Friday, September 17, 2021

Algebra 1 PARCC question: compare job offers


The following multi-part 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:

Part A

Marcella wants a job as a sales representative. She receives two job offers from companies that sell office machines to businesses.

  • Office Essentials offers Marcella a salary of $2,500 per month, plus a commission of $125 for every office machine she sells.
  • Everything Office offers Marcella a salary of $2,000 per month, plus a commission of $150 for every office machine she sells.

Let M represent the total monthly earnings, in dollars, and let n represent the number of office machines sold in a month. For each company, write an equation that represents the relationship between M and n.

Enter your equations in the space provided.

Part B

Marcella wants to earn a total of at least $4,000 per month. For each company, find the least number of office machines she would need to sell each month in order to meet this goal. Show your work.

Enter your answers and your work in the space provided.

Part C

Compare Marcella’s possible earnings at Office Essentials to her possible earnings at Everything Office. How many machines would Marcella have to sell for the earnings at both companies to be the same? Find the interval of machines sold for which the total earnings at Everything Office are greater than the total earnings at Office Essentials. Show your work.

Enter your answers and your work in the space provided.

Answer and references

Correct answers: Part A: Moe = 2500 + 125n and Meo = 2000 + 150n. Part B: oe: 12 machines; eo: 14 machines. Part C: 20 machines or higher.

PARCC evidence statement(s) tested: HS.D.1-1:

Solve multi-step contextual problems with degree of difficulty appropriate to the course, requiring application of knowledge and skills articulated in 7.RP.A, 7.NS.3, 7.EE, and/or 8.EE.

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

[MP.4] Model with mathematics. 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.

It may also reference other standards of mathematical practice, including MP.1 (make sense of problems and persevere in solving them), MP.2 (reason abstractly and quantitatively), and MP.5 (use appropriate tools strategically).

The question requires students’ prior understanding of the eighth-grade Common Core standard 8.EE.C, found under Expressions and Equations (analyze and solve linear equations and pairs of simultaneous linear equations), which states that they should be able to [EE.C.7] “solve linear equations in one variable” and [EE.C.8] “analyze and solve pairs of simultaneous linear equations.”

Note that the PARCC question referenced here simply uses understanding of these standards from eighth grade, considering this understanding to be securely-held knowledge. The PARCC question in Algebra 1 tests students’ ability to model with mathematics, as referenced in Math Practice 4.

Example of a solution strategy (there are others)

Part A does not require work to be shown. Model the situations with an equation.

Marcella’s total earnings each month will be her base pay plus the commission amount times the number of machines she sells.

This is a very common way to describe the linear relationship between two quantities. Think of this type of word problem like the slope-intercept form for the equation of a line, y = mx + b. The y-intercept is the amount Marcella would make if she sold x = 0 office machines. The slope is the rate of change for her earnings based on the number of machines she sells. For Office Essentials, the slope can be expressed like this: 125 dollars per office machine sold.

Don’t let the simple fact that the quantity on the horizontal axis is n and not x and that on the vertical axis is M and not y throw you off.

y = mx + b
M_{oe} = 125n + 2500
M_{eo} = 150n + 2000

Part B requires work. Set M = at least 4000 for each company and solve for n.

M_{oe} = 125n + 2500
4000 \le 125n + 2500
1500 \le 125n
12 \le n
M_{eo} = 150n + 2000
4000 \le 150n + 2000
2000 \le 150n
13\frac{1}{3} \le n

Since Marcella can’t sell ⅓ of an office machine, she has to sell the next whole machine at Everything Office to reach her target earnings for the month. Always round up in a context like this, regardless of which integer is nearest to your actual answer.

Part C requires work. Set Moe = Meo and solve for n.

125n + 2500 = 150n + 2000
500 = 25n
20 = n

So if she sold 20 office machines at Office Essentials, she would make the same total earnings as if she had sold 20 office machines at Everything Office. We can check this by graphing M vs n from the two equations on the same set of axes (OE is blue, EO is orange):

If she thinks she can sell more than 20 office machines every month, she should take the job at Everything Office, despite a lower base pay. Unless she doesn’t like the people there, of course, in which case, the money wouldn’t be worth it, right?

Resources for further study

Purple Math, developed by Elizabeth Stapel, a math teacher from the St Louis area, has a few pages about how to translate word problems into equations in setting up a little mathematical model like the one in this problem. The first page is here.

Chapter 4 of Paul A Foerster’s book Algebra and Trigonometry is devoted to systems of linear equations and inequalities, much like the setup in Part C of this question from PARCC. “Linear functions may be used as mathematical models of the real world,” he writes. “You will learn how to find the intersection of the graphs and how to tell what this intersection can represent. The techniques you develop can be used to find the best way to operate a business…”

Complete reference: Foerster, Paul A. Algebra and Trigonometry: Functions and Applications, revised edition. Addison-Wesley, 1980, 1984. The book is used in several algebra classes taught in middle and high schools in both Illinois and Maryland.

Analysis of this question and online accessibility

The question measures knowledge of the Common Core standard it purports to measure and tests students’ ability model a real-world system using systems of two equations or inequalities in two unknowns. It is considered to have a median cognitive demand.

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 using the equation editor if they are unfamiliar with the tool, which makes them enter math work in paragraph form.

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


Suppose your family is purchasing a new air conditioning unit for your house. Brand A costs $1600 to purchase but will cost $40 a month to operate. Brand B costs $2100 to purchase but is more efficient, costing only $32 a month to operate.

How many months will it take of full operation for Brand B to be a less expensive option in the long run? Based on the temperatures where you live, how old will you be when that “break-even point” occurs?

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.

Paul Katula
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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