Cover Letter
The Baker’s Choice Unit Lesson is one that presents linear programming as a type of problem to solve. In teaching this principle, students are allowed to discover for themselves with facilitation of teachers the best way to solve the problem as well as a deep understanding of why the methods work. The unit problem is one about a couple who own a bakery shop. The couple is trying to determine how many dozens of cookies they should make, both plain and iced. They want to maximize their profit, but must stay within a few constraints. The constraints include amount of cookie dough and icing because they only have 110 pounds of cookie dough and 32 pounds of icing. One dozen iced cookies require 0.7 pounds of cookie dough and 0.4 pounds of icing, while a dozen of plain cookies require no icing and 1 pound of cookie dough. Additionally, they have 15 hours of preparation time available between the two of them and one dozed iced cookies require 0.15 hours of preparation time and one dozen plain cookies require 0.1 hours of preparation time. The last constraint involved the amount of oven space that they have, which only allows for 140 dozen cookies to be baked in total. In terms of cost and profit, plain cookies cost $4.50 a dozen to make and they sell for $6.00 a dozen. Iced cookies cost $5.00 a dozen and sell for $7.00 a dozen. Based on this information, the student needs to determine how many plain and iced cookies they should bake in order to maximize their profit.
This problem is solved by creating linear inequalities, determining a feasible region and calculating where the profit is maximized within this feasible region. In addition to allowing students to learn the way to solve a linear programming problem on their own, they are also introduced to many additional mathematical ideas. In order to create the constraints and understand the profit equation, students must have a firm grasp on what a line is as well as what an equation looks like for a line. Once the line is understood, they must next examine inequalities. Whether students have been introduced to inequalities or not before this, they will have a much better understanding of them after this unit. In addition to just understanding inequalities, they gain a conceptual understanding of why negatives affect the inequality signs. The reasons for the feasible region in a linear programming problem are based on inequalities and students come to a conceptual understanding in the way the lessons in this book are designed. Graphing is another key concept that is used in this lesson. Students also either have to learn to write or may already know how to write algebraic expressions. In addition to just having students write these expressions, they are given homework to have them look at variables in their own lives so that they add background knowledge to the conceptual understanding. In the Broken Eggs problem, students use the idea of a least common multiple to solve it, which is another important mathematical concept that we want students to understand conceptually. If used at the right time in a student’s education, all of these topics could be introduced initially by completing this unit and students would be much further ahead in understanding than those who learn the topics through lecture or book reading in a typical classroom setting. By combining understanding of all of these topics, students are able to complete the core problem, Baker’s Choice of the book as well as multiple other problems introduced.
This problem is solved by creating linear inequalities, determining a feasible region and calculating where the profit is maximized within this feasible region. In addition to allowing students to learn the way to solve a linear programming problem on their own, they are also introduced to many additional mathematical ideas. In order to create the constraints and understand the profit equation, students must have a firm grasp on what a line is as well as what an equation looks like for a line. Once the line is understood, they must next examine inequalities. Whether students have been introduced to inequalities or not before this, they will have a much better understanding of them after this unit. In addition to just understanding inequalities, they gain a conceptual understanding of why negatives affect the inequality signs. The reasons for the feasible region in a linear programming problem are based on inequalities and students come to a conceptual understanding in the way the lessons in this book are designed. Graphing is another key concept that is used in this lesson. Students also either have to learn to write or may already know how to write algebraic expressions. In addition to just having students write these expressions, they are given homework to have them look at variables in their own lives so that they add background knowledge to the conceptual understanding. In the Broken Eggs problem, students use the idea of a least common multiple to solve it, which is another important mathematical concept that we want students to understand conceptually. If used at the right time in a student’s education, all of these topics could be introduced initially by completing this unit and students would be much further ahead in understanding than those who learn the topics through lecture or book reading in a typical classroom setting. By combining understanding of all of these topics, students are able to complete the core problem, Baker’s Choice of the book as well as multiple other problems introduced.
Homework 15: Beginning Portfolio Selection
1. To solve a linear programming problem, you must go through the following steps: Determine what the constraints of the problem are as well as how to determine profit. Create inequalities based on these constraints and an algebraic expression to calculate profit. Graph the inequalities for the constraint and look at where the regions of each inequality come together. This is considered the feasible region of the problem. Any points within this region are possible within the constraints of the problem. However, since the Woos want to maximize their profit, we must determine which point, or combination of plain and iced cookies will do this. By creating an equation for the profit line and looking for which point maximizes this profit, the problem can be solved. We know that the vertices of the feasible region will give the possible maximum profits and the maximum can be determined from looking at the highest of these.
2. By far, the most helpful activity for me was Profitable Pictures to look at how we know which point in the feasible region would give the maximum profit. I had always been told before that one of the vertices would give the maximum and never questioned the reason for this – it just sort of made sense to me, but I couldn’t explain it. By graphing multiple profit lines based on a particular profit that the Woos might want, this enabled me to see that the profit equation has a slope that remains the same no matter what the profit. By sliding the profit lines up along this same slope, I was able to see why the vertex at 6 pastels and 10 watercolors was the one that gave a maximum profit. Both by plotting the different profit lines and using a straw to push up along this slope, there was no longer any confusion about why a maximum profit was at a vertex or along one of the lines of the constraints.
2. By far, the most helpful activity for me was Profitable Pictures to look at how we know which point in the feasible region would give the maximum profit. I had always been told before that one of the vertices would give the maximum and never questioned the reason for this – it just sort of made sense to me, but I couldn’t explain it. By graphing multiple profit lines based on a particular profit that the Woos might want, this enabled me to see that the profit equation has a slope that remains the same no matter what the profit. By sliding the profit lines up along this same slope, I was able to see why the vertex at 6 pastels and 10 watercolors was the one that gave a maximum profit. Both by plotting the different profit lines and using a straw to push up along this slope, there was no longer any confusion about why a maximum profit was at a vertex or along one of the lines of the constraints.
I also liked the way that main problem of the unit was presented and students were given time to think about this problem before going into easier ones to help lead them to a better understanding. The choice of problems and questions that were asked along the way were very well developed. It was helpful for me to talk within a group about the possible combinations that could be made within the constraints on the Baker’s Choice problem. I took the linear programming knowledge that I had and plotted the graphs to look at possibilities, but understanding it through others’ eyes of a guess and check method was useful for me to understand it better conceptually.
Picturing Cookies – Parts 1 and 2 are probably the best other activities to help lead students to understanding linear programming. These problems go into not only looking at the constraints, but creating equations for them, graphing them and creating a feasible region by coloring in each inequality and looking where all fit. This is the key to understanding the basics of this problem. Although there were other assignments that led up to understanding variables, inequalities, etc., this exercise really helps to pull it together for students.
Problem of the Week (POW) - Broken Eggs
This POW did not help me to understand the Baker’s Choice problem or linear programming any better, but it did lead me to understand Least Common Multiples much better. It is an excellent problem and could probably lead students to understand a number of things, but it gives a great questions to have students think mathematically in different ways.
Click here to see my POW Write-Up
Click here to see my POW Write-Up
Baker's Choice Revisited
To: Abby and Bing Woo of PI Bakery
CC: TOBrien Consulting Firm
Re: Maximizing profit for your bakery
In order to maximize profit for your bakery, you should make 75 dozen plain cookies and 50 dozen iced cookies per day. Assuming that you are able to sell this number of cookies, you will make the maximum profit with this combination. With this combination expect the following:
- The total cookies you will make and sell is 125 dozen, so you will not use all of your oven space.
- The total preparation time is the full 15 hours that you have available.
- The amount of dough you will use is the full 110 pounds that you have available.
- The amount of icing you will use is 20 pounds.
- It will cost you $587.50 to make these cookies.
- You will make $800 selling these cookies, with a net profit of $212.50.
We have solved your problem using an intricate mathematical process called linear programming. To summarize for you so that you understand the reason for your maximum profit, we have made a graph too help you visualize this maximum profit.
CC: TOBrien Consulting Firm
Re: Maximizing profit for your bakery
In order to maximize profit for your bakery, you should make 75 dozen plain cookies and 50 dozen iced cookies per day. Assuming that you are able to sell this number of cookies, you will make the maximum profit with this combination. With this combination expect the following:
- The total cookies you will make and sell is 125 dozen, so you will not use all of your oven space.
- The total preparation time is the full 15 hours that you have available.
- The amount of dough you will use is the full 110 pounds that you have available.
- The amount of icing you will use is 20 pounds.
- It will cost you $587.50 to make these cookies.
- You will make $800 selling these cookies, with a net profit of $212.50.
We have solved your problem using an intricate mathematical process called linear programming. To summarize for you so that you understand the reason for your maximum profit, we have made a graph too help you visualize this maximum profit.
- The number of plain cookies is represented by how far to the left and right you go on the graph with the farthest left being no plain cookies and the farthest right being about 150 dozen plain cookies.
- The number of iced cookies is represented by how far up and down you go on the graph with the farthest down being no iced cookies and the farthest up being about 200 dozen iced cookies.
- As a matter of notation, for each point the number of dozen plain cookies is listed first and the number of dozen iced cookies is represented second. So for your maximum profit, you’ll see the point (75, 50).
- The green area that is colored in represents the different combinations that are possible given your constraints of dough, icing, oven space and prep time. You can see each line labeled to show which represents each constraint and by putting all of these together, we can up with this green feasible region.
- In order to show you which of the points within this feasible region give what profits, we have added in the profit lines for you to visualize. The net profit equation was based on the fact that you would make a net profit of $1.50 for each dozen of plain cookies you sell and $2.00 for each dozen of iced cookies you sell.
o The purple line shows you the combinations you could make in order to get a net profit of $100.
o The orange line shows you the combinations you could make in order to get a net profit of $150.
o The green line shows you the combinations you could make in order to get a net profit of $200.
o As you can probably see, to increase profit, this line must be pushed up in the same orientation. If this is done again, we arise to the possibility of 75 plain cookies and 50 iced cookies giving the highest net profit of $212.50.
- The number of iced cookies is represented by how far up and down you go on the graph with the farthest down being no iced cookies and the farthest up being about 200 dozen iced cookies.
- As a matter of notation, for each point the number of dozen plain cookies is listed first and the number of dozen iced cookies is represented second. So for your maximum profit, you’ll see the point (75, 50).
- The green area that is colored in represents the different combinations that are possible given your constraints of dough, icing, oven space and prep time. You can see each line labeled to show which represents each constraint and by putting all of these together, we can up with this green feasible region.
- In order to show you which of the points within this feasible region give what profits, we have added in the profit lines for you to visualize. The net profit equation was based on the fact that you would make a net profit of $1.50 for each dozen of plain cookies you sell and $2.00 for each dozen of iced cookies you sell.
o The purple line shows you the combinations you could make in order to get a net profit of $100.
o The orange line shows you the combinations you could make in order to get a net profit of $150.
o The green line shows you the combinations you could make in order to get a net profit of $200.
o As you can probably see, to increase profit, this line must be pushed up in the same orientation. If this is done again, we arise to the possibility of 75 plain cookies and 50 iced cookies giving the highest net profit of $212.50.
Personal Growth
The Baker’s Choice activities that we completed in class have moved me much further along in my mathematical understanding. I have a much deeper understanding of linear programming problems as well as how to think about problems in a different way. I have never had to write and describe my thought process in terms of a math problem before or presented them, so this was very challenging for me. However, I feel that by having done it, I better know how to both understand my own thinking but even more importantly help to understand my students’ thinking.
I grew leaps and bounds in how to work with others for a math problem through this lesson. Previously in my schooling and life, it had always been more of a competition when I worked with others. When I started this problem, I began like that, immediately going into the steps of solving a linear programming problem and not understanding why no one else was sure of the answer. I quickly learned that I needed to listen to my teammates and that I could actually learn a lot from hearing their thought processes on the problem. If I had just worked the problem based on what I knew about linear programming, I would have gotten the right answer but never would have developed the conceptual understanding that I now have for this type of a problem. I also would not have understood how to really push my students to understand ideas for themselves instead of just being told the procedural steps to make. Through working in groups, I learned an extraordinary amount from my teammates and know now that instead of steamrolling my way through a problem over everyone around me, I need to stop and listen.
I don’t have a difficult time presenting in general because I am comfortable speaking in front of people. However, I did have a hesitation when I presented the POW on broken eggs. I had a difficult time with this POW and didn’t have near the level of proficiency with the problem as David did who spoke right before me. However, by working through this unit and learning about working with others, I was able to do it to present a different way of thinking about a problem. It also helped me to understand my thought process better by presenting it to a group of people who understood the problem already.
The thing that I need to work on the most is how to keep my mouth closed and listen better. Although I have grown in this area, I think I have a ways to go. In my teaching, I try to hold back and not give the answer or a hint that will immediately lead a student to the answer, but I find myself still doing this. Though I also find myself able to hold back, I have a ways to go in improvement in this area. Hopefully by knowing the type of teacher I want to be, I will be more apt to notice my interactions, my student’s responses and fine-tune my teaching style to help my students.
I love the way of teaching in Baker’s Choice and have never seen anything like it before. With the Common Core standards changing education next year, I was able to glimpse some textbooks being considered at my school site. Two of these have similar teaching styles and if administered correctly, it could be very exciting for education. Knowing that this type of problem-based learning is possible in math education, I don’t think I’ll ever be able to only teach in the traditional style again. I know that I may need to do this at times for collaboration and working with a department, but I will always add in problem like this and questioning strategies presented in this book.
I grew leaps and bounds in how to work with others for a math problem through this lesson. Previously in my schooling and life, it had always been more of a competition when I worked with others. When I started this problem, I began like that, immediately going into the steps of solving a linear programming problem and not understanding why no one else was sure of the answer. I quickly learned that I needed to listen to my teammates and that I could actually learn a lot from hearing their thought processes on the problem. If I had just worked the problem based on what I knew about linear programming, I would have gotten the right answer but never would have developed the conceptual understanding that I now have for this type of a problem. I also would not have understood how to really push my students to understand ideas for themselves instead of just being told the procedural steps to make. Through working in groups, I learned an extraordinary amount from my teammates and know now that instead of steamrolling my way through a problem over everyone around me, I need to stop and listen.
I don’t have a difficult time presenting in general because I am comfortable speaking in front of people. However, I did have a hesitation when I presented the POW on broken eggs. I had a difficult time with this POW and didn’t have near the level of proficiency with the problem as David did who spoke right before me. However, by working through this unit and learning about working with others, I was able to do it to present a different way of thinking about a problem. It also helped me to understand my thought process better by presenting it to a group of people who understood the problem already.
The thing that I need to work on the most is how to keep my mouth closed and listen better. Although I have grown in this area, I think I have a ways to go. In my teaching, I try to hold back and not give the answer or a hint that will immediately lead a student to the answer, but I find myself still doing this. Though I also find myself able to hold back, I have a ways to go in improvement in this area. Hopefully by knowing the type of teacher I want to be, I will be more apt to notice my interactions, my student’s responses and fine-tune my teaching style to help my students.
I love the way of teaching in Baker’s Choice and have never seen anything like it before. With the Common Core standards changing education next year, I was able to glimpse some textbooks being considered at my school site. Two of these have similar teaching styles and if administered correctly, it could be very exciting for education. Knowing that this type of problem-based learning is possible in math education, I don’t think I’ll ever be able to only teach in the traditional style again. I know that I may need to do this at times for collaboration and working with a department, but I will always add in problem like this and questioning strategies presented in this book.