From my experience teaching these concepts, many students are familiar with translations and can often move through this part of the module much quicker than others. Typically before learning about congruence transformations, students are proficient in basic algebra skills (simplifying expressions, solving multi-step equations), basic graphing skills (graphing points, graphing figures on the coordinate plane), basic comprehension of rotational and reflection symmetry, and understanding of congruence.

This learning module should take approximately 5 weeks to complete, but can and should be modified based on the students' needs. The outline of the timeframe is provided in detail on the "Student" side of this section of the learning module.

Students will need access to Internet and a device like a laptop or tablet to effectively work through this module. Supplemental materials that would be beneficial for students would include: graphing paper, straightedge/ruler, patty paper, and a mirra. These materials are supplemental because some students may not benefit from these manipulatives, but other students may need to physically interact with the concepts to fully grasp them.

The following Common Core State Standards align with this learning module:

1. CCSS.MATH.CONTENT.HSG.CO.A.2 - Represent transformations in the plane using, e.g., transparencies and geometric software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not.
2. CCSS.MATH.CONTENT.HSG.CO.A.3 - Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
3. CCSS.MATH.CONTENT.HSG.CO.A.5 - Given a geometric figure and a rotation, reflection or translation, draw the transformed figure suing, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
4. CCSS.MATH.CONTENT.HSG.MG.A.3 - Apply geometric methods to solve design problems.

This section on translations, I have found, is usually easier for students to grasp. Because of this, I always introduce congruency transformations with translations. This gives students a boost in their confidence at the beginning of a unit that will help them throughout the entire module. The students have two options to choose from to learn about translations. The first is a flipped classroom approach that is more direct. The students will watch the video lecture and copy down the corresponding note pages provided as a PDF in this section. The second is a more exploratory approach with the help of Geogebra. Tha aplet provided will allow students to manipulate the parameters of a general translation to observe and record their findings from. By allowing students to choose the way they want to learn the content, the principles of universal design for learning are implemented. Erika Patall, Harris Cooper, and Susan R. Wynn discussed the importance of choice in education in their article for the Journal of Educational Psychology saying, "In this study, teachers reported believing that providing students with choices increases student interest, engagement, and learning; that sutdents spend more time and effort on the learning task if they are offered choices" (Patall et al., 2010). They elaborated upon this saying how teachers were able to better motivate students that typically had little interest in a specific task. By providing students with options to choose from, the students had higher engagement with the task. That is the goal with how this section (and subsequent sections) of the learning module are formatted.

In my previous experience teaching this, I have found many students struggle with vector notation. Encourage the students to compare and contrast the coordinate and vector notations of a given translation. This will help them see the similarities between the two to help better understand vector notation altogether. Additionally, if you are unfamiliar with Geogebra and its aplets, I highly suggest exploring through the translations aplet on your own and drawing your own conclusions from it before assigning it to students. Students often strugle to understand how to efficiently manuever within Geogebra, and they will need help with the basics of it.

The Common Core State Standards aligned with this section include:

• CCSS.MATH.CONTENT.HSG.CO.A.2 - Represent transformations in the plane using, e.g., transparencies and geometric software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not.

• CCSS.MATH.CONTENT.HSG.CO.A.5 - Given a geometric figure and a rotation, reflection or translation, draw the transformed figure suing, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

Just as in the translations section, the students have a decision in how they learn the new concepts of reflections. The different aspect in this section comes in the assignment. Students are being put in the role of the teacher by being asked to create an update that models a lesson on reflections. Jessica Lander of Hardard Graduate School of Education discusses how allowing students to teach to their peers sends strong messages that allow for a natural way to strengthen leadership skills, build confidence, and grow empathy. Lander states it relays to students that they "have knowledge worth sharing, you have a teacher's trust, and you have an opportunity to support your friends' learning and growth" (Lander, 2016). This positive impact affects not only the students engaged in teaching, but also the students learning from their peers. From this experience, students see themselves and their peers as "role models with similar experiences and concerns, who can affirm them and also push them to reach higher" (Lander, 2016). Engaging in this update will further build the students' confidence as they continue to grow. The students will grow in their communication skills in an academic way that will support them as they move through life.

I have found that students typically struggle with visualization of reflections and finding the line of reflection given a set of reflected figures. Having access to mirras for the students to use are immensely helpful for the visual students that need to physically see how a line of reflection corresponds to a given graph. In the same sense, many of my past students have benefited from seeing a simulation of a reflection through a platform like Geogebra or Desmos.

The Common Core State Standards assessed in this section include:

• CCSS.MATH.CONTENT.HSG.CO.A.2 - Represent transformations in the plane using, e.g., transparencies and geometric software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not.

• CCSS.MATH.CONTENT.HSG.CO.A.5 - Given a geometric figure and a rotation, reflection or translation, draw the transformed figure suing, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

As with previous sections, the instructional decision of providing students choice in the way they learn new material stays constant for this section as well. The assignment for this section of the module emphasizes a richer dialogue related to rotations. Students are asked to describe the factors of rotations they found unique or interesting, to compare and contrast the concept of rotating a figure to that of translating and reflecting a figure. Enhancing discourse in the mathematics classroom is essential in our ever-evolving technological world. There is no longer a need to produce students that can memorize rote facts and regurgitate formulas and multiplication facts because we live in a society where a calculator that can do all of that fits in the palm of one's hand. Because of this, the National Council of Teachers of Mathematics (NCTM) explains the following:

"Changes in the workplace increasingly demand teamwork, collaboration, and communication. Similarily, college-level mathematics courses are increasingly emphasiing the ability to convey ideas clearly, both orally and in writing. to be prepared for the future, high school students must be able to exchange mathematical ideas effectively with others." (National Council of Teachers of Mathematics, 2005)

In summary, it is more important for students to be able to communicate their thought processes when solving mathematics problems and to be able to explain the underlying conceptual understanding of the mathematical idea. Thus, it is important to include opportunities for students to do so, just as the assignment portion of this section does.

Generally, students strugle to understand the notation involved with the coordinate rules for rotations. Because of this, I try to take time to go through examples of what a transformed point would look like. An example of this is shown in the screenshot below:

The directions of clockwise and counterclockwise are often difficult for students to grasp as well (hello, digital clocks everywhere). Because of this, students struggle with understanding how 90 degrees counterclockwise is the same as 270 degrees clockwise (and 180 degrees clockwise is the same as 180 degrees counterclockwise; 90 degrees clockwise is the same as 270 degrees counterclockwise). If possible having students physicall turn 90 degrees in a clockwise direction and comparing that end result to where they end up after turning 270 degrees in a counterclockwise direction is a great visual that is concrete for students. You can also look into simulations online that virtually express this same concept.

The Common Core State Standards aligend with this section include:

• CCSS.MATH.CONTENT.HSG.CO.A.2 - Represent transformations in the plane using, e.g., transparencies and geometric software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not.
• CCSS.MATH.CONTENT.HSG.CO.A.3 - Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
• CCSS.MATH.CONTENT.HSG.CO.A.5 - Given a geometric figure and a rotation, reflection or translation, draw the transformed figure suing, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

When learning sequences of transformations, students have already learned the basic concepts to do so. Sequences of transformations is a combination of the three previous lessons on translations, reflections, and rotations. Since students are combining multiple transformations to create one final image, the visualization can often be lost in that process. It can be tedious and take away from the underlying concept to have students repetitively graph these sequences of transformations on paper. Thus, incorporating virtual simulations can enhance their learning process. As previously mentioned, the need for students to regurgitate fluency procedures and facts is mute. In today's society, there is a higher need for conceptual understanding of mathematical ideas. NCTM states, "The computational capacity of technological tools extends the range of problems accessible to students and also enables them to execute routine procedures quickly and accurately, thus allowing more time for conceptualizing and modeling" (NCTM, 2005). The Desmos activity does just that. Throughout the Desmos Transformation Golf activity, students are focusing more on the underlying concepts of building a sequence of transformations by repetitively reflecting, translating, and rotating a figure. They are able to do this and easily visualize these processes through the Desmos platform and not be overwhelmed by the repetitive procedure of graphing each step along the way. Doing this places their focus solely on the concept of sequences of transformations.

Students generally struggle to initially understand how a sequence of transformations builds upon each other as opposed to having multiple new images that stem from the original. Providing the visual representation like the YouTube video and the Desmos simulation activity helps those students physically see how the sequence builds.

The Common Core State Standards this section assesses include:

• CCSS.MATH.CONTENT.HSG.CO.A.2 - Represent transformations in the plane using, e.g., transparencies and geometric software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not.

• CCSS.MATH.CONTENT.HSG.CO.A.5 - Given a geometric figure and a rotation, reflection or translation, draw the transformed figure suing, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

The tessellations section of this learning module focuses on connecting mathematics and art. Creating the connection between mathematics and something visual, like art, or something students are more interested in, being creative, can help increase their overall motivation to do so. Ivona Grzegorczyk and Despina Stylianou completed a study on classrooms that integrated mathematics and art curricula. From their study they found that "an art-studio environment and art-based instructions support development of abstract mathematics thinking." (Grzegorczyk & Stylianou, 2006). The students involved in this study were able to engage in deeper conversations related to the underlying mathematics being completed. The goal with including a section on tessellations is to deepen students' understanding of congruence transformations. In this section, they are applying their knowledge of the mathematical concepts learned thus far to an artistic perspective.

Tessellations are difficult to do through a virtual platform in a way that ensures students are using their knowledge of congruency transformations (and not just pushing buttons that create the sequences for them). I have found that many students prefer to complete their tessellations on paper, even if they traditionally do not prefer working out these concepts on paper. However, the very computer-savvy students will prefer to manipulate programs like Geogebra to create their own images which is still a great option.

The Common Core State Standards that align with this section of the module are:

• CCSS.MATH.CONTENT.HSG.CO.A.2 - Represent transformations in the plane using, e.g., transparencies and geometric software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not.
• CCSS.MATH.CONTENT.HSG.CO.A.3 - Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.
• CCSS.MATH.CONTENT.HSG.CO.A.5 - Given a geometric figure and a rotation, reflection or translation, draw the transformed figure suing, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

• CCSS.MATH.CONTENT.HSG.MG.A.3 - Apply geometric methods to solve design problems.

The assessment portion of this learning module is centered around the idea of student choice. Traditionally, students simply memorize formulas and regurgitate them for the test, just to be forgotten the following day after. By implementing a choice board of assessments for students to choose from, their engagement with the content is heightened. Casey Cook of Texas A&M University conducted a study among her mathematics classes that dove into the benefits of implementing choice boards. Her findings were that the majority of her students enjoyed being given a choice in their learning and were thus more motivated to engage in the activity they chose to complete. Cook goes on to discuss how choice boards led to an overall higher motivation among her students (Cook, 2020). For these same reasons, a choice board was implemented in this learning module in the form of an assessment.

The Common Core State Standards aligned with this section of the learning module are:

• CCSS.MATH.CONTENT.HSG.MG.A.3 - Apply geometric methods to solve design problems.

The peer review is its own section for this learning module because it is vital for students to learn how to effectively provide efficient and detailed feedback on others' works. Doing this will help them, the editors, better understand parts of their own project that may need improvement. Ngar-Fun Liu and David Carless discuss the learning element involved with peer feedback in their article for Teaching in Higher Education. They summarize that having students apply a grade to an assessment is only part of the process. When students are engaged in asking questions about how effective an assessment is based on a provided rubric, they are understanding "what one is trying to assess and by what means one comes to an accurate judgement" (Liu & Carless, 2007). By having students engage in the peer editing process, they are engrossed in the underlying conceptual understanding involved with the content taught. Thus, when students are peer editing, they are checking not only that the work follows a given rubric, but also that the concepts discussed are correct. In order to determine if concepts are correctly discussed, the student must be knolwedgeable on those concepts.

Allow students ample time to complete the End of Unit Survey. You can provide this survey as early as section 6 "Assessment" in this module. Analyze and apply the data to adjust and modify the learning module for your future use. If the survey link does not work, a screenshot of the questions in the survey is provided below:

I encourage you to modify these survey questions to fit your own instructional needs. If the survey needs to include more content based questions for your analysis, this survey should include them.