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Names:

Suzanne Brignole, Bethlehem Area Vocational Technical School

Melodee Brosious, Shikellamy High School

David Taylor, South Fayette Township School District

Lois Winslow, Connellsville Area High School

 

Grade Level:  11

 

Content Area:  Geometry

 

PA Standard(s) Addressed:  2.9 - Geometry, 2.10 - Trigonometry

 

NCTM Standard(s) Addressed:  Geometry Standard

 

Problem Name:  Court Comparisons

 

 

Problem:

 

The town of Riverview has two basketball courts, one at their high school and one at their local college.  The high school Geometry teacher asked his class to make a comparison of the two courts.  One group of students discovered that the length of the college basketball court was 10 feet longer than the length of the high school court.  A second group of students was surprised to find that the widths of the two courts were the same.  Another group computed the areas of both courts and found there was a difference of 720 square feet.  The last group found that a diagonal of the high school basketball court measured 120 feet.

 

A.                 Accurately sketch and label the high school and college basketball courts with the given information.

B.                 Determine the width of each court.

C.                 Find the area of each basketball court.  Be sure to consider all given information.

 

Show your work and explain the steps you used to justify your answer.  Do all work for this problem in the box below.  Remember you must show all the steps you used to solve the problem even if you used a calculator.  To receive the highest score, all calculations and steps must be shown and explained in writing.  Numeric answers must always be labeled.

 

For full credit, you must do the following:

 

  1. show OR describe each step of your work, even if you did it in your head (“mental math”) or used a calculator,

 

AND

 

  1. write an explanation stating the mathematical reason(s) why you chose each of your steps.

 

Problem Solution:

 

Part A

l + 10
 

w
 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 


Part B

 

Algebraic Solution:

 


The formula for finding the area of a rectangle is                                           .  Since the difference in the areas is 720 square feet, the equation to find the width of the courts will show that the area of the high school court is smaller than the college court.

 

 

or

 

 

or

 

Non-Algebraic Solution:

 

College Court

High School Court

Difference in Areas

Length

Width

Area

Length

Width

Area

80

40

3,200

70

40

2,800

400

90

60

5,400

80

60

4,800

600

110

80

8,800

100

80

8,000

800

95

60

5,700

85

60

5,100

600

105

70

7,350

95

70

6,650

700

105

71

7,455

95

71

6,745

710

30
 

2,160

20

1,440

720

105
 

7,560

95

6840

720

 

 

 

 

 

 

 

 

 

 

 

Part C

 

Algebraic Solution:

 

To find the areas of the high school and college basketball courts, the length of each court must be found.  Since the diagonal of the high school court is known, we have a right triangle where two of the three sides are known.  The Pythagorean Theorem must be used to find the length of the high school court.  While the student may be able to use a table as above to find lengths that are 10 units apart with areas that have a difference of 720 square units, there is only one length that solves the right triangle below.

 

To find the lengths of the sides of a right triangle, the Pythagorean Formula can be used.  The Pythagorean Formula is .

 

 

 

or

 

 

or

 

 

Since the length of the college court is 10 ft. longer, the length of the college court must be 106 ft.

 


To find the areas of each court, use the formula for the area of a rectangle which is                                             .

 

College
 

High School
 


Specific Rubric:

 

5

Advanced Understanding – Excellent

The response includes:

·         Diagram drawn and labeled accurately.

·         Correct numerical answers with labels (width = 72 ft., area of college court = 7,632 sq. ft., area of high school court = 6,912 ft.)

·         All work shown AND fully explained.

·         All quantities have appropriate labels.

·         Must use Pythagorean Theorem.

 

4

Satisfactory Understanding

The response includes:

·         Diagram drawn and labeled accurately.

·         Correct numerical answers (one label may be missing).

·         Explanation why steps were performed may be incomplete, unclear, or might explain what is being done, but an attempt is made.

·         Work and/or explanation may contain minor blemishes.

·         Must use Pythagorean Theorem.

 

3

Almost Satisfactory Understanding

A.     The response includes:

·         Diagram drawn and labeled accurately.

·         Correct numerical answers (labels may be missing).

·         All work shown with no explanation.

·         Work may contain minor blemishes.

 

OR

 

B.     The response includes:

·         Diagram drawn and labeled accurately.

·         Correct numerical answers (labels may be missing).

·         Some procedures shown and/or explained (some steps are missing but you can follow what is being done).

 

OR

 

C.     The response includes:

·         Diagram drawn and labeled accurately.

·         Incorrect answers with correct procedures and some explanation but with one calculation or copying error carried through.

 

2

Partial Understanding

A.     The response includes:

·         Incomplete diagram.

·         Correct numerical answers.

·         Few procedures/calculations shown or described.

·         Some explanation.

·         Too many steps missing to follow what is being done.

 

B.     The response includes:

·         Incomplete diagram.

·         Incorrect numerical answers.

·         Half or more correct procedures shown.

·         Some or no explanation.

 

C.     The response includes:

·         Incomplete diagram.

·         Incorrect answers.

·         Correct procedures.

·         No explanation.

·         No more than two calculation or copy errors.

 

1

Minimal Understanding

The response includes:

·         Any work appropriate to any part of the problem.

 

0

Incorrect

The response includes:

·         Off task.

·         Blank.

·         Inappropriate responses.

 

 

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