baker's choice portfolio
Cover Letter
In this unit, through multiple assignments and class discussions, we aimed to “deepen [student] understanding of the relationship between equations or inequalities and their graphs and for them to reason and solve problems using graphs” (Baker’s Choice, pp. 1). The depth of understanding was reached through collaboration and strategically placed guided questions. Students found reasoning in meaningful ways to make sense of the bigger picture.
When we refer to a linear programming problem, it is a problem where the purpose is to find some sort of minimum or maximum based on a set of restrictions. These restrictions are referred to as constraints. The linear programming problems we looked at in Baker’s Choice ultimately aim to find the maximum profit. To do this, we write the constraints as inequalities, graph them, and shade. Once all the constraints are represented on the graph, we use the equation for the profit to determine which combination of choice will result in the maximum profit.
As we saw in this unit the maximum profit will occur at one vertex of the shaded region or one entire side of the figure. This is because of the slope of the profit line. No matter what profit you’re looking into, the slope of this line will be the same for the entire problem. So, we see why it is that a vertex (or one entire side) will result in the maximum profit
The Baker’s Choice unit starts off with a big question regarding The Woos making a maximum profit on their cookie sales. They sell plain cookies and iced cookies and are limited in the following ways: the amount of dough they have, the amount of icing they have, the oven space available, and the time they have to make cookies. The constraints are broken down below:
The amount of dough they have: 110lbs
The amount of icing they have: 32lbs
The amount of oven space available: enough room to make 140 dozen cookies
The time they have to make cookies: 15 hours
We then represented the constraints as inequalities and graphed them in Picturing Cookies Part 2. From our graph we were able to determine all possible combinations of plain and iced cookies. Using this information we saw how to find the maximum profit The Woos can make if they profit $1.50 from plain cookies and $2.00 from iced cookies. This is investigated in Picturing Cookies Part 2.
In the assignment Profitable Pictures, we looked at a problem that is similar to Baker’s Choice. In this problem, we met Hassan. Hassan is a painter and wants to maximize his profits from selling watercolor paintings and pastel paintings. Like the Woos, Hassan has limitations on what he can produce. In this problem we found why the slope of the profit line played a vital role in determining the maximum profit and why the maximum profit occurred at a vertex of the shaded region in the graph.
During this unit there were three POWs (Problem of the Week). The first POW was The Broken Eggs. In this problem, we helped the farmer determine how many eggs she had. All her eggs broke when her cart knocked over. Using what she recalled about how she packaged her eggs in the morning, we determined how many eggs she could have had as well as to determine if there were multiple possibilities to how many eggs she could have. During class discussions, it was fascinating to see how many ways there were to go about solving this problem. My classmates’ thought processes differed from mine and we all arrived to the same conclusion. Seeing the variety of ways to work through a problem to still get the correct answer is something that students can really benefit from. They may see a process that makes more sense to them and they will see that mathematics is all interconnected.
This unit focused on critical thinking and communicating that thinking as a way for students to have a deep understanding of the material. This learning process can carry over to virtually all topics in mathematics, not just linear programming.
Selected Papers
This portfolio contains Picturing Cookies Part 2, Profitable Pictures, Baker’s Choice Revisited, Problem of the Week: The Broken Eggs, and Homework 14: Reflections on learning within the Statement of Personal Growth
Picturing Cookies Part 2
In this assignment, we take a look at all the constraints that the Woos have in making cookies and then graph these constraints onto a single graph. With shading we ultimately find the 1 common shaded region.
Given the constraints presented in Baker’s Choice, we came up with the following inequalities where x represents the number of dozen of plain cookies and y represents the number of dozen of iced cookies:
· x + 0.7y ≤ 110 (for the amount of cookie dough)
· x + y ≤ 140 (for the amount of oven space)
· 0.1x + 0.15y ≤ 15 (for the time available)
· 0.4y ≤ 32 (for the amount of icing)
The graph of these inequalities can be seen in the bakerschoice_0003 pdf below.
The figure that is shaded all four colors has vertices (0, 80), (30, 80), (75, 50), and (110, 0). This shaded region is important because it accounts for all the constraints. Each constraint is shaded using a different color. So, when we put all four constraints on the same graph, we can distinguish certain possibilities for specific constraints. More importantly, we can see the set of possible cookie combinations by looking at the part that is shaded with all four colors. Any combination of plain and iced cookies that falls within this shaded region will satisfy all four constrains.
When plugging these vertices into the profit equation, we found that (75, 50) resulted in the greatest profit, $212.50. Later in class, during Profitable Pictures, we see why we focus on the vertices of the common shaded area in order to find the maximum profit.
Profitable Pictures
In this problem, we meet Hassan the Painter and learn about the constraints he’s facing to determine how many watercolors he should paint and how many pastels he should paint. These are the facts:
- Each pastel requires $5 in materials and earns a profit of $40 for Hassan.
- Each watercolor requires $15 in materials and earns a profit of $100 for Hassan.
- Hassan has $180 to spend on materials.
- Hassan can make at most 16 pictures.
This problem takes the approach to have a goal profit in mind and then find what points in the feasible region yield that exact profit. The three combinations my group and I found to give a $1000 profit are (0, 100), (5, 8), and (10, 6). Where the x-coordinate represents the number of pastels and the y-coordinate represents the number watercolors. Then, we found three combinations that gave a $500 profit and labeled them on the graph. Finally, we found three combinations to give $600 and label them on the graph. Finally we found the maximum profit Hassan could make.
The pdf titled BakersChoice_0002 shows the graphed constraints.
The following class we had a discussion as to how we knew that what we found was in fact the maximum. We noticed that the points we found yielding $1000 profit al formed a line, the points that gave $500 profit formed a line, and the same thing happened with the points that result in $600 profit. These three separate lines were all parallel to one another. Using prior knowledge, this told us that these profit lines had the same slope and that as this slope line (as demonstrated with the coffee stirrer) went higher up the graph, the profit it was representing increased.
Maintaining the slope, we slid the coffee stirrer all the way up the graph. The last place that the coffee stirrer touched as we slid it off the feasible region is the highest profit. If that profit line were to be a tad bit higher, meaning an even greater profit, then the coffee stirrer wouldn’t touch anything in the feasible region. Thus, we saw why it must be the vertex that gives the combination yielding maximum profit; based on the slope of the profit line, and the shape of the feasible region, it is at this point of intersection that would provide that final place where the coffee stirrer (the profit line) is on the commonly shaded area.
Baker’s Choice Revisited
The Woos sell plain and iced cookies. They are limited on how many cookies they can make in the following ways:
- They have 110 pounds of cookie dough; 1 dozen iced cookies require 0.7lbs of dough and 1 dozen plain cookies require 1lb of dough.
- They have room to bake a total 140 dozen cookies.
- They have 15 hours for cookie preparation; 1 dozen plain cookies require 0.1 hours and 1 dozen iced cookies require 0.15 hours.
- They have 32 pounds of icing; a dozen iced cookies require 0.4lbs of icing
We can write these restrictions in the form of inequalities and graph them. The graphs will give us all possibilities that are allowed. So, when we graph them all together, the overlap part takes all restrictions into account. The common area gives the Woos all possible options of how many iced and plain cookies they could make. We let x represent the number of dozen of plain cookies and let y represent the number of dozen of iced cookies. So our restrictions (aka constraints) can be written as:
Using these inequalities, we get the following graph found in the BakersChoice_0003 pdf posted below.
In order for the Woos to make the greatest profit, we want to know how many dozens of plain and iced cookies to make. The Woos make $1.50 per dozen of plain cookies and $2.00 per dozen of iced cookies. Putting this information into an equation, we’re given: Profit = 1.5x + 2y. The slope of this linear equation, no matter the profit, is always -3/4. As the profit line slides upward, as can be seen with the coffee stirrer, the profit increases.
So, to find maximum profit we want this profit line to be as high as possible such that part of the line still touches the shaded region. This occurs when the profit line equation is y = (-3/4)x + 106.25. The point (75, 50) is the point this line touches. So, 75 dozen plain cookies and 50 dozen iced cookies must yield the highest profit. This is a $212.50 profit. So, any other combination of plain and iced cookies will yield a profit greater, nor equal, to this.
Problem of the Week: The Broken Eggs
In this POW we are presented with the following problem:
A Farmer’s cart of eggs tipped over. All of the eggs broke. For insurance purposes she needed to know how many eggs she had. Based on how she attempted to package her eggs in the morning, this is what she remembers:
· When she packaged the eggs in pairs, there was 1 egg left over.
· When she packaged the eggs into groups of 3, there was 1 egg left over.
· When she packaged the eggs into groups of 4, there was 1 egg left over.
· When she packaged the eggs into groups of 5, there was 1 egg left over.
· When she packaged the eggs into groups of 6, there was 1 egg left over.
· When she packaged the eggs into groups of 7, there were no eggs left over.
From this information, what can the farmer figure out about how many eggs she had? Is there more than one possibility?
The posted document below provides the full POW write-up.
Statement of Personal Growth/ Homework 14: Reflections on Learning
The method of how math was presented in Baker’s Choice was far different from anything I’ve experienced. This unit really focused on learning math through collaboration. The process and method used in Baker’s Choice focuses on learning the concepts extensively. The depth of material covered in Baker’s Choice ensured conceptual understanding instead of mindless work in order to arrive to a solution.
Prior to Baker’s Choice I already knew the process of writing constraints as inequalities, graphing those inequalities, and plugging in vertices of the shaded region to find the minimum or maximum. Baker’s choice however, showed me why plugging in those vertices would actually lead to the correct answer. In my history of math, none of my teachers taught math by facilitating student discussion. Whenever I collaborated with my peers it was outside of class. I considered this to be the chance for me to clarify topics covered in class. After Baker’s Choice, I now consider those after-class “study-groups” to be where the real learning took place. That is where I had my light bulb moments.
This class put structure on group work where the instructor facilitated and provided “thinker” questions. This style of learning is something I want my students to experience. This interaction left me with confidence in mathematics that is somehow different than the confidence that I already felt. I also gained confidence in my classmate’s mathematical abilities. It was interesting to see how other people went about solving the same problem in ways that differed greatly from my way of thinking. Seeing others succeed through a method that I wouldn’t have thought of showed me their mathematical strengths. Their different methods even reconfirmed how certain parts of mathematics are related.
I presented the Problem of the Week: Kick It! To the class. I enjoyed this presentation because it was about my way of thinking when attacking the problem; it wasn’t about whether or not I got the right answer. Because the emphasis was placed on my thought process, I wasn’t a bit nervous.
Presentations weren’t the only part of this unit that focused on my thought process. The emphasis of this course was to pay attention and take notice to everyone’s thought process. This forced me to actively think, reflect, and describe my own thought process.
The methods I learned in this class are quite beneficial in deeper understanding. This process aligns with the goals of Common Core for students to think critically and to communicate their thinking. I want my classroom to use this model.
In this unit, through multiple assignments and class discussions, we aimed to “deepen [student] understanding of the relationship between equations or inequalities and their graphs and for them to reason and solve problems using graphs” (Baker’s Choice, pp. 1). The depth of understanding was reached through collaboration and strategically placed guided questions. Students found reasoning in meaningful ways to make sense of the bigger picture.
When we refer to a linear programming problem, it is a problem where the purpose is to find some sort of minimum or maximum based on a set of restrictions. These restrictions are referred to as constraints. The linear programming problems we looked at in Baker’s Choice ultimately aim to find the maximum profit. To do this, we write the constraints as inequalities, graph them, and shade. Once all the constraints are represented on the graph, we use the equation for the profit to determine which combination of choice will result in the maximum profit.
As we saw in this unit the maximum profit will occur at one vertex of the shaded region or one entire side of the figure. This is because of the slope of the profit line. No matter what profit you’re looking into, the slope of this line will be the same for the entire problem. So, we see why it is that a vertex (or one entire side) will result in the maximum profit
The Baker’s Choice unit starts off with a big question regarding The Woos making a maximum profit on their cookie sales. They sell plain cookies and iced cookies and are limited in the following ways: the amount of dough they have, the amount of icing they have, the oven space available, and the time they have to make cookies. The constraints are broken down below:
The amount of dough they have: 110lbs
- 1 dozen plain cookies: 1lb
- 1 dozen iced cookies: 0.7lbs
The amount of icing they have: 32lbs
- 1 dozen iced cookies: 0.4lbs
The amount of oven space available: enough room to make 140 dozen cookies
- Altogether 140 dozen cookies can be made.
The time they have to make cookies: 15 hours
- 1 dozen plain cookies: 0.1 hours
- 1 dozen iced cookies: 0.15 hours
We then represented the constraints as inequalities and graphed them in Picturing Cookies Part 2. From our graph we were able to determine all possible combinations of plain and iced cookies. Using this information we saw how to find the maximum profit The Woos can make if they profit $1.50 from plain cookies and $2.00 from iced cookies. This is investigated in Picturing Cookies Part 2.
In the assignment Profitable Pictures, we looked at a problem that is similar to Baker’s Choice. In this problem, we met Hassan. Hassan is a painter and wants to maximize his profits from selling watercolor paintings and pastel paintings. Like the Woos, Hassan has limitations on what he can produce. In this problem we found why the slope of the profit line played a vital role in determining the maximum profit and why the maximum profit occurred at a vertex of the shaded region in the graph.
During this unit there were three POWs (Problem of the Week). The first POW was The Broken Eggs. In this problem, we helped the farmer determine how many eggs she had. All her eggs broke when her cart knocked over. Using what she recalled about how she packaged her eggs in the morning, we determined how many eggs she could have had as well as to determine if there were multiple possibilities to how many eggs she could have. During class discussions, it was fascinating to see how many ways there were to go about solving this problem. My classmates’ thought processes differed from mine and we all arrived to the same conclusion. Seeing the variety of ways to work through a problem to still get the correct answer is something that students can really benefit from. They may see a process that makes more sense to them and they will see that mathematics is all interconnected.
This unit focused on critical thinking and communicating that thinking as a way for students to have a deep understanding of the material. This learning process can carry over to virtually all topics in mathematics, not just linear programming.
Selected Papers
This portfolio contains Picturing Cookies Part 2, Profitable Pictures, Baker’s Choice Revisited, Problem of the Week: The Broken Eggs, and Homework 14: Reflections on learning within the Statement of Personal Growth
Picturing Cookies Part 2
In this assignment, we take a look at all the constraints that the Woos have in making cookies and then graph these constraints onto a single graph. With shading we ultimately find the 1 common shaded region.
Given the constraints presented in Baker’s Choice, we came up with the following inequalities where x represents the number of dozen of plain cookies and y represents the number of dozen of iced cookies:
· x + 0.7y ≤ 110 (for the amount of cookie dough)
· x + y ≤ 140 (for the amount of oven space)
· 0.1x + 0.15y ≤ 15 (for the time available)
· 0.4y ≤ 32 (for the amount of icing)
The graph of these inequalities can be seen in the bakerschoice_0003 pdf below.
The figure that is shaded all four colors has vertices (0, 80), (30, 80), (75, 50), and (110, 0). This shaded region is important because it accounts for all the constraints. Each constraint is shaded using a different color. So, when we put all four constraints on the same graph, we can distinguish certain possibilities for specific constraints. More importantly, we can see the set of possible cookie combinations by looking at the part that is shaded with all four colors. Any combination of plain and iced cookies that falls within this shaded region will satisfy all four constrains.
When plugging these vertices into the profit equation, we found that (75, 50) resulted in the greatest profit, $212.50. Later in class, during Profitable Pictures, we see why we focus on the vertices of the common shaded area in order to find the maximum profit.
Profitable Pictures
In this problem, we meet Hassan the Painter and learn about the constraints he’s facing to determine how many watercolors he should paint and how many pastels he should paint. These are the facts:
- Each pastel requires $5 in materials and earns a profit of $40 for Hassan.
- Each watercolor requires $15 in materials and earns a profit of $100 for Hassan.
- Hassan has $180 to spend on materials.
- Hassan can make at most 16 pictures.
This problem takes the approach to have a goal profit in mind and then find what points in the feasible region yield that exact profit. The three combinations my group and I found to give a $1000 profit are (0, 100), (5, 8), and (10, 6). Where the x-coordinate represents the number of pastels and the y-coordinate represents the number watercolors. Then, we found three combinations that gave a $500 profit and labeled them on the graph. Finally, we found three combinations to give $600 and label them on the graph. Finally we found the maximum profit Hassan could make.
The pdf titled BakersChoice_0002 shows the graphed constraints.
The following class we had a discussion as to how we knew that what we found was in fact the maximum. We noticed that the points we found yielding $1000 profit al formed a line, the points that gave $500 profit formed a line, and the same thing happened with the points that result in $600 profit. These three separate lines were all parallel to one another. Using prior knowledge, this told us that these profit lines had the same slope and that as this slope line (as demonstrated with the coffee stirrer) went higher up the graph, the profit it was representing increased.
Maintaining the slope, we slid the coffee stirrer all the way up the graph. The last place that the coffee stirrer touched as we slid it off the feasible region is the highest profit. If that profit line were to be a tad bit higher, meaning an even greater profit, then the coffee stirrer wouldn’t touch anything in the feasible region. Thus, we saw why it must be the vertex that gives the combination yielding maximum profit; based on the slope of the profit line, and the shape of the feasible region, it is at this point of intersection that would provide that final place where the coffee stirrer (the profit line) is on the commonly shaded area.
Baker’s Choice Revisited
The Woos sell plain and iced cookies. They are limited on how many cookies they can make in the following ways:
- They have 110 pounds of cookie dough; 1 dozen iced cookies require 0.7lbs of dough and 1 dozen plain cookies require 1lb of dough.
- They have room to bake a total 140 dozen cookies.
- They have 15 hours for cookie preparation; 1 dozen plain cookies require 0.1 hours and 1 dozen iced cookies require 0.15 hours.
- They have 32 pounds of icing; a dozen iced cookies require 0.4lbs of icing
We can write these restrictions in the form of inequalities and graph them. The graphs will give us all possibilities that are allowed. So, when we graph them all together, the overlap part takes all restrictions into account. The common area gives the Woos all possible options of how many iced and plain cookies they could make. We let x represent the number of dozen of plain cookies and let y represent the number of dozen of iced cookies. So our restrictions (aka constraints) can be written as:
- x + 0.7y ≤ 110 (for the amount of cookie dough)
- x + y ≤ 140 (for the amount of oven space)
- 0.1x + 0.15y ≤ 15 (for the time available)
- 0.4y ≤ 32 (for the amount of icing)
Using these inequalities, we get the following graph found in the BakersChoice_0003 pdf posted below.
In order for the Woos to make the greatest profit, we want to know how many dozens of plain and iced cookies to make. The Woos make $1.50 per dozen of plain cookies and $2.00 per dozen of iced cookies. Putting this information into an equation, we’re given: Profit = 1.5x + 2y. The slope of this linear equation, no matter the profit, is always -3/4. As the profit line slides upward, as can be seen with the coffee stirrer, the profit increases.
So, to find maximum profit we want this profit line to be as high as possible such that part of the line still touches the shaded region. This occurs when the profit line equation is y = (-3/4)x + 106.25. The point (75, 50) is the point this line touches. So, 75 dozen plain cookies and 50 dozen iced cookies must yield the highest profit. This is a $212.50 profit. So, any other combination of plain and iced cookies will yield a profit greater, nor equal, to this.
Problem of the Week: The Broken Eggs
In this POW we are presented with the following problem:
A Farmer’s cart of eggs tipped over. All of the eggs broke. For insurance purposes she needed to know how many eggs she had. Based on how she attempted to package her eggs in the morning, this is what she remembers:
· When she packaged the eggs in pairs, there was 1 egg left over.
· When she packaged the eggs into groups of 3, there was 1 egg left over.
· When she packaged the eggs into groups of 4, there was 1 egg left over.
· When she packaged the eggs into groups of 5, there was 1 egg left over.
· When she packaged the eggs into groups of 6, there was 1 egg left over.
· When she packaged the eggs into groups of 7, there were no eggs left over.
From this information, what can the farmer figure out about how many eggs she had? Is there more than one possibility?
The posted document below provides the full POW write-up.
Statement of Personal Growth/ Homework 14: Reflections on Learning
The method of how math was presented in Baker’s Choice was far different from anything I’ve experienced. This unit really focused on learning math through collaboration. The process and method used in Baker’s Choice focuses on learning the concepts extensively. The depth of material covered in Baker’s Choice ensured conceptual understanding instead of mindless work in order to arrive to a solution.
Prior to Baker’s Choice I already knew the process of writing constraints as inequalities, graphing those inequalities, and plugging in vertices of the shaded region to find the minimum or maximum. Baker’s choice however, showed me why plugging in those vertices would actually lead to the correct answer. In my history of math, none of my teachers taught math by facilitating student discussion. Whenever I collaborated with my peers it was outside of class. I considered this to be the chance for me to clarify topics covered in class. After Baker’s Choice, I now consider those after-class “study-groups” to be where the real learning took place. That is where I had my light bulb moments.
This class put structure on group work where the instructor facilitated and provided “thinker” questions. This style of learning is something I want my students to experience. This interaction left me with confidence in mathematics that is somehow different than the confidence that I already felt. I also gained confidence in my classmate’s mathematical abilities. It was interesting to see how other people went about solving the same problem in ways that differed greatly from my way of thinking. Seeing others succeed through a method that I wouldn’t have thought of showed me their mathematical strengths. Their different methods even reconfirmed how certain parts of mathematics are related.
I presented the Problem of the Week: Kick It! To the class. I enjoyed this presentation because it was about my way of thinking when attacking the problem; it wasn’t about whether or not I got the right answer. Because the emphasis was placed on my thought process, I wasn’t a bit nervous.
Presentations weren’t the only part of this unit that focused on my thought process. The emphasis of this course was to pay attention and take notice to everyone’s thought process. This forced me to actively think, reflect, and describe my own thought process.
The methods I learned in this class are quite beneficial in deeper understanding. This process aligns with the goals of Common Core for students to think critically and to communicate their thinking. I want my classroom to use this model.
bakerschoice_0003.pdf | |
File Size: | 514 kb |
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bakerschoice_0002.pdf | |
File Size: | 667 kb |
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pow_write-up_broken_eggs.docx | |
File Size: | 16 kb |
File Type: | docx |