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### The Taxman

 Develop strategies based on systematic analysis Present convincing arguments Use uniform introduction for all classes Generalize from specific cases

Suggested time allotment

One class period

Student social organization

Working alone or in pairs, following a videotaped introduction

Assumed background: This task assumes that children are familiar with the concept of factors of whole numbers and, in particular, with prime numbers. It also assumes that they have had some experience in developing multi-step strategies and in articulating those strategies coherently.

Presenting the task: The problem is to analyze a game that we assume is unfamiliar to the children. Hence the first task for the teacher is to introduce the rules of the game. One way of doing this is to show a videotape in which a teacher shows a small group of

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Measuring Up: Prototypes for Mathematics Assessment The Taxman Develop strategies based on systematic analysis Present convincing arguments Use uniform introduction for all classes Generalize from specific cases Suggested time allotment One class period Student social organization Working alone or in pairs, following a videotaped introduction Task Assumed background: This task assumes that children are familiar with the concept of factors of whole numbers and, in particular, with prime numbers. It also assumes that they have had some experience in developing multi-step strategies and in articulating those strategies coherently. Presenting the task: The problem is to analyze a game that we assume is unfamiliar to the children. Hence the first task for the teacher is to introduce the rules of the game. One way of doing this is to show a videotape in which a teacher shows a small group of

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Measuring Up: Prototypes for Mathematics Assessment children how the game of Taxman is played. The script for such a videotape is given below. (If a tape is not available, the classroom teacher can use the script as a guide to introducing the game.) The scene opens in a classroom setting, with a teacher and a group of six students. There is a chalkboard on one wall. Teacher: Today I'm going to show you how to play a new number game, called Taxman. The game is played with a list of numbers starting with the number one. For example, the Taxman game with six numbers would start with this list: [Teacher writes list on chalkboard] There are two players, You and the Taxman. Every time it is your turn, you can take any number in the list, as long as at least some factors of that number are also in the list. You get your number, and the Taxman gets all of the factors of that number that are in the list. For example, if you take 4, … Student A: … then the Taxman would get 2! Teacher: Why? Student A: Because 2 is a factor of 4. Student B: The Taxman'd get 1, too. Student C: Oh, yeah, because 1's a factor of 4. Teacher: So if you took 4, then the list would look like this: [writes] Now you have 4 points … Student D: … and the Taxman has 3 points. [Teacher writes]

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Measuring Up: Prototypes for Mathematics Assessment Teacher: So, the first rule of the Taxman game is: 1. The Taxman must get something every time. That means you can't choose if there aren't any factors of that number still in the list. The only other rule is this: 2. When none of the numbers in the list has any factors left in the list, then the game is over, and the Taxman gets all the numbers that are left in the list. Student E: I don't get the second rule. Teacher: Let's play a game together to see what this second rule means. We'll use the same list: What would you like to start with? Student F: 6. [Teacher gives chalk to Student F, who crosses off the "6" and records 6 points for You.] Student A: So the Taxman gets 3 and 2 [crosses them off the list] … Student B: … and 1 [crosses off 1 and updates the score]. Student F: So now you could take 5. Student E: No you can't! 5 doesn't have any factors that are left on the list. Student F: Oh, OK. The 1's not there any more. How about 4?

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Measuring Up: Prototypes for Mathematics Assessment Student D: That won't work either. The factors of 4 are 1 and 2, and they're both gone already. Student F: Oh, I see. So the game is over? Teacher: Right. At this point, the only the numbers left are 4 and 5. Neither of these has any factors that are still in the list. So the game is over, and the Taxman gets both the 4 and the 5: [writes] Student C: So You lost — 6 to 15. Student B: I bet we can do better than that. Teacher: Well, let's see. We'll start with our list again: [writes] Student F: How about if we start with 5 this time? [Student F crosses off the 5] Student B: Hah! The Taxman only gets 1. [Crosses off 1 and records scores] Student A: Now take 6. Student D: No, wait a minute. If you do that, the Taxman'll get 3 and 2. Student E: And then you won't be able to take the 4 … Student D: … 'cause the 2 and 1'll be gone. Student A: So take the 4 now.

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Measuring Up: Prototypes for Mathematics Assessment Student C: Yeah, that'd be better. Then the Taxman'll get only the 2. Student F: And you'll still be able to take the 6. Teacher: Wait a second. I don't follow you. So what do you want to do now? Student B: Take the 4, so the Taxman gets the 2. [Crosses off and records scores.] Student E: And now you can choose the 6 because the 3 is still left for the Taxman. [Crosses off and records scores.] Student D: So this time, You won, 15 to 6! Teacher: Do you think you could ever do better, starting with the list from 1 to 6? Student A: I don't think so. … Student C: I'm positive there's no way to do any better than that. Teacher: How do you know? Student C: Well, look. Every time you play, the Taxman has to get something, right? So that means … (Fade to black)

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Measuring Up: Prototypes for Mathematics Assessment Following the viewing of the videotape, let students play a game of Taxman using more than six numbers. In small groups they can play Taxman-8: Together the group should choose the numbers to claim for You; then one student removes that number and updates the score, while another member of the group removes the factors and updates the Taxman's score. Remind the students that the Taxman must always get something. Student assessment activity: See the following pages. A Spanish translation is provided after the English version.

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Measuring Up: Prototypes for Mathematics Assessment Name ________________________________________ Date _____________ Play several games of Taxman with these ten numbers: Make a record of your best game. Be sure to show which numbers you took and the order in which you took them, not just the final score. Then answer the following questions: 1. Did you beat the Taxman? ________ 2. What number did you choose first? ________ Why? 3. Do you think anyone could ever play a better game than your best game? ________ Explain why or why not. 4. Suppose you were going to play Taxman with the whole numbers from 1 to 95. What number would you choose first? ________ Why?

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Measuring Up: Prototypes for Mathematics Assessment Nombre ________________________________________ Fecha _____________ Juega varios partidos de el Cobrador de Impuestos con estos diez números: Mantín un registro de tu mejor partido. Asegúrate de escribir los números que escogiste y en el orden en que los escogiste, y no solamente tu puntuación final. A medida que investigas el juego de el Cobrador de Impuestos con los diez números, contesta las siguientes preguntas: 1. ¿Puedes ganarle al cobrador de impuestos? ________ 2. ¿Qué número escogerías primero? ________ ¿Por qué? 3. ¿Crees que alguien pudiera jugar un partido mejor que el tuyo? Explica por qué sí o por qué no. 4. Imaginate que vas a jugar al Cobrador de Impuestos con los números del 1 al 95. ¿Qué número escogerías primero? ________ ¿Por qué?

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Measuring Up: Prototypes for Mathematics Assessment Rationale for the mathematics education community The Taxman task provides a good example of a mathematical situation in which systematic analysis involving several steps is critical, an important characteristic of mathematical power. The task also presents an opportunity for children to generalize what they have learned beyond the particular instance; e.g. analyzing Taxman-10 to describe a strategy for Taxman-95. At the same time, the task asks students to construct convincing arguments about their strategy. These steps are all essential facets of mathematical proofs. Moreover, the task is a situation in which reasoning with prime numbers, composites, and factors plays a vital role; it is up to the children, however, to determine how these ideas should be applied to analyzing the game. It is worth noting that the Taxman game has no pretense of applications to the ''real world" in the usual sense; it is a purely abstract, without grounding in some physical context. But games are a significant aspect of a fourth grader's real world and can be motivational in and of themselves. To ensure that children from different classrooms will get exactly the same introduction to the game, the use of a videotaped introduction is recommended. Uniformity of presentation is important if the results of one classroom are to be compared fairly with results from another. The widespread availability of videocassette recorders in classrooms now makes it feasible to introduce problem settings that would previously have been too complex to replicate reliably from one classroom to another. Certain features of the Taxman task — and other potential assessment tasks — make a uniform introduction even more important. With a videotape, everyone will have the same experiences in exploring optimal and non-optimal choices. A teacher who introduces the games in his or her classroom without the videotape might never consider an opening move of 6, for example, simply because no one in the class happened to offer it as a choice during the explanation.

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