The problem posed in Baker's Choice is about maximizing profit from selling two types of cookies, plain and iced. There are several constraints in the problem including limited cookie dough (110 lbs), icing (32 lbs), preparation time 15 hours, and oven space (holds 140 dan cookies.) Iced cookies cost $5.00/dzn to make and sell for $7.00/dzn yielding a profit of $2.00/dzn. Plain cookies cost $4.50/dzn to make and sell for $6.00/dzn yielding a profit of 1.50/ dzn.
Baker's Choice begins to scaffold student understanding by exploring specific combinations that work and ones that don't work with the constraints. The unit then looks at a simpler problem where only the time is constrained. As inequalities are introduced, manipulation is taught where students create equivalent inequalities using mathematical reasoning.
In Picturing Cookies Part I, Students become familiar with what graphing inequalities actually means by plotting points that satisfy a given constraint in one color and don't satisfy the constraint in another. This clearly shows the border of one inequality dividing the plane into two distinct regions (one that satisfies the constraint and one that doesn't.) Then students graph the constraints of the problem for Picturing Cookies Part II and construct a feasible region which meets all the constraints. It is evident that this region will be some sort of a polygon due to being composed of straight lines with half a plane meeting the constraints.
Finally, the profit function is examined in Profitable Pictures. Students are asked to plot points of a specific profit (first $1,000, then $500, and finally $600) in different colors for different profits. By connecting these points, students are led to discover that each profit produces points that are on the same line. Also, students conjecture that each profit "line" has the same slope (coffee stirrers can be used to move the profit line up.) As the profit increases, the profit function moves up and to the right and the slope remains the same. Consequently students discover that the last point on the feasible region is the one that must have the maximum profit.
After connecting that the maximum profit is at a vertex (or along the constraint line), students revisit the profit maximizing of making dozens of plain and iced cookies. They look at the feasible region and point at which the profit line crosses the last vertex. Here they find that the last place the profit line touches is at (75,50) when the profit=$212.50. Thus 75 dozen plain cookies and 50 dozen iced cookies gives a maximum profit of $212.50.
Selected Papers (Projects)
(Including Homework 15 and Baker's Choice Revisited)
Steps to Solving a Profit Linear Programing Problem
First identify the variables and write the constraint inequalities. Graph the constraint inequalities shading the area that makes the inequality true. Identify the feasible region that meets all the constraints (the part that is shaded by every constraint.) Write the profit equation using the defined variables. Find the vertices of the feasible region and test them in the profit equation to find the maximum profit.
Lesson Summary for Baker's Choice Unit
Picturing Cookies Part I and II
This introduces the discovery of the relationship between an inequality an the the equation of that line. The idea is to plot many points to get the idea of why we shade a region when working with an inequality. This was especially powerful to me because many students don't understand what they are actually doing when they are shading an inequality.
As more and more points are plotted, students begin to see the ones that satisfy the inequality and the ones that don't form a border (the equation of the line.)
Using this information the Cookie constraints are revisited.
Profitable Pictures
Profitable Pictures was the key to understanding why the maximum must be at a vertex of the constrained region. The profit equation (in this case P(x,y) = 40x + 100y, where x = # of pastels and y=# of watercolors sold) is a line. As the desired profit changes the profit line also changes. However, the ratio of pictures sold does not. This rate is the slope of the profit line, no matter what the profit is. As the line moves up the y axis it increases in profit. Therefore, the highest point still in the constrained region is the maximum profit. This example on from Desmos demonstrates profitable pictures. Several coordinate points were graphed at a particular profit. For a profit of $1000 the points (0,10), (10,6) and (5,8) that met this criteria were graphed. These all form a straight line with a slope of -2/5. For a profit of $500 the points (0,5), (10,1), and (5, 3) met this criteria and were graphed (also having a slope of -2/5 but different y intercept). Lastly, the graph shows what the profit line looks like at a $600 profit using the specific points (0,6), (5,4) and (10,2) which again are in a straight line with slope of -2/5 and a different y intercept. The profit equation is the last equation rewritten to allow the graphing program to work as -40x + p= 100y. p is put on a slider to illustrate the changing profit. It stops at the maximum profit of $1240 occurring at the vertex (6,10). So the maximum profit is $1240 with 6 pastels and 10 watercolors paintings.
Click on the link then on the far right line. Push play where it has p=0 to watch the profit line change.
Problem Of the Week: Kick It
Problem Statement: A football game is scored in a different way. Each field goal counts for 5 points and each touchdown counts 3 points. What is the highest impossible score?
Click on link below to see my write up of the problem.
POW Write Up: Kick It
This problem was really interesting to see presented in class. I appreciated the link to a past problem called "Lot's of Squares" that several students made. What especially fascinated me was the fact that both students who presented came up with basically the same way of thinking about the problem.
Baker's Choice Revisited
Baker's Choice allows for students to construct a method to solve linear programing problems rather than given them set ways to solve a problem. Using all of the constructed information, a final examination of the problem can be demonstrated using Desmos below. Click on the link to see the equations and watch the profit line animation by pushing play.
The first inequality on the left represents the condition on cookie dough.
The second inequality represents the condition on icing.
The third inequality represents the condition on oven space.
The last inequality represents the condition on prep time.
This creates a feasible region with vertices at (30, 80) and (75,50), (0,80), and (110,0)
Using the profit equation p(x,y)= 1.5x +2y where x=#of dozens of plain cookies and y = #of dozens of iced cookies, a profit line is constructed with a changing profit (last equation.) Click play to change the profit. By doing this, you can see that the last place the profit line touches is at (75,50) when p=$212.50. Thus 75 dzn plain cookies and 50 dzn iced cookies gives a maximum profit of $212.50.
In Class Assessment
#1
The first inequality represented the condition on cookie dough. Therefore by removing that equation, cookie dough is no longer a constraint and is unlimited.
The second inequality still represents the condition on icing.
The third inequality represents the condition on oven space.
The last inequality represents the condition on prep time.
Using the profit equation p(x,y)= 1.5x +2y where x=#of dozens of plain cookies and y = #of dozens of iced cookies, a profit line is constructed with a changing profit (last equation.)
In this case, a new feasible region is created and the vertex the profit line crosses last is at (120, 20). Therefore when 120 dozen plain cookies and 20 dozen iced cookies are made, the profit is maximized. By using the profit equation with those amounts, the profit is $220. Below is a desmos graph. Press play on the p variable to see the profit line move through the new constraints.
#2 In this case the Woo's can't sell more than 60 dozen plain cookies. This offers a new constraint, because the original highest profit occurred when 75 plain cookies were sold. Re-graphing the constraints creates one additional inequality to create a new feasible region.
The first inequality on the left represents the condition on cookie dough.
The second inequality represents the condition on icing.
The third inequality represents the condition on oven space.
The last inequality represents the condition on prep time.
The last inequality (in red represents the limitation on plain cookies).
Click to follow the new profit line to until it reaches the end of the feasible region (60,60). So this means that to maximize profit the Woos should make 60 plain and 60 iced for a total profit of $210. Below is a graph using desmos to illustrate this. Click on the graph then on the play button next to the variable p.
#3 This time everything is the same in the original problem except the Woos make a different profit. Thus the profit equation changes, but the constraints remain the same.
The first inequality on the left represents the condition on cookie dough.
The second inequality represents the condition on icing.
The third inequality represents the condition on oven space.
The last inequality represents the condition on prep time.
This creates a feasible region with vertices at (30, 80) and (75,50), (0,80), and (110,0)
Changing the profit does not change the feasible region.
Using the new profit equation p(x,y)= 2x +4y where x=#of dozens of plain cookies and y = #of dozens of iced cookies, a profit line is constructed with a changing profit (last equation.) Click play to change the profit. By doing this, you can see that the last place the profit line touches has changed due to the change in slope of the profit line. There for the last place the profit line touches a point in the feasible region is now (30,80) when p=$380. Thus 30 dzn plain cookies and 80 dzn iced cookies gives a maximum profit of $380. Below is a desmos representation of the new profit line. Click on the graph then push play on the variable p for the new profit line.
Homework 14 Reflections on Learning
I have never seen a high school level thematic math unit that was challenging enough in content. The units I have seen often introduce a concept, but then fail to continually relate it using reasonable mathematics. I must say, I was skeptical in seeing the Baker's Choice Book. I thought it would include watered down mathematics. Over this course, I have found this unit to require students to actually do more mathematics at a greater depth than any other math curriculum I have seen. I can't wait to work on the next unit and have looked at IMP website for additional materials.
Working together in a group has been very rewarding. When I was learning mathematics in my formal education, we never worked in groups. Our desks were lined up in rows and we plowed through the problems. I was very good at recognizing patterns and could calculate quickly but had no sort of passion for math. It wasn't until college that I realized math was not just about solving random problems, it was actually about thinking reasonably and that is when I changed my major to math. I am thankful that students get a chance to experience this "real" math before college.
This really allowed me to explore mathematics of a concept I thought I knew well. For the next unit, I want to really put my mind in that of a student. What kind of background would I have? What would confuse me? What would be difficult to see? What would be easy and difficult? I would like to keep this at the forefront of my mind for the next unit.
Baker's Choice begins to scaffold student understanding by exploring specific combinations that work and ones that don't work with the constraints. The unit then looks at a simpler problem where only the time is constrained. As inequalities are introduced, manipulation is taught where students create equivalent inequalities using mathematical reasoning.
In Picturing Cookies Part I, Students become familiar with what graphing inequalities actually means by plotting points that satisfy a given constraint in one color and don't satisfy the constraint in another. This clearly shows the border of one inequality dividing the plane into two distinct regions (one that satisfies the constraint and one that doesn't.) Then students graph the constraints of the problem for Picturing Cookies Part II and construct a feasible region which meets all the constraints. It is evident that this region will be some sort of a polygon due to being composed of straight lines with half a plane meeting the constraints.
Finally, the profit function is examined in Profitable Pictures. Students are asked to plot points of a specific profit (first $1,000, then $500, and finally $600) in different colors for different profits. By connecting these points, students are led to discover that each profit produces points that are on the same line. Also, students conjecture that each profit "line" has the same slope (coffee stirrers can be used to move the profit line up.) As the profit increases, the profit function moves up and to the right and the slope remains the same. Consequently students discover that the last point on the feasible region is the one that must have the maximum profit.
After connecting that the maximum profit is at a vertex (or along the constraint line), students revisit the profit maximizing of making dozens of plain and iced cookies. They look at the feasible region and point at which the profit line crosses the last vertex. Here they find that the last place the profit line touches is at (75,50) when the profit=$212.50. Thus 75 dozen plain cookies and 50 dozen iced cookies gives a maximum profit of $212.50.
Selected Papers (Projects)
(Including Homework 15 and Baker's Choice Revisited)
Steps to Solving a Profit Linear Programing Problem
Homework 15
First identify the variables and write the constraint inequalities. Graph the constraint inequalities shading the area that makes the inequality true. Identify the feasible region that meets all the constraints (the part that is shaded by every constraint.) Write the profit equation using the defined variables. Find the vertices of the feasible region and test them in the profit equation to find the maximum profit.
Lesson Summary for Baker's Choice Unit
Picturing Cookies Part I and II
This introduces the discovery of the relationship between an inequality an the the equation of that line. The idea is to plot many points to get the idea of why we shade a region when working with an inequality. This was especially powerful to me because many students don't understand what they are actually doing when they are shading an inequality.
As more and more points are plotted, students begin to see the ones that satisfy the inequality and the ones that don't form a border (the equation of the line.)
Using this information the Cookie constraints are revisited.
Profitable Pictures
Profitable Pictures was the key to understanding why the maximum must be at a vertex of the constrained region. The profit equation (in this case P(x,y) = 40x + 100y, where x = # of pastels and y=# of watercolors sold) is a line. As the desired profit changes the profit line also changes. However, the ratio of pictures sold does not. This rate is the slope of the profit line, no matter what the profit is. As the line moves up the y axis it increases in profit. Therefore, the highest point still in the constrained region is the maximum profit. This example on from Desmos demonstrates profitable pictures. Several coordinate points were graphed at a particular profit. For a profit of $1000 the points (0,10), (10,6) and (5,8) that met this criteria were graphed. These all form a straight line with a slope of -2/5. For a profit of $500 the points (0,5), (10,1), and (5, 3) met this criteria and were graphed (also having a slope of -2/5 but different y intercept). Lastly, the graph shows what the profit line looks like at a $600 profit using the specific points (0,6), (5,4) and (10,2) which again are in a straight line with slope of -2/5 and a different y intercept. The profit equation is the last equation rewritten to allow the graphing program to work as -40x + p= 100y. p is put on a slider to illustrate the changing profit. It stops at the maximum profit of $1240 occurring at the vertex (6,10). So the maximum profit is $1240 with 6 pastels and 10 watercolors paintings.
Click on the link then on the far right line. Push play where it has p=0 to watch the profit line change.
Problem Of the Week: Kick It
Problem Statement: A football game is scored in a different way. Each field goal counts for 5 points and each touchdown counts 3 points. What is the highest impossible score?
Click on link below to see my write up of the problem.
POW Write Up: Kick It
This problem was really interesting to see presented in class. I appreciated the link to a past problem called "Lot's of Squares" that several students made. What especially fascinated me was the fact that both students who presented came up with basically the same way of thinking about the problem.
Baker's Choice Revisited
The first inequality on the left represents the condition on cookie dough.
The second inequality represents the condition on icing.
The third inequality represents the condition on oven space.
The last inequality represents the condition on prep time.
This creates a feasible region with vertices at (30, 80) and (75,50), (0,80), and (110,0)
Using the profit equation p(x,y)= 1.5x +2y where x=#of dozens of plain cookies and y = #of dozens of iced cookies, a profit line is constructed with a changing profit (last equation.) Click play to change the profit. By doing this, you can see that the last place the profit line touches is at (75,50) when p=$212.50. Thus 75 dzn plain cookies and 50 dzn iced cookies gives a maximum profit of $212.50.
In Class Assessment
#1
The first inequality represented the condition on cookie dough. Therefore by removing that equation, cookie dough is no longer a constraint and is unlimited.
The second inequality still represents the condition on icing.
The third inequality represents the condition on oven space.
The last inequality represents the condition on prep time.
Using the profit equation p(x,y)= 1.5x +2y where x=#of dozens of plain cookies and y = #of dozens of iced cookies, a profit line is constructed with a changing profit (last equation.)
In this case, a new feasible region is created and the vertex the profit line crosses last is at (120, 20). Therefore when 120 dozen plain cookies and 20 dozen iced cookies are made, the profit is maximized. By using the profit equation with those amounts, the profit is $220. Below is a desmos graph. Press play on the p variable to see the profit line move through the new constraints.
#2 In this case the Woo's can't sell more than 60 dozen plain cookies. This offers a new constraint, because the original highest profit occurred when 75 plain cookies were sold. Re-graphing the constraints creates one additional inequality to create a new feasible region.
The first inequality on the left represents the condition on cookie dough.
The second inequality represents the condition on icing.
The third inequality represents the condition on oven space.
The last inequality represents the condition on prep time.
The last inequality (in red represents the limitation on plain cookies).
Click to follow the new profit line to until it reaches the end of the feasible region (60,60). So this means that to maximize profit the Woos should make 60 plain and 60 iced for a total profit of $210. Below is a graph using desmos to illustrate this. Click on the graph then on the play button next to the variable p.
#3 This time everything is the same in the original problem except the Woos make a different profit. Thus the profit equation changes, but the constraints remain the same.
The first inequality on the left represents the condition on cookie dough.
The second inequality represents the condition on icing.
The third inequality represents the condition on oven space.
The last inequality represents the condition on prep time.
This creates a feasible region with vertices at (30, 80) and (75,50), (0,80), and (110,0)
Changing the profit does not change the feasible region.
Using the new profit equation p(x,y)= 2x +4y where x=#of dozens of plain cookies and y = #of dozens of iced cookies, a profit line is constructed with a changing profit (last equation.) Click play to change the profit. By doing this, you can see that the last place the profit line touches has changed due to the change in slope of the profit line. There for the last place the profit line touches a point in the feasible region is now (30,80) when p=$380. Thus 30 dzn plain cookies and 80 dzn iced cookies gives a maximum profit of $380. Below is a desmos representation of the new profit line. Click on the graph then push play on the variable p for the new profit line.
Homework 14 Reflections on Learning
My experience working on Baker's Choice has been very different than any way I have ever been taught math. I often try to take information and connect it to what I already know. Any algorithm I am given, I appreciate knowing why it works and will search that out. I never thought about why teachers don't teach students to figure it out first. I like the phrase "doing math" we heard throughout the process. Attaching "doing math" to thinking through a problem instead of doing a bunch of calculations, is much more accurate description of what true mathematicians do. Even though, I have learned and practiced linear programing, I found new ways of concretely connecting the material to what I intuitively knew. I was especially interested in showing points that fit the constraints of the inequality and ones that didn't with two colors. While I knew what an inequality represented, I appreciated the visual and the increased ability to use this when teaching. I also thought the profit line exploration was valuable in visually explaining why the maximum profit had to be at a vertex. The physical movement of the straw made this very clear to me.
I understood maximizing profit with constraints at a much deeper level during this unit. By taking apart the steps needed to solve a problem with numerous steps, I was able to add greater depth to what I though was a pretty solid understanding.
I very much enjoyed the experience of working together with fellow math colleagues. I appreciated their contributions and especially their questions. Being part of a group held me accountable for any statements I made as well as opened my mind up to new ideas. The ability to make the connections was very satisfying and when I look back I can easily see the concept development of the entire process. I feel more connected to this particular concept than before.
Statement of Personal GrowthI have never seen a high school level thematic math unit that was challenging enough in content. The units I have seen often introduce a concept, but then fail to continually relate it using reasonable mathematics. I must say, I was skeptical in seeing the Baker's Choice Book. I thought it would include watered down mathematics. Over this course, I have found this unit to require students to actually do more mathematics at a greater depth than any other math curriculum I have seen. I can't wait to work on the next unit and have looked at IMP website for additional materials.
Working together in a group has been very rewarding. When I was learning mathematics in my formal education, we never worked in groups. Our desks were lined up in rows and we plowed through the problems. I was very good at recognizing patterns and could calculate quickly but had no sort of passion for math. It wasn't until college that I realized math was not just about solving random problems, it was actually about thinking reasonably and that is when I changed my major to math. I am thankful that students get a chance to experience this "real" math before college.
This really allowed me to explore mathematics of a concept I thought I knew well. For the next unit, I want to really put my mind in that of a student. What kind of background would I have? What would confuse me? What would be difficult to see? What would be easy and difficult? I would like to keep this at the forefront of my mind for the next unit.
