Download Author : Mina Sifain

Author : Mina Sifain
Date : March 7, 2006
Title of Lesson : Normal Distribution
Grade Level(s) : Grade 11
Content Area : Algebra II
Unit Topic : Patterns in Variation
Backward Design Overview
What should students know, understand, and be able to
do by the end of this lesson?
The student will be able to form a conjecture based on given examples of sets of data.
The student will be able to calculate the mean using a formula and approximate it from
the Normal curve. The student will understand the concepts relating the Normal
Distribution curve to mean and standard deviation. The student will be able to explain if
a given set of data follows a Normal curve.
The students will show mastery of the content through working in groups and explaining
what their conjectures are from the performed experiment.
What will you accept as evidence?
After the students have worked individually on an experiment and they have worked on a
different experiment in groups of 4, they will be able to explain in a journal that as the
number of trials in an experiment increase, the data follows a Normal Distribution curve.
Essential Question:
Given a set of data, explain how to identify if the set of data follows a Normal curve?
Standards
New York State Learning Standards and Key Ideas
Mathematics, Science, and Technology- Standard 3
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.
Problem Solving Strand:
Students will build new mathematical knowledge through problem solving.
A2.PS.3
Observe and explain patterns to formulate generalizations and conjectures
Statistics and Probability Strand
Students will collect, organize, display, and analyze data.
A2.S.5
Know and apply the characteristics of the normal distribution
Objectives
Student-Centered Instructional Objectives and Bloom's
Taxonomy
•
•
•
Explain the meaning of "standard deviation" and "mean" (Comprehension).
Given a table of data, the student will be able to graph a histogram and a normal
distribution curve (Application).
Explain the characteristics of a normal distribution curve, such as symmetry
and where the mean and standard deviation lie (Comprehension).
Materials
The materials that the teacher will need are as follow:
•
•
10 Overhead markers: 4 for me (Green, Black, Red, Blue), and 6 for each group
(2 Green, 2 Red, Blue, Black)
24 copies of each activity sheet (1, 2, 3, and dice experiment) for each student
•
•
•
•
6 copies of a grid on overhead paper, one for each group of the students.
2 dice (a pair per group)
24 Dice experiment sheets (one per person)
Homework worksheet for each student and me.
Student Adaptations
Adaptations
One of the students in the class is blind, so one handout will be written in braille, as
well as there will be a note-taker.
Lesson
The Anticipatory Set
"Before we start with today's lesson, I'd like to ask you a quick question to think about as
we discuss today's lesson. In the United States, what do you think are the average heights
of a men compared to the average heights of women? (wait a moment for an answer)
Well, how does that compare with the average height of just men compared to other men,
or just women compared to other women? Now, let's expand a little bit. What do you
think about other countries (such as Japan, France, Australia, Libya)? What's the average
heights of men compared to women? (wait a moment for an answer) How does that
compare with the average height of just men compared to other men, or just women
compared to other women, in other countries? I just want you to think about these few
questions as we go through the lesson."
The Flow of Teaching and Learning Activities
Sorting and Categorizing
As the students enter the room, I will direct their attention to the overhead, where they
will see that they have been put into 6 groups (4 students per group) that have already
assigned, and ask them to get into their groups.
I will hand out to each group a different activity sheet and ask them to complete it. After
each group has completed the activity sheet, each student in the class will
conclude/conjecture what they can see/observe from their activity sheet.
Reflecting and Explaining
After each student has written something, I will ask each student to discuss with his/her
group their observations and to come up with one-group-conjecture. Then, I will ask for
a volunteer from each group to come to the overhead to: show the group's activity sheet
(the data and the histogram) to the class and to reflect on what that group has
observed/conjectured on an overhead which they created when they were working with
their groups.
We will then have a class discussion about each activity sheet, where I will raise some
questions that would lead them to further articulate their reasoning. Some questions will
be: (1) How can you conclude if the graph is normally distributed? (2) If I didn't give
you too much data, would you still get the same results? (3) What's better, if the data
were closer to the mean or further away? (4) Explain the meaning of when your data is
further away from the mean?
Generalizing and Articulating
With the class a whole, we will discuss that each of the data in the activity sheets fits a
normal distribution curve, which will lead into the definition of the Normal Distribution
Curve, its characteristics, and how it relates to the mean and standard deviation.
Verifying and Refining
As a class, perform one experiment, where the class takes 25 trials of the sum of two
dice. I'll call up one person, who will be the dice-thrower. After the 25 trials have been
performed, have each student record the data and graph the histogram on the Dice
Experiment Worksheet. Then, as a class decide whether the experiment is normally
distributed or not; it will be distributed, since all the data should be closer to the mean.
To finish the lesson, I would make sure to tell to the whole class that any experiment
could be normally distributed if there are enough trials.
Conclusion
On the board, I will sketch two histograms, one that is normally distributed, and one that
is not, and classify them as A and B
"As I draw these histograms on the board, raise your right hand if you think that
histogram A is normally distributed or your left hand if you think that histogram B is
normally distributed. What is the mean of the normally distributed curve?"
Homework Assignment
Go around in your neighborhood and take a survey of what are people's favorite
beverages. Tally up your data, draw a histogram, and draw the normal distribution
curve. Does that data look normally distributed? If it does, what is the mean? You must
ask thirty people in your neighborhood.
Name:___________________________
Algebra II
Date:__________
Period:_________
Activity Sheet #1
Directions: The data below give the weights, to the nearest hundredth of a gram, of a
sample of 100 new nickels. Graph the data below in a histogram and then answer the
question that follows.
Nickel Weight Frequency
4.87
2
4.88
1
4.89
1
4.90
2
4.91
2
4.92
4
4.93
4
4.94
4
4.95
5
4.96
5
4.97
6
4.98
6
4.99
6
5.00
7
5.01
8
5.02
7
5.03
7
5.04
6
5.05
3
5.06
4
5.07
3
5.08
2
5.09
2
5.10
1
5.11
2
What are your observations about this graph?
Name:___________________________
Algebra II
Date:__________
Period:_________
Activity Sheet #2
Directions: The table below gives the heights of women in a statistics class at Nazareth
College. Graph the data below in a histogram and answer the question that follows.
Female Heights Frequency
59
2
60
5
61
7
62
10
63
16
64
22
65
20
66
15
67
9
68
6
69
6
70
3
71
1
72
1
What are your observations about this graph?
Name:___________________________
Algebra II
Date:__________
Period:_________
Activity Sheet #3
Directions: For a chemistry experiment, students measured the time for a solute to
dissolve. The experiment was conducted 50 times. Graph the data below in a histogram
and answer the question that follows.
Dissolution Time (s) Frequency
4
1
5
1
6
1
7
1
8
4
9
3
10
6
11
5
12
9
13
5
14
5
15
2
16
2
17
3
19
2
What are your observations about this graph?
Name:____________________________
Algebra II
Date:_________
Period:________
Dice Experiment
Directions: As a whole class, conduct an experiment to check to see if the sum of rolling
2 dice 25 times is normally distributed. Record your data in the table below. Then, draw
a histogram and the Normal Distribution curve, and approximate the mean using the
curve. Finally, calculate the mean using the well-known formula and compare with the
value received from the curve.
Trial Number Sum of Dice
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
Tally
Approximated Mean from Curve:___________
Calculated Mean using the formula:_____________
What’s your final conclusion from the activities seen today?
Name:____________________________
Algebra II- HW
Date:_________
Period:________
Directions: Go around in your neighborhood and take a survey of what are people's
favorite beverages. Tally up your data, draw a histogram, and draw the normal
distribution curve. Does that data look normally distributed? If it does, what is the
mean? You must ask thirty people in your neighborhood.
Trial Number Choice of Beverage
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
Tally
What is the mean?