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The task
Shopping Carts
In this task you are asked to think mathematically about shopping carts. You are asked to create a rule that can be used to predict the length of storage space needed given the number of carts.

The diagram below shows a drawing of a single shopping cart.

It also shows a drawing of 12 shopping carts that have been “nested” together.

The drawings are accurately scaled to the real size.

  1. Create a rule that will tell you the length S of storage space needed for carts when you know the number N of shopping carts to be stored. You will need to show how you built your rule; that is, we will need to know what information you drew upon and how you used it.

  2. Now create a rule that will tell you the number N of shopping carts that will fit in a space S meters long.
The diagram, as reproduced here, is 45% as large as the original task prompt the students worked from.

About the task
This task is designed to see if students can recognize the proportional relationship inherent in this situation (the increase in the length of a nested row of carts is proportional to the number of carts added) and express it in terms of a linear formula or function.

Once students have completed the task as given, it is natural to ask them to look for other examples (in the real world), of structures which, similar to a row of nested shopping carts, can be represented by linear functions of the form y = A + b n.

In their examples, y, A, and b should have a clear geometric meaning that they identify, and n should represent the number of identical components in the structure. Their examples can be represented in a diagram similar to the shopping carts diagram.

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:
k Geometry and Measurement Concepts: Work with geometric measures of length.
n Geometry and Measurement Concepts: Solve problems involving scale.
a Function and Algebra Concepts: Model given situations with formulas and functions.
f Function and Algebra Concepts: Use linear functions as a mathematical representation of proportional relationships.
h Function and Algebra Concepts: Manipulate expressions involving variables.
b Problem Solving and Mathematical Reasoning: Implementation.
l Mathematical Skills and Tools: Use tools in solving problems.
c Mathematical Communication: Organize work and present mathematical procedures and results clearly, systematically, succinctly, and correctly.

Mathematics required by the task
There are two relevant lengths in this task, the full length (call it L) of a single cart, and the amount (call it d) that each new cart in a row sticks out beyond the others. Since the drawing is accurately scaled to th full size, L and d can be found by measuring the drawing and multiplying by 24.

Each new cart added to a row adds the fixed amount d to the length of the row. This means that the length S of a row of carts is a linear function of the number n of carts in the row, and that the slope of this function is d. Since the full length of a single cart is L, this function can be written as:

S = L + d (n-1).

Using the full-size measurements in centimeters of L and d for the shopping cart pictured, the function is:

S = 96 + 28.8 (n-1).

The reason n-1 appears in this formula instead of n is that the contribution of the first cart is contained in the number L. A way of writing the function using n instead of n-1 is:

S = (L-d) + d n = 67.2 + 28.8 n.

It is important to note that the function here is discrete: it is meaningful in this context only for the natural numbers n = 1, 2, 3,…. In particular, n = 0 gives a result, S = L-d, which has no direct meaning in this context; (it would mean the length of a row of 0 carts).

What the work shows

k Geometry and Measurement Concepts: The student works with geometric measures of length….
n Geometry and Measurement Concepts: The student solves problems involving scale…in…
diagrams.

The student recognized the two lengths needed to work the problem, measured them from the diagram, and used the given 1 to 24 scale of the diagram to convert these to full size measurements.
The student said the two answers arrived at were “fairly close.” To be more precise, the answers given in the response agree to two significant digits. Actually, it would have made sense to limit all numbers in the work to two significant digits. After all, the measurements used were made from a small diagram and could not be very accurate.

a Function and Algebra Concepts: The student models given situations with formulas and functions….
The student was clear about interpreting the mathematics in terms of the situation, for example, by saying “The 96 is the length of the first cart and the 28.8 (n-1) is the length added by all the additional carts after the first.”

f Function and Algebra Concepts: The student…uses linear functions as a mathematical representation of proportional relationships.
The student created a simple formula that describes the given situation, and that shows that the length of a row after the first cart is proportional to the number (n-1) of carts after the first.

h Function and Algebra Concepts: The student…manipulates expressions involving variables…in work with formulas, functions, [and] equations….
Having expressed the length S in terms of the number n of carts, using the formula
S = 0.96 + 0.288 (n-1), the student re-expressed this formula to express the number n in terms of the length S. In the language of the student: “…let’s convert the equation.”
This formula should indicate in some way that n must be an integer, perhaps simply saying that any non-integer result must be rounded down to the nearest integer.

b Problem Solving and Mathematical Reasoning: Implementation. The student chooses and employs effective problem solving strategies in dealing with…non-routine problems.
In a situation that was unfamiliar the student chose and applied appropriate mathematics that closely modeled the situation.

l Mathematical Skills and Tools: The student uses tools such as rulers…in solving problems.
The student recognized the two lengths needed to work the problem, measured them from the diagram, and used the given 1 to 24 scale of the diagram to convert these to full size measurements.

c Mathematical Communication: The student organizes work and presents mathematical procedures and results clearly, systematically, succinctly, and correctly.
The student presented an orderly approach to the problem, explained the steps of the solution process clearly and concisely, and arrived at a result that is correct.