Title: Be Fruitful and Multiply
Topic: Back to School: Arithmetic, Mathematics
Author's Notes: For SpeedyHobbit for her birthday. Beta by RiverOtter.
Summary: Pippin has suddenly expressed an inability to appreciate just how the process of multiplication works, and Frodo makes shift to explain.
Word Count: 2,621
As Frodo entered the kitchen at Bag End, he noticed that there was a cup on the table from which someone had been drinking cambric tea. He sighed. “Pippin!” he murmured under his breath. “He’s run away from the Great Smial again!” And sure enough, he found the door to the cool room not fully closed, the bottom of the kettle had been allowed to hang too low over the fire and was charred black, and the milk jug sat upon the kitchen dresser, the square of linen cloth usually kept over it to screen out foreign tastes and possible dust lying crumpled upon the floor where it had been dropped, an easily seen circle of hardening white showing the former level of the liquid within the dark brown pottery, while a strand of light auburn hair floated upon the surface of the milk left in the pitcher.
Frodo frowned down into the jug, torn between Bagginsish thrift (just remove the hair, put a fresh cover over the jug, and return it to the cool room so the remaining milk doesn’t go completely to waste—it will still do well in griddle cakes!) and Brandybuck revulsion (ew—there’s a hair in it!). Today the Brandybuck won out. He poured the milk into a dish and set it out to the side of the pavement by the back door where the hedgehog that dwelt under the lilacs or a visiting cat might enjoy it, washed the jug, and set it on the dresser. He’d have his young miscreant cousin go into the market to bring back a new tin of milk from the dairyhobbit, he decided. After stirring up the fire he checked the teapot and shook his head, cleaning it also; then wiped up a scattering of tea leaves and recapped the tea tin. He noted the lid had been replaced on the biscuit jar, if slightly at a tilt; the jar itself was nearly empty. I just filled it yesterday! he thought to himself, just as the bell by the front door tinkled cheerfully.
“A letter for you,” said the post Hobbit, touching the brim of his blue cap respectfully as he presented the thick envelope. Frodo smiled his thanks and slipped him a couple of brasses before he closed the door and examined the missive he’d just been given.
“From Paladin,” he decided, noting the hand in which the address had been written. He took the letter back to the kitchen. He scoured the bottom of the kettle and rinsed it under the pump before refilling it and setting it to heat. Once he fetched some eggs and a few rashers of bacon to cook for himself and his uninvited guest, he sat upon the corner settle to read the note.
I suspect that you already have Pippin with you, it advised. He and Lalia had quite the confrontation yesterday. It appears that he and a few of the other lads found her chair in a hallway where young Hazel, who is attending upon her aunt at the moment, unwisely left it while Lalia was discussing housekeeping in the back storage rooms. You know Lalia—she refuses to meet with others while seated in her chair—must sit at the table and all, and appear as if she walked there herself.
Anyway, the lads were amusing themselves by wheeling one another about in it, and Pippin had just begun his own turn being seated in it when Ferumbras happened upon them. The other lads had scattered before Pippin could scramble out of the contraption, so it was Pippin who was hauled along by his ear to face Lalia, and he was the one who had to apologize for all of them, although he wouldn’t tell her who else was involved, much to his credit. He’s been much better about that since that last visit you made to the farm before Bilbo left the Shire.
Frodo nodded his head at that. Pippin had been going through a phase of tattling for a time.
Would you mind him staying with you for four days? We will be returning to Whitwell then. And perhaps you can somehow help the lad with his questions about multiplication. He’s been driving our Cousin Turkigard quite mad with his questions about the process, and we don’t begin to appreciate just what the problem is as he appeared to be memorizing the tables quite well until his last visit with Merry a month ago.
Frodo considered this last as he refolded the missive and replaced it in its envelope. But then he heard a stirring from the tunnel leading toward the bedrooms. Tucking the envelope behind the soup tureen on the dresser, he set the skillet on the fire and prepared to start the bacon cooking.
While Pippin was off to the village square to fetch the milk, accompanied by Samwise Gamgee to see to it the lad set off back home in a reasonable time so that the milk would not begin to sour, Frodo delved into a closet in one of the guest rooms where Bilbo and he had stored some of the materials Bilbo had made back in the day the older Hobbit gave lessons to other Bagginses and to Samwise Gamgee to help make some learning more understandable. There had been a time when Sam had found multiplication incomprehensible, he remembered, until Bilbo had brought out a great box of small ceramic cubes done in many colors. By presenting Sam with four sets of cubes, each containing five cubes of a particular color, he was finally able to get Sam to appreciate that multiplication was merely a quick way of counting by a particular multiple, and at last the lad had begun to learn the tables so he could himself multiply swiftly and with creditable accuracy.
Although, as Frodo remembered it, it was Frodo’s own suggestion that Sam use his growing skills to figure out how many daisy plants his father had planted in a particular bed that appeared to confirm in the small Gamgee’s mind how useful multiplication could be. Once Sam realized that nine rows containing eight plants per row should indicate his father had planted seventy-two plants and then confirmed this by carefully counting each and every plant, Sam had finally been satisfied that this was a process that was useful to a gardener to know, and he’d set himself to remembering his multiplication tables well.
They ought to serve well enough, he thought as he dusted off the box with a hastily fetched rag, to teach a young Took his tables, too.
When Pippin returned, he took the lad into the study where he’d set out the box and a few slates, slate pencils, and the like, and began the lesson he’d planned.
“Your father indicated you appear to have been developing some questions about multiplication,” Frodo began.
“Yes. I’d learned it one way, but then it doesn’t appear to work that way with Hobbits, too,” Pippin responded.
Frodo blinked, not certain what the lad meant. “Well,” he began slowly, “perhaps this will help.” He set out two red cubes side by side with two blue ones, “Here are two sets of two cubes each. How many red ones do we have?”
“How many blue ones?”
“How many in all?”
“We say this as two times two equals…?”
“Two times two equals four. I know that.”
Frodo was surprised at the impatient tone his younger cousin took. “All right,” he said slightly cautiously, “let’s try two times three.”
Pippin reached into the box and brought out another blue cube and a red one, setting each with the others of its color. “Two sets of three,” he said. “Two times three equals six.”
Frodo nodded. “How high can you go?” he asked.
“I know through my sevens.”
“Show me seven times six, then.”
Pippin soon had seven sets of six cubes, each set a different color. “Seven times six is forty-two,” he said, laying his hand on each set and counting, “Six, twelve, eighteen, twenty-four, thirty, thirty-six, forty-two.”
Frodo was perplexed. He asked, “Can you write me your seven times?” and handed Pippin a slate and slate pencil.
In a matter of minutes Pippin had the whole list written in his swift and barely legible scrawl, and handed the slate to Frodo. Frodo had to have the lad tell him what two of the numbers were intended to be, and then made him erase them and write them again more clearly. Satisfied that Pippin knew his seven times table, he had him do the same with sixes, then fives, and then fours, threes, and twos in succession. At last he sat back and looked at the young Took with consideration. “I don’t begin to understand what the problem is. Would you like to learn your eights and nines, then?”
Before the mantel clock on the distant parlor had chimed the next hour Pippin could easily recite both his eights and nine times tables, and Frodo was impressed as the lad quickly used the cubes to set up each one and could count by the appropriate numbers through the tables.
“Now for the tens…” Frodo began.
But Pippin interrupted, “I know them already. Ten, twenty, thirty….”
There was no question he knew his tens, and Pippin easily guessed that to do his elevenses he’d count eleven, twenty-two, thirty-three, forty-four….
“I don’t begin to understand what the difficulty is, then,” Frodo finally said. “You know all your tables from ones to elevenses….”
But Pippin was shaking his head. “I can’t do ones, not any more,” he said.
“Because it doesn’t seem to work for Hobbits the way it does for everything else does,” Pippin explained again.
“Multiplication works the same for everything,” Frodo objected.
“Does not,” Pippin said.
“I don’t understand,” Frodo responded. “How is it different for Hobbits?”
“Well,” Pippin began, busying himself by putting the cubes back into the box, “I know that with these little blocks or with apples one goes, one times one equals one.” He dumped the remaining cubes into the box, then reached in and brought one out, setting it on the low table on which he and Frodo had been working. “One set with one block. One.” He brought out another cube and set it by the first. “One set with two blocks. Two.” He brought out another cube and set it with the others. “One set of three—three. One set of four—four.”
He went through up to twelve. “With the blocks, it’s one set of any number equals that number, right?”
Frodo nodded. “You are doing it correctly, Pippin.”
“And it’s the same switched around,” Pippin continued. “Two sets of one—two. Seven sets of one—seven. A hundred sets of one—a hundred.”
“So why isn’t it the same with Hobbits?”
“How is it different with Hobbits?”
Pippin sighed as if he were at a loss to explain himself. Finally he began, “Well, with my mum and dad, one times one equals four—or six, depending how you count it. And with Great-great Granda Gerontius and Great-great Grandmum Adamanta it was one times one equals twelve—or is it fourteen? Now, with you or Merry it almost works out to the same as with the blocks—one times one equaled one in those cases—if it isn’t three. It makes me muddled trying to understand why it’s all different numbers with different families. With Da’s family it was one times one equals six, if it isn’t eight. And with Great Uncle Rory and Great Aunt Gilda it was—“
Frodo’s head was whirling, but he thought he was beginning to understand the pattern. “Wait—that’s not quite the same. If you look at some families there aren’t any children at all,” he hazarded.
Pippin nodded his agreement.
“But having babies isn’t the same as multiplying….”
But again Pippin interrupted, shaking his head. “It is multiplying, so it should work the same as with apples or blocks,” he insisted,
“Why do you think it’s multiplying?” Frodo demanded.
“Because it says so in a book Merry was reading me.”
“Not all stories are exactly correct,” Frodo suggested.
“But it’s a book Bilbo copied out for the Brandybucks and that Merry read to me—a book Merry said Bilbo copied from the Elves. And the Elves wouldn’t be wrong, would they? After all, you said that they live long enough to learn when they were wrong before and get it right.”
Frodo nodded, feeling rather muddled himself. He was certain that if anyone would be precise in what they said and how they said it, Elves would do so. “Do you know what book it was?” he asked. Then he added, “For if Bilbo copied it for the Brandybucks chances are he did a copy for himself first, and we should have it here in this room.”
“It was called Tales of the Beginning Times,” Pippin said.
Frodo smiled. “Yes, although I don’t remember there being anything at all in there about arithmetic or multiplication.” He had already turned to the proper shelf, and quickly found the volume in question—a collection of stories written about the time the Elves first awoke by the Water of Beginning.
The story in question turned out to be the second one, in which Oromë the Hunter first met with the Elves who’d awakened by the shores of Cuiviénen. Frodo quickly scanned the tale and could not find any reference to arithmetic of any sort. “I don’t see how this convinced you that multiplying works differently with Hobbits,” he murmured as he turned back to the beginning of the chapter and began a second scan of the story. “Hobbits don’t appear in it at all. It’s about the earliest days of the Elves, and they were the only speaking creatures living at the time as I understand it.”
“But Merry said it was the same with Elves as it is with Hobbits, how they multiply,” Pippin said, pushing his head under Frodo’s arm to examine the story himself. He shook his head. “Bilbo’s writing is hard to read,” he noted.
Frodo shrugged. “Perhaps it might seem that way to you, but I have no difficulties with it.”
“But you’re used to Bilbo’s writing!”
But just then, as Frodo turned the page, the little Hobbit gave a grunt of recognition. “There—there it is!”
Frodo gave his little cousin a startled glance, then turned his attention back to the page. Yes, the word multiplied appeared there. He went back to the beginning of the sentence. “And the Vala of the Hunt and of the Wild saw that those who had awakened here had multiplied their numbers, and were as numerous as the leaves of the forest that grew beyond their settlement.” And a few lines further on he saw, “Yea, you shall be fruitful and multiply across the lands of Middle Earth, and fill them with your progeny.” As he read the two sentences aloud, he felt Pippin nodding vigorously.
“Yes, and Merry said that that was how people have babies, by multiplying. That when a mummy and a daddy come together they begin multiplying. But one times one should only make one if it worked the same for Hobbits as it does for apples and blocks.”
Frodo shivered as he closed the book and looked down to consider the young face looking up at him so expectantly. It wasn’t just a matter of arithmetic, he realized, that he needed to explain. How does one explain to a Hobbit as young as Peregrin Took the use of euphemisms?