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## UDL Sample Lesson

Exploring Experimental and Theoretical Probability

Description
Students will explore the relationship between experimental and theoretical probability.
They will collect data and compare the experimental data to what would be predicted
(i.e. theoretical probability). This lesson follows an introduction to the basics of
probability with an exposure to the concepts of theoretical and experimental probability.
Prerequisites
-

## Understanding that probability is the likelihood of an event occurring, i.e. what

are the chances of an event occurring?

Given a die, a coin, or an equally segmented 4-color spinner, a student can state
the probability of an occurrence (e.g. a coin landing on tails, a die landing on two,
or spinner landing on certain color).

## Understanding of basic vocabulary related to probability, i.e. probability,

outcome, trial.

Other prerequisites:
-

Familiarity with using a computer and accessing internet sites for the probability
activities.

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Estimated Time
One hour
Standards
Georgia Department of Education - Georgia Standards of Excellence 2015 - 2016
Statistics and Probability 6.SP
Develop understanding of statistical variability.
MGSE6.SP.1 Recognize a statistical question as one that anticipates variability in the
data related to the question and accounts for it in the answers. For example, How old
am I? is not a statistical question, but How old are the students in my school? is a
statistical question because one anticipates variability in students ages.
MGSE6.SP.2 Understand that a set of data collected to answer a statistical question
has a distribution which can be described by its center, spread, and overall shape.
MGSE6.SP.3 Recognize that a measure of center for a numerical data set summarizes
all of its values with a single number, while a measure of variation describes how its
values vary with a single number.
Statistics and Probability 7.SP
Use random sampling to draw inferences about a population.
MGSE7.SP.1 Understand that statistics can be used to gain information about a
population by examining a sample of the population; generalizations about a population
from a sample are valid only if the sample is representative of that population.

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Understand that random sampling tends to produce representative samples and support
valid inferences.
MGSE7.SP.2 Use data from a random sample to draw inferences about a population
with an unknown characteristic of interest. Generate multiple samples (or simulated
samples) of the same size to gauge the variation in estimates or predictions. For
example, estimate the mean word length in a book by randomly sampling words from
the book; predict the winner of a school election based on randomly sampled survey
data. Gauge how far off the estimate or prediction might be.
Statistics and Probability 8.SP
Investigate patterns of association in bivariate data.
MGSE8.SP.1 Construct and interpret scatter plots for bivariate measurement data to
investigate patterns of association between two quantities. Describe patterns such as
clustering, outliers, positive or negative association, linear association, and nonlinear
association.
MGSE8.SP.2 Know that straight lines are widely used to model relationships between
two quantitative variables. For scatter plots that suggest a linear association, informally
fit a straight line, and informally assess the model fit by judging the closeness of the
data points to the line.
MGSE8.SP.3 Use the equation of a linear model to solve problems in the context of
bivariate measurement data, interpreting the slope and intercept. For example, in a
linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an
additional hour of sunlight each day is associated with an additional 1.5 cm in mature

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plant height.
MGSE8.SP.4 Understand that patterns of association can also be seen in bivariate
categorical data by displaying frequencies and relative frequencies in a two-way table.
a. Construct and interpret a two-way table summarizing data on two categorical
variables collected from the same subjects.
b. Use relative frequencies calculated for rows or columns to describe possible
association between the two variables. For example, collect data from students in your
class on whether or not they have a curfew on school nights and whether or not they
have assigned chores at home. Is there evidence that those who have a curfew also
tend to have chores?
Instructional Goals
Students will demonstrate understanding of the relationship between experimental and
theoretical probability.
Objectives

Students will understand that theoretical probability is the likelihood that an event
will happen.

Students will explain that experimental probability of an event is the ratio of the
number of times the event occurs to the total number of trials in the experiment.

## Students will understand that experimental probability approaches theoretical

probability as the number of trials increases.

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Variability UDL Connections
I realize that my class represents three dimensions of variability in terms of the way
students learn and represent information, differences in the way they engage with the
materials, and differences in the way they act on the material and show what they know.
The challenge in designing this probability lesson is to make sure that all students have
sufficient background knowledge in the basics of probability and then to plan for this
variability from the start.
!Variability in Representation and Background Knowledge - To plan for all students, I
present the lesson information in multiple ways. For example, I begin by clearly stating
the goal of this probability lesson and then post the goal on a white board and offer
index cards with the lesson goal printed on it. Throughout the lesson, I remind students
that the goal is to "understand the relationship between theoretical and experimental
probability.
Although there are needed pre-requisites for this probability lesson, I also know that
some students will need repeated exposure to the pre-requisites, therefore I
intentionally use the probability vocabulary defining words as I use them and I give
examples of what I mean by collecting and interpreting data. For example, when I model
the spinner activity, I make explicit how the program is collecting the data and how I
interpret this data that is being collected.
!Variability in Action and Expression - To support the given variability in how students
act on the information, I make sure that I provide varied means of demonstrating
understanding the probability concepts. When students are initially working on the
spinner activity, I walk around and ask students, how they choose to explain what is

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happening. Some choose to talk with their clock buddies and others choose to talk with
me about their findings. I also suggest to students that they can ask other clock buddies
or me for help in using the spinner application and/or in understanding the relationship
between experimental and theoretical probability.
!Variability in Engagement - I find that offering choice is key to engaging students, so I
do this as often as I can. However, the challenge is to know when to give choice and
when to expect all students to engage in the lesson activity as I have found that this
probability activity is typically effective in helping students understand the relationship
between experimental and theoretical probability. The choices that students make are
choices that the program offers: choice of colors for the spinner segments, choice about
the number of spins that the clock buddies make, or choice of number of sectors.
Assessments
Formative Assessments
Use formative assessments as measurable opportunities for learners to practice and for
you to gauge students levels of engagement. Use the results to inform your instruction
as well as to give learners timely feedback on their progress.
Student Learning Objectives

Students will understand that theoretical probability is the likelihood that an event
will happen.

Students will explain that experimental probability of an event is the ratio of the
number of times the event occurs to the total number of trials in the experiment.

## Students will understand that experimental probability approaches theoretical

probability as the number of trials increases.

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Assessment

Walk around the room and check for understanding as the students are doing
their exploration, asking both direct and indirect questions about their
understanding.

Instructional Methods
OPENING
Introduction (15 minutes)
Begin the lesson by clearly stating the lesson goal, i.e. demonstrate understanding of
the relationship between experimental and theoretical probability. In addition to verbally
stating the goal, write the goal on chart paper or on a white board and post in a visible
area of the classroom so that students can readily see the goal. In addition, prepare
index cards with the lesson goal so that students may have a copy at their desk, if they
choose.
Explain that the students will be conducting an experiment to explore the relationship
between theoretical and experimental probability.
Introduce the Spinner from the Shodor (www.shodor.org) website to the class, using the
Smart Board for class demonstration. Highlight how you simulate spinning the spinner
by indicating number of trials and click on spin. Point out the chart at the bottom of the
program that tells you the experimental probability, the theoretical probability, and
the count. Show the results frame that displays experimental results in a pie chart
(different representation).
Model a "think aloud" as you spin the spinner, e.g. if I spin one trial, I predict that the
spinner will land on blue and that the experimental probability will be 100% on blue and

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zero on the three other segments. I then ask myself, what happens to theoretical
probability.
DURING
Pose Leading Questions (30 minutes)
Pair students with their clock buddies and tell them that they will have 30 minutes to
explore the Adjustable Spinner on the computer and respond to the following questions:
What happens when you increase the number of trials in an experiment? What is
theoretical probability? What is experimental probability? What do you think the
relationship is between experimental probability and theoretical probability?
Transition to the classroom computer stations: ideally, one pair of clock buddies for
every station; however, you may need to increase the number of students at a computer
depending on access.
Students may choose to write their responses on the response template that the
teacher provided or they may choose to record their responses on an iPhone or audio
recorder.
(NOTE: Give students time benchmarks so that they know how they are progressing,
e.g. 5 minutes to become familiar with the program, 15 minutes for Exploratory Time, 15
minutes to discuss their findings and 5 minutes to respond to the questions.)
CLOSING Synthesize (10 minutes)
Ask the students to meet with another clock buddy team and discuss their findings,
focusing on the lesson question, "what is relationship between theoretical and
experimental probability?"

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Closing (5 minutes)
Remind students of the lesson objectives and ask each student to complete a short
survey to assess achievement of the goal, using paper/pencil or an online tool, like
survey monkey.
Materials
MATERIALS AND SUPPLIES
Materials for creating "Clock Buddies" (www.readingquest.org/strat/clock_buddies.html),
including a clock handout for each student with a blank line next to each hour with
names of students on each line.
Technology
Classroom computers or iPads for student use to explore the interactive internet
probability activities.
The Spinner from the Shodor website to the class.
Teacher demonstration computer with projection capabilities.
Smart Board
Materials
Paper and pencils
Shodor Website
Clock Buddies Website

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References and Resources
CAST Admin. (2012, October 31). UDL exchange: Exploring experimental and
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