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Process abstract mathematic – Videos

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Luis Radford École des sciences de l'éducation Université Laurentienne

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A teacher talks to her students

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The envelopes have the same number of cards inside. I am asking you a question: “How many cards are in each envelope?” It's the magic question, “How many cards are in each envelope?”

Julie (student)
You take one from Paulette and one from Richard and you put them at the bottom.

Teacher
Ok, we will pretend they disappeared, that there are no more.

Julie
Then you take another one from Paulette and another from Richard.

Teacher
OK

Julie
That leaves five.

Teacher
We took the same number from each side. OK. Go on.

Julie
That leaves five. On one of the envelopes there, on the side, there will be five. Then, everybody will have the same number, there will be five inside.


A student

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Yes, for it to be equal, there should be, as Julie said, an envelope with five cards, but in the other envelope there should be the same number of cards that are in Paulette's envelope. Then you add Richard's two cards, Paulette's two cards, there will be the same number.


A teacher

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A teacher
Go ahead.

A student
I figured that there were five cards per envelope because Paulette has seven cards plus five cards in the envelope, that is twelve, and Richard has two cards plus five in each envelope, that is 2 plus 10 equals twelve.

A teacher
OK, you arrived at the correct answer, you tried it in your head. You substituted the cards for the envelopes and you arrived at the correct answer.


Teacher

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Teacher
I will repeat the step. I think it's Julie who said... I take two from there and I take two from there. Do you agree that I took the same thing from everywhere?

Students
Yes

Teacher
Each side is still equal. What happens if I do this? Do I remove the same thing? Is each side still equal?

Students
Yes

Teacher
So, I ask you. How many cards are in each envelope? Five, five. She has five cards. He has an envelope and it is the same thing because I removed the same thing each time. An envelope has five cards. OK


Students

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Students

  • ....twelve then I give an envelope. Chantal has three cards and four envelopes, right?
  • Yes.
  • One, two, three, four.
  • We can remove a card from each.
  • Remove three cards.
  • She has four cards.
  • Now, remove four envelopes.
  • OK
  • Remove three cards, I will remove three cards. They are there. OK
  • One envelope each.
  • An envelope each, Eric. There, none left.
  • Can I count this? (Est-ce qu'on peut compter ça?)
  • Two, four, six, eight, nine. Nine divided by three, three each.
  • Nine.
  • There are three, three cards in each envelope.
  • OK. There, another envelope.
  • There are three cards in each envelope.
  • Another envelope.
  • That is one, two, three, four, five.
  • One, two, three, four, five. We got it.

Students

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Students

  • OK. You need to do the same thing you do on the other side. You must. Yes. Strange. Let's say you leave six on the other side, if you remove six you must remove six from the other side.
  • You must put this in words, more, something.
  • You cannot write this?
  • What he said is correct, but we need to put it in a sentence, as the vocabulary requires. We must remove the same number of cards from each side.
  • You must remove the same number of objects that you had on one side.....

Students

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Students

  • ....eighty-one. Twelve cards and she gives an envelope, three cards and she gives four envelopes. There are three cards, four envelopes. Yes, thank you.
  • Chantal has the same number of hockey cards as Mario. How many cards are there in each envelope?
  • OK.  I will do the same thing as in the table. Then, Mario has twelve cards, that will make twelve cards.
  • Why, we will do it in real time
  • We could do it before.
  • Ah, OK
  • Chantal has the same number of cards as Mario, how many cards are there in each one?
  • That means that we will have to remove the same number of cards from that one.
  • Yes. We will try …
  • We have to...that the four envelopes it must equal something plus 3 cards so that it makes 17 or something....
  • OK. We will have to remove 3 cards.
  • We remove a card, we remove 3 cards, then we remove another card.
  • Then, you have an envelope and I have three. One envelope.
  • OK. That means you have one, two, three, four, five, six, seven, eight, nine.
  • Nine divided by three.
  • There are three cards in each envelope.
  • Because if you take three cards, you remove an envelope.
  • Three, three, three
  • Then it will be the same thing.
  • OK. That means that Mario with the envelope, will have 15 cards and Chantal with her four envelopes, she will have 15 cards.
  • Yes.
  • Then there are three cards in each envelope.
  • Yes.

Students

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Students

  • OK. .... then there
  • Wait a minute ...
  • Letter I am transcribing....
  • I need to make my piles.
  • Yes.
  • Make your piles...
  • Hey, I need to make

Teacher

  • Is everything OK?

Students

  • Yes.
  • OK. We have to remove three cards from each side.

Teacher

  • Is it because you chose to make your drawing like that?

Students

  • Yes.
  • Ma'am, can we make circles to make the piles?

Teacher

  • Yes.

Students

  • OK.

Students

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Students

  • You take away...
  • You take away the lowest number of cards...
  • The same number of cards...

No, no. You take away the cards, you take away the cards more... OK. You have to take away the same number of cards that the person with the fewest cards has.

  • Yes.
  • And then you remove an envelope on each side until there are no more envelopes on one side. And then, there has to be... you take away, you remove...
  • You take away...
  • You take away the cards of the person who's got the fewest cards until there are two envelopes left.
  • Why until there are two envelopes left?
  • Until only the envelopes are left.

Students

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Students

  • ...Mat, we're going to do seven cards plus one envelope. So this means seven...
  • N equals the number of cards in the envelope.
  • N equals the number of cards in the envelope.
  • (Can you imagine if Éric had started a fight in the middle of class)!!!
  • (It would've been funny.)

Teacher

  • Guys, I just want to remind you that this is group work. I don't want to see anyone working on number 3 or 4 while the others are still at number 2. You're not supposed to work on your own; you're supposed to work together.

Students

  • OK.
  • And then it would be 7 + N...
  • 7 + N equals 3 + 3N.
  • Yes.
  • Yes.
  • And then, we do...
  • It would be...
  • No, no. 7, and then we're going to do a small minus 3.
  • Plus 1N, then 3 minus small 3 + 3N
  • And then, we're going to do 4 + N...
  • ...equals N.

Teacher

  • Patrick, are you following? I can't hear you, are you following?

Students

  • We're here. I did one. The first one, the second one, OK.
  • 4 + N equals 3N. That's enough.
  • So it would be 4 + N minus 1 equals...
  • ...yes, the small n, and then the small minus 1.
  • ...3 + N minus 1.
  • Yes. 4 + N minus small 3 equals 3N minus small 1.
  • No. A small 1 and a small 1.
  • Yeah, yeah. Small 1 on top.
  • So this means that here, it will be 4.