Materials to Visually Demonstrate the Mathematic Concepts of Place Value, Exponents, and
Base, as well as the Vast Spaces in Interstellar and Sub-Atomic Space Using the film Cosmic
Voyage and Resources Available on the Internet
Subject: Science/Mathematics
Ages: Elementary – High School – whenever powers and exponents are studied.
Length: Snippet: 8 minutes in three segments; Lesson: Not Applicable – materials will supplement existing lesson plans on the topics covered.
Note to Teachers: This lesson plan collects resources to supplement several important lessons in math and one in physics.
Another version of the trip to interstellar space and into sub-atomic space similar to what is presented in Cosmic Voyage can be accessed at the “Powers of Ten” website http://www.powersof10.com/ This is a portion of the 1977 film of the same title. It includes an interactive slideshow and the 9 minute film to watch. If Cosmic Voyage is not available, this might be a suitable substitute. It also has a feature that allows the viewer to move in and out of the projection at will.
Learner Outcomes/Objectives: Students will absorb striking visual descriptions of the extent of exponential increase and decrease and of the comparison of spaces in the physical world, both interstellar and atomic. They will also learn about place value and different numerical systems.
Rationale: The meaning of place value and exponential notation as well as their ability to economically describe the physical universe are central to the understanding of math and science. Visual presentation of these concepts is the easiest way to teach them.
Description of the snippet: The documentary feature Cosmic Voyage was made for IMAX theatres but is also striking on a normal screen. 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 and shrink. The film explains 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 in the use of place value and exponents as mathematical tools.
Helpful Background:
Erik – What you have given me is a good verbal explanation, but it can be done better with visual or animated presentations. However, we don’t have the resources to do this, so let’s just recommend web sites for teachers to use. Unfortunately, some have advertisements.
Here are some that I found with a quick we search of google videos search on “decimal place value” or something similar

Please review some web sites and fit them into the following topics. Try for three of each.
One or two for each of the different bases. I’ve made a few suggestions
PLACE VALUE
The fact that we use a numeral system of ten symbols that combined can represent any number is obviously related to the fact that we have ten fingers. This system is called the decimal numeral system (deci being a Latin particle that means “ten”). Having ten basic numbers or symbols, including one for the number zero forces us to start combining from number ten onwards, as by number nine we run out of symbols to continue counting.
Videos available on the web to show place value include: http://www.youtube.com/watch?v=iaSFUPRReow http://www.youtube.com/watch?v=2BFu6fib9sA
EXPONENTS
Bases 2, 16, 20 (Mayan) and 60 (Babylonian).
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://www-groups.dcs.st-and.ac.uk/~history/HistTopics/Babylonian_numerals.html
).
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-system-wom/
Web sites: http://www.metacafe.com/watch/330020/learn_the_trick_change_binary_number_to_decim al_vice_versa/
VAST DISTANCES IN THE INTERSTELLAR UNIVERSE AND WITHIN ATOMIC STRUCTURES
An interactive tool that reproduces the journey through scales of different orders of magnitude: 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.php
I would delete the following. Teachers will already know this. The visuals will be better.
The chosen way to do it is to use a second position to the left and place there the smallest non-zero symbol, “1”, letting the first position run again from 0 to 9, obtaining a representation for numbers ten to nineteen. The second digit then changes to “2”, and so on. By ninety-nine, we have used up all possible combinations of two out of ten possible symbols, and we need a third position where to place a “1” and start all 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.
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 large numbers. 10 2 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: 10 6 is definitely shorter to write than 1,000,000. Note how the exponent corresponds to the number of zeros to the right of the “1”. 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*10, which is
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.
Any number of the decimal numerical system can be represented as a sum of powers of ten multiplied by the value of its digits:
1,354.954 = 1 x 10 3 + 3 x 10 2 + 5 x 10 1 + 4 x 10 0 + 9 x 10 -1 + 5 x 10 -2 + 4 x 10 -3
(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).
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 “1” followed by a “0”. 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 “10” gets a subscript:
10 hex
.

In the hexadecimal system, any number can again be represented as a sum of powers of “ten”, multiplied by its digits:
E,7B3 = E x 10 hex
3 + 7 x 10 hex
2 + B x 10 hex
1 + 3 x 10 hex
0 , which is equivalent to:
14 x 16 3 + 7 x 16 2 + 11 x 16 1 + 3 x 16 0 = 59,315 .
Hexadecimal powers are powers of sixteen.
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 the only two possible states of a 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.
The most common and practical choice of symbols for a binary system is “0” for “off” and “1” for “on”. With just two symbols we already need to introduce a new digit to represent number two! This means that “two” in the binary system is “10”, represented with the subscript “2”: 10
2
. Any binary number can also be represented by
“powers of ten”:
1101 = 1 x 10
2
3 + 1 x 10
2
2 + 0 x 10
2
1 + 1 x 10
2
0
Which is equivalent to:
1 x 2 3 + 1 x 2 2 + 1 x 2 0 = 8 + 4 + 1 = 13
Binary “powers of ten” are powers of two.
Location:
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 ERIK – WHY CAN’T WE COMBINE ONE AND TWO, IT WILL TAKE THE

TEACHER TIME TO FAST FORWARD TO THE START OF TWO; MIGHT AS WELL JUST LET
IT PLAY. WHAT DO YOU THINK?
Segment 3: A recap of both journeys, going from the largest scale to the smallest is shown from 32:24 to 33:09
Using the snippet in class:
This section does not apply because this web page collect resources to visually supplement
existing lesson plans.
Erik, I don’t know what to do with this cite
A discussion of further numerical systems: http://www.math.wichita.edu/history/topics/num-sys.html