The key requirement in the problem statement
is “Explain how much gold you can carry out, and how long it
will take to do it.” Since the amount of gold starts at 1 pound
on the first trip, and is cut in half for each successive trip,
finding the quantity of gold that can be carried out amounts to
finding this sum:
1 +
+
+
+
+….
If this is approached as a practical problem, all that is required
is to compute these terms until they get too small to be practical,
then add them up. “Too small to be practical” might be
interpreted as “too small to be represented on a calculator.”
On a calculator with an 8digit display, this happens after about
25 terms of this series. That is,
comes out as 0.0000001 on the calculator, while
comes out as 0.0000000. When all the terms up to
are added up, the sum comes to 1.9999999, but when all the terms
up to
are added, the sum comes to 2.0000000. All these quantities are
in pounds. (To treat these issues fully, we would need to address
questions such as how many digits are stored but not shown on a
calculator.)
As a practical problem, then, the answer is that a little less than
2 pounds can be carried out, and that after about 25 minutes the
2 pound figure has almost been reached, and the amounts to be carried
out per trip are probably too small to measure. In fact, after just
8 minutes more than 1.99 pounds can be taken out, as can be determined
by summing the terms up to .
In summary, the mathematics required to work the task as this sort
of practical problem is an organized application of arithmetic:
taking powers, reciprocals, and summing.
A mathematically more powerful solution would use the summation
of, say, the first 20 terms of the geometric series with factor
.
This sum is
by a formula often developed in high school texts. Such an approach
would give evidence of g
(Function and Algebra Concepts: Uses…geometric sequences and
their sums…). The student work samples shown here did not actually
make use of such a summation formula.
a
Number and Operation Concepts: The student
uses addition…multiplication, and division…in forming
and working with numerical and algebraic expressions.
c
Number and Operation Concepts: The student
has facility with the mechanics of operations as well as understanding
of their typical meaning and uses in applications.
e
Number and Operation Concepts: The student
represents numbers in decimal…form….
h
Number and Operation Concepts: The student
understands…numbers with specific units of measure, such as
numbers with…rate units.
a
Mathematical Skills and Tools: The student
carries out numerical calculations…effectively, using…pencil
and paper, or other technological aids, as appropriate.
The
table is organized by its first column, the trip number. The second
column is the weight of gold taken out on the trip with that number,
the third column is a running sum of the weights in the second column,
and the last column gives the value in $ of the gold taken out up
to that point. This is formed by multiplying the weight in ounces
given in the third column by the cost of gold per ounce ($350).
The
table is organized by its first column, the trip number. The second
and third columns give the weight in ounces of gold taken out on
the trip with that number, the fourth column is a running sum of
the weights in the second column, and the last column gives the
value in $ of the gold taken out up to that point. It is formed
by multiplying the weight in ounces given in the fourth column by
the cost of gold per ounce ($350).
A
further step appears in the text of the response, though not in
the table; the hourly rate earned at various stages is figured by
dividing the value in $ of the gold taken out by the time in hours
up to that point.
The
student has expressed the 1 pound weight as 16 ounces. This is sensible,
since it means that the numbers obtained in the repeated halving
are larger and hence easier to work with: 16, 8, 4, 2, 1, ,…,
as opposed to 1, ,
, ,….
Note
the misprint here:
should be .5, as it is in the table.
b
Number and Operation Concepts: The student
understands and uses operations such as…
reciprocal.
Notice
that the student explicitly used reciprocals of powers of 2 (,
, ,…)
in representing the repeated halving operation.
h
Number and Operation Concepts: The student
understands…numbers with specific units of measure, such as
numbers with…rate units.
Here
the weight has been converted to its monetary value using the given
price of $350 per ounce. The hourly rate of earnings has
been figured by dividing the monetary value by the time in hours
(first converting 15 minutes to 0.25 hours, etc.).
Notice that
the largest total the student found was 32 ounces, but that there
is no justification given that the total could not go higher. A
justification would require showing that 32 is the sum of the geometric
series with first term 16 and common factor .
The student interpreted the problem in practical terms, and the
references to amounts that are eventually “immeasurable”
or “so small” refer to practicalities, not to the mathematics
of the situation. The student did not deal with the issue of whether
the amounts 32 oz. and $11,200 actually would be reached
on the 26th trip, or whether these are figures that have been rounded
up. See the discussion above in “Mathematics required by the
task.”
The largest
total the student found was 31.999999 ounces, and there is a remark
that “further calculations are useless.” Still, there
is no justification given that the total could not go higher. A
justification would require showing that 32 is the sum of the geometric
series with first term 16 and common factor .
There is
a misprint here. Line 20 should be $11,199.989.
g
Function and Algebra Concepts: The student
uses…geometric sequences.
The student
recognized that the weights on successive trips form a geometric
sequence.
b
Mathematical Communication: The student uses
mathematical representations with appropriate accuracy, including
numerical tables….
e
Mathematical Communication: The student presents
mathematical ideas effectively…in writing.
Both students wrote coherent explanations of the steps taken to
solve the problem and produced
clearly labeled tables.
