Sunday, September 20, 2020

Grade 7 PARCC math: furniture store sale

The following two-part computation-based question, explained here in hopes of helping seventh-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 “EOY” (end-of-year) test for grade 7 math:

A furniture store had the following sale:

Buy one item at the regular price,
get the second item of equal or
lesser value for

$\frac{1}{2}\;\textrm{off!}$

Mr. Davis bought 2 chairs during the sale. The regular price of each chair was $168. What was the total price, in dollars, for both chairs during the sale, not incluidng tax? Part B Ms. Wilcox bought a sofa and a chair during the sale. The regular price of the sofa was$875 and the regular price of the chair was $250. After the discount was applied, a sales tax of 6.25% was charged on the total purchase. How much did Ms. Wilcox pay, in dollars, for the sofa and chair, including tax, during the sale? Friends play on a sofa they just bought on sale, but where are Mr Davis and Ms Wilcox? Answer and references Correct answers: Part A:$232. Part B: 1062.50. PARCC evidence statement(s) tested: 7.EE.3, according to the PARCC alignment document. Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of$27.50. If you want to place a towel bar 9¾ inches long in the center of a door that is 27½ inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.

The evidence statement above references Math Practice 5 in the Common Core: Use appropriate tools strategically.

Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

The question involves computations in the ratios and proportions section of the seventh-grade math Common Core, including standard 7.RP.A.3, which says students should be able to “use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.”

This Common Core standard, though clearly assessed with the problem, does not show up on the PARCC alignment document published with the public release, but it’s fair game on the seventh-grade math test.

Example solution strategy (there are others)

If Mr. Davis bought two chairs that were marked at the same price of $168, he paid$168 for the first chair and half of $168, or$84, for the second chair. Add those up, and you get

$168 + 84 = 232$

Part B is a little more work. First we need to know that the chair was the “lesser” of the two items, since it had the lower price, $250. That item will cost Ms. Wilcox half of the marked price, or (½)($250) = $125. The sofa will be at the original price,$875, which needs to be added to the $125 she’ll pay for the chair. $125 + 875 = 1000$ On top of that total before tax,$1000, we need to add a 6.25 percent sales tax. Since we have a nice round number, note that anything times 1000 just means to move the decimal point to the left three spaces:

• 6.25% = 0.0625
• 0.0625 × 1000 = 62.5
$1000.00 + 62.50 = 1062.50$

Resources for further study

The Khan Academy, developed by Sal Khan, an engineer who has created thousands of video tutorials for students of all ages, has an entire series devoted to solving word problems that involve “discount, tax, and tips.” Purple Math, developed by Elizabeth Stapel, a math teacher from the St Louis area, has several pages on “percent of” word problems, such as discounts, markups, and sales tax.

In addition to the tutorials, several authors have produced worksheets loaded with practice problems that involve discounts, tips, sales tax, and markups:

Analysis of this question and online accessibility

The question measures knowledge of the Common Core math standard I have listed above, in addition to assessing whether students are proficient in the math practice it purports to test.

The question can be delivered online and would yield performance statistics that are as valid, or perhaps more valid, than those obtained from paper-and-pencil test-takers. The reason for this is that converting this question to a paper-and-pencil question would mean coming up with plausible incorrect answers to make at least four options for multiple choice. This format encourages guessing, which reduces the quality of performance statistics to a certain extent.

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

I would be remiss not to note the missing comma in between two independent clauses in Part B, here: “… sofa was $875 and the regular price …” There should be a comma before the “and,” but the error doesn’t affect readability at all. Some people might even say a comma isn’t necessary in this instance, since both clauses are short and have the same structure. In addition to the missing comma in Part B, the advertisement sign also features a comma splice error in which two imperative sentences are joined with only a comma (and no conjunction) between them. In a real classroom Chelsea Katz writes in the Kilgore News Herald in Texas that seventh-grade students of Delina Chitwood set up a supermarket in the library of Kilgore Middle School, with the help of a local grocery store and a little math. “They have to mark up their product, and they are going to resell it to their classmates,” Ms Chitwood was quoted as saying. Half the class became the “owners” of the store and had to purchase goods from “manufacturers” (the teachers). Each student had$100 with which to fill their shelves. Then the customers (the other half of the class) got \$20 to spend in the store.

They are going to do the business part of it first, figure out the profit and the percent increase, and then we’re going to come back and they’ll be the customer. … We’re going to record what items sell and then give them that information, so they can know how much money they’ve made—the profit they’ve made.

We want them to be able to understand tax is an additional amount and as far as the teaching part of the seventh grade part they have to be able to calculate that as far as a percent proportion. We are also looking at constant rate of change that if you buy one of something, two of something, three of something, that that pattern—that constant rate—stays the same for items, for hourly wages.

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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