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One Saturday evening this past summer, I found myself engaged in this text exchange with my then-9-year-old son, who was reaching out to me via his dad’s phone. He was away from home for the weekend, and the wellbeing of his newly adopted dwarf hamster Peanut was clearly front-of-mind.
In the exchange, you can see how important this issue was to him. You also get a flavor for what it’s like to be my child: I will always help you … but we're also going to need to get clear about our units!
The Difference Between Length (inches) and Area (square inches)
The Difference Between Length (inches) and Area (square inches)
450 inches is a measure of length, the distance from one point to another. It’s equal to 37.5 feet. But that is not a sensible way to describe the amount of space a dwarf hamster requires for comfortable living.
(Let’s pause to see what a dwarf hamster is. This is Peanut. She is tiny. She sits easily in the palm of a child’s hand or in a large scallop shell, as you can see. She is about 2 inches long. She is the cutest ever and a really, really delightful, low-maintenance pet.)
In contrast to 450 inches, 450 SQUARE inches is a reasonable way to describe the amount of space a hamster requires. That’s because a SQUARE inch is a measure of area or two-dimensional space, not just of length. So 450 inches in length is 37.5 feet: that's really long, compared to our 2-inch-long hamster. How big is 450 square inches? What kind of space could we set aside for Peanut to ensure she has enough room to be happy and well?
Well, an easy start is a space that’s 10 inches by 45 inches (a rectangle that is not-quite 1 foot by not-quite 4 feet). It would look like a long skinny rectangle. Finding a space in our house for a nearly 4-foot-long bin would be hard. This shape would provide enough space, but it's not a good fit for our home.
What are some other rectangles with an area of 450 square inches?
What are some other rectangles with an area of 450 square inches?
What other dimensions could give us a total area of 450 square inches? We can keep the total area constant, while cutting and rearranging that area (if we think of the problem in a visual way).
I want to pause here to clarify that this is the sort of out-loud reasoning I do with my kids. Often, I’m simply modeling for them that with enough information and creativity, we can reason through most mathematical questions we ask ourselves.
The "best" answer is not always an "exact" answer.
The "best" answer is not always an "exact" answer.
450 square inches is not an exact requirement: it’s a starting point, a reasonable target for how much space Peanut requires to be healthy and well in our care. That means a 20-by-24-inch bin with a total footprint of 480 square inches, like the one we ultimately got for Peanut, is a great solution to the problem of providing our dear hamster with a living space that meets her needs.
Understanding the difference between length and area units (inches versus square inches) and considering the relationship between the dimensions of a rectangle and its area helped us select a bin that was right for Peanut and for the space in our home.
Math gives us opportunities to build relationships and make choices that show care for others.
Math gives us opportunities to build relationships and make choices that show care for others.
When I start mathematical conversations with children based on things they really care about, it always presents opportunities for us to talk about our values. Sure, we use data and information to make good decisions, and it's important to become skilled in doing so. But we also use our values to inform the decisions we make. Mathematics, informed by compassion and a sense of responsibility to others, can be a powerful force for decision making and change.
My children and I also develop stronger relationships with each other when we talk thoughtfully about what matters to us. In the end, I told my son how proud I was of him for researching what his hamster needs, asking for help from me, and making good choices to meet the responsibilities of caring for her.
NOTE: All animals require some optimal amount of space for comfortable living and activity. Actually, we humans do too. And those space requirements (often reported in square or cubic feet) depend upon how many of us will present and what we’re doing in the space. For example, school buildings have a minimum number of square feet in each room, based on the expected number of students, the age of the students, and the way in which the space will be used. For example, a regular classroom requires (in theory) fewer square feet than does a wood shop or art room. Cafeterias, where students spend just a little time, mostly seated, require fewer square feet per student. Naturally, none of these requirements were established with pandemic safety requirements in mind, which is part of why it has been challenging for many of us to return to school with our students, colleagues, and community members.
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