## Sticks and Stones

- Lesson

Students will play *Sticks and Stones*, a game based on the Apache
game "Throw Sticks," which was played at multi-nation celebrations.
Students will collect data, investigate the likelihood of various
moves, and use basic ideas of expected value to determine the average
number of turns needed to win a game.

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:

**Object of the Game:**Be the first player to move your piece around the board past your starting point.**Set-Up:**Each student should place a marker on opposite sides of the circle. The area inside the circle is used for throwing the sticks when playing the game.**Play:**Determine which player will go first. Player 1 throws the three sticks into the center of the circle and moves her piece according to the results:Player 2 then throws the sticks and moves accordingly. Play continues with players alternating turns.

**Special Rule:**If one player’s marker lands on or passes another player’s, the player passed over must move her piece back to the starting point.

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 vertical axis.

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 questions:

- Which is more likely—a throw with one stick decorated or a throw with two sticks decorated? [Neither. They both occur about the same amount.]
- Which is more likely—a throw with three sticks decorated or a throw with no sticks decorated? [Neither. They both occur about the same amount.]
- 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 diagram:

DDD | DDB | DBD | BDD |

BBD | BDB | DBB | BBB |

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 2^{3} = 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.]

- Stones, chips, or other markers (to create a circular game board)
- Popsicle sticks
- Feathers, arrowheads, or other place markers

**Assessment Options**

- Observe students during the lesson and look for the following:
- Can students use correct mathematical vocabulary when disussing their findings? Students might use the following words or phrases: more likely, less likely, equally likely, probable, outcomes, fraction, ratio, tree diagram.
- Can students use a tree diagram to accurately represent outcomes?
- Can students use an organized list to accurately generate all possibilities?
- Do students use modular arithmetic when moving their pieces around the game board?
- Can students compare categories on the graph?
- Can students determine which of two outcomes is more likely and explain why?

- Ask students to use the sticks to create a new game. They should provide complete rules for the game, including how players move and win. Then, they should explain the possible outcomes when a player throws the sticks, and they should use probability to discuss how the game will likely proceed.
- The game of
*Sticks and Stones*uses moves of 10, 5, 3, and 1. However, these moves are not proportional to the likelihood of each throw. (For example, a throw with three decorated sides is 1/3 as likely as a throw with two decorated sides, yet a player moves more than three times as far (10 > 3 × 3).) Ask students to rewrite the rules so that the distance a player moves is in alignment with the probability of each result. [For instance, a proportional distribution would allow a move of 3 stones for throws with zero or three decorated sides and a move of just 1 stone for throws with one or two decorated sides.]

**Extensions**

- Instead of using four groups of 10 stones as the game board, use six groups of 7 or three groups of 12. These other arrangements allow for exploration of modular arithmetic with other bases.
- Use a hundreds boards instead of a circle of stones for extended practice with the place-value system.

**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.]

**Teacher Reflection**

- 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 distraction?
- Did students meet the objectives of the lesson? If not, what should be done in subsequent lessons?

### Learning Objectives

Students will:

- Collect and display data regarding the moves in a game of
*Sticks and Stones*. - Use probability to estimate the average number of turns needed to win a game.
- Modify the rules to create a different game.

### NCTM Standards and Expectations

- Select, create, and use appropriate graphical representations of data, including histograms, box plots, and scatterplots.

- Collect data using observations, surveys, and experiments.

- Represent data using tables and graphs such as line plots, bar graphs, and line graphs.

- Propose and justify conclusions and predictions that are based on data and design studies to further investigate the conclusions or predictions.

- Discuss and understand the correspondence between data sets and their graphical representations, especially histograms, stem-and-leaf plots, box plots, and scatterplots.

- Compute probabilities for simple compound events, using such methods as organized lists, tree diagrams, and area models.

### Common Core State Standards – Mathematics

Grade 7, Stats & Probability

- CCSS.Math.Content.7.SP.C.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.

Grade 7, Stats & Probability

- CCSS.Math.Content.7.SP.C.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.