Download Lesson Plan1

Jeff Rosen
Multimedia Lesson #1
Experimental Probability
Lesson Plan
Introduction to the Lesson:
Historical BackgroundThe traditional beginning of modern probability is the exchange of letters in July
and October 1654 between two French Mathematicians, Blaise Pascal (1623-1662) and
Pierre Fermat (1601-1665). The letters were written in reply to this problem: Two
players, A and B, each stake 32 pistoles on a three-point game. When player A has 2
points and player B has 1 point, the game is stopped and cannot continue. How should
the stakes of 64 pistoles be evenly distributed?
Pascal explanation emphasized expectation and equality between the two players,
the central point of his analysis, instead of on the calculations of the probability outcomes
and their connected values. Therefore, he removed probability from as much of the
problem as possible, using certain gain and equality in its place.
The Pascal- Fermat exchanged was not published until 1679. The credit for the
first published work in mathematical probability, as a result, it went to two Dutch
mathematicians, Christiaan Huygens (1629-1695), who had heard of Pascal- Fermat
problem and solutions, and Johann de Witt (1625-1672). Their work continued to stress
the mathematics of expectations.
Huygens believed that a fair game was one in which each of the players had an
equal expectation, so that the game did not work to the disadvantage of anyone who trust
entry into the risk. If the game was fair, the players should be willingly to exchange
expectations. This thinking once again lead the modern order of reasoning to be
reversed, instead of the game being fair because of probabilities, the risk is equal for all
players, the probabilities are equal because the game is assumed to be fair because of the
willingness to exchange expectations.
The early emphasis on expectations rather and probabilities, inferred in the
absence of sufficient statistics, was not able to deal with more complex situations
involving different degrees of risk.
Anticipatory SetTo introduce and attract the student’s interests, this quick hand on activity will be used:
Label equal sections of each of four spinners with the group member's first names. Find
the probability, introduced in lesson 8-1 (probability of simple events), that the same
name will appear on each spinner when the spinner I spun.
Each group member takes turns in spinning the spinner. As the group spins the spinner
ten times record the results. What is the probability of spinning each group member after
ten spins? Repeat this process for 25, 50 and 100 trials. And record the probability after
number of spins. This information will be used at the conclusion of the lesson.
Also have the students find the meaning of the Latin root for the words experiment,
experience and expert; to try to learn by trying. Also, have the students predict what
experimental probability means; probability that is determined by trying something, such
as tossing a coin.
After the introductory activity ask the students, if they know anything in there lives deal
with probability or games of chance. Have a conversation with the students and discuss
in the specific instances where they have experienced probability.
Use a KWL chart; fill in what the students know, and what they would like to know about
probability.
Explanation of the mathematics involved:
In lesson 8-1 the idea of probability of an event, a ratio that compares the number
of favorable outcomes to the number of possible outcomes, is learned and practice. In
this lesson the student will discuss the differences between theoretical probability,
probabilities based on known characteristics or facts and experimental probability,
probabilities that are based on frequencies obtained by conducting experiments. Then
after discussing the two types of probabilities, you can use past performances to predict
future events.
Instructional Methods:
The teacher will used multiple representations to teach to find experimental probability.
The students will roll number cubes and use the random function on Microsoft Excel and
record results to simulate this event.
Teacher-Centered ApproachesDemonstration- The teacher will demonstrate the different types of hands-on and
technology based instruction. This will help the students to understand the
different methods that are being used to get across the objective of the lessons.
Direct Instruction- The teacher will use direct instruction if groups of small
students have difficulty in demonstrating mastery in the stated objective.
Student-Centered ApproachesCooperative Learning- The teacher will use this method for the warm-up exercise.
This allows the students to interact and have rich mathematical conversation.
Simulation- The students will simulate rolling a number cube on a Microsoft
Excel worksheet.
Act out the Problem- The students will act out a real life situation in the warm-up
and use the data gathered in future problems.
Discussion- The students will have a discussion where they have seen probability
in their everyday lives.
Discovery Learning- The students will use this method for most of the instruction,
the teacher will be there as a facilitator.
Know- What to Know- Learned- Use a KWL chart; fill in what the students
know, and what they would like to know about probability.
Step by Step Procedure:
1) Warm-up Exercise- 5- Minute Check
2) Use a KWL chart; fill in what the students know, and what they would like to know
about probability.
3) Label equal sections of each of four spinners with the group member's first names.
Label equal sections of each of four sections of the spinner with the group member's first
names. Find the probability, introduced in lesson 8-1 (probability of simple events), that
the same name will appear on each spinner when the spinner is spun.
Each group member takes turns in spinning the spinner. As the group spins the spinner
ten times record the results. What is the probability of spinning each group member after
ten spins? Repeat this process for 25, 50 and 100 trials. And record the probability after
number of spins. This information will be used at the conclusion of the lesson.
4) Review the concept of probability of an event (lesson 8-1), the measure of the
likelihood that an event will occur. What will be the probability of an event that it is
impossible to occur; occurs half the time; and certain to occur?
5) With using the context of the spinner introduced in the warm-up, talk about that one
possible result of a probability event is called an outcome and an event is a collection of
outcomes.
6) In the context of coin flipping, ask the students what will be the probability of flipping
a tails. Use this situation to talk about favorable and unfavorable outcomes, and
theoretical probability, a type of probability that is based on the number of favorable
outcome divided by the total number of outcomes.
7) Refer to the beginning exercise in spinning a spinner introduce the concept of
experimental probability, probabilities that are based on frequencies obtained by
conducting an experiment. Make sure that the students understand what the difference
between theoretical and experimental probability is.
8) Using the information presented above, have the students complete the following Mini
Project worksheet.
9) Use the textbook to discuss and teach how to use probability
to predict. Make sure that the students review the concept of
proportions. Have the students come up with their own
situation where they use probability to predict future behavior.
10) Use the PowerPoint; have the student answer the questions
to the right. Give the students time to answer the questions in
small groups and debrief the answers and concentrate on how
different groups answer the questions using different methods.
11) The next representation of experimental probability is to use Microsoft Excel in
virtual coin flipping. First start by introducing the students to random number function
in excel, RAND(), this returns an evenly distributed random number greater than or equal
to 0 and less than 1. Also discuss the integer function, INT(), which rounds a number
down to the nearest integer functions.
12) We need to make function that allows us to generate random number that is 0 or 1.
=INT(2*RAND()). We use this function because since the RAND function gives a
number greater than or equal to 0 and less than 1, and then twice that will be between 0
and 2. Then if you take the integer part, half the time you will get 0’s and the other half
1’s. After defining the function take the formula and extend it down the column. In this
case, to produce a different set of random numbers you can use the F9 key on the PC or
the APPLE= on the MAC, these are instructions for the spreadsheet to recalculate the
random number, which gives you different samples of them.
13) On this spreadsheet, we will represent the 1’s as heads and the 0’s as tails. In order to
count the number of heads that were generated we can take the sum of the column that
has the random numbers in it. In this situation you can use the SUM function, which
adds all the numbers in a range of cells. [=SUM(C3.. B12)]. To find the number of tails,
take the total number of trails and subtract it from the sum you found before. . After
finding this information we can calculate the probability of flipping a head. You can now
use the cells were the following information is.
# of Heads / (# of Heads + # of Tails)
=B14/ (B13+B14)
14) To facilitate getting better statistics we can extend the function to 100 trials, and
remember to change the area of your sum from 10 to 100 samples.
15) Revisit spinner activity- as a concluding activity have the students take the spinner
and split into a number of sections of their choice and find the theoretical probability and
experimental probability using different amounts of trials. Have them compare and
contrast the two different probabilities and which is more accurate.
16) Conclusion
a) Use a Venn diagram, and fill it in with the similarities and differences of
experimental and theoretical probability.
b) Fill in the “L” column of the KWL chart with the concepts and knowledge the
students learned during the lesson.
Key Questions to be asked:
Some key questions that should be asked:
What is the difference between theoretical and experimental probability?
How does the number of trails change the experimental probability of an event?
How does the future behavior change if you use theoretical or experimental
probability?
Which is the better method of predicting future behavior?
Which probability, theoretical or experimental is more accurate?
Description of Class Activities:
Spinner ActivityBeginning of LessonLabel equal sections of each of four sections of the spinner with the group
member's first names. Find the probability, introduced in lesson 8-1 (probability
of simple events), that the same name will appear on each spinner when the
spinner is spun.
Each group member takes turns in spinning the spinner. As the group spins the
spinner ten times record the results. What is the probability of spinning each
group member after ten spins? Repeat this process for 25, 50 and 100 trials. And
record the probability after number of spins.
End of LessonAs a concluding activity have the students take the spinner and split into a number
of sections of their choice and find the theoretical probability and experimental
probability using different amounts of trials. Have them compare and contrast the
two different probabilities and which is more accurate.
Coin Flipping WorksheetHave the students answer the worksheet on their own pace in their groups.
Debrief answers and look for different methods of solving the problems.
Virtual Coin FlippingThis activity uses Microsoft Excel to simulate flipping a coin using different
number of trails. See instructions in procedure.
Closure to Lesson:
Revisit spinner activity: as a concluding activity have the students take the
spinner and split into a number of sections of their choice and find the theoretical
probability and experimental probability using different amounts of trials. Have
them compare and contrast the two different probabilities and which is more
accurate.
Venn Diagram: Use a Venn diagram, and fill it in with the similarities and
differences of experimental and theoretical probability.
KWL Chart: Fill in the “L” column of the KWL chart with the concepts and
the students learned during the lesson.
Relation to Future Lesson: The next lesson uses Sampling to Predict, which uses
probability and proportions to predict how a larger sample will answer.
Assignment for the Students:
Glencoe Textbook: p.402-403, 1-31
Microsoft Excel: Change given situation from flipping coins to rolling a number
cube.
Assessments:
Probability Quiz:
Student Made Quiz: Have the student create their own 5-10 question quiz on
Probability. This allows the students to show their understanding in a different
manner.
Plan for providing feedback:
I plan to provide feedback in rubric form. A rubric will be created that will have
the different requirements for the Microsoft Excel worksheet for the rolling of the
number cube assignment. For example, the students must include two printouts;
the finished product and a sheet that shows the formulas (toolsformula
auditing formula auditing mode).
The students can also ask questions during class, before class, or email. Having
these three options the student should have multiple methods to ask questions.
Answer Keys to assignments and assessments:
Selected answers to assignment from textbook-
Answers to Probability Quiz-
Supplemental Activities Lesson:
Higher-Level Thinking- How often would you expect even numbers to come up
when rolling a number cube? (About half the time) Roll a number cube 20 times.
Do even numbers come up as often as you expect? Roll it 50 times, and then 100
times, recording the percent of time that even numbers come up. Are these results
getting closer to what you expected? Explain. (Yes; as the number of tries
increases, the experimental probability will get closer to the theoretical
probability)
Enrichment- This worksheet will help the students to have some higher level
thinking and expand their knowledge.
Study Guide- This worksheet will be useful for students who need extra help and
further explanation.
Practice Skills- This worksheet has extra problems and examples that can help
students to apply their new knowledge.
Practice: Word Problems- This worksheet give the students gives the students
extra practice on a critical FCAT skill of word problems. This will help on the
short and extended response questions.
Reading to Learn Mathematics- This worksheet allows ESOL students to follow
along and fill in important information the as the lesson progresses.
Connection Section:
Have students brainstorm and quick write several ideas how probability relate to
other subjects and mathematics topics. One integrated idea is Science, especially in the
field of meteorology. The weather casters say many times that there is 30% chance of
rain which is a prediction. Find and discuss other uses of Probability in everyday life.