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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.
- 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.
- 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.
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.
| 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
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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. |
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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).
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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.”
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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.
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