The "Green Machine"
For students to discover the mechanics of the mathematics, start off
with the simplest form of the birthday-guessing devices, which is the
"green machine." This activity involves a green-construction-paper
packet containing an assortment of numbers from 1 to 31 on each page.
The "magician" shows each of the five pages, asks whether the secret
birthday appears on that page, and receives a "yes" or "no" response.
After the fifth card, the magician has gained enough information to
announce the mystery date. Although the students want to ascribe the
process to advanced memorization skills, they will find that patterns
of numbers promise to be a more workable strategy. More important, this
revelation encourages them to suspect that they may have the skills to
guess a hidden birth date.
The students examine individual cards in the packet to identify any numerical patterns.
They begin to recognize and talk about patterns by noting such
characteristics as that all numbers on card 1 are odd; that the numbers
increase by 4 in both columns on cards 1 and 2; or that card 5 starts
at the number 16 and counts up by 1.
Card 1 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31
| | Card 2 2 3 6 7 10 11 14 15 18 19 22 23 26 27 30 31
| | Card 3 4 5 6 7 12 13 14 15 20 21 22 23 28 29 30 31
| |
Card 4 8 9 10 11 12 13 14 15 24 25 26 27 28 29 30 31
| | | Card 5 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31
| |
The class works on detecting patterns for each card as well as
synthesizing individual regularities and arrangements into rules that
encompass all the cards. Then they might be asked to work backward and
focus on how the activity is actually done, examining the relationship
between the yes-no responses and the outcome. If they do not come to
the conclusion themselves, a prompt that the yes responses are a key to
the ability to guess the birthday might supply a clue. Using several
sample numbers, one at a time, students isolate the cards that have
generated an answer of yes for a particular secret number and inspect
them to see if any patterns can be discerned. As students collect data
mentally and search for relationships between the numbers on the cards
and the final results, they develop possible rules that describe what
is occurring. Each rule suggested is tested to examine whether it works
with other numbers.
Eventually one or more students notice that adding the
top-leftmost number of each yes card generates the secret number.
Therefore, to guess the birthday, when the audience says yes to a card,
the magician mentally stores the number appearing in the upper-left
corner. As additional cards are shown, from those receiving a yes
answer the magician continues to add the top-leftmost numbers mentally
and then announces the sum, which is the secret number. Students can
test this procedure for several different birthday numbers to check the
validity of the solution. Of course, students find that some
memorization skills are used, but not to the degree that they
originally imagined, as they need remember only one number, the
accumulated sum, as each new card is considered. They discover that
through the use of mathematics, a seemingly magical ability is within
their grasp.
You may use the following link to help you create your own set of Green Machine cards.
"Sorting Cards"
The next birthday-guessing activity involves a set of
four-inch-by-six-inch index cards with holes punched across the top.
The students examine an overhead-projector display consisting of five
boxes filled with various combinations of the numbers from 1 to 31 and
report whether a particular student's birthday is present. The magician
responds to their yes and no answers by placing a large open paper
clip, or other thin metal rod, into the holes and sorting the cards.
The response regarding the last box of numbers on the display sheet and
the last sort leave one index card, which reveals the student's
birthday.
This card-sorting activity works very much in the same way as the
"green machine," but it recognizes the yes-no values through a system
of punched holes across the top of the index cards to symbolize the
numbers 1 through 31. Each hole represents a place-value column of the
binary system, that is, 1, 2, 4, 8, and 16.
Each student needs thirty-one index cards, which should be numbered
from 1 through 31 and identically punched with five holes along the
top. Then the cut on some of the holes is extended to match the value
of the particular number. If the number contains a value of 1 in any
given binary place value, the hole is left intact. If the value in that
place is 0, the hole is opened with a full cut so that the paper clip
will pass through.
For example, on card number 7, a value of 1 appears in the 1,
2, and 4 columns (1 + 2 + 4 = 7), and a value of 0 appears in the 8 and
16 columns.
Therefore, the 1, 2, and 4 holes are left intact and the 8 and 16
holes are cut through. Readers will notice that the number sheet used
on the overhead projector with this activity is actually a
rearrangement of the numbers on the five cards incorporated in the
"green machine" activity.
To guess the secret number correctly, the person handling the cards
begins by asking whether the secret number is in the box labeled I on
the overhead transparency. The first box on the transparency
corresponds with the hole in the upper-right corner on the cards. If
the response is yes, the cards that are lifted by the open paper clip
are kept for further sorting with the remaining questions. The other
cards are put aside. If the response is no, the cards that fall out of
the sort are kept and the cards that are lifted are put aside. The
sorts continue accordingly, moving to each hole in order from right to
left, as the students respond to boxes II-V on the transparency.
The sorting cards are similar to keypunched cards used with
early computer systems. The keypunched holes represented binary data
that were read by a machine. Instead of sorting the cards through the
cut-versus-uncut-hole process, the keypunched holes enabled electrical
or mechanical impulses to be sent through, signaling specific
information for the computer to interpret. Computers still operate on
the binary principle, but current software conceals this "machine
language" from most users.
The "sorting cards" also more distinctly represent the
complete translation of a base-ten number to a base-two number. For
example, 7 in base ten is represented by 111_{2}, and the number 4 in base ten is represented by 100_{2}.
The value of working in base two comes not so much from the content
knowledge acquired or understanding the working of the
base-two-place-value system but from the process of encoding and
decoding information; understanding, clarifying, and using number
patterns; and developing a better understanding of place value.
"Window Cards"
This activity involves a set of posterboard cards with small windows.
The windows are shown in the figures by the shaded spaces that are
marked with an X. Each card contains two columns of numbers. Again,
through a yes or no response from the class as to the appearance of a
birthday on the card and a strategic placement of the cards, the final
card reveals the student's birthday through a window created by the
other cards. If an observer is watching the cards in use when a
participant selects the secret number 11, the following interactions
would be seen. The magician holds up the first card as the participant
looks for the number 11. As the participant examines card 1 and sees
the number 11, he or she responds yes, so the card is placed faceup
with the numbers right-side up (a). Card 2 also receives a yes response and is placed on top of card 1, again faceup with the numbers right-side up (b). Because the number 11 is not on card 3, it receives a no response and
is placed faceup on top of card 2, but with the numbers upside down (c). Card 4 receives a yes response and is placed on top of card 3 faceup with the numbers right-side up (d). Card 5 is set aside. Card 6 receives a yes and is placed on top of card 4 facedown and upright (e).
Finally the entire stack is picked up, made flush, and held so that the
face of the bottom card in the stack is shown to the participant. The
secret number, 11, is revealed in the window (f).
The "window cards" carry out the same process of elimination as the
"sorting cards" except that the filtering is done through spatial
patterns. Each time the students report a yes or no response, the
different slots in the cards are turned and thereby eliminate or
include the appropriate numbers on card 6 in the final position. The
placement of the slots matches patterns related to the binary system as
discussed subsequently and displayed below.
On the one hand, when a response to a card is a yes, the card should
be placed faceup with the numbers right-side up for the magician to
read. On the other hand, when a response to a card is no, the card
should be placed faceup with the numbers in a completely upside-down
position if the magician tried to read them. These cards are stacked
one by one in the appropriate direction as the activity proceeds. The
blank card in the fifth position is put in place to keep card 6 from
view during the process, so it should be removed and put aside when the
magician arrives at that point. Then the front side of card 6 should be
shown to the audience. If the number is on that side of the card, it
should be placed facedown with that side touching the pile of cards. If
the number is not
on that side, the card should be turned over with the back side
touching the pile of cards. The magician can then pick up the stack and
with a little adjustment to make the cards flush, hold them up so that
the audience can view the secret number.
Usually students are given models of the cards to trace the
cutouts and then use scissors to create the openings. The numbers can
readily be placed on the cards with reproduced grids and glue. Card 6
must have the numbers carefully placed to match the slots exactly so
that the secret number will be easily visible at the conclusion of the
activity.
At the point where the process behind the activities becomes
familiar, the mathematics relating to the secret numbers and the
connections among the three guessing devices can be introduced. The
patterns in the binary representation of decimal numbers are the key to
linking the three activities. The previous figure may help explain this
connection.
Students are quick to note that in the first column from the
right, every other position is shaded; and many will note that in the
second column, two positions are shaded and then two positions are not
shaded. Without the previous diagram, few students are able to discover
these patterns. Fewer still will note that the pattern in each column
to the left doubles, unless a visual model is used. Yet this crucial
concept of the binary system is the basis for the activities. Students
must be able to note that the pattern in the previous diagram is 1, 2,
4, 8, 16. The pattern of numbers is the same for all three guessing
devices except that the "window cards" require the columns to be folded
over so that they fit on the sheets. To test for generalization,
students can be asked to extend the pattern to include larger numbers.
For instance, what would be the pattern in the next column and why?
After the mystery of all the activities is uncovered, students
are required to take the official oath of secrecy. This promise
requires that the students never reveal the process of the guessing
activities, but they are allowed to confirm the correct solution if
someone describes the mathematics accurately. Students often decide to
try their tricks with other members of the school community, which can
include everyone from the principal to the physical-education teacher
to the nurse.