Download MAFS.912.S-IC.1 - Understand and evaluate random processes

Standard 1 : Understand and evaluate random
processes underlying statistical experiments
This document was generated on CPALMS - www.cpalms.org
Algebra 2 - Supporting Cluster
Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity
to enhance the major work of the grade with the supporting clusters.
Number: MAFS.912.S-IC.1
Title: Understand and evaluate random processes
underlying statistical experiments
Type: Cluster
Subject: Mathematics
Grade: 912
Domain: Statistics & Probability: Making Inferences &
Justifying Conclusions
Related Standards
Code
MAFS.912.S-IC.1.1
MAFS.912.S-IC.1.2
Description
Understand statistics as a process for making inferences about population parameters based on a random sample
from that population. ★
Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation. For
example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to
question the model?
Related Resources
Virtual Manipulative
Name
Advanced Fire Simulator Shodor:
Description
In this online activity, students burn a simulated forest and adjust the probability that the fire spreads from one tree
to the other. This simulation also records data for each trial including the burn probability, where the fire started, the
percent of trees burned, and how long the fire lasted. This activity allows students to explore the idea of chaos in a
simulation of a realistic scenario. Supplemental materials, including background information about the topics covered, a
description of how to use the application, and exploration questions for use with the java applet are linked to the
applet.
Interactive Marbles:
This online manipulative allows the student to simulate placing marbles into a bag and finding the probability of pulling
out certain combinations of marbles. This allows exploration of probabilities of multiple events as well as probability with
and without replacement. The tabs above the applet provide access to supplemental materials, including background
information about the topics covered, a description of how to use the application, and exploration questions for use
with the Java applet.
Simple Monty Hall:
In this activity, students select one of three doors in an attempt to find a prize that is hidden behind one of them.
After their first selection, one of the doors that doesn't have the prize behind it is revealed and the student has to
decide whether to switch to the one remaining door or stay on the door of their first choice. This situation, referred to
as the Monty Hall problem, was made famous on the show "Let's Make A Deal" with host Monty Hall. This activity
allows students to explore the idea of conditional probability as well as unexpected probability. This activity includes
supplemental materials, including background information about the topics covered, a description of how to use the
application, and exploration questions for use with the java applet.
Spinner:
In this activity, students adjust how many sections there are on a fair spinner then run simulated trials on that
spinner as a way to develop concepts of probability. A table next to the spinner displays the theoretical probability for
each color section of the spinner and records the experimental probability from the spinning trials. This activity allows
students to explore the topics of experimental and theoretical probability by seeing them displayed side by side for the
spinner they have created. This activity includes supplemental materials, including background information about the
topics covered, a description of how to use the application, and exploration questions for use with the java applet.
page 1 of 5 Perspectives Video: Expert
Name
Birdsong Series: Statistical
Analysis of Birdsong:
Birdsong Series:
Mathematically Modeling
Birdsong:
Description
Wei Wu discusses his statistical contributions to the Birdsong project which help to quantify the differences in the
changes of the zebra finch's song.
Richard Bertram discusses his mathematical modeling contribution to the Birdsong project that helps the progress of
neuron and ion channel research.
Jens Foell discusses how statistical noise reduction is used in fMRI brain imaging to be able to determine which
fMRI, Phantom Limb Pain and specifics parts of the brain are related to certain activities and how this relates to patients that suffer from phantom
Statistical Noise:
limb pain.
History of Probability and the What was the first question that started probability theory?
Problem of Points:
How Math Models Help
Hurricanes can hit at any time! How do insurance companies use math and weather data to help to restore the
Insurance Companies After a community?
Hurricane Hits:
COAPS oceanographer Steve Morey describes how math is used to help research hurricanes and strong deep ocean
Hurricanes & Strong Deep
currents that could effect deep water oil rigs.
Ocean Currents:
Mathematically Exploring the
Wakulla Caves:
The tide is high! How can we statistically prove there is a relationship between the tides on the Gulf Coast and in a
fresh water spring 20 miles from each other?
MicroGravity Sensors &
Statistics:
Statistical analysis played an essential role in using microgravity sensors to determine location of caves in Wakulla
County.
Probabilistic Weather Modeling: Meteorologist from Risk Management discusses the use of probability in predicting hurricane tracks.
Statistical Inferences and
Confidence Intervals :
Florida State University Counseling Psychologist discusses how he uses confidence intervals to make inferences on
college students' experiences on campus based on a sample of students.
Lesson Plan
Name
Birthday Paradox:
CollegeReview.com:
Hot Coffee Coming Through:
Description
Students explore the basic concepts of probability in this activity that demonstrates the Birthday Paradox. Students will
use a graphing calculator to run a Monte Carlo simulation and perform a graphical analysis of the paradox.
This is a model-eliciting activity where students have been asked by a new website, CollegeReview.com, to come up
with a system to rank various colleges based on five categories; tuition cost, social life, athletics, education, city
population and starting salary upon graduation.
In this lesson, students will explore data collection using the temperature probe sensor and perform statistical analysis
of the data. Students will use a scientific method of inquiry to plan an investigation to determine which coffee mug is
the best. This activity is meant to allow students to use a variety of skills they have acquired throughout a statistics
unit in a problem based STEM challenge. Due to the multiple skills there are many standards that are covered.
There are two options for this lab. The first student handout is for students at an average high school statistics level
(Algebra 1) and will allow for standard deviation and graphical analyses of the data. The second option is for
advanced students that have been exposed to hypothesis testing of claims (Algebra 2 or AP Stats).
This lesson shows students how to produce a survey and display their results. The lesson takes the students
through:
Inferences:
1.
2.
3.
4.
5.
What is a statistical question?
General population verses sample population.
What is a hypothesis?
What is a survey?
How to make inferences.
This lesson is intended to help you assess how well students are able to:
Interpret data and evaluate statistical summaries.
Interpreting Statistics: A Case
Critique someone else's interpretations of data and evaluations of statistical summaries.
of Muddying the Waters:
The lesson also introduces students to the dangers of misapplying simple statistics in real-world contexts, and illustrates
some of the common abuses of statistics and charts found in the media.
This lesson is designed to teach the implementation of random sampling that leads to inference to a larger group or
Is It a Guess or Statistics?:
population.
Sampling Lionfish:
In this lesson, students will develop a sampling method to make inferences about the invasive lionfish in the Atlantic
Ocean. Students will carry out their investigation, create histograms, and calculate quantitative data like standard
deviation to help make conjectures about the lionfish. Students will then analyze their sampling methodology by
repeating the procedure with the population data. This investigation allows students the opportunity not only to
simulate and improve their own methodologies but provides a current and real-life scientific issue to be examined.
Sensoring Data:
In this follow up lesson, students will explore data collection using the weather station sensor and perform statistical
analysis of the data. Students will use a scientific method of inquiry to plan an investigation of their own. This activity
is meant to allow students to use a variety of skills they have acquired throughout a statistics unit in a personally
meaningful way.
In this follow up lesson, students will explore data collection using the weather station sensor and perform statistical
page 2 of 5 Sensoring Data:
analysis of the data. Students will use a scientific method of inquiry to plan an investigation of their own. This activity
is meant to allow students to use a variety of skills they have acquired throughout a statistics unit in a personally
meaningful way.
This lesson is designed to develop students' ability to create tree diagrams and figure probabilities of events based on
those diagrams. This lesson provides links to discussions and activities related to tree diagrams as well as suggested
Tree Diagrams and Probability:
ways to work them into the lesson. Finally, the lesson provides links to follow-up lessons designed for use in succession
with the current one.
Problem-Solving Task
Name
Block Scheduling:
Description
In this task, output is given from a computer-generated simulation, generating size-100 samples of data from an
assumed school population of 2000 students under hypotheses about the true distribution of yes/no voters. Students
are asked to draw conclusions about reliability using simulated distributions.
Musical Preferences:
This problem solving task asks students to make deductions about what kind of music students like by examining a
table with data.
Sarah, the Chimpanzee:
The purpose of this task is to give students experience in using simulation to determine if observed results are
consistent with a given model (in this case, the “just guessing” model).
School Advisory Panel:
Students are asked to choose the best sampling method for choosing the new School Advisory Panel.
Strict Parents:
This task challenges students to describe parameter of interest for the given context, and design a sample survey.
Unexpected Answers:
Why Randomize?:
This lesson is designed to introduce students to statistical situations where the probabilities or outcomes might not be
what is first expected. The lesson provides links to discussions and activities motivated by the idea of unexpected
answers. Finally, the lesson provides links to follow-up lessons designed for use in succession with an introduction to
probability and unexpected answers in probability.
This task requires students to estimate the mean (average) area of the population of 100 rectangles using the
average area of a sample of 5 rectangles. Students are asked to make one estimate using a judgement sample and
another using a random sample of the population. Finally, students are asked to consider bias in sampling methods.
Perspectives Video: Professional/Enthusiast
Name
Camera versus Trap
Sampling: Improving how
NOAA Samples Fish :
Description
Underwater sampling with cameras has made fishery management more accurate for NOAA scientists.
NOAA Scientist Doug Devries discusses the differences between fishery independent surveys and fishery independent
Fishery Independent vs
Dependent Sampling Methods surveys. Discussion includes trap sampling as well as camera sampling. Using graphs to show changes in population
of red snapper.
for Fishery Management:
Hurricane Dennis & Failed
Math Models:
What happens when math models go wrong in forecasting hurricanes?
Hydrologic Modeling and
Reservoir Simulation:
Dr. Tom Van Lent and Rajendra Paudel describe how modeling and simulation of water reservoirs are used to inform
decisions about regulation of water flow in the Everglades.
Making Inferences about
Wetland Population Sizes:
This ecologist from the Coastal Plains Institute discusses sampling techniques that are used to gather data to make
statistical inferences about amphibian populations in the wetlands of the Apalachicola National Forest.
Modeling the Everglades with Dr. Tom Van Lent and Rajendra Paudel describe how hydrologic modeling is used to evaluate environmental
conditions in the Everglades.
Mathematics:
Population Sampling and
Beekeeping:
This buzzworthy video features statistics, sampling, and how scientists make inferences about populations.
Teaching Idea
Name
Conditional Probability and
Probability of Simultaneous
Events:
The Birthday Paradox:
Description
This lesson is designed to further students' practice with probability as well as introduce them to conditional probability
and probabilities of simultaneous independent events. The lesson provides links to discussions and activities related to
conditional and simultaneous probabilities as well as suggested ways to integrate them into the lesson. Finally, this
lesson provides links to follow-up lessons designed for use in succession with this one.
The exercise detailed in this resource poses a paradox regarding birthdays. The question raised: how large does a
group have to be in order to have a 50% or better probability that two or more people in the group share the same
birthday? Intuitions about probability are challenged as students use their calculators to simulate random values to
represent the birthdays and reach an answer.
Text Resource
Name
How to Win at Rock-PaperScissors:
Description
This informational text resource is intended to support reading in the content area. This article describes a new study
about the game rock-paper-scissors. The study reveals that people do not play randomly; there are patterns and
hidden psychology players frequently use. Understanding these potential moves can help a player increase their winning
edge. As part of interpreting the results of the study, the article references the Nash equilibrium and the "win-stay loseshift" strategy.
This informational text resource is intended to support reading in the content area. This article describes the
important process used when setting up trials for statistical investigation. The article explains each parameter that is
page 3 of 5 Sample Size Calculation:
needed to calculate the sample size, then provides examples and illustrates the process. This article will enhance an
upper level math course's study of statistics after significance levels and basic inferential statistics concepts have been
taught.
Scientists See the World
Differently:
This informational text resource is intended to support reading in the content area. Pew Research Center surveyed
scientists and the general public on 12 science oriented issues, including genetically modified foods, vaccines, nuclear
power and evolution. Results of the survey showed large discrepancies between the thoughts, causes and
recommendations on the issues of the scientists and the general public. Sample sizes and margins of errors are given
on the survey results which are represented in percent form. The overall survey showed that the public and the
scientists see the world very differently.
Understanding Uncertainty:
What Was the Probability of
Obama Winning?:
This informational text resource is intended to support reading in the content area. The article examines various
factors that changed the uncertainty of whether Barack Obama would win the 2008 election. Specifically,the article
discusses probability, the science of quantifying uncertainty. The article questions common methods for assessing
probability where symmetrical outcomes are assumed. Finally, the author explains how to use past evidence to
assess the chances of future events.
Video/Audio/Animation
Name
MIT BLOSSOMS - Flu Math
Games:
Description
This video lesson shows students that math can play a role in understanding how an infectious disease spreads and
how it can be controlled. During this lesson, students will see and use both deterministic and probabilistic models and will
learn by doing through role-playing exercises. There are no formal prerequisites, as students in any high school or even
middle school math class could enjoy this learning video. But more advanced classes can go into the optional applied
probability modeling that accompanies the module in a downloadable pdf file. The primary exercises between video
segments of this lesson are class-intensive simulation games in which members of the class 'infect' each other under
alternative math modeling assumptions about disease progression. Also there is an occasional class discussion and local
discussion with nearby classmates.
Student Center Activity
Name
New York Balance:
Description
In this lesson, students will create a mathematical model that quantitatively explains the relationship between varying
mass and distance from the pivot point on a balanced lever. Students will make modifications to their models as they
use it to predict the results of various scenarios. This activity is an excellent springboard into many subjects and is
appropriate for use in grades 6 – 12. Worksheet
Name
Description
This lesson is designed to develop students' understanding of sampling with and without replacement and its effects on
the probability of drawing a desired object. The lesson provides links to discussions and activities related to replacement
Replacement and Probability:
and probability as well as suggested ways to work them into the lesson. Finally, the lesson provides links to follow-up
lessons that are designed to be used in succession with the current one.
Assessment
Name
Sample 4 - High School
Algebra 2 State Interim
Assessment:
Description
This is a State Interim Assessment for 9th-12th grade.
Student Resources
Title
Advanced Fire Simulator Shodor:
Description
In this online activity, students burn a simulated forest and adjust the probability that the fire spreads from one tree
to the other. This simulation also records data for each trial including the burn probability, where the fire started, the
percent of trees burned, and how long the fire lasted. This activity allows students to explore the idea of chaos in a
simulation of a realistic scenario. Supplemental materials, including background information about the topics covered, a
description of how to use the application, and exploration questions for use with the java applet are linked to the
applet.
Interactive Marbles:
This online manipulative allows the student to simulate placing marbles into a bag and finding the probability of pulling
out certain combinations of marbles. This allows exploration of probabilities of multiple events as well as probability with
and without replacement. The tabs above the applet provide access to supplemental materials, including background
information about the topics covered, a description of how to use the application, and exploration questions for use
with the Java applet.
Mathematically Exploring the
Wakulla Caves:
The tide is high! How can we statistically prove there is a relationship between the tides on the Gulf Coast and in a
fresh water spring 20 miles from each other?
MicroGravity Sensors &
Statistics:
Statistical analysis played an essential role in using microgravity sensors to determine location of caves in Wakulla
County.
Musical Preferences:
This problem solving task asks students to make deductions about what kind of music students like by examining a
table with data.
page 4 of 5 Population Sampling and
Beekeeping:
This buzzworthy video features statistics, sampling, and how scientists make inferences about populations.
School Advisory Panel:
Students are asked to choose the best sampling method for choosing the new School Advisory Panel.
Simple Monty Hall:
In this activity, students select one of three doors in an attempt to find a prize that is hidden behind one of them.
After their first selection, one of the doors that doesn't have the prize behind it is revealed and the student has to
decide whether to switch to the one remaining door or stay on the door of their first choice. This situation, referred to
as the Monty Hall problem, was made famous on the show "Let's Make A Deal" with host Monty Hall. This activity
allows students to explore the idea of conditional probability as well as unexpected probability. This activity includes
supplemental materials, including background information about the topics covered, a description of how to use the
application, and exploration questions for use with the java applet.
Spinner:
In this activity, students adjust how many sections there are on a fair spinner then run simulated trials on that
spinner as a way to develop concepts of probability. A table next to the spinner displays the theoretical probability for
each color section of the spinner and records the experimental probability from the spinning trials. This activity allows
students to explore the topics of experimental and theoretical probability by seeing them displayed side by side for the
spinner they have created. This activity includes supplemental materials, including background information about the
topics covered, a description of how to use the application, and exploration questions for use with the java applet.
Parent Resources
Title
Musical Preferences:
Description
This problem solving task asks students to make deductions about what kind of music students like by examining a
table with data.
Population Sampling and
Beekeeping:
This buzzworthy video features statistics, sampling, and how scientists make inferences about populations.
School Advisory Panel:
Students are asked to choose the best sampling method for choosing the new School Advisory Panel.
page 5 of 5