# Deep Algebra Projects

Table of Contents

*Deep Algebra Projects* are rich, complex mathematical and real-world investigations

that stretch advanced learners out of their comfort zones! The projects enhance students'

abilities to think independently, flexibly, and with deep understanding.

**Tips for using the projects**

Be patient. This is a new way of thinking about math for most students and adults. It takes time to get used to it!

As the teacher or mentor, join in the spirit of adventure with the student! Expect to be surprised by your students' ideas, and don't worry when you can't answer some of their questions. Enjoy getting lost and finding your way back!

Don't worry about finishing. These projects are challenging. It's better to spend time digging deeply into one or two questions than rushing to complete everything. This is one of those "it's more about the journey than the destination" situations.

Break it up if necessary. You don't have to do the activity all at once. If students get bogged down for a very long time, come back to it again later in the school year.

Just click on the title to download a free pdf of a project.

## Pre-Algebra / Algebra 1

Develop number sense; find and extend patterns in repeating decimals; analyze and practice fraction / decimal conversions (especially for repeating decimals); compare and analyze definitions of rational numbers

Modular Arithmetic

Explore properties of operations via modular ("clock") arithmetic; analyze and extend complex patterns; connect algebra to concepts from number theory; discover and prove Fermat's Little Theorem (Note: I am still writing this project. Thank you for your patience.)

Use number lines to explore properties of operations on negative and positive numbers; think abstractly about *relationships *as opposed to focusing on calculation rules

The Stories That Formulas Tell

Interpret algebraic formulas from science, finance, sports, and geometry; make sense of negative numbers; evaluate algebraic expressions; apply order of operations; apply proportional reasoning to analyze algebraic equations; work with units and dimensional analysis

FTG Template

Use a template that brings out connections between formulas, tables, graphs, and real-world and mathematical contexts; recognize and explore connections to concepts across the curriculum (Note: I am still writing this project. Thank you for your patience.)

Create and analyze models using absolute value and piecewise-linear functions; represent functions using formulas, tables, and graphs; interpret slope; analyze complex patterns in rates of change

Discover a variety of patterns in the multiplication table; express the patterns algebraically and prove them; discover formulas for differences of squares and the sum of cubic numbers

Use linear equations to discover and describe patterns in areas of triangles in the coordinate plane; discover or apply knowledge of slopes and y-intercepts; find intersection points of graphs of linear functions; create general formulas to describe families of linear equations

Write a variety of inequalities that "capture" selected sets of points on a number line while avoiding others; explore linear, polynomial, radical, and absolute value inequalities informally

## Algebra 1 / Algebra 2

Explore step functions with formulas, tables, and graphs; modify equations to stretch and compress graphs vertically and horizontally; create a mathematical model

Create mathematical models to maintain habitable environments in aquariums; use function notation; understand iteration and fixed points of functions; analyze linear functions using tables and graphs; interpret slopes and y-intercepts; solve linear equations; analyze arithmetic and geometric sequences

Create stories to match linear data; interpret slopes and y-intercepts; solve linear equations and inequalities graphically and symbolically; find intersection points of graphs; represent solutions to inequalities on a number line

Create and use mathematical models for payments on long-term loans; apply knowledge of geometric sequences; discover and apply a formula for geometric series

Discover a formula for the unit circle; make connections between the unit circle and the Pythagorean theorem; find formulas for linear graphs; solve linear equations; solve systems of two equations (one linear and one quadratic); discover formulas for generating Pythagorean triples; understand stereographic projection

Create and interpret mathematical models for traffic flow; invent and apply methods to solve systems of linear equations that have multiple (often an infinite number of) solutions; use real-world constraints to determine bounds on solutions

Add, subtract, and multiply polynomial expressions; rewrite polynomial expressions in multiple equivalent forms; connect algebra to geometric patterns; use algebraic manipulations to prove claims about patterns; discover formulas for polygonal numbers; discover formulas for sums of whole numbers, square numbers, and cube numbers

Maximizing Revenue

Create a mathematical model to maximize revenue; write and solve linear and quadratic equations; understand the significance of the factoring process in a real-world context; generalize equations in order to understand deeper connections (Note: I am still writing this project. Thank you for your patience.)

Surprise!

Apply knowledge of probability to investigate three classic surprising results: the birthday problem, Buffon's needle problem, and the Monty Hall dilemma (Note: I am still writing this project. Thank you for your patience.)