Cover Letter
Baker’s Choice is a themed unit about linear programming that comes from the Interactive Mathematics Program (IMP). This unit focuses on using graphs of linear equations and inequalities to solve problems. The central problem of the unit is about the Woo family who owns a bakery shop that makes only two kinds of cookies. The Woo’s want to know how many plain and iced cookies they need to sell in order to maximize profit. The Woo’s face the following constraints: preparation time, amount of ingredients, oven space and cost. The activities in the Baker’s Choice unit help students work toward a graphical solution of the problem. By graphing the linear inequalities that represent the constraints, students discover the optimal solution occurs inside or along the border of the feasible region. There are a few activities included to help students focus on the profit line. Students discover that the profit line for this problem is straight line that shifts in a parallel fashion depending on the value of the function. At this time students realize that the maximal profit occurs at the intersection of the profit line and the border of feasible region. This point allows the Woo family to make the precise amount of plain and iced cookies that will maximize their profit.
This portfolio contains my work on selected problems from Baker’s Choice. The pieces I have chosen highlight the work I found most meaningful and does not include everything I worked on during the unit.
This portfolio contains my work on selected problems from Baker’s Choice. The pieces I have chosen highlight the work I found most meaningful and does not include everything I worked on during the unit.
Selected Problems
Homework 8: Picturing Pictures:
Picturing Pictures is a linear programming problem similar to the Baker’s Choice problem. The problem involves artist, named Hassan, who wants to maximize profit by selling watercolor paintings and pastel paintings. He makes a profit of $40 for each pastel painting and a profit of $100 for each watercolor painting. There are two constraints: materials and total amount of paintings. Hassan can paint at most 16 pictures. He has $180 to spend on materials; it costs $5 for each pastel picture and $15 for each watercolor picture. Given the constraints and profit function, the problem asks for a graphical representation of the feasible region. It also asks us to identify 5 possible combinations of pastels and watercolors and the profit that combination yields.
The following is the work for the problem I did in my interactive notebook:
Let x represent pastel paintings
Let y represent watercolor paintings.
The two constraints are:
x + y ≤ 16 (total amount of pictures to make)
5x+15y ≤ 180 (cost of materials)
I used these constraints to create my graph of the feasible region. I picked the following 5 points along the boundary of the feasible region in order to calculate profit using the profit function:
40x+100y = P (let P=profit).
(16,0) yields a profit of $640
(6,10) yields a profit of $1,240
(0,12) yields a profit of $1,200
(10,6) yields a profit of $1,000
(8,8) yields a profit of $1,120
Picturing Pictures is a linear programming problem similar to the Baker’s Choice problem. The problem involves artist, named Hassan, who wants to maximize profit by selling watercolor paintings and pastel paintings. He makes a profit of $40 for each pastel painting and a profit of $100 for each watercolor painting. There are two constraints: materials and total amount of paintings. Hassan can paint at most 16 pictures. He has $180 to spend on materials; it costs $5 for each pastel picture and $15 for each watercolor picture. Given the constraints and profit function, the problem asks for a graphical representation of the feasible region. It also asks us to identify 5 possible combinations of pastels and watercolors and the profit that combination yields.
The following is the work for the problem I did in my interactive notebook:
Let x represent pastel paintings
Let y represent watercolor paintings.
The two constraints are:
x + y ≤ 16 (total amount of pictures to make)
5x+15y ≤ 180 (cost of materials)
I used these constraints to create my graph of the feasible region. I picked the following 5 points along the boundary of the feasible region in order to calculate profit using the profit function:
40x+100y = P (let P=profit).
(16,0) yields a profit of $640
(6,10) yields a profit of $1,240
(0,12) yields a profit of $1,200
(10,6) yields a profit of $1,000
(8,8) yields a profit of $1,120
I concluded that at making 6 pastel paintings and 10 watercolor paintings would yield the maximum profit. I realized that this occurred at the intersection of the two constraints, but at this time I could not prove why this combination was in fact the maximum profit.This homework activity helped me understand the relationship between the feasible region and maximum profit. Along the boundary of the feasible region, all of the constraints are maximized such that no “paintings” are left unsold. This showed me that somewhere along the edges of the feasible region lies the maximum profit.
Rock 'n Rap
This problem is another linear programming problem about maximizing profit. This one is about a music company, Hits on a Shoestring, that’s trying to decide how many rock and rap CD’s to make. According to this problem, the company makes $20,000 in profit for a rock CD, and $30,000 on each rap CD. The constraints include: production time, production costs, and number of CD’s released. Apparently, the company cannot release more rap CD’s than rock. The following are the constraints I created for the problem:
Let x = rock CD’s
Let y = rap CD’s
1,500x + 1,200y ≤ 15,000(production costs)
18x + 25y ≤ 175(production time)
y ≤ x (cannot produce more rap CD’s than rock)
The profit function is: 20,000x + 30,000y = P
Let x = rock CD’s
Let y = rap CD’s
1,500x + 1,200y ≤ 15,000(production costs)
18x + 25y ≤ 175(production time)
y ≤ x (cannot produce more rap CD’s than rock)
The profit function is: 20,000x + 30,000y = P
To maximize the profit, I created a list of 5 possible combinations of rap and rock CDs and listed their profits. I found that the producing 5 59 rock CD’s and 5 59rap CD’s would maximize profit. I noticed that the maximum profit occurred at an intersection of two constraints and along the boundary of the feasible region. This time of the profit max occurred at the vertex where the cost constraint and the quantity-produced constraint intersect. Thus, setting these inequalities equal to each other will result in the maximum profit.
The third question asks how the maximum profit would change if the company made a profit of $30,000 for a rock CD and $20,000 for a rap CD. I used points along the same graph because the constraints are the same. This time I found that making 10 rock CD’s and not producing any rap CD’s would yield maximum profit of $300,000. Similarly, the maximum profit occurred along the boundary of the feasible region and also along an outer vertex. This activity helped me realize that the maximum profit most likely occurs at a vertex of the feasible region.
The third question asks how the maximum profit would change if the company made a profit of $30,000 for a rock CD and $20,000 for a rap CD. I used points along the same graph because the constraints are the same. This time I found that making 10 rock CD’s and not producing any rap CD’s would yield maximum profit of $300,000. Similarly, the maximum profit occurred along the boundary of the feasible region and also along an outer vertex. This activity helped me realize that the maximum profit most likely occurs at a vertex of the feasible region.
POW: Kick It
Problem Statement:
The Free Thinkers Football League has a different scoring system than traditional football. They score 5 points for each field goal and 3 points for each touchdown. For each play the league can score either a combination of a field goal and touchdown (worth 8 points), a single touchdown (worth 3 points), or a field goal (worth 5 points). A member of the Free Thinkers has just realized that not every numerical score is possible. For example, a score of 1 point is not possible because scoring a field goal grants 5 points and scoring a touchdown grants 3 points. This team member believes that beyond a certain score, all scores are possible. The Free Thinkers League wants to know what the highest impossible score is that teams will not be able to get. The League wants you to figure out the highest impossible score. Is there a highest impossible score, and if so, what is it? Next, they ask you to come up with your own scoring systems (using integers only). Are there scores that are impossible to make? Is there always a highest impossible score? If you think there is, explain why. Do you notice any patterns or rules to use to figure out the highest impossible socre? If you don’t think there is always a highest impossible score, write a rule for when there is and when there isn’t a highest impossible score.
Process:
When solving this problem the first thing I did was label field goals as worth 5 points and label touchdowns as worth 3 points. From there I began listing possible scores. I started my list at 3 points because I knew this had to be the lowest possible score (aside from 0 indicating no points were scored). This score resulted from a single touchdown. Then I listed the multiples of 3 to indicate the possible scores if only touchdowns were scored. Here, I listed 6, 9, 12, 15, etc. Next I identified the possible scores that result from scoring only touchdowns. These scores are multiples of 5 such as 5, 10, 15, 20, etc. The next step was to identify various combinations of touchdowns and field goals. For example if one touchdown was scored and one field goal was scored, the team would have 8 points. If 2 field goals were scored and 2 touchdowns were scored the result would be 16 points. I quickly realized that if the same amount of touchdowns and field goals were scored, the resulting score would be a multiple of 8 (such as 8, 16, 24 etc). The most difficult part was to identify the possible scores that result from a different amount of field goals and touchdowns. For example if one field goal and 2 touchdowns were scored then the score would be 11. If one field goal and 3 touchdowns were scored then the score would be 14. I then switched the numbers of field goals and touchdowns, such as one touchdown and 2 field goals results in 13. One touchdown and 3 field goals results in 18. I continued to list other possible combinations like 2 field goals and 3 touchdowns is 19 points. And 3 field goals and 2 touchdowns is 21 points. I listed the results on my paper then arranged them into consecutive order. However, I did not list every possible score, as there could be infinite scores. I looked at my list and noticed that the numbers: 1, 2, 4 and 7 were not listed. Consecutive numbers up until my stopping point (25) were listed. This made me believe that 7 was the highest impossible score, but why?
This is a table of scores I listed:
The Free Thinkers Football League has a different scoring system than traditional football. They score 5 points for each field goal and 3 points for each touchdown. For each play the league can score either a combination of a field goal and touchdown (worth 8 points), a single touchdown (worth 3 points), or a field goal (worth 5 points). A member of the Free Thinkers has just realized that not every numerical score is possible. For example, a score of 1 point is not possible because scoring a field goal grants 5 points and scoring a touchdown grants 3 points. This team member believes that beyond a certain score, all scores are possible. The Free Thinkers League wants to know what the highest impossible score is that teams will not be able to get. The League wants you to figure out the highest impossible score. Is there a highest impossible score, and if so, what is it? Next, they ask you to come up with your own scoring systems (using integers only). Are there scores that are impossible to make? Is there always a highest impossible score? If you think there is, explain why. Do you notice any patterns or rules to use to figure out the highest impossible socre? If you don’t think there is always a highest impossible score, write a rule for when there is and when there isn’t a highest impossible score.
Process:
When solving this problem the first thing I did was label field goals as worth 5 points and label touchdowns as worth 3 points. From there I began listing possible scores. I started my list at 3 points because I knew this had to be the lowest possible score (aside from 0 indicating no points were scored). This score resulted from a single touchdown. Then I listed the multiples of 3 to indicate the possible scores if only touchdowns were scored. Here, I listed 6, 9, 12, 15, etc. Next I identified the possible scores that result from scoring only touchdowns. These scores are multiples of 5 such as 5, 10, 15, 20, etc. The next step was to identify various combinations of touchdowns and field goals. For example if one touchdown was scored and one field goal was scored, the team would have 8 points. If 2 field goals were scored and 2 touchdowns were scored the result would be 16 points. I quickly realized that if the same amount of touchdowns and field goals were scored, the resulting score would be a multiple of 8 (such as 8, 16, 24 etc). The most difficult part was to identify the possible scores that result from a different amount of field goals and touchdowns. For example if one field goal and 2 touchdowns were scored then the score would be 11. If one field goal and 3 touchdowns were scored then the score would be 14. I then switched the numbers of field goals and touchdowns, such as one touchdown and 2 field goals results in 13. One touchdown and 3 field goals results in 18. I continued to list other possible combinations like 2 field goals and 3 touchdowns is 19 points. And 3 field goals and 2 touchdowns is 21 points. I listed the results on my paper then arranged them into consecutive order. However, I did not list every possible score, as there could be infinite scores. I looked at my list and noticed that the numbers: 1, 2, 4 and 7 were not listed. Consecutive numbers up until my stopping point (25) were listed. This made me believe that 7 was the highest impossible score, but why?
This is a table of scores I listed:
Next I created my own scoring system: a field goal worth 7 points and a touchdown worth 4 points. First, I labeled touchdowns as worth 4 points and field goals as worth 7 points. Then I began to list the possible scores. I started at 4 points because this is the lowest possible score that could occur if only one touchdown was scored. The next possible score was 7, occurring if only a field goal was scored. Then 11 points if one field goal and one touchdown were scored. If only touchdowns were scored the possible scores include: 4, 8, 12, 16, 20 etc. If only field goals were scored the possible scores were 7, 14, 21, 28, 35, etc. I created a chart like this to represent possible scores:
Solution:
My solution is only partial. I concluded that the highest impossible score the Free Thinkers League is 7 points. I created an equation to represent the possible scores: p = 3t+5f, where t,f ≥ 0 and t represents number of touchdowns and f represents number of field goals. I believe this is correct because the points scored depends on the number of touchdowns and field goals scored during a game. Since each touchdown is worth 3 points and each field goal is worth 5 points, I multiplied 3 by the number of field goals, f, and I multiplied 5 by the number of touchdowns, t. However I did not develop a rule for the impossible scores. I created a similar equation to represent the possible scores for my scoring guide: p= 4t+7f, where t and f ≥ 0.
Extensions and Variations:
To scaffold this problem for those who need support, ask them to fill out the following chart for these ten combinations. Next, ask them to list any numbers that do not appear in the possible score column and reflect on what they notice.
Evaluation Option B:
When I began this problem it seemed almost too easy. I began listing the possible scores up to 20. I looked back at my list and realized 1, 2, 4, and 7 were impossible. It was not difficult for me to come up with rule for the possible scores. But in the end I did not come up with a rule for the impossible. That frustrated me. This problem was educationally worthwhile for me because it challenged my need to conclude with an answer. I really like following procedures, and usually this gets me to an answer. I did not follow any procedures when solving this problem. I made sense of it in a way that worked for me. I still do not know how to prove a rule for the impossible scores, but I forth effort and persevered through it (CCSS.MATH.PRACTICE.MP1). I ultimately learned that the mathematical process is much more valuable than obtaining the end result. What I mean is that I thought a great deal about it. Although I don’t have a rule to show, I am satisfied with my partial solution because I obtained it without procedures. This problem required me to think deeper than I typically do.
My solution is only partial. I concluded that the highest impossible score the Free Thinkers League is 7 points. I created an equation to represent the possible scores: p = 3t+5f, where t,f ≥ 0 and t represents number of touchdowns and f represents number of field goals. I believe this is correct because the points scored depends on the number of touchdowns and field goals scored during a game. Since each touchdown is worth 3 points and each field goal is worth 5 points, I multiplied 3 by the number of field goals, f, and I multiplied 5 by the number of touchdowns, t. However I did not develop a rule for the impossible scores. I created a similar equation to represent the possible scores for my scoring guide: p= 4t+7f, where t and f ≥ 0.
Extensions and Variations:
To scaffold this problem for those who need support, ask them to fill out the following chart for these ten combinations. Next, ask them to list any numbers that do not appear in the possible score column and reflect on what they notice.
Evaluation Option B:
When I began this problem it seemed almost too easy. I began listing the possible scores up to 20. I looked back at my list and realized 1, 2, 4, and 7 were impossible. It was not difficult for me to come up with rule for the possible scores. But in the end I did not come up with a rule for the impossible. That frustrated me. This problem was educationally worthwhile for me because it challenged my need to conclude with an answer. I really like following procedures, and usually this gets me to an answer. I did not follow any procedures when solving this problem. I made sense of it in a way that worked for me. I still do not know how to prove a rule for the impossible scores, but I forth effort and persevered through it (CCSS.MATH.PRACTICE.MP1). I ultimately learned that the mathematical process is much more valuable than obtaining the end result. What I mean is that I thought a great deal about it. Although I don’t have a rule to show, I am satisfied with my partial solution because I obtained it without procedures. This problem required me to think deeper than I typically do.
Baker’s Choice Revisited:
After carefully investigating the Woo’s constraints in order to maximize profit, they should make 75 plain cookies and 50 iced cookies. If they make this combination of cookies their profit will be $212.50.
Here’s Why:
Let x = plain cookies
Let y = iced cookies
x+y ≤ 140 (amount of oven space)
x+0.7y ≤ 110 (amount of cookie dough)
0.4y ≤ 32 (amount of icing)
10x+15y ≤ 1500 (amount of time)
1.5x+2y= P (profit function)
Here’s Why:
Let x = plain cookies
Let y = iced cookies
x+y ≤ 140 (amount of oven space)
x+0.7y ≤ 110 (amount of cookie dough)
0.4y ≤ 32 (amount of icing)
10x+15y ≤ 1500 (amount of time)
1.5x+2y= P (profit function)
Statement of Personal Growth
The Baker’s Choice unit was very different from the type of mathematical learning I am used to. My previous experiences of learning math consisted a teacher/professor lecturing and explaining the procedures to solve a problem. That traditional learning style happened to work for me but after working on Baker’s Choice I realize the need to practice thinking deeply about the math. The Baker’s Choice never gave explicit procedures to solve the problems. Instead, it guided me through the unit by asking questions like “why” instead of “what”. This forced me to apply what I already knew about writing, solving, and graphing inequalities and to think deeply about these questions.
The Baker’s Choice unit was more hands-on than most of my previous mathematics experiences. I had to experiment with different ways to problem solve. I really liked that the tasks in this unit were based on real-life situations unlike most problems I have done. It makes the math more relatable and worthwhile to show the real world applications of the concepts. I will try to incorporate real world applications in my future classes.
The Baker’s Choice also emphasized the importance of collaboration and discussions in the math class. Discussing the mathematics this semester challenged me to explore the content and helped me gain a deeper understanding. Collaborative mathematical conversation is something I will carry forward in my own teaching practice.
Finally, the Baker’s Choice never gave solutions and that helped me focus on the process instead of the product. Overall, I enjoyed this unit and this course because it challenged my need to follow procedures and more importantly, it made me think deeper about mathematics.
The Baker’s Choice unit was more hands-on than most of my previous mathematics experiences. I had to experiment with different ways to problem solve. I really liked that the tasks in this unit were based on real-life situations unlike most problems I have done. It makes the math more relatable and worthwhile to show the real world applications of the concepts. I will try to incorporate real world applications in my future classes.
The Baker’s Choice also emphasized the importance of collaboration and discussions in the math class. Discussing the mathematics this semester challenged me to explore the content and helped me gain a deeper understanding. Collaborative mathematical conversation is something I will carry forward in my own teaching practice.
Finally, the Baker’s Choice never gave solutions and that helped me focus on the process instead of the product. Overall, I enjoyed this unit and this course because it challenged my need to follow procedures and more importantly, it made me think deeper about mathematics.