|
|
|
|
Home
Forums Grades Pre K-2
Grades 3-5
Grades 6-8
Grades 9-12
Instructional Plan Template
Mathematics Governor’s
Institute 2004
(Download as Microsoft
word document: Exploring Experimental and Theoretical Probability.doc,
bench fair or unfair game problem.doc, Coin toss
data sheet.doc, Homework Problem.doc, Rolling
a Number Cube data sheet.doc, Simulating
Rolling a Number Cube.doc)
Names of group
members: Rick Adamsky
Mary Flanagan
Jennifer
Repetto
Linda Stewart
Topic/Theme: Exploring Experimental and Theoretical Probability
Level: 6-8th grade
Time Element: 2-3 days
NCTM Standards
Addressed:
·
Data Analysis and Probability
·
Communication
·
Connections
PA Math Standards
Addressed:
·
2.5.8.B Verify and interpret results using precise
mathematical language, notation and representations, including numerical tables
and equations, simple algebraic equations and formulas, charts, graphs and
diagrams.
·
2.7.8.B Present the results using visual
representations.
·
2.7.8.D Compare and contrast results from
observations and mathematical models.
·
2.7.8.E Make valid inferences, predictions and
arguments based on probability.
Math Assessment
Anchors Addressed:
·
M6.E.1.1 Interpret data shown in histograms.
·
M6.E.3.1 Determine all possible combinations, outcomes and/or calculate the probability of
a simple event.
·
M7.E.3.1 Determine theoretical or experimental probability.
·
M7.E.4.1 Draw conclusions
and/or make predictions based on data displays.
·
M8.E.1.1 Choose, display or
interpret data.
·
M8.E.3.1 Calculate the probability of an event.
·
M8.E.4.1 Draw conclusions,
make inferences and/or evaluate hypotheses based on statistical and data
displays.
·
R8.A.2.3
Make inferences and draw conclusions based on text.
Objectives:
Students will:
·
predict, obtain and record the outcomes of
tossing a coin
·
create a histogram from experimental data
·
calculate theoretical probability
·
use a graphing calculator to simulate rolling
a number cube
·
interpret data to account for the differences
between theoretical versus experimental probability
·
determine whether a game is fair
·
create a “fair game”
Instructional
Strategies and Plan (include strategies used to help different types of
learners, i.e. auditory, visual, etc):
INTRODUCTION
While the Monday night football theme is playing, teacher walks in with a
Nerf football, throws it to one of the students and describes the Superbowl—the
WORLD championship of football. Teacher
explains that everything is choreographed with the Superbowl, even the toss of
the coin. Teacher points out that in a
recently played Superbowl game heads was tossed. One television commentator remarked to the
other that this was the fifth consecutive time heads had been tossed in that
game. He wondered what were the chances
of that occurring.
The
teacher poses that question to the class.
Teacher asks, “What are the chances of tossing a coin and having it come
up heads?” Class will discuss the
probability when tossing a coin.
IMPLEMENTATION
1.
Students will be broken into pairs. Each student will receive a penny, data
collection sheet and a small post-it note.
Students will be instructed to toss the penny 10 times each and record
the results on the data sheet. Then, students will be called up to the board
where the axes for a histogram are displayed.
Students will form the histogram by using their post-it notes. They will place their post-it notes in the
appropriate column that stands for the number of heads they obtained in their
trials. At this point, the class will
discuss the results and attempt to account for any differences. Teacher will ask students if the results
deviated from the results they expected.
2.
Next, teacher will develop the concept of
probability with the students. The teacher
will introduce the concept of theoretical probability. This is the number of favorable outcomes that
are supposed to occur compared the number of possible outcomes. Specifically, in terms of the penny, the
result is either heads or tails. Therefore,
the theoretical probability of obtaining heads is 1 out of 2. Likewise, the theoretical probability of
obtaining tails is 1 out of 2. Teacher
will ask the class, does this mean that if a penny is tossed twice, one result
will be heads and the other tails? When
the class comes to the conclusion that this does not always happen, teacher
will talk about experimental probability.
That is, the number of desired outcomes that result out of the total
number of trials. Teacher will inform
students that this is the probability that results from an actual
experiment.
3.
Now, teacher asks the students to predict how
many times heads will appear in an additional ten trials per student. After discussion, students will once again
conduct ten trials. Next, the teacher
will model how to mathematically find the experimental probability and the
theoretical probability. Students will
find the experimental probability from their data.
4.
Students are each given a number cube. The teacher asks how many possible results
there are when a number cube is tossed.
Together as a class, students will find the theoretical probability of
getting a one on a number cube. Then,
each student will conduct an experiment by tossing a number cube ten times and
recording the results on the data sheet.
They will be prompted on their data sheet to find the experimental
probability for their ten trials. Then,
they will combine their data with their partner and find the experimental
probability out of the twenty trials.
5.
Now, the teacher will demonstrate how to
simulate rolling a number cube by using the random integer generator function
on the graphing calculator. Walk
students through simulating rolling the number cube ten times and have them
record their data as they go along. Then, once again, they will record the number
of times they obtained a one out of ten trials.
They combine their data with their partner and find out how many times
they obtained a one out of their twenty trials.
Finally, they will find out how many ones they obtained out of all forty
trials. Then, the teacher will call on
the pairs to state the number of ones they tossed out of the forty trials. The experimental probability for the whole
class will be found. The class will
compare the experimental probability and the theoretical probability. The teacher may want to question the students
as to any differences they saw in the experimental probability when the number
of trials increased.
CLOSURE
Students will be given
a handout with a story of a game.
Students are asked to read the story and respond to the questions after
the story. Students will explore the
idea of a fair game. Class discussion
will follow.
ASSESSMENT
Students
will complete a worksheet for homework that is similar to the closure
activity.
FOLLOW-UP ACTIVITY
As a follow-up assignment, students will be instructed to design a fair game. They have to clearly describe the game and how the winners and losers are determined. Also, they have to mathematically show that the game is fair by finding the theoretical probability of each player winning.
Materials/Resources: pennies, number cubes, graphing calculators, post-it notes, student data
sheet, “fair game” worksheet, “fair game” homework sheet, Nerf football, music
Interdisciplinary
Connections:
·
·
Technology- Students will use the graphing calculator to generate random integers.
·
Writing
o
Students will write a paragraph responding to
the fair/unfair game.
o
Students will write a description of their
own fair game.
Assessment Strategies:
·
Formative Evaluation (checking
student understanding during the lesson):
Teacher will question and focus students during the
lesson.
·
Summative Evaluation
(How will it be determined that the objectives were achieved?): At the conclusion of
the lesson, students will demonstrate their knowledge of probability by
analyzing a game for fairness. Models of
theoretical probability will support the conclusion. Independently, students will demonstrate mastery
by creating games and defending the mathematical fairness of the games.
Correctives/Remediation: If a pair is struggling, students can form a small group for more support. Teacher can demonstrate a directed activity if necessary.
Extensions/Enrichment:
·
Students may compute the theoretical
probability of the introductory story.
·
Students could calculate the theoretical
probability of outcomes involving multiple materials.
·
Students could create/play the game they
described.
Special Accommodations
(special needs students)
·
Description of the
Special Needs student selected:
“Thomas” receives Emotional Support Services in a Part time Learning and
Emotional Support classroom. His IEP
includes annual goals to develop reading skills to build reading fluency and
comprehension, to develop math skills in the four basic processes (he is two
grade levels behind in the district math curriculum), and to improve social
skills by acquiring conversational skills, recognizing and expressing feelings,
and solving problems in conflict situations.
Through an informal Functional Behavior Assessment (FBA) the Team has
ascertained that Thomas engages in noncompliant behaviors such as arguing,
talking out, and destruction of learning materials as a means to escape
completing his assigned school work, especially in his regular (inclusion)
classes. Incidents of these challenging
behaviors occur at least three times a week in his regular math class, although
his teacher reports that it appears he is motivated to be in the class (Thomas
will ask her for help when other children are not present.)
·
Accommodations to use
with Thomas:
-Pre-teach vocabulary terms.
-Pair with an average reader.
-Add picture cues on worksheets.
-Allow this student’s partner group to be
removed from the other groups and in close proximity to the teacher.
-Allow Thomas to assist the teacher in counting
post-it notes and compiling the totals for the histograms. Also, Thomas could model the experiments for
the students.
-During the graphing calculator activity,
Thomas can aid the teacher with the calculator at the overhead instead of
having his own.
-Teacher will make an overhead of the data
sheet and model how to record data on the data sheet.
-At the beginning of the lesson, clarify to the
class expectations for partner activities.
-At the end of class, teacher will ask Thomas,
“Can you tell us what you have written in your assignment book so that everyone
can make sure they have the homework copied down correctly?”
-Teacher can add picture cues to homework paper
as necessary.
-Teacher may want to meet with Thomas at the
end of class or after class to make sure Thomas understands the homework
assignment. Teacher may even want to
begin homework assignment with Thomas to ensure success.
Name ______________________ Date _____________
Fair Game?
Juan and Sue were waiting for their friends when they found a number cube under the park bench. Being competitive, they decided to play a game of chance. The loser would buy the winner an ice cream cone.
The
rules of the game were:
a) Sue would get a point each time the die
showed a
number higher than 4.
b) Juan would get a point each
time an even number
was
tossed.
c) The winner would be the
person who obtained ten
points
first.
When
the score was seven to one, Sue grabbed the die and refused to continue
playing, stating the game was simply not fair. Juan insisted the game was fair,
and that she was just a poor loser. Which person was right and why?
Use
what you learned about theoretically probability to mathematically show whether
the game is fair or unfair. Also, write
a short paragraph explaining your answer.
DATA SHEET
COIN TOSS
Name ____________________________________________
DIRECTIONS: Toss the penny 10 times. Record H for heads, T for tails.
