LESSON PLAN

Name:                                 Amber Blakley and Philip Kromer
Title of lesson:                      Probability: Counting Outcomes
Date of lesson:                       March 8th, 2004
Length of lesson:                     50 minutes
Description of the class
              Name of course:         7th Grade Math
              Grade level:            7th Grade
              Honors or regular:      Regular

Source of the lesson:
Mathematics: Applications and Connections, Course 3, Chapter 12, 518-19.
Practice Masters, 93.
Study Guide Masters, 93.

TEKS addressed:
   TEKS: 14A, 15A, 15B
     7.14 (A) Communicate mathematical ideas using language, efficient tools, appropriate units, and
     graphical, numerical, physical, or algebraic mathematical models.
     7.15 (A) Make conjectures from patterns or sets of examples and non examples;
     7.15 (B) Validate his/her conclusions using mathematical properties and relationships.

I.   Overview
     Students should understand how to count outcomes by making a tree diagram or by applying the
     counting principle (multiplication of outcomes rule).
     Students should be able to do examples on their own and feel comfortable that they did so correctly.
     On the small scale, counting outcomes is the first step in many problems in probability. Students
     have to be able to evaluate how many possible outcomes exist in order to find the probability of a
     specific event. More broadly, probability is fundamental to making informed decisions in voting,
     games, finance, and other real-life situations involving expectation, risk, and reward.

II. Performance or learner outcomes
    Students will be able to:
     Construct a tree enumerating all possible outcomes of a given scenario.
     Invent such a tree to find the probability of a simple outcome or set of outcomes.
     Apply the counting principle (multiplication rule) to find the number of outcomes for a given
     Explain how the tree diagram and the counting principle yield the same number of outcomes
       and how the counting principle encapsulates the information in the tree diagram.

III. Resources, materials and supplies needed
     Pencils, Paper
     Giant Dice
     Chart for MASH game* (done on chalkboard)

IV. Supplementary materials, handouts.
    A handout with practice problems and an outline of ideas covered.

                                        Five-E Organization
       Teacher Does                Probing Questions             Student Does

Play MASH. MASH is a                 How many places can you end          Students should understand that
children’s game to predict the       up living?                           there are four outcomes for
future: the player chooses where                                          where to live.
they will live (usually Mansion,     What is the probability of living    Depending, of course, on where
Apartment, Shack, and House –        where you want?                      the student wants to live, the
hence MASH), what they will                                               student should understand that
drive, # or kids. (Students                                               the probability is 1/4 (2/4, 3/4).
nominate the possibilities for the                                        The student realized that this
last categories). A simple           What is the probability of getting   question requires a way to
scheme eliminates choices until      what you want in each case?          enumerate the outcomes.
the player’s future is divined.
The teacher shows the class how                                           Students see the basic idea of
to use an outcome tree to solve                                           making a tree.
this problem:
     List all the possibilities
        for the first choice
     For each of those, list the
        possibilities for the
        second choice (Ferrari,
        Honda, Bike), and so on.
The teacher starts off with an     How many possible outfits can           Students may offer guesses or
outfit and how many ways can       we make?                                wonder how to solve the
we put that outfit together.                                               problem.
(Outfit = shirt + shorts) Teacher
gives appropriate information on
board. List that we have
     Red or Blue shirts
     Black, Gray, or White
The solution: Teacher will show                                            Students may be confused on
that the students that no matter                                           how the red shirt may use and of
what they do they will always                                              the shorts and the blue shirt can
need to start with either a red or                                         use all the shorts. Clarify that we
blue shirt, and can put any shorts                                         were not wearing these clothes;
with any shirt.                                                            we just want to know how many
                                                                           different ways we can wear
We again show the students how                                             The students may not first
to form a tree diagram.                                                    understand how to draw the tree.
Show the students that for each      How many outcomes are there           3 outcomes for blue shirt
path you can follow, it will         for blue shirt?
make a different outcome.            How many outcomes are there           1 outcome for the white shorts
                                     for the white shorts and red shirt?   and red shirt.
Leave the diagram up; step out       How many different outfits are        Students should be able to
of the way so the students may       there?                                answer this now by counting the
see the board clearly.                                                     different paths and adding them
                                                                           together; by doing this they come
                                                                           up with six different outfits.
Explain to the students that you     Do you think this method is           Will pause to think about the
start your tree diagram with your    slow? Can you think of a faster       given questions.
first choices and the tree           way? Is there even a faster way?
diagrams end with the different
types of outcomes. Also remind
them to be careful drawing their
trees and to make sure not to
repeat any outcomes.
Moving on…first we’ll start off      How many outcomes with one            By seeing the example students
with the probability throwing of     die?                                  should be able to see that dice
a die and dice. Show an              Can you explain to me how you         resemble a cube and it has six
example of die and dice.             got that answer?                      sides along with six different
Then roll the die for the first      What’s the probability of getting     numbers on each face. Students
question. Roll the die a couple      a six?                                should already understand that
of times to help the students see                                          with six faces there are six equal
that only one value can appear at                                          outcomes. So probability of
one time.                                                                  rolling each of the numbers is
Next move on to two die. Roll        How many outcomes with two            Some students may already
the dice so the students may see     dice?                                 know, that’s great, others that
that every time they roll the dice   How many outcomes for the first       don’t will get refreshed and
they will get two numbers. Here      die?                                  understand how the other
we want to use the tree diagram     How many outcomes for the             students got their answer.
and then introduce the Counting     second die?
Principle.                          Do you think there is an easier
Go through tree diagram with        why to get the outcome?
the children. Then show how         Do you know what might work?
they can multiply the number of
outcomes for each specific case.

Teacher does                       Probing Questions                      Students Reponses
Explain that we can multiply to    Is this a faster way to count          Student may not understand how
find 36 outcomes. This is called   outcomes?                              you are able to do this; they may
the “Counting Principle.”          Can we see all of the possible         want a better explanation.
                                   outcomes using this method?
Next give the example of tossing If you flip three coins, how will
three coins.                       the tree look?
                                   Would any one like to try to           One student gets up to the board
                                   make the tree diagram of this          and draws the diagram, other
                                   outcome?                               students decide if he/she is right.
The diagram of the tree should     How many outcomes are there?           Students will be able to counts
split up into the first coin being How many outcomes have just            these and decide upon eight.
head or tails. The next coin will heads or just tails?                    Look at the diagram and see just
either come up head or tails and Why is it ok to have both HTH            two have heads or tails.
so will the last one. So we have and THH?                                 They will have to think about the
two separate tree diagrams and                                            last question and decide if the
there outcomes.                                                           order the coins come out matter.
Go back to an outfit and use the How many outcomes are there?             Will take the two shirts and three
counting principle of math to      Was this method faster?                shorts and multiple 3 * 2 to get
find answer.                                                              six the same number of outcomes
                                                                          as before.
Another example: there is a test    For this one is it better to draw a   Students may try to answer ten,
of five true or false questions,    tree diagram or use the Counting      clarify their confusion and make
how many outcomes are there?        Principle?                            sure they do 2*2*2*2*2 and not
                                                                          just 2*10.
The teacher shows an example        What are the advantages of using      Students should spot the pattern
of a tree diagram and the           the tree diagram?                     between making the tree and
Counting Principle. Example         What are the advantages of using      multiplying the outcomes
You have Dr. Pepper, Coke, and      the Counting Principle?
Big Red. That you can pour on
two ice creams (vanilla and
Explain which methods are           Is one method better then the
better to use in which questions.   other?
Shows that the Counting             Do you really need to know          Depending on the question
Principle is a short cut and if     which is better for different case? students should be able to decide
you need to know the specific                                           which method would be better.
outcomes, then the tree diagram
would be better.

Extend / Elaborate:
Recall the game that we played     How many outcome were there?       Calculate the number of
at the beginning of the class.     Which method did you use to        outcomes, by themselves and
Have the students find the total   find the number of outcomes?       have them give their answers.
number of outcomes for the


                                   Could the Tree Method and the
                                   Multiplication Rule ever give
                                   different answers?
                                   Which is better, the tree method   OK: Multiplying: it’s easier
                                   or the multiplication rule?        OK: Tree: it’s simpler
                                                                      Better: Either, they both give the
                                                                      right answer.
                                                                      Best: Tree is fine for small
                                                                      problems; multiplication rule is
                                                                      best for large # of outcomes or
Ask class if there are any         Any about the tree diagram or      Ask their question or remain
questions.                         the Counting principle?            quite.
Hand out worksheets and walk                                          Start to work on the worksheets.
around room as students fill
them out.
Go over the worksheet.             Ask the whole class, what the      Students give their answers.
                                   right answer to one? Pick a
                                   Maybe go up and down the rows
                                   and ask for their answers.

      Percent effort each team member contributed to this lesson plan:
      _50_%___       ____Name of group member_Amber Blakely_____

      _50_%___        ____Name of group member_Philip Flip Kromer__

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