VI. Probability and Discrete Mathematics

1. Probability
2. Discrete Mathematics

Additional Links to General Mathematics Sites

• Mathematics Games and Student Activities
• Mathematics Lessons Plans
• Interactive Learning and Problem Solving Activities
• Word Searches and Puzzles in Mathematics
• Miscellaneous Sites

Middle School

1. Benchmark 1. "Describe events as likely or unlikely and give qualitative and quantitative descriptions of the degree of likelihood."
   1. Certain and Impossible Events - Probability (VI.1.MS.1)
      This site defines certain and impossible events. The student will identify as certain or impossible, and to compute the probabilities of these events. Interactive spinners and dice are included. The student takes a short quiz after reading and working with the interactive spinners and dice.
   2. Introduction to Probability - Probability (VI.1.MS.1)
      This site defines experiment, outcome, event, and probability. The site helps to understand the formula for finding the probability of an event and to find the probabilities of simple events with equally likely and with non-equally likely outcomes. Interactive spinners and dice are included. The student takes a short quiz after reading and working with the interactive spinners and dice.
   3. Simulating Probability Situations - Probability (VI.1.MS.1,2,3,4&5)
      Students can use interactive box models to explore probability. Open-ended questions are included. Bar graphs of the experiments are automatically generated.
   4. The Cereal Box Problem - A Lesson in Expected Value - Probability (VI.1.MS.1,4&5)
      This activity addresses the NCTM Standard for Probability for grades 5-8 through it's use of modeling situations and asking students to make predictions. It includes online simulation of the problem in addition to a hands-on activity using dice and a pencil and paper. Also includes teacher notes, references and related questions.
   5. What Are Your Chances to Win? $Lottery - Probability (VI.1.MS.1)
      The student is taken through a short explanation of how to determine their chances of winning a 3 digit lottery draw.  They use this information to determine probably for other lottery games.
2. Benchmark 2. "Describe probability as a measure of certainty ranging from 0 to 1 and conduct activities that allow them to express probabilities of simple events in mathematical terms."
   1. Dice Table - Probability (vI.1.MS.2)
      This activity allows the user to calculate the probability of each number (2-12) appearing when two dice are rolled. They also get practice converting probabilities into fractions, decimals, and percentages. This site gives why, what, and how to tips for the user.  Student centered activity.
   2. Fish Tank - Math Files - Probability (VI.1.MS.2,)
      Outstanding - This is a student interactive site, where students must add or remove colored fish to a tank to achieve the correct answer, finding the probability of removing a red fish from a group of yellow and red fish in the tank. Several levels of difficulty, to challenge all learners. Levels two and three are much harder. In these levels, students must remove or add fish to get the desired probability (listed in either decimal or fraction form). Students have to really use their problem-solving skills in level three.
   3. Simulating Probability Situations - Probability (VI.1.MS.1,2,3,4&5)
      Students can use interactive box models to explore probability. Open-ended questions are included. Bar graphs of the experiments are automatically generated.
3. Benchmark 3. "Conduct experiments and give examples to illustrate the difference between dependent and independent events."
   1. Independent and Dependent Events  - Probability (VI.1.MS.3)
      After a brief explanation and example of independent and dependent events, the student conducts experiments, tally results, and determine probability to find the number of yellow chips in a bag.
   2. Math Goodies: Probability Independent Events - Probability (VI.1.MS.3)
      Introduces probability by giving examples and interactive samples. Introduction to independent events with challenging exercises, as well as interactive games and puzzles are included.  Challenging quiz for students is found at the end of the lesson.
   3. Replacement and Probability - Probability (VI.1.MS.3)
      This site is a teacher lesson plan which goes through a step by step description of the differences between dependent and independent events. It allows the teacher to use hands-on activities to show students the difference. It then has follow up questions for the students.
   4. Simulating Probability Situations - Probability (VI.1.MS.1,2,3,4&5)
      Students can use interactive box models to explore probability. Open-ended questions are included. Bar graphs of the experiments are automatically generated.
4. Benchmark 4. "Explain the difference between probabilities determined from experiments or chance events (empirical) and probabilities derived mathematically (theoretical), and explain how the empirical probability changes for a large number of trials."
   1. Area Probability (Throw Darts!) - Probability (VI.1.MS.4&5)
      Students manipulate the size of a circle and a rectangle to explore the probability that a dart thrown into a rectangle will land in the circle.
   2. Crazy Choices Game - Probability (VI.1.MS.4)
      This activity allows the user to run up to three different games of chance at once, allowing for comparison of experimental and theoretical probabilities. This site includes exploration questions, discussion, and a lesson that introduces the concept of probability.
   3. Marlowe the Spy, Problems with Odds - Probability  (VI.1.MS.4&5)
      The students follows Marlowe the spy on a journey to capture the attack plans of the wicked Overumberians.  At every turn of the path, the probability of capture threatens.  Both empirically and theoretically, probabilities are determined in this real-life situation.
   4. Simulating Probability Situations Using Box Models - Probability (VI.1.MS.1,2,3,4&5)
      This site focuses on probability by having the students actually choose numbers and showing simple events in mathematical terms. The interactive tool is a "box model" that allows students to explore the relationship between theoretical and experimental probabilities. It relates back to flipping coins and rolling dice as a range of certainty. Activity based for students.
   5. The Cereal Box Problem - A Lesson in Expected Value - Probability (VI.1.MS.1,4&5)
      This activity addresses the NCTM Standard for Probability for grades 5-8 through it's use of modeling situations and asking students to make predictions. It includes online simulation of the problem in addition to a hands-on activity using dice and a pencil and paper. Also includes teacher notes, references and related questions.
5. Benchmark 5. "Conduct probability experiments and simulations to model and solve problems."
   1. Area Probability (Throw Darts!) - Probability (VI.1.MS.4&5)
      Students manipulate the size of a circle and a rectangle to explore the probability that a dart thrown into a rectangle will land in the circle.
   2. The Birthday Problem: A short lesson in probability - Probability (VI.1.MS.5)
      This activity surveys the possibility of birthdays occurring on the same day in as small a group as 28 students. It also extends the problem to using a computer to simulate this problem to see what the probability generated by a random list of birthdays would be.
   3. The Cereal Box Problem - A Lesson in Expected Value - Probability (VI.1.MS.1,4&5)
      This activity addresses the NCTM Standard for Probability for grades 5-8 through it's use of modeling situations and asking students to make predictions. It includes online simulation of the problem in addition to a hands-on activity using dice and a pencil and paper. Also includes teacher notes, references and related questions.
   4. Figure This! Math Challenges for Families - Probability (VI.1.MS.5)
      Figure This! Mathematics Challenges for Families provides interesting math challenges that middle-school students can do at home with their families. Each challenge features a description of the important math involved, a note on where the math is used in the real world, a hint to get started, complete solutions, a "Try This" section, additional related problems with answers, questions to think about, fun facts related to math, and resources for further exploration. Math Challenge #26 is an example of a problem that would help teach this particular benchmark.
   5. Simulating Probability Situations - Probability (VI.1.MS.1,2,3,4&5)
      Students can use interactive box models to explore probability. Open-ended questions are included. Bar graphs of the experiments are automatically generated.