Casa di Mir Summer Mathematics: Adventure #3

Symmetry Salad

“Mommy, I’m eating symmetry salad – and it’s DEEE-licious!” Truthfully, I’m not sure my kid really cared all that much about the symmetry part – she was starving and that’s what I was making at the time. What, you may ask, is symmetry salad? It’s a version of fruit salad (a fruit platter really) that highlights the natural symmetry in most fruits.

Making this salad with your kids is a great way to explore symmetry. Start with a variety of different fruits, at least two of each. Try a few new ones, so the opportunity for exploration is there for everyone, including the grown-ups. For each piece of fruit, imagine there is an axis through the center that stretches from the “north pole” (the stem end) to the “south pole” (the blossom end).  Set two pieces of the same kind of fruit on a cutting board. You’re going to cut one of them in half through the axis and cut the other one in half through the “equator.”

Before you cut, ask your kids, “What do you think it will look like inside when I cut it?” Predictions may include color, texture, and shape among other ideas.  (Symmetry isn’t usually the first thing kids think about.)

Cut the first two pieces of fruit and take a look. Ask your kids, “What do you notice?” (If you have kids of different ages, let the younger kids respond first.) Let them ponder a bit and flounder through their musings about what they see. Even the older kids probably won’t use the formal mathematical language of symmetry. They might say, “This part looks like that part,” or “This side looks like a mirror image of that side.” The sharing of observations out loud is what helps people see things from many different perspectives.

It turns out that slicing most fruit through its axis results in slices that have reflectional symmetry across a line, with the axis being the line of reflection. Sometimes there is more than one line of reflection. (You can explore this further using a small rectangular mirror – lay an edge of the mirror along what you think is a line of reflection to see if the reflected image really does look like the other half.).  Strawberries, apples, citrus fruit, kiwi, and some stone fruit are a few good examples.

There are many fruits for which slices through the equator show rotational symmetry – that is, the slice has (more-or-less) identical sectors that radiate from the center – you could rotate the slice around the center by an amount that is less than a full turn and it will look the same as it did in its original position.  If you cut those sectors apart and stack them on top of each other, it’s pretty amazing how close the central angle measures are to one another. Examples include citrus fruits, apples, watermelon, star fruit, persimmon, and bananas among others.

Once you and your kids have looked at many different slices of many different fruits, you’ll have quite a lot to compare and contrast. You can finish your symmetry salad by cutting true slices of each fruit, anywhere from ½ to 1 inch thick and arranging them on a platter – although if your kids are anything like my kids, the fruit will get wolfed before it ever makes it to the platter!

About the author: Trisha Bergthold has been the middle school math teacher at Casa di Mir Montessori School since 2014. She holds a PhD in mathematics with emphases in curriculum and pedagogy. Prior to her work at Casa di Mir, she designed curriculum for kindergarten through college level. She also taught university-level mathematics courses for sixteen years.

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