End of unit project for middle or high school students based around geometric transformations and student self-efficacy.

Geometry, Mathematics, Self-Efficacy, Project-Based Learning

This learning module aligns with the ideas of student self-efficacy and productive struggle in the mathematics classroom. Self-efficacy is "the beliefs students hold about their academic capabilities" (Usher & Pajares, 2009, p. 89). This can have a positive or negative effect on the outcomes of their academic courses. Students who have greater confidence in their academic ability are more likely to be successful in problem-solving and other academic tasks, compared to those with lower self-esteem (Pajares & Miller, 1997, p. 214). Students' self-efficacy is influenced by four different factors: mastery experience, vicarious experience, social persuasions, and emotional and psychological states (Usher & Pajares, 2009, pp. 89-90). The video below gives examples of these factors in a standard student experience. These include getting a good grade on a test, seeing peers being successful, and being able to verbalize feelings and emotions (Transforming Education, 2017).

Video 1: Importance of Self-Efficacy (Transforming Education, 2017)

In order for students to build their self-efficacy, they must overcome challenges, which is where productive struggle becomes important. Productive struggle is the balance between students understanding a solution to a problem right away and the problem is too challenging for students to grasp. When students experience productive struggle, they have the opportunity to deepen their understanding of mathematics while still being able to achieve the success necessary to build self-efficacy (Warshauer, 2015, p. 393). The figure below shows several ways that teachers can support productive struggle during a task.

In a task that encourages productive struggle, the teacher acts as a guide, asking scaffolded questions to probe deeper into student solutions. Student solutions should include both procedural and conceptual reasoning, as well as opportunities for feedback from the teacher and peers. This formative feedback helps build student self-efficacy through social persuasion (Rakoczy et al., 2019, p. 156). This learning module helps to assess and build student-self efficacy through a project meant to induce productive struggle in students, as well as encourage feedback and mastery in the form of a peer review process.

I currently teach 10th-grade Geometry, and previously taught 8th-grade pre-Algebra, where geometric transformations were introduced. The pre-Algebra course focuses mostly on learning the different transformations, while Geometry focuses on using the transformations to prove various theories and properties about functions and figures. While I have not taught this particular project before, it is one I am considering for my 10th-grade students.

This project is based on the three-act task by Robert Kaplinsky. The original task is attached below.

This task was originally meant to be done on paper but has been modified by myself and several other teachers to have virtual and hybrid options as well.

*This module is designed for* Geometry students

*Prior Knowledge:*

- Ability to graph on the coordinate plane
- Apply the four geometric transformations: translation, rotation, reflection, and dilation

*Materials needed:*

- Pencil
- Graph paper
- Laptop or tablet
- Tracing paper (optional)

*By the end of this module, you will be able to:*

- Apply different geometric transformations to solve real-world puzzles
- Create a puzzle on a coordinate grid by understanding how to apply different transformations
- Critique a peer through both positive and critical comments

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: *I will be able to describe my thoughts and feelings towards math and determine at least one strategy that I will use to help me persevere in the upcoming task.

Math can be a difficult subject, and it is important to be able to communicate your thoughts and feelings about math. To start this project, the instructor wants to determine your initial perceptions of math. You will then take this assessment again at the end of the project.

*Self-Assessment:* Please take this self-assessment. There are no correct answers, so please be honest!

*Lesson:*

Math is not always easy, and it is important to be able to voice your thoughts and feelings about math in order to grow. Watch the video below, where students talk about their experiences in math.

Video 1: With Math I Can (With Math I Can, 2016)

In the rest of this learning module, you will be completing a multi-part Geometry task. There may be moments when you get stuck or frustrated. Below is a chart of different problem-solving strategies.

*Comment: *Identify one of the problem-solving strategies that you think you might use during this project. Why does this strategy appeal to you?

*Update:* Describe your previous experience in a math course. Was is positive or negative? How did it impact your idea of your abilities in math in future courses? Your response should be at least one paragraph long and include at least one media element. Then respond to at least two other student's posts.

*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: *I will be able to describe the three geometric transformations and apply them to show the congruence of two figures.

*Vocabulary:*

- Congruent figures: two figures that are the same size, same shape, and have the same corresponding angle measures
- Similar figures: two figures that have congruent corresponding angles, are the same shape and have sides multiplied by the same scale factor.
- Rotation: turn
- Reflection: flip
- Translation: slide
- Dilation: grow or shrink by a scale factor

*Lesson:*

Watch the video below to review the different kinds of transformations.

Video 3: Geometry- Unit 2 Review - Transformations (Math Mindy, 2019)

*Practice: *Practice the different geometric transformations here.

*Comment: *Choose one transformation. Describe how to perform the transformation to a friend who may have been absent.

*Update: *Find a real-world example of one or more of the transformations. This can be in art, architecture, science, etc. Describe the transformation(s) taking place and why it is important to the example you chose. Include at least one media element and respond to at least two peers' posts.

*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: *I will be able to describe Ms. Pac-Man's movements using the three geometric transformations.

*Warm Up: *Watch the video below. What seems to be the objective of the game? How can you describe Ms. Pac-Man's movements?

Video 6: Ms. Pac-Man (Robert Kaplinsky, 2013)

*Lesson:*

Based on your answer to the warm-up question, watch the video that matches your response below. As you watch, consider the following questions:

- What information is needed for each kind of transformation?
- What units are we measuring?

Translations only:

Video 7: Ms. Pac-Man - Part 1 (Translations only) (Robert Kaplinsky, 2013)

Translations and reflections:

Video 8: Ms. Pac-Man - Part 2 (Translations and Reflections only) (Robert Kaplinsky, 2013)

Translations, reflections, and rotations:

Video 9: Ms. Pac-Man - Part 3 (Translations, Reflections and Rotations) (Robert Kaplinsky, 2013)

*Comment: *Why is it important to include so much information when performing a transformation? What happens when you don't include this information?

*Update: *Find another video game that uses at least one geometric transformation to move a character. Include a video of the game and describe the character(s) movements using the different transformations. Be as specific as possible!

*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:* I will be able to describe transformations on a coordinate plane a use them to travel to specific points on the grid.

*Warm Up*: What do you notice about the picture below? What do you wonder?

*Lesson:*

Today, you will apply the transformations we discussed yesterday and apply it to the game board on a coordinate plane. Your instructor will post a link to the Desmos activity for today.

As you write your different transformations, keep these ideas from the last update in mind:

- Translation: Need to know the direction and units
- Reflection: Need to know the line of reflection
- Rotation: Need to know the degree, direction, and center of rotation

*Comment: *What was the hardest part of this activity? When you got stuck, what is a strategy that you tried?

*Update: *On the final Desmos slide, there is a final challenge. Watch the video, and write at least 10 transformations that you see occurring. Be as specific as possible.

*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.

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

*Project:*

Today, you will be designing your own Ms. Pac-Man game! Use the board below to create your game. Your game should include:

- At least 3 fruits
- At least 3 power-ups
- At least 3 ghosts
- Labeled starting position of Ms. Pac-Man

You will also write a 20-move solution to your game. Using the chart below, record your different transformations. You should use at least four translations, four rotations, and four reflections. Make sure to include all the information for each transformation!

See the attached rubric here for this project.

*Comment: *Did you notice any patterns in your transformations?

*Update: *Post a picture of your Ms. Pac-Man game board. Then comment on three other posts with a reflection, rotation, and translation you would use to solve part of the puzzle.

*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: *I will be able to critique another student's project by playing the puzzle they created. I will be able to express how I feel about mathematics.

*Lesson:*

Today, you are going to peer review another student's Ms. Pac-Man game. First, watch this video about how to leave constructive comments on a math project.

Video 11: How to make peer review comments on your math projects (Katya Boone, 2015)

You will peer review one other student's work. Start by filling out the rubric. Leave at least two positive comments and two comments on something that can be improved. Then, play through the solutions found by the original author of the work. Leave at least two positive comments and two comments on something that can be improved.

Once you receive your peer review comments, you will have two class periods to revise your work.

*Self-Assessment:* Please take this self-assessment.

*Comment:* Has your perception of mathematics changed from the start of this project to now? Why or why not?

*Update: *Make a post about a topic in mathematics that you would be interested in exploring moving forward in this course. Include at least one media element. Then respond to at least two other student's responses.

*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.

For this learning module, you will create your own Ms. Pac Man game using the game board below.

You will also write a 20-move solution to your game. Using the chart below, record your different transformations. You should use at least four translations, four rotations, and four reflections. Make sure to include all the information for each transformation!

The rubric for this project is attached below.

After turning in your project, you will peer review one other student's work. Start by filling out the rubric. Leave at least two positive comments and two comments on something that can be improved. Then, play through the solutions found by the original author of the work. Leave at least two positive comments and two comments on something that can be improved.

Once you receive your peer review comments, you will have two class periods to revise your work.

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.

Common Core State Standards Initiative. (2010). *High school: Geometry*. http://www.corestandards.org/Math/Content/HSF/IF/

Darren Best. (2018, July 9). *Desmos Teacher Dashboard Running an Activity* [Video]. YouTube. https://www.youtube.com/watch?v=NvKUrOice30

IXL. (2020, January 31). *IXL is personalized learning!* [Video]. YouTube. https://www.youtube.com/watch?v=9tCmvn-FUeQ

Kaplinsky, R. (2013, April 10). *How Did They Make Ms. Pac-Man?* https://robertkaplinsky.com/work/ms-pac-man/

Katya Boone. (2015, December 31). *How to make peer review comments on your math projects* [Video]. YouTube. https://www.youtube.com/watch?v=XyQ56uMV4mU

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Math Mindy. (2019, October 6). *Geometry - Unit 2 review - Transformations* [Video]. YouTube. https://www.youtube.com/watch?v=pfYTBsc1WJU

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*Ohio Department of Education. (2017). *Standards of mathematical practice*. https://education.ohio.gov/getattachment/Topics/Learning-in-Ohio/Mathematics/Model-Curricula-in-Mathematics/Standards-for-Mathematical-Practice/Standards-for-Mathematical-Practice.pdf.aspx

*Pajares, F., & Miller, M. D. (1997). Mathematics self-efficacy and mathematical problem solving: Implications of using different forms of assessment. *The Journal of Experimental Education, 65*(3), 213–228. https://doi.org/10.1080/00220973.1997.9943455

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*Rakoczy, K., Pinger, P., Hochweber, J., Klieme, E., Schütze, B., & Besser, M. (2019). Formative assessment in mathematics: Mediated by feedback’s perceived usefulness and students’ self-efficacy. *Learning and Instruction, 60*, 154–165. https://doi.org/10.1016/j.learninstruc.2018.01.004

Robert Kaplinsky. (2013, April 5). *Ms. Pac-Man* [Video]. YouTube. https://www.youtube.com/watch?v=wgd9rSsWt2Y

TEDx Talks. (2016, May 23). *How you can be good at math, and other surprising facts about learning | Jo Boaler | TEDxStanford *[Video]. YouTube. https://www.youtube.com/watch?v=3icoSeGqQtY

Transforming Education. (2017, June 7). *Importance of self-efficacy* [Video]. YouTube. https://www.youtube.com/watch?v=VW5v6PQ5PEc

*Usher, E. L., & Pajares, F. (2009). Sources of self-efficacy in mathematics: A validation study. *Contemporary Educational Psychology, 34*(1), 89–101. https://doi.org/10.1016/j.cedpsych.2008.09.002

*Warshauer, H. K. (2015). Strategies to support productive struggle. *Mathematics Teaching in the Middle School, 20*(7), 390–393. https://doi.org/10.5951/mathteacmiddscho.20.7.0390

Winter, B. (n.d.). *Transformation record sheet*. Teachingthelatest. https://sites.google.com/view/teachingthelatest/shape-and-space/position-and-direction

With Math I Can. (2016, February 2). *With Math I Can* [Video]. YouTube. https://www.youtube.com/watch?v=sLPFaOvhlKw