##
* *Reflection: Student Grouping
Translations, Reflections, and Rotations - Section 1: Pair Practice: Translations and More Rotations

I was glad that I asked students to practice in pairs because this created a natural opportunity for me to informally assess students' understanding while holding them accountable to the work. On this assignment, students needed to show their understanding of translations by describing them with words and with an ordered pair rule. As students worked in pairs, they checked in with each other, attending to precision as they carefully considered the words and notation they would use. Students, for the most part, were able to work through their confusion and catch their mistakes on their own, which helped foster the idea that they are mathematical authorities--this is a notion I try to impress on my students all year long. For this reason, when pairs called me over when they were genuinely stumped, I tried very hard *not* to answer their questions, but instead, to train them to utilize the resources around them so they could see how they could solve their own problems.

A change I made this year was to have students write an exit ticket reflection to their partner. It is important to me that my students reflect on how they and others worked, especially because I want them to see the connection between *how* they work and *what *they learn. Because this lesson occurred at the beginning of the year, I thought I might provide students with some structure by providing them with some sentence frames:

- "Something I learned from my partner was..."
- "My partner helped me understand..."
- "A question I asked my partner was..."
- "A goal I have for working with a partner in the future is..."

I was pleased to read these exit tickets and to see that students, overall, recognized that they could learn more when working with others.

*Increasing Accountability Through Pair Practice*

*Student Grouping: Increasing Accountability Through Pair Practice*

# Translations, Reflections, and Rotations

Lesson 2 of 4

## Objective: Students will discover and apply characteristics of translation, reflection, and rotation and will explain (using words and symbols) the position and orientation of 2-D shapes after transforming them.

## Big Idea: Building conceptual understanding and fluency with transformations requires lots of practice.

*75 minutes*

I ask students to work in pairs Translations Practice. In this task, students make connections between the coordinates, graphs, and verbal and symbolic representations of shapes under translation. Working together encourages risk-taking. I also think it promotes individual understanding and accountability for learning. Pairs of students perform translations and see that while translations change the position of a figure, the pre-image and image are congruent because corresponding angles and sides are congruent. Giving students tracing paper in this activity (and throughout the unit) helps reinforce observations that translations, reflections, and rotations produce congruent figures.

To keep the learning on track I have pairs of students check in with me after each section. As they check in, I look over their work and ask questions like “**how do you see the translation in the graph and why does it make sense with your ordered pair rules?**” or “**how do you know the figures are congruent?**” which require them to justify their answers (**MP3)**. When pairs have sufficiently demonstrated their understanding to me, I ask them to work on reflection and translation practice.

In this practice, students also discover reflecting a figure over the x-axis and then the y-axis is equivalent to rotating the figure 180^{o} about the origin—again, tracing paper can really help students to understand this equivalence. What is important is that this discovery highlights different transformations or combinations of transformations producing the same result, a understanding which is pivotal to students’ ability to engage in algebraic manipulation and symbolic sense making **(MP1)**.

When pairs have sufficiently demonstrated their understanding to me, I ask them to work on Transformations Page 1, which gives them opportunities to make these transformations concrete. In this practice, students also discover that reflecting a figure over the x-axis and then the y-axis is equivalent to them rotating the figure 180^{o} about the origin—again, tracing paper can really help students to understand this equivalence—which highlights the idea that different transformations or combinations of transformations can produce the same result, an understanding that is pivotal to students’ ability to engage in algebraic manipulation and symbolic sense making, which is critical to their success in future math classes.

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#### Debrief Notes: Rotations

*10 min*

As typical for all of our lessons, we debrief the lesson’s main ideas in our Notetakers at the end of the lesson. Since students have now played around with translation, reflection, and rotation, it is an opportune time to discuss what stays the same and what changes under translation, reflection, and rotation and to organize our ideas in a table.

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- UNIT 1: Creating Classroom Culture to Develop the Math Practices
- UNIT 2: Introducing Geometry
- UNIT 3: Transformations
- UNIT 4: Discovering and Proving Angle Relationships
- UNIT 5: Constructions
- UNIT 6: Midterm Exam Review
- UNIT 7: Discovering and Proving Triangle Properties
- UNIT 8: Discovering and Proving Polygon Properties
- UNIT 9: Discovering and Proving Circles Properties
- UNIT 10: Geometric Measurement and Dimension
- UNIT 11: The Pythagorean Theorem
- UNIT 12: Triangle Similarity and Trigonometric Ratios
- UNIT 13: Final Exam Review