Scoring Guide for the Farmer Problem

The total score will be the sum of the 5 scores below. (maximum score = 20)

POINTS

CORRECT ANSWER - See chart following this section

4

2 or more correct answers are given (see answer table below)

3

1 correct answer is given (see answer table below)

0

No correct answer is given (not in the table below)

 

POINTS

PROBLEM UNDERSTANDING
Understand the patterns, concepts, and approach

4

  • Notices that the area covered one sack of each vegetable is a fraction of the acre (corn=1/3,potatoes = 1/4, squash = 1/5, beans = 1/6, carrots = 1/12), or (conversely) that an whole number of sacks of each vegetable covers 1 acre (corn = 3 sacks,potatoes = 4 sacks, etc.).
  • Notices at least one other pattern. Several patterns that can be noticed are listed at the bottom of this rubric.
  • Understands that the total area covered by all sacks of vegetables must equal 43,560 sq. ft.

3

  • Understands that the total area covered by all sacks of vegetables must equal 43,560 sq. ft.
  • Notices that a whole number of sacks of each vegetable covers 1 acre.

2

  • Understands that the total area covered by all sacks of vegetables must equal 43,560 sq. ft.

0

No attempt at response, or misunderstood problem.

 

POINTS

STRATEGY
Choose and apply a valid strategy (Some valid strategies are given in the table below.)

4

Choice of a valid strategy: visual representation, make a table, guess and check, or analyze using fractions
Applied the strategy correctly and completely

2

A good strategy was chosen, but was partially or incorrectly completed. For example, they made a table but the table had errors.

1

Strategy did not apply to this problem.

0

No attempted strategy

 

POINTS

COMMUNICATION
Explain the reasoning at each step, using pictures, symbols, labels, and correct vocabulary

4

Described strategy and solution clearly and completely, step-by-step
Correct presentation of solution, including appropriate labels, terminology, and symbols. For example, if they made a table, the table has column descriptions.

2

Strategy was described, but steps may be missing
Labels may be missing or directions may not have completely followed

1

Strategy was not explained according to directions or reasoning is confusing.

0

No attempt to explain

 

POINTS

REASONABLE RESULT
Review the work and show why the result is reasonable

4

Review used a different approach to support the result
Explained why the result makes sense.

3

The problem was reworked using the same approach to check its accuracy.

2

The check for reasonableness was not complete, or they said something like "we checked our result."

0

No attempt was made to demonstrate the reasonableness of the result



POSSIBLE STRATEGIES:
ANALYZE USING FRACTIONS The team that uses this approach abstracts the problem into fractions of an acre, and then solves the problem by using fraction arithmetic.
MAKE A TABLE Made a table with labelled columns showing vegetable, number of sacks, total area for this vegetable, and totals at the bottom.
MAKE A PICTURE Made a picture breaking the acre up into 12ths (the common fraction of 1/3,1/4,and 1/6).
GUESS-AND-CHECK Randomly guessed at numbers of sacks of various vegetables and stumbled upon a valid result. This is a valid but time-consuming strategy for this problem.







Here are all 17 possible solutions to the Farmer Problem:

Coverage=> (1/3)   (1/4)     (1/5)    (1/6)    (1/12)
	    acre    acre      acre     acre     acre
Sol.       #Sacks  #Sacks    #Sacks   #Sacks   #Sacks 
No.        Corn    Potatoes  Squash   Beans    Carrots
 1           0        1        0        1        7        
 2           0        1        0        2        5        
 3           0        1        0        3        3        
 4           0        1        0        4        1        
 5           0        2        0        1        4        
 6           0        2        0        2        2        
 7           0        3        0        1        1        
 8           1        0        0        1        6        
 9           1        0        0        2        4        
10           1        0        0        3        2        
11           1        1        0        0        5        
12           1        1        0        1        3        
13           1        1        0        2        1        
14           1        2        0        0        2        
15           1        2        0        1        0        
16           2        0        0        1        2        
17           2        1        0        0        1        


Patterns that teams may notice:
  1. There is a whole number of sacks of each vegetable that cover 1 acre.(3 sacks of corn, 4 sacks of potatoes, etc.)
  2. One sack of each vegetable covers a fraction of the whole acre. (corn covers 1/3 acre, potatoes cover 1/4 acre, etc.) (This is the first pattern restated).
  3. No solution uses squash! That is because 5 doesn't have a common factor with any of the other denominators. The requirement that they must plant at least 3 different vegetables means that if any squash is used, the other 2 vegetables must add up to some number of fifths. With thirds, fourths, sixths and 12ths this is not possible if the acreage is limited to 1. If the requirement were to plant 2 acres, then squash could be used (5 sacks of squash, 2 sacks of potatoes, 1 sack of corn and 1 sack of beans, for example.)
  4. You can choose any 2 vegetables other than carrots (or squash), (for example corn and potatoes), pick any number of sacks of each that do not add up to 1 (for example 2 sacks of corn and 1 sack of potatoes) and then the difference can always be made up with carrots, (add 1 sack carrots) because the common denominator of 1/3 and 1/4 and 1/6 is 12 (the denominator of the carrot sack coverage).