**In the Triangle Centers Portfolio Project** students are tasked with solving six mini-projects which are organized into a portfolio and combine algebra, geometry, and literacy skills. Portfolio Items #1-5 take place as semi-real-world scenarios in which students will:

- Calculate the cheapest location for an airport,
- Find the center of a circular garden when they only know the location of 3 trees,
- Craft an evidence based argument for where a cell tower should be placed to restore service,
- Determine the likelihood of landing in different regions on a dart board and come up with a sequence of throws to earn a specified number of points,
- Locate the center of a fountain inside a koi pond AND find out how long each option really is at baseball conditioning practice

Item #6 is a chance for students to be creative and design their own scenario which includes a map, storyline, and construction. This piece encourages creativity! To round it out, they are also asked to complete an overarching reflection piece to help them practice an analysis of their experience and practices they engaged in.

**Academic Skills and Content:**

- Triangle center constructions,
- creating and solving linear equations,
- applying properties of midsegments,
- calculating and comparing: costs, distances, areas;
- reading comprehension and interpretation,
- written communication of student ideas with mathematical vocabulary

**Final Product: **Students submit all six parts as a portfolio demonstrating their learning in these topics. The last item has the option of being hard-copy, electronic, or a hybrid.

**Suggested Duration: **2 weeks

Created with the support of the California Department of Education California Career Pathways Trust

Students are tasked with using their knowledge of linear equations and properties

of geometric constructions to determine the cheapest location to place an airport

that services three major towns. In this analysis, they must factor in the cost of

building new roads and resurfacing old roads to connect the airport to the major

cities and roads. The first half of this activity serves as a model students could

use to guide their thinking in the second half of the activity as well as in future

parts of the project. This Airport Problem also gives students their first chance to

apply their book learning to a real-world scenario and they should be encouraged to

actively search through their work to find connections with the project!

Students are tasked with building a circular garden for the recuperation of hospital patients. If they wish to use the 3 trees that have already been planted as part of the circular boundary around the garden, where must the center of the garden be? Students again have to use their knowledge of linear equations, unit conversion, and properties of geometric constructions to determine the cost of building the circular boundary with visually engaging river rocks and small shrubs. Students get to practice their visualization techniques to help them determine how the 3 trees fit into the overall garden! This is a skill important for most geometric experiences (e.g. how much paint is needed, which path is quickest factoring in gradient, cost of landscaping, etc) that students will engage with in academia and the real world!

Students are told there is a region in Southern Georgia that has patchy cell service. They are asked to determine where to place a single cell tower in order to restore service to that one region. They will write up a plan which cites evidence from their constructions which are superimposed with the map and their understanding of how cell towers work. This is the first project piece that emphasizes the importance of applying literacy skills in a math context! This includes being able to interpret maps and legends, apply reading strategies to understand that task laid before them, and the ability to communicate their thought process with evidence.

Students are told that a carnival game has a dart board that is triangular with a center located at its center of gravity. They must find the location of the center and then calculate the area of the dart board in order to determine the likelihood of their dart landing in each part of the board. Then, they will figure out a possible sequence of throws to earn their desired score! The area portion of this activity calls on students to employ creative thinking where they are encouraged to practice shape decomposition as an alternative to lengthy formulas. This creative process of “seeing” shapes that are not explicitly outlined is useful in real world applications such as city planning and rearranging furniture to make space for future objects. In addition, finding probability identifies the underlying mathematics that are part of our unconscious decision making process. By coming up with a sequence of possible throws, students practice their number fluency and learn that there are multiple ways of arriving at the same answer!

This project piece allows students to connect their geometric properties with scenarios and imagery they may have encountered in outdoor spaces! This activity asks students to again employ their visualization and literacy techniques to connect the written description and provided pictures with the properties of geometric constructions. The first two scenarios ask students to determine where to place a fountain in a koi pond and where to place the center of the largest possible community pool given constraints of the community center features already in place. The last scenario supposes that it is conditioning practice and the athletes are given two possible routes they can run. They have to determine how long each route is and then explain which route they would personally pick and why! This overall process of analyzing information, finding similarity with real life experiences, and coherently explaining their rationale helps students develop their problem solving and communication skills!

This Project portion ties the whole experience together and gives students the chance to curate a map, storyline, and construction just like in the rest of the project. Students are encouraged to be creative and borrow ideas from fiction, fantasy, and the real world. The reflection portion helps students practice communicating their thoughts while making sure they have had time to look back at their work and analyze their time management and resource utilization. They are also asked to leave feedback with the understanding that their comments will help shape the project for future years. Altogether, Item #6 allows students to put their own unique stamp on the project and make their own voice heard!

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__Math__PBL Essentials- Electioneering
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