Thursday, April 22, 2021

Algebra 1 PARCC question: rational-irrational


The following constructed-response 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:

Pi, transcendental number, by Tom Blackwell via Flickr Creative Commons
Pi is an irrational number: the digits never end or repeat (Tom Blackwell via Flickr Creative Commons)

Two real numbers are defined as:

  • a = 0.444444444444 . . .
  • b = 0.354355435554 . . .

Determine whether each number is rational or irrational. Is the product of a and b rational or irrational?

Justify your answers.

Enter your answers and your justifications in the space provided.

Answer and references

Correct answers: a is rational, b is irrational, and the product of a and b is irrational.

PARCC evidence statement(s) tested: HS.C.2.1:

Base explanations/reasoning on the properties of rational and irrational numbers.

For rational solutions, exact values are required. For irrational solutions, exact or decimal approximations may be required. Simplifying or rewriting radicals is not required; however, students will not be penalized if they simplify the radicals correctly.

The evidence statement above references Math Practice 3 in the Common Core: Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. …

The question tests students’ understanding of the high school Common Core algebra standard HSN.RN.B.3, found under high school algebra (Number and quantity: the real number system), which states, under “use properties of rational and irrational numbers,” that they should be able to “explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.”

Example of a solution strategy (there are others)

Construct an argument to justify your answers.

To justify that a is rational, you need only state that it can be represented as an integer divided by another integer. That’s the definition of a rational number.

0.44\bar{4} = \frac{4}{9}

To justify that b is irrational, you need only state that it cannot be represented as an integer divided by another integer. It is a non-repeating, non-terminating decimal. Those are never rational.

When it comes to the product of a and b, you could show that the actual product of these two numbers, each truncated at the 12th decimal place, is a non-repeating, non-terminating decimal that can’t be represented as an integer divided by another integer.

0.444444444444 \times 0.354355435554 = 0.157491304690\textrm{...}

That doesn’t look like it’s ever going to end or repeat. But a better way to do it (although no algebra 1 test-taker on last year’s PARCC sample student responses did it this way) would be to show that, in the general case, the product of a nonzero rational number, such as a, and an irrational number is an irrational number.

(Anything times 0 is 0, which is a rational number, so that’s why we have to be clear that we’re talking about the product of the irrational number with a rational number that is not 0.)

On with the reasoning. If we assume R is a rational number not equal to 0 and S is an irrational number, then R can be represented as some integer divided by another integer. Let’s use p and q for our argument.

R = \frac{p}{q}

The irrational number S on the other hand cannot be represented as one integer divided by another number, because no irrational number can be represented that way, by definition.

Multiply R × S and we get

R S = \frac{p}{q}S

Now the trick. Ask yourself: Is there any possibility that the product can be a rational number? Whether there is or isn’t, we’re going to assert that it can be rational. That would mean it could be represented as one integer divided by another integer, let’s say u/v.

\frac{p}{q}S = \frac{u}{v}

Solve that for S and we get

S = \frac{u q}{v p}

That’s a problem, since the product of two integers, u and q, is an integer, as is the product of v and p. That means S is equal to one integer divided by another integer, which contradicts our initial assumption that S was an irrational number.

This is called a proof by contradiction, and it means that, in general, any rational number not equal to 0 (note that if v or p = 0, the fraction above is undefined) times an irrational number will result in an irrational number as the product.

Resources for further study

The Oswego City Schools in New York developed the Regents Exam prep center, courtesy of Lisa Schultzkie. It has a very nice page explaining the difference between rational and irrational numbers, here.

The Khan Academy, developed by Sal Khan, an engineer who has created a library of thousands of video lessons, has a landing page dealing with rational and irrational numbers. The site features more than a dozen lessons on the subject, including some interesting algebraic proofs.

Purple Math, developed by Elizabeth Stapel, a math teacher from the St Louis area, has a page or two, starting here, to explain the different types of numbers, including a lesson about differentiating rational and irrational numbers.

Analysis of this question and online accessibility

The question measures knowledge of the Common Core standard it purports to measure and tests students’ ability to explain why the product of an irrational number and a nonzero rational number is an irrational number, as well as identifying decimal representations of numbers as rational or irrational.

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 and may not be able to achieve full credit (3 points) for this question if they are too unfamiliar with the operation of the tool in the online test delivery system.

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

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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