Lesson Designers: Katie Bookbinder, Amy Clark, Megan Clay,
Brynn Cody, Valerie Parr, Krista Roop
Due: April 14th, 2009
Math Education Lesson Plan Template
Target Audience of the Lesson:
4626
Overall Goal of the Lesson:
To discuss statistics and statistical models using sets of data from real world applications.
Student Learning Objectives:
Students will demonstrate that they will find the interquartile range and outliers with 85% accurracy.
Students will demonstrate that they can find the standard deviation, variance, and z-score with 85% accuracy.
Assessment: (Go back to your learning objectives. What will the students do to demonstrate they have met the
learning objectives? Make sure each objective is paired to an assessment measure)
The class will be observed on basis of classroom participation during activity.
Virginia Standards of Learning addressed:
http://www.pen.k12.va.us/VDOE/Superintendent/Sols/home.shtml
*PS.1
The student will analyze graphical displays of data, including dotplots, stemplots, and histograms, to identify and describe
patterns and departures from patterns, using central tendency, spread, clusters, gaps, and outliers. Appropriate technology
will be used to create graphical displays
*PS.2
The student will analyze numerical characteristics of univariate data sets to describe patterns and departure from patterns,
using mean, median, mode, variance, standard deviation, interquartile range, range, and outliers. Appropriate technology
will be used to calculate statistics.
*PS.14 The student will develop, interpret, and apply the binomial probability distribution for discrete random variables, including
computing the mean and standard deviation for the binomial variable
A.17
The student will compare and contrast multiple one-variable data sets, using statistical techniques that include measures of
central tendency, range, and box-and-whisker graphs.
NCTM Process Standards: (List targeted standards and briefly describe how the lesson will elicit them)
Communication will be exhibited through class discussion on the topics.
Representation will be carried out when depicting the box-and-whisker plots.
Reasoning and Proof will be made when the standard deviation and variance are shown how they work through equations.
Problem Solving will occur while working through the book problem.
Length of Lesson in Minutes:
Lesson Designers: Katie Bookbinder, Amy Clark, Megan Clay,
Brynn Cody, Valerie Parr, Krista Roop
Due: April 14th, 2009
45 minutes
Schedule of Activities: (Focus on what students will be doing and what teachers will be doing during each part of
the activity, accounting for the amount of time required)
Launch: Amy – Explain and carry out Penny Stacking Activity (5 minutes)
-Stack as many pennies as possible in 30 seconds with dominant hand. Record number
-Repeat the same experiment with nondominant hand.
-Ask: What do you think the results from everyone's dominant hands will look like? (Higher numbers, closer together, smaller
range)
-Ask: What do you think the results from everyone's nondominant hands will look like? (Lower numbers, more spread out,
higher range)
-Record the results of the whole class's penny stacks of both hands on the board
Investigate: Katie – Input dominant and non- dominant hand information in calculator with class
5 number summary on calculator
Box – whisker plot (go over interquartile range, etc.) (5 minutes)
-Begin discussion by inputting the results of the experiment: go around the room and have the class input the results from the
dominant hand in L1 and the input the results from the non dominant hand in L2. After inputting the data, begin
investigation by making two box and whisker plots (one for L1 and one for L2).
Teacher: To describe a set of data, we use the five number summary to gain insight on a set of data. The summary consists of
the lower and upper extremes, the lower and upper quartiles which cuts off the lowest and highest 25% of the data,
respectively, and the median.
-First we will find the five number summary of L1:
Teacher: How do we find the median?
Student (should answer): List the data in ascending order and then find the middle number
Teacher: How do we find the lower quartile?
Student (should answer): Find the median of the lower half of the data.
Teacher: How do we find the upper quartile?
Student (should answer): Find the median of the upper half of the data.
Teacher: Now let’s find the interquartile range. (Algorithm on board)
Q3-Q1
-Now that we know how to find the five number summary by hand, let’s use the calculator to find the five number summary
Lesson Designers: Katie Bookbinder, Amy Clark, Megan Clay,
Brynn Cody, Valerie Parr, Krista Roop
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Due: April 14th, 2009
with the calculator:
-Instruct them on how to find the summary using the calculator:
STAT  CALC  1  ENTER
-Now find the interquartile range (Q3-Q1).
Teacher: What is the difference between the range and the interquartile range?
Student (should answer): The range shows the variability of the entire set of data and the interquartile range shows the
variability of the central 50% of the data.
-Instruct them on how to make box and whisker plots on the graphing calculator:
KEY STROKES: 2nd  STATPLOT  Plot1...On  Enter  On  Enter  Type : 5th choice  Xlist  Enter  L1  Enter Then hit
“graph.”
Perform same process, but this time use L2 instead of L1
Compare the two box and whisker plots and describe the differences between the two sets of data
Teacher: Is one plot more skewed/more normal than the other? How can you tell?
Megan – Lectured based beginning explaining outliers. (5 minutes)
Teacher: An Outlier is a data value that lies far away from the bulk of other values. Although these values usually stand out
we must use an algorithm to prove that an observation is an outlier as some numbers that are far away from the bulk do
not fall into the category of an outlier.
Student: If we look at the box and whiskers plot that we just created, what values do you think will be considered outliers?
Teacher: Algorithm (on board)
1. Find the lower (Q1) and upper (Q3) quartiles and then subtract them to get the inter-quartile range. (In an ordered data set
where N represents the number of data points)
a. Q1 = .25*(N+1)
i. If Q1 = 10.5 then Q1 = the 10th observation + .5*(10th obs. + 11th obs.)
b. Q3 = .75*(N+1)
i. Follows similarly
c. (Q3-Q1)
2. Multiply the IQR by 1.5.
a. 1.5*(Q3-Q1)
3. Add the value in Step two to the third quartile.
Due: April 14th, 2009
Lesson Designers: Katie Bookbinder, Amy Clark, Megan Clay,
Brynn Cody, Valerie Parr, Krista Roop
a. Q3 + 1.5*(Q3-Q1)
4. Check if any values are larger than the number in step 3.
a. Outlier ≥ Q3 + 1.5*(Q3-Q1)
5. Subtract the value in step 2 from the first quartile.
a. Q1 - 1.5*(Q3-Q1)
6. Check if any values are smaller than the number in step 5.
a. Outlier ≤ Q1 + 1.5*(Q3-Q1)
Student: Follow along side using the information from the class.
- Discussion:
 Which measure of center (mean or median) is more affected by an elimination of an outlier?
 When should you remove an outlier?
Student: Recalculate mean and median without the outliers.
Valerie – Lead class through doing standard deviation by hand and how the calculator does it for you (10 minutes)
Teacher: What is Variability?
Students: tells us how much the data is separated (defined last time with the 5-number summary and box and whiskers)
 ( x  x )2
n 1
Teacher: What is standard deviation?
Students: a distance that is used to describe the variability in a distribution
-
 (x  x )
2
n 1
When you start at the middle of the standard curve and go the distance of one standard deviation to the left and one
deviation to the right, you will enclose the middle 68% of the values in the data. Show on the standard curve.
Teacher: How does the standard deviation relate to the standard curve?
Lesson Designers: Katie Bookbinder, Amy Clark, Megan Clay,
Brynn Cody, Valerie Parr, Krista Roop
Due: April 14th, 2009
Students: When you start at the middle of the standard curve and go the distance of one standard deviation to the left and
one standard deviation to the right, you will enclose the middle 68% of the values in the data. Show on the standard curve
Teacher: How is variability and standard deviation related?
Students: standard deviation is the square root of the variability
Teacher: What is more useful standard deviation or variability? Why?
Students: standard deviation because something squared does not do us any good
Teacher: At this point I will introduce the equation for the standard deviation if it has not been introduced.
-
 (x  x )
2
n 1
Teacher: What does each part of the equation mean?
Students: ( x  x ) - The difference of each point to the mean, we have to square it because when we sum the un-squared
values we will get zero. We divide by n-1 because we are dealing with a sample of the population.
We now we have the standard deviation squared which is the variability. This does not do us any good. We need the unsquared amount so we need to take the square root.
Teacher: Now that we have standard deviation defined let us find the standard deviation of our data set. We will use the
right hand data. Now I want everyone to take their number of pennies minus the mean and then square it. Once all of the
members in your table have completed this add up all of the numbers. Once you have this new number have one member in
your group come write it up on the board.
Teacher: As a class we will add these four numbers together. We have just summed all of the ( x  x ) 2. Then as a class we
will divide this number by the number of data points minus one. This is the variability. Then we will take the square root.
This is the standard deviation. Did everyone get the same answer?
Teacher: This number is how much change there is from one standard deviation to the next. .
Teacher: Does everyone understand how we got this number?
Students: yes, if no we will go over where the confusion is.
Teacher: The calculator can find the standard deviation for you. The right hand data set is in list one. Stat, arrow over to
calc, enter on 1-var stats, enter. (Students will do this on the calculator as I do it.)
Teacher: Sx is the sample standard deviation (this should be the same as the number we got)
Teacher: Why is standard deviation important?
Students: it tells us how far apart the data is
Krista – Explain z-score, whether there is a significance or not in the data (5 minutes)
Lesson Designers: Katie Bookbinder, Amy Clark, Megan Clay,
Brynn Cody, Valerie Parr, Krista Roop
Due: April 14th, 2009
- The z-score measures how many standard deviations away from the mean a particular observation is. The z-score is
value  mean
also called a standardized value. The formula is: z 
.
sd
Teacher: What is this formula close to? And why do you think we use it?
Students: should be able to explain that it is useful process to see how far away from the average a particular score is, as well
as notice similarities with the standard deviation formula and concept.
Teacher:What are some of the relationships between standard deviation, z-scores, mean, and quartiles?
Students: Measures of how far away from the mean the data is.
Teacher: So to check to see if you were slower or faster than the class normal stacking pennies than the rest of the class,
you could take your number of pennies stacked in 30 seconds for your dominant hand and subtract the mean value of
the classes dominant hand and divide the total by zero to see how far away you were from everyone else.
- The class will then find the z-score of their non-dominant hand and see if they were closer or farther away from the class
average.
- Students can then compare their dominant and non-dominant z-scores and talk about what they think they mean.
- Students will then be asked to demonstrate their knowledge of z-scores by answering the following question.
Teacher: In a certain city the mean price of a quart of milk is $.63 and the standard deviation is 8 cents. The average prize of
a package of bacon is $1.80 and the standard deviation is $.15. If we pay $.89 for a quart of milk and $2.19 for a package
Lesson Designers: Katie Bookbinder, Amy Clark, Megan Clay,
Brynn Cody, Valerie Parr, Krista Roop
of bacon, which is relatively more expensive?
Students: will solve the problem by finding the z-score of the milk: z 1 
Due: April 14th, 2009
.89  .63
 3.25 and find the z-score of the bacon:
.08
2.19  1.80
 2.6 .
.15
Teacher: So which one is relatively more expensive? What does relatively mean?
Students: Milk because the z-score is higher, it is further away from the normal price of milk than the bacon is.
z 2
Summarize: Brynn – similar to pg. 132 #6
- We will use the TI-nspires to summarize the use and meaning of the 5-number summary and graphic representations of the
data sets. First, students will open a new document, adding a List and Spreadsheets page. They will enter in the following
data for two Virginia Tech basketball players (the game dates will not be entered):
Two shooting stars
Vasallo
Game
Nov 14
Nov 17
Nov 20
Nov 21
Nov 23
Nov 26
Dec 1
Dec 7
Dec 9
Dec 14
Dec 20
Dec 21
Dec 29
Jan 4
Points Scored
15
18
10
15
25
15
12
21
11
12
25
22
16
12
Delaney
Points Scored
19
15
24
4
22
16
30
17
23
21
16
24
16
7
Due: April 14th, 2009
Lesson Designers: Katie Bookbinder, Amy Clark, Megan Clay,
Brynn Cody, Valerie Parr, Krista Roop
Jan 10
Jan 14
Jan 17
Jan 21
Jan 25
Jan 29
Jan 31
Feb 8
Feb 11
Feb 14
Feb 18
Feb 21
Feb 25
Feb 28
Mar 4
Mar 8
Mar 12
Mar 13
Mar 18
Mar 21
-
24
12
19
21
29
37
18
18
23
16
11
25
26
7
29
13
17
17
20
14
29
6
16
16
28
10
20
24
18
20
21
15
21
26
25
14
14
26
33
12
Students will then be led to producing histograms adjacent to the spread sheet using a split screen technique. (menu-dataquick graph, menu-plot type-histogram). Students will then be asked to write down estimates for the 5 number summaries.
Students will be shown how to change between graph types (menu-plot type – box / histogram). Using the graphs, students
will answer the following questions.
Teacher: What do you think the mean number of points is for each player?
Student: about 18 for Vasallo, 19 for Delaney.
a. Which player had the higher mean number of points per game? Delaney
b. What summary statistics would you use to measure consistency in a player? How can you use the graphs to figure it out?
Which player was more consistent according to your estimates? Least range? Least IQR? What would be the difference in using
Lesson Designers: Katie Bookbinder, Amy Clark, Megan Clay,
Brynn Cody, Valerie Parr, Krista Roop
Due: April 14th, 2009
each? (Getting rid of outliers)
c. Do you think there are any outliers for the players? How do you know?
d. Why do you think the outliers have occurred? If you ignore the outliers, how do your answers for questions a & b change?
- Next students will add a new Calculator page and shown how to find the 5-number summaries and standard deviations for the
data set..
(menu-statistics-stat calculations-one variable statistics, then number of lists = 2).
Teacher: How do those calculations compare to our estimations. Do any of our answers change for the questions above?
- If there is time left at the end, we will open up a Data & Statistics page. Click the x-axis to add a variable. Then switch plot
type to histogram. Then hit menu-analyze-show normal pdf. Discuss what properties this is showing.
Adaptations: (How might the lesson be geared up or down? How might the lesson need to be adapted for
students with special needs?)
Adaptations for Special Needs Students:
Work in heterogeneous groups so that students can help with investigative process and calculator work.
Use TI-Navigator to work through problems.
If there are students with physical disabilities then have students use a clicker and count how many times they can click it in 30
seconds.
Blocks can be used to help visualize median value.
Adaptations for Gifted Students:
Have students find a way to use standard deviation to look for outliers.
Finding z-score and probablility of occurrence using normal distribution table.
Find a line of best fit for the data in a dot plot.
Materials Needed: (Go through each activity and identify what items and technology are needed to complete this
lesson.)
Pennies (as many as 40 per person)
Graphing Calculators