Reason for Alignment: The From Probability to Combinatorics lesson shows how the use of tables and trees can be utilized to compute and understand probability. These types of data structures are a good tie to the textbook lesson.
Reason for Alignment: This lesson goes into probability theory, along with tables and trees. It ties with the combinations portion of this section. The word â€ścombinatoricsâ€? is used in the lesson, but it is explained in the discussions.
Arithmetic: Student must be able to:
use division to count outcomes in probability problems
use multiplication in working with data structures
Technological: Students must be able to:
perform basic mouse manipulations such as point, click and drag
use a browser for experimenting with the activities
The science that studies the numbers of different combinations, which are groupings of numbers. Combinatorics is often part of the study of probability and statistics
The chances of something happening, based on repeated testing and observing results. It is the ratio of the number of times an event occurred to the number of times tested. For example, to find the experimental probability of winning a game, one must play the game many times, then divide the number of games won by the total number of games played
The measure of how likely it is for an event to occur. The probability of an event is always a number between zero and 100%. The meaning (interpretation) of probability is the subject of theories of probability. However, any rule for assigning probabilities to events has to satisfy the axioms of probability
In mathematics, superscripts are numbers or letters written above and to the right of other numbers or letters or symbols indicating how many times the latter is to be used as a factor. When typing, one can represent a superscript by using the ^ symbol to indicate raising the number. For example, x3 is the same as x^3, which equals x * x * x
The chances of events happening as determined by calculating results that would occur under ideal circumstances. For example, the theoretical probability of rolling a 4 on a four-sided die is 1/4 or 25%, because there is one chance in four to roll a 4, and under ideal circumstances one out of every four rolls would be a 4. Contrast with experimental probability
Focus and Review
Remind students of what they learned in previous lessons that will be pertinent to this lesson
and/or have them begin to think about the words and ideas of this lesson.
Let the students know what they will be doing and learning today. Say something like this:
Today, class, we will learn how to use division to solve probability problems.
We are going to use the computers to help us, but please do not turn your computers on until I
ask you to. I want to show you a little about this activity first.
Racing Game with Two Dice , which will introduce the concept of data structures and computing particular probabilities.
Have the students begin with the
Racing Game with Two Dice Several players "race to the finish" using the software or on paper. For every round, each
player makes either one or two steps depending on the outcome of the roll of two dice. Each
group of students can come up with its own way of randomly choosing which players make one or
Have a discussion about
divisibility as it can be used in probability. The discussion is based on the
Dice Table activity.
Lead the discussion:
Tree as a data structure. This discussion introduces and develops the idea of trees as data structures. It is based on
all the other parts of the lesson. Plan it as a "live" discussion where students have an
opportunity to ask their own questions, because the topic tends to be interesting to many
people and it can lead to various investigations in math and computer science.
Have the students work alone or in small groups and play the
Dice Table activity, where students research tables as data structures and use tables to count outcomes
and compute probabilities.
You may wish to bring the class back together for a discussion of the findings. Once the
students have been allowed to share what they found, summarize the results of the lesson.
Have students read the
divisibility discussion independently, or use the text to prepare for a "live" discussion.
After these discussions and activities, the students will have seen how data structures such as
tables and trees can be used when solving probability problems. You may want to introduce student
to the idea of expected value next.