Download Native American Games

Culture: Northeastern American Indian - Potawatomi
Activity: Kwezage’win – Two-sided dice game of the
Potawatomi
Photo of the dice game above is from the Hannahville
Indian Community and is part of the Virtual Museum
created by students. This game is at the National
Museum of the American Indian archives and the photo
was taken and interactive video was developed by Lance
Sagataw. View the video at
http://www.potawatomilanguage.org/virtualmuseum/dicegame.php
“It is important for American Indian students to see the mathematics in the traditional
activities of their culture. The two-sided dice game has been played for hundreds of
years by most tribes in North America and students continue to enjoy the game, and
learning the mathematics incorporated in it.”
Richard Sgarlotti
Context – Games, including games of chance and the associated wagers, is one of the
categories of ethnomathematics listed by Alan Bishop. Bishop says that the six universal
mathematical behaviors practiced by any culture are counting, measuring, designing,
locating, explaining, and playing (Mathematical Enculturation: A Cultural Perspective
on Mathematics Education, Alan J. Bishop, Springer, 1991)
Most tribes in North America used two-sided dice, sometimes during healing ceremonies.
Because of tribal environmental differences, the dice were made of sticks, stones, peach
pits, walnut shells, and antler buttons; usually one side plain and one side decorated. In
one variation, four to ten dice were shaken in a moccasin, basket, or other container,
tossed onto a surface, with a winner determined to be an equal number up and down.
A good explanation of the variations of the game is given in the following document.
Ecological Importance of Native Americans Culture to the Kentucky Coffee Tree
(Gymnocladus dioicus), Andrew R. VanNatta
http://www.uwsp.edu/forestry/StuJournals/Documents/NA/avannatta.pdf
Specifically, the Potawatomi game (Kwezage’win) was played only by women, and
mostly in the winter in place of double-ball. (For our use here allowances are made so
that all students, boys and girls, can explore probability in this cultural activity). Like the
two preceding games, a woman in honor of her guardian spirit also sponsors this one, and
similar ceremonial preliminaries are held. After the feast, a blanket is spread out on the
floor and the women sit in a circle, but divided into two teams with each side sitting in a
semi-circle facing the other. As many women can play as want to, but there are only four
prizes: yard goods of red, blue, green, and white. The gaming equipment consists of a
wooden bowl and eight dice, six of which are thin, circular discs; one is carved in the
form of a turtle and one represents a horse’s head. Dice were formerly made of buffalo
rib, but horse’s ribs are used at present. One surface of each die is colored blue (red may
also be used). Thus each die has a colored and a white side. The bowl is held with both
hands, and the dice shaken to the far side of the bowl which is given one flip, set on the
floor, and the score counted, as follows:
All of similar color except 2, counts 1 point
All of similar color except 1, counts 3 points
All of similar color except turtle, counts 5 points
All of similar color except horse, counts 10 points
All of similar color counts 8 points
All of similar color except turtle and horse counts 10 points
The score is tallied by the woman laying out the correct number of bean counters in front
of her. Each woman shakes until she misses twice, and passes on the bowl in a clockwise
rotation. The first to score ten points wins the game, and a piece of the yard goods is
given to one of the men spectators, who returns a gift of equal value in the future.
From The Potawatomi Indians of Wisconsin by Robert E. Rizenthaler
In addition, information from other Native tribes can be found at
Oneida Oral History http://www.mpm.edu/wirp/ICW-136.html
"Wa'lade hama'gan", a Penobscot bowl and dice game:
http://www.nativetech.org/games/bowl&dice.html
An assignment for students could be to find information on the game from other tribes in
books or online.
The above information can be used as background for each of the grade level activities.
Students can play a version of the game online at
http://www.nativetech.org/games/dicegame/
Brief description of the mathematical content and activities (please very briefly list
the activities for each grade band – K/3, 4-8, and 9-12)
The Principles and Standards for School Mathematics of the National Council of
Teachers of Mathematics has the following statement on Probability:
“A subject in its own right, probability is connected to other areas of mathematics,
especially number and geometry. Ideas from probability serve as a foundation to the
collection, description, and interpretation of data.
In prekindergarten through grade 2, the treatment of probability ideas should be
informal. Teachers should build on children's developing vocabulary to introduce and
highlight probability notions, for example, We'll probably have recess this afternoon, or
It's unlikely to rain today. Young children can begin building an understanding of chance
and randomness by doing experiments with concrete objects, such as choosing colored
chips from a bag.
In grades 3–5 students can consider ideas of chance through experiments—using coins,
dice, or spinners—with known theoretical outcomes or through designating familiar
events as impossible, unlikely, likely, or certain.
Middle-grades students should learn and use appropriate terminology and should be able
to compute probabilities for simple compound events, such as the number of expected
occurrences of two heads when two coins are tossed 100 times.
In high school, students should compute probabilities of compound events and
understand conditional and independent events. Through the grades, students should be
able to move from situations for which the probability of an event can readily be
determined to situations in which sampling and simulations help them quantify the
likelihood of an uncertain outcome.
Many of the phenomena that students encounter, especially in school, have predictable
outcomes. When a fair coin is flipped, it is equally likely to come up heads or tails.
Which outcome will result on a given flip is uncertain—even if ten flips in a row have
resulted in heads, for many people it is counterintuitive that the eleventh flip has only a
50 percent likelihood of being tails. If an event is random and if it is repeated many,
many times, then the distribution of outcomes forms a pattern. The idea that individual
events are not predictable in such a situation but that a pattern of outcomes can be
predicted is an important concept that serves as a foundation for the study of inferential
statistics.”
The two sided dice activities provide the same kind of results as tossing a fair coin, with
the added novelty of incorporating a traditional American Indian activity. In most cases,
students can make the dice from a variety of materials. One question for students: “How
can you be sure that the dice are fair?
All activities can be adapted as individual activities that students do on their own as
discovery lessons, or a teacher directed lessons.
K-3 – Probability and Data
Even though the new Common Core State Standards do not list any K-3 standards for
probability, many of the students’ activities in this lesson involve counting,
fractions/ratios, organizing and graphing data, and other concepts important for these
grade levels.
1. In one variation of the game, only one two sided die (plain and colored) is tossed ten
times and results are recorded in a table, with class results totaled. The results are also
put on individual student’s graphs, and a class total graph. Students observe the results
and hypothesize on the ratio of plain to colored tosses, and the fraction of plain and
colored tosses.
2. In this variation of the game, students have four two sided dice. The dice can be made
from a variety of materials, or two sided counters work well. Shake and toss them at
least ten times, record your results, and determine how likely it is that would get at least
one pair of colored sides up on any toss.
Make a check mark for the result on each toss.
Trial
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
No Red
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One Red
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Two Red
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Three Red
________
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Four Red
________
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Totals
Make a graph of the number of “red” dice results.
How many of your tosses resulted in at least one pair of colored sides up? _______
If you tossed the dice ten more times, how many times would you expect to get at least
one pair of colored sides up? _______
Try it. What happened? Will this always happen?
How would you determine the theoretical probabilities associated with this game?
4-8 - Probability and Data
The standards below are CCSS for grade 7 and the two sided dice games which follow
are good models for these standards.
Investigate chance processes and develop, use, and evaluate probability models.
•
7.SP.5.7.SP.5. Understand that the probability of a chance event is a number
between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers
indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability
around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1
indicates a likely event.
•
7.SP.6. Approximate the probability of a chance event by collecting data on the
chance process that produces it and observing its long-run relative frequency, and predict
the approximate relative frequency given the probability. For example, when rolling a
number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but
probably not exactly 200 times.
•
7.SP.7. Develop a probability model and use it to find probabilities of events.
Compare probabilities from a model to observed frequencies; if the agreement is not
good, explain possible sources of the discrepancy.
o
Develop a uniform probability model by assigning equal probability to all
outcomes, and use the model to determine probabilities of events. For example, if a
student is selected at random from a class, find the probability that Jane will be selected
and the probability that a girl will be selected.
o
Develop a probability model (which may not be uniform) by observing
frequencies in data generated from a chance process. For example, find the approximate
probability that a spinning penny will land heads up or that a tossed paper cup will land
open-end down. Do the outcomes for the spinning penny appear to be equally likely
based on the observed frequencies?
•
7.SP.8. Find probabilities of compound events using organized lists, tables, tree
diagrams, and simulation.
o
Understand that, just as with simple events, the probability of a compound
event is the fraction of outcomes in the sample space for which the compound event
occurs.
o
Represent sample spaces for compound events using methods such as
organized lists, tables and tree diagrams. For an event described in everyday language
(e.g., “rolling double sixes”), identify the outcomes in the sample space which compose
the event.
Activities:
1. Use student made dice. Students can decorate each side, maybe red and white. Each
student will toss 4 two-sided dice at least five times and record the number of distinct pairs
he/she gets. For example, 4 red are two pair of red.
Make a check mark for the result on each toss.
Trial
1.
2.
3.
4.
5.
No pairs
______
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1 Red pair
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2 Red pairs
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1 Red/1 white
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1 white pr.
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________
Which outcome (number of pairs) came up the most often? ________
Explain why that should happen.
Which outcome came up the least often? ________
Explain why that should happen.
Will those results always happen?
2 white
_______
_______
_______
_______
_______
2. Combine the students’ results above, either as small groups (5 students) or a whole
class. Make tables and bar graphs of the results.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Student
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# of tosses # of red pairs # of white pairs
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Which outcome (number of pairs) came up the most often with the whole class?
________
Explain why that should happen.
Which outcome came up the least often? ________
Explain why that should happen.
Will those results always happen?
3. Multiplying probability with independent events
Flip a two sided die and record the outcome. Flip it again. If you think of it as a coin,
what is the probability of getting heads both times?
These two flips of the coin are independent because they don’t have any influence on
each other. Whatever the first outcome is does not affect the next flip of the coin.
A tree diagram shows outcomes for two flips.
First flip
Second flip
Outcomes
H
H H
T
H T
H
T H
H
T
T
T T
There are four outcomes, but only one is a favorable one (H, H). So the probability of
(H, H) is ¼.
Use the Tree diagram above.
1. How many ways can we get (T, T)?
2. What is the probability of (T, T)?
3. How many ways can we get a head and a tail? (It doesn’t matter which one is
first)
4. What is the probability of getting a head and a tail in any order?
We could use the counting method to find the number of outcomes for getting heads both
times. What numbers fill in the boxes?
First flip
Number of
favorableoutcomes
Number of
possible outcomes
Second flip
?
X
?
=
?
?
X
?
=
?
What is the probability of head / tail (in either order)? _______
Using the same method, what is the probability of two heads? ______
Using the same method, what is the probability of two tails? ______
4. Students develop their own game based on two sided dice.
How many dice?
What questions to ask?
9-12 - Probability
From CCSS
Mathematics » High School: Statistics & Probability » Introduction
Randomization has two important uses in drawing statistical conclusions. First, collecting
data from a random sample of a population makes it possible to draw valid conclusions
about the whole population, taking variability into account. Second, randomly assigning
individuals to different treatments allows a fair comparison of the effectiveness of those
treatments. A statistically significant outcome is one that is unlikely to be due to chance
alone, and this can be evaluated only under the condition of randomness. The conditions
under which data are collected are important in drawing conclusions from the data; in
critically reviewing uses of statistics in public media and other reports, it is important to
consider the study design, how the data were gathered, and the analyses employed as well
as the data summaries and the conclusions drawn.
Random processes can be described mathematically by using a probability model: a list
or description of the possible outcomes (the sample space), each of which is assigned a
probability. In situations such as flipping a coin, rolling a number cube, or drawing a
card, it might be reasonable to assume various outcomes are equally likely. In a
probability model, sample points represent outcomes and combine to make up events;
probabilities of events can be computed by applying the Addition and Multiplication
Rules. Interpreting these probabilities relies on an understanding of independence and
conditional probability, which can be approached through the analysis of two-way tables.
Some of the specific standards relate to these activities
Use the rules of probability to compute probabilities of compound events in a uniform
probability model
•
S-CP.6. Find the conditional probability of A given B as the fraction of B’s
outcomes that also belong to A, and interpret the answer in terms of the model.
•
S-CP.7. Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and
interpret the answer in terms of the model.
•
S-CP.8. (+) Apply the general Multiplication Rule in a uniform probability model,
P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.
Activities:
1. Read the information on Kwezage’win – Two-sided dice game of the Potawatomi,
written above, and using similar dice, play the game. Use the scoring table as listed.
2. With a two sided die (one colored, one plain), you know that the odds of one die
coming up on the colored side is one out of two (1/2). Answer the questions and fill out
the chart below with the theoretical probabilities.
With just one die, the probability of no colored die is ½.
With two dice, the probability of no colored is _____
With three dice, the probability of one colored is _____
With four dice, the probability of two colored is _____
With five dice, the probability of three colored is _____
With six dice, the probability of four colored is _____
With seven dice, the probability of five colored is _____
With eight dice, the probability of six colored is _____
Note: The six situations above are the same as “All of similar color except 2”,
and count 1 point according to the rules.
Now toss the dice ten times and record the number of colored dice each time.
Number of Colored Dice
Trial
0
1
2
3
4
5
6
7
8
1.
____ ____ ____ ____ ____ ____ ____ ____ ____
2.
____ ____ ____ ____ ____ ____ ____ ____ ____
3.
____ ____ ____ ____ ____ ____ ____ ____ ____
4.
____ ____ ____ ____ ____ ____ ____ ____ ____
5.
____ ____ ____ ____ ____ ____ ____ ____ ____
6.
____ ____ ____ ____ ____ ____ ____ ____ ____
7.
____ ____ ____ ____ ____ ____ ____ ____ ____
8.
____ ____ ____ ____ ____ ____ ____ ____ ____
9.
____ ____ ____ ____ ____ ____ ____ ____ ____
10.
____ ____ ____ ____ ____ ____ ____ ____ ____
What do your experimental results show?
Combine the results from all students in class.
Find the experimental probabilities for each of these events using the table above.
All of similar color except 1, counts 3 points
All of similar color except turtle, counts 5 points
All of similar color except horse, counts 10 points
All of similar color counts 8 points
All of similar color except turtle and horse counts 10 points
Activity 3.
Find the theoretical probabilities for each of these events using the any methods.
All of similar color except 1, counts 3 points
All of similar color except turtle, counts 5 points
All of similar color except horse, counts 10 points
All of similar color counts 8 points
All of similar color except turtle and horse counts 10 points
Activity 4.
Design a scoring system based on the theoretical probabilities you found.
Activity 5
Archaeologists have found evidence of games of chance on prehistoric digs, showing
that gaming and gambling have been a major pastime for different peoples since the dawn
of civilization. Given the Greek, Egyptian, Chinese, and Indian dynasties' other great
mathematical discoveries (many of which predated the more often quoted European
works) and the propensity of people to gamble, one would expect the mathematics of
chance to have been one of the earliest developed. Surprisingly, it wasn't until the 17th
century that a rigorous mathematics of probability was developed by French
mathematicians Pierre de Fermat and Blaise Pascal. Although it is not directly related to
this game, a nice explanation of the mathematics of two sided dice is given on the
website mathforum.org/isaac/problems/prob1.html
Students can research the American Indian games in terms of Pascals work.
Incorporation of Common Core Math Standards
K/3 - 2.MD.10. Draw a picture graph and a bar graph (with single-unit scale) to represent
a data set with up to four categories. Solve simple put-together, take-apart, and compare
problems using information presented in a bar graph.
4-8 - 7.SP.6. Approximate the probability of a chance event by collecting data on the
chance process that produces it and observing its long-run relative frequency, and predict
the approximate relative frequency given the probability. For example, when rolling a
number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but
probably not exactly 200 times.
9-12 - S-CP.8. (+) Apply the general Multiplication Rule in a uniform probability model,
P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.
Rational for inclusion of this culture in the book:
Numerous games of chance involving concepts of probability permeate cultures around
the world. American Indian cultures often used ‘stick dice’, peach pits, or crafted twosided disks from locally gathered materials to construct objects for their games. Students
explore probability concepts in this culturally situated game simulation as they connect a
traditional game (and one the still may be played at cultural gatherings) with
contemporary mathematics concepts.