Optimization & Rewriting with Constraints

One of the joys of teaching a Non-AP Calculus class is that I can spend some time digging into a topic that's glossed over far too quickly in AP Calculus: Optimization. It's one of the most intuitive topics for students- they understand the frustrations of trial and error when a better mathematical process exists and the applications of trying to find the most or the least something.

I am anticipating that the algebra behind optimization will be the real issue for my students, so I created this scaffolded assignment to get them thinking about using constraints to write equations in terms of one variable. It builds from being very apparent about which formulas to use and how to solve them to being completely open ended- the way they will actually appear in a Calculus questions. Note that this would be appropriate for an Algebra 2 or a Pre-Calculus class as they study optimization from non-calculus perspective as well! Even an advanced Algebra I class would be able to tackle many of these! 

Link to the File: 

Rewriting with Constraints Scaffolded Assignment 

I thought about it in 3 phases...

Phase 1- Working with the Algebra

In each of these questions, the equations are given. Students need to isolate variables and use substitution to rewrite. 

Phase 2- Basic Modeling

In each of these questions, a basic scenario is given that can be translated easily into an equation. Constraints are labelled as such. Students need to generate their own equations, then use substitution to rewrite. 

Phase 3- Removing the Scaffolding

These questions progressively remove the scaffolding, ending in a problem that is worded in the way an optimization problem would be. Students should be able to rely on their experience form the previous questions to solve these. 

Feel free to adapt and let me know your thoughts! Happy Optimizing!