The following constructed response question, explained here in hopes of helping fifth-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” (performance-based assessment) for grade 5 math:
Joshua planted carrots and peas in his garden.
Use the model to write and solve an equation that shows how much larger the pea section of the garden is than the carrot section of the garden. Enter your equation and your solution in the space provided.
What color do you think the carrots were in Joshua’s garden?
Because the left side of the garden measures 1 yard and it’s divided into 4 equal sections, the length of each section is 9 inches, and each little square represents 9 × 9 = 81 square inches in area in the model as drawn. Since the carrot section has 3 little squares, its area, represented by C, is
Turning to the pea section of Joshua’s garden, I see each side is the same length as two sides of the little squares, which I know are 9 inches each. The length of a side of the pea section is therefore 18 inches, and the area of the pea section, represented by P, is
I am asked how much bigger the pea section is than the carrot section, which means I have to subtract C from P to find the answer:
Resources for further study
Materials on working with fractions can be found almost anywhere on the Internet. Purple Math, developed by Elizabeth Stapel, a math teacher from the St Louis area, has an entire series devoted to working with fractions, as intended in this problem.
Analysis of this question and online accessibility
Students don’t need knowledge of substantial aspects of the math standards in the Grade 5, Number & Operations—Fractions section of the Common Core to solve the problem, unless they decide to use fractions, which isn’t the only approach. Because knowledge of those specific fifth-grade standards isn’t necessary to solve the problem as given—see my sample solution, which doesn’t have a fraction anywhere in sight—and because knowledge of the standards must be both necessary and sufficient to address the question if we’re claiming the question is aligned to the learning standard, this question is largely out of alignment with the Common Core.
As a result, a parent or teacher can base no useful conclusions about how well students understand modeling with fractions, which is what the evidence statement says this question measures, on any data that includes scores on this question.
The question can be delivered online and would yield performance statistics that are as valid as those obtained from paper-and-pencil test-takers. However, online test-takers may have difficulty with the equation editor, and if they have such difficulty, they may not be able to type in all the logic and reasoning needed to receive full credit (3 points) for this question.
I remind students, as I have in the past, when the equation editor pops up for a question, take a little extra time to make sure you:
- Type in all the logic and reasoning you used when solving the problem
- Transfer all your work from scratch paper to the computer
No special accommodation challenges can be identified with this question, so the question is considered fair.
Real-world setting
Rachel Weill and her husband, David Levitt, in San Anselmo, California, have two children, one 11- and the other 7-years-old. When those kids get home from school, they can’t wait to get fresh vegetables and fruit from the garden they’ve planted in their own backyard, Sunset reports.
“They constantly bring in food at random times, wanting to make a salad,” Ms Weill says. “And when they have friends over, they like to show off the garden.”
Among the veggies she recommends planting are carrots and snap peas, just like Joshua in the math problem here.
She hired a contractor to build four raised beds of untreated redwood into the hillside and secured those boxes with long bolts. Then she anchored the beds to the ground with cement. She grew the carrots from seed, and everything else was started in a nursery, she said.
