# PBL Pathways

##### Content Area

College Algebra, Finite Math

##### Time Frame

2 to 3 weeks with mini-lectures

June 1, 2011

##### Keywords

System of Linear Equations, matrices, row echelon form, systems with an infinite number of solutions

#### Project Title

Blending Scotch Whisky

#### Author

David Graser, Yavapai College, Prescott, AZ (David_Graser@yc.edu)

#### Abstract

This project follows the pattern that I have established in tow other projects involving systems of linear equations, Arsenic Removal from Water and Gas Prices. Like those projects, students set up a system of two equations in three variables. In this case, students blend different malt whiskeys instead of water or gasoline. Although the content of this project might be offensive to some students, others may find it fascinating. The project background document contains information about how whisky is made and what alcohol by volume (abv) is.

#### Project Content

Project Background - DOC | PDF

Project Letter - DOC | PDF

#### Scaffolding Resources

Technology Assignment: Substitution DOC | PDF - This assignment introduces students to documenting their work with Mathtype.

Technology Assignment: Gauss-Jordan Method DOC | PDF | VIDEO- The goal of this tech assignment is to help students put a matrix into row echelon form. Once they have done this, they document their work usinfg Mathtype.

Link - What is Proof? PDF

Technology Assignment: Row Operations on the Ti-83 and in Mathtype DOC | PDF - In this assignment, students create their own system of equations and then use row operations to solve the system. This assignment is almost identical to the one above, but this version includes instructions for carrying out the row operations on a TI graphing calculator and documenting the steps using Mathtype.

Technology Assignment: Write and Solve a System Using Excel DOC | PDF - This assignment is designed to get students to write down their system of equations for the project so that they can be checked before turning in the final assessment for the project.

Dependent Systems Handout - PDF

Reduced Row Echelon Form on a TI Graphing Calculator PDF - This document takes a student through the steps of putting an augmented matrix into a TI calculator and using the rref command to find the reduced row echelon form for the augmented matrix.

Reduced Row Echelon Form in Excel PDF - This document shows how to do row operations in Excel in order to end up with the reduced row echelon form of an augmented matrix.

#### Notes

You can drop the number single malt whiskeys down to 2 to make the problem a bit easier or increase the number of single malts to make the problem more difficult. With more equations, you could also add information for finding more equations...perhaps the cost to ship the single malt whiskeys to the blending facility or information about the color of the individual malts and the color of the final blend.

I have compiled a list of cask strength whiskies that you can use to vary the numbers in the project. This PDF document contains a couple hundred cask strength whiskies that are commercially available (I got this list from http://www.sohowhisky.com/ ). Unless noted, each price is for a 70 centiliter bottle. You might be able to modify this project to include the prices as I did in the Gas Prices Project.