Unit Plan Cover Sheet
Name(s): Jessica Pace, David Burgstrom / Date: March 8, 2006Unit Title: Probability of Yahtzee
Fundamental Mathematical Concepts: We would like the students to understand the difference between combinations and permutations and how to it relates to probability. We plan to do this by experimenting with the game Yahtzee.
q Combination describes the number of different sets that exist, while permutation includes every possible way of ordering each of those different sets.
q The “back-door approach” is recognizing that sometimes it is easier to find the probability of the compliment and subtracting that from one. Approaching a problem directly might be much more complicated and much easier to miss some of those “successes” in your counting.
q There are many ways to count the same occurrence. For example, the product rule, nCr, etc. During certain circumstances, using combinations and permutations will result in different answers. To speak of combinations with regard to the outcomes of rolled dice is inappropriate because it disregards the independence of each die. Furthermore, a “combination” of outcomes of dice may not have the same chance of occurrence as another outcome, such as doubles.
q The theoretical concept of probability involves the number of favorable outcomes divided by the number of possible outcomes, when each outcome is equally likely.
Describe how state core Standards, NCTM Standards, and course readings are reflected in this unit.
1) State Core Standards
a) Standard 2: Students will represent and analyze mathematical situations and properties using patterns, relations, functions, and algebraic symbols.
i) Use patterns, relations, and functions to represent mathematical situations.
(1) Represent a variety of relations and functions using tables, graphs, manipulatives, verbal rules, or algebraic rules. During this unit, students will be required to count multiple events. This will require the use of patterns, relations, functions and algebraic symbols. Students will be able to represent the outcomes in numerous ways.
b) Standard 5: Students will draw conclusions using concepts of probability after collecting, organizing, and analyzing a data set.
i) Formulate and answer questions by collecting, organizing and analyzing data.
(1) Conduct a survey or experiment to collect data. Make predictions and describe the limitations of the predictions when using data samples. Students will make predictions of the probability of achieving a five of a kind on one roll. They will understand that it has to be greater than zero, even though they will probably never roll one during their scenario.
(2) Evaluate reported inferences or predictions based on a data set. Students will assess their data and make predictions on how close their data is to the actual number.
ii) Apply basic concepts of probability.
(1) Conduct experiments to approximate the probability of simple events. Students will run an experiment to have some raw data to compare the calculated results to, although this is a very small part of the unit.
(2) Recognize that results of an experiment more closely approximate the actual or theoretical probability of an event as the number of trials increases. This is the law of large numbers which students will use as they assess the validity of their findings. Students will come away with a more thorough understanding of this principle.
(3) Derive the probability of an event mathematically, e.g. building a table or tree diagram, creating an area model, making a list, or using the basic counting principle. Students will calculate the probability of achieving a five of a kind, a four of a kind, and a three of a kind. This will involve either using a diagram or using the basic counting principle. The diagram method might be easier for some students then the counting principle method, thus both methods will be taught.
(4) Represent the probability of an event as a fraction, percent, ratio, or decimal. The probabilities found will most likely be represented as a fraction and students will understand what that fraction means.
(5) Recognize that the sum of the probability of an event and the probability of its complement is equal to one. For some probabilities, it is easier to count the number of not-successes then the number of successes. Thus this knowledge will help them arrive at the correct answer.
2) Grades 6-8 NCTM Standards
a) Understand and use ratios and proportions to represent quantitative relationships. Students will understand that the probabilities arrived at represent the number of success divided by the number of total outcomes, which is a quantitative relationship between the two.
b) Develop analyze and explain methods for solving problems involving proportions such as scaling and finding equivalent ratios. Students will solve the problem of deriving the probability of achieving certain outcomes when rolling five dice. They will need to explain their methods to the class and be able to articulate their thinking in a way that their peers will understand.
c) Represent, analyze, and generalize a variety of patterns with tables, graphs, words, and when possible, symbolic rules. Students will be required to present their findings to the class which explanation will involve a variety of representations.
d) Model and solve contextualized problems using various representations such as graphs, tables and equations. Some of the probabilities will be more easily solved using tables. Those tables will need to be organized and readable. The students will also have to explain their tables which require a higher level of thinking.
e) Use proportionality and a basic understanding of probability to make and test conjectures about the results of experiments and stimulations. Students will compare their raw data from their experiments to the data found when calculating the actual probability. They will understand why there is a difference between the two.
f) Compute probabilities for simple compound events, using such methods as organized lists, tree diagrams, and area models. In order to find the probability of four of a kind, or three of a kind, students will combine their knowledge of simple events numerous times. Each way of counting successes will involve at least one of the simple methods of counting.
3) Course Readings
a) Talking about Math Talk
i) Students will be given the opportunity to discuss in small groups probabilities explored in each task. Students will be given numerous tasks that they will discuss in groups. This will require students to talk about mathematics.
ii) Students will also be given the opportunity to articulate the discussions they had in the groups to the entire class. Some groups will be asked to present their findings to the class which will require them to articulate their findings in a clear way.
b) Become a Reflective Mathematics Teacher
i) While the students are working in groups, I will be able to see how the students approach the tasks at hand which will show me what concepts need further understanding. This will help me learn how to better anticipate student responses. This will also allow me to assess what I need to explain further.
ii) I will be able to see if the students can articulate their understanding during the presentations of their ideas to the class at the end of each task. Again, I will be able to assess students true understanding as they participate in a higher level of understanding.
Explanation: During this unit of discovering probabilities of outcomes of a set of 5 dice, students will be required to discuss their findings in groups and to the class. This discussion will require the use of a variety of patterns, relations, and functions to describe their findings. Finding probabilities requires the use of the counting principle. Although it sounds elementary, the counting principle will require numerous ways to represent, analyze, and generalize patterns occurring in the outcomes. Certain activities will require students to list outcomes to certain circumstances. This will build students understanding of the importance of being able to articulate thought processes.
Outline of Unit Plan :
Day1: Introduce probability by using the Birthday Problem. This will get students to start thinking about probability which will prepare them for Day 2.
Day 2: Students will have hands-on experiences with the Yahtzee. They will determine an experimental probability and an actual probability for certain hands. Students will gain an understanding of the Law of Large Numbers. This will get the students thinking about strategies and connect inductive and deductive logic and reasoning.
Day 3: Students will calculate the probability of rolling a four of a kind with five dice on one roll. They will then move onto three of a kind and two of a kind, in that order with the idea that I do not expect them to finish this task in one day.
Day 4: The class will continue the discussion which began on Day 3 regarding combinations, permutations, independence and equally likely events.
Day 5: Students will calculate the probability of rolling a three of a kind and a two of a kind.
Tools: 5 dice per person to allow each person to experiment.
Unit Plan Sequence: Learning activities, tasks and key questions (What you will do and say, what you will ask the students to do, how you will accommodate to unexpected students’ mathematics) / Time / Anticipated Student Thinking and Responses
(What mathematics you will look for in student work and interactions.) / Formative Assessment (to inform instruction and evaluate learning in progress)
Miscellaneous things to remember
LESSON 1: THE BIRTHDAY PROBLEM AND INTRODUCTION TO PROBABILITY
Launching Student Inquiry
Introduce the Birthday Problem and break into groups. / 5
Min / · We hope that students have not seen this problem before. If they have, then they will be asked to guide their group.
Supporting Productive Student Exploration of the Task (Students working in groups or individually)
Walk around the room and listen to students’ conversations. / 10
Min / · We anticipate that students will understand the idea that the probability of having a birthday on a specific day is 1/365.
· We think that students will understand the idea of multiplying many probabilities together.
· We think that students will understand the principle of finding the probability of the requirements not happening to arrive at the solution. / Ask them questions to set them up for a solution.
Facilitating Discourse and Public Performances (Active Presentations by Students to Entire Class)
Ask groups to present their answers and explain their strategies. / 10
Min / · Anticipate that students will give general strategies but not the right answer.
· Some will know the answer but will have guiding each other.
Unpacking and Analyzing Students’ Mathematics (Clarity, Reasoning, Justification, Elegance, Efficiency, Generalization)
Go over the terms probability and combinatorics. Ask for student ideas on definitions. Introduce idea that probability is not always intuitive. / 5
Min / · Anticipate that students will understand the idea that probability won’t be always what they think it will be.
· Anticipate that students will be able to come up with the definitions of probability and combinatorics. / Ask the students for the big ideas regarding probability and combinatorics.
LESSON 2: EXPERIMENTAL PROBABILITY AND THE LAW OF LARGE NUMBERS
Launching Student Inquiry
1. Open lesson by reminding class of what we did last week. Ask them how they approached the birthday problem. Did anyone ask a single person when their birthday was? So how did they solve it instead? Bring up experimental probability.
2. Use the 100% dice example. Discuss pros and cons of sampling and modeling. Mention relation to quantum physics.
3. Split class into two groups. / 5
Min / · I am sure students will already be aware of what sampling and modeling are. I will simply facilitate a short discussion comparing the two. Tie in with last week’s lesson.
· Point out that last week no one suggested asking increasingly larger groups of people for their birthdays until we finally got two that matched. Yet it is a perfectly valid approach. Instead everyone went straight to paper and pencil. Sometimes we get stuck in equations and forget that there are alternative approaches to the “real world”. / Lead discussion. Call on specific people to answer.
Supporting Productive Student Exploration of the Task (Students working in groups or individually)
1. Give the first group a box of dice. Ask them to determine experimentally the probability of rolling doubles on their first roll. Tell them they have 15 minutes and let them go at it.
2. Ask the second group to solve for the theoretical probability of rolling doubles on their first roll. Tell them they have 15 minutes to work on it. / 15
Min / · I think most students are already familiar with the game of Yahtzee. I might have to remind them that the game involved five dice.
· Hopefully group one will realize that having each of them sampling at the same time would be the best way to obtain a large sample size. If not, I will drop hints. (“Is there a faster way to do this? We only have 15 minutes.”)
· Group two should be able to see the connections with the Birthday problem we did in our previous lesson and will hopefully be able to create a model in a similar fashion. Again I could drop hints. (“How did we solve the birthday problem? Are there any similarities to this one?”) / I will monitor the two groups. Get involved in their discussions and ask them questions about what they are doing.