Ages: Elementary High School Whenever Powers and Exponents Are Studied

Subject: Science/Mathematics

Ages: Elementary – High School – whenever powers and exponents are studied.

Length: Snippet: 8 minutes in three segments; Lesson: One class period.

Learner Outcomes/Objectives: Students will absorb striking visual descriptions of (1) the power of exponential increase and decrease, (2) the comparison of spaces both atomic and interstellar in the physical world, and (3) 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 and the relationship of spaces in the physical world.

Description of the snippet: The documentary feature Cosmic Voyage, made for IMAX theatres but also striking on a normal screen, 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:

In any case, the fact that we use a numeral system of ten symbols that combined can represent any number is obviously related to humans having ten fingers. This system is called Decimal Numeral System (deci being a Latin particle that means “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.

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: 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 “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 103 + 3 x 102 + 5 x 101 + 4 x 100 + 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: 10hex.

In the hexadecimal system, any number can again be represented as a sum of powers of “ten”, multiplied by its digits:

3 2 1 0 E,7B3 = E x 10hex + 7 x 10hex + B x 10hex + 3 x 10hex , which is equivalent to:

14 x 163 + 7 x 162 + 11 x 161 + 3 x 160 = 59,315 .

Hexadecimal “powers of ten” 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”:

102. Any binary number can also be represented by “powers of ten”:

3 2 1 0 1101 = 1 x 102 + 1 x 102 + 0 x 102 + 1 x 102

Which is equivalent to:

1 x 23 + 1 x 22 + 1 x 20 = 8 + 4 + 1 = 13

Binary “powers of ten” are powers of two.

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/

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 ).

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

“Powers of Ten” website by the author of the 1977 film of the same title. It includes an interactive slideshow and the 9 minute film to watch: http://www.powersof10.com/

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

A discussion of further numerical systems: http://www.math.wichita.edu/history/topics/num-sys.html