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Open-ended Template

 

Names:  Becky Spurlock, Ellen Skradski, Lori Pawluck, Vince Bondi

 

Grade level:  9-12                                                        Content Area:  Mathematics

 

PA standard(s) addressed:              NCTM standard(s) addressed:

 

Probability and Predictions                                                           Data Analysis and Probability

 

 

Problem Name:  Gratuitous Grade Grubber

 

Problem:  A geometry student was very unhappy with the grade of a C on a recent test on finding areas.  The student approached the mathematics teacher to see if it was possible to take a retest to improve the grade.  Instead the teacher proposed a game to play in which the student could get either an A or an F.  In the game, the player must spin the pointer two times on the board illustrated below.  If after those two spins the accumulated area of the figures shown is greater than thirty square cm., the student will get an A.  If that area is not greater than thirty, the student will receive an F for the test.  Should the student play the game?

 

 

Directions:  Using the diagram showing the board game described in the statement above, decide whether this student should play this game.  Show all of the work involved in your calculations as well as a complete explanation for your procedure.  Also, explain why you make the decision for your final response to the question.

 

 

 

 

 

 

 

SOLUTION – 9/12 PROBABILITY

 

Step 1:  We need to find the area of each figure.

 

A)  Square:  When the diagonal is drawn, two 45-45-90 triangles are formed.  Since the length of the diagonal is 6, the formula to find the length of a side is h = lÖ2      

H = 6 so 6 = l  Ö2     , l = 6/  Ö2     , l = 3  Ö2       .

      The formula for the area of a square is A = side squared, therefore A = ( 3 Ö2)^2 = 18 square centimeters.

 

B)                 Triangle:  This is an equilateral triangle.  We need to find the height of the figure since the area formula is A = ½ bh.  The two smaller triangles formed are 30-60-90 triangles.  The side opposite the 30-degree angle is 2 (or ½ of 4).  Therefore, The height is 2 Ö3.  So A = ½ (4)(2Ö3) = 4 Ö3 approximately6.93 square centimeters.

 

C)  Trapezoid:  The formula for area of a trapezoid is A = ½ h ( base 1 + base 2).  We need to find the height.  When the height is drawn a right triangle is formed.  Since the top base is 4 and the bottom base is 10, the side of the triangles on the bottom base becomes 3 ( ( 10-4)divided by 2).

      Next, the height is 4 because the Pythagorean theorem is used.  5^2 = 3^2 + x^2.

      Now , A = ½ (4) ( 10+4) = 28 square centimeters.

 

 

 

 

Step 2: Combinations

 

Two spins are permitted. The following 9 combinations with area may result:

 

SQUARE     SQUARE     18 + 18 = 36

SQUARE     TRIANGLE     18 + 4Ö3 = approximately 24.93

SQUARE     TRAPEZOID     18 + 28 = 46

TRIANGLE     TRIANGLE     4Ö3 + 4Ö3 is approximately 13.86

TRIANGLE     TRAPEZOID     4Ö3 + 28 is approximately 34.93

TRAPEZOID     TRAPEZOID     28 + 28 = 56

TRIANGLE    SQUARE     4Ö3 + 18 is approximately 24.93

TRAPEZOID     SQUARE     28 + 18 = 46

TRAPEZOID     TRIANGLE     28 + 4Ö3 is approximately 34.93

 

 

 

 

 

Step 3:  Probability

 

The total areas are listed in order so that we can see which are below 30 and which are above:  13.86,  24.93,  24.93,  34.93,  34.93,  36,  46,  46,  56.

Note that 6 of the 9 areas have sums above 30 sq units.  Therefore, the probability of getting an area sum greater than 30 is 6/9 or 2/3.

This represents a better than even chance that the student will win the game. 

 

Note that the decision to play the game or not is up to the student.  Some students may decline because they believe that the chance of winning is less than desired.  For instance, they may prefer at least ¾ chance of winning.

 

 

 

 

 

RUBRIC – 9-12 PROBABILITY/PREDICTIONS

 

5                    ADVANCED UNDERSTANDING – EXCELLENT

 

Response includes correct numerical answers:

A)     Areas:  square = 18 sq cm

Triangle = 4Ö3 = 6.93 sq cm

Trapezoid = 28 sq cm

B)      Sample Space:  9 combinations possible.  S = square,  E = equilateral triangle, T = trapezoid.  Combinations:  SS, SE, ST, EE, ET, TT, ES, TS, TE,

C)      Combined areas from sample space:  36, 24.93, 46, 13.86, 34.93, 56, 24.93, 46, 34.93 sq cm.

D)      Probability (area sum>30) = 6/9 = 2/3

 

All work shown and fully explained.

Explanation includes why steps were performed.

Nothing incorrect (all quantities have appropriate labels).

 

4                    SATISFACTORY UNDERSTANDING

 

Response includes correct numerical answers.

All work shown and/or explained.

Some explanation required.

Explanation of 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.

 

3                    ALMOST SATISFACTORY UNDERSTANDING

 

A)      Response includes correct answers.

All work shown with no explanation.

Work may contain minor blemishes.

                                Or

B)       Response includes correct answers.

Adequate work shown or explained ( some steps are missing but you can follow what is being done – all appropriate mathematical procedures are used.)

                                                               Or

C)       Response includes incorrect numerical answer due to one calculation,

copying, or rounding error.

Adequate work shown and/or explained.

 

2                    PARTIAL UNDERSTANDING

 

A)      Response includes correct numerical answers.

Some work and/or explanation.

Appropriate mathematical procedures used.

                                 Or

B)       Response includes incorrect numerical answer due to more than one

calculation or copying errors.

Adequate work shown or explained.

                                 Or

C)       Response includes procedural error (finding incomplete sample space).

Or

D)      Response shows correct answers to two parts with some work/explanation.

                                                             i.      Correct solution with minimal work or some explanation.

                                                            ii.      Incorrect solution due to multiple calculations errors (incorrect squaring of radicals when computing area).

                                                          iii.      Incorrect answer due to a conceptual error such as the failure to recognize special right triangles, area formulas, or a complete sample space.

                                                          iv.      Finding 3 out of 5 solutions and values correct within rounding, some work and explanation, may contain a calculation error.

                                                           v.      Find 2 out of 5 correct solutions with all work shown, explanation and sketch.

 

1                     MINIMAL UNDERSTANDING

 

A)     Correct solution with no work or explanation.

B)      Calculations of the 3 areas are correct only.

C)      Two out of 5 correct answers (no explanation and no work but formulas).

D)      One correct answer and an attempt at another part (sample space = 6). 

 

0                     OFF TASK OR NO ATTEMPT MADE

 

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