#### 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

*represent the number of office machines sold in a month. For*

**n****each**company, write an equation that represents the relationship between

**and**

*M***.**

*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.

## 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.

## Challenge

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.