RENE DESCARTES

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Random pairs line up? How is that possible?
I think, therefore I am? Was Descartes a figment of his own imagination? Click To Tweet

THE PLEASURE OF MAKING 10

story by George Gadanidis & Janette M. Hughes

1. MISSING GRANDPA

“Will Grandpa ever come back?” asks Alexander, turning to his sister.

“He will. I really miss him too,” responds Molly quietly. 

Alexander holds Molly’s gaze for a moment and then shares the information he’s been anxious to tell her. “I found his pocket watch.” 

Molly’s eyes widen.  “Where?” she asks urgently.

“In his room. Under his desk.” They both consider the implications of this quietly for a few moments and then Molly breaks the silence.

“That’s strange. I’ve never seen him without it.”

“I think he dropped it. The chain is broken, and the glass is cracked.” Alexander hands the pocket watch to Molly.

She puts it to her ear. “It’s stopped at 9:15,” says Molly as she winds it. “It’s not working,” she adds.

“I know,” says Alexander sadly.

Molly hands the pocket watch back to Alexander.

“I also found something else,” adds Alexander.

“What?” asks Molly.

“Come. I’ll show you,” says Alexander as he motions Molly towards Grandpa’s office.

Molly follows Alexander as he bounds up the stairs, turns right, and enters the dimly lit office.

“Here,” he says, pointing to the bookshelf beside the oak desk.

Alexander reaches up and pulls down a rectangular wooden box. “This is really neat,” he says excitedly.

Alexander opens the box and pulls out a booklet, and a die. “Grandpa wrote this. It’s a math adventure,” he says handing the booklet to Molly.

The booklet’s title, written on the cover, is Adventure #1. Molly opens it. “For Molly and Alexander, the Spring in my life,” Molly reads the dedication.

“He wrote it for us,” nods Alexander.

“What’s it about?” asks Molly.

“It’s a math surprise,” smiles Alexander. “And, it tells some of the history about René Descartes.”

Alexander opens the booklet to the first activity, The Pleasure of Making 10. “Try this,” he says. “Start with the equation __ + __ = 10. Roll the die to get the first number. Then calculate the second number.”

Molly rolls the die and gets a 3. “OK,” she says. “So the two numbers are 3 and 7. What’s the surprise?”

“Keep rolling,” adds Alexander.

Molly shrugs and continues. She records her results in the table provided in the booklet. After the third roll, she comments: “There are only 6 possibilities because the die only has 6 faces. I don’t have to roll the die anymore. I know that the other numbers I need to roll are 2, 5 and 6.” She writes these numbers in the first column in the table.

“Now what?” asks Molly.

“Now imagine the pairs of numbers as ordered pairs graphed on a grid. For example, for 3 and 7, we go 3 across and 7 up, and plot the ordered pair (3,7).”

Molly plots the six pairs of numbers on the grid, provided in the booklet. “Hey! They line up!” she says. “You’re right,” she adds. “That is a surprise!”

“I told you so.”

“Rolling the die to get the first number makes it seem random,” says Molly.

“But because the sum is always 10 we get a pattern.”

Molly looks up at the bookshelf. “There are more wooden boxes,” she notices. “Did Grandpa write more math adventures?”

“Not yet. The rest of the boxes are empty, except for Adventure #2.”

“Let’s take a look,” suggests Molly.

“But we haven’t finished this one yet. There are more math surprises!”

“OK,” concedes Molly, still gazing at the wooden boxes on Grandpa’s shelf. “Tell me about the next math activity.”

2. THE GATE

Alexander turns the page to the next activity. The page is not staying flat, so he places the stopwatch on the page to weigh it down.

“Wait!  The watch is ticking,” exclaims Molly breathlessly.

“What?”

“Grandpa’s pocket watch.  It’s ticking,”

Alexander picks up the pocket watch and puts it up to his ear. “No, it’s not.

You’re hearing things,” he says, as he puts it back down on the booklet.

“There! It’s ticking again,” says Molly pointing.

Alexander puts his ear down to the pocket watch. “It is ticking!” he says excitedly.

Alexander picks up the watch, then he puts it down on the booklet, then he puts it on the table but off the booklet, then on the wooden box, then to his ear, then down again.

“This is weird, Molly. Grandpa’s watch only ticks when it’s on the math booklet or on the wooden box.”

“How can that be?” asks Molly, puzzled.

“I don’t know,” he responds. “I’m going to set it to the correct time.”

“It’s 3:45,” says Molly glancing at the clock on the wall.

Alexander sets the pocket watch to the correct time. As he places the pocket watch back on the booklet, a thick fog quickly descends around them. Then an opening appears, like a gate. Through the gate, Molly and Alexander see an evening scene of what appears to be a military camp.

“Wow! What’s that?” blurts out Alexander.

“It’s scary,” adds Molly, sliding the pocket watch off of the math booklet. As she does so, the fog fades and disperses. The gate to the military camp remains open for a few more seconds. Then it too wanes and disappears.

“That must be where Grandpa is!” exclaims Alexander. “He needs his watch to get back.”

“What?”

“He must have opened the gate with his pocket watch. Then he must have dropped it just before he stepped through.”

“That explains the broken chain and the cracked glass,” adds Molly.

Alexander slides the watch back onto the booklet. The watch starts ticking, the fog and gate to the military camp reappear.

“Let’s go and find Grandpa,” urges Alexander as he grabs the pocket watch and jumps through the gate.

“No. Wait!” says Molly, as the fog starts dispersing.

But Alexander is already gone.

Molly grabs the math booklet, pauses in front of the waning gate, then jumps through herself.

3. THE MILITARY CAMP

Molly lands beside Alexander on a patch of damp grass. They find themselves on the side of a hill, overlooking a grid of tents. A half moon illuminates  the scene. Scattered torches mark points between tents. A few soldiers are milling about. Sentries can be seen pacing the periphery.

“This can’t be happening,” says Molly. “It’s like we’re in a science fiction movie.”

“Let’s look for Grandpa,” suggests Alexander.

Alexander starts walking down the hill. Molly hesitates then follows him. They enter the grid of tents.

“Let’s try that bigger tent,” suggests Alexander, pointing to a tent up ahead.

“Who goes there?” a voice says behind them.

Alexander turns to see a sentry running toward them.

Molly grabs Alexander’s hand and pulls him to the right. They run between tents, zigzagging as they go. Right, left, right, left, then left again. When they reach the big tent, Molly lifts the side and they crawl inside. They glance around from behind an armoire. It’s a spacious tent, lit by 3 torches. Diagonally across, a man with long black, wavy hair is standing with his back to them, looking at the opposite tent wall. He seems lost in thought.

Molly and Alexander listen as the sentry’s heavy footsteps approach. They can hear his laboured breathing as he runs by. Then he stops.

To their right, the tent flap opens and the sentry enters. Sweat is running down his face.

“Monsieur Descartes,” he says.

The man with the long curly black hair waves him away.

“We have intruders,” the sentry stutters.

“Go away,” responds the man absorbed in thought, and waves him off again.

The sentry backs out of the tent. His footsteps pause then disappear into the night.

“They are not speaking English,” whispers Alexander, confused.

“I know,” concurs Molly.

“But I understood everything they said.”

“Me too,” whispers Molly. Then, opening the booklet to the last section, she adds, “That must be René Descartes.” She shows Alexander the painting of René Descartes depicted in the booklet.

Alexander nods. “Look at the tent wall,” he adds. “Look at what he has written.”

“It’s x + y = 10!” says Molly excitedly. “Like Grandpa’s booklet!”

4. RENE DESCARTES

Crouched behind the armoire, Molly turns to the back of the booklet and reads in a whisper. “René Descartes was born in La Haye, France, in 1596. While a student at the Jesuit college of La Fleche, his teachers allowed him to sleep in in the morning because of his fragile health. Staying late in bed in the morning was a habit that he kept for his entire adult life. In 1619, he joined the Bavarian Army for nine years.”

“This means we’ve traveled back to the seventeenth century,” comments Alexander.

Molly continues reading. “Then he settled in Holland and spent the rest of his life as a mathematician and philosopher. Late in his life, he was invited to Sweden by Queen Christina. Her demands of studying early in the morning may have helped cause Descartes’ death. The lack of sleep may have compromised his immune system. He caught pneumonia and died in 1650. René Descartes is also well known as the mathematician and philosopher who proved his own existence by stating, ‘I think, therefore I am’.”

Molly and Alexander do not notice René Descartes standing over them until he grabs them by their arms and raises them to their feet.

Molly screams. Alexander swings his free arm, trying to hit Descartes.

“Don’t struggle,” says Descartes. “I don’t mean to harm you,” he says smiling. “Who are you?”

“I’m Molly,” says Molly bravely. “This is my brother, Alexander.”

“And you are René Descartes,” adds Alexander.

“I am. How do you know of me?”

“You are famous. You are the mathematician who said ‘I think, therefore I am’.”

René Descartes lets go of Molly and Alexander. “Cogito, ergo sum,” says Descartes in Latin. “Yes. This is the proof of my existence.”

“Unless you are a figment of your own imagination,” adds Alexander.

René Descartes breaks out in laughter. “You have a sense of humour, my little friend. But it’s more likely that you and your sister are figments of my imagination. It’s late at night, I am weary from today’s battle, and you are the most peculiar children I’ve ever met.”

“We exist too,” says Molly. “But we are not from your time.”

“This would explain your peculiar clothing.”

“We’re from the twenty-first century,” offers Alexander. 

“And you say that I am famous in your century?”

“Yes. Here’s a painting of you,” says Molly as she opens the math booklet.

“Ah, yes. That is a good likeness of me,” he says as he studies his image. What does it say beside the painting?”

“It says that you helped link geometry and algebra, by developing a way of representing algebraic equations geometrically,” says Molly.

“Hmm. I haven’t made this discovery yet. But I think I am close.”

René Descartes walks over to the opposite tent wall, where he has been writing with charcoal. “You see, I have this equation, x + y = 10. And I have created a table to show some of the values of x and y that satisfy the equation. I have been trying to map the values on this grid.”

“Molly and I were working on this today too. If you plot the x values along the horizontal axis and the y values along the vertical axis, you get a nice pattern.”

René Descartes follows Alexander’s instructions and plots some of the points. “This is wonderful. Here we have the algebraic equation x + y = 10 and here, on the grid, we have a geometric representation of it, as a line of points. There is so much beauty in mathematics! Don’t you agree?”

“I do,” concurs Alexander, smiling.

“And surprises, too,” adds Molly.

5. THE PARISIAN

“Thank you, Alexander and Molly,” says René Descartes.

“These are your ideas, monsieur Descartes,” says Molly.

“That might be, but you’ve helped me discover them. How can I repay you?”

“We came to find our grandfather. Have you seen him?”

“I’ve been in this treacherous battlefield for three months. All I have seen is soldiers. Soldiers and death.”

“His name is Timothy Harley.”

“Timothy Harley,” repeats René Descartes.

“He is a little taller than you. White hair. Glasses. This is his pocket watch.”

René Descartes looks closely at the watch. “Oh yes. I remember this fine instrument. I have met your grandfather. A fine gentleman.”

“Is he here?” asks Alexander excitedly.

“I am afraid not. I met monsieur Harley over a year ago, in Paris. We had dinner together and discussed mathematics. He is a great mathematician.”

“Have you seen him since?” asks Molly.

“I saw him again the next day, briefly. That was the last time.”

“How was he dressed?” asks Alexander.

Descartes considers the question for a moment, then responds, “He was dressed like a Parisian.”

“We were hoping to find him here,” says Molly sadly.

“Perhaps he returned to your time,” suggests Descartes.

“He has been missing for three weeks. He dropped his pocket watch before he left. I don’t think he can get back without it.”

“The pocket watch helps him travel through time?” asks René Descartes.

“Yes. We used it to come to your time,” says Molly.

Alexander puts the watch to his ear. “It’s ticking. Maybe if we set it to the correct time, the gate might open again.”

“The gate?” asks René Descartes.

“The gate back to our home,” explains Molly.

“It’s almost ten o’clock,” says René Descartes, looking at his own pocket watch.

As Alexander sets the watch to ten, the thick fog descends around them once again. Then the gate appears.

“It was very nice to meet you, monsieur Descartes,” says Molly.

“Thank you for remembering our grandfather,” says Alexander.

“Give him my regards when you find him.”

“Good-bye,” say Molly and Alexander in unison, as they wave and jump through the gate.

6. HOME

Molly and Alexander land in Grandpa’s office with a thud. As they look back they see René Descartes turning his attention to the mathematics on his tent wall. Then the gate closes and the fog dissipates.

Molly glances at the clock. “It’s still 3:45,” she says, looking confused.

“That’s interesting,” says Alexander. “It’s as if we never left.”

“Where is Grandpa?” says Molly forlornly. “I wish we had found him.”

“Maybe there are some clues here in his office. Maybe we can find out where he is and bring him back.”

Molly’s mood brightens a bit.  “Let’s take a look,” she suggests.

7. EXTENSIONS

Try the following activities along with Molly and Alexander. Have fun!

7.1 Making 8

“I wonder what will happen if we change the equation to __ + __ = 8?” asks Molly.

“Do you think the points will line up again?” asks Alexander.

“Let’s try it!” says Molly excitedly.


7.2 Making 6 and making 4

“Wow!” exclaims Alexander.

“Let’s try a few more,” suggests Molly.

“How about __ + __ = 6 and __ + __ = 4?”

“OK! Let’s plot them on the same grid as __ + __ = 8.”

7.3 X – Y = 4

“That’s awesome!” exclaims Molly.

“I wonder if we can write an equation whose graph slopes in a different direction?” asks Alexander.

“What if we change the plus sign to a minus sign, like   __ – __ = 4,” suggests Molly.

“Let’s try it!” says Alexander excitedly.


7.4 Is there a different way?

“It worked!” says Molly.

“Let’s look for another equation that slopes in a different direction,” suggests Alexander.

“I wonder if we can get a pattern of points that’s horizontal?”

“Or vertical,” adds Alexander.

“What if we multiply x or y by another number, like 2, or 3, or 0?” suggests Molly.

“Let’s try 2x + y = 10,” says Alexander.

“And maybe also 0x + y = 10,” adds Molly.

7.5 Making it curve

“I have an idea,” says Molly. “Let’s try to make an equation whose graph is curved.”

“Is that possible?” asks Alexander.

“I’m not sure,” says Molly. “I’ve seen graphs that curve,” she recalls.

“Maybe if we multiply x and y, like xy = 10,” suggests Alexander.

“Or use exponents, like x2 + y2 = 25,” adds Molly.

7.6 Performing it

Script a math story about one of the Making Ten activities to share with your family and friends.

Good math stories:

  • take us to a math world that we do not know;
  • offer mathematical surprises;
  • communicate feelings and emotions about doing mathematics;
  • involve doing some mathematics.

You want your family or friends to feel excited and surprised, and to engage emotionally with the math experience.

You can write your math story in a variety of ways:

  • A mystery or an adventure story.
  • A math ad.
  • A poem.
  • A song.
  • A skit.
  • A tableau or a human sculpture.
  • A work of art, like a drawing, a painting or a sculpture.

Or, you might want to use a combination of the above to tell your story.

Have fun!

8. RANDOM PAIRS LINE UP?

An animated video summarizing some the key math ideas in the above activities.

9. IN A GRADE 4 CLASSROOM

Grade 4 students investigated the above activities.

They built a coordinate grid on the classroom floor. In groups they designed challenges such as the following:

Human graphs of __ + __ = 10 and __ = __.
  • The named a number sentence, such as __ + __ = 7, and challenged other groups to stand on the coordinate grid to represent its graph.
  • Or, they stood on the coordinate grid themselves, and asked other groups to identify the number sentence that they were representing.

A painting was created by artist Ann Langeman, depicting key experiences of the Grade 4 students. The painting was donated to their school. Here is a PDF poster of the painting.

10. MODELLING WITH CODE

Linear and nonlinear relationships

View, edit and investigate modelling linear and non-linear relationships with code at: https://colab.research.google.com/drive/12qzXANGGr-pYx6T6ImvsJuOzAFgvUXig?usp=sharing