Distribution Dilemma
Task
Steve was hired to give out free movie tickets to customers during the grand opening of the new pizza parlor. On Friday, he gave out 1/2 of his supply; on Saturday, he gave out 1/3 of what was left; on Sunday, he gave out 1/4 of the remaining amount; on Monday, he distributed 1/3 of what was left; on Tuesday, he gave away 1/6 of what remained and had 60 tickets left.
How many tickets did he have when he began to give them away?
Alternate Versions of Task
| More Accessible Version
Steve was hired to give out free movie tickets to customers during the grand opening of the new pizza parlor. On Friday, he started with 120 free tickets to give away, and he gave out 1/2 of them; on Saturday, he gave out 1/3 of what was left; and on Sunday, he gave out 1/4 of the remaining tickets.
How many of the 120 tickets were not given away?
More Challenging Version
As part of a sales promotion for his new pizza business, Steve decided to give away movie coupons to customers who ordered large pizzas during his grand opening. Steve contacted the local theater who agreed to sell him movie coupons at half their coupon’s value. Movie tickets typically cost $5.00.
On Friday, he gave out 1/2 of his supply that consisted of free movie coupons. On Saturday, he gave out 1/3 of the remaining supply that consisted of 50% off coupons. On Sunday, he gave out 1/4 of the remaining supply that consisted of 25% off coupons. On Monday, he gave out 1/3 of the remaining supply that consisted of 20% off coupons. On Tuesday, he gave out 1/6 of the remaining supply that consisted of 10% off coupons. Steve then had 60 coupons left that he returned to the movie theater.
How many coupons did he have when he began to give them away? How much money did he have to reimburse the movie theater for the coupons?
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Context
I was looking for a task that would encourage students to improve their mathematical representations. We have worked on visual models for problem solving this year and I hoped they would use such modeling in solving this task.
What This Task Accomplishes
This task requires the recognition of fractions and using common denominators to add and subtract fractions. It also provides the opportunity to use a visual model for problem solving. Students will encounter many problems of this type and having a workable strategy for this type of problem will serve them well.
What the Student Will Do
I had the students work alone on this task and encouraged them to work on their mathematical representations. Some students drew rectangles and divided them to represent each day's distributions. Other students worked backwards, starting with the 60 tickets that remained.
Time Required for Task
45 minutes
Interdisciplinary Links
There are no obvious interdisciplinary content links for this task. This problem-solving technique could be used to solve tasks written to complement any content area - historical topics, science units or literature connections.
Teaching Tips
We had studied area models of fraction representation before attempting this task. I had graph paper available to students. They are better able to subdivide rectangles using graph paper than using blank paper. They use a grid square to represent one unit more easily and accurately than estimating one unit on blank paper.
Being familiar with working backwards as a strategy will also help students experience success with this task.
Suggested Materials- Grid paper
- Calculators
- Counting pieces of any sort (for acting out the problem)
Possible Solutions
Steve distributed 432 tickets in all. See Expert benchmark for details of solution.
| More Accessible Version Solution
120 x 1/2 = 60 given away with 60 left
60 x 1/3 = 20 given away with 40 left
40 x 1/4 = 10 given away with 30 left
More Challenging Version Solution
Steve distributed 432 tickets in all. See Expert benchmark for details of solution.
Steve began with 432 coupons.
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Task Specific Assessment Notes
Novice:
This student had the right idea with the rectangular area model. The fractional subdivision is accurate. The student fails to remember the fact that the final eight units represent 60 tickets rather than eight tickets. Since there is no attempt to verify the solution, the error is not picked up and the solution is incorrect. Willingness to invest more time in this activity would have undoubtedly rectified the error.
Apprentice:
This student understood what was to be done to solve the problem. There appears to be no particular reason for the choice of a "12 x 12" box for the model. When 60 tickets remain in a space containing 20 unit squares, the student fails to recognize the connection to three tickets/square and instead tries to subdivide the rectangle into 20 square subdivisions. This strategy unfortunately falls apart as the student attempts to deal with the extra 11 squares. Had the student checked the work with computation, the error would have been discovered.
Practitioner:
This student understands the task, uses good mathematical representations and good mathematical language. The solution is accurate and well documented with diagrams. Solid mathematical reasoning was used throughout and a connection to past mathematical experiences is mentioned - though not expanded upon enough. Providing some form of verification of the solution would move this response to the Expert level.
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Expert:
This student worked backwards to find the solution. S/he worked both computationally and with a visual model to verify the solution. Confident use of mathematical language appears throughout the solution. Mathematical representations are clearly labeled (and color coded) and work to verify the solution. This student goes on to an extension, which adds depth to the solution. This clever extension gives the opportunity to demonstrate further mathematical understanding.
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