IfYouBuildItProject.pdf

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UVA-SCPS Office of Mathematics Outreach with support from VADOE Mathematics and 1 Mathematics Senior Level Capstone Course

Unit Overview

Title of Unit: If you Build it…They will Come Unit Designers:

Rebecca Bienvenue-Stafford Co. Chris DuBois-Stafford Co.

Denise Glassford-Spotsylvania Co. Kim Riddle-Spotsylvania Co. Pam Bailey-Spotsylvania Co. Context:

Summary of the issue, challenge, investigation, or problem.

Students will be given a plot of land to design a community. Students will play the role of a developer and will have a goal of designing an optimum healthy community.

Number of Class Hours:

18 hours Unit

Design: ___Task Based

_X__Project Based Other Subject

Areas/Disciplines Addressed:

Drafting/Technical Drawing

Driving Question: How do land developers build the best community possible with restrictions? Mathematics Content

Addressed:

Area, Perimeter, Surface Area, Ratios and Proportions, Percents, Similar 3D figures, Systems of Linear Equations and Inequalities, Linear Programming

MPE Addressed:

Problem Solving, Decision Making and Integration

Understanding and Applying Functions Assumption of Prior

Knowledge:

Measurement and Conversions, Area/Perimeter/Surface Area, Graphing Linear Equations and Inequalities

College and Career Readiness/21st Century

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UVA-SCPS Office of Mathematics Outreach with support from VADOE Mathematics and 2 Skills to be taught (T)

during this unit or expectation (E) for student use during this unit and assessed (A): BIE Page 35-37

Communication (Oral and/or Written) E, A Technology

Critical Thinking/Decision Making E, A Other: (Describe)

Major Products and/or Performances:

Group: Presentation of Community Plan Presentation Audience:

X Class

School

Individual X Expert

Community Other: Launch: Event or

experience used to engage the students interest and inquiry:

Designing a Great Neighborhood http://www.youtube.com/watch?v=hf_gFXJ3WGI Shows the construction of a neighborhood from beginning to end.

a homeowner’s association.

ENGAGE How will student’s interested be piqued so they want to engage in the inquiry in this unit? 20 minutes

Teacher shows the Designing a Neighborhood video. http://www.youtube.com/watch?v=hf_gFXJ3WGI

The students discuss their own neighborhoods and communities.

Mathematician Journal Prompts What characteristics or aspects of a community do you think are the most important and why?

EXPLORE Teacher provides

guidance for the explorations to prepare students with the

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

4 hours

Exploration 1: Ratios and Proportions

See HO #1a-c (student copy) and HO #2a-e (teacher key) The goal is for students to investigate the use of ratios and proportions in practical applications. Teachers give students HO #1a-c to work on in small groups and assist as needed. Encourage alternative methods to determine the solution including pictures or using manipulatives. Consider assigning a problem to a group and have them be the expert and presenter.

Exploration 2: Similar Solids

See HO #3a-b (student copy) and HO #4a-b (teacher key) The goal is for students to investigate the properties of similar figures. Teachers give students HO #3a-b and assist as needed. Students need 1 cm cubes. Individual or Groups of two.

Exploration 3: Linear Programming

See HO #5a-b (student copy) and HO #6a-f (teacher key) The goal is for students to investigate practical uses of linear programming. Teachers give students HO #5a-b and assist as needed. Encourage multiple methods of determining the solution(s).

Self Assessments and Tutorials See HO #7

The goal is for students to self assess and complete tutorials as needed to ensure mastery of prerequisite skills. Websites for the self assessments and tutorials are on the handout.

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May use at the beginning of the “Explore” section or as needed as the class proceeds through Exploration 1-3.

EXPLAIN Teacher introduces the main task of the unit and

prepares students to in small group independent work... 20 minutes

The teacher introduces the Plots and Plats project (See HO #8a-c) and the rubric (See HO #9a-b) so students are aware of all expectations. In particular, the teacher addresses HO #8a-b to go over details on the timeline requirements and the final product, with student input on the intermediate deadlines. Teacher reminder to students: The planning commission wants the students to justify their decisions; so the groups should use graphs and other visuals as they present the information that is highlighted on handout #8b.

Mathematician Journal Prompts ELABORATE The student groups are working independently with teacher consultations. 9 hours

The teacher monitors and checks in with each group during each class period.

The students research the project and make adjustments to their timeline to map out how they plan to complete their task. Changes in the timeline must be approved by the “Planning Commission”.

Teacher NOTE: Stress that student need to be punctual with the schedule they create

Details of project may be found on HO #8a-b.

Groups schedule each of the items on the timeline to be turned in to the “investors”. This enables the teacher to guide them on their path to the final project.

Mathematician Journal Prompts How are you collaborating as a group? How is the work being split up and how are decisions being made? EVALUATE Working groups submit products or make presentations 4 hours

Each of the timeline items turned in to the “investors” is a way for the teacher to guide, question, and insure that the groups are on the right path.

Students give a formal presentation of their community. Select a group of students, and/or administrators and teachers to serve as the planning commission. Be sure to alert the “commission members” to any portions of the Municipal Code that you enforced. Their presentation is to include graphical representations to justify number of homes needed to maintain open spaces (this may take relating several graphs) along with their plot and plat maps, brochure, and homeowner’s association overview. See HO #8a-b for details.

The teacher uses the Plot and Plat Project Rubric (See HO #9a-b) to evaluate the students.

Mathematician Journal Prompts Now that you have watched other groups’ presentations, what community

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Map the Unit

What do students need to know and be able to do to complete the task/project/problem

successfully? How and when will they assess their own necessary knowledge and skills? How will they remediate their own gaps or weaknesses in knowledge and skills? Look at each major task for the unit and analyze the tasks necessary to produce a high-quality product.

Task: To create a desirable healthy community that will be sustainable through a

homeowners association.

KNOWLEDGE AND SKILLS NEEDED Assumed

already learned

Students will self-assess

Will be taught during the unit

1. Area, Perimeter, Surface Area X X

2. Measurement/Conversions X X

3. Graphing Linear Equations/Inequalities X X

4. Similar Figures X X

5. Ratios and Proportions X

6. Linear Programming X

7.

8.

9.

10.

11.

What project tools will student’s use?

 Know/need to know lists

 Daily goal sheet

 Mathematician’s Journals

 Briefs/Memos  Task lists - timeline

 Planning Calendar

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HO #1a

Ratios and Proportions Lab

A ‘Work’ Word Problem is one that involves a number of people or machines working together to complete a task. We are usually given individual rates of completion and are asked to find out how long it would take if they work together. Consider different methods such as drawings to help you understand and work the problems.

Consider these “work” problems. Please show all work and/or pictures. Be ready to share your justification of procedures.

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Jack can paint a wall in 3 hours. John can do the same job in 5 hours. How long will it take if they work together?

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Working together, printer A and printer B would finish a task in 24 minutes. Printer A alone would finish the task in 60 minutes. How many pages does the task contain if printer B prints 5 pages a minute more than printer A?

(3)

When the tub faucet is on full, it can fill the tub to overflowing in 20 minutes (ignore the

existence of the overflow drain). The drain can empty the tub in 15 minutes. Your four-year-old brother has managed to turn the faucet on full, and the drain was closed. Just as the tub starts to overflow, you run in and discover the mess. You grab the faucet handle, and it comes off in your hand, leaving the water running at full power. You yank the drain open, and run for towels to clean up the overflow. How long will it take for the tub to empty, with the faucet still on but the drain now open?

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HO #1b

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Ben takes 2 hours to wash 500 dishes, and Frank takes 3 hours to wash 450 dishes. How long will they take, working together, to wash 1000 dishes?

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Suppose that it takes Tom and Dick 2 hours to do a certain job. It takes Tom and Harry 3 hours to do the same job. It takes Dick and Harry 4 hours to do the same job. How long would it take Tom, Dick and Harry to do the same job if all 3 men worked together?

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A plot of land is set aside for a family garden. The following information is known about the plot of land and it usage:



The total land dimension is 24 feet by 20 feet.



30% will be used for Tomatoes.



20% will be used for Roses.



15% will be used for Lettuce.



10% will be used for Cucumbers.



10% will be used for Pumpkins.



15% will be used for other flowers.

a) Using the grid, how much land does each square represent?

b) Using the grid, how many squares represent 50% of the total plot of land?

c) Using the grid, how many squares represent 35% of the total plot of land?

d) As stated above, “30% will be used for tomatoes”. How many squares will represent 30% ?



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HO #1c

e) Draw three different configures representing how tomatoes can be planted.

f) Using the constraints listed, design a diagram that meets all of the constraints. 30% will be used for Tomatoes. ( T )

20% will be used for Roses. ( R ) 15% will be used for Lettuce. ( L )

10% will be used for Cucumbers. ( C ) 10% will be used for Pumpkins. ( P ) 15% will be used for other flowers. ( F )

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Re-Boxing If the length and width of a rectangular box are both decreased by 50% while the box’s

height is increased by 50%, what is the overall effect on the volume of the box? e) f) g) h) i) j)

k) l) m) n) o) p) q) r) s) t) u) v) w) x) y) z) aa) bb) cc) dd) ee) ff) gg) hh)

ii) jj) kk) ll) mm) nn) oo) pp) qq) rr) ss) tt) uu) vv) ww) xx) yy) zz) aaa) bbb) ccc) ddd) eee) fff) ggg) hhh) iii) jjj) kkk) lll)

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HO #2a

Ratios and Proportions Lab - Key

"The fact that many aspects of our world operate according to proportional rules makes proportional reasoning abilities extremely useful in the interpretation of real world phenomena." 1

A ‘Work’ Word Problem is one that involves a number of people or machines working together to complete a task. We are usually given individual rates of completion and are asked to find out how long it would take if they work together. Consider different methods such as drawings to help you understand and work the problems.

Consider these “work” problems. Please show all work and/or pictures. Be ready to share your justification of procedures.

Teacher’s Notes:

Algebraic Approach to The ‘Work’ Problems.

STEP 1 : Calculate how much work each person/machine does in one unit of time. For example:- If we are given that A completes a certain amount of work in X hours, simply reciprocate the number of hours to get the per hour work. Thus in one hour, A would complete (1/X) of the work.

STEP 2 : Add up the amount of work done by each person/machine in that one unit of time. This would give us the total amount of work completed by both of them in one hour. For example, if A completes (1/X) of the work in one hour and B completes (1/Y) of the work in one hour, then TOGETHER, they can complete [(1/X) + (1/Y)] of the work in one hour.

STEP 3 : Calculate total amount of time taken for work to be completed when all persons/machines are working together. The logic is similar to one we used in STEP 1, the only difference being that we use it in reverse order. Suppose [(1/X) + (1/Y)] = (1/Z). This means that in one hour, A and B working together will complete (1/Z) of the work. Therefore, working together, they will complete the work in Z hours.

Graphical or Pictorial Approach to The “Work” Problems.

Students may draw or use some type of concrete materials to represent the problem and determine how much of the job is done in one unit of time. Tables are another choice to record data, see patterns, and determine the solution.

(1) Jack can paint a wall in 3 hours. John can do the same job in 5 hours. How long will it take if they work together?

STEP 1: Calculate how much work each person does in one hour. Jack (1/3) John (1/5) STEP 2: Work done in one hour when both are working together = (1/3)+(1/5) = (8/15)

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HO #2b

#1 Adapted from L. POST, T, BEHR, M., & LESH, R. (1988). Proportionality and the development of prealgebra Understanding. In A. Coxford (Ed.), Algebraic concepts in the curriculum K-12 (1988 Yearbook, pp. 78-90). Reston, VA: National Council of Teachers of Mathematics.

(2) Working together, printer A and printer B would finish a task in 24 minutes. Printer A alone would finish the task in 60 minutes. How many pages does the task contain if printer B prints 5 pages a minute more than printer A?

STEP 1: ‘Working together, printer A and printer B would finish a task in 24 minutes’, tells us that A and B combined would work at the rate of (1/24) per minute.

STEP 2: ‘Printer A alone would finish the task in 60 minutes’, tells us that A works at a rate of (1/60) per minute.

STEP 3: At this point, it is possible to calculate the rate at which B works. Rate at which B works = (1/24) – (1/60) = (1/40)

STEP 4: ‘B prints 5 pages a minute more than printer A’ , means that the difference between the amount of work B and A complete in one minute corresponds to 5 pages. It will be



(1/40) – (1/60) = (1/120) STEP 5: ‘How many pages does the task contain?’ If (1/120) of the job consists of 5 pages, then the 1 job will consist of (5*1)/(1/120) = 600 pages.

(3) When the tub faucet is on full, it can fill the tub to overflowing in 20 minutes (ignore the existence of the overflow drain). The drain can empty the tub in 15 minutes. Your four-year-old brother has managed to turn the faucet on full, and the drain was closed. Just as the tub starts to overflow, you run in and discover the mess. You grab the faucet handle, and it comes off in your hand, leaving the water running at full power. You yank the drain open, and run for towels to clean up the overflow. How long will it take for the tub to empty, with the faucet still on but the drain now open?

STEP 1: We see that the drain can empty 1/15 of the tub per minute. The faucet can fill 1/20 of the tub per minute. Then, working together, they can empty 1/15 – 1/20 of the tub per minute. The subtraction indicates that the faucet is actually working against the drain.

STEP 2: Let "t" indicate how long it takes to drain the whole tub. Then, 1/t is drained per minute. STEP 3: Therefore, 1/15 – 1/20 = 1/t



1/60 = 1/t



t = 60 minutes (1 hour).

(4) Two mechanics were working on your car. One can complete the given job in six hours, but the new guy takes eight hours. They worked together for the first two hours, but then the first guy left to help another mechanic on a different job. How long will it take the new guy to finish your car?

STEP 1: The first guy can do 1/6 per hour. The new guy can do 1/8 per hour. STEP 2: Together, they can do 1/6 + 1/8 = 7/24 per hour.

STEP 3: They worked for two hours



they got 2( 7/24 ) = 7/12 of the job done. STEP 4: That leaves 5/12 of the job to do.

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HO #2c

= 5/12



h/8 = 5/12



h/8 = 5/12



h = 10/3 = 3 1/3. It takes the new guy another 3 hours and twenty minutes to finish fixing your car.

(5) Ben takes 2 hours to wash 500 dishes, and Frank takes 3 hours to wash 450 dishes. How long will they take, working together, to wash 1000 dishes?

For this exercise, we are given how many can be done in one time unit, rather than how much of a job can be completed. But the thinking process is otherwise the same.

STEP 1: Ben can do 250 dishes per hour. Frank can do 150 dishes per hour. STEP 2: Working together, they can do 250 + 150 = 400 dishes an hour.

STEP 3: To find the number of hours that it takes to wash 1000 dishes, we set things up so units cancel and you're left with "hours": (1000 dishes) × (1 hour / 400 dishes) = (1000 / 400) hours = 2.5 hours.

(6) Suppose that it takes Tom and Dick 2 hours to do a certain job. It takes Tom and Harry 3 hours to do the same job. It takes Dick and Harry 4 hours to do the same job. How long would it take Tom, Dick and Harry to do the same job if all 3 men worked together?

Step 1: Using logic and simple observations estimate or bound the answer. A simple observation: Tom, Dick and Harry working together can do the job in less time than Tom and Dick can do the job (without Harry's help). Since Tom and Dick can do the job in 2 hours, the three of them together should be able to do the job in less than 2 hours. (This is a tip off that 4.5 hours must be a wrong answer.)

Step 2: Write down verbal equations. The information stated in the problem is:

a)

[Work done by Tom in 2 hours] + [Work done by Dick in 2 hours] = 1 job

b)

[Work done by Tom in 3 hours] + [Work done by Harry in 3 hours] = 1 job

c)

[Work done by Dick in 4 hours] + [Work done by Dick in 4 hours] = 1 job

Step 3: If x is the number of hours it takes the three of them to do the job, then:

[Tom’s Hourly Work Rate] + [Dick’s Hourly Work Rate] + [Harry’s Hourly Work Rate] = 1 job xT + xD + xH = 1

STEP 4: With these variables, translate the verbal equations into algebraic equations: 2T + 2D = 1 3T + 3H = 1 4D + 4H = 1 xT + xD + xH = 1

STEP 5: Solves the first three linear equations for T, D and H. T = 7/24 D = 5/24 H = 1/24

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HO #2d

(7) A plot of land is set aside for a family garden. The following information is known about the plot of land and it usage:



The total land dimension is 24 feet by 20 feet.



30% will be used for Tomatoes.



20% will be used for Roses.



15% will be used for Lettuce.



10% will be used for Cucumbers.



10% will be used for Pumpkins.



15% will be used for other flowers.

a) Using the grid, how much land does each square represent? 4 feet by 4 feet

b) Using the grid, how many squares represent 50% of the total plot of land? 15 squares c) Using the grid, how many squares represent 35% of the total plot of land? 10.5 squares d) As stated above, “30% will be used for tomatoes”. How many squares will represent 30% ?

9 squares

e) Draw three different configures representing how tomatoes can be planted. Numerous diagrams

f) Using the constraints listed, design a diagram that meets all of the constraints. 30% will be used for Tomatoes. (T) [9 squares]

20% will be used for Roses. ( R ) [6 squares] 15% will be used for Lettuce. ( L ) [4.5 squares] 10% will be used for Cucumbers. ( C ) [3 squares] 10% will be used for Pumpkins. ( P ) [3 squares] 15% will be used for other flowers. ( F ) [4.5 squares]

X X X

X X X X X X

X X X

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HO #2e

Numerous Possible Diagrams

(8) Re-Boxing If the length and width of a rectangular box are both decreased by 50% while the box’s

height is increased by 50%, what is the overall effect on the volume of the box?

STEP 1: The volume of the box is its length, width and height multiplied together: V = lwh.

STEP 2: We want to know the relative change in V given proportional changes in l, w and h. Since both l and w are decreased by 50%, the new length and width can be represented as: lnew = l – .5l = .5l wnew = w – .5w = .5w

STEP 3: The new height is represented as: hnew = h + .5h = 1.5h

STEP 4: The total volume becomes: Vnew = lnew wnew hnew = (.5l)(.5w)(1.5h) = 0.375lwh

STEP 5: The volume is 37.5% of its original amount. However, we are asked what percentage change this represents from the original volume. This proportion represents a decrease of 62.5% since lwh – 0.375lwh = 0.625lwh. Answer: It is decreased by 62.5%

T T T R R R

T T T C C C

L L L / F F F P

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HO #3a

Similar Solids

(1) Use 1 cm cubes to build three figures with dimensions A: 1 x 1 x 1; B: 2 x 2 x 2; and C: 3 x 3 x 3. (2) Determine the area of one face of each cube and then calculate the total surface area of each cube. (3) Fill in the table to make comparisons between different cubes:

Comparing Ratio of edge lengths Ratio of surface areas A : B

A : C B : C

(4) How is the ratio of surface areas related to the ratio of edge lengths?

(5) Use 1 cm cubes to build a rectangular prism where each dimension of the figure is a different value. Place it resting on its largest face. Then build a second rectangular prism with dimensions that are double the first prism and place it on its largest face.

Dimensions of first prism_________Dimensions of second prism__________ (6) Predict the ratio of surface areas based on what you found for the cubes.

(7) Fill in the table to make comparisons between the two prisms:

Face Area of face on prism 1 Area of face on prism 2 Ratio of areas of prism 1 to prism 2 Top

Bottom Front Back Right Side

Left Side Total Surface

Area

(8) What is the similarity ratio of corresponding lengths of similar prisms?

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HO #3b

(10) The cylinders shown are similar. What is the similarity ratio of the dimensions of the larger cylinder to the smaller cylinder?

(11) Predict the ratio of surface areas between the larger and smaller cylinder.

(12) Check your prediction by calculating the surface area of each cylinder and making a comparison.

(13) The ratio of corresponding linear measures of two similar cans of fruit is 4 to 7. The smaller can has a surface area of 220 square centimeters. Find the surface area of the larger can.

(14) Julia is painting the playhouse she made for her sister. She is going to paint only the outside of the house, which does not have a floor. The paint she is using can only be purchased in full gallons. Each gallon covers 150 ft2 and costs $25.40 a gallon. How much will it cost Julia to buy the paint she will need?

(15) Jose and Richard are planning to paint the exterior walls of their farmhouse and to put new shingles on the roof. The paint costs $25 per gallon and covers 250 square feet per gallon. The wood shingles cost $65 per bundle and each bundle covers 100 square feet. How much will this project cost? All measurements are in feet.

15 6.5

End View 12

38.5 12

24

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HO #4a

Similar Solids - Key

(1) Use 1 cm cubes to build three figures with dimensions A: 1 x 1 x 1; B: 2 x 2 x 2; and C: 3 x 3 x 3. (2) Determine the area of one face of each cube and then calculate the total surface area of each cube. (3) Fill in the table to make comparisons between different cubes:

Comparing Ratio of edge lengths Ratio of surface areas

A : B 1:2 6:24

A : C 1:3 6:54

B : C 2:3 24:54

(4) How is the ratio of surface areas related to the ratio of edge lengths?

Ratio of the surface areas is equal to the square of the ratio of the edge lengths.

(5) Use 1 cm cubes to build a rectangular prism where each dimension of the figure is a different value. Place it resting on its largest face. Then build a second rectangular prism with dimensions that are double the first prism and place it on its largest face.

Dimensions of first prism__2 x 3 x 4______ Dimensions of second prism___4 x 6 x 8_______ (6) Predict the ratio of surface areas based on what you found for the cubes.

Square of the ratio of the edge lengths

(7) Fill in the table to make comparisons between the two prisms:

Face Area of face on prism 1 Area of face on prism 2 Ratio of areas of prism 1 to prism 2

Top 6 24 6:24

Bottom 6 24 6:24

Front 8 32 8:32

Back 8 32 8:32

Right Side 12 48 12:48

Left Side 12 48 12:48

Total Surface Area

52 208 52:208

(8) What is the similarity ratio of corresponding lengths of similar prisms? 1:2

(9) How does the ratio of surface areas compare to the ratio of edge lengths of prisms?

(18)

HO #4b

(10) The cylinders shown are similar. What is the similarity ratio of the dimensions of the larger cylinder to the smaller cylinder?

3:1

(11) Predict the ratio of surface areas between the larger and smaller cylinder. 9:1

(12) Check your prediction by calculating the surface area of each cylinder and making a comparison. Surface area of larger cylinder = 13,734.36

Surface area of smaller cylinder = 1526.04

Ratio of surface areas = 9:1

(13) The ratio of corresponding linear measures of two similar cans of fruit is 4 to 7. The smaller can has a surface area of 220 square centimeters. Find the surface area of the larger can.

673.75 square centimeters

(14) Julia is painting the playhouse she made for her sister. She is going to paint only the outside of the house, which does not have a floor. The paint she is using can only be purchased in full gallons. Each gallon covers 150 ft2 and costs $25.40 a gallon. How much will it cost Julia to buy the paint she will need?

Total surface area = 377 square feet She will need to purchase 3 gallons of paint Total cost = $76.20

(15) Jose and Richard are planning to paint the exterior walls of their farmhouse and to put new shingles on the roof. The paint costs $25 per gallon and covers 250 square feet per gallon. The wood shingles cost $65 per bundle and each bundle covers 100 square feet. How much will this project cost? All measurements are in feet.

Total surface area to paint = 3894 square feet Need to purchase 16 cans of paint

Total paint cost = $400

Total surface area for shingles = 1720 square feet Need to purchase 18 bundles

Total shingle cost = $1170 Total project cost = $1570

15 6.5

End View 12

38.5 12

24

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HO #5a

Linear Programming Explorations

1. a) You have $2.50 in your pocket. The 22 coins in your pocket are a combination of nickels and quarters. How many of each type of coin do you have?

b) You have 22 coins in your pocket totaling $2.50. The coins are a combination of nickels, dimes, and quarters, and you have at least one of each type of coin. How many of each type of coin do you have?

2. A student is taking a test that contains computation problems worth 6 points each and word problems worth 10 points each. A computation problem can be completed in 2 minutes and word problems can be completed in 4 minutes. The student has 40 minutes to complete the test and may answer no more than 12 problems.

 Assuming the student answers all the problems attempted correctly, how many of each type of problem must you do to maximize your score?

 What is the maximum score the student can receive on the test?

3. Food and clothing are being sent by commercial airplanes to hurricane victims in Louisiana. The relief effort is bound by the following constraints:

 Each container of food is estimated to feed 12 people. In addition, a food container weighs 50 pounds and occupies 20 cubic feet of space.

 Each container of clothing is intended to help 5 people. A clothing container weighs 20 pounds and has a volume of 10 cubic feet.

 The total weight of the supplies cannot exceed 19,000 pounds.

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HO #5b

a)

How many containers should be sent in each plane shipment if the shipment contains

 food containers only

 clothing containers only

 food and clothing containers

b)

What is the maximum number of people that each plane shipment can help if the hurricane victims are

 fed only?

 clothed only?

 fed or clothed?

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HO #6a

Linear Programming Explorations - Key

1. a) You have $2.50 in your pocket. The 22 coins in your pocket are a combination of nickels and quarters. How many of each type of coin do you have?

This problem may be solved either by a system of equations or a table of values. Using a system of equations Let q = the number of quarters

Let n = the number of nickels

The total number of coins is represented by q + n = 22

The value of the coins is represented by 0.25q + 0.05n = 2.50 Solving the system results in q = 7 and n = 15

Using a table to find every possible combination of coins

Quarters Nickels # of coins

10 0 10

9 5 14

8 10 18

7 15 22

6 20 26

5 25 30

4 30 34

3 35 38

2 40 42

1 45 46

0 50 50

b) You have 22 coins in your pocket totaling $2.50. The coins are a combination of nickels, dimes, and quarters, and you have at least one of each type of coin. How many of each type of coin do you have?

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HO #6b

Quarters Nickels Dimes # of coins Quarters Nickels Dimes # of coins

10 0 0 10 5 1 12 18

9 5 0 14 4 30 0 34

9 3 1 13 4 28 1 33

9 1 2 12 4 26 2 32

8 10 0 18 4 24 3 31

8 8 1 17 4 22 4 30

8 6 2 16 4 20 5 29

8 4 3 15 4 18 6 28

8 2 4 14 4 16 7 27

8 0 5 13 4 14 8 26

7 15 0 22 4 12 9 25

7 13 1 21 4 10 10 24

7 11 2 20 4 8 11 23

7 9 3 19 4 8 11 23

7 7 4 18 4 6 12 22

7 5 5 17 4 4 13 21

7 3 6 16 4 2 14 20

7 1 7 15 4 0 15 19

6 20 0 26 3 35 0 38

6 18 1 25 3 33 1 37

6 16 2 24 3 31 2 36

6 14 3 23 3 29 3 35

6 12 4 22 3 27 4 34

6 10 5 21 3 25 5 33

6 8 6 20 3 23 6 32

6 6 7 19 3 21 7 31

6 4 8 18 3 18 8 30

6 2 9 17 3 17 9 29

6 0 10 16 3 15 10 28

5 25 0 30 3 13 11 27

5 23 1 29 3 11 12 26

5 21 2 28 3 9 13 25

5 19 3 27 3 7 14 24

5 17 4 26 3 5 15 23

5 15 5 25 3 3 16 22

5 13 6 24 3 1 17 21

5 11 7 23 2 40 0 42

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HO #6c

Quarters Nickels Dimes # of coins Quarters Nickels Dimes # of coins

5 7 9 21 2 36 2 40

5 5 10 20 2 34 3 39

5 3 11 19 2 32 4 38

2 30 5 37 1 25 10 36

2 28 6 36 1 23 11 35

2 26 7 35 1 21 12 34

2 24 8 34 1 19 13 33

2 22 9 33 1 17 14 32

2 20 10 32 1 15 15 31

2 18 11 31 1 13 16 30

2 16 12 30 1 11 17 29

2 14 13 29 1 9 18 28

2 12 14 28 1 7 19 27

2 10 15 27 1 5 20 26

2 8 16 26 1 3 21 25

2 6 17 25 1 1 22 24

2 4 18 24 0 50 0 50

2 2 19 23 - - - -

2 0 20 22 - - - -

1 45 0 46 - - - -

1 43 1 45 0 0 25 25

1 41 2 44

1 39 3 43

1 37 4 42

1 35 5 41

1 33 6 40

1 31 7 39

1 29 8 38

1 27 9 37

**NOTES TO TEACHER: It is recommended that you do not tell students there are multiple answers. Once students find an answer, compare answers of the entire class and incorporate different answers in class discussion. You can ask questions such as “This group has a different answer, are they right and why?”

2. A student is taking a test that contains computation problems worth 6 points each and word problems worth 10 points each. A computation problem can be completed in 2 minutes and word problems can be completed in 4 minutes. The student has 40 minutes to complete the test and may answer no more than 12 problems.

None of the values in this range meet the criteria that you must have at

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HO #6d

 Assuming the student answers all the problems attempted correctly, how many of each type of problem must you do to maximize your score?

Let c = the number of computation problems Let w = the number of word problems

Students cannot answer more than 12 problems. Therefore, c + w ≤ 12

Students need 2 minutes for computation problems and 4 minutes for word problems and have a maximum of 40 minutes to complete the test. 2c + 4w ≤ 40

Solve the systems to find w ≤ 8 and c ≤ 4

 What is the maximum score the student can receive on the test?

Computation problems are worth 6 points and word problems are worth 10 points 6c + 10w

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Substitute in 8 for w and 4 for c to find the maximum points

6(4) + 10(8) = 24 + 80 = 104

3. Food and clothing are being sent by commercial airplanes to hurricane victims in Louisiana. The relief effort is bound by the following constraints:

 Each container of food is estimated to feed 12 people. In addition, a food container weighs 50 pounds and occupies 20 cubic feet of space.

 Each container of clothing is intended to help 5 people. A clothing container weighs 20 pounds and has a volume of 10 cubic feet.

 The total weight of the supplies cannot exceed 19,000 pounds.

 The total volume must be no more than 8,000 cubic feet.

Let f = the number of food containers

Let c = the number of clothing containers

There are two constraints on the containers. First the weight of any shipment cannot exceed 19,000 pounds. Second, the plane only has a cargo volume of 8,000 ft3.

a) How many containers should be sent in each plane shipment if the shipment contains

 food containers only

Addressing the plane’s weight limitation results in the inequality 50f ≤ 19000

Solving for f: f ≤ 380

Addressing the volume restrictions results in the inequality 20f ≤ 8000 Solving for f: f ≤ 400

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HO #6e

 clothing containers only

Addressing the plane’s weight limitation results in the inequality 20c ≤ 19000 Solving for c: c ≤ 950

Addressing the volume restrictions results in the inequality 10c ≤ 8000 Solving for c: c ≤ 800

Since we have to adhere to both restrictions c ≤ 800

 food and clothing containers

The plane’s weight cannot exceed 19000 pounds. Each food container weighs 50 lbs and each clothing container weighs 20 lbs. 50f + 20c ≤ 19000

The maximum volume the plane can hold is 8000 ft3. The volume of a food container is 20 ft3, and the volume of a clothing container is 10 ft3. 20f + 10c ≤ 8000

Solving the system of equations results in f ≤ 300 and c ≤ 200.

b) What is the maximum number of people that each plane shipment can help if the hurricane victims are

 fed only?

As discovered earlier, any shipment of only food containers would have 380 food containers. Each food container feeds 12, therefore 380 x 12 = 4560 people fed

 clothed only?

As discovered earlier, any shipment of only clothing containers would have 800 clothing containers. Each clothing container aids 5 people, therefore 800 x 5 = 4000 people clothed

 fed or clothed?

Each food container feeds 12 people. Using the solution from earlier, there can be no more than 300 food containers. 300 containers x 12 people = 3600 people fed

Each clothing container clothes 5 people. Again using the solution from earlier, there can be no more than 200 clothing containers. 200 containers x 5 people = 1000 people clothed

If people only get one type of assistance or the other, then 3600 + 1000 = 4600 people will receive aid

 fed and clothed?

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HO #6f

Each clothing container clothes 5 people. Again using the solution above, there can be no more than 200 clothing containers. 200 containers x 5 people = 1000 people clothed

If people can receive both types of aid (there can be overlap in assistance), then 3600 – 1000 = 2600 people will receive both food and clothing

(27)

HO #7

Self Assessments and Tutorials Self Assessments

 Measurement/Conv:http://www.regentsprep.org/Regents/math/ALGEBRA/MultipleChoiceReview/M easurement.html

 Percents: http://www.regentsprep.org/Regents/math/ALGEBRA/AO3/pracPer.htm

 Proportions: http://www.regentsprep.org/Regents/math/ALGEBRA/AO3/pracProp.htm

 Similar Figures: (Triangles) http://www.regentsprep.org/Regents/math/geometry/GP11/PracSim.htm

 Area of Polygons and Circles:

http://www.regentsprep.org/Regents/math/ALGEBRA/AS1/PracArea.htm

 Perimeter and Circumference:

http://www.regentsprep.org/Regents/math/ALGEBRA/AS1/PracPer.htm

 Surface Area and Volume: http://www.regentsprep.org/Regents/math/ALGEBRA/AS2/PracSol.htm

 Systems of Eq/Ineq: http://www.regentsprep.org/Regents/math/ALGEBRA/AE9/PracGr.htm

Tutorials

 Measurement/Conversions:

o

http://www.khanacademy.org/video/scale-and-indirect-measurement?playlist=ck12.org%20Algebra%201%20Examples o

http://www.khanacademy.org/video/perimeter-and-unit-conversion?playlist=Developmental%20Math

o http://www.regentsprep.org/Regents/math/ALGEBRA/AM2/LesEng.htm

 Percents:

o http://www.khanacademy.org/video/more-percent-problems?playlist=Algebra o http://www.khanacademy.org/video/taking-percentages?playlist=Algebra o http://www.regentsprep.org/Regents/math/ALGEBRA/AO3/Lpercent.htm

 Proportions:

o http://www.khanacademy.org/video/understanding-proportions?playlist=Developmental%20Math

o http://www.khanacademy.org/video/find-an-unknown-in-a-proportion-2?playlist=Developmental%20Math

 Similar Figures: http://www.khanacademy.org/video/similar-triangles?playlist=Geometry

 Area/Perimeter: http://www.khanacademy.org/video/area-and-perimeter?playlist=Geometry

 Surface Area: http://www.regentsprep.org/Regents/math/ALGEBRA/AS2/Solids2.htm

 Linear Systems of Inequalities:

o http://www.khanacademy.org/video/graphing-systems-of-inequalities?playlist=Algebra%20I%20Worked%20Examples

o http://www.khanacademy.org/video/graphing-systems-of-inequalities-2?playlist=Algebra%20I%20Worked%20Examples

o http://www.khanacademy.org/video/u06-l3-t1-we3-graphing-systems-of-inequalities?playlist=Algebra%20I%20Worked%20Examples

 Non Linear Systems:

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HO #8a

PLOT AND PLAT CONSTRUCTION COMPANY

Plot and Plat Construction Company is working with a group of investors on some property in Spotsylvania County, Virginia. Your task as a team is to create a desirable healthy community that will be sustainable through a homeowners association.

A map of the property is attached showing the existing road and a lake on the property. To guide you with the project expectations a rubric is also attached. You will have ____ days to prepare your project for the final presentation to the county planning board.

PROJECT EXPECTATIONS:

 Minimum requirements:

o Residential homes – enough home sites to fund open spaces

o Open spaces to meet the healthy living idea you have for your community o Lot sizes (> 10,000 sq ft with specific criteria – see municipal code) o Homes – ranch style homes, 1000 sq ft. < homes < 5,000 sq.ft.

 Timeline of how the team will approach completing project, see groupings below. Timeline is to be turned in to the “investors” for approval in the groupings shown. They do not have to be in the order given but the groupings should stay together. Any changes in timeline must be approved by the investors. Timeline to include but limited to decisions made on

1. Percentages of land use (residential, open space, roads)

2. Type of open space(s) and what will be included on the property

3. More specific residential home requirements (possible exterior requirements, number of bedrooms or bathrooms, lawn specifics, garage specifics, …).

4. Minimum of two house plans (one floor plans only) 5. Homeowners association (HOA) rules and regulations 6. Show how open space and roads will be maintained 7. Plat map of subdivision

8. Plat map of two lots 9. Brochure

10. Final presentation to planning board

Groupings of items to turn in to investors are given below.  #1, 2

 #3, 4  #5, 6  #7, 8  #9, 10

 Subdivision plat map to include:

o Residential lots and dimensions (property line lengths and area) o Open space (property line lengths and area)

o Roads (named)

o Lots labeled for identification

 Individual plat maps (2) to include:

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HO #8b

o Placement of house and setbacks o Placement of driveway and setbacks o Adjacent road for ingress and egress

o The plat maps of lots should correlate with the two homes chosen below.

 Two homes

o Dimensions

o What is included in the home (BR, BA, …)

o Exterior choices (Ex: Percentage of foundation surface area to be exposed, types of coverage on home….)

 Be able to defend all of the above (mathematically, regulations, and aesthetically)

 Investors would like a theme for the subdivision that will guide you as the contractor in developing the open spaces and the types of homes to be built. Provide a name for your subdivision.

FINAL PRODUCT:

 Plat map of subdivision drawn to scale showing lots, open spaces, and roads.

 Plat map of two lots chosen from the subdivision map showing placement of house and driveway.

 Show how open spaces and roads will be maintained by homeowner’s association fees. Justify the minimum number of homes to be built in order to maintain the open spaces. Representation should allow for the investor to look at various numbers of homes built and determine income for HOA and maintenance fees for open spaces.

 Brochure to advertise subdivision.

 Overview of homeowner’s association rules and regulations.

 Two possible homes and any possible selections a buyer may select.

 Open space explanation – what is to be included.

 Present your subdivision to the planning board. Be able to justify choices made.

RESOURCES:

Municipal Code for Spotsylvania County, VA. You may also look up other areas using this site. http://library.municode.com/indes.aspx?clientID=12105&stateID=46&statename=Virginia Key areas/sections of the code that may be considered are: 20-2; 20-5.2.1; 23-2; 23-6.6.4

Comments for Facilitator:

Map contains about 1379 usable “squares” which is about 28.5 acres or 1,241,100 sq. ft. Each “square” is 900 sq.ft.

0.5 acre is about 24 squares

Lake is about 0.98 acre (42,750 sq ft.)

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HO #8c

State Road

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HO #9a

Plot and Plat Construction Company Rubric

4

3

2

1

UNDERSTANDING

Final documents and presentation shows a deep understanding of the mathematics needed to obtain their conclusion.

Final documents and presentation is complete but understanding needed to obtain their conclusion is missing or lacking in explanation.

Final documents and presentation is attempted. Some of the mathematics applied is the correct concept but lacking completion.

Final documents and presentation is not completed or not meeting the expectation.

REASONING

Appropriate

strategies/thinking skills used create the timeline leads to a final product.

Some strategies/thinking skills used to create the timeline that leads to the final product.

Needed help in creating timeline that leads to the final product.

Unable to create a timeline to progress toward the final product.

Justifications of their conclusion are appropriate and mathematical sound.

Some of the justifications of their conclusion are

appropriate and mathematical sound.

Justification of the conclusion is given but lacking in mathematical understanding.

No justification is given for the mathematics.

COMMUNICATION

All submissions are clear and well-organized.

Some of the submissions are clear and well-organized.

Portions of submissions are unclear or confusing.

Submissions are unclear and confusing.

All necessary concepts are presented in writing and in the presentation.

All necessary concepts are presented in the writing but the presentation was missing explanation.

Written submissions and presentation is missing concepts and explanations.

Written submission OR presentation is missing.

Presentation of conclusion to the investors and planning commission in an appropriate manner (professionalism).

Presentation of conclusion to the investors and planning commission was done but some may be in an unprofessional manner.

Presentation of conclusion to the investors and planning commission were unprofessional.

No presentation of

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HO #9b

4

3

2

1

CONNECTIONS

Connections were made between the mathematics and the problem to reach a conclusion.

Connections between the mathematics and the

problem were not clear in all explanations.

Connections between the mathematics and the problem were unclear.

No connections were made between the mathematics and the problem.

REPRESENTATIONS

All mathematical representations and

symbols were complete and correct.

Some of the mathematical representations and symbols were complete but incorrect.

Mathematical representations were complete and incorrect.

Mathematical representations were incomplete and incorrect.

MATHEMATICAL SKILLS AND PROCEDURES

Execution of algorithms and computations are complete and accurate.

Execution of algorithms is complete with minor errors in computations.

Execution of algorithms and computations contain major errors.

Execution of algorithms and computations are attempted.

TIMELINESS

Met all deadlines according to project guidelines and timeline created by the team.

Had to request extensions to deadlines according to project guidelines and timeline created by the team.

Did not request extensions to some of the deadlines according to project guidelines and timeline created by the team.

Did not request extensions to deadlines according to project guidelines and timeline created by the team.

COLLABORATION

All team members participated equally in the project.

Team members need encouragement to participate equally in the project.

Team members did not equally share the work of the project.

IfYouBuildItProject.pdf

a homeowner’s association.

homeowners association.

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a)

b)

c)

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HO #4a

a)

b)

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Categories

4

3

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Categories

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