For this lesson for my 10th grade students, I thought it would be fun to bring something that was in the news that I knew they had heard about (the Chinese spy balloon floating over the US, Spring 2023) and show them how the tools we learned about in class (quadratics) could be used to better understand it.  My students had little to no experience with modeling, so I decided that I would make a scaffolded learning experience that would help them weave together prior non-math classes, prior lessons from our class, and new skill sets they would need in another class.  

To ensure a lesson that was broadly accessible to all students, but specifically with regards to my EML students and IEP/504 students, I wanted to make sure the structure of the lesson was clear and that every student would be able to access it and feel successful as they went through it.  The lesson utilizes the knowledge students gained as 9th graders in their conceptual Physics class and applies it to our work this year on Quadratics.  It also reminds students of conversions and how we are able to move through different types of units to compare things in the physical world (this is an important skill they were building concurrently in their work in Chemistry class).  Over the course of the lesson, we had many opportunities for students to express their understanding of the world around them and connect it to a mathematical representation.

At the end of this process, my students were able to recognize how they might apply mathematical modeling in other scenarios to answer broad questions.  Though this particular model was more light-hearted, the intent of the lesson was to prepare kids to create models on their own by understanding the modeling cycle and how to identify the pieces and types of information that will be helpful for them to build it.

I have chosen this lesson to highlight because it not only showcases my ability to use a range of creation tools to design an engaging lesson, it is also an example tied to one of my essential beliefs: students should be presented with ways to use math meaningfully out in the world.  I feel like there are a lot of things that worked really well with this lesson, and a handful of things that I would like to update for future iterations.

Combined with the slide show, this lesson created a dynamic experience that kept a high level of engagement.  I built it in a way that always helped students know exactly where we were in the lesson (numbers across all sections) and I encouraged groups to push through sections at their own pace.  While doing this, I was still able to keep the slides moving along at the speed that I wanted to the class to move without losing any students, and a range of students were able to explain the how and why of each part to their classmates.  Because it was tied to a slideshow and blocked out into pieces, students were able to grasp how long the lesson would last and recognize when they needed to refer back to other parts of the workbook.  The final part of the workbook is a reflection on the model itself, and I particularly chose the model to assume certain variables to be negligible in order to facilitate a discussion with students about the impact of those choices.

While there were many things that worked well with this lesson, there are also a small number of things that I would want to adjust or re-imagine for future iterations.  In the beginning of the lesson, I need to more eloquently craft the essential question with an understanding that students are likely to just "Google" an answer and assume that it is correct.  It might be useful to incorporate that into the lesson itself to recognize that would be the first choice, and then adjust the essential question to ask "what would it take" rather than "can it happen."  I taught this across multiple classes, and I found that in one of my classes I got lower engagement and some of the lesson felt like it was "on rails" - a lesson that just kept going and students were writing things down but not really understanding.  External to the lesson, I think I could do a better job of engaging students prior knowledge and the learning in other classrooms to help them make sense of the modeling process.  Finally, this lesson is meant to be an introduction which worked well for my students now, but I would want to have a structure in place where more advanced or math-confident students could push the model further on their own.

The project shown here is my math classroom's contribution to a grade-wide project called Border | Lines, where students focused on researching the various borders, barriers, and walls around the world that divide communities.  I collaborated with all of the teachers on my team for this project.  Early on, I knew that I didn't want to do "boring math," despite the fact that geometry involving solids is part of the content standards for Integrated Math 2.  I wanted to create a project that would allow students to look at math in a more creative way, utilizing the tools they had already learned this year and experiencing other ways of practicing mathematics.

I decided that what I wanted the project to focus on was using the tools of mathematics to develop codes and to teach students the early forms of cryptography.  Modern cryptography, of course, is a much more complicated subject and is heavily built into computer science curriculum.  However, the thought process involved in designing and implementing techniques for obfuscating are inherently mathematical and lend themselves to discussion in a math classroom.  Through the design of this project, I created multiple failed prototypes before arriving at the product idea shown.  Early versions had various challenges:  some wound up using techniques that were far behind the expected skill level of my students, others did not seem feasible to implement in the 7 week timeline available for our project. Through collaboration with other teachers on my team, I learned how to use the school's laser cutter and created prototypes for a cipher disc tool that allowed me to design a lesson set that wove together students' understandings of functions (specifically quadratics), new learnings in the geometry of circles, and a number of mathematical practices.

I developed a series of lessons that slowly introduced students to cryptography and gave them time to explore how messages can be hidden.  We began with simple alphabetic shift ciphers and worked our way up to numeric substitution ciphers.  We looked at how functions can be used to take in numeric representations of letters (place in the alphabet) and return a number that can be the basis of our secret code.  Prior to starting this project officially, I led several lessons of "Desmos Scavenger Hunts" that got students to creatively meet conditions they were given in Desmos - an important skill that they would need to create a viable function for their code in this project.  Students also returned to using compasses to diagram circles and consider optimization as they tried to fit their rough draft cipher discs on 11x17 paper.  Every step of this project built on the prior one to ensure that students would ultimately have a secret code built from functions they had learned about during math this year, implemented on a tool built through their understanding of geometry.

After students learned about cryptography, keys, and secret codes, it was important that they could effectively build their cipher discs to be laser cut for usage and exhibition.  Working with my team's multimedia teacher, we found time were students would be able to work on their designs in Illustrator (the mathy-est of art programs) but they needed guidance on how to actually build this tool.  I designed a lesson series on Loom that walked through students through building an example disc, noting how we can use all of our knowledge and understand of geometry and Cartesian planes to construct these tools.  Students were able to use the video series to create their cipher discs with over a 95% success rate.  Designing a lesson that had a video component was immensely helpful for my students who need extra support and my EML students because they were able to slow the video down, turn on subtitles, and repeat sections that they needed to hear and see again.

As students learned how to create this tool in Illustrator, I trained a 12th grade intern to use the laser cutter and cut all of the discs for this project which took a huge burden off of me for this.  This student followed my original set of instructions to make an initial cipher disc himself (shown in the slide show) which then gave me a good perspective on what students would need to do as "finishing touches" to ensure they had a readable, usable cipher disc.  I created a few final videos that addressed those challenges.  The intern that I trained was able to be a vital part of the production pipeline for this project and I was able to train him to effectively error check and correct issues to ensure all student work came out correctly.  

Once discs were in process, I taught students how to create a zine that would serve as a guide for users to set up and use the cipher disc correctly.  This zine showcased the understanding students had of the functions used to create their code, the geometry of their discs, and a sample message for a reader to decipher.  Students also implemented their secret code as messages inside of artwork for other classes (steganography).  This is visible in the tweets they did in history class, the propaganda they did in multimedia class, and the design for their poetry from humanities class.

Overall, I think this project was a great success for me and the goals that I had for the project.  I want students to see that the tools of mathematics and mathematical thinking can be applied in creative ways, both functionally and artistically.  This project did both of those things.  I scaffolded the process of exploring in Desmos and pushed student understanding of how quadratic functions work and what impact the coefficients have on these types of functions.  The Illustrator portion of this offered new insights into how artistic tools can be used to express mathematical understanding.

For students, I think this project was a mixed success.  I had some students that this project resonated with immensely, as shown by the amount of time that they spent trying to find elegant solutions to the code requirements, the care they put into optimizing their cipher disc size and design, and the meticulous construction of their cipher guides.  For some others, this project was very difficult because their core understanding of functions was not as strong as it should have been for their grade level and they faced challenges shifting to a precision tasks while in their multimedia class.  Most of their tasks prior to this in that classroom setting were more free form, versus the mathematically minded precision task of creating an optimized set of cipher discs.  Despite this, the vast majority of students were successful at completing this project and showcasing strong mathematical work at exhibition.

Feedback from students about this project overall was very positive.  While it was difficult and challenged them, in discussions with students I found that broadly they understood the techniques used for creating their codes, grew in their knowledge of quadratic functions, improved their ability to represent mathematical ideas artistically, and were able to communicate all of this effectively.  The student ownership in this project, particularly once it all came together with the laser cut cipher discs and disc guides, was overwhelming and wonderful to see.  Many students expressed to me how proud they are of their work and plan to take their disc home and hang it up as a representation of their accomplishments in math for the year.

As I reflect on how this project went, there are a few things that I would have like have gone differently.  As my team decided on the Border | Lines project idea, our launch date was pushed back twice due to internal scheduling conflicts.  I wanted to start 2 weeks earlier, but due to scheduling challenges in my classroom and within my team, that was not possible.  An extra week would have been incredibly useful for me to rebuild the foundational function knowledge with my students to help increase understanding of the secret code portion of this task.  Another week would have benefited students immensely during the creation of the cipher discs.  Based on our timeline, most students chose to paint or decorate their cipher discs rather than laser cut designs simply because they did not have the time to implement more laser cutting work.  As a digital artist, I would have loved to have had more students able to laser cut artwork into their disc rather than using other physical art.  Also, this project would have been easier for me if I had known earlier in the year that this is what was coming.  If I were teaching 10th grade again and planning to do this project, I would have spent a lot more time earlier in the year during the quadratics units having students practice skill sets that would help them at the end of the year on this project.

This project was an immense undertaking.  It required me to: 

• Create an entire lesson sequence that appropriately scaffolded student understanding of cryptogaphy

• Devise ways to implement our grade level learning into the project

• Learn new a new tool and teach a student to use this tool after I learned it (laser cutter)

• Construct a video lesson series to teach students how to build cipher discs

• Prototype, through multiple iterations, this entire project so that students had source material to reference (huge benefit)

• Work with teacher teammates to determine how messages could be hidden in work from their class, then build demos of this

• Troubleshoot student work in mathematics, design, art, and technology

I learned a great deal from this process and I look forward to using the skills I've developed here on future projects with students.