Sunday, December 15, 2019

#### The following multi-part constructed-response question, explained here in hopes of helping fourth-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) test for fourth-grade math:

Jessica shades two grids that each equal one whole to represent and compare the fractions

$\frac{3}{10} \textrm{ and } \frac{29}{100}$

## Part A

From the choices, select the decimal that represents $\frac{3}{10}$, and the decimal that represents $\frac{29}{100}$. Then drag and drop each decimal into a box to create a true comparison.

• 0.03
• 0.3
• 3.1
• 0.29
• 0.92
• 2.9

_____ < _____

## Part B

Jessica says that $\frac{3}{10} + \frac{29}{100} = \frac{32}{100}$ because 3 + 29 = 32 and there are 100 squares in each of the grids. Explain how you know Jessica is incorrect by using the grids or the decimal inequality you created. Then find the correct sum.

Correct answers: Part A: 0.29 < 0.3. Part B: Correct sum is $\frac{59}{100}$.

PARCC evidence statement(s) tested: 4.C.4-5, according to the PARCC alignment document.

Base arithmetic explanations/reasoning on concrete referents such as diagrams (whether provided in the prompt or constructed by the student in her response), connecting the diagrams to a written (symbolic) method. Content Scope: Knowledge and skills articulated in 4.NF.C

Tasks have “thin context” or no context. Tasks are limited to denominators 2, 3, 4, 5, 6, 8, 10, 12, and 100.

The evidence statement above references Math Practice 2 (reason abstractly and quantitatively), Math Practice 3 (construct viable arguments and critique the reasoning of others), Math Practice 5 (use appropriate tools strategically), and Math Practice 6 (attend to precision) in the Common Core.

The question involves computations in the Number & Operations — Fractions section of the fourth-grade math Common Core, including standard 4.NF.C.7, which says students should “understand decimal notation for fractions and compare decimal fractions” by “comparing two decimals to hundredths by reasoning about their size; recognizing that comparisons are valid only when the two decimals refer to the same whole; and recording the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model.”

Example solution strategy (there are others)

Part B requires an explanation and a computation. You can do the computation by expressing both fractions with 100 in the denominator. That is,

$\frac{3}{10} = \frac{30}{100}$
$\frac{30}{100} + \frac{29}{100} = \frac{59}{100} = 0.59$

To “explain how you know Jessica is incorrect” with her answer of 0.32, is pretty easy, given that it’s just the wrong number. However, the problem tells you that you have to refer to the grids or decimal inequality from Part A in your explanation. This is confusing for an adult, perhaps, but the Common Core asserts that students should know that

$\frac{3}{10} = \frac{1}{10} + \frac{1}{10} + \frac{1}{10}$

Likewise, 29/100 equals 29 × 1/100 and 3/10 = 30 × 1/100, which is greater than 29 × 1/100. That’s how you can use the grids and the inequality from Part A in your explanation, although the request to do so doesn’t definitively steer fourth graders in that direction and does indicate a minor deficiency in the design of the question itself.

## Resources for further study

Dan Johnson at Washington Elementary School in Wenatchee, Washington, has developed a compact tutorial about this particular set of skills and knowledge from the section of the Common Core. The page includes links, all the way down at the bottom, to other resources, including a practice test that serves as a little worksheet to let you practice on problems just like this.

## Analysis of this question and online accessibility

The question measures knowledge of the Common Core math standard it purports to measure, in addition to assessing whether students are proficient in several math practices associated with the PARCC evidence statement.

The question can be delivered online and would yield performance statistics that are as valid as those obtained from paper-and-pencil test-takers. Note that some students who take the test online may encounter difficulties using the equation editor that is part of the online test-delivery system. We have written about this at great length and won’t belabor the point here, except to remind students that when you use the equation editor for a constructed-response problem like this, make sure you:

• Transfer all your work from scratch paper to the computer
• Include every step in your logic or explanation

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

## In a real classroom

Students can use graph paper and make grids of all sizes. The Common Core limits fourth-grade fractions to those with certain denominators, so that’s as far as testing can go, but give kids some graph paper and let them explore.

(from the Top Drawer Teacher Resource in Australia (website)
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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