Tree Diagrams

A tree diagram can be used in many disciplines for many different things. When it comes to probability, a tree diagram is used illustrates all the possible outcomes from an experiment (sample space). It can also be used to help determine the probability of individual outcomes within the sample space. It consists of arcs and nodes and each branch of the tree represents a possible outcome of one event. I can personally remember using tree diagrams in school when dealing with probability and found them very helpful because they allow you to better visualize and organize the problem you are working on. Many students will better understand probability and the problem that they are working with if they are able to have a visual representation in front of them. It is for this reason that I feel tree diagrams have a valid place in teaching probability to primary/elementary children. The following are sample probability problems that can be figured out through the use of a tree diagram:

1. Show the sample space for tossing one penny and rolling one die. (H = heads, T = tails)

By following the different paths in the tree diagram, you can arrive at the sample space which is { H1, H2, H3, H4, H5, H6, T1, T2, T3, T4, T5, T6 }. The probability of each of these outcomes is 1/2 x 1/6 = 1/12.


2. A family has three children. How many outcomes are in the sample space that indicates the sex of the children? Assume that the probability of male (M) and the probability of female (F) are each 1/2.

There are 8 outcomes in the sample space. The probability of each outcome is
1/2 x 1/2 x 1/2 = 1/8.

I also found a web site that provides some great probability problems that involve the use of tree diagrams that students could use for practice. Have a look!
Practice Page

The following are references used while creating this blog:

Lesson Plans: Tree Diagrams

Tiscali.Reference: Tree Diagram

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The Not so Random Coin Toss

While searching up some information about chance and probability, I came across this interesting article that gives a different side to the concept of flipping a coin. It states that a coin toss may not be not as fair as once thought. Apparently, the randomness is due to the fact that when a human flips a coin, it lands randomly because of the way in which it was tossed. The article can be found at the following link: The Not so Random Coin Toss. It really makes you think about how fair a coin toss really is.

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Probability in the Real World

In doing my research, I have come to realize that probability is one of the most prominent uses of mathematics in our everyday lives. Knowing the probability of a certain event happening or not happening can be very important to us in the real world. The following are examples of situations that take place in our everyday lives that involve the use of probability:

Weather Forecasting
Suppose you have some outdoor plans made for a particular day and the weather report says that the chance of rain is 70%. Should you still go ahead with your plans or should you cancel them for another day? Where does this forecast comes from? Meteorologist are able to calculate the likelihood of what the weather may be on a particular day by looking back in a historical database and examining all the other days in the past that had the same weather characteristics and then determine that on 70% of those similar days it rained. The mathematical formula for probability can be used to demonstrate these findings. When looking for the chance it will rain, this will be the number of days in the database that it rained is divided by the total number of similar days. For example, if there is data for 100 days with similar weather conditions (the sample space), and on 70 of these days it rained (a favorable outcome), the probability of rain on the next similar day is 70/100 or 70%. Since a 50% probability means that an event is just as likely to happen as not to happen, a 70% chance means that it is more likely to rain than not. Therefore, perhaps it is best that you stay home and reschedule your plans for another day!

Batting Average
A batting average involves calculating the probability of a player hitting the ball. The sample space is the total number of time a player has had at bat and each hit is a favorable outcome. Therefore, in 10 at-bats a player gets 3 hits, his or her batting average is 3/10 or 30%. For baseball stats, all the percentages are multiplied by 10, so a 30% probability translates to a 300 batting average. So let's say your favorite baseball player is batting 300. This means that when he or she goes up to the plate, they only have a 30% chance of hitting the ball!

Winning the Lottery
Millions of people around the world spend their money on lottery tickets in hopes of winning the big jackpot and become millionaires. But do these people realize how low their chances of winning actually are? Determining the probability of winning the lottery will allow you to see what the likelihood of winning truly is. The following formula is used to figure out the probability of winning Lotto 6/49:

The number of winning lottery numbers
The total number of possible lottery numbers

The Canadian 6/49 Lottery has 6 numbers drawn from a total of 49 balls with the numbers 1 through 49 on them. You must use the formula for permutations and combinations to figure out the probability of getting all 6 numbers in the correct order. The answer would be that the number of ways of choosing 6 numbers from 49 is 49C6 = 13, 983, 816 . Therefore the probability of winning the lottery is 1/13983816 = 0.000 000 071 5 (3sf), which is about a 1 in 14 million chance! The website

Permutations and Combinations helps to explain this area of probability in more detail. I came across the following link that provides a sampling of the odds of some of the World's Great Lotteries. Have a look: The Lottery Site

The Futures Channel
While doing my research I found a website that contained a short five minute video that really helped to illustrate probability at work in a real life situation. The film is about a probability map that was constructed by a mathematician and used to locate a sunken U.S. ship with the largest sunken gold treasure in U.S. history. It would be a great film to show in your classroom when teaching probability because it will help children realize that what they learn in math class and be applied and used in the real world. The following is the link to the film: Movie Title: Undersea Treasure

The following are references used while creating this blog:

The Math Forum

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Atlantic Canada Mathematics Curriculum Guide

I feel it is very important to have a good understanding of the curriculum guides and outcomes for the province that you are teaching in. These documents are a great resource for teachers and they provide many suggestions as to how to teach the required curriculum. It is for this reason that I had a look through The Atlantic Canada Mathematics Curriculum Guide and choose several activities that they suggested to use when teaching probability for each primary/elementary grade level.

Grade One: Provide an opaque bag and coloured cubes for the student. Ask the student to put 10 cubes in the bag so that red will never be chosen. Have the student repeat the task, this time putting in cubes so that red will always be chosen. Finally, the student repeats the task so that red will sometimes be chosen.

Grade Two: Ask the child to design a spinner so that spinning red is more likely than spinning green, but spinning red is less likely than spinning yellow.

Grade Three: Ask pairs of students to think of what might happen about half the time when a die is rolled. Students should experiment with the die, record outcomes, and later present their findings to classmates.

Grade Four: Teachers could have students station themselves within or near the school, where they can see passing cars. They record the colours of the first 10 cars they see. They then describe the probability that a passing car will be blue. They might then explain why they might get a different probability the next time they perform the experiment, and check to see if they do.

Grade Five: Tell the student that you rolled a pair of dice 25 times and the sum of the numbers was 8 on 4 of the rolls. Ask: What is the experimental probability that the sum is 8? Does that seem reasonable?

Grade Six: Tell students that a particular baseball player has an average of .250, i.e., he gets 1 hit in 4 times at bat, on average. Ask them to conduct a simulation to determine the probability that the player will get a hit each time at bat in a particular game. [This can be simulated by creating a spinner with four equal sections, one of which is labelled ‘hit’ and the other three labelled ‘miss’. Spin the spinner 4 times recording each outcome (batting average) and repeat several times.]

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Lesson Plans and Worksheets

The Internet is a network filled with information that can be extremely helpful to teachers as they try to come up with ideas on how to create a lesson plan and teach a unit. When you are getting information from any public forum, it is important to review the material that you take from the Internet and ensure that it is accurate information. In the case of lesson plans, the teacher should conduct the actual lesson on their own before trying it out in the classroom. The following are links to sample lesson plans that incorporate hands-on experiences that could be used when teaching probability in a primary/elementary classroom.

Sticks and Stones
Probability and Statistics K-6
Risks and Limits
Playing with Probability

Probability

Many websites also provide printable worksheets that can be used to accompany your lesson plan. The following are sample worksheets that can be printed off and used in addition to hands-on activities in a primary/elementary classroom:

Heads or Tails: Penny Math
Make Predictions

What is Probability?
Skillswise: Probability 1

Skillswise: Probability 2

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Teaching Probability to Young Children

Many people assume that probability is a mathematical concept that is introduced and taught in junior high and high school, however, after reading and reviewing much research, I now realize that this idea is untrue. Young children seem to have an intuitive understanding of chance and allowing them to explore this in the early grades simply enhances their probabilistic thinking, number sense, and mathematical connections. Principles and Standards emphasizes that activities in the early grades should take the form of answering questions about the likelihood of events, using such vocabulary as more likely or less likely. Children must be able to recognize that although individual events may not be predictable, there exists trends and patterns that can often occur.

When doing my research it became quite clear that problems dealing with probability should be taught through hands-on experiences. It is extremely important that children participate in hands-on activities such as playing with dice, spinners, or coins as a means to investigate, make predictions, and engage in probabilistic thinking. As the Atlantic Canada Mathematical Curriculum Guide points outs, it is also important to use informal experiences that can be built on and expaned during the primary grades. These experiences should involve real-world situations both inside and outside the classroom where math is used. In doing this, mathematical learning becomes relevant to students. Templates for sample spinners and dice cutouts that can be used in the classroom can be printed off the following website:

Probability and Statistics K to 6

In today's world computer simulations can also be very useful in assiting in the teaching of probability. Many websites and computer programs allow students to watch as a spinner spins or a coin flips and random outcomes are produced with the press of a button. I think the use of computers would be a great addition to a lesson on probability because it provies the hands-on experience that is essentail to learning and then also combines it with an animated visual graphic which is very appealing to children. It enables students to investigate more realistic situations than were previously possible. Having the opportunity to complete problems using a computer allows children to get a break from the usual paper and pencil tasks and actually be interested and have fun while learning their math. There are hundreds of interavtive online activites that can be used to assist in the learning of probability, however, it is important to ensure that they provide accurate information and are age appropriate for the students in your classroom. The following are several computer activites that I found online that I thought would be useful when teaching probability:

Interactivite: Experimental Probability (Spinner Game)

Coin Tossing

Illuimations: Adjustable Spinner

Colour Spinner

Racing Game with one Dice

The following are references that I used while creating this blog:


National Council of Teachers of Mathematics

The Atlantic Canada Mathematical Curriculum Guide

Van De Walle, John. and Sandra Folk. Elementary And Middle School Mathematics: Teaching Developmentally. Canada: Pearson Education Inc, 2005

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Children's Understanding of Probability

Understanding how students' think about a particular topic is an important component of instruction because this kind of knowledge is useful to teachers as they plan and implement instruction. It is for this reason that I choose to research information about how children think and comprehend the topic of probability and chance.

There often exists many misconceptions about chance and probability in the minds of young primary children. When reading an article written by Jeffery A. Frykholm, he says that, “children, roughly up to the age of seven, often struggle with concepts of uncertainty and are generally unable to distinguish events that will necessarily occur from those that may or may not occur.” Many times they believe that an event will happen because they are lucky or it is their favorite colour. This can be seen when little children get very excited while playing a game of Snakes and Ladders because they simply do not understand that the outcome of the game is completely random. In order to help deal with such misconceptions, teachers must allow time for children to be explore and experiment with the concept of probability over a long period of time. They need to develop the idea that some events are more, less likely or equally likely to occur. I will get into more detail about ways to teach children probability in a future blog.

Although there does exist some misconceptions about probability, most young children seem to have some type of understanding of the concept. Frykholm says in his article that young children have an intuitive understanding of chance and that this should be developed through explorations in the early grades that will enhance children's probabailistic thinking, number sense, mathmatical connections. The understandings that children bring to school are typically learned from games that they play at home in informal settings, An example of such a game is the childhood rhythm “Eenie, Meenie, Minie Moe” used when choosing a person to be “it”. There seems to be an acceptance of this form of random chance and even young children seem to understand the process. One article that I read describes student's understanding of probability and probabilistic thinking in terms of levels that range from one to four. Level one involves students who use only subjective reasoning and have a narrow way of thinking that is early distracted and misled by irrelevant information. A student who is functioning in this prestructural level would probably reason that the color red has selected because it is their favorite color. This level is often associated with concrete-symbolic thinking that often involves myths and mental imaging. Level two includes students who are in a transitional period that is characterized by naive and inflexible attempts to quantify probabilities. In this unistructural level, children become involved in probability tasks in relevant ways, however usually only one aspect is pursued. Students are beginning to function in concrete-symbolic mode and are able to use mathematical language and symbols to represent their understandings. The third level of thinking involves children who exhibit both the multistructural and relational levels of thinking and uses quantitative reasoning when dealing with probability tasks. Unlike students in level two, children in this level recognize that there are more than one relevant feature in a probability tasks and they can coordinate and quantify thinking about sample space in relation to probability. Students who reach the fourth and final level of thinking are able to make connections between sample space, outcomes, and probability. Using this relational level, they can use valid numerical measures to describe the probabilities of events and conditional events. Children are able to make meaning from abstract ideas of probability by relating them to their own experiences.

In doing research about children's understanding of probability, I found it really hard to find information and I still have some questions that I would liked answered. I feel as though there are many contradicting ideas out there involving this topic and I must do a little more research in order to get a better understanding. Perhaps getting some first hand experience while working with children on probability would help clear up many of my remaining questions.


The following are references that I used while creating this blog:

Van De Walle, John. and Sandra Folk. Elementary And Middle School Mathematics: Teaching Developmentally. Canada: Pearson Education Inc, 2005


Fykholm, Jeffery A. “Eenie, Meenie, Minie, Moe: Building on intuitive notionsof chance.” Teaching Children Mathematics 8.2 (2001): 112 - 118

Graham A. Jones, Cynthia W. Langrall, Carol A. Thornton, and A. Timothy Mogill.
“Student's Probabilistic Thinking in Instruction.” Journal for Research in Mathematics Education 30.5 (1999): 487 - 519

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