The task
How Much Gold Can You Carry Out?

 A vault contains a large amount of gold and you are told that you may keep as much as you can carry out, under the following conditions: On the first trip you may only take one pound. On each successive trip you may take out half the amount you carried out on the previous trip. You take one minute to complete each trip. Explain how much gold you can carry out, and how long it will take to do it. Also, determine your hourly rate of earnings if you work only fifteen minutes. Use the current value of gold, \$350 per ounce. What would be your hourly rate if you work for twenty minutes? What if you worked for an entire hour?

This task, in different variants, is commonly seen in classrooms. It is designed to show how very rapidly a quantity shrinks if it is halved over and over. Similar tasks show how very rapidly a quantity grows if it is doubled over and over. These tasks illustrate exponential decay and growth. The fanciful context makes the task memorable. Still, the
context is easily stripped away to get at the underlying mathematics required to answer the questions.

 Circumstances of performance These samples of student work were produced under the following conditions: - alone in a group in class - as homework with teacher feedback with peer feedback timed opportunity for revision with manipulatives - with calculator

Sample 1

 Sample 2 Sample 2
 These work samples illustrate standard-setting performances for the following parts of the standards: a Number and Operation Concepts: Use addition, multiplication, and division in forming and working with numerical and algebraic expressions. b Number and Operation Concepts: Understand and use opposite, reciprocal, raising to a power, taking a root, and taking a logarithm. c Number and Operation Concepts: Have facility with the mechanics of operations as well as understanding of their typical meaning and uses in applications. e Number and Operation Concepts: Represent numbers in decimal form. h Number and Operation Concepts: Understand numbers with specific units of measure, such as numbers with rate units. g Function and Algebra Concepts: Use arithmetic sequences and geometric sequences and their sums, and see these as the discrete forms of linear and exponential functions, respectively. a Mathematical Skills and Tools: Carry out numerical calculations effectively. b Mathematical Communication: Use mathematical representations with appropriate accuracy. e Mathematical Communication: Present mathematical ideas effectively.

Sample 2 Translation

Sample 2 was completed in Russian by a bilingual student. The translation was provided by the teacher.

 Mathematics required by the task 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 8-digit 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. What the work shows 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.