ANOTHER MATHEMATICS LESSON

 Basic Information

Teacher

Milton Francis

Summary

In this lesson, students will expand on the concept of probability previously learned.  This expansion will see them learning and applying the Bernoulli's Theorem.  In applying this theorem, they will use the Bernoulli's Formula, commonly called the Binomial Probability Formula.

11-12

Time Frame

90-minute (Double period)

Subject

Mathematics

Topic

Finding the probability of an event occurring either exactly, or at least / at most in an experiment.

Notes

After the students surf the Internet by looking at the web sites given by the teacher, they will then use the formula discovered to answer the last three questions of the handout, as the first one would have been answered by them since it was not necessary to apply the formula to answer this question.  The teacher would render assistance where necessary.

 Standards and Key Concepts

Standards

NY- New York State Standards

Subject: Mathematics, Science, and Technology (1996, 2005 Math update)

Learning Standard 3: Mathematics (2005 update)
Students will:
•understand the concepts of and become proficient with the skills of mathematics;
•communicate and reason mathematically;
•become problem solvers by using appropriate tools and strategies;
through the integrated study of number sense and operations, algebra, geometry, measurement, and statistics and probability.

Grade/Subject : Algebra 2 and Trigonometry

Area : Content Strands

Strand : Statistics and Probability Strand

Standard : Students will understand and apply concepts of probability.

 Performance Indicator : A2.S.15 Know and apply the binomial probability formula to events involving the terms exactly, at least, and at most

Understandings

Students should realize that real life probability experiments can be easily solved by applying a mathematical formula.  They will be expected to learn this formula and apply it to the solution of probability questions that involve the following terms:

- Exactly

- At least

- At most

Essential Questions

1.  If a fair coin is tossed, what is the probability that it falls tail?

2.  If that same fair coin is tossed 10 times, what is the probability that it falls tails exactly six times?

3.  If the same fair coin is tossed the same number of times (10), what is the probability that it falls

(i) at least 6 times?

(ii) at most 6 times?

Are the answers to these questions the same?  Why? or Why not?

Knowledge and Skills

Students should bring the following concepts to the lesson:-

* Probability  P(event A) =   _# of outcomes in event A

formula:                    # of outcomes in the sample space

* P(Ac)    =   1 – P(A), where Ac is the complement of A

* Combination  nC =___n!__ =  n(n –1)(n –2)...(n – r + 1)

formula:                    r!(n – r)!                        r!

* Exponential expressions : ax   and  a y–x .

Concept of "at most" with its symbol (≤)

Concept of "at least" with its symbol (³)

Concept of "exactly" with its familiar symbol (=)

These concepts will be reinforced during the conduction of the lesson, as they are the requirements needed for the students to fully comprehend this lesson.

The teacher will distribute the handout Coin Experiment to each group of students.  [10 minutes will be given for this exercise]

On completion of the handout task, a 5-minute discussion of the students' results will take place, without the teacher giving any of the correct answers.

The students will then go to the Internet, given a set of URLs, to discover what to do to solve the problems.  Among the URLs will be the Binomial Probability Calculator that shows exact, fewer and more.  The URLs are shown below:

Attachments

1.               Coin Experiment

1.               Binomial Probability 1

2.               Binomial Probability 2

3.               Binomial Probability Calculator

4.               Binomial Probability 3

Performance Prompt

The students, working in groups, will do the following mini-worksheet and submit it before leaving the classroom:

Attachments

Assessment/Rubrics

The students will be assessed based on their performance on the mini-worksheet given near the end of the class.

 Learning Experiences and Resources

Sequence of Activities

(i)  With the students seated in groups upon entering the classroom, the teacher will distribute the handout titled Coin Experiment.  They will be given about 10 mins to attempt this exercise.

(ii)  The teacher will then solicit answers from the students on the exercise, without sharing his answers.

(iii)  They will then be given a short list of web site addresses to 'look up' the answers to the questions on the handout.

(iv)  Surprisingly, to them, the exact answers are not there, instead they will find the method used to solve these problems.  This would evoke a discussion in the class about the topic, which the teacher will entertain.

(v)  During the discussion, the web site about Binomial Probability Calculator will be highlighted and be used for a mini practice of the formula.

(vi)  The students, still working in groups, will now use the formula learned to answer the questions of the experiments.

(vii)  The mini-worksheet on the topic will be given as the culminating activity of the class.

Differentiated Instruction

My special needs students will be treated as follows:

Special Education – Personal assistance rendered by the teacher during the lesson, and receiving assistance from members of the group who are able to give assistance.

English Learners – In addition to working closely with their group members, I would ensure that my weakest ESL students also work closely with the stronger ESL students who will aid in the interpretation.

At-Risk of Failing – These students will also work closely with their group members, but each of them will definitely be buddied with my advanced learners on an ongoing basis, not just for this lesson.

Advanced Learners – In addition to being the student-teacher in my classroom, where they reinforced their learning by helping their classmates, they will be given additional work so that they can reinforced their own learning.  For example, instead of giving them three questions to do as the culminating activity, I would add another two or three.

Resources

• Materials and resources:
Notebooks
Pens/Pencils
Experiment Sheet
Coins
Mini-Worksheet
• Technology resources:
Internet Explorer
• The number of computers required is 1 per student.
• Students Familiarity with Software Tool:
The students are familiar with IE as they used it regularly in class.
• Experiment sheet and coins will be used first. This will be followed by the use of the laptops (Internet Explorer), and finally the mini-worksheet. The notebooks and pens or pencils will be used throughout the lesson.