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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.

d Function 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.

a Statistics and Probability Concepts: The student collects data, organizes data, and displays data with tables…and graphs that are appropriate….

a Mathematical 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.


f Mathematical Skills and Tools: The student uses equations, formulas, and simple algebraic notation appropriately.

g Mathematical 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.