Download Arithmetic Research Project

Alyssa Borton
C&M in Arithmetic
April 20, 2012
What is Probability?
The definition of probability is the likelihood or chance of an event or circumstance
occurring. The basic purpose of probability is to predict the likelihood that something will or will
not happen, and is computed by observing the number of actual outcomes compared to the
number of possible outcomes (Hatfield, pg. 442). Students are faced with the ideas of probability
outside of school, and teaching probability will provide opportunities for them to understand the
concepts in a useful and applicable way. There are differing opinions on the scope and sequence
for probability, but the activities used to teach it are consistent. There are a multitude of activities
and SMARTBoard lessons and activities available for teachers to use to teach probability.
Scope and Sequence
In early grades, Reys (2009) suggests that probability is not taught formally, but that the
proper language for probability is being taught. In the primary grades terms such as “certain”,
“uncertain”, “impossible”, “likely” and “unlikely” are used and the building blocks for
probability. Probability is normally preceded by data analysis and intertwined with statistics.
Probability incorporates both those concepts and takes it to the next level to the “what if” stage.
However, the reverse is also true. The foundational concepts of probability are used to analyze
data and statistics (Reys, 2009). The Common Core Standards (Common Core) have this
standard for first grade: “Students will be able to organize, represent, and interpret data with up
to three categories; ask and answer questions about the total number of data points, how many in
each category, and how many more or less are in one category than in another.” This leads to the
standard of being able to draw a picture and bar graph to represent a data set with up to four
categories; solve problems using information presented in a bar graph. Common Core does not
include statistics and probability until sixth grade, and even then, what adults would think of as
probability is not broached until seventh grade. The Saxon text (Larson, 1991) states that
children must learn to pick out patterns before learning to graph and therefore before learning
about probability. Several curriculums (Reys, 2009; Larson, 1991) however, suggested that the
beginnings of probability be taught in second grade, with graphing in first grade.
Therefore, in first grade, children learn how to record data and transfer that data onto a
graph. Larson (1991) suggests doing this by having a weather, birthday, and lunch/attendance
graph. Larson (1991) also incorporates other subjects or concepts the children are learning about
to create the graphs. For example, if the students are learning the days of the week, students can
create a classroom graph of everyone’s favorite day. As the year progresses, Larson raises the
level of difficulty of graphing. By the end of the year, students create their own surveys to find
data and then graph that information. Children are then able to find the likelihood of an event.
Sample space, or the set of all possible outcomes of a specific experiment, is a
fundamental concept that is needed before the probabilities of specific events can be determined.
Reys (2009) suggests children create their own events and determine the possible outcomes to
help understand sample space. As children study probability, it will help develop their critical
thinking skills and interpret the probability that is commonly used.
Activities and Materials
To understand how probability is used outside of school, students can be required to
create a list (daily or weekly) of probability statements they have seen (Reys, 2009). This will
encourage students to observe outside of the classroom and make connections between the world
and mathematics. These are used to toss, roll, etc. Events and outcomes are large uses of
probability, such as birthdays or weather. Some examples are that there is 0% chance of having a
birthday on July 32nd, or that there is 40% chance of rain. Hatfield (2005) suggests that students
learn probability by doing experiments, predicting the outcome, and recording the results.
Through this, the students will learn of samples, a small number that represents a large group
called population.
The main materials used for probability, particularly for elementary grades are coins,
dice, or game chips. Using coins and dice creates a limited number of choices for students to
work with, which allows probability to be taught at a younger age. Hatfield (2005) gives
examples of a few activities for children to do such as record the outcome of tossing a coin 30
times, then write the results as a fraction, then as a ratio. This also helps with representation and
connections. Reys (2009, pg. 406) uses an activity to teach about randomness that uses a drawing
or pool. Each person puts their name in, and one name is drawn as winner. What the activity
does, is have several statements about the winner, and place them on the number line 0:
impossible to 1: certain to happen. One example for 0 (zero) would be that the winner will be
someone from the other class; an example for 1 (one) would be that it will be either a boy or girl.
This helps with the connections standard. Burns (2007) relates real world experiences to teach
graphs. Permutations can also be introduced with probability, and Heddens (1997) suggests
adding a new coin to the groupings. This helps teach probability because by adding a coin it
changes the odds. Heddens (1997) also uses a bag of candy to help learn the probability and
graphing by having students draw candy out of a bag at random. Students first figure the
probability of drawing a certain kind, and then as they pick the candy, they record the data.
Students then take the data and make a graph.
One activity that could be used to introduce probability is Runts Race. This activity uses
a worksheet and a spinner. Students begin by getting the worksheet and putting an ‘x’ in the box
next to the picture of the fruit that they predict will be selected most; they continue to spin the
spinner and draw and color the corresponding fruit in said fruits’ row. This also incorporates fine
motor skill as students will be coloring the fruit (Runts Race, 2005).
Another activity is one called Free the Animals Game. In this game, each student is given
a game mat with 2 dice, 6 markers, and the recording sheet. Players place their markers (animals)
in any of the six cages. They may have more than one animal in any cage. Players use the
recording sheet to record how many animals they placed in each cage. Player A rolls the dice and
finds the difference of the numbers. Player A then frees one animal from that number cage, if
there is an animal in the cage (for example, if player A tosses a 6 and a 2, the difference is 4.
Player A would free one animal from Cage #4). Player B then does the same. They continue to
play until one player frees all of his/her animals. This helps teach probability because the
students are taking the chance of which cage to place his/her animals in. The class then can
discuss and calculate the probability of the different ways to place the animals in the cages (Free
the Animals, 2005).
Another game is M & M probability. This game calls for each team (two players)
receiving 24 counters, and a number line and set of dice that they share. Both players then place
all 12 counters along their number line between 2 and 12. One student then rolls the dice. Using
the sum of the numbers that appear students remove any markers off the number line that were
on that number. Students continue playing until all the markers are off the number line; students
are then required to answer questions after round one and two. The questions incorporate
probability by asking if they will place the counters differently, and if the changes made a
difference in the outcome (M & M Probability, 2005).
The last activity is a simple one involving a paper bag and ten suckers (making sure to
have enough for each student at the end of the lesson). The class counts the suckers as they go
into the bag while recording how many of each color were put inside. The teacher asks a few
students what their favorite flavor is; then students find what the probability is of them receiving
that flavor if they draw out of the bag blindly. As suckers are drawn out, the denominator
changes, therefore changing the probability (Probability Lesson, 2003).
SMARTBoard Lessons
The SMARTBoard is very useful in teaching probability. Dice, coins, spinners, and
graphs are all easily available to download. Rolling a die, or flipping a coin is as easy as flicking
a finger over the screen. It also allows everything to be spread out, so more students will be able
to participate at once. The board allows children to conduct their experiment, record the data, and
then directly produce a graph showing the results.
Some SMARTBoard lessons dealing with probability involve marbles or other markers in
a sack that the students were shown and had to find the probability by drawing out one color or
another. The lessons incorporated the language used when dealing with probability, which is
helpful when beginning to teach probability. One lesson shows a corridor with five different
doors, and children have to guess which door the teacher is behind; then discuss the odds or
probability that they guessed correctly (1 out of 5).
Works Cited
Burns, Marilyn. About Teaching Mathematics. 3rd ed. Sausalito, CA: Math Solutions
Publications, 2007. 85-106. Print.
Common Core State Standards Mathematics. Common Core State Standards Initiative , 2010.
Web. 1 May 2012. http://www.corestandards.org/the-standards/mathematics.
"Free the Animals Game." Math Wire . Math Wire, 2005. Web. 2 May 2012.
http://www.mathwire.com/games/freeanimalsgame.pdf .
Hatfield, Mary M., Nancy T. Edwards, Gary G. Bitter, and Jean Morrow. Mathematics Methods
for Elementary and Middle School Teachers. 5th ed. USA: John Wiley & Sons, Inc.,
2005. 442-59. Print.
Heddens, James W., and William R. Speer. Today's Mathematics. 9th ed. Columbus: PrenticeHall, Inc., 1997. 224-37. Print.
Larson, Nancy. Math 2: An Incremental Development. Norman, OK: Saxon Publishers, Inc.,
1991. Print.
"M &M Probability." Math Wire . Math Wire, 2005. Web. 2 May 2012.
www.mathwire.com/games/mmprobgame.pdf .
"Probability Lesson." ProTeacher. ProTeacher Collection, 8 Dec. 2003. Web. 2 May 2012.
http://www.proteacher.org/org/a/66792_Probability_lessons_the_kids_brag_about.html .
Reys, Robert, Mary M. Lundquist, Diana V. Lambdin, and Nancy L. Smith. Helping Children
Learn Mathematics. 9th ed. USA: John Wiley & Sons, Inc., 2009. 382-410. Print.
"Runts Race." Let the Probability Games Begin. Utah Education Network, 22 Sept. 2005. Web.
18 Apr. 2012. http://www.uen.org/Lessonplan/preview?LPid=14395 .