SchoolParadiseProject.pdf

(1)

UVA-SCPS Office of Mathematics Outreach with support from VADOE Mathematics and Science Partnership Grant Program NCLB Title II Part B 21st Century Teaching and Learning: Project-based Unit

Snapshot of Unit Content and Student Expectations

Title of Unit: School Paradise Project Unit Designers:

Tom Metzger (Staunton City) Meredith Ward (Staunton City) Brooke Sullivan (Spotsylvania County)

Context of the Project: Your school district is opening to the public design considerations for a new High School. Convince the Council of your site plan and prepare to defend all aspects of construction, site location and other important factors.

Number of Class Hours:

8 to 10 hours Unit

Design: Project-based Unit Other Subject

Areas/Disciplines Addressed in the Unit:

World Geography (Topography), Environmental Science, Drafting

Driving Question: Where is the best plot of land to build the new High School? Mathematics Content

Addressed:

Area, Perimeter, Surface Area, Ratios and Proportions, Slope, Optimization, Linear Regressions, Drawing to Scale, Linear Equations, Systems of Linear Equations Inequalities, Estimation

Mathematical Process Goals Addressed

x Problem Solving __Communication _x_Reasoning x_Connections _x_Representations

Assumption of Prior Knowledge:

Linear Equations, Linear Inequalities, Estimation, Area, Perimeter, Budget, Regressions, Unit Conversion

Courses for Which the Unit is Appropriate

Algebra I, Algebra II

College and Career Readiness/21st Century Skills

BIE Page 35-37

T for skills to be taught and expected to use during the unit,

E for skills student are expected to know and be able to use during this unit A for skills that will be assessed during this unit

E_Collaboration EA_Research EA_Communication (Oral and/or Written) EA_Technology TEACritical Thinking/Decision Making

(2)

UVA-SCPS Office of Mathematics Outreach with support from VADOE Mathematics and Science Partnership Grant Program NCLB Title II Part B Template adapted from Buck Institute for Education: Project Based Learning for the 21st Century Major Student

Products and/or Performances:

Group Presentation Audience:

x Class

School

Individual Expert

x Community

Other:

Engage the students interest and inquiry:

Evaluation: Formative Assessments (During the Unit)

Interview x Practice Presentations

Mathematicians Journal x Notes

Preliminary

Plans/Outlines/Prototypes

x Checklists

Rough Drafts Concept maps

Field Tests Other:

Summative Assessment (End of Project)

Written Products, with a rubric x Peer Evaluation, with a rubric

x Oral Presentation with a rubric x Self Evaluation, with a

rubric

x Other Product(s) or

Performance(s), with a rubric

Other:

Resources Needed: On-site people, facilities: Transportation representative, school board member(s) Equipment/Technology: Computers with access to internet and excel

Materials: Graph Paper, Calculators, Rulers, Chart paper or Poster Board, Computers

Community Resources: Local Realtors, Contractors, & Engineers, Planning Commission or Town Council

Reflection Methods: Individual, Group, and/or Whole Class

Mathematicians Journal x Small/Focus Groups x Whole Class Discussions x Fishbowl Discussions

Survey Other:

(3)

UVA-SCPS Office of Mathematics Outreach with support from VADOE Mathematics and Science Partnership Grant Program NCLB Title II Part B

Quick Snapshot for the Sequence of Unit Activities

UNIT TITLE: School Paradise Project 45 – 60 MINUTES PER CLASS

D a y 1 D a y 2 D a y 3 D a y 4 D a y 5

P R O J E C T W E E K O N E

Notes

Propose Project

Students brainstorm what information will be helpful to know

Groups are formed

Students determine how many students/staff will attend new school

Students look at data and make predictions about future enrollment and how long the building will last using population growth/regression and life-span of a structure. Regression analysis activity attached.

Students will use Excel to create graphs/tables to model growth for student population. Introduce students to systems and share with class how several factors can limit an overall outcome. Systems activity attached.

Students determine the square footage of the school, the number of parking spaces needed and the field/court requirements. Then relate to their predicted population growth and make assumptions for sizing the new structure.

Determine important factors in deciding a location

Consider Bus Routing and Transportation Concerns Research and decide on 3 possible land plots

Internet access needed to utilize local GPS.

D a y 6 D a y 7 D a y 8 D a y 9 D a y 1 0

P R O J E C T W E E K T W O

Notes

Students determine Max and Min Rectangular Area of Building

Determine a comprehensive Plan and determine other considerations in their groups

Students research “other “ factors to help them determine which Lot will be the best choice for the new school  Construction Costs  Environmental Impact  Access

 Layout

Students make a decision about the lot to purchase and

build/create a site plan  Compile supporting

information & calcs  Prepare to discuss their

motivating factors

Students work on final paper and Presentation

(4)

Overview of Student Knowledge and Skills

What do students need to know and be able to do to complete the unit successfully? Project: School Paradise Project

Analyze each major task and the final project for the unit for the knowledge and skills necessary to produce a high-quality product.

How and when will students assess their own necessary prior knowledge and skills?

How will they remediate their own gaps or weaknesses in expected prior knowledge and skills?

KNOWLEDGE AND SKILLS NEEDED Assumed already learned Students will self-assess Will be learned and assessed during the unit 1. Area/Perimeter/Surface Area X

2. Ratios/Proportions X

3. Slope X

4. Optimization X

5.Linear/Quadratic Regressions X

6. Scale Drawing X

7. Systems of Inequalities X

8. Estimation/Budget X

What project tools will student’s use to monitor their progress through the unit and especially the project?

 Student developed Know/need to know lists

 Student developed Daily goal sheet X Student Mathematician’s Journals  Student developed Briefs/Memos X Student developed Task lists

 Student developed Planning Calendar  Teacher developed Rubrics

(5)

Unit Title: School Paradise Project

Driving Question: What kind of school would you want and where? Project Description:

ENGAGE

How will student’s interested be piqued so they want to engage in the inquiry in this project-based unit? Number of hours: 30 mins

Superintendent comes to school or sends a letter with proposed task. (This can be “manufactured”)

Students discuss information they feel is important to know to be able to complete the task.

Students discuss in groups factors they feel are important to them in a school.

Mathematician Journal Prompts What

information is helpful to know to complete this project? How could we determine the number of students that attend the new school?

EXPLORE

Teacher provides guidance for the explorations to prepare students with the

knowledge and skills to engage in the project-based unit. Students will self-assess on the prior knowledge and skills assumed for the unit

Number of hours___

A. Collect data about the current school’s make-up and discuss importance, any obsolete features, any future improvements:

1. Student body population 2. Building size

3. Site Layout (Parking, Athletic Fields,…) 4. Provide students with a means to measure

(Long measuring tape/wheel, methods of estimation, blueprints on file with locality) B. Regression Analysis Activity & Linear Systems Activity

1. Use real data to derive a formula and make predictions 2. Regression problems & systems problems with some

exploration involved as a formative assessment then followed by a comprehensive summative assessment 3. Regression Analysis Activity (HO #1)

Linear Systems Activity (HO # 2 & HO #3)

4. Answer sheet to Regression Analysis Activity located at end of document (HO #1b)

Answer sheet to Linear Systems Activity located at end of document (HO #2b & HO #3b)

5. Materials/Equipment/Resources Needed: Graph paper, HO # 1, HO # 2, HO #3, graphing calculators, computers with Excel, straight edge C. Students now use the skills from the modeling and systems

Mathematician Journal Prompts How long do you think a school building will last? What factors

contribute to the life-span of a building? 21 Century Teaching and Learning Inquiry Learning

Project-based Learning Unit

(6)

activities to arrive at real predictions for this project. This pre-assessment gives the student the opportunity to use whatever tool best suits them to predict a student population for an assumed year.

6. Student Direction(s) Sheet Attached identified with a Title and Handout number (HO #4)

D. Students present their prediction to the teacher who will then compare the predictions and come to a consensus on the best prediction and revisit population modeling. Teacher may want to perform a statistical analysis on the predictions to choose the best answer.

E. Rubrics are attached and identified with a title and handout number (HO #5)

EXPLAIN

Teacher introduces the project and prepares

students to work independently in small groups

Number of Hours___

A. Introduce the Project: Notes to Teacher

1. Provoke questions that are site specific – what does the school site need to look like and how should it function. Do not box the teams into one design approach but encourage them to use their personal interests in guiding their design (i.e. environmental impact, location, contribution to community,…) 2. Design teams need to determine what their focus is

along with the necessities. Using poster paper, list things that must be considered in their design, then identify their focus to guide their design.

3. Have them share their design focus with another team and discuss critical factors they need to consider using the focus they have chosen. By the end of this the teams should have a clear idea of what their site achieves.

B. The final product contains a variety of approaches but all should incorporate predictions for size, population, and spatial orientation on an actual site using an effective means of presentation.

1. Project description and expectations attached identified with a title and handout number (HO #6) 2. Students create a final design that includes all

necessary components of the new school in digital form, hand sketch, 3-d model, or other types of media. All drawings/models must be drawn to scale and a summary with all supporting calculations must be included.

3. Students use previously learned and new skills learned throughout project to support their design. Previously learned material include slope, finding trends in data, linear functions, and use of t-tables. New material includes modeling and systems.

(7)

4. Have internet, poster paper, locality ordinances, scales for drawings, measuring tape or wheels, and graphing calculator available to students.

5. Rubric(s) attached identified with a title and handout number (HO #5)

6. Due to the nature of this project, an answer sheet is impossible to provide. It is suggested that the teacher compare results among the groups. Answers will vary, but should be in close proximity to one another. C. Suggestions for what is necessary to preparing students

1. Students will be working on a universal goal, but must communicate to the group and teacher what they plan to have in their final project and who is assuming responsibility for each part of the project.

2. When the team decides on the components to their project, students create a time-line with responsibilities labeled and assigned.

3. Rubric(s) attached identified with a title and handout number (HO #5)

ELABORATE The student groups are working independently with teacher consultations. Number of Hours_____

A. The teacher occasionally aids the groups along the design path of the teams’ choosing. For example, suggest resources for finding sites, analyzing terrain, supplying contacts for construction costs, etc.

B. Throughout the project, the teacher may assign mini-tasks that are project specific to help aid the students in

individual components of their design. For example, calculate the slope of a selected property and determine if it is safe for a school site.

C. Students self-monitoring

1. Tools for group planning attached identified with a title and handout number (HO #6)

2. Tools for student self-monitoring of individual participation in group work attached identified with a title and handout number (HO #6 & HO #7)

3. Rubric(s) for self-evaluation of product and/or presentation attached identified with a title and handout number (HO #5)

(8)

EVALUATE

Working groups submit products or make

presentations Number of Hours___

A. Again, this unique and individualized project will produce a unique solution; therefore, the rubric is essential in determining a grade as well as daily monitoring to check work as the team progresses.

B. Rubric(s) for teacher and/or audience for presentation attached identified with a title and handout number (HO #5)

Mathematician Journal Prompts What are important factors to consider in determining a location for the new school?

(9)

HO#1 Wolves Task: Linear Modeling

Before you begin:

Define interpolate: _________________________________________________________________

Define extrapolate: _________________________________________________________________

The problem: Grey wolves are an integral part of the ecosystem in the American Rocky Mountain states. In the 1980s, however, grey wolves were so endangered in Montana that only 14 remained in the entire state. The Wildlife Fund put grey wolves on the endangered species list and began efforts to repopulate the state. Today, there are 219 wolves in Montana. You are on the Endangered Species committee, and are evaluating whether the wolf should remain on the endangered species list, or whether things have improved enough to begin focusing the Wildlife Fund’s resources on other endangered species.

Montana’s Wolf Population, 2000-2011

Year Population

2000 48

2001 52

2002 68

2003 71

2004 101

2005 130

2006 163

2007 178

2008 165

2009 209

2010 240

(10)

1. Use Excel to aid your data analysis. Begin by creating the necessary and helpful data.

Step i: Enter the years, 2000-2011, in Column A. Cell A1 should be the title (Year), and 2000 should be just below. Below 2000, use a simple addition formula, =A1+1, and copy through cell A13, to enter the years.

a)

Explain in words what Excel is doing with the formula you entered.

Step ii: Enter the population in Column B. Cell B1 should be the title (Population), and the data should be entered in B2 through B13, matching up with the year.

Step iii: Enter the total change from one year to another, in Column C. Cell C1 should be the title (Total Change). Beginning in Cell C3, use a formula, =B3-B2, and copy through cell C13.

b)

Explain in words what Excel is doing with the formula you entered.

c)

Explain why the formulas begin in C3, instead of C2.

(11)

Step iv: In cell C14, find the average change in wolf population using the formula =AVERAGE(C3:C13).

Step v: Find the percent change in each year in column D. Cell D1 should be the title (Percent Change), and, beginning in cell D3, use the formula =(B3-B2)/B2. Copy the formula through cell D13.

e)

Explain in words what Excel is doing with the formula you entered.

Step vi: In cell D14, find the average percent change by using the formula =AVERAGE(D3:D13).

2. Create an Excel graph of your data

Step i: Highlight cells A1:B13. Be sure to include the titles (Year and Population) in the data field.

Step ii: Click Insert, and choose Scatterplot from the menu.

(12)

3. Analyze the Data

a)

How has the wolf population changed over the past decade? Describe the change in

absolute numbers of wolves, and in the total percent change in population.

b)

Were some years better than others in terms of population growth? Is the number of

wolves growing by a similar amount or by a similar percentage each month?

c)

Which year was the “best” for the wolves? Might there be more than one possible way to

express this growth? Give two different perspectives on the question.

(13)

e)

Extrapolate

the number of wolves there will be in 2012. Will the population increase or

decrease, and by how many? Explain your reasoning.

4. Run regressions by using a TI-83 or TI-84 calculator to find an appropriate model of your data.

Step i: Using either input method, enter the year values into L1 and the population values into L2 as shown. However, unlike in Excel, be sure to enter 0 for 2000, 1 for 2001, 2 for 2002, etc.

Method using the Stat Edit… menu Method using braces { } and STOL1

(14)

a)

Compare the shape here with the graph you created in Excel. Recall your answer from

question 2a. Do you still agree with your previous estimate of a linear, quadratic, or

exponential model?

Step iii: Active the Diagnostic function in your calculator. Under the Catalog menu, an alphabetical listing of all of your calculator’s functions, scroll down or jump to “D” to find “DiagnosticOn”.

(15)

Stat Calc menu LinReg(ax+b) L1,L2,Y1

Regression equation Graph

Step v: Repeat the process to create quadratic and exponential models of your data. This time, when running the regressions, end the commands with Y2 for the quadratic model and Y3 for the exponential model so that you do not overwrite the previous equations.

(16)

Linear: y = _______________________________________ R2 = __________________

Quadratic: y = _______________________________________ R2 = __________________

Exponential y = _______________________________________ R2 = __________________

b)

Which equation do you believe best models the data? How are you determining your

answer? Are there limitations to the method you used to determine the best model?

5. Analyze your regressions.

a)

Examine the linear equation you have found, which is in slope-intercept form. What does

the slope represent in the context of the wolf population? What does the y-intercept

represent in the context of the problem? Compare your Excel data in cells B2 and C14

with the values in your equation.

(17)

c)

How does cell D14 compare to the values in your equation?

d)

Using the model you chose as best in 4b, make an estimate for the number of wolves you

expect there to be between 2008 and 2009. Compare your answer to the interpolation you

made in question 3d.

Original estimate: _________________________ Model estimate: _________________________

(18)

f)

Using your model, make an estimate for the number of wolves you expect there to be in

2050. Do you think your estimate is reliable? Why or why not? Do you need to rethink

the model you chose for your estimation? Compare your estimate to another student or

group who chose a model different from yours.

g)

When entering L

1

into your calculator, you entered 2000 as 0, 2001 as 1, etc. Why was

this necessary and why did it not affect your analysis? Try rerunning the exponential

regression with L

1

as 2000, 2001, etc. if you are unsure of the reason.

(19)

HO #1b

Wolves Task: Teacher Answer Key Before you begin:

Define interpolate: To estimate unknown data point(s) contained inside (between) known data points Define extrapolate: To estimate unknown data point(s) which fall outside a range of known data The problem: Grey wolves are an integral part of the ecosystem in the American Rocky Mountain states. In the 1980s, however, grey wolves were so endangered in Montana that only 14 remained in the entire state. The Wildlife Fund put grey wolves on the endangered species list and began efforts to repopulate the state. Today, there are 219 wolves in Montana. You are on the Endangered Species committee, and are evaluating whether the wolf should remain on the endangered species list, or whether things have improved enough to begin focusing the Wildlife Fund’s resources on other endangered species.

Montana’s Wolf Population, 2000-2011

Year Population

2000 48

2001 52

2002 68

2003 71

2004 101

2005 130

2006 163

2007 178

2008 165

2009 209

2010 240

(20)

1. Use Excel to aid your data analysis. Begin by creating the necessary and helpful data.

Step i: Enter the years, 2000-2011, in Column A. Cell A1 should be the title (Year), and 2000 should be just below. Below 2000, use a simple addition formula, =A1+1, and copy through cell A13, to enter the years.

f)

Explain in words what Excel is doing with the formula you entered.

Excel begins the first year, 2000, and adds 1 to the year in each next cell. It always adds 1 to the year that came before it.

Step ii: Enter the population in Column B. Cell B1 should be the title (Population), and the data should be entered in B2 through B13, matching up with the year.

Step iii: Enter the total change from one year to another, in Column C. Cell C1 should be the title (Total Change). Beginning in Cell C3, use a formula, =B3-B2, and copy through cell C13.

g)

Explain in words what Excel is doing with the formula you entered.

Excel finds the difference between population in a certain year and the population in the previous year.

h)

Explain why the formulas begin in C3, instead of C2.

The data from the year before 2000, 1999, is not known. The previous year cannot be compared to 2000.

i)

Explain why some of the values are positive, and some of the values are negative.

A positive value indicates the wolf population increased between the years indicated. A negative value indicates the wolf population declined between the years indicated. Although the population increased overall from 2000 to 2011, the net change each year was not always positive.

Step iv: In cell C14, find the average change in wolf population using the formula =AVERAGE(C3:C13). Step v: Find the percent change in each year in column D. Cell D1 should be the title (Percent Change), and, beginning in cell D3, use the formula =(B3-B2)/B2. Copy the formula through cell D13.

j)

Explain in words what Excel is doing with the formula you entered.

Excel finds the difference between two years and dividing by the total population of the first year, finding the percent by which the population changed from the first year to the second year.

Step vi: In cell D14, find the average percent change by using the formula =AVERAGE(D3:D13). 2. Create an Excel graph of your data

(21)

Step ii: Click Insert, and choose Scatterplot from the menu.

b)

Examine the overall shape of the scatterplot points. Does it appear to be a linear,

quadratic, or exponential shape?

Answers may vary. Based solely on the shape of the data given, an argument could be made for linear, quadratic, and exponential models. However, students may have already discussed in biology class, for example, that populations often follow exponential models of growth.

3. Analyze the Data

f)

How has the wolf population changed over the past decade? Describe the change in

absolute numbers of wolves, and in the total percent change in population.

In ten years, the population increased by 171 wolves, or an increase of +356%.

g)

Were some years better than others in terms of population growth? Is the number of

wolves growing by a similar amount or by a similar percentage each month?

Answers may vary. Earlier years saw increases of between 3 and 16, while later years saw increases of 30-40. 2007-2008 and 2010-2011 saw a decrease in population. The percentage increase tends to hover between 20-30%.

h)

Which year was the “best” for the wolves? Might there be more than one possible way to

express this growth? Give two different perspectives on the question.

Answers may vary. 2008-2009 saw the biggest net increase in population, an increase of 44 wolves, but 2003-2004 saw the biggest percent change, +42%.

i)

Interpolate

the number of wolves there were midway between 2008 and 2009. Explain

how you found your answer.

(22)

j)

Extrapolate

the number of wolves there will be in 2012. Will the population increase or

decrease, and by how many? Explain your reasoning.

Answers may vary. Estimates between 200 and 260 are reasonable, assuming the change in population is a decrease similar to those seen in recent years or an increase similar to those seen in recent years.

4. Run regressions by using a TI-83 or TI-84 calculator to find an appropriate model of your data. Step i: Using either input method, enter the year values into L1 and the population values into L2 as shown. However, unlike in Excel, be sure to enter 0 for 2000, 1 for 2001, 2 for 2002, etc.

Method using the Stat Edit… menu Method using braces { } and STOL1

Step ii: Create a scatterplot of the data on your calculator. Under the StatPlot menu, turn on Plot1 as shown. Resize your window using appropriate x- and y-axis scaling, or ZoomStat. A sample window scale is shown below, but yours may be slightly different.

(23)

Step iii: Active the Diagnostic function in your calculator. Under the Catalog menu, an alphabetical listing of all of your calculator’s functions, scroll down or jump to “D” to find “DiagnosticOn”.

Step iv: Create a linear model of your data under the Stat menu in your calculator. Under the VARS menu, Y-Vars, Function, enter Y1 at the end of the command to automatically enter your equation into the graph mode of your calculator.

(24)

Regression equation Graph

Step v: Repeat the process to create quadratic and exponential models of your data. This time, when running the regressions, end the commands with Y2 for the quadratic model and Y3 for the exponential model so that you do not overwrite the previous equations.

Step vi: Copy the equations created by your calculator, along with the R2 value, to four decimal places.

Linear: y = _______________________________________ R2 = __________________

Quadratic: y = _______________________________________ R2 = __________________

Exponential y = _______________________________________ R2 = __________________

(25)

According to the R2 value of each equation, the quadratic model is the best. However, just because the R2 value for the quadratic model is closest to 1, it may not actually be the most accurate model for data further from the 11 year sample provided.

Linear:

Quadratic:

Exponential

5. Analyze your regressions.

h)

Examine the linear equation you have found, which is in slope-intercept form. What does

the slope represent in the context of the wolf population? What does the y-intercept

represent in the context of the problem? Compare your Excel data in cells B2 and C14

with the values in your equation.

The slope, 18.51, represents the average change in wolf population, which cell C14 shows is actually 15.55. The y-intercept, 35.19, represents the initial wolf population, which cell B2 shows is actually 48.

i)

Examine the exponential equation you have found. What does “a” represent in the

problem? What does “b” represent in the problem? Compare your Excel data in cells B2

and D14 with the values in your equation.

The “a” value, 50.74, represents the initial wolf population, which cell B2 shows is actually 48. The “b” value represents the percent increase in wolf population, +117%, which cell D14 shows is actually +15.87%.

j)

How does cell D14 compare to the values in your equation?

The equation contains one plus the percent increase to represent an increase in that amount each year.

k)

Using the model you chose as best in 4b, make an estimate for the number of wolves you

expect there to be between 2008 and 2009. Compare your answer to the interpolation you

made in question 3d.

Original estimate: _________________________ Model estimate: _________________________ Linear estimate: By using x = 5.5, y = 193.

Quadratic estimate: By using x = 5.5, y = 193.

(26)

l)

Using the same model you chose as best in 4b, and used in the previous question, make

an estimate for the number of wolves you expect there to be in 2012. Compare your

answer to the extrapolation you made in question 3e.

Linear estimate: By using x = 12, y = 257.

Quadratic estimate: By using x = 12, y = 257.

Exponential estimate: By using x = 12, y = 328.

m)

Using your model, make an estimate for the number of wolves you expect there to be in

2050. Do you think your estimate is reliable? Why or why not? Do you need to rethink

the model you chose for your estimation? Compare your estimate to another student or

group who chose a model different from yours.

Linear estimate: By using x = 50, y = 961.

Quadratic estimate: By using x = 50, y = 914.

Exponential estimate: By using x = 50, y = 120,572

Answers may vary. The linear and quadratic models seem reasonable but the exponential estimate appears to be an overestimate of what is reasonable.

n)

When entering L

1

into your calculator, you entered 2000 as 0, 2001 as 1, etc. Why was

this necessary and why did it not affect your analysis? Try rerunning the exponential

regression with L

1

as 2000, 2001, etc. if you are unsure of the reason.

The limited processing capability of the TI-83 and TI-84 prevent exponentiation with numbers in the thousands. By adjusting years to 2000 less, the model does not change but the overflow error is avoided. As long as the adjustment to input year is also made (for example, 2050 becomes 50), then the estimate remains the same as if the exponential model with years in the 2000s could have been correctly run.

(27)

HO#2 Linear Programming Word Problems – PRACTICE / EXPLORE Part

1. A manufacturer of ski clothing makes ski pants and ski jackets. The profit on a pair of ski pants is $2.00 and on a jacket is $1.50. Both pants and jackets require the work of sewing operators and cutters. There are 60 minutes of sewing operator time and 48 minutes of cutter time available. It takes 8 minutes to sew one pair of ski pants and 4 minutes to sew one jacket. Cutters take 4 minutes on pants and 8 minutes on a jacket. Find the maximum profit and the amount of pants and jackets to maximize the profit.

a. Let x = ski pants and y = ski jackets. Since there cannot be negative pants or jackets, write two inequalities to represent that situation.

b. Express the cutters’ time to make pants and jackets as an inequality.

c. Express the sewing operators’ time to make pants and jackets as an inequality.

d. Write an equation for the anticipated profit.

e. Graph the constraints.

(28)

h. How many ski pants and ski jackets have to be made to maximize profit?

2. The automotive plant in Rockaway makes the Topaz and the Mustang. The plant has a maximum production capacity of 1200 cars per week. During the spring, a dealer orders up to 600 Topaz cars and 800 Mustangs each week. If the profit on a Topaz is $500 and on a Mustang it is $800.

a. Let x = y =

Since you cannot have negative cars, write two inequalities to represent the situation.

b. Since the plant has a capacity of 1200 cars, write an inequality to represent the situation.

c. Since the dealer orders up to 600 Topaz and 800 Mustangs, write two inequalities to represent the situation.

(29)

e. Graph the constraints.

f. Use the corner points to find maximum profit.

g. How many types of each car are needed to maximize the profit?

(30)

3. A farmer has a field of 70 acres in which he plants potatoes and corn. The seed for potatoes costs $20/acre, the seed for corn costs $60/acre and the farmer has set aside $3000 to spend on seed. The profit per acre of potatoes is $150 and the profit for corn is $50 an acre. Find the optimal solution for the farmer.

a. Write the constraints for the problem.

b. Write the profit equation.

c. Graph the constraints and find the corner points.

d. To find the optimal solution you are looking for the maximum. Use your corner points to find the maximum profit.

(31)

4. Impact Printing makes two kinds of computer paper using premium or ordinary quality stock. They have a contract to supply at least 5000 cases of paper. There is only enough stock to make 4000 cases of premium paper, but ample stock for ordinary paper. Both kinds are made with the same machine and 1200 hours of machine time are available. Premium paper takes 18 minutes per case to make and ordinary paper takes 12 minutes per case. The profit on each is $4/case and $3/case, respectively.

a. Write the constraints for the problem. (HINT: you need to convert the hours to minutes so that the inequality has all minutes).

b. Write the profit equation.

(32)

HO #2b Linear Programming Word Problems – PRACTICE / EXPLORE Part ANSWER KEY

1. A manufacturer of ski clothing makes ski pants and ski jackets. The profit on a pair of ski pants is $2.00 and on a jacket is $1.50. Both pants and jackets require the work of sewing operators and cutters. There are 60 minutes of sewing operator time and 48 minutes of cutter time available. It takes 8 minutes to sew one pair of ski pants and 4 minutes to sew one jacket. Cutters take 4 minutes on pants and 8 minutes on a jacket. Find the maximum profit and the amount of pants and jackets to maximize the profit.

a. Let x = ski pants and y = ski jackets. Since there cannot be negative pants or jackets, write two inequalities to represent that situation. x ≥ 0, y ≥ 0

b. Express the cutters’ time to make pants and jackets as an inequality. 4x + 8y ≤ 48

c. Express the sewing operators’ time to make pants and jackets as an inequality. 8x + 4y ≤ 60

d. Write an equation for the anticipated profit. 2x + 1.5y = P

(33)

f. Use the corner points to find the maximum profit.

(7.5, 0) (6, 3) (0, 6)

2(7.5) + 1.5(0) = $15 2(6) + 1.5(3) = $16.5 2(0) + 1.5(6) = $9

g. What is the maximum profit?

$16.50

h. How many ski pants and ski jackets have to be made to maximize profit?

6 ski pants and 3 ski jackets

2. The automotive plant in Rockaway makes the Topaz and the Mustang. The plant has a maximum production capacity of 1200 cars per week. During the spring, a dealer orders up to 600 Topaz cars and 800 Mustangs each week. The profit on a Topaz is $500 and on a Mustang it is $800.

a. Let x = Topaz y = Mustang

Since you cannot have negative cars, write two inequalities to represent the situation. x ≥ 0, y ≥ 0

b. Since the plant has a capacity of 1200 cars, write an inequality to represent the situation. x + y ≤ 1200

c. Since the dealer orders up to 600 Topaz and 800 Mustangs, write two inequalities to represent the situation.

x ≤ 600 y ≤ 800

d. Write an equation for the profit. 500x + 800y = P

(34)

f. Use the corner points to find maximum profit.

(600, 0) (600, 600)

500(600) + 800(0) = $300,000 500(600) + 800(600) = $780,000

(400, 800) (0, 800)

500(400) + 800(800) = $840,000 500(0) + 800(800) = $640,000

g. How many types of each car are needed to maximize the profit? 400 Topazes and 800 Mustangs

h. What is the maximum profit? $840,000

3. A farmer has a field of 70 acres in which he plants potatoes and corn. The seed for potatoes costs $20/acre, the seed for corn costs $60/acre and the farmer has set aside $3000 to spend on seed. The profit per acre of potatoes is $150 and the profit for corn is $50 an acre. Find the optimal solution for the farmer.

x = acres of potatoes y = acres of corn

a. Write the constraints for the problem.

x ≥ 0 y ≥ 0 20x + 60y ≤ 3,000 x + y ≤ 70

b. Write the profit equation. 150x + 50y = P

(35)

d. To find the optimal solution you are looking for the maximum. Use your corner points to find the maximum profit.

(30, 40) (70, 0) (0, 50)

150(30) + 50(40) = $6,500 150(70) + 50(0) = $10,500 150(0) + 50(50) = $2,500

e. What is the optimal solution (the max profit and the amount of corn and potatoes it take to get it)?

The greatest profit occurs when planting 70 acres of potatoes and no corn. The profit for this scenario is $10,500.

4. Impact Printing makes two kinds of computer paper using premium or ordinary quality stock. They have a contract to supply at least 5000 cases of paper. There is only enough stock to make 4000 cases of premium paper, but ample stock for ordinary paper. Both kinds are made with the same machine and 1200 hours of machine time are available. Premium paper takes 18 minutes per case to make and ordinary paper takes 12 minutes per case. The profit on each is $4/case and $3/case, respectively.

x = Cases Premium y = Cases Ordinary

a. Write the constraints for the problem. (HINT: you need to convert the hours to minutes so that the inequality has all minutes).

x + y ≥ 5000 x ≤ 4000 x ≥ 0 y ≥ 0 0.3x + 0.2y ≤ 1200

(36)

d. Find the optimal solution.

Corner Points: (0, 5000) (2000, 3000) (0, 6000)

Profit Eq.: 4(0) + 3(5000) = $15,000 4(2000) + 3(3000) = $17,000 4(0) + 3(6000) = $18,000

(37)

HO # 3 SYSTEMS ASSESSMENT

http://www.flickr.com/photos/rickbogacz/28821978/

What is the big idea? What are the unknowns?

What are the named restrictions (constraints) represent algebraically.

Verbally: Algebraically:

1)

2)

3)

(38)

Graphically:

What are possible combinations of buses and cars?

(39)

HO #3b SYSTEMS ASSESSMENT – Teacher Answer Key

http://www.flickr.com/photos/rickbogacz/28821978/

What is the big idea? What are the unknowns?

Optimize income based on rates and How many cars and buses make the most money for the

and sizes of different types of vehicles. attendant?

What are the named restrictions (constraints) represent algebraically. Let x = # cars y = # buses

Verbally: Algebraically:

4) Size of Vehicles 65x + 325y = 6500

5) Amount manageable x + y = 60

6) Income per vehicle 2.5x + 7.5y = P

Graphically:

What are possible combinations of buses and cars?

(40)

60 cars and 0 buses

50 cars and 10 buses

0 cars and 20 buses

What combination of buses and cars addresses the Big Idea? 2.5(60) + 7.5(0) = $150

2.5(50) + 7.5(10) = $200

2.5(0) + 7.5(20) = $150

(41)

HO#4

memo

School Paradise Project

To: Design Team From:

CC: Design Board Date:

Re: Population Modeling

Comments: The Design Board is requiring we send them some preliminary figures regarding the student population the new high school is going to provide for in the coming years. This is based on your discussions on how long the new building should last and how many students and staff it provides for? Please send your estimate within twenty-four hours so that we can plan for this in our budget.

Online Resources: Metropolitan Statistical Data Demographics http://www.yesvirginia.org/

(42)

HO #5

Grading Rubric

Inadequate Adequate Exemplary

Completeness and Accuracy of Required Information:

- Population considerations - Topography considerations - Efficiency considerations

- Environmental and energy considerations - Aesthetic considerations

- Practicality considerations - Community considerations

Some required information is not included.

Significant information is incorrect and/or important details are omitted or inaccurate.

All required and pertinent information is included, but there are minor errors and/or missing details.

All required and pertinent information is included, and is correct and

discussed in detail.

Concepts and Understanding:

- Logical conclusions drawn from information - Relevant research done

- Appropriate methods used - Accurate calculations used - Work shown for support

- Technology used to aid in calculations -

Explanations are incomplete,

unclear, inaccurate, and/or lack detail. Work is incorrect and/or not provided.

Explanations are generally complete, clear, accurate, and detailed. Work is generally correct.

Explanations are complete, clear, accurate, and detailed. Work is accurate and correct.

Communication and Presentation: - Clear, concise, and thorough - Engaging

- Neat and well put-together - Collaborative

- Oral and written portions of report

- Graphs, charts, and models used to support proposals

- Appropriate vocabulary and technical terminology used - Professional Some important details or information lacking from presentation; presentation is unprofessional, sloppy, and/or lacks contribution from all group members

Presentation covers all required topics and effectively delivers the group’s research Presentation “wows” the selection committee; all group members contribute to a sophisticated and professional presentation

(43)

HO#6: School Paradise Project

Overall Goal

Create a design that represents your ideal future school that would function as a school of the future.

Team Design Focus

As your team builds a design for your school, choose a focus that drives your design. Examples include athletics, environmental impact, community contribution, location, cost minimization, and aesthetics.

Use of Mathematical Knowledge

Use previously learned math skills to support your design as well as skills learned throughout the project.

Final Product

(44)

HO#7: Design Team Building Guide and Resources

Team Name & Members: _______________________________________________________________

Necessary Design Components: 1.

2. 3. 4. 5.

Design Team Focus: 1.

What our Final Product looks like: 1.

2.

(45)

What our Final Product Includes: Visuals

Calculations

Summary

Online Resources

Metropolitan Statistical Data

Demographics

http://www.yesvirginia.org/

Topo Maps http://www.spotsylvania.va.us/content/2614/147/2742/189/default.aspx

www.staunton.va.us

Google Earth http://www.google.com/earth/index.html

MLS (real estate listings)

http://www.mls.com/

Quadratic Max and Min Applications Practice