Algebra 1 PARCC question: area of rug equation

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:

Tonya has a rectangular rug with an area of 21 square feet. The rug is 4 feet longer than it is wide.

Part A

Create an equation that can be used to determine the length and width of the rug. Justify your answer.

Enter your answer and your justification in the space provided.

Part B

Tonya adds a 1.5-foot border all the way around the rug. What is the area of the enlarged rug? Show all your work.

Enter your answer and your work in the space provided.

Answer and references

Correct answers: Part A: You can use the system l = w – 4 and l × w = 21. Part B: 60 ft². Both Part A and Part B are human-scored.

PARCC evidence statement(s) tested: HS.D.2-5:

Solve multi-step contextual word problems with degree of difficulty appropriate to the course, requiring application of course-level knowledge and skills articulated in A-CED, N-Q, A-SSE.3, A-REI.6, A-REI.12, A-REI.11-1, limited to linear equations and exponential equations with integer exponents.

A-CED (A-CED.4-2, e.g., rearrange formulas that are quadratic in the quantity of interest to highlight the quantity of interest, using the same reasoning as in solving equations) is the primary content; other listed content elements may be involved in tasks as well.

The evidence statement above references Math Practice 2 and Math Practice 4 in the Common Core:

[2] Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

[4] 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 algebra standard HSA.CED.A.1, found under high school algebra (creating equations), which states that they should be able to “Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.”

Example of a solution strategy (there are others)

Part A: Use the area formula for a rectangle, A = w × l.

Set the longer dimension of the rectangle to the variable w, the area of the rectangle to 21, and the shorter side of the rectangle to w – 4.

$\begin{array}{rcl} A = w (w-4) & = & 21 \\ w^2 - 4w - 21 & = & 0 \end{array}$

I can factor the left side by noting that –7 × 3 = –21 and –7 + 3 = –4.

$(w-7)(w+3) = 0$

Once we get to this point, we know from the multiplicative property of 0 that if a × b = 0, then either a = 0, b = 0, or both a and b = 0.

If (w–7) = 0, then w is 7, and if (w+3) = 0, then w is –3. The width can’t be less than zero, so the –3 isn’t useful in the context of the problem. That leaves a width of 7 feet. The other dimension is 4 feet less than that, so 3 feet.

Part B: Use the area formula for a rectangle, A = w × l.

The new dimension of the rug, with a 1.5-foot border added on each of the four sides is (7 + 1.5 + 1.5) feet by (3 + 1.5 + 1.5) feet.

$A = (10)(6) = 60$

Resources for further study

Purple Math, developed by Elizabeth Stapel, a math teacher from the St Louis area, has a six-part series on solving quadratic equations by factoring, as I did above. The series starts here.

The Khan Academy, developed by Sal Khan, an engineer who has created a library of thousands of video lessons, has a series of video lessons that demonstrate how to factor quadratics, starting here. He starts by factoring x² – 14x + 40 as (x–4)(x–10), even though the text on his site says the second factor is (x–1). Despite the minor error in marketing, which would be unforgivable if done on the PARCC test, the video’s fine. Mr Khan clearly understands completely how to factor quadratics, even if he typed it incorrectly on his site.

Chapter 4, Section 4.4, of the book Algebra 2, Illinois edition by Ron Larson et al deals with solving quadratic equations by factoring. Students are trained to spot the structure in polynomial expressions of the general form

$ax^2 + bx + c = (kx+m)(lx+n) = klx^2 + (kn+lm)x + mn$

where k and l must be factors of a and m and n must be factors of c.

Complete reference: Ron Larson, Laurie Boswell, Timothy D Kanold, Lee Stiff. Algebra 2, Illinois edition. Evanston, Ill.: McDougal Littell, a division of Houghton Mifflin Company, 2008. The book is used in several algebra classes taught in Illinois high schools.

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 create an equation to model a real-world situation. The knowledge of the area of a rectangle is both given to students on a formula sheet and considered securely-held knowledge, and so that is not tested here, just used. It is considered to have a median cognitive demand.

(We note that most ninth graders can probably figure out what the length and width of the original rug is in their heads. What number, when multiplied by a number that is 4 less, gives 21? That’s a fairly simple logic they can do by trial and error or “guess and check.” Try 6: 6 × 2 = 12; nope. Try 7: 7 × 3 = 21; bingo! But Part A assesses the Common Core skill cited here, which is students’ ability to construct a mathematical model for a real-world situation. Just getting the 7 and the 3 will earn a point out of 6 total given to this problem. The bulk of the points come from the mathematical modeling—coming up with the equation.)

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 with the equation editor, as the use of this online tool is required to receive full credit in both Parts A and B.

If students are unfamiliar with the tool—which requires them to enter math work in paragraph form by selecting math symbols from a series of drop-down palettes and does not in any way resemble the way they would do the work if given a pencil and paper—their score will be in jeopardy. Typos are not forgiven in the PARCC scoring rubrics, and students are advised to take a little extra time when using the equation editor tool to make sure they have

Entered all the work or logic necessary
Transferred all work from scratch paper to the computer

(I realize using the equation editor is difficult, but you’re not alone. And while people at PARCC are trying to figure out how to make this tool usable for an online test, that’s not going to happen anytime soon. If PARCC people were here, they would be apologizing profusely for this poorly conceived online monstrosity. But they’re not; you’re here, and you have to do this test if you live in a PARCC state.)

The Maryland and Illinois governments have passed laws giving the respective state boards of education the responsibility of adopting standards of learning for students. Good or bad, both states have adopted customized versions of the Common Core, which, in math, incorporates what are known as “math practices.”

The principles are considered overarching. In spirit, then, no one solution strategy is endorsed or disallowed, but students are expected to be creative and use whatever tool or solution strategy they feel is appropriate to solve a given math problem. This question makes it very difficult for students to achieve full credit if they solved Part B by making a drawing of the rug with the border, a perfectly valid and plausible solution strategy for the question as asked. The equation editor online doesn’t allow them to show this work and therefore robs them of point-earning potential in the scoring, especially in Part B.

This clearly violates the Common Core’s prevailing math principles, which both states have duly adopted by willful action of the appropriate governing bodies.

(The use of technology to show “all” work, in paragraph form, is an artifact of PARCC and results from the consortium’s interpretation of and self-imposed restrictions on what the Common Core math practices refer to as the use of technology. This artificial limitation imposed by the test penalizes creative students, who are unable to show their work on this problem in a way that would be appropriate to the task at hand: area. The question, as delivered and scored online, does not align to several key aspects of the Common Core and is therefore invalid. On paper, the question is completely valid, though.)

Note that the writing tasks on the PARCC test and especially the various multiple-choice formats, despite being enhanced by technology, fail to represent the rigor required by the Common Core, as Maryland and Illinois have adopted the standards. The Hechinger Report had this to say:

The Common Core tests contain multiple-choice questions and some writing tasks that don’t measure up to the ambitious Common Core education goals with which they are supposed to be aligned. … If students are taught to question what they’ve learned and reflect on the source of their knowledge, why should they be judged by a test on which they must choose from among several pre-fab answers?

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

Challenge

Explain why, using the form of a two-column mathematical proof, the quadratic formula would yield exactly the same solutions as the factoring method used above. For reference the quadratic formula is

$x = \frac{b \pm \sqrt{b^2 - 4ac}}{2a}$

where a is the coefficient of x², b is the coefficient of x, and c is the constant in a quadratic equation of the form

$ax^2 + bx + c = 0$

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.