The Sticks and Stones game is based on the Apache game "Throw
Sticks." To play the game, students throw three sticks, each decorated
on one side. Students move their pieces around the game board based on
the results of the throw, as described below.
Allow students to decorate three sticks on one side only;
the other side should be blank. (If playing this game as part of a
larger unit about Native American culture, you can allow students to
decorate the sticks with tribal symbols.) Students will use these
sticks to determine how far they move when playing the game.
To create the game board, arrange 40 stones in a circle,
preferably divided into four groups of 10. (In groups of 10, a side
benefit of this game is that it helps to develop student understanding
of the place-value system. For instance, if a student is currently on
the seventh stone in one group of 10 and rolls a 5, she gets to move to
the second stone in the next group of 10. This demonstrates modular
arithmetic, because 7 + 5 = 12, which has remainder 2 when divided
by 10.) As an alternative, you can use a Monopoly® game board, which consists of 10 squares on each of four sides.
The rules of the game are as follows:
Pair students together, and let them play the game once, for
fun. Then, before playing a second time, have students make a chart of
all throws that are possible. During a second game, have them keep
track of their throws while playing. How many of each occurred?
As an alternative, students can use the demonstration below to generate random throws.
After tallying their throws during the second game, have kids use
sticky notes to build a bar graph. Place a large piece of paper on the
wall, or draw a graph on the chalkboard, which shows the possible
throws on the horizontal axis and the number of occurrences on the
For each time a particular throw occurred during their games,
students should place a sticky note on the graph. For instance, if a
student had three throws with zero sides decorated, the student should
place three sticky notes in that category. Allow 4-6 students to place
sticky notes on the same graph. Compiling the data in this way will
give a larger sample size and should yield experimental results that
are close to the theoretical probabilities; if only 1-2 students place
their data on a graph, the results are more likely to be skewed. As
necessary, create a new graph for each group of 4-6 students. (If
possible, you can put all of the data from the entire class on one
graph, but if there is too much data, the bars will get too tall.) A
completed graph may look something like the following:
Allow students to compare the relative heights of the bars on
the graph. [The bars for one or two sides decorated are much taller,
meaning that those results are more likely when the sticks are thrown.
It also means that the probability of having a throw with three sides
the same is less likely.]
To facilitate a discussion about what the graph means, have
students compare just two categories. You may want to ask the following
- Which is more likely—a throw with one stick decorated or a throw
with two sticks decorated? [Neither. They both occur about the same
- Which is more likely—a throw with three sticks decorated or a
throw with no sticks decorated? [Neither. They both occur about the
- Which is more likely—a throw with three sticks decorated or a
throw with two sticks decorated? [A throw with two sticks decorated is
about three times as likely as a throw with all three decorated.]
- Which is more likely—a throw with no sticks decorated or a
throw with one stick decorated? [A throw with one stick decorated is
about three times as likely as a throw with no sticks decorated.]
Be sure to use mathematical terms during this discussion, such as likely and probability.
For instance, you may want to ask students, "How much more likely is it
to throw three decorated sides than to throw only two decorated sides?
Is it twice as likely? More than twice as likely?" [From the graph, it
appears to be about three times as likely, because the bar is
three times as tall.]
Return to the context of the game. Ask students, "Why do you
think you get to move more spaces when all three sticks land on the
same side?" [Throws with zero or three sides decorated are less likely
than throws with one or two sides decorated. Since they are more rare,
the reward for those throws is greater. On the other hand, a throw with
three sides decorated is just as likely as a throw with no sides
decorated, yet the reward for three sides decorated is greater; this is
not a mathematical decision, but it probably has to do with human
appreciation of art.]
The bar graph allows student to use experimental results to
discuss probability, but they should also consider the theoretical
probability of each result. This can be accomplished by constructing a
tree diagram that shows the results after three throws; a D represents
a decorated side, and a B represents a blank side:
There are eight possible outcomes, as indicated by the number of
elements in the third row. The path to each of those elements indicates
one possible outcome; for example, the highlighted path shows a first
throw of D, a second throw of B, and a third throw of B.
An organized list could also be created. The list below shows
the eight possible outcomes, which verify the results of the tree
Because three sticks are thrown, and because there are two possible
results with each stick (D or B), it makes sense that there would be 23 = 8 outcomes.
To promote conceptual understanding, be sure to compare the
items on the list to the outcomes from the tree diagram. For instance,
show that the highlighted path is equivalent to DBB in the list.
Based on the list and tree diagram, students should realize
that three decorated sides or no decorated sides occur, on average,
only once out of every eight throws, whereas one or two decorated sides
occur three times every eight throws. Ask students to compare these
theoretical probabilities to the experimental results they obtained
when playing the game.
Finally, ask students, "On average, how many turns do you think
it will take to complete a game?" Students can investigate this
question by playing again and recording the number of turns, and then
comparing their results with the rest of the class. Alternatively, if
students are prepared for the mathematics, they can reason through the
solution using basic ideas about expected value. [In eight turns, a
player would be expected to get three decorated sides on one throw, two
decorated sides on three throws, one decorated sides on three throws,
and no decorated sides on one throw.
Consequently, the player will move
1(10) + 3(3) + 3(1) + 1(5) = 27 stones in eight turns, or approximately
27 ÷ 8 = 3.375 stones per turn. At that rate, it will take
40 ÷ 3.375 = 11.85, or about 12, turns for a player to complete the
circle. Of course, it will take more if the player is passed over and
sent back to the starting point.]
Questions for Students
1. What are the possible outcomes when three sticks are thrown?
[There can be 0, 1, 2, or 3 sides decorated.]
2. What is the likelihood of each outcome?
[Throws with zero or three sides decorated are less likely than throws with one or two sides decorated. Specifically, P(0) = P(3) = 1/8, and P(1) = P(2) = 3/8.]
3. On average, how many turns will be necessary to complete a game?
[As shown above, it will take about 12-13 turns for a player to make it around the board. Since there are two players, a complete game will take approximately 25 turns.]
- How did you ensure that students understood the relationship
between the experimental results that they collected and the
theoretical probability of each outcome?
- Were students actively engaged in this lesson?
- Did the game provide motivation for the mathematics, or did it
provide a distraction from the objectives that were to be learned? What
modifications could you make for next time so that the game is not a
- Did students meet the objectives of the lesson? If not, what should be done in subsequent lessons?