Living Algebra, Living Wage

Summer 2007

	Illustration: J.D. King

8th graders learn some real-world math lessons

By Jana Dean

I know people don't usually ask. But do you mind telling me how much you make an hour?"

I felt a little uneasy asking this question at the checkout stand at the big-box store next to the freeway. It's not something you usually talk about when exchanging money for groceries.

The checker was happy to answer. "Well, I started at $9.12," she said, then added. "Last month I got a 22-cent raise. I thought it would make a big difference, but I can hardly tell. My paycheck's almost the same size."

I explained the reason for my question. "I teach math at the middle school down the street, and I want to use wages to teach them algebra."

"Well, you can tell your students there's no way to make it on less than $10 an hour," she said. "Costco's where they want to work. They start you at $10."

I thanked her and paid for my groceries.

In using wages from our community to teach about linear relationships—mathematical relationships in which the rate of change is steady and graphs as a straight line—I had two goals.

First, I wanted my students to engage their skills in math class to inform an ethical stance on a social issue close to their own lives: working for a decent wage. They would calculate daily and monthly incomes in service sector jobs, discover hidden costs associated with being employed, research local housing prices, compare state and national minimum wages, and learn about the activism and organizing effort behind Washington State's highest-in-the-nation minimum wage. Ultimately, they would use their knowledge of economics and mathematics to develop a point of view about the minimum wage.

My second goal had to do with motivating students to stick with algebra. Giving all students access to higher math regardless of family background is one reason I teach middle school math. Socioeconomic status informs student experience both in and out of school. In my school, students are tracked starting in 8th grade, which is when algebra comes around. The placement of students tends to fall along class lines. A few weeks ago, when I asked how many of my advanced track students know someone who has taken out a payday loan, a few timid hands went up. Most didn't know what I was talking about. When I asked my other three classes the same question, they had plenty of experience to draw on: almost everyone knew someone. Decent, middle-class paychecks keep most people from having to borrow at 375 percent annual interest to make it to payday. In using the topic of a living wage to teach algebra, I hoped to build a bridge between my students' lives and algebra.

A Living Wage

Advocates for the working poor set living wages for U.S. communities by researching typical housing, childcare, food, medical, and transportation expenses in a given community. According to the Economic Policy Institute, the living wage for the Tumwater, Wash., area, which includes no "extras" such as new shoes, big screen TVs or birthday presents, is $39,000 for a family of three, or about $19 an hour full time. This is about twice the federal poverty level. Agencies use the poverty level to determine government assistance and as an economic indicator. While 12.6 percent of our state's population lives below the federal poverty level, 26.9 percent of all people live in households whose income is below a living wage, as determined by the EPI. Most of those households have heads of household who are employed. This mirrors the nation.

Thinking to launch our study with an engaging story, I read out loud from Barbara Ehrenreich's Nickel and Dimed. I chose a section in the middle of the chapter "Selling in Minnesota" in which Ehrenriech describes her struggle to pay for the clothes she needs for her new job at Wal-Mart and her frustration at how tedious and difficult the work is. I chose it because I thought it illustrated the complexities and compromises that come with accepting a low-wage job.

My students didn't respond at all. Rather than the open-eyed engagement I'd expected, they zeroed in on the fine art of finding and tracing nicks and scratches in the surfaces of their desks. Either I'd completely missed the mark, or I'd struck too close to home.

Tapping Students' Opinions

Luckily, I had a planning period before I was to teach this lesson to my next class. It was possible my students didn't need Barbara Ehrenreich. Maybe they had some personal experience I could draw on instead. To my next class, I read the following statements out loud:

1. Those who work should be paid.
2. No one who works full time should live in poverty.
3. Wages should be high enough to support a family on one income.
4. The legal minimum wage should be high enough to get by.

I then asked students to talk to their partner about each statement and to say whether they agreed or disagreed with it. Afterward, they wrote their responses in their notebooks.

The contrast with the previous class was stunning. As the room filled with their voices, I knew I'd hit the jackpot. They spoke from experience. They all connected work with income: everyone thought that anyone who works should be paid. No one thought that people who work full time should have to live in poverty. They had mixed responses to my prompt about whether or not a family should be able to cover basic needs on one income, and revealed that the community norm is that at least two people in a household work.

Daryl's response was typical: "The legal minimum wage should be high enough to get by because if people aren't making enough working full time, they need to be paid more."

Stephanie's comment sounded personal, "People work hard for their money...It's expensive to pay bills, buy food, pay rent and all the extras."

I knew I had students engaged: ethical consideration of fair pay interested them. But I wasn't sure yet if the head of steam we'd gathered would carry into the hard work of learning algebra. Sadly, by middle school, students expect math class to be disconnected from their lives. This gives an extra boost to any lesson that occurs within math class that appears to contain no math. "Thank goodness," students seem to say to themselves. "Finally we're doing something that matters!"

So before the bell that day, I threw in some numbers. I wanted to take advantage of my students' interest in the social issue of fair pay and connect it right away with mathematics. I asked them to estimate the federal minimum wage. Most students guessed on the order of $10 an hour. When I told them it was $5.15 an hour, they were shocked. Jay exclaimed, "Wait a minute! I make $4 an hour splitting firewood and all I have to do is pay my mom back for my iPod." I then asked them to write down Washington State's minimum wage: $7.93 per hour.

Graphing Wages

My students' task the next day was to graph four linear relationships on the same coordinate grid and write equations for each. Again, a linear relationship is one in which change is steady, and it graphs as a straight line. They would graph a day's and then a month's full-time wages for four service industry occupations.

I chose service sector occupations to reflect our community's job market. We do have skilled jobs available in our community. For example, local sheetmetal workers can apprentice at more than $20 an hour with benefits, but the work isn't always steady. Like many places, part-time work is common and the low-wage service sector is growing, as corporations outsource higher-paying manufacturing and skilled jobs to Mexico or China.  During the 1970s and '80s, many of the locally owned manufacturing operations were sold to multinational corporations.  Subsequently, many of them have moved on.  In the last five years alone, the Miller Brewing Co. shut down the 106-year-old Tumwater brewery leaving nearly 400 workers out of a job. Oregon-based Tree-Source closed shop at one of the town's last sawmills and laid off more than 100 workers.

I gave students cards with these service sector job titles:

Retail Clerk at Wal-mart

Security Guard

Retail Clerk at Costco

Home Nursing Aide

I wrote up role cards as though my students had just landed these jobs. For each job, I wanted to paint a picture that showed the importance of and the value of the work. I worded the cards carefully because I knew that many students would have family members making meager livings in these occupations. I didn't want in any way to contribute to the devaluing of human beings and their daily labor.

The Costco card reads:

In high school you went out for basketball. You've always been really strong and light on your feet. In school, teachers asked you to run errands because they knew you'd do the job quickly and well. In your job interview you talked about how you've always loved working as part of a team. It worked. You were hired. Your duties take you all over the store, nearly at a run. You check prices for customers at the register, you return unwanted items, you break down boxes and you restock with the pneumatic pallet jack.

Students worked in groups of four, with each occupation represented. I told them to read the cards aloud and try to estimate hourly wages for each, given the state and national minimum wages we had discussed the previous day. The task brought up questions. Cody, in typical stream-of-questioning style, asked: "What is a minimum wage? Who gets to decide? What does it mean? I don't get it." Questions like his gave me an opportunity to talk about the meaning of the minimum wage and share the story of the labor activism that led to Washington State's indexed-to-inflation minimum wage.

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-intercept When 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.

Impact

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 (jdean@reachone.com) teaches algebra to 8th-grade students at Bush Middle School in Tumwater, Wash.

Summer 2007