The following constructed response question, explained here in hopes of helping third-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 2015 “PBA” test for grade 3 math:
Andre visits the library. It takes Andre 26 minutes to walk from his house to the library. He stays at the library 45 minutes. His mother drives him home, which takes 15 minutes. How many more minutes does Andre spend at the library than traveling to and from the library?
Show all the steps for solving the problem. Explain each step and give the final answer.
Enter your answer, your work, and your explanation in the space provided.
Andre goes VROOM on the way home from the library, but mom’s a real slow driver.
Correct answer: 4 minutes. Time at the library is given as 45 minutes. We need to find the total time Andre spends traveling to and from the library. We find that by adding the time he spends traveling to the library, given as 26 minutes, and the time he spends traveling home, given as 15 minutes.
The total traveling time is 26 + 15 minutes, which is 41 minutes. We subtract that from the time he spends at the library, which is 45 minutes:
So he spends 4 minutes longer sitting in the library than he spends traveling to and from the library.
PARCC evidence statement(s) tested: 3.D.1:
Solve multi-step contextual word problems with degree of difficulty appropriate to Grade 3, requiring application of knowledge and skills articulated [elsewhere].
Multi-step problems must have at least 3 steps.
The evidence statement above references 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. In early grades, this might be as simple as writing an addition equation to describe a situation. … 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 relies on students’ understanding of a math operations standard in second grade (Common Core math 2.OA.A.1) in order to test students in third grade on their ability to reason with that knowledge in solving a real-world problem. The second-grade standard itself says students should be able to “use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.”
Resources for further study
K5 Learning, a website developed “by parents for parents,” has several good worksheets about finding the answers to word problems that involve addition and subtraction. Also, the site Dad’s Worksheets.com prides itself on developing more than 8,000 free worksheets by “one dad” for “two daughters” and has several worksheets available that involve solving word problems with a mixture of addition and subtraction.
The Khan Academy, developed by Sal Khan, an engineer who has created a library of thousands of video lessons, has developed several video tutorials for third-grade students to learn how to solve “challenging addition and subtraction word problems with ‘more’ and ‘fewer’,” as we solved this problem (it had ‘more’ in it). Multi-step problems are also included, and the numbers used are within 100 or less. Here’s a tip: When used as they are in this problem, the words ‘more’ and ‘fewer’ should automatically make you think that you’re going to subtract one number from another at some point in the problem.
Analysis of this question and online accessibility
The question measures knowledge of the Common Core math practice it purports to measure and tests students’ ability to show reasoning and mathematical modeling in solving a two-step word problem. I don’t see a third step here, as the evidence statement calls for, but that does little to affect the quality of the problem.
The wording of the question contains a faulty parallelism in the comparison of the time Andre spends “at the library,” a prepositional phrase used to describe Andre’s location, and the time he spends “traveling to and from the library,” a gerund clause, not prefaced, used to describe his actions. This shouldn’t cause native English speakers too much difficulty in determining what the question is asking (because they will likely put the parallel structure branch point after “Andre” instead of after “spend”), but it is substandard English usage.
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. It is important for students to write out, nearly in paragraph form, the reasoning they used in solving the problem.
Why did you subtract 41 from 45? Why did you add 15 and 26? All of this has to be explained in order for the student to receive full credit.
The scoring rubric for this problem indicates that only one point is given for coming up with a correct answer of 4 by the right calculation. In order to get the remaining two points for the problem, students must explain their reasoning. Make sure they know not to be shy about typing in the logic and reasoning used in solving the problem. More than likely, they’ll get more points for that on the test than they will for the right answer.
Finally, 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.)
No special accommodation challenges can be identified with this question, so the question is considered fair.