Thursday, September 28, 2023

Algebra 2 PARCC: imaginary number arithmetic


The following fill-in-the-number question, explained here in hopes of helping algebra 2 students and their parents in Maryland prepare for the PARCC test near the end of this school year, appears on the released version of the PARCC Algebra 2 sample test released following the 2016 test (#5).

In the table shown, i represents the imaginary unit. Select all cells in the table for which the product of the row value and the column value is a real number.

Value –3 –2i 5 i2
i __ __ __ __
–4 __ __ __ __

Correct answer and references

Correct answer: Real numbers are –3, 5, and i2 × –4, but only –2i × i.

The question is aligned to Common Core High School Math standard HSN (Number and Quantity) » CN (The Complex Number System) » A.2, which requires that algebra 2 students be able to “use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.”

Sample solution strategy (there are others)

Replace every i2 with the real number –1 and figure out if an i remains in the product.

Let’s consider each product one at a time. For each result, if an i remains in the product, the number is not a real number and would not be checked in the table above.

For the first row, we plan to multiply i × –3, –2i, 5, and i2.

(i)(-3) = -3i \textrm{ ... complex}
(i)(-2i) = (-2)(i^2) = (-2)(-1) = 2 \textrm{ ... real}
(i)(5) = 5i \textrm{ ... complex}
(i)(i^2) = (i)(-1) = -1i \textrm{ ... complex}

For the second row, we plan to multiply –4 × –3, –2i, 5, and i2.

(-4)(-3) = 12 \textrm{ ... real}
(-4)(-2i) = (8)(i) = 8i \textrm{ ... complex}
(-4)(5) = -20 \textrm{ ... real}
(-4)(i^2) = (-4)(-1) = 4 \textrm{ ... real}

Analysis of this question and online accessibility

The question can be delivered online or on paper. However, because students have to make multiple selections, a multiple-choice presentation might have slightly different performance statistics from the online version of the item.

The question is aligned to the Common Core math standard to which it purports to align, but the use of the word “cell” to describe a table element may make the question less accessible for some students who have a limited proficiency in English, as this word has multiple meanings with very different translations into whatever may be their native language. These errors are considered minor, since the interaction is clear, but a better word choice would have made the question more accessible to the broader population.

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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