None of the suspected small animals visited that night, and Miko’s hand fared well. The tide was rising again as they ate breakfast, so they quickly gathered more firewood, and grass for Tera, before the beach was flooded.
Ilika took the morning, around the fire in the cavern, to introduce the arithmetic for multiplication. They rejoiced, freed at last from making tedious and boring rectangles of dots.
Soon after mid-day, the tide began to retreat and the sun came out to warm the beach. Ilika went into town with Mati and Sata to sell the two unneeded saddlebags. Mati easily found a crock of molasses for sale, but Ilika had to ask in several shops before locating a small stack of writing paper.
By the time they came to the cheese maker on the northern edge of town, the trio knew they were being followed. Ilika remained calm as they acquired a wheel of hard cheese, then added bread and a berry pie from the bakery.
Sata glanced around constantly, and mumbled something about how safe she felt in the wilderness.
When the shoppers returned to the cave, the tide was low and everyone was outside seeing what the ocean had left behind. At many places on the beach, laughter and squeals of delight came from the students as they discovered strange shells and mysterious creatures with tentacles. Sata helped Mati dismount, and the two friends went down to the water’s edge to see what they could find.
Ilika joined Kibi, prying mussels from a rock.
NEBADOR BookTwo: Journey 107
“We’ve had fun here,” she began, “but we might have our own thief. I’m sure there were more carrots last night, and Buna can’t find her comb.”
“Sounds like a small thief.”
“That’s what I was thinking. Some critter now has a comb in its nest somewhere, and a belly full of carrot.”
They both looked at each other in silence for a long moment, half-smiles on their faces . . . then turned their attention to some gulls squabbling over a morsel nearby.
“Addition and subtraction are opposites. Multiplication has an opposite too. Can anyone guess what it might be?”
Toli fidgeted to buy some time. Ilika could see a sparkle in Rini’s eyes.
“Rini?”
“I imagine a loaf of bread. When we multiply, it’s like stacking up several slices. So the opposite would be taking a whole loaf and slicing it.”
“Can everyone picture that?” Ilika asked.
Most heads nodded, Toli and Sata most vigorously.
“Slicing is a good name. It’s actually called division. It is not commutative, cannot be done in either direction, just like subtraction.”
“The first number is the size of the whole, the second is how many equal slices we want, and the answer is the size of the slice, in the same unit as the whole.”
The rest of the evening around the fire was devoted to drawing and discussing pies — plum pies divided into two, four, and eight equal parts, apple pies divided into three, six, and twelve, and finally berry pies divided into five and then ten. The last division into tenths would be just right for their group, Neti pointed out.
With a grin, Sata brought out the real pie.
After getting their bedrolls ready, everyone carefully packed all their small
belongings into rucksacks or saddlebags. Ilika sat by the fire a little longer, then began looking around with a worried expression.
Boro noticed. “Sata? She’s out looking at the ocean,” he said, putting another piece of driftwood on the fire.
Ilika wandered outside. The tide was most of the way in, but not yet high.
By the light of the waxing moon, he could see his youngest student sitting on the sand just above the reach of the waves. He joined her.
She smiled at him but didn’t say anything.
After a few minutes, he said, “The ocean is very powerful, has many dangers, and some funny smells. Different, but also similar to the steam vent and hot spring terraces. How are you doing with places like this?”
She rolled the question around in her mind for a moment. “Slowly making peace with them. I’ve always had walls around me, and parents nearby. You do some of the things my parents did, and Kibi too, but you know when to let me . . . be grown up and take care of myself. It’s scary sometimes.”
They both took a moment to scoot farther up the beach to avoid a wave.
“Personal power is the ability to stand on your own two feet, with a smile on your face, in the middle of a universe that contains a million ways to crush you.”
Sata cocked her head thoughtfully. “Please say that again.”
He did, and added, “I didn’t make it up. It’s a well known saying in my country, especially in the Transport Service.”
“I’m working on the two feet. Maybe when they don’t feel so shaky, I’ll try the smile too. Is that the real million, or just a whole bunch?”
“I think in this case, it’s closer to the real million.”
They sat in silence for a few more minutes.
“I’m going in,” Ilika said. “Good night.”
“Good night. I’ll be in soon . . . or else the ocean will push me in.”
Ilika chuckled and headed for his bedroll.
“My pack is gone!” Neti gasped as soon as she crawled out of bed the next morning.
Everyone scrambled to check on their own, then help Neti look for hers.
After a quarter hour of searching, even with Ilika’s bracelet light, they
remained clueless. Not even a scrape mark in the sand, where it might have been dragged, could be found.
“So much for thinking it was just a small animal,” Miko grumbled as they gathered back at the fire ring.
Kibi and Rini started passing out things to eat.
“Only a person could make off with something that big without a sign,”
Boro stated.
No one disagreed.
“All your stuff was in it?” Rini asked.
“Yep. Clothes, comb, ointment . . . one pencil, and my hat. Luckily I keep my coin pouch in my bedroll at night!”
“Cloak?” Boro asked.
“No, it’s my pillow. And I have the clothes I’m wearing. Hey! That’s still more than I ever had as a slave!”
Ilika smiled. “Let’s consider our options.”
“Leave today?” Sata suggested, but quickly scrunched her face, clearly not happy with her own idea.
“Go back to having someone on watch all night,” Kibi proposed.
“Set a trap and catch the thief!” Buna asserted dramatically.
No one could think of other possibilities, so they discussed the pros and cons of each. Sata spoke against her own plan. In the end, feelings were split between setting a watch and setting a trap.
“I’d like to try Buna’s idea,” Ilika finally said.
Buna looked almost shocked, then grinned with happiness.
“It has the advantage,” he continued, “of possibly getting Neti’s stuff back, if the thief just has them hidden in one of the crawl-holes nearby. I have a strong hunch our thief thinks we are easy prey and will be back. And I have the means to render him harmless.” He tapped on his bracelet.
“I
knew that did more tricks!” Buna said with sparkling eyes.
Ilika put his finger to his lips. “So I’d like to ask Neti to wait a couple of days to replace her stuff, see if we have any luck.”
“Okay. Anyway, this is exciting!”
“We’ll talk about our plans more out on the beach this afternoon. I don’t want to talk in here because you-know-who could be listening. Tide is coming
in now, and it’s Boro’s turn to lead a shopping trip this afternoon. Toli and Buna, you want to go?”
“Sure!” they both agreed.
“We need potatoes,” Kibi said.
“Carrots,”
Neti
added.
“Pie,” Mati suggested. “Or sweet biscuits.”
“Yeah!” many voices agreed.
After they finished eating, Ilika had paper and pencil out.
“You’re all getting good with numbers. I’ll teach you the arithmetic for division soon, but let’s play with it now that you know about both multiplication and division.
“First of all, it’s important to remember that numbers, all by themselves, are just abstract concepts. To be of use, they need units attached to them. If I say ‘five,’ you know how many that is, but we aren’t talking about anything yet. If I say ‘five apples,’ all of a sudden it’s something real, something interesting.”
“Especially if you’re hungry,” Mati noted.
Ilika smiled. “But you have to be careful when adding or subtracting, and make sure the units are compatible. If I have five apples plus three plums, I have eight something, but it’s not apples or plums.”
“Fruit?” Kibi suggested.
“Good thinking. Sometimes you can find a word that applies to everything. What if I had three feathers plus two mushrooms?”
After the chuckling died down, Rini said, “Five things?”
“Yes, sometimes we have to resort to very vague words. With addition, we can still do it. Not with subtraction. If I have five apples and I subtract three plums . . .”
They all looked at each other.
Miko grinned. “If you can do that, you’ve been drinking too much ale.”
Everyone snickered as Neti elbowed him to be quiet. “So for subtraction, the unit has to be the same?” she asked.
“Yes. Now let’s switch to multiplication and division. They can actually combine units in useful ways. Today you are going to learn your first
NEBADOR BookTwo: Journey 111
mathematical formula that describes how reality works and allows us to measure, analyze, and predict real events in the world.”
He looked around. Seeing that he had their complete attention, he picked up paper and wrote.
“Speed, multiplied by Time, equals Distance.”
All his students stared with big round eyes as if he was revealing a magical incantation that would allow them to rule the world.
“Each variable in the formula has a unit, and they have to make sense.
Speed is in knots, Time is in hours, and Distance is in nautical miles. These make sense together because knots means nautical miles per hour, so all the units match.”
He looked around. Rini nodded understanding, but the rest needed a few examples.
Once they were comfortable with the formula and the correct units, Ilika’s eyes sparkled with mischief and he gave them a few more examples. The students enjoyed pointing out to their teacher that apples were NOT a unit of time, and cups were NOT a unit of distance.
When Boro, Toli, and Buna returned from town, they reported being questioned by the guards. Boro had remained very quiet, since he had already had a similar experience. A couple of silver pieces from Toli’s pouch, with trembling hands, made the guards bid them farewell. Buna hugged Toli again, as she had done on the road, pride showing on her face.
The three shoppers unpacked vegetables and sweet biscuits, and Boro held up the extra knife he had purchased, just in case something happened to the one they had. Buna had a new comb, and passed out lengths of colored ribbon to the other girls, with hers already shining from her tangled hair.
During afternoon lessons, all their belongings came out to the beach with them. Smoothed sand, with a bedroll cover and several sheets of paper, made
a workable writing surface.
Knowing the dagger-grip his students used in the capital city would not allow them to truly learn to write, Ilika showed them a more delicate hold.
They tried, but the pencils tended to spring away as if they had minds of their own. After many tries, Miko and Buna were nearly in tears. Rini had no better luck and couldn’t stop laughing.
With the passing of the first hour, some control was achieved. With finger muscles quickly cramping, most had to rest their hands after each letter.
Suddenly a snapping sound came from Boro’s direction. The large boy sighed.
Ilika handed him another pencil.
By the time they came to the letter Z, they were ready to burn their papers and throw their pencils into the ocean. Ilika collected the supplies and shooed them away to run on the beach or collect driftwood. Kibi stayed behind with Ilika, as others were in charge of dinner that evening.
“Will it get any easier?” she asked with a sad face.
He started massaging her shoulders. “Of course. It’s a physical skill, and our bodies take time learning new things.”
She relaxed and let her mind drift over the sea. “Is there a job on your ship I can learn to do?” she asked with her eyes closed.
“Oh, yes. I can already see which job each of you would do best.”
“Thank you. I know it will be a hard decision.”
“Actually, after the experiences we’ve had together, I could almost pick my five right now. I’m sure I’ll have no trouble by the end of summer.” In the silence that followed, he kissed her neck.
Kibi glowed with happiness. She knew Ilika couldn’t yet announce a final decision. She also knew he was telling her anyway, in hints and touches that her heart could understand clearly, even if her mind was still unsure.
Deep Learning Notes
As the author thought back over his own public-school education, he realized that none of his teachers ever told him what the word “arithmetic” meant. It was always presented monolithically, as if it was too huge to define: “Do
arithmetic.” For this reason, Ilika teaches his students “the arithmetic for multiplication,” and thereby reveals that it just means “calculation method,”
and is not so scary after all.
When Ilika talked about division, he described that two of the numbers needed units, and one number didn’t: N1u / N2 = N3u, where N1 is the dividend, N2 the divisor, N3 the quotient, and u the unit. Ilika was, of course, emphasizing understanding, instead of rote arithmetic.
The commutative property states that an operation of two terms can be done in either direction. Addition and multiplication are commutative: A + B = B +
A and A x B = B x A. Neither subtraction nor division are commutative.
In the Transport Service saying that Ilika shared with Sata, both parts are important: standing on your own two feet (being competent), and smiling (being happy). Many schools of thought emphasize only one of these. If competence is over-emphasized, a cold, military-like functioning results. If happiness is over-emphasized, a selfish attitude results.
When Neti’s pack disappeared, what thought process did Sata go through?
When Ilika proposed adding 5 apples + 3 plums, the students took their first step into set theory, searching for words that could describe the mixed total.
We can now be sure that the “mile” in that kingdom was the nautical mile, since Speed (knots) x Time (hours) = Distance (miles). “Knots” is short for
“nautical miles per hour.”
What values, involving apples and cups, might Ilika have put into the Speed x Time = Distance formula to see if his students were awake?
Ilika told Kibi he could already see what job each person could do best on his ship. What would you guess, at this point in the story, for each of the nine students?
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