standard_description: Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
problem_activity: Activity
Students have seen images showing a sequence of transformations in the first lesson of this unit, however they have not heard the term sequence of transformations. They have also not been asked to describe the moves in the sequence using precise language. The launch of this activity introduces this term and gives students an opportunity to describe the sequence of more than one transformation.
For the second problem, encourage students to find a different sequence of transformations than the one shown in the image. Each time a reflection is mentioned, ask students where the line of reflection is located and when a rotation is mentioned, ask for the center of the rotation and the number of degrees. Monitor for students who apply different transformations (or apply transformations in a different order).
Digital
Launch
Arrange students in groups of 2, and provide access to their geometry toolkits. Display the image for all to see. Ask students if they can imagine a single translation, rotation, or reflection that would take one bird to another? After a minute, verify that this is not possible.
Ask students to describe how we could use translations, rotations, and reflections to take one bird to another. Collect a few different responses. (One way would be to take the bird on the left, translate it up, and then reflect it over a vertical line.) Tell students when we do one or more transformations in a row to take one figure to another, it is called a
sequence of transformations
.
[IMG]
If using the digital activity, you may want to review the transformation tools in the applet. (The instructions are repeated in the activity for students' reference.)
Give students 2 minutes of quiet work time to engage in the task followed by 3 minutes to discuss their responses with a partner and complete any unfinished questions. Follow with a whole-class discussion.
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner. Display sentence frames to support student conversation such as “To take Figure
\(A\)
to Figure
\(B\)
, I _____ because…”, “I noticed _____ so I...”, “Why did you...?”, or “I agree/disagree because….”
Supports accessibility for: Language; Social-emotional skills
Student Facing
Here are some figures on an isometric grid. Explore the transformation tools in the tool bar. (Directions are below the applet if you need them.)
Name a transformation that takes Figure A to Figure B. Name a transformation that takes Figure B to Figure C.
What is one
sequence of transformations
that takes Figure A to Figure C? Explain how you know.
Translate
Select the Vector tool.
[IMG]
Click on the original point and then the new point. You should see a vector.
Select the Translate by Vector tool.
[IMG]
Click on the figure to translate, and then click on the vector.
Rotate
Select the Rotate around Point tool.
[IMG]
Click on the figure to rotate, and then click on the center point.
A dialog box will open. Type the angle by which to rotate and select the direction of rotation.
Reflect
Select the Reflect about Line tool.
[IMG]
Click on the figure to reflect, and then click on the line of reflection.
Student Response
Teachers with a valid work email address can
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for free access to Student Response.
Print
Launch
Arrange students in groups of 2, and provide access to their geometry toolkits. Display the image for all to see. Ask students if they can imagine a single translation, rotation, or reflection that would take one bird to another? After a minute, verify that this is not possible.
Ask students to describe how we could use translations, rotations, and reflections to take one bird to another. Collect a few different responses. (One way would be to take the bird on the left, translate it up, and then reflect it over a vertical line.) Tell students when we do one or more transformations in a row to take one figure to another, it is called a
sequence of transformations
.
[IMG]
If using the digital activity, you may want to review the transformation tools in the applet. (The instructions are repeated in the activity for students' reference.)
Give students 2 minutes of quiet work time to engage in the task followed by 3 minutes to discuss their responses with a partner and complete any unfinished questions. Follow with a whole-class discussion.
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner. Display sentence frames to support student conversation such as “To take Figure
\(A\)
to Figure
\(B\)
, I _____ because…”, “I noticed _____ so I...”, “Why did you...?”, or “I agree/disagree because….”
Supports accessibility for: Language; Social-emotional skills
Student Facing
Here are some figures on an isometric grid.
[IMG]
Name a transformation that takes Figure
\(A\)
to Figure
\(B\)
. Name a transformation that takes Figure
\(B\)
to Figure
\(C\)
.
What is one
sequence of transformations
that takes Figure
\(A\)
to Figure
\(C\)
? Explain how you know.
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?
Experiment with some other ways to take Figure
\(A\)
to Figure
\(C\)
. For example, can you do it with. . .
No rotations?
No reflections?
No translations?
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 students with different correct responses to show their solutions. Be sure to highlight at least one rotation. If no students mention that, demonstrate a way to take A to C that involves a rotation. Whether or not students use the geogebra applet, it may be helpful to display the applet to facilitate discussion:
ggbm.at/jqvTEgsj
Emphasize that there are many ways to describe the translation that takes figure A to figure B. All one needs is to identify a pair of corresponding points and name them in the correct order (and to use the word “translate”).
For students who used a reflection to take B to C, emphasize that reflections are determined by lines and we should name the line when we want to communicate about it.
After a student or the teacher uses a rotation, emphasize that a rotation is defined by a center point and an angle (with a direction). The center point needs to be named and the angle measure or an angle with the correct measure needs to be named as well (as does the direction). Reinforce to students that when we do more than one transformation in a row, we call this a sequence of transformations.
Speaking: Math Language Routine 7 Compare and Connect.
This is the first time Math Language Routine 7 is suggested as a support in this course. In this routine, students are given a problem that can be approached using multiple strategies or representations and are asked to prepare a visual display of their method. Students then engage in investigating the strategies (by means of a teacher-led gallery walk, partner exchange, group presentation, etc.), comparing approaches, and identifying correspondences between different representations. A typical discussion prompt is: “What is the same and what is different?” regarding their own strategy and that of the others. The purpose of this routine is to allow students to make sense of mathematical strategies by identifying, comparing, contrasting, and connecting other approaches to their own, and to develop students’ awareness of the language used through constructive conversations.
Design Principle(s): Maximize meta-awareness: Support sense-making
How It Happens:
Use this routine to compare and contrast different strategies for transforming Figure A to Figure C. Invite students to create a visual display showing how they made sense of the problem and why their solution makes sense for transforming Figure A to Figure C.
Students should include these features on their display:
a sketch of the figures (not necessary if using the applet)
a sketch of the figures after each transformation (not necessary if using the applet)
a written sequence of transformations with an explanation
Before selecting students to show their solutions to the class, first give students an opportunity to do this in a group of 3–4. Ask students to exchange and investigate each other’s work. Allow 1 minute for each display and signal when it is time to switch.
While investigating each other’s work, ask students to consider what is the same and what is different about each approach. Next, give each student the opportunity to add detail to their own display for 1–2 minutes.
As groups are presenting, circulate the room and select 2–3 students to share their sequence of transformations taking Figure A to Figure C. Be sure to select a variety of approaches, including one that involves a rotation.
Draw students’ attention to the different ways the figures were transformed (e.g., rotations, reflections, and translations) and how the sequence of transformation is expressed in their explanation. Also, use the bullet points in the Activity Synthesis to emphasize specific features of translations, reflections, and rotations.
After the selected students have finished sharing with the whole class, lead a discussion comparing, contrasting, and connecting the different approaches.
Consider using these prompts to amplify student language while comparing and contrasting the different approaches: “Why did the approaches lead to the same outcome?”, “What worked well in __’s approach? What did not work well?”, and “What would make __’s strategy more complete or easy to understand?”
Consider using these prompts to amplify student language while connecting the different approaches: “What role does a translation play in each approach?” “Is it possible to use the all three types of transformations?”, and “What transformation do you see present in all the strategies?”
Close the discussion by inviting 3 students to revoice the strategies used in the presentations, and then transition back to the Lesson Synthesis and Cool-Down.


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