Aquarium Equations
A Deep Algebra Project
Algebra 1/Algebra 2

In Aquarium Equations, students use a real-world situation involving salt concentrations in an aquarium as a springboard to explore the concept of iteration of mathematical functions. Iteration involves choosing an initial input and applying a mathematical function repeatedly with each output becoming the next input. The project also enhances students’ understanding of numerous standard algebra concepts: function notation, linear functions (using tables, graphs, and formulas); slopes and y-intercepts, solving linear equations, arithmetic and geometric sequences (brief experience with the latter), and mathematical modeling.

About Deep Algebra Projects

The Deep Algebra Projects are rich, complex mathematical and real-world investigations designed specifically to stretch advanced learners beyond their comfort zones and to enhance their ability to think independently, flexibly, and with deep understanding. The projects are ideal for use in self-contained pre-algebra and algebra courses for highly capable or motivated students and in pullout or push-in environments for these students. The projects may also be offered as options in mixed-ability settings if students are given extended time, appropriate expectations, and sufficient support.

Each project contains a set of approximately five to twelve problems organized into three stages that dig progressively deeper into a targeted set of algebra concepts. The projects (the third stages especially) have very high ceilings in order to stretch the thinking of even the most advanced learners. Consequently, most students will not attempt every problem but will progress as far as they are able. Projects may often take weeks rather than hours or days to complete.

Each project includes:

• A general Introduction including Common Core content connections
• A focused introduction for each of the three stages
• Descriptions of What Students Should Know and What Students Will Learn in each stage
• Student Handouts (one problem per page) with plenty of room to write
• Conversation Starters for each problem to help you guide discussion
• Detailed Solutions showing multiple strategies and ways of thinking
• Diving Deeper suggestions for extending selected problems even further

The Deep Algebra Projects are not generally meant as handouts to fill time for students who finish other work, because conversation is a very important part of every project. Students will need to talk with each other and with you in order to experience the full depth and richness of the mathematical ideas. Collaboration is strongly encouraged! It energizes the problem-solving process, exposes students to more ideas, and increases opportunities for success.