Snippet Lesson Plan on the Powers of Ten Using the
Powers of Ten or Cosmic Voyage
There are two sources for the video on which this Snippet
Lesson Plan is based. The first is a clip of the 1977 short
subject entitled <a
href=“http://www.powersof10.com/”><em>Powers of
Ten</em>, by Charles and Ray Eames. The short subject
has been placed on Youtube. It was the first film to
employ the concept of using a movie to demonstrate the
powers of exponential increase and decrease. The
Eames website also features an interactive tool that
allows the viewer to select an order of magnitude and
then see the picture corresponding to that order of
magnitude.
<br><br>
The makers of <em>Cosmic Voyage</em> copied and
adapted the concept from <em>Powers of Ten</em> and
three short clips from the movie provide similar visuals.
<br>
Subject: Mathematics – Powers of Ten, Numerical
Systems: Decimal, Hexadecimal, Binary, Duodecimal.
Ages: Powers and exponents are first introduced in Grade
Five. Powers of 10 as such are studied in Grade Seven.
Length: Snippet: <em>Powers of Ten</em>: The short
subject is 9 minutes long; <em>Cosmic Voyage</em>: 8
minutes in three clips. Lesson: One 45 to 55 minute class
period.
Learner Outcomes/Objectives: Students will have a
vibrant graphic sense of the meaning of exponential
increase and decrease. They will also become familiar
with the use of exponents and powers and learn about
different numerical systems.
Rationale: Cosmic Voyage takes viewers on a journey
through space and into the heart of matter explaining
along the way how distance and viewing perspective
grow. In order to do so the film introduces the concept of
increasing and decreasing the distance and viewing scale
in steps that are factors of ten. It is an interesting way to
spark students’ interest and understanding of the use of
exponents and powers as a mathematical tool.
Description of the snippet: The documentary feature
Cosmic Voyage, made for IMAX theatres, takes viewers
through space and time. The journey in space is done in
two directions, towards the large scales of the universe
and into the small scales inside the constituents of
matter. In both cases, progress is shown by drawn circles
that are explained to represent an increase or decrease in
scale by a factor of 10.
Helpful Background:
There are many things in everyday life that we take for
granted and without asking why they are as they are.
Why do we count with a system with precisely ten
symbols? Widespread belief relates this to the number of
fingers in our hands. Children tend to use their fingers to
count, add and subtract. In fact, humans have done just
that since counting became part of human culture. The
most accepted theory on Roman Numerals is that they
have evolved from Etruscan symbols. However there are
some authors that have given serious consideration to a
folk etymology for Roman numerals explains that “I”s
represent single fingers and, consistently, the number 5 is
a &quot;V&quot;, representing a hand with five
outstretched fingers. Two such hands united by crossed
wrists make an &quot;X&quot;, which represents the
number 10. (The other Roman symbols would then still
need different explanations).
In any case, the fact that we use a numeral system of ten
symbols that combined can represent any number is
certainly related to humans having ten fingers. This
system is called Decimal Numeral System (em>deci</em>
being Latin for ten). Having ten basic numbers or symbols,
including one for the number zero (which the Romans did
not have, by the way) forces us to start combining from
number ten onwards, as by number nine we run out of
symbols to continue counting.
<br><br>
The chosen way to do it is to use a second position to the
left and place there the smallest non-zero symbol,
&quot;1&quot;, letting the first position run again from 0
to 9, obtaining a representation for numbers ten to
nineteen. The second digit then changes to
&quot;2&quot;, and so on. By ninety-nine, we have used
up all possible combinations of symbols in two positions,
and we need a third position where to place a
&quot;1&quot; and start over again with the other two
positions. We all know how this works, but it is important
to recall the underlying rationale, in order to understand
different numerical systems that will be discussed below.
<br><br>
Knowing how exponents work – the exponent indicates
how many times to multiply a number by itself – allows us
to introduce a new notation that is especially useful for
very large or very small numbers. 102 is the same as 100,
without any obvious advantage in writing it either way
(we need to use three digits in both cases). But take one
million: 106 is definitely shorter to write than 1,000,000.
Note how the exponent corresponds to the number of
zeros to the right of the &quot;1.&quot; Numbers smaller
than one can also be represented by powers of ten, using
negative exponents, as a negative power means how
often the number one is divided by that number: 10-2
equals 1/10 times 1/10, which is 1/100 or 0.01. So, this
notation becomes also an advantage for very small
numbers: 10-6 is a short way to represent one millionth,
or 0.000001. In Cosmic Voyage, both the outward and the
inward journeys quickly reach scales which would be
rather complicated to express in writing without this
notation of &quot;powers of ten. &quot;
<br><br>
Any number of the decimal numerical system can be
represented as a sum of powers of ten multiplied by the
value of its digits:
<br><br>
1,354.954 = 1 x 103 + 3 x 102 + 5 x 101 + 4 x 100 + 9 x 10-1 +
5 x 10-2 + 4 x 10-3
<br><br>
(Recalling that any number to the power of 1 is equal to
itself, and any number to the power of 0 is equal to 1).
<br><br>
With these rules one could imagine and construct an
alternative numerical system with either more or less
symbols. One that is in use in particular areas of
mathematics and computing is the hexadecimal system,
built upon sixteen symbols: the numbers 0 to 9 and the
letters A to F. Now, we can count to fifteen using single
digits: 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F, and it is only when
we get to sixteen that we need a new position, where, as
before, we place the number &quot;1&quot; followed by
a &quot;0&quot;. Note that the number 10 in the
hexadecimal system has now the value of sixteen in the
decimal system. In order to distinguish both notations,
the hexadecimal &quot;10&quot; gets a subscript: 10hex.
<br><br>
In the hexadecimal system, any number can again be
represented as a sum of powers of “ten”, multiplied by its
digits:
<br><br>
E,7B3 = E x 10hex3 + 7 x 10hex2 + B x 10hex1 + 3
x 10hex0 ,
<br><br>
which is equivalent to:
<br><br>
14 x 163 + 7 x 162 + 11 x 161 + 3 x 160 = 59,315 .
<br><br>
Hexadecimal &quot;powers of ten&quot; are powers of
sixteen. See more examples explained with graphics at
http://homepage.smc.edu/morgan_david/cs40/hexsystem.htm Because higher values can be expressed
with fewer digits than in the decimal system, the
hexadecimal system is much used in programming
languages for computers and internet communications,
where the tightest packaging of information is
advantageous. There are even iPhone applications that
will display a clock in the hexadecimal system!
http://itunes.apple.com/us/app/hexclock/id300748897?mt=8
<br><br>
A very interesting way to count is the binary numerical
system, which uses only two symbols. It is the basis of any
computing language, because computers are based on
tiny electrical currents and there are only two possible
states of an electrical switch: on or off. The first
computers were built with switches that were valves
(open/closed). Later, electric and the current electronic
switches (transistors) were introduced. Quantum
computers replace the concept of switches with that of
quantum states of a particular property of electrons such
as spin, but there are still only two possible states thereof
(up/down) that will be represented with a binary
numerical system. Binary encoded information is
translated into decimal or hexadecimal formats, for
display or transmission purposes once the computer has
processed the information in the only format it can
handle: binary.
<br><br>
The most common and practical choice of symbols for a
binary system is &quot;0&quot; for &quot;off&quot; and
&quot;1&quot; for &quot;on. &quot; With just two
symbols we already need to introduce a new digit to
represent
number
two!
This
means
that
&quot;two&quot;
in
the
binary
system
is
&quot;10&quot;, represented with the subscript
&quot;2&quot;: 102. Any binary number can also be
represented by &quot;powers of ten&quot; notation:
<br><br>
1101 = 1 x 1023 + 1 x 1022 + 0 x 1021 + 1 x 1020
<br><br>
Which is equivalent to:
<br><br>
1 x 23 + 1 x 22 + 1 x 20 = 8 + 4 + 1 = 13
Binary &quot;powers of ten&quot; are really powers of
two.
See more examples explained with graphics at
http://britton.disted.camosun.bc.ca/jbbinary.htm
<br><br>
There is no limit to the numeral systems one could devise.
Our culture has settled for the decimal system, but there
is another one that is still deeply rooted in our society
since Roman times and before. There are 12 months in a
year and two 12-hour periods in a day, eggs are sold by
the dozen, there are 12 inches to a foot and 12 Pence in
an old British Schilling. The choice of 12 is not random, as
it is the smallest number that can be divided in halves,
thirds and quarters, making it especially useful in trade
and storage. The two extra symbols to complete the set
of 12 are most commonly represented by A and B, but
there are other alternatives. See more on the duodecimal
system
at
http://www.bookrags.com/research/duodecimal-systemwom/
<br><br>
Historically, there are more complex systems, such as the
Mayan
20-based
numerals
(http://www.mayacalendar.com/mayamath.html)
and
the
Babylonian
base-60
system
(http://wwwgroups.dcs.stand.ac.uk/~history/HistTopics/Babylonian_numerals.html
).
Using the snippet in class:
Preparation
1. Be familiar with the location of the segments
Segment 1: The outward journey from Venice to the
largest scale in the universe is shown from minute 7 to
minute 12.
Segment 2: The inward journey from a water drop to the
smallest scale goes from 13:12 to 15:41
Segment 3: A recap of both journeys, going from the
largest scale to the smallest is shown from 32:24 to 33:09
Step by Step
1. Introduce the topic of exponents and powers
according to the level of your students.
2. Play segments 1 and 2, commenting on their use of
the concept of powers of ten.
3. Allow for a time of discussion on the topic of powers
of ten, so as to clarify any questions and doubts.
4. Explain different numerical systems.
5. Play segment 3 as a recap of the snippet and the
lesson.
Supplemental Materials and Links
http://micro.magnet.fsu.edu/primer/java/scienceopticsu/
powersof10/
Another interactive tool to play and become familiar with
the scale of things:
http://primaxstudio.com/stuff/scale_of_universe/index.p
hp
A quick overview on numerical systems:
http://knowgramming.com/nanosemaphore/a_bit_about
_binary.htm
A detailed explanation on numerical systems can be
found here and the links tehrein:
http://www.mathsisfun.com/binary-decimalhexadecimal.html
On the hexadecimal system:
http://www.wisconline.com/objects/ViewObject.aspx?ID=DIG1102
http://www.youtube.com/watch?v=q1FTRIG5j8k
A discussion of further numerical systems:
http://www.math.wichita.edu/history/topics/numsys.html
http://mathforum.org/alejandre/numerals.html