotating it 180 degrees. This means the new shape will have two pairs of parallel sides. Quadrilaterals that have two pairs of parallel sides are called parallelograms.”
If time allows, have students compare their first and second drafts. If not, have the students move on by discussing other aspects of the activity.


aligns: no