# Best way to teach concept of real numbers using a hands-on activity?

I know a middle school math teacher looking for some suggestions for hands-on activities to teach the concept of real numbers. I'm new to this site, so this may be a little off topic.

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I retagged this with math-ed and real-analysis, but I'm not sure that the r-a tag is appropriate. –  Harry Gindi Mar 8 '10 at 20:18
We need more information here. What do the real numbers mean in a middle school algebra course? Infinite not necessarily repeating decimals? What is it about the real numbers that your teacher friend (or his/her curriculum) desires middle school students to know? –  Pete L. Clark Mar 8 '10 at 20:38
I think many of the answers here are far beyond middle school. Here is a true story. It is from maybe 20 years ago, so perhaps things have changed. One of my university colleagues gave me the story. There was a course for future teachers. It included a unit on use of a calculator. Then one day they came to some problem with a decimal answer. The students objected: "We will be teaching middle school, so we are not required to know decimals!" The professor tried to suggest that they might want to know decimals for their own personal use, even if they didn't teach them. No good. –  Gerald Edgar Mar 8 '10 at 21:33
@Gerald Edgar: thanks for backing me up on this. As to how things have changed from 20 years ago (when I was in middle school!): no one seems to think that they have changed for the better. –  Pete L. Clark Mar 8 '10 at 21:38

What do you mean exactly? Show them that not all numbers are rational? That not all numbers are algebraic? I think that you could explain historically how the Pythagoreans thought all numbers were rational. Then challenge them to find a number which is not rational. You might be surprised with the creativity you get. Maybe they know rational numbers have repeating decimal expansions and so they will invent one which does not repeat. Challenge them to explain why rational numbers have repeating decimal expansions. If you like geometry, you could lead them to the essential step in the geometric argument here:

http://blog.plover.com/math/sqrt-2-new.html

that the square root of two is irrational.

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The questioner's user page identifies him as an American, so he is presumably asking about the American middle school (grades 6-8) system. (In other words, most entering middle school students will be 11 years old, and most departing middle school students will be either 13 or 14.) Some of the responses so far seem to me to be pretty far away from this, so I am enclosing a link to the NCTM standards for grades 6-8:

http://standards.nctm.org/document/chapter6/index.htm

Note: I didn't see anything about "real numbers" in the above link, although that doesn't mean that a discussion of real numbers would necessarily be inappropriate. These are intended to be minimal standards. (Still, realistically speaking, even these minimal standards are, on average, a long way from being met.)

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There is an argument to be made that the real numbers, by which I mean the completed reals, does not belong in an algebra course. In the answer above, all of the motivation comes from the real-algebraic-closure of $\mathbb{Q}$ (the largest algebraic field extension not containing $\sqrt{-1}$).

The reason one might want to introduce the whole continuum comes from numbers like $e$ and $\pi$, which are transcendental (the fact that these numbers are transcendental is not immediate and requires a proof that I would consider past middle-school level). If you're willing to state those facts without proof, you can give a moral argument for $e$ by showing that it is the limit of the sequence $((1+1/n)^n)_{n\in \mathbb{N}}$, which is Cauchy, and its inclusion in the real numbers follows from the completeness of $\mathbb{R}$. However, this argument may still be somewhat sophisticated for a middle-school algebra course.

Edit: On Prof. Clark's suggestion, I've copied my comments into the body text (with an additional section as well):

The notion of a sequence converging to a limiting value has a very intuitive geometric interpretation, so it wouldn't be hard to give a geometric argument (say on a graph, for example) that $e$ is a real number, since after relatively few iterations, the graph does level out. Showing that it is not the solution to a polynomial is effectively proving that it is transcendental, and I can't think of an informal argument showing this, but since this answer is community wiki, if someone has an idea, this would give a "moral" argument for the study of the "whole" continuum.

I find this approach useful because it can be introduced using the compound interest formula, which is often taught in an introductory algebra class.

We can see this as follows $$A=P(1+r/n)^{nt}.$$

Let $N:=n/r$. Then we have $$A=P(1+1/N)^{Nrt},$$ which we can rewrite as $$A=P((1+1/N)^N)^{rt}.$$

If we increase $N$ and leave $r$ fixed, this is equivalent to increasing $n$ (this is obvious because $r>0$), which amounts to increasing the number of compounding periods per unit time. Taking the limit (in some informal geometric sense), we can see that as we increase the number of compounding periods, we approach the continuously compounded interest formula $$A=Pe^{rt}.$$

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"May" be "somewhat" sophisticated? So you think it's possible that discussion of limits, Cauchy sequences and completeness of the real numbers might be appropriate for some middle-school algebra course?? –  Pete L. Clark Mar 8 '10 at 20:35
@fpqc: What you say in the comments sounds much more reasonable than in the answer itself. You might want to edit your answer accordingly. In general, when mathematicians are asked for advice on pre-collegiate math ed, it is a big problem if what we suggest is years above the abilities of the average student. This makes us look completely out of touch with the problems that educators are facing. –  Pete L. Clark Mar 8 '10 at 21:08
@fpqc: +1; what you have now looks helpful. (Note that by "the real numbers", everybody means "the completed reals".) –  Pete L. Clark Mar 8 '10 at 21:29
fqpc, what does &ldquo;the algebraic reals&rdquo; versus &ldquo;the completed reals&rdquo; mean? Is it a conceptual difference, an intuitionistic difference, or something else&mdash;or do you maybe just mean whether we are regarding the real numbers as forming a field or a topological space? –  L Spice Mar 8 '10 at 21:53
Well, this might not exactly be Real Analysis but possibly interesting to a school child. (I was fascinated as a kid) -- The (binomial) approximation of first two terms to the compound interest formula gives the formula for simple interest. One realizes that higher powers of small numbers can be neglected. –  Abhishek Parab Mar 9 '10 at 14:47

My mind immediately jumps to grabbing a ruler and some string and trying to determine the ratio of the circumference of a circle to its diameter, or the hypotenuse of the right triangle with unit legs. In this way, the most accessible irrational numbers to most non-math majors (i.e. up to college even for most freshman) can be explored.

It should be easy to see that numbers can be built up from the humble unit 1, and quickly overtake the natural numbers with addition, the integers with subtraction, and the fractions with division. Informing someone that there are numbers outside these operations should be the first step.

I would then construct some irrational numbers and argue (likely without proof) that they are not rational. 0.1234567891011121314... seems like a good candidate for this. This also gives a powerful technique to the students to construct other irrational numbers and notice how many there actually are.

We can also explore the number Pi. Cut out in paper a square, a pentagon, a hexagon, etc, to have several shapes of about the same area. In some sense you can consider the ratio of a length to the perimeter in relation to the number of sides each shape has (and assign it as homework). As the number of sides goes up, you can argue that this number is approaching Pi.

I am of the impression that this question counts as community wiki since the question is subjective.

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I guess the "hands-on" version of the real numbers is the Euclidean line.

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The book “Mathematics for high-school teachers” by Usiskin et al. has what I find a hideously over-complicated approach, at the nutshell of which is a nice idea: How does the notion of the collection of real numbers as a geometric object (a line) mesh up with its realisation as an algebraic object (at the middle-school level, decimal expansions)?

Some things to think about in this connection: If real numbers are directed lengths, how do we add and subtract them? OK, that's not so bad; how about multiplication and division? Now how about square roots? One can draw a nice picture to show $\sqrt2$ as the length of the diagonal of a unit square (constructed on a unit length by ruler and compass) being ‘unfolded’ (by a compass) down to the real line; now we've taken square roots geometrically. Can we do the same thing with the circumference of a circle?

If one wants to think purely in terms of decimals, there's the age-old “Are $1$ and $0.\overline9$ really the same number (and why)?”. It's easy for middle schoolers, and good practice in long division, to find decimal expansions of some unfamiliar (for them) fractions, like $1/7$ and $1/13$; which ones terminate? Which ones repeat? Will one of these always happen? Can you predict in advance which is which, and how long it'll take before it terminates or repeats?

UPDATE: On further reflection, I suppose that the latter is “hands on” only in a very rough sense. Perhaps one could consider the analogous sum $\sum_{i = 1}^\infty \frac1{2^i}$ by (‘building up’) laying down a strip, then another strip of half the length, then another of half that length, and so on; or (‘tearing down’) by cutting off from a long strip half its width, then half of the remaining width, and so forth.

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Thanks to Pete for the link on what the question means. I was misled by the other responses.

Here are two thoughts:

i. What is the square root of 14? Well, it's between 3 and 4. Then we can do better by taking the average of 3 and 14/3. Then we can repeat this and do even better.

ii. Take the Fibonnaci sequence 1,1,2,3,5,8,13,... Take ratios of successive terms 1/1,1/2,2/3,3/5,5/8,8/13,... then these are approximations to the "golden ratio".

A personal anecdote: My daughter is 11 yrs old, bright and interested in maths. I tried i. on her and drew a blank look. I conclude from this that either it is a mistake to teach your own children or that she was not ready for this.

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Sin(x) in middle-school algebra? Again, this is off by several years from actual middle-school curricula. –  Pete L. Clark Mar 8 '10 at 21:10
Could someone say what "middle school" means? Thanks. –  Bruce Westbury Mar 9 '10 at 2:28
@BW: Please see my response. –  Pete L. Clark Mar 9 '10 at 3:39
@Bruce: your edit and new reponse is appreciated. I was speaking out against a sort of strange behavior that mathematicians sometimes exhibit, which is to grossly exaggerate how early certain concepts could/should be taught and learned. This is really not helpful. Let me try a little honesty: I did not know one iota of calculus until I was 16 years old. When I was in middle school I had my hands full learning about division of polynomials. Any talk of limiting processes would have sailed right over my head. (And I was a strong student.) –  Pete L. Clark Mar 9 '10 at 4:07
I too was going to suggest calculating square roots by means of successive approximation. I wouldn't go as far as to discuss the subtleties of the real number system, but this would allow them to feel it at an intuitive level. –  gowers Mar 9 '10 at 8:12

I would suggest looking at materials provided by the Art of Problem Solving. The exposition is generally very high quality.

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