Problem(Please view attachment: “Blacktop Battle” Entry Doc.)Standards/Big Ideas AddressedMultiplying BinomialsFind a common monomial factor in a polynomialFactoring Quadratic ExpressionsLikely units/big ideas that came before this problem

Likely units/big ideas that come after this problem

Polynomials: Add and subtract polynomials Multiply and divide polynomials

Solving Quadratic Equations by FactoringSolving Quadratic Equations by GraphingSolving Quadratic Equations using the Quadratic FormulaSolving Quadratic Equations by Completing the Square

Assumptions about Student Prior KnowledgeStudents know how to calculate the area of a rectangleStudents can apply the Distributive PropertyStudents know how to factor linear expressions

Problem OverviewProblem Title: Blacktop BattleCourse: Algebra 1Author(s): Sarah Gaynor

Facilitation NotesPhase Anticipated Student

ActionNotes/tips – including time for phase

Assessment

Roll out (k/ntk/next steps)

Students will likely identify these knows:

We’re designing a layout for a b-ball tournament

There are 24 total rows of bleachers

Areas for seating are same length as the b-ball court.

Land for court is square with length x.

Concession Stand is between 2 sections of bleachers.

There’s a date set for our meeting

We need a written letter with:o Visualso Possible

configurations for bleachers & concessions

o Min. & max. area for concession space.

o Dimensions for total

Total Roll-out time: 25 min.

...this is important, as the area of the 2 bleacher sections will become the ___x values. …this is imp., as the area of the court space will become the x2 value.…this is imp., as it will allow students to use the # of bleachers per section to calculate the area of the concession stand space.

Informal assessment of oral communication skills as students share out their thoughts on K/NTK’s and Next Steps.

area of layout. Submit plans & dimensions Submit 3-4 examples of

work that show we have no miscalculations.

And if they have prior knowledge about calculating area of rectangles, they might identify:

The area of the court space is x x = x∙ 2

Likely NTK’s: Do we NTK the

dimensions for a row of bleachers?

How do you calculate the min. & max. area of the concession stand space?

Are there certain arrangements that we can/can’t use?

How do you write a letter to include all of the requirements?

What are “samples of supporting evidence?”

Likely Next Steps: Determine how many

bleachers we want/need in each designated space.

Calculate area of concession stand space

Find dimensions of total land area.

Making connections: For students who visualize this and state it as a “know”…ask them to demonstrate their thought process so that “everyone can be in the know”! This will help others begin to label parts of the rectangle & think about finding the area of the sections.

…min. & max. area: When this NTK is asked, you may use the attached “Area Workshop” page.…arrangements: THIS will be where their critical thinking comes in! When they’ve found all possibilities, they should make a selection based upon what they feel works best for b-ball!…supporting evid.: This could be an array of practice problems/homework which they have SUCCESSFULLY completed to demonstrate mastery of these skills.

Student work time As students arrive at the NTK and/or next step of “calculating area”, provide them the “Area workshop” page in addition to a set of algebra tiles for those who prefer the hands-on approach. (See attached for cut-outs).

Student Work Time: 20 min.

Students should complete the workshop in teams of 2, such that one takes on the role of Record Keeper (for supporting evidence), while the other becomes the Liaison (to address team questions with the facilitator and/or other teams).

Informal/formative assessment of team collaboration & content skills (through practice problems).

Sharing out As students complete workshop (with algebra tiles), randomly select groups to give an update

Share-outs throughout this workshop should be minimal in time length to provide each team

Informal assessment on oral communication.

on the status of their findings.

Example:To team A: “Tell us your status on the 3rd configuration. Is it possible? Not possible? Why or why not?”To team B: “Have you determined the same conclusion? Why or why not?”Etc.

the opportunity for investigation, however, be sure to especially select teams to share out on the sample problems where they have said “No possible configuration”.

There are many times where a team cannot find a configuration & will give up trying, but it actually may be possible. The competitive feel of another team finding the solution before them in this workshop will a motivator for some during the share-outs.

Work time* Once students are proficient with calculating area as a sum and area as a product (which will be the dimensions of their land layout), they will begin to generate all of the possible solutions available for the basketball court.

Students should begin writing their letter with the requirements mentioned in the Entry Doc., in addition to their final selection for the layout.

Student Work Time: 25 min.

As teams begin to determine the possible configurations, be sure to encourage them to RECORD the dimensions of the land layout (i.e. “area as a product”) and the area of each individual section (i.e. “area as a sum”).

If technology is available… students may select to use Microsoft Word, Geometer’s Sketchpad, Paint program, Adobe Indesign, etc. to create their letter with the visual of their selected configuration included.

Direct instruction * (this can occur at different times, as long as students work on the problem first)

Using “area of a rectangle” as a means for multiplying polynomials.

Using F.O.I.L. to multiply binomials.

Factoring quadratic expressions.

For teams which do not seem to make the connection between the area as a sum and area as a product, you may find it necessary to do a workshop on any or all of these skills.

This problem based unit guides students through factoring and multiplying polynomials using the area of a rectangle method, HOWEVER, as students mature in their mathematical careers, many will need/want to know what “F.O.I.L.” means. This is a good time to introduce that acronym.

Informal/formative assessment on content skills and work ethic (i.e. participation in workshops).

Final Action on Problem

Students will present their findings to a “construction team” (aka Small Panel ).

If multiple persons are available to assess student presentations on a small panel, teams can present their findings simultaneously, with the panel using the rubric to assess.

Team Presentations: 5 min.Q & A time from panel: 3-5 min.

The presentation is the time to ensure that students can articulate the methods used (F.O.I.L./ area of a rectangle/ factoring/ etc.) for calculating total area and finding the dimensions of the land layout.

Panel possibilities:

Content or presentation assessed using rubric.

Work ethic assessed on the submission of supporting evidence.

Oral Communication assessed using rubric.

Collaboration assessed

Groups should submit their letter with the recommended configuration and other required components.

If your town/city has a local b-ball tournament, ask the planners if they will be present.

Algebra 1 “Alumni” Parent Volunteers Yourself (teams will simply

share their findings one at a time.)

using collaboration evaluation (completed by team members).

*These are typically optional, depending upon the nature/quality of the initial sharing out of students.

Follow UpExtension Designers,

It seems as though your configuration possibilities are coming along quite well. What you should know, however, is that we have recently been provided with information regarding the dimensions of the bleachers. Although we still don’t know the length of a single row, we do know that they have a 2.5 foot width. Additionally, we have had some questions asked of us about the dimensions of the concession stand. All we currently know is that the concession stand itself (i.e. just the building, no extra space around it included) is 575 ft2.

Please ensure that the configuration you select will allow enough area for the concession stand with some additional space around it for customers to comfortably stand and purchase their concessions.

Many thanks,

Cori J. MyersDirector of the Blacktop Battle

Practice Problems 1. Use these rectangles to multiply or factor in order to rewrite these expressions.a) b) c)

2. Write an algebraic equation for the area of each of the following rectangles.a) b) c) d)

3. Identify the following polynomial expressions in parts (a) through (h) as either a sum or a product.

a) 2x + 1 b) x2 + 7x + 12 c) (x+1)(x+4)d) 3x + 9

e) (x+5)(x+2) f) x(2x+5) g) 5x3 + 8x2 + 10x h) 2x(x2 – 3x + 5)

i) Which of the expressions above are quadratics?

4. Make a drawing to represent each equation. Label each part to show why the following equations are true. Write the area equation to accompany each drawing.

a) x2 + 7x + 6 = (x+6)(x+1) b) x2 + 4x + 4 = (x+2)(x+2)

c) x2 + 3x + 2 = (x+2)(x+1) d) 2x2 + 5x + 3 = (2x+3)(x+1)

5. Find the dimensions for each of the following rectangles.

a) b) c)

6. Multiply the following binomials.a) (y – 8)(y + 3) b) (20p + 4)(5p + 3) c) (8k – 7)(-5k – 3)

7. Find the factors of the following trinomials.a) x2 – x – 2 b) t2 + 3t – 10 c) w2 + 13w + 36

Solution to ProblemStudent teams may select a configuration for seats which is different than the 12 x 12 (max. configuration). Regardless of the configuration selected, students should support their final selection with reasoning for why they made this choice, which should include a comparison of all possibilities (as you find in the table below).

Bleacher/Seating Possibilities

(units)

Area for Concession Stand

(units2)

Bleacher/Seating Possibilities

(units)

Area for Concession Stand

(units2)

1 by 23 23 (minimum) 7 by 17 119

2 by 22 44 8 by 16 128

3 by 21 63 9 by 15 135

4 by 20 80 10 by 14 140

5 by 19 95 11 by 13 143

6 by 18 108 12 by 12 144 (maximum)

Solution to ExtensionWe Know:

Bleachers:

Total Concession Stand Area: 550 ft2

Based on our previous data, we can calculate:

Rows (units) Rows (feet)Concession

Stand Area (ft2)Rows (units) Rows (feet)

Concession Stand Area (ft2)

1 by 23 2.5 x 57.5 143.75 7 by 17 17.5 x 42.5 743.75

2 by 22 5 x 55 275 8 by 16 20 x 40 800

3 by 21 7.5 x 52.5 393.75 9 by 15 22.5 x 37.5 843.75

4 by 20 10 x 50 500 10 by 14 25 x 35 875

5 by 19 12.5 x 47.5 593.75 11 by 13 27.5 x 32.5 893.75

6 by 18 15 x 45 675 12 by 12 30 x 30 900

Due to the area given for the Concession Stand (the building itself), students may select any option EXCEPT: 1x23, 2x22, 3x21, 4x20, 5x19, 6x18. Students should also state (or you may ask them during their presentation), if they calculated in area for the concession stand lot which allows customers a place to stand while purchasing concessions (since 550 ft2 is the building only)!

2.5 ft.

x ft.

Dear Recreational Facilities Designers,

The Recreational Facilities Commission is currently reviewing the master plan for the Blacktop Battle, basketball tournament layout at a nearby park. We’re asking for your thoughts and input as we continue to look for optimal designs for our first annual Blacktop Battle tournament. These facility design alterations, as based on your input, will not take place until final revisions from our construction team have taken place. For a visual of this area, please view the blue prints below.

Based on the projected attendance for the tournament, the Facilities Commission is considering 24 total rows of bleachers* for seating. The rectangular areas allotted for the bleachers will have the same length as the land for the basketball court. Depending on funding, the court size may change, which means that the construction team is assuming the land available for the court will be square and have length of x. The land area for the bleachers will be adjacent to the two sides of the basketball court area, with the Concession Stand located in the corner, directly between the two sections of bleachers.

As valued Facilities Designers, we’re asking for your participation in our Blacktop Battle Planning meeting on (insert date here). We plan to discuss the ways in which this potential facility design plan, as showcased by your supporting evidence, may or may not affect the outcomes of the tournament. To have a successful meeting, we ask that you come prepared with a written letter including visuals of land space, to our Recreational Facilities Commission and construction team from the park, with thoughts on topics in the following areas:

All of the possible configurations for bleacher seating The area allotted for the Concession Stand within each configuration The area needed for the Concession Stand to be at its minimum and maximum allotment of land Support of the potential designs with mathematical evidence, for these minimum and maximum land

areas, including the dimensions and the total area for the entire facility design (i.e. court, bleachers, and Concession Stand).

In order to participate in our Blacktop Battle Planning meeting, you must show evidence of your progress in an interview between your Facility Designers and the construction team on (insert date here) . This initial interview will include, but is not limited to:

Your plans and dimensions for configurations Three or four samples of supporting evidence to verify that there are no miscalculations in your final plans/

thoughts.

We will be in touch with further information later in the design phases.

Sincerely,

Cori J. MyersDirector of the Blacktop Battle

*Note: We do not yet know what the length of a section of bleachers will be, so we will currently need to use the identical, unknown, dimensions for the length of the square land allotted for the court.

To whom it may concern,

The adult basketball league has recently had issues formalizing plans for a similar basketball tournament. In an effort to assist in their construction dilemmas, planning teams have used a method similar to the rectangles indicated below. The beauty of this method is that it has allowed them to calculate the area of an entire land space as a sum of its components, while also utilizing the dimensions of available land to calculate the area as a product.

The first phase in any planning process of this nature is to try many possible combinations knowing that successes and failures will arise. We would like to provide you with the chart below to further assist in your planning phase. (When attempting areas given multiple dimensions, use the examples below as your guide.)

Sincerely,

Cori J. Myers

Number of x2’s

Number of x’s

Number of 1’s

Is it possible? Rectangle Area as a sum and a product

1 3 2 Yes

x2 + 2x + x + 2

sum = x2 + 3x + 2product = (x + 1)(x + 2)

So, x2 + 3x + 2 = (x + 1)(x + 2)

1 5 3 No (not possible, so no rectangle can be drawn) –

1 4 4

1 6 5

1 3 9

1 4 3

1 7 10

Collaborative Work Skills Assessment:Project: Blacktop Battle

Please complete this assessment by highlighting, or circling the box which BEST represents this teammate's collaborative skills.

Teacher Name: ____________________________________ Date Completed: _____________________Teammate's Name:     ________________________________________Evaluator's Name(That's YOU!):     ________________________________________

CATEGORY 4 3 2 1Time-management Routinely uses time

well throughout the project to ensure things get done on time. Group does not have to adjust deadlines or work responsibilities because of this person's procrastination.

Usually uses time well throughout the project, but may have procrastinated on one thing. Group does not have to adjust deadlines or work responsibilities because of this person's procrastination.

Tends to procrastinate, but always gets things done by the deadlines. Group does not have to adjust deadlines or work responsibilities because of this person's procrastination.

Rarely gets things done by the deadlines AND group has to adjust deadlines or work responsibilities because of this person's inadequate time management.

Problem-solving Actively looks for and suggests solutions to problems.

Refines solutions suggested by others.

Does not suggest or refine solutions, but is willing to try out solutions suggested by others.

Does not try to solve problems or help others solve problems. Lets others do the work.

Attitude Never is publicly critical of the project or the work of others. Always has a positive attitude about the task(s).

Rarely is publicly critical of the project or the work of others. Often has a positive attitude about the task(s).

Occasionally is publicly critical of the project or the work of other members of the group. Usually has a positive attitude about the task(s).

Often is publicly critical of the project or the work of other members of the group. Often has a negative attitude about the task(s).

Focus on the task Consistently stays focused on the task and what needs to be done. Very self-directed.

Focuses on the task and what needs to be done most of the time. Other group members can count on this person.

Focuses on the task and what needs to be done some of the time. Other group members must sometimes nag, prod, and remind to keep this person on-task.

Rarely focuses on the task and what needs to be done. Lets others do the work.

Preparedness Brings needed materials to class and is always ready to work.

Almost always brings needed materials to class and is ready to work.

Almost always brings needed materials but sometimes needs to settle down and get to work

Often forgets needed materials or is rarely ready to get to work.

Monitors Group Effectiveness

Routinely monitors the effectiveness of the group, and makes suggestions to make it more effective.

Routinely monitors the effectiveness of the group and works to make the group more effective.

Occasionally monitors the effectiveness of the group and works to make the group more effective.

Rarely monitors the effectiveness of the group and does not work to make it more effective.

Working with Others

Almost always listens to, shares with, and supports the efforts of others. Tries to keep people working well together.

Usually listens to, shares, with, and supports the efforts of others. Does not cause "waves" in the group.

Often listens to, shares with, and supports the efforts of others, but sometimes is not a good team member.

Rarely listens to, shares with, and supports the efforts of others. Often is not a good team player.

Course: Project: Blacktop Battle

STUDENTS:______________________________________________

EVALUATOR: ____________________________ DATE: _______

CriteriaDeveloping/ Unsatisfactory

(Under Construction)

Emerging

(Making an Attempt)

Proficient

(Taking it to the next level)

Advanced

(Analyzing every component!)

Land Area Proposal

Does not mention any other total land area configurations beyond the selected choice, and min. and max. Concession Stand area options.

Does not provide Basketball Court Area (in sq. units)

Does not provide Bleacher Seating Areas (in sq. units)

Does not provide Concession Stand Area (in sq. units)

Does not provide Basketball Court Area, Bleacher Seating Areas, or Concession Stand Area (in sq. feet) – from Advanced column.

Does not provide equation for calculating total area.

Does not describe significance of area equation.

Does not compare and contrast the possible land area configurations generated.

Fails to compare and contrast configurations in square feet.

Does not take customer standing room into consideration when making configuration selection.

States/Shows possible land area configurations

States the Basketball Court area (in sq. units)

States the total area of Bleacher Seating (in sq. units)

States the Concession Stand Area for selected configuration (in sq. units)

States equation for calculating total area.

States methods which can be used for calculating total land area.

Describes how the Basketball Court Area (in sq. units) was calculated.

Interprets the methods used to calculate Bleacher Seating Areas (in sq. units).

Describes how the Concession Stand area (in sq. units) was calculated for selected configuration.

Describes significance of area equation (using area as a sum AND product)

Describes multiple methods which can be used for calculating the total land area.

In addition to meeting Proficient Criteria…

Team analyzes possible land area configurations (including dimensions and total area)

Compares and contrasts configurations in square feet.

Team reflects upon the ability for customers to have standing room for the Concession Stand.

10 - - - 12- - - 16- - - 20 - - - 24- - - 28 - - - 34

(30% - 68%)

1 point = 1 bullet point

35 - - - 36 - - - 38 - - -40

(70% - 80%)

1 point = 1 bullet points

42 - - - 44 - - -46 - - -48

(84% - 96%)

2 point = 1 bullet point

49 - - -49.5 - - - 50

(98% - 100%)

0.25 point = 1 bullet point