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It's easy for natural numbers: $3\times 5=5\times 3$

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but how do you explain that $x.y=y.x$ for any real numbers $x$ and $y$.

Moreover, in $\Bbb{N}$, do you prefer to define $n\times m=\underbrace{n+n+\cdots+n}_{m\text{ times}}$ or $n\times m=\underbrace{m+m+\cdots+m}_{n\text{ times}}$.

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    $\begingroup$ How will you explain real numbers to middle school students? $\endgroup$
    – GregRos
    Jul 2, 2013 at 16:55
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    $\begingroup$ This and this. $\endgroup$
    – Git Gud
    Jul 2, 2013 at 17:07
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    $\begingroup$ @Greg: Can't you take your ruler and use it to measure the diagonal of a square? $\endgroup$
    – MJD
    Jul 2, 2013 at 17:17
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    $\begingroup$ @GregRos I explain the existence of irrational numbers (in fact algebraic numbers and $\pi$) and then I define the real numbers to be their union. $\endgroup$
    – user5402
    Jul 2, 2013 at 17:30
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    $\begingroup$ Dear @GregRos : I didn't say that a ruler is a perfect model of a real line, I just meant that this is middle school's first approximation for introducing the real line to students. They are taught (or at least I was taught) that "in theory" you can read beyond the markings on the ruler by adding more, and that "if you keep dividing and reading more accurately you get closer to the length of that thing." It's not as if students get to highschool with no conception of real numbers at all, that's all I mean. $\endgroup$
    – rschwieb
    Jul 2, 2013 at 18:17

2 Answers 2

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First do it with integers: make a rectangular array of dots, then turn the rectangle ninety degrees. Now instead of an array of $n$ rows, each with $m$ dots, it's an array of $m$ rows, each with $n$ dots. But the number of dots didn't change, only the way they were arranged.

More generally, a rectangle with real-length sides doesn't change its area when you rotate it a quarter-turn.

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    $\begingroup$ That's pretty neat. I guess you don't really need to explain real numbers at all. $\endgroup$
    – GregRos
    Jul 2, 2013 at 17:00
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    $\begingroup$ That's probably a good thing, since real numbers are incredibly complicated and bizarre. $\endgroup$
    – MJD
    Jul 2, 2013 at 17:00
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    $\begingroup$ Yeah, direct analogy works really well in this case, and I think that a lot of them have a picture of real numbers as "like a measurement you get off a ruler" which meshes well with this. $\endgroup$
    – rschwieb
    Jul 2, 2013 at 17:05
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    $\begingroup$ While intuitively is a very nice explanation, isn't in reality backwards: area of rectangles makes sense just because multiplication is commutative. $\endgroup$
    – N. S.
    Jul 2, 2013 at 18:36
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    $\begingroup$ I strongly disagree. Multiplication is a model of the areas of rectangles, and not vice versa; there were rectangles before there was multiplication. We define multiplication the way we do because we want it to have certain properties, and the reason we want it to have those particular properties is that those are the properties that we observe of certain objects in the physical universe, in particular rectangular objects. $\endgroup$
    – MJD
    Jul 2, 2013 at 19:29
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One option, which my teacher did with us (8th grade, so may not work so well with 6th graders), is to show the class a system where multiplication (or perhaps even addition) is not commutative, like matricies or 3D vectors. After showing them that that type of multiplication is not commutative, it becomes much easier to understand that there is something special about multiplication on the reals.

The entire rest of the field axioms can be handled the same way, if you can just show systems that don't have all the properties that they take for granted.

  • commutativity of addition
  • distributivity of multiplication over addition
  • associativity of addition/multiplication
  • existence of identity/zero
  • existence of an equivalence relation
  • existence of an order relation
  • trichotomy (for any $a,b$, exactly one of $a < b$, $a = b$, or $a> b$)
  • transience of equality
  • substitution ($a=b \oplus c \wedge c=d \to a=b \oplus d$)

Although most middle schoolers might not be willing to learn this sort of stuff.

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    $\begingroup$ Well, 8th graders don't know anything about 3D vectors much less about matrices so I think it should be the other way around: students will appreciate the commutativity of multiplication on $\mathbb{R}$ when they learn 3D vectors and the cross product not learning 3D vectors and the cross product to appreciate the commutativity of multiplication on $\mathbb{R}$. $\endgroup$
    – user5402
    Jul 2, 2013 at 20:32
  • $\begingroup$ @metacompactness "8th graders dont know anything about 3D vectors much less about matricies." That's what you are for. "Students will appreciate the commutativity of multiplication on $\mathbb{R}$ when they learn 3D vectors and matricies." That is exactly the point I was making. $\endgroup$ Jul 2, 2013 at 20:43
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    $\begingroup$ +1 by middle school, they understand that multiplication is commutative pretty well. But when I was taught the terms "commutativity," "associativity" etc. in Middle School, I didn't understand why we gave such complicated names to such trivial ideas (and neither did my teacher, which just made it more frustrating - she had the attitude that "you have to learn it because the state says so"). It wasn't until I started to learn about linear algebra in High School that I understood why "commutativity" even has a name... $\endgroup$ Jul 2, 2013 at 22:33
  • $\begingroup$ @BlueRaja-DannyPflughoeft Maybe you're right; I can explain why the multiplication of two decimal numbers $\frac{a}{10^n}$ and $\frac{b}{10^m}$ is commutative (as a consequence of the commutativity of the multiplication on $\mathbb{Z}$). When $n,m\to +\infty$, we get this property on $\mathbb{R}$. $\endgroup$
    – user5402
    Jul 7, 2013 at 16:34

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