Equi and Stretch

Stretch crouched down by the chalk lines. "Here's what I'd start with," she said. "Each group has the same outside boundary. One-quarter of the whole plot, right? Their strip. So when we look INSIDE each strip, every group is slicing their own quarter into sub-portions."

"Right."

"Herbs group's strip is one-quarter of the plot. They need to slice their quarter into three equal sub-portions. Each sub-portion is one-third of their quarter. Which means each sub-portion is one-twelfth of the whole plot, because thirds-of-quarters is twelfths."

Equi blinked. "Did you just multiply?"

"I just multiplied. One-third times one-quarter equals one-twelfth. Same idea as your equivalence rule, but applied to a slicing problem."

"That's not multiplication, that's scaling. You took the herbs group's quarter and you stretched it down into a finer scale. The quarter became three twelfths."

"Yes," Stretch said. "That's what I do. I stretch fractions to a common scale so we can compare them across groups."

Equi thought about this. "So if I want to know how much space basil gets relative to one whole row of lettuce..."

"You stretch them both to the same scale. The basil takes one-twelfth of the plot. One row of lettuce takes one-sixth of the salad-greens strip, which is one-twenty-fourth of the whole plot. So basil and one row of lettuce, both on the same scale: one-twelfth versus one-twenty-fourth."

"Which means basil is twice as big as one row of lettuce."

"Right."

Equi was quiet for a long second. Then she said: "I have been teaching kids that 2/3 equals 4/6 for two years and I have never thought about it as stretching before. I have always thought about it as naming the same number differently. Like a kid having a nickname."

"Both are right," Stretch said. "Same idea seen from two angles. You see the equivalence. I see the scaling. Together we get the answer."

Equi picked up a fresh piece of chalk. "Okay," she said. "Let me try restating what you're saying so I'm sure I have it. We can't just give each group their strip and call it a day, because each group will slice their strip differently. So we need to scale every group's sub-portions to a COMMON scale, so we can see who has more space and who has less and whether everything still adds up."

"Right."

"And the common scale is — what? Twelfths? Twenty-fourths?"

Stretch took the chalk from Equi and drew a small grid in the dirt. "We need a common denominator. Something all four groups' sub-portions can be expressed in. Herbs needs thirds-of-the-strip. Salad-greens needs sixths-of-the-strip. Root-vegetables needs halves-of-the-strip. Sunflowers needs the whole strip — which is just one."

"Three, six, two, one," Equi said.

"What's the smallest number that all four of those divide into evenly?"

"Six."

"So if we slice each strip into sixths, then:"

She drew on the dirt:

Herbs group's three sub-portions take up TWO sixths each. Salad-greens group's six sub-portions take up ONE sixth each. Root-vegetables group's two sub-portions take up THREE sixths each. Sunflowers group's one sub-portion takes up SIX sixths (the whole strip).

"Now everyone is on the same scale," Stretch said. "Sixths-of-a-strip."

Equi looked at the diagram in the dirt. "And this is what you mean by common denominator."

"Yes."

"And this is also what I mean by equivalent fractions. Every group's portion has been renamed in sixths."

"Yes."

"They are the same idea."

"They are the same idea."

Equi stood up and brushed the dirt off her knees. "All right," she said. "Let me also restate one more thing, because I want to be sure. The thing we just did is: we took four groups with four different ways of slicing, and we scaled them ALL to a common base — sixths — so we could plan the whole garden as one shared scale. Each group still slices their own strip however they like. But on PAPER, on this diagram, everything is in sixths."

"Right."

"So when the herbs group says 'basil gets one of our three sub-portions,' we can translate that to 'basil gets TWO sixths of the herbs strip.' And when the salad-greens group says 'arugula gets one of our six sub-portions,' we translate that to 'arugula gets ONE sixth of the salad-greens strip.' And both of those measures are now comparable, even though basil and arugula are in different strips."

"You just defined the whole point of common denominators in one sentence," Stretch said.

"I just restated the whole point of common denominators," Equi said. "And the only reason I could restate it is because you showed me what scaling looks like in dirt."

She smiled.

"My equivalence rule and your scaling rule are the same trick, told two different ways."

"And applied to garden plots, they let us share space fairly."