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Take a “Chance”:
Connecting Probability
to Rational Number
Reasoning
Megan H. Wickstrom
Nicole M. Wessman-Enzinger
Illinois State University
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What will we discuss?
 Example
Lesson
 “A New Spin on Fair Sharing” – A
lesson using probability to elicit
students’ rational number reasoning
 Play, Participate
and Discuss Games
 Dice Difference
 Modified Roller Derby
 Twister
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Rational Numbers in Context
 Probability
is often integrated into curriculum to
explore rational numbers or to extend student
thinking.
 Rational
Number thinking is considered
fundamental in the comprehension of probability
 It’s
important for students to have different kinds
of experiences with rational numbers so that they
extend their thinking.
 Probability
 Misconceptions can challenge students thinking
 Helps to place rational numbers in a real context
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A New Spin on Fair Sharing

Classroom lesson taught with 2 fourth grade classes.

Objective: Provide students opportunities to make connections
to the equivalence of rational numbers through play with
spinners typically utilized to teach probability.
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The Problem:
 Fair
Sharing Problem:
 The principal has bought candy for the fourth
grade class because they have been so good.
When she gets to school, she realizes that she
miscounted and she didn’t buy enough candy for
everyone. A fourth grade student suggests that
maybe the candy should go to only the boys or
only the girls. The principals thinks this is a good
idea and suggests using a spinner to help decide
who gets the candy. The red color represents the
girls and the blue color represents the boys.
 Alternate: Race
Problem
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The Spinners
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What did the students see initially?
Students were asked, which spinner the
teacher should pick and why?
 Some
students focused on which spinner was their
favorite (superstitious beliefs)
 Some
focused on the layout of the spinner (boy, girl,
boy, girl versus mixed up pieces)
 Only
3 of the students from both classes said they
were all the same
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Sample of Student Response
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Sample of Student Response
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Students play with spinners and
record findings
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Focus on Fairness
 After
the students played they submitted their
results and we compiled them (200 spins per
spinner or you could use tinkerplots).
 Heard
the word “fair” several times by
students.
 Fairness
issue.
helped us to focus on the equivalence
 Students
discussed fairness and if they thought
each of the spinners were fair.


What would the results look like if the spinners were fair?
What would they look like if they were unfair?
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Students Perceptions of Fairness
 Fairness
means 50/50.
 “I
will win about half of the time and my friend will
win about half of the time.”
 Students
thought that most of the spinners were
probably fair except for:
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The Reveal
Girls
Boys
82 Spins
81 Spins
Spinner F
50%
50%
87 Spins
92 Spins
48%
51%
Spinner D
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Different But The Same??
 After
seeing all the results, students were
asked how could the spinners look
different but have the same results?
 Spinner
A as a benchmark to compare
 Focus on number of pieces in each spinner
 Focus on number of pieces and same size
pieces
 Using math/equivalent fractions to show they
are the same.
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Student Work: Expressing Similar
Fractions
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Student Work: Spinner A as
benchmark
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Lessons Learned
 Always


It’s important to have students make these connections, but
they are good at redirecting the focus.
Have a plan to help them focus (for us that was the focus on
fairness and equivalence).
 Two

on your teaching toes!
Contexts = Double the Misconceptions
This lesson is a nice extension/introduction to either topic
but it shouldn’t be the only way students see fractions or
probability.
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It’s Your Turn!

We are going to have you play some probability games.

As you play the games, talk about how you think the students
will use rational numbers (if at all) and what kind of use
rational numbers have in the task.
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Dice Difference
Rules:

You will play with partners.

The oldest player is the “high player.”

The youngest player is the “low player.”

Toss the dice and take the difference.

The “low player” wins with a difference of 0, 1, and 2. The
“high player” wins with a difference of 3, 4, and 5.
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Dice Difference

Play the game 25 times and record the number of times “high
player” wins and the number of times the “low player” wins.
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Dice Difference Discussion

What do you think?

What kinds of mathematical ideas would you talk about with
your class?
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Dice Difference

We have played this with a variety of students. No student has
ever stated that they thought the game was unfair before the
game begun. Similarly, most students have been surprised
when they may lose by a lot.

After the game has been played and the student record their
results on the board, the students’ discussion start buzzing.
Students notice something is going on. But, when we
questioned them their answers lack precision
mathematically.
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Dice Difference: Our Students
We challenged our students to consider the sample space, or
all of the possible outcomes of this game.

Our students struggled to organize their data into a chart.

Most of our student organized their results in organized lists
or order pairs.

Most students said the sample space was 21, rather than 36.
Why do you think that happened?
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Dice Difference Change Challenge
1
2
3
4
5
6
1
0
1
2
3
4
5
2
1
0
1
2
3
4
3
2
1
0
1
2
3
4
3
2
1
0
1
2
5
4
3
2
1
0
1
6
5
4
3
2
1
0
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Dice Difference: Our Students

As the students accepted the organization of the chart and
the sample space of 36, we challenged our students to
consider if this game was fair.

With the introduction of “fairness” we began to approach
ideas of rational numbers.

How do you think the students discussed fairness?
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Dice Difference Sample Space
1
2
3
4
5
6
1
0
1
2
3
4
5
2
1
0
1
2
3
4
3
2
1
0
1
2
3
4
3
2
1
0
1
2
5
4
3
2
1
0
1
6
5
4
3
2
1
0
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Dice Difference Change Challenge

How can you change the rules of Dice Difference to make the
game fair?
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Roller Derby with a Twist

Play with a partner

Each player receives 10 chips that they will then place on the
board

During the game, you will roll both dice and add the two
numbers together to get a sum.

If you have a chip(s) on that sum you may remove one chip
from the board

The first person who removes all their chips is the winner!
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Students at Play

Common Student Misconceptions



Challenge students to write down initial strategies and how
their strategy changed each time




Placing chips evenly across all the spaces
Placing chips on the space marked 1
Matching another student (initial strategy)
Seeing sums that seem to come up more often than others
Learning from prior games
Challenge the students to model the game using rational
number reasoning. Questions like why are we seeing a sum of
6 more than a sum of 2?

Students can begin to articulate this such as there is only one way
to make a 2 but there are several ways to make a 6
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1
2
3
4
1
2
3
4
5
2
3
4
5
6
3
4
5
6
7
4
5
6
7
8
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1
2
3
4
1
3
4
5
6
2
3
4
5
6
3
7
4
5
6
4
7
2
5
8
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Your Turn!

Use Dice Difference and Roller Derby to help you come up
with your own twist. What kind of game with dice could you
use to talk about probability and rational numbers?
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Twister

In the popular game of Twister® people
spin the following spinner to determine
where to place their body parts (i.e.,
left hand, right hand, left foot, right foot)
on various colored dots (i.e., yellow,
green, blue, and red).

Using Twister as a context, brainstorm
some probabilistic and rational number
topics that you could discuss in your
own classroom.
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A Twister Task:
What do you think?

What kinds of probability topics and rational number topics
would you talk about? What kind of questions would you
pose?
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Task #1

Suppose that you are making a homemade Twister® game.
You want to account for all of the rules and possibilities that
are part of the original game; however, you want to use
multiple spinners. How can you use multiple spinners, where
you need to spin at least twice, to generate the same sample
space as the traditional Twister above? Draw your new
spinners below. Describe how you would use these spinners
and how these generate the same sample space as the
spinner above.
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Task #2

You spun the original Twister spinner above, 20 times. You
landed on “left hand, yellow” twice. What is the experimental
probability? What is the theoretical probability? Explain why
the probabilities are similar or different.
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Student Responses

Students struggled coordinating the difference between
experimental and theoretical probabilities.

This struggle highlights the need to recognized our “whole.”
Is the whole the number of times we spun? Or, is the whole
the sample space (or number of possible outcomes)?
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Task #3

While playing the game, Kyle argues that the Twister game
board affects the probabilities because, while playing, he
notices that more people have their hands and feet on the
yellow dots compared to the other dots on the board. How
would you respond to Kyle?
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Task #4

Daric said that if he made a new board, this would also
change the probabilities. Daric’s “new” board is shown
below. How would you respond to Daric? Is the game still
fair?
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Student Responses

Students struggled coordinating separating that the board is
independent of the spinner.

Many students held the misconceptions that the changing the
board changed the probabilities.

Descriptions of past experiences playing the game helped
the students overcome their misconceptions.
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THANK YOU!

(THANK YOU!)2013

Please feel free to contact us:

Megan:
[email protected]
www.meganwickstrom.com

Nicole:
[email protected]
www.nicoleenzinger.com
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