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El Boleto Premiado ("The Winning Ticket")

Spanish Translation

Anoche camino a casa de la escuela, me paré en la tienda a comprar un boleto de lotería. Aunque normalmente no hago este tipo de cosa había sido un buen dia y decidí probar mi suerte. ¡¡Pues suerte tuve!! ¡¡Gané!! El único problema es que tuve que escoger entre dos premios: A) Mil dólares de golpe o B) Dos dólares el primer día, cuarto dólares el segundo día, ocho dólares el tercer día, dieciséis el cuarto día, y así por diez días seguidos.

¿Cúal premio crees que debería escoger? ¿Cómo llegaste a tu decisión? ¿Puedes probar que tu decisión es la mejor?

P.D. Si tu solución es lo suficientemente clara, ¡tal vez compartiré mi buena fortuna contigo!

English Translation

Last night on my way home from school I stopped at the store to buy a lottery ticket. Although I usually don't do this sort of thing, it had been a great day so I thought I'd try my luck. Well, lucky I was!! I won!! The only problem was I had to pick between two prizes: A) One thousand dollars all at once or B) Two dollars on the first day, four dollars on the second day, eight dollars on the third day, sixteen on the fourth day, and so on for ten days in a row.

Which prize do you think I should choose? How did you make your decision? Can you prove to me that your choice is the best choice?"

Context

A student had mentioned during our morning class meeting that his father's friend had won one thousand dollars on a scratch-off lottery ticket. This statement led us into a brief discussion about gambling, instant scratch games, and lottery tickets.

What this task accomplishes

This task requires children to solve several parts of a problem, discover and continue patterns, add large numbers, and make reasoned decisions.

What the student will do

Most students started off trying to discover the pattern in choice B. While most students discovered and continued the pattern, many of them did not solve the problem. Many students just looked at the amount received on the tenth day and figured it wasn't worth it to wait ten days. They did not take into account that the total amount of money received was the sum of the amounts received on each of the ten days.

Time required for task

1-2 45 minute periods

Interdisciplinary links

Games of chance, Language Arts: play on words (many tickets have clever names that have more then one meaning). Students could figure out the double meanings of current games, or create their own game names and submit them to the lottery commission for consideration!

Teaching Tips

In order to adapt this task to make it easier or more complicated, the pattern or money amounts that are used can be changed to be made simpler or more complex. This problem provides a good opportunity to make sure students know how to make accurate tables or charts.

Concepts to be assessed and skills to be developed

Concepts of Whole Number Operations

Developing meaning for operations through problem solving.
Relating mathematical language to problem situations.
Developing operation sense.

Whole Number Computation

Developing reasonable proficiency with basic facts and algorithms.
Using calculators in appropriate computational situations.
Selecting and using computational techniques to determine if results are reasonable.

Patterns and Relationships

Recognizing, describing and extending patterns.
Representing and describing mathematical relationships.

Mathematics As Problem Solving

Using problem solving approaches to investigate and understand mathematical content.
Developing and applying strategies to solve everyday problems
Verifying and interpreting results with respect to the original problem.
Acquiring confidence in using mathematics meaningfully

Mathematics as Communication

Reflecting on and clarifying thinking about mathematical ideas and situations.
Relating everyday language to mathematical language and symbols.
Relating mathematical ideas to accurate representations.

Mathematics As Reasoning

Drawing logical conclusions about mathematics.
Using known facts and mathematical relationships to explain thinking.
Justifying answers and solution processes.
Using patterns and relationships to analyze mathematical situations.
Believing that mathematics makes sense.

Suggested Materials

Graph paper, calculators, money manipulatives, markers

Possible Solutions

The amount of money received in choice B is $2,046.00. This amount is the sum of the amount received each day. Each day for ten days, the recipient receives twice as much as s/he did the day before.

Task Specific Rubric/Benchmark Descriptors Click on a level for student example.
Novice	This student shows no understanding of the task. There is no evidence of conceptual understanding, no evidence of mathematical reasoning, no relevant explanation, and no math language or representation.
Apprentice	This student clearly understands part of the problem. S/he recognizes the pattern and accurately applies a strategy of doubling the amount of money received the previous day. This is solid evidence of mathematical reasoning. However, this student does not seem to understand that the amount of money received on day 10 is not the same as the total amount of money received after the ten days. There is some appropriate use of math language and representation.
Practitioner	This solution shows a broad understanding of the problem. The student uses an effective strategy that leads to a correct solution. The student has a brief yet clear explanation, some correct notation, and uses a key to link her/his computation to representation.
Expert	This student shows a deep understanding of the task and clearly identifies the concepts and information needed for the solution. S/he applies procedures correctly, uses a clear and effective explanation, and uses math representation correctly.

Author

Amy Morse teaches a multi-age 3-4 at the Warren Elementary School in Warren, Vermont. She has a Master's degree in curriculum and instruction from the University of Vermont. Amy is a consultant for the Vermont Portfolio Assessment Program.