Reprinted from Barbara Moses, ed.,Algebraic Thinking, Grades K-12:Readingsfrom NCTM's Scbool-Based fournals and Other Publications (Reston,
Va.:National Council of Teachers of Mathematics, 2000), pp. 146-49. Originally appeared in Mathematics Teacher 90 (February 1997): 92-95.
© 1997 by NCTM.
(Naturally) Numbers
Are Fun
Don Crossfield
A
bout six years ago, I put in my classroom
large colored
banners
of the sequences
shown here. This article lists some ways that
these sequences
have gradually been incorporated into my lessons, as well as my observations
about the caliber of thinking and questioning that
my students are developing as they learn appropriate ways to use these numbers.
A teacher once impressed me with the statement
that quality mathematics
occurs by "thinking
deeply about simple things." These sequences are
certainly simple things; it does not get much simpler than the natural numbers.
Here are some
questions that have come up in our classes and
the paths down which they have taken us.
1. What is the next number in each sequence? (This
question is better asked early in the course)
We usually all get the same answers on this question but not for the same reasons. Frequently my
students notice the recursive nature of the red
numbers, that is, adding consecutively larger odd
numbers, before they notice that they are perfect
squares. That approach is fine. In fact, if you explore the recursive nature of the green sequence as
well, adding 7, then 19, then 37, then 61, ... (Wow!
Look at the differences in those numbers!), then
before you know it, you are looking at finite differences, which contain the seeds for differentiating
Don Crossfield, doncrsfldsuaol.com, teaches at
Roseburg High School, Roseburg, OR 97470. His
mathematical interests include combinatorics,
number theory, and the history of mathematics.
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Pink
Numbers
2
3
5
7
II
13
17
19
23
29
31
37
41
43
47
53
59
61
67
71
Red
Numbers
I
Green
Numbers
I
4
9
16
25
36
49
64
81
100
121
144
169
196
225
256
289
324
361
400
8
27
64
125
216
343
512
729
1000
1331
1728
2197
2744
3375
4096
4913
5832
6859
8000
Blue
Numbers
2
4
8
16
32
64
128
256
512
1024
2048
4096
8192
16384
32768
65536
131072
262144
524288
1048576
polynomials.
It is easy to bring up relationships
among odd, square, triangular,
and polygonal
numbers in general, which will pay future dividends in subsequent questions and explorations.
2. Can you find two numbers of the same color
whose sum is also that color?
The avenues for exploration here are twin primes
(the pinks), the Pythagorean theorem (the reds),
Fermat's last theorem (the greens), and the question of "distinct" (the blues)-all
quality avenues to
explore. A class that has previously noticed the recursive nature of the red numbers and now notices
that the Pythagorean theorem is linked with sums
of red numbers has a simple verification that an infinite number of primitive Pythagorean
triples exists. Simply, red numbers are 9 apart (16 and 25),
25 apart (144 and 169), 49 apart (576 and 625),
and so on, every odd red number producing
a
Pythagorean
triple. A class that discovers several
primitive triples early in its high school career has
an easier time in several later topics. For instance,
finding cos x when sin x = 7/25 is much easier if
the triple (7, 24, 25) is known. The ideas are simple when freed from the arithmetic.
3. Can you find tuio numbers of the same color
whose product is also that color?
We reinforce the definition of prime with the pink
numbers. We have introductions to various laws of
exponents with the red, green, and blue numbers.
Depending
on the color, note that the product of
the third and fourth numbers in three of the sequences is either the twelfth or the seventh number in that sequence. The blue numbers give us an
excellent insight into logarithms. Not only are the
products of blue numbers also blue, but multiplication problems can be recast as addition problems.
Historically,
logarithms were first perceived
as a
way of relating
geometric
and arithmetic
sequences. Another two hundred years passed before Euler popularized our present concept of logarithms as being inverses of exponential functions.
Another way to introduce some laws of exponents
is to ask, "When you square a colored number,
does it change color?" Obviously, if you square a
pink number, it can no longer remain pink but must
by definition now be red. If you square a blue number, it not only remains blue but shows up twice as
far down in the sequence. That nice observation
leads directly to the relationship
(ax)y = aXY• If you
square a green number, it remains green, but where
along the green sequence the square appears is a
little harder for students to predict. I leave that exploration for your classes to pursue; our frequent
"aha" hint is that exponentiation is commutative.
4. Can you find two numbers of the same color
whose difference or sum is a pink number?
I have had-to
my and to their embarrassmentclasses carefully scrutinize the green numbers for a
pink sum two days after an assignment
in which
one of the requisite skills was how to factor a3 +
b': If they do not see that factoring a sum of cubes
is related to nonpink totals of green numbers, then
algebra for them is just manipulation
of symbols
and I need to do some serious backtracking.
It is interesting, however, that consecutive reds or
greens may produce pink differences. This revelation is an "aha" for my students who realize that
just because (a - b) is a factor of (an- bn) does not
mean that (an - bn) is necessarily composite. The
number 212 - 202 is prime because one of its factors is just 1, written as 21 - 20, and the other factor, 21 + 20, is prime. Invariably, phrasing a mathematics concept using the respective colors makes it
more concrete
to my students,
and they find it
much easier to recall the concept by referring to
the appropriate
color and operation. Yes, we have
a very colorful vocabulary in my classroom.
5. Can you find a number whose double, or twice
that number, is the same color?
After immediate success with the blues and immediate failure with the pinks, both by definition, the others are a bit more challenging. I like this question because determining that twice a red will never be red
means deciding that no natural-number solutions are
possible to a2 = 2b2, which means that Ca/b)' will
never equal 2 for natural values of a and b, which, in
turn, implies that J2 is irrational. Can twice a green
be green? No, because the cube root of 2 is also irrational. Another nice related thought is that determining which red numbers come closest to being twice
another (for example, 72 is practically double 52) is
equivalent to finding rational approximations (7/5) for
J2. By what number can you multiply a red number
to ensure that the product is red? Good question.
6. What percent of the numbers in one color also
belong to another color?
"If 50 percent of the blue numbers are red, why
are not 50 percent of the red numbers blue? Who
can tell about percents when you have an infinite
number of elements
anyway?" These questions
open up some interesting
doors, not the least of
which is the concept of "limit." It is also a good
opportunity
to review properties of exponents
as
you explain why your observations
on the numbers that are visible will continue to hold for those
terms not listed on the wall. Note that the use of
inductive reasoning establishes a conjecture and a
natural reason to apply the language
of mathematics to verify the conjecture deductively.
, One class asked what percent of the natural numbers were prime. We noted that 25 percent,
or
25/100, of the first 100; 23 percent, or 46/200, of
the first 200; and 21 percent, or 62/300, of the first
300 natural numbers were prime and surmised that
although the set of primes was infinite, gradually
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this percent was heading toward a limit of 0 percent. We came very close to discovering the primenumber theorem of Gauss, that the reciprocals of
these percents grow logarithmically, which gives
me pause for thought. Should I have stepped in, as
their guide, and pointed it out, or were they not
quite ready? Such is the dilemma that every teacher
faces several times a year. Oh well, you can always
show it to the mathematics club.
7. Can you make some observations involuing binomial products?
What teacher has not been plagued by the sloppy use
of a2 ± b2 for (a ± b)2? The number sequences provide some natural examples to clarifythis confusion.
I can square 1 000 mentally, and I can square 24
with my sequences (576). So shall I conclude that
the square of 1 024 is 1 000 576? That conjecture
can be checked and disproved quite easily with the
blue numbers, in particular, noting that 1 024 is the
tenth blue number. Consequently, its square is the
twentieth blue number, which is 1 048 576. Why
the "middle term" of 48 OOO?
Oh yes, it is because
2
0000 + 24)2 = 1000 + 20 000)(24) + 242
Each term of your trinomial answer can be seen in
the digits of 1048 576. Since 1 0002 indicates the 1
in the millions column, 242 is red and is the final
three digits, and the middle term is 48000, which
is represented by the digits 048. It is a nice example to remind students that squaring a binomial
produces a trinomial, since the middle term is sitting there in plain blue sight.
Factoring a difference of squares and squaring a
binomial both tie nicely into an observation about
consecutive red numbers. When we say that 212is
41 larger than 202,we are observing an instance of
factoring the difference of squares, 212- 202, into
(21 + 20)(21 - 20). Further notice that the equality
(n + 1)2 = n2 + 2n + 1 can be interpreted as adding
2n + 1, which is always an odd number (remember their observation about the recursive nature of
the red sequencer), to a red number, n»; to produce the next larger red number, (n + 1)2
8. Can you make some observations regarding the
units digits in each sequence?
The units digits in the red, green, and blue sequences are all cyclical, with a nice bonus of symmetry 0, 4, 9, 6, 5, 6, 9, 4, 1) in the red sequence
around each multiple of 5. Also note that whereas
the pink, red, and blue numbers are choosy about
which digits they will permit in the unit position, the
green numbers are all equal opportunity employers.
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9. Can you approximate irrational numbers, for
example, find .J630 to the nearest tenth?
With these sequences available on the wall, we can
reach a consensus that .J630 , which is greater than
.J 625 , is roughly 25.1 in the time it takes to turn on
a calculator and with more of an understanding
than simply parroting a readout. Furthermore,
.J6.30 must be about 2.51 with only a little more introspection. Do you see how .J53 is very close to
2.3, by noting that 529 = 232?Most square and cube
roots can be estimated rapidly and with reasonable
accuracy (note the key word, "estimate"). My students estimate base-two logarithms from the chart
before they think to grab a calculator (the better
students use the sequences to estimate logs for almost any base), and they develop a good sense of
linear versus nonlinear interpolation. I think that it
gives them a comfortable feeling for the Richter
scale, decibels, inflation and consumer price indexes, and related logarithmic scales.
Do you see how easy it is to evaluate 324/5by staring at the blue sequence? Just go 4/5 of the way to
32 along the terms of that sequence. By bouncing
between the red and green sequences, students
find that 343.213 is just as easy to evaluate, and from
there students can approximate related but irrational expressions like 7203/4• In their language,
"you fourth root 720" by "double reading it down,"
from 720 down to slightly under 27 and then down
to a little over 5. Cubing this number gets you
somewhere in the 130 to 140 range, since 5
"greened" is 125. Yes, a calculator will give you
about 139 in about the same length of time, but the
mental gymnastics that students do appears to pay
far greater dividends in building number sense.
Even my geometry classes get involved. For example, consider two similar three-dimensional shapes
whose linear dimensions are in a ratio of a.b. How
are the ratios of their surface areas or volumes related? This concept has traditionally been difficult for
those classes, especially when worded as "Two similar shapes have a surface area ratio of 9:1. What is
the ratio of their volumes?" The answer is 27:1. Now
my students just bounce around the appropriate sequences-red = surface area, and green = volumeand they do not realize that it is a hard question.
By the time we evaluate logarithms, most students
are using their calculators only to verify their previous, sequence-generated guesses. For example, solving the equation 5x = 11 takes only a few seconds of
mental effort, since 53yields a green number (25)
very close to 1P (21). The obvious' conclusion is
that to change a 5 into an 11, you must cube 5 and
then take the square root of the cube so that the necessary exponent is about 3/2. Since 121 is less than
125, our guess would actually be slightly under 3/2.
Check your calculator to see how close we came.
if we sum our sequences or-a
bit more difficult-their reciprocals?
10. What happens
You have heard the story of the king of Persia's rewarding the inventor of chess with wheat? The
amount of wheat the king paid him filled a chessboard in the following manner: 1 grain was placed on
the first square, 2 on the second, 4 on the third, 8 on
the fourth, 2n on the nth, until each of the 64 squares
was filled. This problem involves the blue numbers.
In our class the idea of its sum is expressed by the
sentence "Summingblue numbers gets you practically
the next blue number." I have been tempted to start
my blue sequence with a 1, but it causes more trouble than it is worth. Summing blue reciprocals leads
to the discovery of a/O - r) and of the relationship
between repeating decimals and infinite series.
Can you describe what happens when you sum the
first k green numbers? If you have the sequence of triangular numbers posted lI, 3, 6, 10, 15, 21, ... I, which
a few of my classes have demanded be added to the
other sequences, you get a pretty excited group of
students trying to explain how the green sums are related to the triangular numbers and also to the square
numbers. If you have not checked out those partial
sums, your clue is that all totals are red, but only certain special red numbers. It is worth looking into.
I have not yet had a class observe wonderful things
on their own about summing red numbers or pink
numbers. If anyone's class discovers something in this
area and communicates it to me, great would be my
gratitude. Also, no class of mine has yet found anything remarkable about the reciprocals of the reds,
greens, or pinks, but I strongly suspect that some
beautiful observations are waiting to be discovered.
Closing Observations
Before I posted these sequences on my wall, even
my top second-year-algebra students seemed to have
a significant lack of number sense. Their knowledge
of primes seemed to stop around 31, squares at 144,
and cubes at around 8. Their fear of a problem was
proportional to both the number of lines in the explanation of the question and the number of digits in
the related data. Numbers over 100 were deemed to
be "not friendly." Calculators and brains were used in
parallel rather than in series.
I think that staring at these numbers has given my
classes a "securityblanket." As we wander down algebraic paths, we always have concrete examples readily at hand. Also, by allowing them constant access to
some of the most common yet fruitful number patterns, I believe that I am providing for my students
the same sort of catalyst that Euler, Gauss, Newton,
Bernoulli, Fermat, Descartes, and Pascal had for their
explorations. Many of these people are renowned for
their prodigious memories, and lists like these were
surely a part of their mental repertoire.
I like what is happening.
I frequently hear
thoughts like "Of course the roots are rational ...
it's got a red discriminant" or "Excuse me, you're
trying to partition a blue number of people equally
into a pink number of committees! Blues don't
have pinks in them!" When we try to pin down
our patterns algebraically, the expressions make
more sense. They reflect concrete observations
rather than stand alone without numeric support.
By the time we formally discuss number-theory
concepts like divisibility and congruence, the students have a much better concrete background
about the properties-we call them the "personalities"-of the natural numbers. Can you find any
red numbers that are congruent to 1, mod 8? The
pink numbers appear to do a strange grouping,
mod 6. Can you explain why? Very few blue numbers exist in which the sum of the digits is divisible
by 3. Congruence and divisibility lend themselves
to a lot of interesting questions. Try some.
Catalogs are full of books that supply the teacher
with numerous warm-up problems with which to
start a class period or to fill a five-minute void near
the end of class. We have as many warm-up questions as we need, posted on our wall. I frequently
ask the class to produce a question. Will it be an
interesting question? That in itself is an interesting
question; what percent of the questions they ask
will be "interesting"? The more we play with the
sequences, the more I am convinced that no uninteresting questions are possible.
On a mathematics-club outing a few years ago,
one student was playing with her calculator,
checking out the decimal equivalents of some
common fractions. She knew the decimals for 1/9
and 1/99 and 1/999 but did not know the decimals
for 1/98 or 1/998. Do you? Check them on a calculator. You will like the color, and if this article has
done its job, you now have several new and interesting questions to think about. Have fun.
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