Cover Letter

The Bakers Choice unit covers problems that allow students to see the effects and implications of linear equations and linear inequalities and how they can be applied to linear programming. It is important for students to be able to understand how to graph linear equations and inequalities. Using the graphs of linear equations and inequalities, students will be able to compare the graphs to develop meaningful solutions.
Picturing Cookies was the first problem for students to tackle regarding linear programming. The problem required students to investigate how many combinations of plain and iced cookies can be created with given constraints. Students were given four constraints to work with; the amount of dough that can be used, the amount of icing that can be used, the amount of time that they are given to make the cookies, and the amount of oven space. These four constraints can be represented using inequalities:

Dough: (1)x + (0.7)y is less than or equal to 110.
Icing: (0)x + (0.4)y is less than or equal to 32
Time: (.1)x + (.15)y is less than or equal to 15
Oven: x+y is less than or equal to 140.
Using these inequalities, x represents the amount of plain cookies while y represents the amount of iced cookies.
Graphing these constraints will give present a region that satisfies all 4 equations, which is also known as a feasible region. From there, students can experiment with the amount sold and the amount of profit to help determine the best combination that will yield the highest profit with the least amount of cost.

Profitable Pictures was another problem to help demonstrate linear programming and utilizing given constraints. Profitable pictures provides a problem with less constraints in comparison to Picturing Cookies (Part 1 and 2). Profitable pictures provides students with two separate pictures to sell; pastels and watercolors. The pastels and watercolors present different profits. Pastels can yield a profit of 40$ while watercolors can yield a profit of 100$. The cost of each pastel is 5$ and each watercolor costs 15$ to make. There are two constraints in this problem: Only 16 pictures can be made and there is only 180$ to spend on materials. The two constraints can be written as follows:

Number of pictures: p+w is less than or equal to 16
Money for materials: 5p + 15w is less than or equal to 180.
Using these constraints, p represents the pastels and w represents the watercolors. After graphing these two inequalities students will again see a feasible region that can represent possibilities for how many pastels and how many watercolors can be created. After this feasible region is defined, students can experiment with the feasible region to find possibilities of the amount of pastels and watercolors that can be made. Students can then further their thinking by testing out possible values to find the combination that will yield the highest profit overall.

Linear programming problems are important not only in mathematics but in everyday life. The problems in Bakers choice center on finding the best profit region. This can be applied to everyday life, in the fact that everyone wants to spend the least amount of money possible to gain the highest profit in return. Linear programming is an effective method to help determine cost constraints to ensure that you are gaining the highest amount of profit possible.