Living Algebra, Living Wage

By Jana Dean

The Dependent Variable

Following an example that I had prepared in advance, students worked together to draw an xand y-axis on 11- by 17-inch graph paper. I told them that they would be graphing one day’s wages, or eight hours, and that the size of one’s paycheck depends on the number of hours worked; therefore, money—the dependent variable—belongs on the y-axis. I said, “I’m not going to tell you what the scale of that axis will be. That depends on your wages, and you’ll have to discuss together how high it needs to go. The independent variable, however, will be the same for everyone: You are all going to work an eight hour day.”

Having students graph all four different wages on the same axes forced them to see that higher wages meant steeper lines. It also helped them see how quickly an extra dollar per hour adds up. After students completed the graphs, I introduced the variables x and y. In this case, y represents your paycheck, and x stands for the number of hours you work. I challenged them to write an equation for each line that would show the relationship between time and earnings. Every group arrived at equations to represent the lines on their graphs.

Later, they would repeat the exercise by graphing a month’s pay at the same wage.

— J.D.

Back in 1986, the minimum wage here was $2.30 an hour. The Washington State Labor Council, in collaboration with churches and women’s groups, began advocating for an increase. By 1993, after three successive legislative victories, it had risen to $4.90—still too low for a single person to live independently, let alone support a family. Rather than continue to fight for each successive increase, the groups joined forces to lobby to have the minimum wage indexed to inflation. That way future increases would come annually, without expensive and time-consuming lobbying efforts. In 1998, two-thirds of the state’s voters passed a measure that set the minimum for the following year at $6.50 and guaranteed a yearly increase, indexed to inflation.

As students talked and asked questions, I learned the word on the street in Tumwater is that Costco is the place to work. Several students had parents who worked as security guards, and many reported relatives in low-paying healthcare roles.

“My mom works at Wal-Mart and she’s always working,” Stephanie exclaimed. “We never have enough money.”

Their discussions assured me that putting algebra into this context would connect with students’ lives outside school.

After small group discussions, I passed out approximate hourly wages to go with each occupation. For graphing ease, I rounded wages to the nearest dollar. I set the Wal-Mart wage at $7.00 an hour, which is lower than the company-reported average national wage of $8.23. In many states, however, Wal-Mart starts workers as low as $6.25. While setting it lower than our state’s minimum risked confusing students, I did so to expose them to the idea that two different employers such as Costco and Wal-Mart can have different policies that profoundly affect the quality of workers’ lives. Reports from the U.S. Department of Labor placed the security guard and nursing aide at $11 and $8 respectively. My grocery store cashier provided the source for Costco: $10.

After students presented their graphs, I brought their attention to the algebra involved by asking them to respond in writing to the following prompt:

How does the rate of pay affect the shape and steepness of the lines on your coordinate grid? Describe the shape of a graph for the wage of a job at $20 per hour. Describe the shape of a graph for the wage of a job at the federal minimum of $5.15 per hour.

The prompt led students to observe that the steeper the line, the higher the wage, and that each of the situations produced a straight line. Both observations paved the way for introducing “slope”—the rate of increase—and “linear”—a relationship that graphs as a straight line. The prompt also served to identify the coefficient of x or the number that multiplies x as the value that determines the steepness of the line.

Students expressed dismay at the federal minimum wage of $5.15 per hour. The nearly $40-a-day gap between Costco’s $10 per hour and the federal minimum looked enormous. However, they didn’t yet have any inkling of how much money it takes to maintain a household; later I would help them back up their outrage by providing that information.

‘It Sounds Like My Family’

Next, we spent several days practicing recognizing linear patterns in tables and graphs and writing equations from them.

Once students could recognize linear relationships, it was time to broaden their understanding of the fairness of a given wage. A recent film shown on the PBS documentary series POV titled Waging a Living served my purpose. The online teacher’s guide for the program comes with downloadable footage of Jerry, who struggles to get by as a San Francisco security guard. I discovered later that many students identified with Jerry. Jade wrote: “I remember sitting and listening to Jerry’s story and thinking, ‘My goodness! It sounds like my family.'”

After hearing Jerry talk about the expense of dressing to work in the fancy lobby of the building he guards, I introduced a problem so my students could experience the costs that come with employment. I proposed that my “lucky” students had just landed jobs. The bad news was that each job required a uniform. To allow students to compare equations, and to emphasize that not everyone’s circumstances are the same, I wrote two different uniform descriptions for each occupation. For example, Security Guard A drew the following:

The company wants you in uniform. Even though you already have nice clothes, they issue you the following, at your expense: hat $15, belt $20, pants $36, and three shirts — $25 each.

Security Guard B met with different circumstances:

You’re expected to look nice. Luckily you’re an average size and you can make it to Goodwill before your first day on the job. You pick up two jackets for $12 each, slacks for $7 and three dress shirts for $5 each. Shoes you have to buy new. You have foot trouble, so you decide you’d better get good ones. They cost you $70.

I assigned occupations to pairs of students and then had each of them draw a different card.

When Angie announced, “Dang! I’ve been working all day, and I still haven’t broken even!” I knew that students were beginning to realize, through the math, that working for a living meant more than paying your mom back for your iPod. They were ready for a bigger problem: rent.

I showed more footage of Jerry in a segment where he described the challenge of living in a long-term occupancy hotel in order to be close to work. Affordable housing outside the city would make him late to work. This time, I had students work in groups of four. I took a page of the classified ads for local rentals and challenged them to find affordable housing.

The Y-interceptWhen I felt comfortable that students were able to recognize linear equations that began at zero, I knew it was time to introduce the concept of y-intercept. This time I had students start by making a table showing pay minus expenses for the first ten hours of work. This gave students a real-world context for operating with negative numbers.After they finished the tables, I introduced the standard form for linear equations: y = mx + b, in which m represents the pay per hour, and b stands for the expenses. In general terms, m corresponds to the relationship between x and y, and b corresponds the value of y when x equals 0. I asked students to graph pay per hour less expenses and then to write an equation that would describe earnings minus expenses at any given hour.— J.D.

Heated discussions ensued about whether or not it was fair for a coed group to select the “women-only, no smoking, no drinking” house to share at $250 per month. After they made their housing choices, I introduced additional factors, again on cards. Some were positive, others negative. One card read: “Lucky you! When you moved your grandma’s old couch into the apartment, you found $25 in coins beneath the cushions. You start the month $25 ahead.” Then I had students use the minimum wage to write an equation that would tell them how many hours they would have to work to make the rent.

For most groups, the number of hours to make the rent ranged from 40 to 80. One group decided that they’d go for the house at $1,500 per month. Given their relatively high Costco wage and the requirement that they pay a month’s rent in advance, their calculations revealed that they’d have to work 300 hours to make the rent.

So far, I hadn’t complicated our calculations with expenses such as social security and employee insurance contributions that shrink a paycheck before it’s even cut. I hoped to get to that later. I wanted students to have a solid understanding of how prededuction rates of pay are linear. I also wanted to bring students back to their families’ and each other’s experiences before exposing them to additional expenses.

I framed a discussion around two questions: What happens in families when there’s not enough money; and what can happen in a family that makes it so there’s not enough money? Again, talk was lively.  Students shared family experiences of itinerant homelessness and living with various friends and relatives; struggles to pay the bills; absent, hard-working parents; families split not by rancor but by economic necessity, job loss, death, and disability.

The next day I provided some typical expenses such as $230 for new tires, a $1,500 trip to the dentist for a cracked tooth, $30 for new shoes for a growing adolescent, and $405 each month to feed a family of three. I also discussed the common experience of not getting enough hours of work in a week. Wal-Mart workers, for instance, average 30 hours a week. When I asked students if they knew anyone who has a job but complains about not getting enough hours, most hands went up.

I reminded students of Washington State’s minimum wage of $7.93 an hour. We then entered the equation y = 7.93x to our graphing calculators. We looked at both the graphs and tables generated from the equation to answer such questions as, “How many hours do you have to work at the minimum wage to pay for the dentist? For new tires? For new shoes? For food?”

Then I provided students with a printout of the living wage information for our community. Because the size of household determines expenses, I asked students to use the graphs and tables in their calculators to determine the number of hours at minimum wage it would take to support families of various sizes. They discovered that three people have to work full time at the minimum wage to support a family of two adults and one child. When I asked if this was possible, several students besides Stephanie reported that they never see either parent because they work all the time.

I then asked them to use the mathematical evidence to attack or defend the statement: “Washington State’s minimum wage is high enough.” Four fifths of my students determined that it wasn’t high enough. Mark wrote: “The minimum wage is not high enough. You have to pay the rent, then buy food and what if you have to go to the doctor? How are you going to have enough money to kill some bills?

On the minimum wage, you’re going to run out of money two weeks before your paycheck and then the rent will be due.” Although Mark didn’t put math in his writing, his graph had a big black arrow pointing to the gap between income and expenses. On the other hand, Chance thought that the minimum wage was plenty high. He knew money would be tight, but suggested credit cards could help with expenses. He carefully calculated the estimated shortfall during a month requiring new tires: $253. “Just don’t have kids until you have a job good enough to pay for it,” he advised.


Several months after our study of wages I asked students about the impact of the unit. Some students reported that they began planning their futures, including working on good grades and staying in school. Other students wrote about increased awareness of socioeconomic class. “What was interesting was using mathematics to see how other people are in the same status as me,” Evan wrote. “My mom works for just above the minimum and we never have money for extra things.”

Lizzie reflected on the teaching and learning as a whole. “It was interesting to learn algebra that way because we learn more than just one thing,” she wrote. “It unshields us from the safety of our home to be ready for the outside world.”

Next time, I’d like to teach students even more about the world. While I turned my students’ attention to the circumstances of people in their own community, those circumstances exist within a larger system permeated by assumptions about the value of labor and the role of the rich and the poor, both locally and globally. I will prompt students to consider factors that make some work worth more than other work and ask them to consider the fairness or unfairness of those factors. I will also challenge them to graph the $1.00 to $2.00 a day that most of the world’s poor earn and have them consider the hourly wage of the top American CEOs who, according to a Washington Post report, bring in up to $40 million per year. 

While it’s a victory to have students recognize that schooling can improve their personal prospects, next time we’ll examine the role that union organizing and labor struggles have played in advancing the right of workers to a living wage.  For example, according to an EPI report, on average, workers covered by a union contract have 14.7 percent higher wages, are 28.2 percent more likely to have health coverage, and are 53.9 percent more likely to have pension benefits than nonunion workers.

In their reflections, some students considered their own personal circumstances and others considered the world around them. Their responses let me know that I had met my goals of building a bridge between algebra and the world of wages and work, and of showing them that math can be a tool for reading their world.

My students have learned that algebra matters — and so do they.

Jana Dean ( teaches algebra to 8th-grade students at Bush Middle School in Tumwater, Wash.