Transcript Big Ideas of PROBABILITY

Probability and
Statistics (Grades 3-5)
Workshop
DAY 1
Dr. Leah Shilling-Traina
Community of Learners
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Complete 3 X 5 notecard:
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Name
Email
Where do you teach?
Number of years teaching & grade levels
Favorite mathematics topic
Why are you here?
Introductions – Introduce another
person in our class to everyone!
2
Books Used in Workshop
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Navigating through Data Analysis and
Probability in grades 3-5 by Chapin, Koziol,
MacPherson, Rezba (ISBN 978-0-87353521-2) published by NCTM
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Exploring Statistics in Elementary Grades
Book 1 by Bereska, Bolster, Bolster,
Schaffer (ISBN 1-57232-344-2) published
by Dale Seymore in 1998
3
Books Used in Workshop
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Math By All Means by Marilyn Burns
(ISBN 0-941355-12-8)
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Mathematics for Elementary Teachers
Activity Manual by Sybilla Beckman
(ISBN 978-0-321-64696-5) published by
Addison-Wesley
4
Virginia Department of
Education Resources
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Probability and Statistics for Elementary
and Middle School Teachers: A Staff
Development Training to Implement the
2001 Virginia Standards of Learning
(PSEMT)
(http://www.doe.virginia.gov/testing/sol/s
tandards_docs/mathematics/2001/resou
rces/elementary/probability_module/mpr
obstatentire.pdf)
5
Virginia Department of
Education Resources
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2009 Mathematics Standards of
Learning (SOL)
(http://www.doe.virginia.gov/testing/sol/s
tandards_docs/mathematics/2009/stds_
math.pdf)
Mathematics Curriculum Framework for
grades 3 - 5
6
Manipulatives for You
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Set of blank dice
Overhead dice
Overhead color tiles
Overhead circle spinners
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Extras for You—Put in Your
Binder!
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The packet of print-out materials include:
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Documents and materials related to the NCTM
Standards and VA SOLs
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VDOE’s Mathematics Word Wall Vocabulary Cards for Grades
3-5 Probability and Statistics (all cards can be found at
http://www.doe.virginia.gov/instruction/mathematics/resources/v
ocab_cards/index.shtml)
A copy of activities from Exploring Statistics in the
Elementary Grades by Bereska, Bolster, Bolster, and
Schaeffer (ISBN 1-57232-344-2)
Some additional lesson plans and handouts
Copies of overheads/handouts for all activities in the
NCTM and Burns texts
You should also take notes in your binder!
8
First…Let’s Take a PreWorkshop Content Assessment!
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Remember, you are not be graded
during this workshop!
Please answer the questions to the best
of your ability.
At the end of our third day together, you
will take a post-workshop assessment to
see how this workshop has impacted
your knowledge of Grades 3-5
probability and statistics!
9
Standards in Mathematics
Standards in mathematics education represent the
goals we set for our students. They are value
judgments about what we would like our students to
know and be able to do.
They are chosen through a complex process that is
fed by societal expectations, past practice, research
information, and visions of the professionals in the
field… They represent our priorities.
--- Hiebert, J. (2003). What Research Says About the NCTM Standards
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2009 VA SOL: Grade 3
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2009 VA SOL: Grade 4
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2009 VA SOL: Grade 5
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VA SOL: Probability or
Statistics?
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In small groups, review the VA
probability and Statistics Stands for
grades 3-5. Classify each as addressing
probability OR statistics.
Why did you make your choices? In your
own words, what is probability? What is
statistics? How do you distinguish them
in your mind?
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VA Statistics SOL (Gr. 3-5)
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3.17 The student will
a) collect and organize data, using observations, measurements, surveys, or
experiments;
b) construct a line plot, a picture graph, or a bar graph to represent the data;
and
c) read and interpret the data represented in line plots, bar graphs, and picture
graphs and write a sentence analyzing the data.
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4.14 The student will collect, organize, display, and interpret data from a
variety of graphs.
5.15 The student, given a problem situation, will collect, organize, and
interpret data in a variety of forms, using stem-and-leaf plots and line
graphs.
5.16 The student will
a) describe mean, median, and mode as measures of center;
b) describe mean as fair share;
c) find the mean, median, mode, and range of a set of data; and
d) describe the range of a set of data as a measure of variation.
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VA Probability SOL (Gr. 3-5)
3.18 The student will investigate and describe the concept of
probability as chance and list possible results of a given
situation.
4.13 The student will
a) predict the likelihood of an outcome of a simple event; and
b) represent probability as a number between 0 and 1,
inclusive.
5.14 The student will make predictions and determine the
probability of an outcome by constructing a sample space.
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NCTM Data Analysis and
Probability Standards (Gr. 3-5)
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NCTM Data Analysis and
Probability Standards
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The VA SOL vs. The NCTM
Standards
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The NCTM Standards tell us what the
government is expecting us to do in the
classroom—how do the SOL compare?
What aren’t we doing that we SHOLD be
doing? Where should we be going?
Together, can we list the NCTM’s 5
Process Standards? How do they relate
to probability and statistics?
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NCTM Expectations: By
Grade-Level
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Although our focus is on the probability
and statistics learned in Grades 3-5, as
teachers it is important to know what
has come before and what will come
after!
Let’s look at these expectations
together—PSEMT, pages 15-18.
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Big Ideas
Focusing on the big ideas also means that teachers use
strategies for advancing all students’ mathematical
thinking (Fraivillig, 2001) by:
 eliciting from students a variety of solution methods
through appropriate prompts, collaborative learning,
and a positive, supportive classroom environment;
 helping students develop conceptual understanding
by attending to relationships among concepts;
 extending students’ mathematical thinking by (a)
encouraging them to try alternative ways of finding
solutions and to generalize, and (b) setting high
standards of mathematical performance for all
students.
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Probability AND Statistics
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Probability and statistics are NOT THE
SAME…but they are closely related and
each depends on the other in a number
of different ways. They have been
traditionally studied together
(stochastics) and justifiably so.
The relationship cuts both ways –
statistical analyses makes use of
probability and probability calculations
makes use of statistical analyses.
22
Activity: What is Statistics?
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Break into small groups of 3 or 4; each
group should get one piece of chart paper
and a marker.
Come up with a list of words and phrases
that you (and potentially, your students)
associate with the term statistics.
When you’re done, hang them in front of
the room? What commonalities do we see?
Can we agree on some of the BIG IDEAS
of statistics?
23
Statistics is…
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A problem-solving process that has 4
major components:
1.
2.
3.
4.
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Ask a question
Collect the appropriate data
Analyze the data.
Interpret the results.
In short, using data to answer questions!
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Big Ideas of Statistics (from
GAISE*)—Look Familiar? 
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GAISE offers as a central part of their
framework 4 components of statistical
problem-solving that include:
1. formulating questions
2. collecting data
3. analyzing data
4. interpreting results
*Guidelines for Assessment and Instruction in Statistics Education for Pre K12 Education
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Activity: Questions, Please?
(NCTM, p. 13)
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In statistics, it all begins with THE
QUESTION! It’s always where the process
begins. Think of a question that you would
like to know the answer to! Write it on your
post-it!
Let’s think about the purposes/reasons for
conducting an investigation, as well
practice formulating questions that have
meaning related to those purposes.
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Summary of Reasons to
Conduct an Investigation
1.
2.
3.
4.
To describe or summarize what was
learned from a set of data
To determine preferences or opinions
from a set of data
To compare and contrast two or more
sets of data
To generalize or make predictions from
a set of data
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Activity: How Long is the
Classroom?
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In attempts to answer this question, let’s
break into small groups. Use 3 different
measuring tools to collect your data: (1) a
ruler, (2) a shoe, and (3) an arm span
(stretch arms out like a “T” and it’s the
length from fingertip to fingertip).
Once collected, add your data to the
whole class data at the front of the room.
Which tool was best? Why? What
happened with the other tools?
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What is the lesson here?
What is Random Sampling?
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Question: What TV shows do Americans watch the
most?
In most circumstances, collecting information from every
member of a population is impossible. Therefore, we
collect data from a sample of the population and use the
sample to make inferences about the population.
Samples can be very accurate in describing the
population characteristics. However, for samples to be
accurate, they must represent the population.
If the sample is not representative of the entire
population, the sample is considered biased because it
does not accurately reflect the population being studied.
30
Activity: Ice Cream Preferences
(Bereska, p. 2)
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It’s summertime, and there is really only
one snack to help beat the heat—ice
cream! Before you start dreaming about
your favorite flavor, let’s do an activity
together. As I pass out some papers, read
over the description. Don’t turn over your
paper until I say to!
Let’s answer some questions!
Now…what is YOUR favorite flavor?? Let’s
collect data, and then think of some ways
to represent it!
31
Activity: What Color are Your Eyes?
What Month Were You Born?
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I want to know the eye colors of the students
in the classroom-how can we figure this out?
Now, suppose you can’t use any written
communication—how can we organize our
data?
My curiosity is unbounded! Now I want to
know during what month you were born! Let’s
get data from the whole class and make some
graphs and observations!
What do these questions have in common?
32
Categorical Data
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The past two activities have used categorical datavalues that correspond to a particular category or
label. Can we think of other questions that would
require the use of categorical data to answer?
Graphical representations of categorical data include:
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Pictograph-- uses a picture or symbol to represent an object.
If there is more than one of the objects, multiple
representations of the symbol are used. A key should be
included that states the value of the symbol.
Bar graph -- uses parallel horizontal or vertical bars to
represent counts for several categories. One bar is used for
each category, with the length of the bar representing the
count for that category
Circle graph -- shows the relationship of the parts to the
whole
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Examples of Graphs
34
Activity: Interpreting Graphs
(Bereska, p. 10)
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Together, let’s interpret these graphs
representing categorical data.
For each graph we discuss, let’s also
identify the graph type!
35
Activity: What’s Wrong with
This Graph?
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Activity: If the Shoe Fits (Bereska,
p. 34)
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I’m still curious about you all! You can tell a lot about
a person from their shoes…everyone take off your
right shoe and put it on the table. In small groups, first
come up with some questions we could ask about the
shoes worn by people in the class.
Which questions will we answer using categorical
data? What graphs could you use to organize this
data?
What kind of data is needed to answer the other
questions?
Let’s look at some of these other kinds of questions
together….
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Numerical Data
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The last activity elicited some questions that used
numerical data- data that consist of numerical
measures or counts. Can we think of other questions
that would require the use of numerical data to
answer?
Graphical representations of numerical data include:
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Line plot-- uses stacked x ’s to show the distribution of values
in a data set.
Line graph-- uses a line to show how data changes over time.
Stem-and-leaf plot-- a table showing the distribution of values
in a data set by splitting each value into a “stem” and a “leaf”;
this is a useful way to display data that range over several
tens (or hundreds).
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Why Don’t We Have an Activity
to Create a Line Graph?
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Is it because we just don’t like it as
much as the other graphs? 
VDOE’s Enhanced Scope and
Sequence Sample Lesson Plans:
http://www.doe.virginia.gov/testing/sol/st
andards_docs/mathematics/index.shtml
Hot or Cold Lesson and
www.weather.com!
39
Examples of Graphs
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Activity: What’s Wrong with
This Graph? (Part 2)
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Activity: If the Shoe Fits (Part 2)
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Let’s try to determine the typical length of
shoe worn by people in this class! There are a
couple ways we could do this…suggestions?
For now, let’s all just measure our shoe. What
kind of questions do we need to consider
first?
Put your name and measurement on a post-it.
How can we organize and display this data?
How would this activity be different if we
asked about the typical shoe size of students
in the class?
42
Activity: Counting on You
(Bereska, p. 22)
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Look at the “Can We Count on You”
Survey—what do you notice?
Would you like to add any questions?
Break into small groups; each of you are
responsible for answering 1-2 of these
survey questions!
Display your data in a way that makes
sense based upon your data type to
share with the class.
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End of Day 1 Grade-Band
Break Out!
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For the last 30 minutes (or so) of our day,
let’s break up into small groups according
to the grade we teach! Where are the 3rd
grade teachers? 4th grade? 5th grade?
Reflect on the different activities in which
you engaged today. Which activities
could/would you do with your students?
How would you implement those activities?
Be sure to refer to all of the documents (i.e.
the Curriculum Framework)!
44
Probability and
Statistics (Grades 3-5)
Workshop
DAY 2
Dr. Leah Shilling-Traina
Welcome Back! 
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How are you feeling today? What kind of
graph could we use to represent our
data if we asked for responses to be
“happy,” “sad,” “angry,” or “indifferent”?
What kind of data is this?
Any questions/comments about what
we’ve done so far?
Let’s begin where we left off…
46
Quick Review: Categorical or
Numerical Data?
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Think about possible responses to the following
questions and decide which responses will provide
categorical data and which will provide numerical
data:
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How many pets do students in our class have?
How many hours a week do we spend watching TV?
What is the typical monthly rainfall in Seattle?
What kind of music do we like best?
How many hours a week do we talk on the phone?
What kinds of snacks do we like?
How many kids eat in the cafeteria each day?
How much candy do we eat each week?
What kind(s) of graphs would you use to display each
data set?
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Quick Review: Name that
Graph!
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Activity: Graph Detective
(PSEMT, p.97)
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Look at the three graphs and
conclusions. In your group, discuss
whether the conclusion is accurate and
what factors about the graph may have
lead to inaccurate conclusions.
Be prepared to discuss your thoughts
with the whole class!
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Thinking back…
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We answered questions like “How many
colors are you wearing?” , “What is the
length of your foot?” and “How tall are
you?”and looked at how to display the
data for the entire class.
What other questions might we ask
about the different colors we are all
wearing, or the length of our feet, or how
tall we are? What else might be good to
know?
50
Measures of Center
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There are three measures of center:
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Mean-- the sum of the values in a data set divided by the
number of values; also known as the average. The mean can
also be interpreted as fair-share.
Median-- the middle value in equal distribution a data set
when the values are arranged in order; if there is no “middle”
value, the average of the two middle values.
Mode-- the value(s) that occur most often in a data set.
Other important terms
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Range-- the difference between the least and greatest
numbers in a data set.
Outliers-- a value widely separated from the others in a data
set.
51
Activity: How Many Letters in
Your First Name?
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Take one unifix cube for each letter of
your first name and make a vertical
stack. Also, write your number on a
post-it.
What kind of graph(s) can we make with
our data? Let’s use our post-its AND
ourselves to graph this data!
What is the range of our data? Are there
any outliers? What is the median?
Mode? Mean?
52
Activity: More Mean as a
Fair- Share
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Let’s explore more the idea of mean as
fair-share—everyone look to the front at
the balance! Let’s think through some
questions.
Cut out your own scale and take some
paperclips. Let’s balance!
In pairs, discuss why mean is called a
center of measure, and why it can be
thought of as fair-share (reflecting on
this activity)!
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How Can We Fairly Share?
What is the Mean?
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What is the Mean?
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What is the Mean?
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The Mean and the Sum of
Distances
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Revisiting Our Past Activities
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Think of everything we’ve done so
far…can we go back and describe the
mean, median, and mode of our data?
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Ice Cream Preferences
Eye Color and Birth Month
Length of Foot
What if we have categorical data? Can
you find all three measures of center?
What if we have numerical data?
58
Which Measure of Center to
Use?
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Mean works well for sets of data with no
very high or very low numbers (no outliers).
Median is a good choice when data sets
have a couple of values much higher or
much lower than most of the others (a data
set with outliers).
Mode is a good descriptor in two
situations:
1.
2.
When the set of data has some identical
values.
When using categorical data.
59
Activity: Use Mean, Median, or
Mode?
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Maddie scored 7, 15, 5, 6, 3, 15, and 9
points in seven basketball games. She
wants to show that she is a valuable player.
Mr. and Mrs. Rodriquez collected donations
of $50, $125, $10, $50, and $175 for
charity. They want to show that they are
good fundraisers.
Andy's last 5 phone calls lasted 20 min.,
7min., 9 min., 12min., and 7min. He wants
to show how long a typical phone call is.
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Activity: Outside Work—How Many
Hours a Week are Typical (NCTM, p. 56)
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How much time do you think teachers typically spend
on school work outside of school? More or less time
than you do? Make a prediction, then let’s find out!
What kinds of things do we need to consider to
answer this question? Who do we ask? What unit of
time do we use? Do we round?
Using the data collected with our survey, create a
stem-and-leaf plot to represent the data.
Let’s analyze and interpret our findings! What do you
see? What is the range for the number of hours spent
working outside of class? What is the mean, median,
and mode? Are there any outliers?
61
What We’ve Done So Far…
Categorical Data
Measures of central
tendency
Graph types
Numerical Data
Mean,
Mode
Median,
Mode
Bar graphs
Stem and Leaf plots
Circle graphs
Line plots
Pictographs
Line graphs
62
And Now…Onward to
Probability!
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Up until this point, we have been
focusing primarily on statistical
concepts…but we can’t ignore the other
half of the title for the workshop!
How are statistics and probability
connected? How are they different?
These are questions for us to ponder as
we engage in the upcoming activities!
63
Activity: What is Probability?
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Break into small groups of 3 or 4; each
group should get one piece of chart paper
and a marker.
As we did with statistics, come up with a
list of words and phrases that you (and
potentially, your students) associate with
the term probability.
When you’re done, hang them in front of
the room? What commonalities do we see?
Can we agree on some BIG IDEAS?
64
Sample Responses for
“Probability Words/Phrases”
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Likely
Unlikely
Chance
Probable
Possible
Impossible
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Counting
Predictions
Dice, cards
Weather, lottery
Experiments
Fairness
• GAMES! 
65
Probability in the Classroom
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Although there is often an omission from
the curriculum, probability has great
relevance to society/life!
A major advantage to teaching
probability: children intuitively have
some understanding of the concept from
their real-life experiences!!

Ex.Weather reports (i.e. what is the chance
of rain today?)
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Probability in Real-life
*
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Probability in Real-life (con’t)
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(More) Probability in the
Classroom
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As teachers, we can build on our students’
past experiences by engaging them in games,
activities, and hands-on investigations that
allow them to predict outcomes and test
predictions.
The aim should be to provide investigations
that engage the student to think and reason
mathematically, and to have them ENJOY
doing it!
69
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Experimental vs. Theoretical
Probability
Experimental probability is a probability based on the
statistical results of an experiment.
 The larger the number of trials, the better the
estimate will be. The results for a small number of
trials may be quite different than those experienced
in the long run.
 This is a focus in Grades 3-5.
Theoretical probability is the ratio of the number of
ways an event can happen to the total number of
possible outcomes.
 What is the probability of getting a heads when
flipping a coin? Of rolling a 2 with a regular die?
 This is not a major focus in Grades 3-5, but we can
start to think about it!
70
Recall: Statistics is…

A problem-solving process that has 4
major components:
1.
2.
3.
4.

Ask a question
Collect the appropriate data
Analyze the data.
Interpret the results.
How does probability relate to this
process of statistics?
71
NCTM Data Analysis and
Probability Standards
72
NCTM Data Analysis and
Probability Standards
73
NCTM Expectations: By
Grade-Level


Although our focus is on the probability
and statistics learned in Grades 3-5, as
teachers it is important to know what
has come before and what will come
after! WHY??
Let’s look at these expectations
together—PSEMT pages 15-18.
74
The Big Ideas of Our
Probability Activities (Burns, p. 2)
1.
2.
3.
4.
5.
Some events are more likely than others, while
some events are equally likely.
It’s possible to measure the likelihood of
events.
A sample set of data can be useful for making
a prediction about an outcome.
Larger sample sizes of data give more reliable
information than do smaller sample sizes.
Sometimes an experiment produces data that
do not exactly match a theoretical probability.
75
Comments about the Burns
Book
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Together, let’s take some time to
familiarize ourselves with the text and its
organization.
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Whole-class activities (WC)
Small-group activities (SG)
From the Classroom
Assessment
76
Activity: What are the
Chances? (Bereska, p. 68)
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So when describing what we think probability is, many
of us described the chance of various events
occurring, or the likelihood of the events.
Do we know what we mean when we talk about
events that are certain? Or impossible? Or likely? Or
unlikely? Can we think of some examples of each?
Let’s look at the list of events—where would you
place each of these activities on the scale at the
bottom?
Looking at this scale, can we describe what “0”
means in terms of the probability? What does “1”
mean? Can we determine some other numerical
values?
77
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Can you give an example of an event
that would have probability 0?
Probability 1?
Major take-aways:
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
Sometimes people use numbers instead of words
to describe the chance of an event occurring; this
number is between 0 and 1.
1 means the same thing as 100% on the line, and
0 means the same thing as 0%.
“Impossible” events have a probability of 0 and
“Certain” events have a probability of 1!
78
WC Activity: The 1-2-3 Spinner
Experiment (Part 1) (Burns, p. 24)
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Now that we have a general idea about probability,
let’s start playing some games!
Let’s make some spinners—let me show you how.
Also make sure you have a Spinner Recording Sheet!
Which number do you think has the best chance (in
other words, is more likely) of reaching the top first?
Why?
Do 4 spins. What did you find? Have we done enough
experiments (i.e. collected enough data) to answer
our question? Form groups of 4 and compare your
data—what did you find?
Do our results support our prediction? How did your
results compare to the whole class data?
79
WC Activity: 1-2-3 Spinner
Experiment (Pt. 2) (Burns, p. 67)
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What Big Ideas have we addressed?
Together, let’s discuss the “From the
Classroom” section.
The Importance of Sample Size in
Data Collection & More Likely Events
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Larger samples are more reliable than
smaller samples for making predictions!
 If not already there, let’s include the terms
data and sample on our Probability List
In this case, we found that one event was
more likely than others—can we look at the
spinner and reason why this was the case?
Can you design a different spinner that would
yield an event that is more likely than others?
Let’s look at what happened in Burns’
classroom—p. 28.
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Activity: Revisiting the
Probability Line

Our last activity looked at how we could
use spinners to explore the probability of
events. Can we create some spinner
faces that would correspond to
“impossible,” “very unlikely,” “equally
likely,” “very unlikely,” and “certain” on
the probability line?
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Example: Chances of
Spinning Blue
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End of Day 2 Grade-Band Break
Out!
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For the last 30 minutes (or so) of our day,
let’s break up into small groups according
to the grade we teach! Where are the 3rd
grade teachers? 4th grade? 5th grade?
Reflect on the different activities in which
you engaged today. Which activities
could/would you do with your students?
How would you implement those activities?
Be sure to refer to all of the documents (i.e.
the Curriculum Framework)!
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Why so Much Difficulty with
Probability? Think About This
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Counterintuitive results in probability are found even at
very elementary levels. For example, the fact that
having obtained a run of four consecutive heads when
tossing a coin does not affect the probability that the
following coin will result in heads is counterintuitive.
(More about coin flipping tomorrow!)
In arithmetic or geometry, elementary operations (like
addition) can be reversed and this reversibility can be
represented with concrete materials. In the case of
random experiment, we obtain different results each
time the experiment is carried out and the experiment
cannot be reversed (we cannot get the first result again
when repeating the experiment). This makes the
learning of probability comparatively harder for children.
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Probability and
Statistics (Grades 3-5)
Workshop
DAY 3
Dr. Leah Shilling-Traina
Welcome Back! 

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What is the probability that you are
excited that today is the last day of the
workshop??
Any questions/comments about what
we’ve done so far?
Let’s quickly discuss where we left off…
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WC Activity: The Game of
Pig (Burns, p. 38)
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Break into pairs and be sure you have a
pair of dice! What do we observe about
these dice?
Let’s play Pig—here are the rules. Let me
demonstrate with some help. Play a round
in your pair!
Now, I wonder…what do you think the
chances are of rolling a 1 with one die?
Do at least 5 experiments each and record
your data on our class chart.
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SG Activity: Roll Two Dice
(Burns, p. 92)
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This relates to the Game of Pig activity
we’ve just done!
If time, let’s play!
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WC Activity: Tiles in a Bag
(Burns, p. 54)
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Here I have a bag 12 colored tiles in it,
some red and some yellow. Can you tell
me what combination might be in here?
Can we be sure we found them all?
Let’s gather some data…here’s what
we’re going to do.
Write your predictions!
Okay, what if I told you that now this bag
has 12 tiles, but they are red, yellow,
and blue? Let’s do the activity again!
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Sampling with Replacement

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The last activity used what is called
sampling with replacement– when you
take out a sample, record it, then return
it to the population.
How would the activity have been
different if replacement was not used?
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SG Activity: Tiles in Three
Bags (Burns, p. 105)

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This activity provides further experience
in making predictions based on a
sampling of information.
Read through on your own; use if it
seems that your students could use
more experience!
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WC Activity: 1-2-3 Spinner
Experiment (Pt. 2) (Burns, p. 67)
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What do we notice about the outcomes of the games
we’ve played so far?
Let’s return to playing with a different spinner—
compare this new one to our old one. How are they
the same? How are they different?
Let’s once again record the outcomes of our spin
experiments. Make a prediction—which number do
you think will be landed on most frequently?
Do 4 experiments and post your data at the front of
the class.
How do our results compare to our first spinner
activity? To our prediction? Does a larger sample size
really make a difference?
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SG Activity: Spinner Puzzles
(Burns, p. 122)
 Let’s extend what we’ve done with our other
spinner activities to think about how different
spinner faces can affect outcomes!
 I’ve given you a sheet of paper with 6 blank
circles on it. Turn your attention to the
“Spinner Puzzles” overhead. Let’s read
through this together, and I’ll show you an
example.
 Now that you have the hang of it, try it! Be
sure to attend to #2! Hold on to your sheet
until everyone is done, then we’ll exchange.
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Activity: Is There Such a Things
as a Lucky Coin? (NCTM, p. 68)
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Have you ever heard of a “lucky coin”? What does it
mean for it to be “lucky”?
We can think of these coins as being “fair” or “unfair.”
Now, we are going to toss some coins (our penny,
nickel, dime, and quarter) to check their “fairness.”
What do you expect to happen?
Toss each of your four coins 10 times each and
record the number of heads (H) and tails (T) for each
coin. Once you’ve done your tosses, come add your
data to the table in front of the class.
What do we notice? Is there anything surprising in the
results? How does one group’s probabilities relate to
the whole group’s probabilities (look at ratios)?
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Well, is There??

All US coins ARE fair coins!
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Did size or weight impact the probability of
tossing a heads (or tails)?
Why might somebody consider a coin to be
“lucky”?
If you toss a dime 200 times, how many
heads would you expect? 500 times?
If we toss a quarter 10 times, will we get 5
heads and 5 tails?
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Activity: Is the Sum Game
Fair or Not?
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If I roll two dice, what are all the possible sums we
can get? Let’s list them!
In pairs, let’s play the Sum Game: Student 1 gets a
point if a sum of 2, 3, 4, 5, or 6 is rolled. Student 2
gets a point if a sum of 7, 8, 9, 10, or 11 is rolled. No
points for a sum of 12. Roll the dice 30 times and
keep track of the points for each of you. Do you think
the game is fair? Why or why not?
Can we create a line plot to help us understand this?
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Is the Game Fair or Not?

Your friend wants to use a pair of dice to determine
which of you gets the last cookie in the bag. Which of
the following situations will ensure that the selection is
made fairly? Explain your thinking.
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You both roll 1 die. If the sum of the two dice is even,
he wins the cookie. If the sum is odd, you win the
cookie.
You both roll 1 die. If the sum of the two dice is prime,
he wins the cookie. If the sum is composite, you win the
cookie.
You both roll 1 die. If the product of the two dice is
even, he wins the cookie. If the product is odd, you win
the cookie.
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Activity: Matching Line Plots
with Spinners (NCTM, p. 116)

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Review the lesson “Spin City” (NCTM, p.
73) in your small group and discuss.
Now, turn to p. 116. Take some time to
independently try to match each line plot
with a spinner.
Now, in small groups, compare your
answers and explain how you know they
match.
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Categorical or Numerical
Data?
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In our probability experiments, we’ve
used spinners, dice, and coins.
Do these tools result in categorical data
or numerical data? What kinds of graphs
could we use to display our data?
We want to MAKE CONNECTIONS as
often as possible!
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Students’ Probability
Misconceptions (Beckman, p. 388)

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Now that we’ve covered many of the big
ideas in probability, let’s look at some
situations that address common student
misconceptions.
Taken from Beckmann textbook!
In small groups, reflect on your
experiences thus far to answer the
questions in Activity 16B.
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Some Ideas for Data in the
Classroom
102
Assessment



Assessment, while likely not your
favorite part of teaching, is a necessary
part of instruction.
In her text, Burns provides 7 different
assessments that are placed near the
activities from which they evolved
(activities with which we’ve ourselves
engaged!)
Let’s take a look at a couple of these
together and discuss.
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Burns’ Assessments
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Equally Likely, p. 83
Spinner Statements, p. 136
Probability Vocabulary, p. 197
Favorite Activity, p. 200
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Assessment: Favorite Activity
(Burns, p. 200)

Think about all of the activities we’ve
done during this workshop (feel free to
flip back through your notes to help you).
What was your favorite activity in the
probability (or statistics) unit? Why did
you like it so much?
What did you learn from these activities?
 How will you use these activities with your own
students?
 In general, how do you feel about teaching
probability and statistics to your students?

105
End of Day 3 Grade-Band Break
Out!



For the last 30 minutes (or so) of our
workshop, let’s break up into small groups
according to the grade we teach! Where are
the 3rd grade teachers? 4th grade? 5th grade?
Reflect on the different activities in which you
engaged today (and over the course of the
workshop). Which activities could/would you
do with your students? How would you
implement those activities? Be sure to refer to
all of the documents (i.e. the Curriculum
Framework)!
Share general thoughts/questions!
106
Time for the Post-Workshop
Assessment!


How quickly three days pass when
you’re having fun!  Remember, you
are not be graded during this workshop!
Please answer the questions to the best
of your ability so that we can see how
this workshop has impacted your
knowledge of Grades 3-5 probability and
statistics!
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Things to Remember

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Visit the VDOE website to explore all of
the great resources available (i.e. Word
Wall Vocabulary Cards, ESS Sample
Lesson Plans)
Also check out the NCTM Illuminations
website for great lesson ideas
Probability and statistics are FUN topics
to teach and learn!
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Thank You! 
Thank you for coming to this workshop
and sharing with us your thoughts,
ideas, and experiences—but moreover,
THANK YOU for everything that you do
in the classroom each and every day!
And remember… "Good teaching is more
a giving of right questions than a giving
of right answers." - Josef Albers
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