This learning module is intended for middle and high school students. This learning module is meant as a project for the end of a unit on geometric transformations, which is typically taught between grades 7-10. 

Before starting this learning module, students should be able to:

• Plot points on the coordinate plane and describe a given point using its coordinates (x,y) 
• Identify and apply the different geometric transformations (translation, reflection, rotation, and dilation)
• Determine whether two figures are similar, congruent, or neither 

Time needed to complete this module:

• The first and third updates can be completed in one 60-minute period.
• Updates two, four, five, and six can each be done over two or three 60-minute periods, depending on student progress.

Common Core State Standards addressed:

• ​CCSS 8.G.1 Verify experimentally the properties of rotations, reflections, and translations.
• CCSS 8.G.2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.
• CCSS 8.G.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
• CCSS 8.G.4 Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.
• CCSS G-SRT.2 Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.
• CCSS G-CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
• CCSS G-CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.
• CCSS G-CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.
• CCSS G-CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

(Common Core State Standards Initiative, 2010)

Learning Target: Students will be able to describe their self-efficacy in math and determine at least one strategy that they will use to them persevere in the upcoming task.

The first Standard of Mathematical Practice is to "make sense of problems & persevere in solving them" (Ohio Department of Education, 2017).  Students' self-efficacy is a big key to their ability to persevere in challenging situations. This lesson helps the instructor to understand students' current perceptions of their mathematical abilities through an initial self-assessment, which will be repeated at the end of the course. This self-assessment is based on the work by May (2009).

The lesson then has students take a deeper dive into their feelings about math and how to persevere through problems. This culminates in the prompts for a comment and update, which asks students to reflect on problem-solving strategies and past experiences in math courses. It is important that students feel comfortable communicating their emotions about math in order to break the stigma that if one is not good at math right away, one will never be good at math. For more about the stigma and strategies for helping students change their mindsets, see the video by Jo Boaler below.

Video 2: How you can be good at math, and other surprising facts about learning (TEDx Talks, 2016)

Learning Target: Students will be able to describe the three geometric transformations and apply them to show the congruence of two figures. Students will be able to use transformations to describe different properties of congruent figures. 

Since this project takes place at the end of a unit on geometric transformations, this update serves as a review, rather than individual lessons on each transformation. A modification of this update would be to spend two or three days reviewing and practicing the transformations, rather than just one class period. This module could be done virtually or in a physical space. 

While the video reviews conceptual knowledge from the unit, the IXL practice problems promote procedural fluency. This project involves a lot of applications of transformations on the coordinate plane, so it is important that students are comfortable with each kind of transformation. IXL is a virtual practice option that provides harder or easier problems based on student work, as well as examples and worked-out problems. See more about IXL in the video below.

Video 4: IXL is personalized learning! (IXL, 2020)

IXL does require a school subscription. If your school does not have this option, Khan Academy has similar individual practices at no cost for each transformation in the playlist linked here (Khan Academy, 2022). 

Learning Target: Students will be able to describe Ms. Pac-Man's movements using the three geometric transformations and specific details. 

This lesson serves as the start of the video game project. Students are able to see the different geometric transformations applied to a familiar concept. This also opens up discussions about the precise use of mathematical language, which is another of the Standards of Mathematical Practice (Ohio Department of Education, 2017).

This update is best done in person, to allow for more flexible groupings and discussions. In this lesson, the instructor is the facilitator, and the activity is scaffolded based on students' answers to the initial warm-up question. As students describe more transformations, the instructor can give the groups the appropriate video to analyze. By the end of the lesson, students should be including the following information for each transformation:

• Translation: Direction of the translation and the distance of the translation (estimated)
• Reflection: The line of reflection
• Rotation: The center of rotation, the direction of rotation, and the degree of rotation

During the lesson, facilitate and push students' thinking by asking the following questions:

• What movement did Ms. Pac-Man make?
• What was the very first thing Ms. Pac-Man did?
• Does anyone have the same answer but a different way to explain it?
• How did you reach that conclusion?
• How can you demonstrate what you are saying is correct?
• What assumptions are you making?

(Kaplinsky, 2013)

Learning Target: Students will be able to describe specific transformations on a coordinate plane.

This lesson is a continuation of update 3, where students identified translations, reflections, and rotations in the game "Ms. Pac-Man." This lesson builds on this knowledge by having students apply and test different transformations on a coordinate plane. The goal of collecting dots and fruit adds a game element to this part of the module. 

This lesson uses Desmos as a platform where students can test their transformations and share their work with their classmates. The teacher dashboard for the Desmos lesson is by John Rowe, and can be found here (2014). A student-facing link can be posted to the learning platform used by your class. The video below gives a basic overview of the Desmos platform for instructors:

Video 10: Desmos Teacher Dashboard Running an Activity (Best, 2018)

If your classroom does not have access to one-to-one technology or technology that supports Desmos, you may also use the paper version of this activity instead. 

Figure 4: Ms. Pac-Man Game Board (Miller, 2016)

Learning Target: I will be able to create a Ms. Pac-Man game board and come up with at least one viable solution.

This is the project for this unit. Students will create their own Ms. Pac-Man maze and then write a solution to their maze. The rubric for this project is here.

This update can be done over the course of two to three days, depending on student progress. The first day can be spent creating the game board, and the second day can be spent finding the solutions. Both parts of the project must be completed in order to start the peer review.

By leaving this project open-ended, it provides multiple entry points for students. This helps build students' feelings of confidence by letting them take control of the project. In the next update, students will get to play each other's game board, giving the project a fun, game-like element. 

Learning Target: Students will be able to leave positive and constructive critiques on another student's project by playing with the puzzle they created. They will be able to express how they feel about mathematics using the same assessment as the pre-survey.

This final update gives students a chance to view other students' work and critique it. It is important that peer review does not only happen in English and Social Studies classes. Critiquing each others' ideas helps students practice using precise mathematical language and analyze thinking that might be different than their own. Students are then given time to revise their work based on the peer review comments. The amount of revision time can be modified to fit the needs of the class. 

This update may take one or two class periods, depending on the complexity of student projects and if students are taking the self-efficacy survey in class. Since students are testing their peers' designs, the instructor may want to provide one day for students to leave comments on the rubric and one day where students analyze the solution to the game. 

Finally, students will fill out the same self-efficacy survey that they took at the start of the course. As the instructor, you can compare their responses from before and after this project. This survey can also be given throughout a course in order to provide more data on students' confidence in mathematics. 

This is the project for this unit. Students will create their own Ms. Pac-Man maze and then write a solution to their maze. The rubric for this project is here.

This update can be done over the course of two to three days, depending on student progress. The first day can be spent creating the game board, and the second day can be spent finding the solutions. Both parts of the project must be completed in order to start the peer review.

The peer review may take one or two class periods, depending on the complexity of student projects. Since students are testing their peers' designs, the instructor may want to provide one day for students to leave comments on the rubric and one day where students analyze the solution to the game.