# The Square Root of a Fair Share

Illustrator: Michael Duffy

Man-o-man, were they crowded. When I asked my 8th-grade class how much personal space they got at school, their eyes rolled and — no surprise — they started talking to each other. They felt cramped in the hall, leaving class, and pressing against the door in the cold in the morning. And pity the poor souls who get a middle locker on the bottom tier. In-class conditions weren’t much better. At their desks, the bother was their stuff. Prepared students pack a binder, a planner, hopefully two pencils, a pen, a textbook, a notebook and a book for silent reading. Students soon learn that part of getting along in class means taking up as little space as possible. While 8th graders don’t usually make the connection, how much room they get is mathematics: it’s a matter of area.

After a few years of having my middle school students struggle to grasp square roots, I had decided to start with listening to how they experience area in order to connect squares and square roots to their everyday experience. I hoped that giving my students multiple opportunities to see that the side length of a square is the square root of its area, and conversely, that the area of a square is its side length squared, would help them understand this fascinating mathematical concept.

After my students’ ready response to my query about personal space, I decided they would compare the area of their desks to the area taken up by their supplies. From there, I would invite them to think critically about how and where people experience luxurious amounts of space or pressing closeness every day. The class would compare the size of the average newly built suburban house to houses built a generation ago and to dwellings in other parts of the world.

Any time students made comparisons of area they made square scale representations. As unconventional as it was, using squares made it easy to both compare the area of dissimilar shapes and see side lengths of squares as square roots. It also provided a link to the convention of measuring area in square units.

### New Desks Anyone?

The previous spring, our vice principal had come around and asked if anyone would like new desks. The offer lasted about 25 minutes and I missed it. I started my students on area with the story. “I didn’t jump on the opportunity,” I told them, “because I hadn’t been well enough prepared to know whether the new desks would give you any more work space than you’ve already got. In case he comes around again, I want to be prepared and I need your help. You are the best people to figure out how much space 8th graders need on their desks.”

I passed out rulers, and issued a challenge: “You have 10 minutes to measure and calculate the total desktop area you need for your books and supplies.”

As students finished measuring and began reporting their results to the class, Terra, with quiet drama, zipped up her binder to make the foundation of an elaborate tower on her desk. Ever prepared to learn, Terra always has several pencils, all sharp, and pens and highlighters, and she loves to read so she’s got a reading book and a back-up just in case. Once she’d finished her creation, she raised her hand with urgency. “Ms. Dean, Ms. Dean, look at all my stuff. There’s no way I have space.”

I turned to her decidedly undramatic desk partner Shane and asked, “Shane, ever feel crowded?” He nodded gravely.

It turned out that Terra needed more space than most: she required 640 square inches, while the class averaged 568 square inches.

### Seeing Squares and Roots

In our school, students sit in pairs and share eight square feet of desktop. That’s four square feet each. To introduce modeling area with squares, I made a masking tape square on the floor with side lengths of two feet to represent the four square feet allotted to each student. After fielding questions before class started about the unusual square on the floor, I asked students to make a scale drawing of the square in their notebooks and use it to determine the number of square inches in a two-foot by two-foot square. Because my primary focus was the new concept that the side length of a square is the square root of its area rather than the previously learned concept of scale, I provided the scale. One unit on the graph paper was to equal one linear inch.

Once their squares were drawn, they could see that a sketch of a square with sides of two feet encloses 576 square inches. Some students laboriously counted squares, but most saw that 24 rows of 24 yielded 576 in total. As a class, students then compared the amount of space taken up by their supplies to the area of their desks and determined that they had on average eight square inches of desktop elbow-room to spare. Seeing how small that really was would wait until the next day.

I then asked them to select three integers between three and 10, and sketch squares using those side lengths. Most students quickly saw the relationship I wanted them to see, that is, the product of a number multiplied by itself can be represented by the area of a square.

Before school the next day, I taped off three eight-square-inch squares. I would use these little squares as a means to work back from squares to square roots: to find the side length of a square, you find the square root of its area. What’s hard about square roots is that most numbers don’t have a square root in the family of rational numbers, which includes fractions, decimals and integers:

We can only estimate the square root of 6, 7 or 8 unless we leave all or part of it under the radical sign. And leaving a number under a sign like that seems terribly unfinished: it just doesn’t look like “an answer.”

I put one of the eight square-inch squares on Shane’s desk, one on the floor at the front and one in the corner of the white board at the back. They looked pretty small. During our discussion of the relative size of these squares, Shane tried fitting his elbow into the one on his desk. No luck. With his bulky sweatshirt, it was too small. His gesture elicited a chorus of “That’s not fair!” In order to build a bridge from their indignation to the math I wanted students to extract from their experience, I asked, “How do you think I was able to know how long the sides of those eight-square-inch squares needed to be so that we could visually compare eight square inches to the 576 square inches of your desk?”

So far, I had kept silent on a little secret: the square root button. This little item on the calculator makes magic of finding square roots, and with its apparent ease keeps students from truly understanding a concept that at first is as new as division was when they’d never seen it before. Before handing out calculators, the most common mistakes surfaced, including dividing eight by two and by four. Two times four is indeed eight, but we needed the same number multiplied by itself to result in an eight-square-inch square. Someone suggested that since two was too small and four was too big, we ought to try three. No luck. A square with a side length of three has an area of nine. As I handed out calculators, I kept my carefully guarded secret, and privately thanked Texas Instruments for hiding the square root function in a second key. Five almost silent minutes followed as students used their calculators to try to find a number between two and three that would multiply by itself to get eight. Lexi got the closest with 2.83 squared coming to 8.0089.

Following this careful introduction, students spent about five class periods modeling squares on graph paper and finding the lengths of their sides. This exposed them to the difference between perfect squares like 9, 16, and 576, which have integers for the lengths of their sides, and squares such as 5, 8, and 17 whose side length can only be expressed exactly using a radical sign. Students became increasingly comfortable with both the radical sign and decimal approximations for side length. For this part of their learning I relied on our district’s *Connected Mathematics* textbook, and a visual mathematics curriculum called *Math and the Mind’s Eye* (see resources).

### Room for the Stuff You’re Going to Buy

By now, I knew my students had a good initial grasp of square roots, but I knew that asking them to apply their understanding in a new context would provide more practice for those who needed it and an opportunity to make interdisciplinary connections for everyone else.

I began setting the stage to analyze an area issue that I knew they had experienced right in their neighborhoods: the super-sizing of new homes. The small cities surrounding Seattle have some of the highest population growth in the country. More new subdivisions punch into forests and farms every month. As though this pressure to house a lot of people weren’t enough, it’s amplified by an advertising-fueled drive to want more, which means house sizes have increased dramatically in the last few generations. Prior to the 2008 mortgage crisis, one of the largest regional builders, Quadrant Homes, boasted a home-building rate of seven houses per day and a slogan, “More home for less money.” On the internet, Quadrant Homes and others featured prominent links to mortgage deals that seemed too good to be true. On the street, Saturday temp workers waved signs to the nearest show model. Since then, the building frenzy has slowed.

On a recent trip to Home Depot, which serves contractors as well as homeowners, I experienced a 5 to 1 salesperson to customer ratio in what had been a bustling, largely self-serve warehouse store. The super-sized houses remain, however, and new subdivisions were very much a part of my students’ experience.

### It Hasn’t Always Been This Way

I wanted to place the phenomenon of super-sized single-family houses into an historical context. Until the 1970s, the average new home in the U.S. measured no more than 1,700 square feet. Just before and after the Second World War, they were even smaller. The 1940 average was 1100 square feet. In contrast, new homes in 2005 averaged 2,400 square feet. And the size of the average household hasn’t grown: it has become smaller as houses have grown. If you put the U.S. average household of 2.3 people into a brand new 2,400 square foot house, everyone gets about 1,000 square feet. Compare that to the 1,100 square feet of the 1940s, typically shared four ways. Of course, averages don’t necessarily reflect conditions on the ground. According to a quick in-class hand-raising poll, only about a fourth of my students lived in households with three or fewer people, with the majority living in households of four to seven. All the same, the national trend is more personal space.

In order to assess local home sizes, students tallied the square footage of the houses for sale in one day’s classifieds. They ranged from as small as 1,200 square feet to as large as 5,900 square feet, with most of them falling between 1,600 and 2,400 square feet. Clearly homes in our town were bigger than the national average of two or three generations ago.

I asked students, “Why do you think houses have gotten so much bigger over time?”

Their responses were astute. Maya said, “It’s that luxury thing people see on TV. They want status.” Rickie echoed, “Yeah. They want to impress their friends.” Alex said, “All that space is for the stuff.” “What stuff?” I asked. “A couch? A bed? A dresser?” “No,” he said. “Not what you already have. The room is for everything you’re going to buy.” Grant said the space was for the technology. “You know, we need room for our computers, our TVs, stereos, and everyone wants their own so we need separate rooms so we don’t bother each other.”

### My Big Dream Home

Great big brand new houses weren’t new to most of my students. Whether or not they’d ever been in one, they’d seen them. When I showed the class a picture of a housing development downloaded from the internet to get them thinking about what they see every day, Sean said, “I live there!”

In my haste to be ready for class, I’d downloaded the first image I found. It was one from the South. “What do you mean, Sean? This picture’s from Georgia.”

“No. I live there. That house is across the street from mine. There are seven of them just like that.” He went on to describe a whole row of brand new houses with huge brick entry facades. After Sean described the cul-de-sac facing his house, I asked, “Who do you think gains from these big new houses?”

Samantha snapped right to. “I think the builders of the houses gain because the bigger the house gets, the more it will cost, so the builder gets more money.” Katy carried her thinking to who might lose from having such big houses. Like many of her classmates, she’s seen woods consumed by development. She said, “Sometimes just two people will build a house big enough to fit fifteen people… We knock down trees that give us fresh air… like over by my house they probably knocked down one hundred trees for more huge houses.”

Soon Nate had his hand high in the air with an offer. “Ms. Dean, I draw houses all the time. Here, I’ve got plans right here in my binder.” He pulled out detailed house plans for his dream home. It had two stories, complete with a library, media room, bonus room, sky bridge, breakfast nook, five bedrooms, and private walk-in closets and bathrooms for everyone. Nate hadn’t drawn his plans to any formal scale, so there was no way for us to find the square footage, but all the same, it provided one representation of what the American dream home has become. After Nate shared his plans, I asked the class if they’d like to live in a house like the one he’d designed. Most of them thought it’d be pretty nice, especially the media room. Then I asked how many thought their current home was as big and fancy as Nate’s dream. None did.

### Traveling the Material World

I wanted students to see that not only have house sizes increased in the United States, but they are far larger than homes anywhere else in the world. Nowhere else on earth do people expect in such large numbers to have so much indoor living space.

Photographer Peter Menzel’s book, *Material World*, drives this point home with grace and compassion. The book consists of family portraits and statistics about life in 30 nations. Menzel created the book by moving in with each family for a week for photographs and interviews. Each visit culminated with the household moving all of its belongings outside the dwelling and posing for a portrait. When I mentioned his research method, “No way, I’d never do that!” rounded the room. When I asked why, my students said that apart from being too much work, it was too private. Accordingly, Menzel’s photographs show a rare combination of pride and vulnerability on the part of his subjects. Even though the book dates from more than a decade ago, the power of the portraits endures.

I hated to do it, but I had to cut up the book, two books in fact. I found two used copies for less than $15 each. Working on our living room floor, my daughter and I mounted all the complete two-page portraits and statistics we could cull from the two dismantled books on construction paper and laminated them. Each set of statistics included the square footage of the family’s dwelling and number of people living in it. This yielded 21 tabletop posters that students could examine in pairs.

The next day, I asked my students if they ever felt crowded at home. I planned to use their stories to tie our study as tightly as I could to students’ own experience and to bring their attention to another layer of the mathematics: comparing home sizes wouldn’t have much meaning unless they considered the number of people sharing the space.

Lizzie told the story of a few years spent living in the city. She said, “Here in Washington we had a house. We moved to Oakland and we didn’t even have room in the kitchen for a cutting board.”

“Did you have the same number of people in the household?” I asked.

“Yes, and I had to share a room with my sister. We got rid of almost everything.”

“So your living space was smaller and with the same number of people you had less space per person,” I confirmed.

She nodded.

Marcos told about how his house used to have four people living in it, and now has eight. “That’s half as much area for each of us,” he said.

I then asked students to talk with one another about the relationship between the area of a home and the number of people living in it.

Zach summarized for the class, “If the area stays the same with more people, you have less space per person. Both the size of a home and the number of people in it decide how crowded it will be.”

I then introduced the portraits from Menzel’s book by proposing that looking at how people in other parts of the world live might make help us understand what’s happening with housing in our own country. After I passed out the posters, I directed students to record the square footage per person for each portrait they examined. Introducing square footage per person and making that the basis of comparison would keep students engaged in making meaning mathematically while engrossed in the social issues raised by the portraits. They were to come to me for a quick notebook check between each poster. This allowed me to assess their understanding of the math. I also got to have brief individual conversations with students about what they had observed.

At first, most of their observations had to do with the pictures. The 1994 photograph of Bosnia features a bombed apartment building and a father with a machine gun. Iraq proved popular, and perplexing to students. Instead of an image of war, Menzel’s book featured a peaceful pre-invasion extended family posed on the roof of a spacious home. The portraits gave rise to such interpretations as: “The Vietnamese family loves their house very much, ” and “In South Africa, a good home equals a good life,” in spite of the cramped quarters in these places.

Gradually, the numbers began to tell a story as well. The South African family of 7 shared 400 square feet. That’s 57 square feet per person. In Vietnam, spacious by comparison, each person got 172 square feet. In a moment between checking notebooks, I sought out Lisa and shared with her another student’s comment, “None of these people are rich. They are just like us.” I asked her if she agreed.

“Well, yes in a way,” she said thoughtfully. “Their circumstances have more to do with the country they’re in than with themselves. But none of these houses are anywhere near as big as the U.S. average, so I also don’t agree.”

### Squaring the U.S. Against the Globe

After my class had spent a little over two class periods examining the portraits and calculating the square footage of living space per person, I heard them discussing with each other the difference in size between the United States dream home and homes in the rest of the world. I wanted to reinforce just how different that was, and I also wanted to bring students’ observations into a discussion that could prompt some written reflection. On our last day with the posters, I announced that students would give presentations, and that in those presentations they’d convey the story told by the pictures and the story told by the numbers, and make a visual representation from which to draw some conclusions.

I told students to take notes in the form of squares representing the square footage per person, all drawn inside a scale square representation of the modern American 2,400 square foot house. Between presentations students made efficient use of the square root function on their calculators and their rulers to draw little squares inside their representation of an average American new house. By the time they’d heard 15 presentations, and drawn in 15 squares to represent the living space allotted to 15 people living in other parts of the world, they still had lots of room to spare in their house. Before students left class that day, I asked them to write down one thought about what they had learned based on the picture in their notebooks. Two sentiments dominated: Americans are spoiled and lucky, and people in other places are poor and unlucky.

I felt disappointed. I wanted them to think more deeply, draw on their own experiences and those of their classmates, and include mathematical reasoning in their responses. The next day I asked students to respond in writing to some more pointed questions. I asked, “What do you think Americans give up in order to have big new houses?” and “Can everyone in the United States get a big house if they want one?” I also asked explicitly that students use mathematics to support their reasoning.

Students commonly reached the conclusion that people trade time with their families in order to have more space, more money, and more stuff, and that having a small house could actually improve family life because people would be closer to one another.

Angel, who I knew had been homeless, wrote, “I’d never really thought about the size of our homes… I’d always wanted a big, big house… but it doesn’t benefit us to build such big houses. They take up space we may need in the future. People may even go homeless because they can’t afford one of the oversized houses. If 57 square feet per person is enough in South Africa, why can’t we be satisfied with 500 square feet which is almost 10 times as much.”

### Falling Short

Not as many of my 8th graders incorporated math into their written reflections as I’d have liked. It’s possible I hadn’t provided enough nuanced background from which to draw. While comparing homes from around the world spurred thinking on the part of my students, I skirted the issue of unequal resource distribution within nations. A day spent on the range of living arrangements in the nations featured in Menzel’s book would have taken advantage of students’ enthusiasm for the portraits as well as paved the way for discussion of inequities in housing arrangements here at home. Big new houses, while ubiquitous, are available to only a small minority of people. Expanding this background might have supported more students to make math connections in their writing. It would also have integrated statistics concepts into students’ learning.

I also tiptoed around having students describe their own housing arrangements. Their families’ housing is something middle school students have no control over: it’s purely a chance of birth. In the part of the county my school serves, housing ranges from small damp apartments and leaky trailers to luxurious gated communities. As one student said early on, “A house shows your wealth, just like clothes and shoes.” On one of the last days of our study, I overheard Luke tease Ally for her enormous house “with servants,” while Josh told me privately that his house was only 400 square feet, and he didn’t want anyone to know. I would like to have supported my students in considering how our economic systems perpetuate the “good luck” and “bad luck” of social class in their own lives. The next time, I will do more to gently surface these issues from the outset.

Even with these shortcomings, teaching square roots by looking at living space connected a difficult topic in middle school math to real human concerns both inside and outside the classroom. On their exam, my students showed far better mastery of square roots than in past years and it opened their eyes to how the rest of the world might perceive Americans. As Ryan wrote, “It makes me sad because of how much we have compared to others.” Caleb echoed, “They could resent that we’re all rich, even though we’re not.”

For me, learning that students have only eight square inches of desktop to spare helped me see the classroom from an 8th grader’s point of view. I’ve also taken a cue from my young students and started celebrating the tight living quarters of my own family more than I complain about the mess. And I’ll be ready when the next offer of surplus desks rolls by.n

### Resources

Lappan, Glenda, (2007). *Looking for Pythagoras: Connected Mathematics 2*, Boston: Pearson-Prentice Hall.

The Math Learning Center. *Math and the Mind’s Eye*, Salem, OR. Available at www.mathlearningcenter.org.

Menzel, Peter, (1994). *Material World: A Global Family Portrait*, San Francisco: Sierra Club Books. (A set of twelve posters made with images from the book along with a curriculum guide is also available from Social Studies School Service at www.socialstudies.com.)

Wilson, Alex and Jessica Boehland, (2005). “Small Is Beautiful: U.S. House Size, Resource Use and the Environment,” *Journal of Industrial Ecology*, 9: 1-2. 277-287.