Responses to the Darts, Anyone? Problem
The problem appearing in the May 1999 "Problem Solvers" section was stated as follows:
Ted and Kate were making a dart game for their children. They wanted to have three rings marked with number values. Ted made a board like the one shown. When three darts are thrown, what is the lowest score possible? What is the highest score possible? How many different point totals are possible using three darts?
Kate did not like this dart board because it permitted only ten point totals. That is, using three darts, a person could score 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9 points. Show how each total can be obtained with three darts.
Kate wondered whether it was possible to label the rings so that a player could reach a higher score with three darts. She wondered whether all the numbers from 1 to 10--or even higher--were possible. What suggestions would you give for labeling the rings?
The student work submitted for Darts, Anyone? demonstrates that children frequently need work on the understanding-the-problem phase of problem solving. Although it was easy for students to determine what scores were possible on the given dart board, clarification and discussions were necessary for them to understand whether it was possible to design the board to allow scores greater than 9 and still be able to obtain all the numbers, beginning at 1.
Most of the students in Gina Simpson's fifth-grade, Kathy Pickett's fourth-grade, and Amy Benedict's sixth-grade classes were able to show that the given dart board would allow all the scores from 0 to 9 (fig. 1). Furthermore, many students Showed multiple ways that a sum could be obtained. As might be expected, a few students did not adhere to the three-darts constraint. Other students either ignored the constraints of the problem or just made up number combinations that added to a given number. For example, some students with the numbers 0, 1, 5, 8 on the dart board showed 4 + 4 + 0 = 8 as a way to obtain 8. (See fig.2.)
Benedict noted, "Most students could verbalize why the 1 was needed." Knowing that a 0 region and a region with 1 are necessary means that only two more numbers can be used to find a "better" arrangement than the one given.
Benedict also stated, "I asked the students to write a note to Kate suggesting numbers to use and why she should use them." (See fig. 3.) Although the students did not produce a full list of possible dart boards, many students did find other combinations that were better than the one given. (See fig. 4.) The fact that students were able to find several possible solutions demonstrates that they are able to pursue alternatives to the problem. However, no individual or group was able to show all the possible solutions and provide a rationale.
A brief description of the reasoning to find a better dart board follows. As Benedict's students noted, finding all possible solutions is made easier by noting that 0 and 1 must be used. In addition, we can also discount possibilities in which numbers repeat, for instance, (0, 1, 1, 3), since a repeated number does not add potential scores. …