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The task
A clothing store offers a 50% discount at the end of each week that an item remains unsold. Patrick wants to buy a shirt at the store and he says. “I’ve got a great idea! I’ll wait two weeks, have 100% off, and get it for free!” Explain to your friend Patrick why he is incorrect and find the correct percent of discount on the original price of a shirt.

Circumstances of performance
This sample of student work was produced under the following conditions:
- alone in a group
- in class as homework
with teacher feedback with peer feedback
- timed opportunity for revision
with manipulatives with calculator

This work sample illustrates a standard-setting
performance for the following parts of the standards:
f Number and Operation Concepts: Compare numbers using order relations, differences, ratios, proportions, percents, and proportional change.
g Number and Operation Concepts: Carry out proportional reasoning.
b Problem Solving and Mathematical Reasoning: Implementation.
c Problem Solving and Mathematical Reasoning: Conclusion.
d Problem Solving and Mathematical Reasoning: Mathematical Reasoning.
a Mathematical Skills and Tools: Carry out numerical calculations effectively.
d Mathematical Communication: Communicate logical arguments clearly, showing sensibility and validity.
e Mathematical Communication: Present mathematical ideas effectively.

Mathematics required by the task
The relationship that underlies a correct interpretation of the discount offer is:
(price at the end of n weeks) = P (0.5)
(P = original price)

After 2 weeks this yields the price (0.25)P, which is one quarter of the original price not zero.
Notice that the incorrect interpretation that Patrick had amounted to this:
(price at the end of n weeks) = P(1 - 0.5n)
(P = original price)

After n = 2 weeks this does yield a zero price.

What the work shows

f Number and Operation Concepts: The student compares numbers using percents.
The student correctly used percent and converted it to dollars.

g Number and Operation Concepts: The student carries out proportional reasoning in cases...involving...contractions.

b Problem Solving and Mathematical Reasoning: Implementation. The student…
• chooses and employs effective problem solving strategies in dealing with non-routine and multi-step problems;
• …uses mathematics to model real world situations….

The student used an example to disprove Patrick’s conjecture. The student applied the successive discount to show that the price of the shirt was not zero dollars after two weeks.

c Problem Solving and Mathematical Reasoning: Conclusion. The student…concludes a solution process with a useful summary of results….
The student concluded the process by stating that Patrick will not get the shirt for free.
The student wrote $30, which represents 25% of the original price, therefore concluded the correct amount of discount.

d Problem Solving and Mathematical Reasoning: Conclusion. The student employs forms of mathematical reasoning including…using counter examples.
The student disproved the conjecture by showing the shirt would never be free but would cost $30 at the end of two weeks.

a Mathematical Skills and Tools: The student carries out numerical calculations….
The student determined that $60 is 50% of $120 and that $30 is 25% of $120.

d Mathematical Communication: The student communicates logical arguments clearly, showing why a result makes sense and why the reasoning is valid.
The student specifically explained why the reasoning is valid and gave numerical answers.

e Mathematical Communication: The student presents mathematical ideas effectively…in writing.
The student “explaination” [sic] clearly stated the reasoning that the shirt would never be free.

The errors of punctuation and spelling do not detract from the overall quality of the work.