Sunday, August 1, 2021

Grade 8 PARCC math: scientific notation


The following multiple-choice question, explained here in hopes of helping eighth-grade students and their parents 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 2016 test for grade eight math:

Laurie entered the mass, in kilograms, of four substances into a spreadsheet. Her spreadsheet automatically converted the masses into scientific notation:

table with 2 columns and 4 rows, first column has a heading of substance, second column has a heading of mass in kilograms, first row, substance A, 2.45 E negative 4, next row, substance B, 6.8 E negative 3, next row, substance C, 7.125 E negative 5, last row, substance D, 9.0 E negative 4

Which list shows the four substances in order from least mass to greatest mass?

A. Substance A, Substance B, Substance C, Substance D
B. Substance B, Substance A, Substance D, Substance C
C. Substance C, Substance A, Substance D, Substance B
D. Substance C, Substance D, Substance A, Substance B

Answer and references

Correct answer: (C) Substance C, Substance A, Substance D, Substance B.

Common Core Math Content 8th grade, Expressions and Equations, Expressions and Equations Work with radicals and integer exponents

(8.EE.A.3) Use numbers expressed in the form of a single digit times a whole-number power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 × 108 and the population of the world as 7 × 109, and determine that the world population is more than 20 times larger.

(8.EE.A.4) Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.

Solution strategy (there are others)

Convert each number in the Mass column to standard form and put in order.

Using standard form will make the values easier to see, and in this case, it’ll just create numbers with a few zeroes (the exponents have a small enough absolute value).

  • 2.45 × 10–4 becomes 0.000 245 00
  • 6.8 × 10–3 becomes 0.006 800 00
  • 7.125 × 10–5 becomes 0.000 071 25
  • 9.0 × 10–4 becomes 0.000 900 00

With scientific notation where the exponent on the 10 is negative, the exponents with the smallest absolute value actually go with the bigger numbers. So, in the case of B and C, we can see the –3 exponent on B makes the number bigger than C with an exponent of –5 (or any of the others), and C is the substance with the least mass.

As for A and D, which have the same exponent (–4) on the 10, we then look at the number in front of the multiplication sign (or, in this problem, before the E), and rank them normally. Since 2.45 is smaller than 9.0, A is smaller than D. And both of these are in between B and C.

Analysis of this question and online accessibility

Scientific notation is one of 29 topics that are new to eighth graders in the Common Core math standards, according to researcher Jessica K Griffin at the College at Brockport. It is our hope, as well as hers, that students take the information their teachers present about scientific notation to learn not just how to put numbers in order from least to greatest, but also why scientific notation is helpful to scientists and other people who work with very large or small numbers.

When Voxitatis used Microsoft Excel to show 0.0009 in scientific notation with one decimal place, we got 9.0E–04. This differs from the format shown in PARCC’s math problem in that the spreadsheet program omitted the space before the “E” and a zero is shown in between the negative sign and the number 4. The text of the problem makes no reference to the specific spreadsheet program used, though, so maybe PARCC used a different spreadsheet program.

But then, the problem as published shows a different format for most cells. Three of the four differ in terms of the number of decimal places shown. This would require setting the format on each cell individually, which I suppose is possible, but the “automatic conversion” told to the students by the text of the problem does not reflect the operation of any spreadsheet program currently available on the commercial market.

Voxitatis diligently checked six independent references for information about scientific notation:

Only NYU even lists the spreadsheet notation on the same page with “scientific notation.” NYU refers to the spreadsheet notation not as “scientific notation” but by saying that “exponents are often expressed using other notations.” The other math references universally say “scientific notation” refers only to something like this:

9.0 \times 10^{-3}

Voxitatis concludes, therefore, based on a preponderance of the evidence, as well as typical worksheets used by eighth-grade teachers across the country, that the question is invalid in that it does not measure students’ understanding of the Common Core standard it purports to measure. If you can’t do what the problem says you should do and get what the problem says you should get, the question itself is invalid and does not test students’ ability to “interpret scientific notation that has been generated by technology.”

(The labeling of the second row from the top as “1” is incorrect, given the functionality of commercially available spreadsheet programs. This is considered a minor or even an irrelevant editorial error, however, which affects neither students’ ability to answer the question nor the fitness of the question for the test.)

A mitigating factor in our analysis comes as we acknowledge the likelihood that eighth graders would understand the spreadsheet notation used in the problem and convert it, in their heads, to the scientific notation PARCC pretends to assess with this problem.

If the fault, however, lies with the wording of the Common Core, which constantly refers only to “scientific notation” and not “spreadsheet” or “calculator” notation, then the problems PARCC presents to students should reflect the actual spreadsheet notation a student would see. If, on the other hand, the fault lies with technology companies that coin a new meaning for an existing mathematical term, we as a society shouldn’t compromise our understanding of the underlying mathematics in order to accommodate spreadsheet manufacturers who are either too lazy to program or too uninformed to comprehend “scientific notation.”

Multiple-choice questions like this one can be delivered easily online for any device students may use. Validity, reliability, and fairness measures should not differ significantly from those found when the question is used on a paper-based test from PARCC.

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

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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