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To Gather
Math Community Chart
This Math Talk focuses on describing rigid motions without a grid. It encourages students to think about the labeled points and to rely on the structure of the figures to mentally solve problems. The strategies elicited here will be helpful later in the lesson when students will need to be able to put together rigid transformations that take one polygon to another without making reference to a grid.
In describing the transformations, students need to be precise in their word choice and use of language (MP6).
Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:
Keep all previous problems and work displayed throughout the talk.
Segment
To involve more students in the conversation, consider asking:
“Who can restate
“Did anyone use the same strategy but would explain it differently?”
“Did anyone solve the problem in a different way?”
“Does anyone want to add on to
“Do you agree or disagree? Why?”
“What connections to previous problems do you see?”
If no student brings it up, ask students how a rotation may be used to take
Math Community
After the Warm-up, display the revisions to the class Math Community Chart that were made from student suggestions in an earlier exercise. Tell students that over the next few exercises, this chart will help the class decide on community norms—how they as a class hope to work and interact together over the year. To get ready for making those decisions, students are invited at the end of today’s lesson to share which “Doing Math” action on the chart is most important to them personally.Display two congruent triangles for all to see:
Ask students:
Display another image for all to see:
Ask students:
Tell students that defining these sequences and looking at the results is enough to conjecture that the sequences take one figure onto another. To be convincing, it is necessary to be able to explain why each vertex lands exactly where they think it should land. Explaining why each point lands exactly where it should is something they will work on over time.
If two figures are congruent, we can always find a rigid transformation that takes one onto the other.
Look at congruent figures
Our goal is to take
We can start with translation: Translate triangle
Now, a pair of corresponding points coincides. Is there a transformation we could use to take
Now, two pairs of corresponding points coincide. Reflecting across line
It is always possible to describe transformations using existing points, angles, and segments. It could take an extra step, but we can be confident that transformations work if we don't guess where the line of reflection or center of rotation might be.
Help us improve by sharing suggestions or reporting issues.
Some students may think that
To Copy (from Blackline Masters)
How Did This Get There Cards
To Gather
Geometry toolkits (HS)
The purpose of this activity is to activate students’ prior knowledge about rigid and nonrigid transformations. This is the first Card Sort activity of the course. An important aspect of this routine is to allow students time at the start to sort the cards into categories of their choosing. This step gives students the opportunity to familiarize themselves with the content of the cards without the additional pressure of organizing them in a specific fashion. It also provides insight into the aspects of each card students attend to and the language they have to describe their observations.
Students also begin to practice the skill of defining a rigid transformation that takes one figure onto the other without using a grid to estimate or define centers, angles, lines of reflection, or directed line segments. If students struggle to define rigid motions precisely, they may benefit from doing the optional lesson on point-by-point transformations.
Students may struggle to describe the location of the line of reflection. There is no need to address this during this activity as it will be covered in a subsequent activity.
Monitor for students who first translate to line up one pair of corresponding vertices and then rotate for the card with triangle
Arrange students in groups of 2, and distribute pre-cut cards. Tell them that in this activity, they will sort some cards into categories of their choosing. When they sort the images, they should work with their partner to come up with categories.
Allow students to familiarize themselves with the representations on the cards:
Attend to the language that students use to describe their categories and images, giving them opportunities to describe their images more precisely. Highlight the use of terms like “congruent,” “polygon,” and “transformation.” After a brief discussion, invite students to complete the remaining questions.
Your teacher will give you a set of cards that show transformations of figures. Sort the cards into categories of your choosing. Be prepared to explain the meaning of your categories.
For each card with a rigid transformation, write a sequence of rotations, translations, and reflections to get from the original figure to the image. Be precise.
The purpose of this discussion is to re-emphasize that translations, rotations, and reflections are rigid transformations and to work on describing those transformations without the structure of a grid.
Invite students to share their strategies for the card with triangles
Wait on a detailed discussion of the transformation of the card with
To Gather
Geometry toolkits (HS)
In this activity, students determine whether a reflection can take one figure to another and how to find the precise line of reflection for disjoint figures.
Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).
Invite a few students to share their responses with the class. Highlight any improvements over previous justifications such as good use of a well-labeled diagram. For each transformation a student suggests, ask them “What points coincide now?”
Diego says, “I see why a reflection could take