 |
|
|
Winning Ticket
Task
Last night on my way home from school I stopped at the store to buy a lottery ticket. Although I usually do not do this sort of thing, it had been a great day so I thought I would try my luck. Well, lucky I was! I won! The only problem was, I had to pick between 2 prizes: A) $1,000 all at once or B) $2 on the 1st day, $4 on the 2nd day, $8 on the 3rd day, $16 on the 4th day and so on for 10 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?
P.S. If your solution is clear enough, maybe I will share my good fortune with you!
Alternate Versions of Task
| More Accessible Version:
Last night on my way home from school I stopped at the store to buy a lottery ticket. Although I usually do not do this sort of thing, it had been a great day so I thought I would try my luck. Well, lucky I was! I won! The only problem was, I had to pick between 2 prizes: A) $2,000 all at once or B) $2 on the 1st day, $4 on the 2nd day, $8 on the 3rd day, $16 on the 4th day, $32 on the 5th day, $64 on the 6th day, $128 on the 7th day, $256 on the 8th day, $512 on the 9th day and $1024 the 10th day.
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?
P.S. If your solution is clear enough, maybe I will share my good fortune with you!
More Challenging Version:
Last night on my way home from school I stopped at the store to buy a lottery ticket. Although I usually do not do this sort of thing, it had been a great day so I thought I would try my luck. Well, lucky I was! I won! The only problem was, I had to pick between 2 prizes: A) $1,000 all at once or B) $2 on the 1st day, $4 on the 2nd day, $8 on the 3rd day, $16 on the 4th day and so on for 10 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?
What if the prize was A) one million dollars, or B) $2 on the 1st day, $4 on the 2nd day, $8 on the 3rd day, $16 on the 4th day, and so on? When does prize B become better than prize A?
P.S. If your solution is clear enough, maybe I will share my good fortune with you!
|
Context
A student had mentioned during our morning class meeting that his father's friend had won $1,000 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 10th day and figured it was not worth it to wait 10 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 10 days.
Time Required for Task
One to two, 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 made simpler or more complex. This problem provides a good opportunity to make sure students know how to make accurate tables or charts.
Suggested Materials- Graph paper
- Calculators
- Money manipulatives
- Markers
Possible Solutions
The amount of money received in choice B is $2,046. This amount is the sum of the amount received each day. Each day for 10 days, the recipient receives twice as much as s/he did the day before.
| More Accessible Version Solution:
She should choose choice two which totals $2046, which is $46 more than choice one.
More Challenging Version Solution:
Prize B becomes better than prize A on day 19.
|
Task Specific Assessment Notes
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 10 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 his/her 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.
|
|
|
|
