This unit emphasizes intuitive reasoning with fractions and proportions. It also illustrates how to perform mental computations by simplifying problems. Lastly, it explores the notions of compassion and empathy.
All the characters in the story are dogs. If pictures are available of Yorkshire Terriers, Labradors, Poodles, and Pit Bulls they can be shown to children as the story’s characters are introduced.
Several discussion questions and exercises are provided at the end of the story to facilitate and enhance the learning experience.
_________________________________________________________________
El Pequeño Cocodrilo Peludo (The Little Furry Crocodile)
Bowwow and Tinkle were happy! It was family day. Their father Arfy, a Yorkshire terrier, had brought them to work with him. Their mother FiFi, a lovely, sweet, white toy poodle, had given each of her boys an extra dog biscuit to take with them!
Bowwow and Tinkle knew that Daddy was the local bookie. But they really didn’t know what he did. They were watching Arfy while he was talking to Rudy, a Golden Labrador who worked for him.
“O.K.” Arfy said to Rudy, his Lab assistant. “Raise the odds on Chihuahuas over Pit Bulls to five thousand to one. That should equalize the betting.”
“Daddy,” Bowwow asked, “what does that mean? What are odds?”
Arfy looked at Bowwow and smiled. Bowwow was a smaller version of him. Bowwow was confident, quite talkative, and always asked questions. Arfy walked over and licked Bowwow’s nose.
“You’re wet but cold, Bowwow!” Arfy said.
“Is that good, Daddy?” Bowwow asked.
“For you, yes. Arfy said. “For your Mother it’s another story, but I’m not talking noses there. Anyway, what did you want to know?”
“What does five thousand to one odds mean, Daddy?” Bowwow asked.
“Well,” Arfy explained, “it means that if you bet one dog biscuit on a Chihuahua to win and he does, then you will win five thousand biscuits!”
“Wow!” said Bowwow. “Five thousand biscuits!”
“You would be one little fat diabetic doggy!” Arfy observed.
Bowwow giggled.
“And what if I bet two biscuits, Daddy?” Bowwow asked.
“See if you can reason it through,” Arfy said. “Think it out.”
“Well,” Bowwow replied, “if I bet twice as much I should win twice as much. I would get ten thousand biscuits!”
“That’s my boy!” Arfy said. “You would be one little fat diabetic doggy on insulin!”
Bowwow giggled again. “You are so funny when you make medical jokes, Daddy!” he said.
Then Tinkle spoke up. When Arfy looked at Tinkle he definitely saw his wife’s child. Tinkle had his mother’s curly coat, long ears, and expressive eyes. Tinkle was the more timid one and got scared easily.
“Daddy,” Tinkle asked, “what do the Chihuahuas have to win at when they go against the Pit Bulls?”
“Cage fighting,” Arfy explained. “A Chihuahua and a Pit Bull are locked in a cage and fight to the death, or until someone gives up. It’s kind of like when you and Bowwow are locked in your kennel and he bites your ear until you say ‘uncle.’”
Tinkle’s tail dropped between his legs and he peed on the floor.
“Cleanup over here!” Arfy shouted.
Bowwow was curious.
“Daddy,” he asked, “what chance does a Chihuahua have against a Pit Bull?”
“Well,” Arfy explained, “it’s what we call a classic matchup of quickness versus size. The Pit Bull can swallow the Chihuahua in one gulp. But he has to catch him first! If the Chihuahua can stay low and gain what we call ‘leverage from the rear,’ he can grab onto the Pit Bull’s nuts and hang on tight. Eventually the Pit Bull will bleed out if he doesn’t say ‘uncle.’”
Tinkle’s tail dropped and he peed on the floor again.
“Rudy! Bring lots of paper over here!” Arfy ordered.
“You don’t have to bark at me!” Rudy responded. “I’m coming.” Rudy did not like being yelled at.
Rudy came over with a big roll of newspaper in his mouth. He dropped it and unrolled it with his nose so it could absorb Tinkle’s pee. Then he spread lots of paper around in other places in little piles. Tinkle looked up sheepishly at Arfy.
“I do not like it when you do that,” Arfy said to Tinkle.
“Do what, Daddy?” Tinkle asked. “Pee on the floor?”
“No.” Arfy said. “When you look at me like a sheep. Don’t be a ewe. Be you!”
Tinkle giggled. “You are so funny when you make homonym puns, Daddy,” he said.
Then Tinkle sat down and looked at the floor. He was obviously bothered by something.
“Daddy,” Tinkle asked, “what about old poor dogs?”
“What about them?” Arfy responded.
“Well,” Tinkle said, “a lot of them are so poor they might not even have one extra biscuit to bet. Shouldn’t they have a chance to bet too?”
“Wow, Tinkle!” Arfy said. “You are so much like your Mother!”
Tinkle wagged his tail and giggled again.
“Yes, they should have a chance,” Arfy responded. “And that is why we have Senior Thursdays! We allow senior dogs to bet halves or even one quarter of a biscuit!”
“That is so neat, Daddy!” Tinkle said.
“Actually, it was your Mommy’s idea,” Arfy explained. “In fact, she cut me off once until I promised to start Senior Thursdays.”
“What did she cut off?” Bowwow asked.
“That’s not important,” Arfy replied. “One day you’ll understand. But take it from me. Your mother can be a bitch!”
“So what if a senior bets one quarter of a biscuit and wins?” Bowwow asked. “What would he get?”
“Reason it out, Bowwow,” Arfy prompted.
“Well,” Bowwow said, “if I bet one biscuit I win five thousand. If I break a biscuit into four equal pieces and bet one piece, I should win five thousand of the same size piece back.”
“That’s a great way to model the problem!” Arfy said. “A wonderful geometrical interpretation, just like they teach in schools these days. So what’s your answer? A big pile of biscuit pieces?”
“Oh, Daddy! You are such a disgruntled traditionalist!” Bowwow said. Then he thought for a moment.
“Don’t let the fraction throw you,” Arfy suggested. “Just use the same reasoning that you did the last time.”
“Well,” Bowwow continued, “if I bet one fourth as much, I should win one fourth as much. So I should win five thousand divided by four biscuits!”
“Excellent!” Arfy said. “Now see if you can do that division without using a calculator.”
“Well,” Bowwow said, “I know that four goes into four thousand one thousand times. That’s easy. And four goes into one thousand two hundred fifty times. That’s easy too. So if I just add them I get one thousand two hundred fifty?”
“Right!” Arfy said. “You have a future in this business!”
“Would that be enough biscuits to make a senior dog diabetic?” Bowwow asked.
“Most of them are already,” Arfy said. “Don’t worry about it.”
“Wow!” Bowwow said. “You do stuff like this all day? You are smart Daddy!”
“Tell your Mommy that,” Arfy said. “Mommy thinks biscuits grow on trees.”
Tinkle looked very sad. Arfy noticed.
“What is it, Tinkle?” Arfy asked.
“Daddy,” he asked, “what about the Chihuahuas that fight the Pit Bulls?”
“What about them, Tinkle?” Arfy responded as he made a cross on his chest.
“Have you ever seen a Chihuahua beat a Pit Bull?” Tinkle asked.
Suddenly Rudy’s voice boomed out!
“El pequeño cocodrilo peludo,” he said. Both he and Arfy stopped to give a moment of silent respect.
“What does that mean, Daddy?” Tinkle asked.
“That was what they called him in Spanish,” Arfy explained. “The little furry crocodile! Or just Little Croc for short. He was the last Chihuahua that ever won.”
Rudy walked over and joined the group.
“It was twenty dog-years ago,” Rudy said. “His real name was Miguel. He was out of Tijuana. The best pound-for-pound fighter I ever saw.”
“Yes,” Arfy continued. “Tinkle, you better go stand on some paper.”
Tinkle stepped onto one of the dry paper piles.
“One day,” Arfy described, “Little Croc was up against Jaws. Jaws was the fiercest and meanest Pit Bull to ever enter a cage!”
Tinkle started trembling.
“What was Jaw’s real name?” Bowwow asked.
“Brucie,” Arfy explained. “But no one dared called him that.”
Both Bowwow and a trembling Tinkle stared at Arfy.
“The cage door opened,” Arfy continued, “and the two dogs entered. The starting buzzer went off! Jaws snarled and tried to bite Little Croc!”
Tinkle peed.
“But he lunged and missed!” Arfy related. “Jaws got nothing but air! Tinkle, stay on the paper.”
Arfy continued. “Little Croc was so fast that had run between Jaw’s legs and gotten behind him without Jaw’s knowing it! He jumped up and chomped down on you know what! Then he started twisting like a crocodile in his death spiral! Go ahead, Tinkle. We’ll wait.”
Tinkle peed again. Then he went over to stand on another pile of paper.
“Jaws howled and jumped way up in the air with Little Croc hanging on!” Arfy explained. “At the top of the leap Jaws cried ‘Uncle!’ Then he came back down and hit the ground.”
“So Little Croc won!” Bowwow said.
“Yes,” Arfy explained. “But Jaws landed on Little Croc’s head when he came down. Little Croc was declared the winner, but he was unconscious and he was never the same after that. He never fought again. Go ahead, Tinkle.”
Tinkle peed one more time.
Tinkle finished and got off of the newspaper. “Daddy, where is Little Croc now?” he asked.
Arfy looked at Tinkle. “Uhhhhh…..he coaches fighting hamsters,” Bowwow explained. “He has a Dojo down on Fifth Street.”
Tinkle got off of the paper, wagged his tail, and smiled. Standing behind Tinkle, Bowwow gave Arfy a knowing wink.
_______________________________________________________________
Questions and Exercises
(1) The payoffs in the story all vary in direct proportion to the bet size. Show other positive and negative examples of proportionality and how similar reasoning can be applied. Possible examples might include working twice as many hours to earn twice as much money, or tripling a dose of heroin and dying.
(2) Recall that Little Croc wins the fight, but is knocked out when Jaws falls on his head. Relate this to current controversies regarding concussions in professional sports, especially with regards to the National Football League in the United States. With younger kids talk about the need and limitations of protective gear in T-ball.
(3) Bowwow solves the division problem by reducing it to a series of simpler problems that he can handle. Highlight this and give examples about how this approach can be generalized to other areas in life. Consider, for example, getting even with a bully. First one can learn how to take digital photos, which is easy, then how to sneak into the teacher’s bathroom which is also easy, then how to set up a Facebook page in the bully’s name and post photos on them.
(4) EMPATHY EXERCISE: Have children pair off, with one pretending to be Bowwow and the other Tinkle. Put each pair on a mat on all fours. Let Bowwow bite Tinkle’s ear until he says ‘Uncle!” Then disinfect the ear and have the children reverse roles. Then disinfect the other child’s ear.
a. Have the children address the following questions: How did you feel when you were Bowwow? How did you feel when you were Tinkle? How do you think Bowwow feels about Tinkle, and Tinkle about Bowwow?
b. Identify all the children that cried during the exercise. Have them sit together in the middle of a circle and let the other kids torment them. You can do this just for fun.
(5) In the story, Bowwow is said to be like his father, Arfy. He is fearless, intelligent, and inquisitive. Tinkle is said to be more like his mother, FiFi. He is compassionate but also more fearful and has less control. Have the children consider whether or not the story is just another sad example of the insidious rampant sexism embedded throughout our society. If any of the children have a mother serving in a combat unit, ask them to demonstrate some of the killing techniques that she taught them.
All the characters in the story are dogs. If pictures are available of Yorkshire Terriers, Labradors, Poodles, and Pit Bulls they can be shown to children as the story’s characters are introduced.
Several discussion questions and exercises are provided at the end of the story to facilitate and enhance the learning experience.
_________________________________________________________________
El Pequeño Cocodrilo Peludo (The Little Furry Crocodile)
Bowwow and Tinkle were happy! It was family day. Their father Arfy, a Yorkshire terrier, had brought them to work with him. Their mother FiFi, a lovely, sweet, white toy poodle, had given each of her boys an extra dog biscuit to take with them!
Bowwow and Tinkle knew that Daddy was the local bookie. But they really didn’t know what he did. They were watching Arfy while he was talking to Rudy, a Golden Labrador who worked for him.
“O.K.” Arfy said to Rudy, his Lab assistant. “Raise the odds on Chihuahuas over Pit Bulls to five thousand to one. That should equalize the betting.”
“Daddy,” Bowwow asked, “what does that mean? What are odds?”
Arfy looked at Bowwow and smiled. Bowwow was a smaller version of him. Bowwow was confident, quite talkative, and always asked questions. Arfy walked over and licked Bowwow’s nose.
“You’re wet but cold, Bowwow!” Arfy said.
“Is that good, Daddy?” Bowwow asked.
“For you, yes. Arfy said. “For your Mother it’s another story, but I’m not talking noses there. Anyway, what did you want to know?”
“What does five thousand to one odds mean, Daddy?” Bowwow asked.
“Well,” Arfy explained, “it means that if you bet one dog biscuit on a Chihuahua to win and he does, then you will win five thousand biscuits!”
“Wow!” said Bowwow. “Five thousand biscuits!”
“You would be one little fat diabetic doggy!” Arfy observed.
Bowwow giggled.
“And what if I bet two biscuits, Daddy?” Bowwow asked.
“See if you can reason it through,” Arfy said. “Think it out.”
“Well,” Bowwow replied, “if I bet twice as much I should win twice as much. I would get ten thousand biscuits!”
“That’s my boy!” Arfy said. “You would be one little fat diabetic doggy on insulin!”
Bowwow giggled again. “You are so funny when you make medical jokes, Daddy!” he said.
Then Tinkle spoke up. When Arfy looked at Tinkle he definitely saw his wife’s child. Tinkle had his mother’s curly coat, long ears, and expressive eyes. Tinkle was the more timid one and got scared easily.
“Daddy,” Tinkle asked, “what do the Chihuahuas have to win at when they go against the Pit Bulls?”
“Cage fighting,” Arfy explained. “A Chihuahua and a Pit Bull are locked in a cage and fight to the death, or until someone gives up. It’s kind of like when you and Bowwow are locked in your kennel and he bites your ear until you say ‘uncle.’”
Tinkle’s tail dropped between his legs and he peed on the floor.
“Cleanup over here!” Arfy shouted.
Bowwow was curious.
“Daddy,” he asked, “what chance does a Chihuahua have against a Pit Bull?”
“Well,” Arfy explained, “it’s what we call a classic matchup of quickness versus size. The Pit Bull can swallow the Chihuahua in one gulp. But he has to catch him first! If the Chihuahua can stay low and gain what we call ‘leverage from the rear,’ he can grab onto the Pit Bull’s nuts and hang on tight. Eventually the Pit Bull will bleed out if he doesn’t say ‘uncle.’”
Tinkle’s tail dropped and he peed on the floor again.
“Rudy! Bring lots of paper over here!” Arfy ordered.
“You don’t have to bark at me!” Rudy responded. “I’m coming.” Rudy did not like being yelled at.
Rudy came over with a big roll of newspaper in his mouth. He dropped it and unrolled it with his nose so it could absorb Tinkle’s pee. Then he spread lots of paper around in other places in little piles. Tinkle looked up sheepishly at Arfy.
“I do not like it when you do that,” Arfy said to Tinkle.
“Do what, Daddy?” Tinkle asked. “Pee on the floor?”
“No.” Arfy said. “When you look at me like a sheep. Don’t be a ewe. Be you!”
Tinkle giggled. “You are so funny when you make homonym puns, Daddy,” he said.
Then Tinkle sat down and looked at the floor. He was obviously bothered by something.
“Daddy,” Tinkle asked, “what about old poor dogs?”
“What about them?” Arfy responded.
“Well,” Tinkle said, “a lot of them are so poor they might not even have one extra biscuit to bet. Shouldn’t they have a chance to bet too?”
“Wow, Tinkle!” Arfy said. “You are so much like your Mother!”
Tinkle wagged his tail and giggled again.
“Yes, they should have a chance,” Arfy responded. “And that is why we have Senior Thursdays! We allow senior dogs to bet halves or even one quarter of a biscuit!”
“That is so neat, Daddy!” Tinkle said.
“Actually, it was your Mommy’s idea,” Arfy explained. “In fact, she cut me off once until I promised to start Senior Thursdays.”
“What did she cut off?” Bowwow asked.
“That’s not important,” Arfy replied. “One day you’ll understand. But take it from me. Your mother can be a bitch!”
“So what if a senior bets one quarter of a biscuit and wins?” Bowwow asked. “What would he get?”
“Reason it out, Bowwow,” Arfy prompted.
“Well,” Bowwow said, “if I bet one biscuit I win five thousand. If I break a biscuit into four equal pieces and bet one piece, I should win five thousand of the same size piece back.”
“That’s a great way to model the problem!” Arfy said. “A wonderful geometrical interpretation, just like they teach in schools these days. So what’s your answer? A big pile of biscuit pieces?”
“Oh, Daddy! You are such a disgruntled traditionalist!” Bowwow said. Then he thought for a moment.
“Don’t let the fraction throw you,” Arfy suggested. “Just use the same reasoning that you did the last time.”
“Well,” Bowwow continued, “if I bet one fourth as much, I should win one fourth as much. So I should win five thousand divided by four biscuits!”
“Excellent!” Arfy said. “Now see if you can do that division without using a calculator.”
“Well,” Bowwow said, “I know that four goes into four thousand one thousand times. That’s easy. And four goes into one thousand two hundred fifty times. That’s easy too. So if I just add them I get one thousand two hundred fifty?”
“Right!” Arfy said. “You have a future in this business!”
“Would that be enough biscuits to make a senior dog diabetic?” Bowwow asked.
“Most of them are already,” Arfy said. “Don’t worry about it.”
“Wow!” Bowwow said. “You do stuff like this all day? You are smart Daddy!”
“Tell your Mommy that,” Arfy said. “Mommy thinks biscuits grow on trees.”
Tinkle looked very sad. Arfy noticed.
“What is it, Tinkle?” Arfy asked.
“Daddy,” he asked, “what about the Chihuahuas that fight the Pit Bulls?”
“What about them, Tinkle?” Arfy responded as he made a cross on his chest.
“Have you ever seen a Chihuahua beat a Pit Bull?” Tinkle asked.
Suddenly Rudy’s voice boomed out!
“El pequeño cocodrilo peludo,” he said. Both he and Arfy stopped to give a moment of silent respect.
“What does that mean, Daddy?” Tinkle asked.
“That was what they called him in Spanish,” Arfy explained. “The little furry crocodile! Or just Little Croc for short. He was the last Chihuahua that ever won.”
Rudy walked over and joined the group.
“It was twenty dog-years ago,” Rudy said. “His real name was Miguel. He was out of Tijuana. The best pound-for-pound fighter I ever saw.”
“Yes,” Arfy continued. “Tinkle, you better go stand on some paper.”
Tinkle stepped onto one of the dry paper piles.
“One day,” Arfy described, “Little Croc was up against Jaws. Jaws was the fiercest and meanest Pit Bull to ever enter a cage!”
Tinkle started trembling.
“What was Jaw’s real name?” Bowwow asked.
“Brucie,” Arfy explained. “But no one dared called him that.”
Both Bowwow and a trembling Tinkle stared at Arfy.
“The cage door opened,” Arfy continued, “and the two dogs entered. The starting buzzer went off! Jaws snarled and tried to bite Little Croc!”
Tinkle peed.
“But he lunged and missed!” Arfy related. “Jaws got nothing but air! Tinkle, stay on the paper.”
Arfy continued. “Little Croc was so fast that had run between Jaw’s legs and gotten behind him without Jaw’s knowing it! He jumped up and chomped down on you know what! Then he started twisting like a crocodile in his death spiral! Go ahead, Tinkle. We’ll wait.”
Tinkle peed again. Then he went over to stand on another pile of paper.
“Jaws howled and jumped way up in the air with Little Croc hanging on!” Arfy explained. “At the top of the leap Jaws cried ‘Uncle!’ Then he came back down and hit the ground.”
“So Little Croc won!” Bowwow said.
“Yes,” Arfy explained. “But Jaws landed on Little Croc’s head when he came down. Little Croc was declared the winner, but he was unconscious and he was never the same after that. He never fought again. Go ahead, Tinkle.”
Tinkle peed one more time.
Tinkle finished and got off of the newspaper. “Daddy, where is Little Croc now?” he asked.
Arfy looked at Tinkle. “Uhhhhh…..he coaches fighting hamsters,” Bowwow explained. “He has a Dojo down on Fifth Street.”
Tinkle got off of the paper, wagged his tail, and smiled. Standing behind Tinkle, Bowwow gave Arfy a knowing wink.
_______________________________________________________________
Questions and Exercises
(1) The payoffs in the story all vary in direct proportion to the bet size. Show other positive and negative examples of proportionality and how similar reasoning can be applied. Possible examples might include working twice as many hours to earn twice as much money, or tripling a dose of heroin and dying.
(2) Recall that Little Croc wins the fight, but is knocked out when Jaws falls on his head. Relate this to current controversies regarding concussions in professional sports, especially with regards to the National Football League in the United States. With younger kids talk about the need and limitations of protective gear in T-ball.
(3) Bowwow solves the division problem by reducing it to a series of simpler problems that he can handle. Highlight this and give examples about how this approach can be generalized to other areas in life. Consider, for example, getting even with a bully. First one can learn how to take digital photos, which is easy, then how to sneak into the teacher’s bathroom which is also easy, then how to set up a Facebook page in the bully’s name and post photos on them.
(4) EMPATHY EXERCISE: Have children pair off, with one pretending to be Bowwow and the other Tinkle. Put each pair on a mat on all fours. Let Bowwow bite Tinkle’s ear until he says ‘Uncle!” Then disinfect the ear and have the children reverse roles. Then disinfect the other child’s ear.
a. Have the children address the following questions: How did you feel when you were Bowwow? How did you feel when you were Tinkle? How do you think Bowwow feels about Tinkle, and Tinkle about Bowwow?
b. Identify all the children that cried during the exercise. Have them sit together in the middle of a circle and let the other kids torment them. You can do this just for fun.
(5) In the story, Bowwow is said to be like his father, Arfy. He is fearless, intelligent, and inquisitive. Tinkle is said to be more like his mother, FiFi. He is compassionate but also more fearful and has less control. Have the children consider whether or not the story is just another sad example of the insidious rampant sexism embedded throughout our society. If any of the children have a mother serving in a combat unit, ask them to demonstrate some of the killing techniques that she taught them.
