Probability Activities/Resources on the Internet

Some web sites on teaching probability and how we would use them in our classrooms to meet the above stated goals:

• How Large is Pennsylvania?  This site includes an activity which meets the fourth goal of the probality curriculum at the 5-8 grade levels.  Students will make predictions and be asked to rethink their predictions based on new information.  We would use this activity at the beginning of class to create interest and discussion about what it means for an event to be more likely than another.  The new information in the activity would be given intermittently, perhaps every 5 minutes.  The students would discuss the problem in small groups.  Whole class discussion would follow at the end regarding the relevance of the activity.   To further increase the relevance of this activity to the lives of students (meeting the first curricular feature standard) we would change the activity to investigate the size of a familiar geographic area--for example, Wisconsin, Eau Claire, or the school.  The actual area would be provided at the end of the class.  The focus of the activity, however, is not on getting the right answer but on the reasoning process involved. Back to top
• Cereal box problem using TI-83   This site provides an activity which can meet all of the goals of the probability curriculum at the 5-8 grade levels.  It also integrates the use of technology in problem solving and reasoning (meeting the fourth curricular feature standard).  Students will use the TI-83 to simulate a discrete random distribution.   The instructions on this page are clear enough to minimize teacher and student frustration.  The students would work in groups of two.  One student will run the simulation and the other will record the results.  The students will switch roles so they both have experience running a simulation with the calculator.  Day 1 would end after compiling the frequency table and the suggested questions would be assigned as homework.  After this part of the exercise, the activity becomes too advanced for the 5-8 curricullum.   Further work and discussion can include creating a frequency histogram and using it to begin answering some of the questions for further exploration.   Students would also explore the descriptive statistics and analyze the meaning of those statistics.   Back to top
• Rock,Paper,Scissors As an introductory activity to fundamentals in probability, this web page and the lesson contained within address the first four goals of the probability curriculum at the 5-8 grade levels. Objectives include introducing students to probability, having students construct a theoretical tree diagram, and comparting their tree diagram to experimental data. Discussion can also include "what constitutes a fair game?" The second curricular feature standard pertaining to math as communication is addressed in this activity as well. At first, we would split students into groups of two. Through playing the "rock,paper,scissors" game and recording the outcomes, students obtain experimental data. Data collection would follow; both individual group data and whole class data. Discussion would include the possible outcomes, comparison of theoretical data to experimental data--why are there differences?--what does it mean to have a "1 in 3" chance of winning? Real world discussion--contests with a "1 in 12" chance of winnning--what does this mean? To create further discussion, rule changes are provided for a game involving groups of 3. Again, we would instruct students to create tree diagrams and then play the game to obtain additional experimental data. Discussion would focus on how the rules affected the theoretical probabilities and experimental data.  Back to top
• Take a Chance Probability Project This site provides a lot of great materials that can be used to introduce probability at the middle school level.  Included is a lesson plan very similar to the rock, paper, scissors activity described above.  The bulk of this site includes a number of printable worksheets that are helpful to introducing probability.  Some of these worksheets are simply tally sheets for penny tossing, die tossing, and spinner activities.

Some of the other worksheets contain activities with follow-up questions.  These questions would be great to assign as group work.  They could also be assigned as homework and then discussed the following day in class.  The worksheets involve conducting experiments, predicting outcomes, and finding theoretical probabilities. These worksheets help to meet the first four goals of the 5-8 probability curricullum.

An activity titled Perry Probability is a great activity that helps students see some possible uses of probability in the real world, meeting the fifth goal of the 5-8 probability curricullum.  The worksheets for this activity could be printed, and a day spent on reading and discussing the problem. The follow up questions could be used during group work or as an assignment.

Towards the end of the web page are some more difficult activities, like urn problems, that introduce the concept of dependent and independent events.  There is a lot of room for class discussion in these activities involving mathematical reasoning, helping to meet the second 5-8 curricular feature.   An entire unit could be developed for a middle level grade based on this site alone.  The activities and worksheets progress as you scroll down the page.   This site would also be great to use for a source of supplementary materials/worksheets for lessons already created.  Back to top

• Tootsie Pop Pull This is a short lesson plan that introduces the idea of sampling without replacement.  It is a fun activity where the class gets a treat!  The lesson plan is well written, just be prepared with the tootsie pops.  This activity meets Standard 1-4 of the NCTM 5-8 Probability Standards.  Back to top
• Going in circles This is a worksheet that requires students to match problems with the correct probabilities.  Try having the students get the answers experimentally first, then theoretically.  By doing this, you would meet Standard 1& 3 of the NCTM 5-8 Probability Standards. It could also be used as a review assignment.  This activity also meets Standard 2 of NCTM 5-8 Probability Standards.  Back to top
• Do you want to Bet? This worksheet is very similar to Going in Circles except it uses probabilities of cards and dice, helping students develop an appreciation for probability.  Thus, this workheet can be used to meet Standards 1-5 of the NCTM 5-8 Probability Standards. Here is a similar worksheet  Back to top
• Conditional Probability This site includes a worksheet that addresses a conditional probabilities using mostly theory.  This would be an effective group assignment to meet Standard 2 of the NCTM 5-8 Probability Standards. Back to top
• Proabability Experiments Here are some ideas for experiments to run.  These would be used effectively as group projects during class time.  This site is great at meeting Standards 1-4 of the NCTM 5-8 Probability Standards.  This site includes a printable worksheet which contains tables to write in predictions and actual results. Back to top
• Chance News This site is part of the Chance homepage.  This section includes lesson plans and articles with discussion questions from the Chance magazine current issues and backissues.  This is a great source for supplementary material for teaching or introducing probability at 5-12 levels.  Some of the topics are more detailed and ready to go while others need some work before being introduced in the classroom.  This site is and excellent source that will meet the fifth goal of the 5-8 probability curricullum as well as the first three curricullar features.  Materials would probably best be chosen from the latest issue and be something that is pertenent to the lives of the students.  The information and questions will be used to provoke discussion about the topics and the probability issues underlying in the topics.  This site should be great in creating students' interest.  Back to top
• Cut-the-knot Probability Page This page has numerous activities relating to probability like the Monte Hall problem.  A description of the problem is given, the theoretical answer, and an interactive simulation for the solution as well.  Any of the problems would be good for the 5-12 curricullum.  Some work would need to be done before introducing these problems to the class.  One of the activities could be introduced to the class, they could work on creating a simulation for the solution (or using the simulation provided in the website if computers are available), and then some discussion question could be asked leading them to finding the theoretical solutions.  Then they could compare their theoretical solutions to their simulated solutions and discuss differences.  The activities provided in this site could be adapted to meet any of the first four goals of the 5-8 probability curricullum.  Some of the problems pertain to real-world situations, meeting the fifth goal and first curricullar feature as well.  These activities could be easily used as supplements as the class is beginning to learn about probability.  A major objective of using these activities would be invoking mathematical reasoning.  Technology could also be implemented through the use of the simulations if access is available.  These are fun activities that help students begin to use mathematical reasoning.  Back to top

Available activities on the "Cut-the-knot" page:

Pauling's jokeThis is a good problem for 5-12 students.  It helps to introduce the idea of dependent and independent events.  It would be a good question to pose at the beginning of a section about dependenct to get students to begin thinking about such things.  The solution could be discussed with the whole class, then definitions of dependence could be provided relating to the joke.   Back to top

Simpson's paradox :  This is an urn problem that has a counter-intuitive solution.  It would be a good exercise  for practice on finding theoretical probabilities and could provoke some good discussion on the solution.  In order to recognize why the solution is counter-intuitive, ask the students what they think the solution is before they attempt to work through the problem by calculating the probabilities.  Back to top

Buffon’s Needle Problem would be good in a geometry class to include some probability.  Begin by asking if students had ever tried avoiding stepping on the cracks in sidewalks or discussing a penny toss game that might be found at a carnival.  Once the idea of the problem is introduced, it might be better to start by solving the simpler problem of the coin landing between two lines rather than inside a square.  Then you could advance to solving the original problem.  Be sure to discuss the notion of "geometric probabilty" and how this problem relates to both geometry and probability.  Also discuss how the probability in this problem differs (compares measurements) from other types of probabilites (like die rolling).  It is probably only reasonable to use the first part of this problem since it gets a little too advanced towards the middle and end of the page.  Back to top

• The New "New" Math - Internet Style  An extensive resource of internet materials for all areas of math education.  An essential "bookmark" for any mathematics educator interested in using the internet for teaching and learning mathematics.  Activities to meet all of the standards for grades 5-12 can be found here.     Back to top
• "Crazy Choices" Game   This is an excellent resource for interactive games, proposed lesson outlines incorporating these games, and sample teacher-student discussions.  A  java-capable browser is required to run the games.  However, the games can be adapted to a classroom without this technology; the only drawback would be the speed of the simulation, quality of the simulation, and the ability to run large numbers of trials.  Back to top