The task
Students were given worksheets from Algebra Experiments, Book 1: Exploring
Linear Functions. The worksheets guide students through an experiment
with analysis to determine the relationship between the volume of a container
and the length of time a candle will burn when covered by it. Students
must gather, record, and graph data in the coordinate plane, determine
an equation and characteristics of a line, and use the equation of the
line to determine one coordinate of a point on the line given the other
coordinate. The questions, while they lead the student through the mathematics,
are informative and instructional.
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
This work sample illustrates a standard-setting
performance for the following parts of the standards:
b
Function and Algebra
Concepts: Represent relationships.
d
Function and Algebra
Concepts: Find solutions for unknown quantities in linear equations.
a
Statistics
and Probability Concepts: Collect, organize, and display data.
a
Mathematical Skills
and Tools: Compute accurately with arithmetic operations on rational
numbers.
c
Mathematical Skills
and Tools: Estimate numerically.
f
Mathematical Skills
and Tools: Use equations, formulas, and simple algebraic notation
appropriately.
g
Mathematical Skills
and Tools: Read and organize data on charts and graphs.
What the work shows
b
Function and Algebra Concepts: The student represents relationships
with…verbal or symbolic rules.
The determination of slope and y-intercept provides strong evidence
that the student can manipulate algebraic expressions and equations
of lines.
dFunction and Algebra Concepts: The student
finds solutions for unknown quantities in linear equations….
This is
a good interpretation of the slope of the line. The meaning ascribed
to the y-intercept suffices only if detached from the scientific
context of this task. The intercept is small (near zero), but it
would be reasonable to expect the line to pass through the origin
(0,0)—no volume, no time! The task’s
use of the phrase “best fit” is undefined here. Presumably,
this phrase means “best fit to the naked eye,” which is
appropriate enough detail of this linear regression idea for middle
school students. The student did not use two data points, as instructed,
but the approach was sensible.
aStatistics and Probability Concepts: The
student collects data, organizes data, and displays data with tables…and
graphs that are appropriate….
aMathematical Skills and Tools: The student
computes accurately with arithmetic operations on rational numbers.
c Mathematical Skills and Tools: The student
estimates numerically…. After collecting
data over three trials for each container, the student determined
the “time to extinguish [the] candle” by choosing the
intermediate value from each set of three trials. She did not compute
an exact average, as encouraged by the task. Instead she opted to
approximate the time needed to extinguish the candle by choosing
an integer between, and often close to, two of the three trial values.
She did this for each container, with one curious exception. This
method of choosing intermediate values for this specific task is
as appropriate as the ones offered in the instructions.
fMathematical Skills and Tools: The student
uses equations, formulas, and simple algebraic notation appropriately.
gMathematical Skills and Tools: The student
reads and organizes data on charts and graphs, including scatter
plots,…[and] line… graphs…. In the data
collection table, the student correctly assigned names and units
of measurement to her independent and dependent variables. Here,
she incorrectly used only units of measurement. A corrected entry
would read something like this:
“time (in sec.) = 0.032 · size (in ml) + 2.” Because
the air is at room temperature and standard pressure, measuring
the air in ml is equivalent to measuring in . The 960,002
seconds are not appropriate units for this question. The student
should have converted this length of time into units that would
give the answer more meaning, e.g., “a little over 11 days.” The trouble
with this claim is that smaller volumes will not result in longer
times. The data already suggest specific extinguishment times for
candles in covered spaces of small and large volumes.
The determination of slope and y-intercept provides strong evidence
that the student can manipulate algebraic expressions and equations
of lines.
This is
a good interpretation of the slope of the line. The meaning ascribed
to the y-intercept suffices only if detached from the scientific
context of this task. The intercept is small (near zero), but it
would be reasonable to expect the line to pass through the origin
(0,0)—no volume, no time!
The task’s
use of the phrase “best fit” is undefined here. Presumably,
this phrase means “best fit to the naked eye,” which is
appropriate enough detail of this linear regression idea for middle
school students. The student did not use two data points, as instructed,
but the approach was sensible.
In the data
collection table, the student correctly assigned names and units
of measurement to her independent and dependent variables. Here,
she incorrectly used only units of measurement. A corrected entry
would read something like this: 
.
The 960,002
seconds are not appropriate units for this question. The student
should have converted this length of time into units that would
give the answer more meaning, e.g., “a little over 11 days.”
The trouble
with this claim is that smaller volumes will not result in longer
times. The data already suggest specific extinguishment times for
candles in covered spaces of small and large volumes.
