Monday, October 18, 2021

Algebra 1 PARCC question: predict from data


The following multiple-droplist question, explained here in hopes of helping algebra students in Maryland and Illinois prepare for the PARCC test near the end of this school year, appears on the released version of PARCC’s Spring 2015 test in algebra 1, here:

A quality-control technician at a candle factory tested eight 16-ounce candles, each 3 inches in diameter. These candles came from the same production run. The table shows the decrease in weight of each candle after burning for 3 hours. Candle makers believe that the rate at which the candles burn is constant.

Candle 1 2 3 4 5 6 7 8
Weight loss (ounces) 0.5 0.6 0.5 0.7 0.7 0.5 0.5 0.6

Write an equation that can be used to model the weight, w, of such a candle as a function of the number, h, of hours burning. Then, explain how the equation can be used to predict the weight of a candle that has burned for 5 hours.

Enter your equation and your explanation in the space provided.

Answer and references

Correct answers: w = 16 – (0.2)(h), with appropriate rounding. 15.0 ounces. Explanations and any math work are human-scored.

PARCC evidence statement(s) tested: HS.D.3-1:

Micro-models: Autonomously apply a technique from pure mathematics to a real-world situation in which the technique yields valuable results even though it is obviously not applicable in a strict mathematical sense (e.g., profitably applying proportional relationships to a phenomenon that is obviously nonlinear or statistical in nature). … Reasoned estimates: Use reasonable estimates of known quantities in a chain of reasoning that yields an estimate of an unknown quantity.

The evidence statement above references mainly Math Practice 4 in the Common Core: Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. … By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

The question tests students’ understanding of the high school Common Core high school modeling standards, which are descriptive in nature and don’t describe discrete or isolated knowledge of skills. For example, “A model can be very simple, such as writing total cost as a product of unit price and number bought, or using a geometric shape to describe a physical object like a coin. Even such simple models involve making choices. It is up to us whether to model a coin as a three-dimensional cylinder, or whether a two-dimensional disk works well enough for our purposes.”

Example of a solution strategy (there are others)

Determine the average rate of weight loss for a burning candle of this type; use it to predict how much a candle will decrease in weight after burning for five hours.

Assuming the candles used for the technician’s investigation were randomly selected from a representative sample of all candles made by this factory, we can just find the mean weight loss for these eight candles and assume, despite not being told anything about the sampling method used in the investigation for which data are reported, that other candles in the factory will burn at about the same rate.

The mean is an appropriate measure of central tendency to use for this data, since there are no extreme outliers apparent upon inspection of the data. If there were outliers that would have a significant effect on the mean, another measure of central tendency might be more appropriate.

\frac{0.5 + 0.6 + 0.5 + 0.7 + 0.7 +  0.5 + 0.5 + 0.6}{8} = 0.575

The average burning candle from this factory, then, loses 0.575 ÷ 3 ounces per hour (the table showed the weight lost by each of the eight candles after three hours of burning, so we need to divide that by 3 to get an hourly rate). That’s 0.191667 ≈ 0.2 ounces per hour. To write that in an equation, as the problem requests, we have to begin with the starting weight, 16 ounces, and subtract the weight loss per hour:

w = 16 - (0.2)(h)

To predict how much weight a candle will lose in five hours, just plug in the numbers.

w = 16 - (0.2)(5) = 16 - 1 = 15

An average candle can be predicted to lose 1 ounce of weight in five hours. Since the starting weight of the candles is 16 ounces, the average weight of candles that have been burning for five hours can be predicted to be 15 ounces, unless there was some flaw in the sampling method used by the technician.

Resources for further study

Finding the average from a set of data is not a high school mathematics skill or an algebra 1 skill. But developing a mathematical model that requires you to use the mean of a set of data and interpret the investigation performed by the technician is. I therefore refer you to some discussion of selecting an appropriate modeling technique, which comes from an eighth-grade math class in the UK, developed by Australian author GS Rehill, here.

The Khan Academy, developed by Sal Khan, an engineer who has created a library of thousands of video lessons, has an entire series devoted to the basics of statistics and how to use them appropriately. For mean, median, and mode, see the video here, and use the navigation at the left to move around to different lessons in the series.

Analysis of this question and online accessibility

The question does not test students’ understanding of the Common Core standard it purports to measure; it tests students’ ability develop a model that may or may not be appropriate, given the insufficiency of information presented, to describe a real-world situation and to use that model in predicting future behavior. Students are required to do this, despite the fact that the context as explained in the text of the problem fails to warrant the use of the model that PARCC provides as the correct answer. The question is considered to have a median cognitive demand.

By leaving out important information about the investigation conducted by the technician and the actual source of the data collected and reported, it’s illogical to assume students can use any mathematical model at all to describe the data or to predict future behavior of the candles. The problem setup fails to align to the modeling standard in the Common Core.

The question can be tested online and should yield results that are as valid and reliable as those obtained on paper. Students online may experience difficulties using the equation editor to show their math in determining the slope. We have previously described problems with the user interface of the equation editor tool (see here), and we will not repeat that argument here. However, we will use the opportunity to remind those of you taking the PARCC math test to spend a little extra time on problems with the equation editor to:

  • Make sure you have typed in all the work necessary to make your case
  • Transfer all your work from scratch paper to the computer so you can receive credit for it

No special accommodation challenges can be identified with this question, so the question is considered fair.


Nine geography students received the following scores on a test.

Student Number 1 2 3 4 5 6 7 8 9
Test Score (%) 47 35 67 32 38 39 36 34 35

It was a really hard test. Assuming all students have about the same level of understanding of geography, predict the score for the 10th student on the test, and explain why you chose to use the model you did.

Purpose of this series of posts

Voxitatis is developing blog posts that address every algebra 1 question released to the public by the Partnership for Assessment of Readiness for College and Careers, or PARCC, in order to help students prepare to take the test this spring.

Our total release will run from February 27 through March 15, with one or two questions discussed per day. Then we’ll move to geometry at the end of March, algebra 2 during the first half of April, and eighth grade during the last half of April.

Paul Katula
Paul Katula is the executive editor of the Voxitatis Research Foundation, which publishes this blog. For more information, see the About page.

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