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Unit 12 – Table of Contents
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Unit 12: Extensions and Applications
Video Overview
Learning Objectives
12.2
Media Run Times
12.3
Instructor Notes
12.4
• The Mathematics of Logic and Probability
• Teaching Tips: Conceptual Challenges and Approaches
• Teaching Tips: Algorithmic Challenges and Approaches
Instructor Overview
• Tutor Simulation: Probability Game Design
12.8
Instructor Overview
• Puzzle: It’s Bargain Time
12.9
Instructor Overview
• Project: What are the Chances of That?
12.11
Glossary
12.18
Common Core Standards
12.20
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Monterey Institute for Technology and Education 2011 V1.1
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Unit 12 – Learning Objectives
Unit 12: Extensions and Applications
Unit Table of Contents
Lesson 1: Logical Reasoning and Number Sets
Topic 1: Number Sets
Learning Objectives
• Identify and define counting, natural, whole, integer, rational,
irrational, and real numbers.
Topic 2: Understanding Logical Statements
Learning Objectives
• Identify the hypothesis and conclusion in a logical statement.
• Determine whether mathematical statements involving linear,
quadratic, absolute value expressions, equations, or inequalities
are always, sometimes, or never true.
• Use counter examples to show that a statement is false, and
recognize that a single counter example is sufficient.
Topic 3: Inductive Reasoning
Learning Objectives
• Identity and provide examples of inductive reasoning.
Topic 4: Deductive Reasoning
Learning Objectives
• Use properties of numbers to construct simple logical arguments.
• Identify and provide examples of deductive reasoning.
Lesson 2: Probability
Topic 1: Events and Outcomes (Counting)
Learning Objectives
• Use the Fundamental Counting Principle to determine the size of
the sample space for simple and compound events.
Topic 2: Probability of Independent Events
Learning Objectives
• Calculate the probability of independent events.
Topic 3: Permutations and Combinations
Learning Objectives
• Use the Fundamental Counting Principle to compute permutations
and combinations.
Topic 4: Probability of Dependent Events
Learning Objectives
• Calculate the probability of dependent events.
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Unit 12 – Media Run Times
Unit 12
Lesson 1
Topic 1, Presentation – 3.2 minutes
Topic 1, Worked Example 1 – 6.2 minutes
Topic 1, Worked Example 2 – 6.7 minutes
Topic 1, Worked Example 3 – 3.1 minutes
Topic 2, Presentation – 5.2 minutes
Topic 2, Worked Example 1 – 3.8 minutes
Topic 2, Worked Example 2 – 6.7 minutes
Topic 2, Worked Example 3 – 3.1 minutes
Topic 2, Worked Example 4 – 1.8 minutes
Topic 3, Presentation – 4.1 minutes
Topic 3, Worked Example 1 – 2.1 minutes
Topic 3, Worked Example 2 – 2.3 minutes
Topic 3, Worked Example 3 – 3 minutes
Topic 4, Presentation – 4 minutes
Topic 4, Worked Example 1 – 1.9 minutes
Topic 4, Worked Example 2 – 3.9 minutes
Topic 4, Worked Example 3 – 4.1 minutes
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Unit 12 – Instructor Notes
Unit 12: Extensions and Applications
Instructor Notes
The Mathematics of Logic and Probability
Unit 12 concludes this course by expanding students' understanding of the scope of
algebra. The unit builds on mathematical ideas from earlier in the course, such as
numeric principles and counting techniques, and extends them into the more complex
areas of probabilities, permutations, and combinations. It also takes concepts that
students may only know from non-mathematical settings, like reasoning and logic, and
puts them into an algebraic framework.
These ideas will help students appreciate the range and application of algebra. They are
also concepts likely to come up in later math courses, so picking up some familiarity with
them here could help students later on.
Teaching Tips: Conceptual Challenges and Approaches
Notions of logic and reasoning contain subtleties that can perplex many students.
Algebra students are used to solving problems and finding numerical answers that are
either right or wrong. In this unit they're asked to find hypotheses and conclusions and
exceptions instead, and that can be confusing. They'll need very careful explanations
and lots of examples of inductive versus deductive reasoning and
always/sometimes/never true statements.
An effective way of teaching students to apply these ideas to algebra is to begin by
introducing them in a non-mathematical context. Use simple stories to define the
meaning and use of terms like logical statements, events, and counterexamples. Then,
when students are comfortable with those concepts, apply them to mathematical
problems.
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Example
Let's consider counterexamples. Here's an image from the presentation for Lesson 1,
Topic 2, which explores the logic statement “If I jump into the lake, I will get wet":
We see that for several cases this statement is true. However, a counterexample is
introduced where someone enters the lake inside a waterproof bubble:
In this case, the statement “If I jump into the lake, I will get wet” is NOT true, and it is
proven not true by this single counterexample.
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Once students understand the meaning and implications of a counterexample, help them
apply this knowledge to a mathematical situation. For example, give them the statement
“|x| > 0 for all real numbers x.” Work with them to turn this into the if-then statement “If x
is a real number, then |x| > 0.” Then ask whether this statement is always true,
sometimes true, or never true and let them discuss the problem. If needed, guide them
towards the counterexample of “x = 0." Be sure students understand that to prove a
statement is false, they need only one counterexample.
Formalized ideas of logic and reasoning, especially in mathematics, will be new ideas for
most students, and they will need time to build their own context and understanding.
Once you introduce these terms, give students statements to analyze and provide a lot
of time for discussion in small groups. This is a very powerful way of helping them test
and strengthen their ideas.
Teaching Tips: Algorithmic Challenges and Approaches
Students are likely to have worked with probability in previous grades, probably since
upper elementary school, but the second half of Unit 12 extends and formalizes these
concepts and ideas. Their biggest hurdle is likely to be procedural—while they
understand what probabilities are, it's often a long road between reading a story problem
and calculating the correct answer. They need to determine whether events are simple
or compound, dependent or independent, permutations or combinations. Once that's
sorted out, they have to choose between using the fundamental counting principal or
factorial formulas, and whether to calculate event probabilities separately or as a single
sequence. Only after all that do they start plugging in numbers.
Students will have an easier time if you help them develop a plan for approaching and
organizing probability problems. Begin with the use of visual aids like sketches and
tables for the simplest cases. As problems become more complex, be sure to follow a
step-by-step procedure that emphasizes the importance of breaking down a situation
into individual events, identifying their nature, and defining event and sample spaces
before doing any calculations. Probability problems lend themselves especially well to
classroom and small group work where students can work together to figure out how to
frame and solve problems.
Example
Here's a sample problem from Lesson 2, Topic 3, which follows the recommended focus
on laying out a problem question by question:
Beth has 10 pairs of socks: 2 black, 2 brown, 3 white, 1 red, 1 blue, and 1 green. Today
she wants a white pair, but she’s in a hurry to get to work, so she grabs a pair randomly
without looking. If it’s not white, she’ll throw it back in the drawer. If she continues to try
grabbing a pair randomly, what’s the probability that she’ll get a white pair on her third
try?
First, define the events. Because we want her to get white on her third try, she must not
get white on the first try or the second try:
Event A: a pair of socks that are not white
Event B: a pair of socks that are not white
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Event C: a pair of socks that is white.
Now check if they’re independent:
Beth removes an outcome when she pulls out a pair of socks, but then she
replaces it if it’s not a white pair, so the probabilities won’t change. The events
are independent, because each removed outcome is replaced. The earlier events
don’t change the probabilities for later events.
We could find the sample and event space for the entire trial and use the ratio. However,
since the events are independent, it’s easier to find the sample and event space for the
individual events and multiply them:
The size of the sample space for each event is 10 (There are 10 pairs of socks to
choose from).
The size of Event A and Event B’s event spaces are both 7. (There are 7 pairs
that are not white.)
The size of Event C’s event space is 3. (There are 3 pairs that are white.)
Calculate the answer:
Summary
This unit covers mathematical logic and reasoning as well as the application of counting
procedures to probability problems. These two topics may seem unrelated, but they do
share some important features. Both ask students to step back from numbers and
calculations to consider context and meaning. Both include concepts that will be
fundamental in many upcoming mathematics classes. The opportunity to discuss and
test their understanding of these ideas will help students form a strong foundation they
can carry into later courses.
This unit has both conceptual and algorithmic challenges. These are best overcome by
teaching students how to frame problems carefully, making sure they are comfortable
with the meaning of the terms used, and encouraging them to work in groups to debate
and solidify ideas and techniques.
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Unit 12 – Tutor Simulation
Unit 12: Extensions and Applications
Instructor Overview
Tutor Simulation: Probability Game Design
Purpose
This simulation is designed to challenge a student’s understanding of probabilities.
Students will be asked to apply what they have learned to solve a real world problem by
demonstrating understanding of the following areas:
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Events and Outcomes
Probabilities
Sample Space
Problem
Students are given the following problem:
Some friends are creating a game that uses two spinners instead of dice to decide the
next move.
They've already made the spinners, but they need to know the probabilities of different
outcomes to help them make the game better.
It's up to you to help them figure out the odds.
Recommendations
Tutor simulations are designed to give students a chance to assess their understanding
of unit material in a personal, risk-free situation. Before directing students to the
simulation,
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make sure they have completed all other unit material.
explain the mechanics of tutor simulations
o Students will be given a problem and then guided through its solution by a
video tutor;
o After each answer is chosen, students should wait for tutor feedback
before continuing;
o After the simulation is completed, students will be given an assessment of
their efforts. If areas of concern are found, the students should review unit
materials or seek help from their instructor.
emphasize that this is an exploration, not an exam.
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Unit 12 – Puzzle
Unit 12: Extensions and Applications
Instructor Overview
Puzzle: It's Bargain Time
Objective
It's Bargain Time is a probability puzzle that will challenge a student's ideas about
games of chance and odds. The game is based on the "Monty Hall Paradox," in which
players are asked to pick among several doors to find a prize. Then, after one or more of
the doors is shown to be wrong, they're given the option to switch their pick. Because
most players won't realize that opening one door changes the probabilities for the other
doors, most players will find that the best strategy is counterintuitive. This puzzle will
encourage students to think very carefully about probability and dependent events. It
may even pique the curiosity of some learners enough to lead them to consider
mathematics a suitable activity for their imagination.
Figure 1. Learners are drawn into the paradox with a simple choice.
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Figure 2. After playing the game for a while, players can ask for an explanation of the paradox,
and see if it holds true in a more complicated game.
Description
This puzzle begins with a 3-door scenario. Players are invited to pick one of the doors in
hopes of revealing a prize. One of the other doors then opens to reveal a goat, and
players are asked whether they want to stick with their original choice, or door, or switch
to the last closed door. Finally, all the doors open to show the winning prize, and a new
set of doors appears.
It's Bargain Time keeps a running total of the winning percentages of both "Sticking" and
"Switching." Eventually, it becomes clear that switching is the more successful strategy
over time. After several rounds, students can ask for an explanation of this paradox, and
also switch to a 6-door version of the game. At each level, they can play as many times
as they wish.
This game can be played alone, but it is especially suited for a classroom setting
because a number of students will be insistent that the solution is simply not the
solution. Group play and debate can lead to a more thorough exploration of probability
than students would get on their own.
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Unit 12 – Project
Unit 12: Extensions and Applications
Instructor Overview
Project: What Are The Chances of That?
Student Instructions
Introduction
Probability is the unseen factor behind the hundreds of decisions you make every day.
In daily decision-making you use experimental probability without even realizing it.
Based on your past experiences, you calculate whether taking a certain action is worth
the risk, or in algebraic terms, what the chance is of having a favorable outcome. You
use theoretical probability when you either haven’t experienced a certain situation before
or you are looking for the likelihood of something occurring in a perfect world.
When calculating probability in your mind, most times you don’t even think of it as math;
still, it’s there just the same. Probability asks you, “What are the chances of that?”
Task
Working together with your group, you will first design a probability experiment. You will
want to design something that will be fun to test and interesting to watch on video. Your
end product will be a video made on either Microsoft Movie Maker or iMovie.
Experiments that require some degree of silliness can be fun to perform and to watch.
Consider experimenting both on your classmates and your teachers. Be sure to also
consider the mathematics behind your experiment, as you will be required to calculate
the theoretical probability.
Instructions
Complete each problem in order, keeping detailed notes along the way. Don’t forget to
capture video of the action.
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First problem: Design Experiment
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You have encountered probability questions about pulling marbles
from a bag, selecting cards (with and without replacement), and
rolling die. These are all situations that you may want to include in
your experiment, but they are certainly not the only materials that
can be used. Think creatively! What fun materials could you use
to perform a probability experiment? Make a list and discuss the
materials with your group.
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Choose one or two materials from the list and begin discussing
the sample spaces that would occur with the given materials. For
instance, if I had a red, blue and green tile in a bag, and I wanted
to select two tiles, then the sample space would be:
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RG
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BG
Once you have defined your sample space, discuss what outcome
you would like to choose as your desired outcome for the
experiment. If my desired outcome is choosing a red tile and then
a green tile, the theoretical probability would be
probability would be written P (red, then green).
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Hint: If you choose two materials to work with, you will also need
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to calculate the probability of the second
event.
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Write down a list of necessary materials, the sample space, and
your desired outcome for the experiment. Make sure to get
approval from your teacher before moving on to the next step.
Second problem: Calculate Theoretical Probability
1.
Now calculate the theoretical probability of the desired outcome of
your experiment. Remember to account for replacement and nonreplacement and multiply the probabilities of compound events.
Show your work carefully as you will display the theoretical
probability in your finished movie.
Hint: If I wanted to add flipping a coin to my experiment with the
colored tiles in the bag, I would need to determine if I was looking
for a heads or a tails. Let’s say I wanted heads. The probability
would then be written as follows:
P (red, then green, heads) =
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1 1 1
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6 2 12
Third problem: Gather Experimental Data
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Now that your theoretical
probability has been calculated, you
should know how many times your experiment would need to be
performed in order to achieve your desired outcome. In the case
of the tiles and the coin, if everything works perfectly, I should get
the outcome once every twelve times. Knowing that not
everything is perfect, it will be important to perform the experiment
more than twelve times. In fact, you will want to perform your
experiment 50 times. Fifty will allow you to collect a substantial
amount of data and it is an easy number to work with when
tabulating your data.
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Create a data table to record the outcome of each of your 50
experiments. Once your table is completed, begin experimenting.
Don’t forget to capture video clips of each of your subjects
performing the experiment.
Fourth problem: Compare
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Sort your data and compare each experimental outcome to the
theoretical outcome. You will need to create a data table to
compare the probabilities. A sample data table follows:
Experimental Results
RG heads 4/50 = 8%
RB heads 5/50 = 10%
BG heads 6/50 = 12%
BR heads 3/50 = 6%
GR heads 4/50 = 8%
GB heads 2/50 = 4%
Experimental Results
RG tails 2/50 = 4%
RB tails 4/50 = 8%
BG tails 6/50 = 12%
BR tails 5/50 = 10%
GR tails 1/50 = 2%
GB tails 8/50 = 16%
2.
According to the table above, my desired outcome of drawing a
red marble, followed by a green marble, and then getting a heads
on the coin flip, was only achieved 4 out of 50 times. This gave a
percentage of 8%. The theoretical probability for that outcome is
1 out of 12 or approximately 8.3%. In this case, the experimental
probability and theoretical probability were very close. This is not
always the case.
3.
Compare your experimental probability to the theoretical
probability. Were some of the other outcomes substantially lower
or higher than you expected?
4.
If you were to perform the experiment over again, what changes
might you make to have your experimental data more closely
mirror the theoretical probability?
Conclusion
For the conclusion of the project, you will make a video on Movie Maker or iMovie. Use
the rubric as a guide to what should be included in your video. A successful project will
mix the mathematics with video clips of students and teachers performing the
experiment. Don’t forget to clearly define what you are studying at the beginning of the
video and wrap up the video with a data table and discussion of your results.
The following are basic directions for Movie Maker:
Capturing Video from Digital Video Camera
1. Open Windows Movie Maker
2. Attach the camera to the computer using a firewire or USB
cable.
3. Choose “capture from video device”
Making a Movie with Movie Maker
On the bottom you can view your work as a “storyboard” or as a
“timeline”. The “storyboard” view is easy to use for adding the
titles, pictures, videos and transitions. When you want to add
audio, music or narration, you must be in “timeline” view.
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Playing the video clips
1. Notice the “split” button on the right. You can split
video clips to take off unnecessary sections of your video.
2. Once you have the video clips the way you want them, you can
drag them down onto the timeline or storyboard view.
Adding a Title
1. Under Tools, Titles and Credit, choose “title at the beginning”
2. Enter the text of the title
3. Now, on the same page where you entered the text for the title,
click the link “Change the title animation”
4. Click “Change the text, font and color” link
5. Finally choose “Done, add title to movie”
Adding music to your video
1. Now in Movie Maker – choose “Import audio or music” (on the
left task bar)
2. Go to the “my music” folder and choose the music you want
3. Now you can drag your music to the correct place on the
timeline (again you can shorten the music – by resizing it from the
right end – if the song is too long)
Now you are ready to make the movie
1. Get everything just how you want it in Movie Maker
2. Choose File “Save Move File”
Additional help on Movie Maker can be found at:
http://www.microsoft.com/windowsxp/using/moviemaker/default.m
spx
http://www.saskschools.ca/resources/techref/moviemaker/movie.h
tml
Help for iMovie can be found at:
http://www.bgsu.edu/downloads/cio/file17748.pdf
http://www.apple.com/ilife/tutorials/ - imovie
Hint: One of the best ways to learn the programs is to play around
with the features. Work together and help each other out. If you
know how to do transitions, help a neighboring group. If they
know how to import music, they can then help you.
Collaboration
Show your project-in-the-making to a neighboring group. Allow the neighboring group to
give constructive feedback about changes that could be made before finalizing the
product. Then watch the other group’s project and give feedback. The goal is to help
enhance the other group’s project before it is time to turn it in. Sometimes just seeing
another group’s product can give you ideas to enhance your own project.
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Instructor Notes
Assignment Procedures
Problem 1
It would be helpful to monitor group progress during the design stage to ensure that they
are developing an experiment in which they can obtain a favorable outcome in 50 trials.
Overly complex experiments will either require additional trials or run the risk of having
an experimental probability of zero.
Problem 2
Consider building in an opportunity for the groups to check each other’s theoretical
probabilities. It will ensure more correct calculations and give the students more
practice.
Problem 3
With administrator permission, consider allowing students time before and after school to
experiment on teachers. Students enjoy seeing the final product, especially if their
teachers are included in the fun.
Recommendations:
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assign a specific timeline for completion of the project that includes milestone
dates.
provide students feedback as they complete each milestone.
ensure that each member of student groups has a specific job.
Technology Integration
This project provides abundant opportunities for technology integration, and gives
students the chance to research and collaborate using online technology.
The student instructions suggest using MovieMaker or iMovie to create a video
describing their experiment.
The following are other examples of free internet resources that can be used to support
this project:
http://www.moodle.org
An Open Source Course Management System (CMS), also known as a Learning
Management System (LMS) or a Virtual Learning Environment (VLE). Moodle has
become very popular among educators around the world as a tool for creating online
dynamic websites for their students.
http://www.wikispaces.com/site/for/teachers or http://pbworks.com/content/edu+overview
Lets you create a secure online Wiki workspace in about 60 seconds. Encourage
classroom participation with interactive Wiki pages that students can view and edit from
any computer. Share class resources and completed student work with parents.
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http://www.docs.google.com
Allows students to collaborate in real-time from any computer. Google Docs provides
free access and storage for word processing, spreadsheets, presentations, and surveys.
This is ideal for group projects.
http://why.openoffice.org/
The leading open-source office software suite for word processing, spreadsheets,
presentations, graphics, databases and more. It can read and write files from other
common office software packages like Microsoft Word or Excel and MacWorks. It can be
downloaded and used completely free of charge for any purpose.
Rubric
Score
4
3
2
Content
The desired outcome to be studied
during the project is clearly identified
and stated. Professional looking
and accurate representation of the
data in tables, including labels and
titles is present.
The theoretical probability of the
desired outcome is correctly figured
with all math work shown.
Conclusion includes discussion of
the comparison between
experimental and theoretical
probabilities.
The desired outcome to be studied
during the project is identified, but is
stated in an unclear manner.
Accurate representation of the data
in tables, including labels and titles
is present.
The theoretical probability of the
desired outcome is figured with all
math work shown. Minor errors may
be noted. Conclusion compares
experimental and theoretical
probabilities.
The desired outcome to be studied
during the project is partially
identified, and/or is stated in an
unclear manner. Accurate
representation of the data is in
written form, but no tables are
presented.
The theoretical probability of the
desired outcome is given, but no
work is shown. Major errors may
also be noted. Conclusion
compares experimental and
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Presentation
Your project contains information presented in a
logical and interesting sequence that is easy to
follow. Makes excellent use of font, color,
graphics, effects, etc. to enhance the presentation.
Many different "takes", camera angles, sound
effects, and/or careful of use of zoom provides
variety in the video. Product shows a large amount
of original thought. Ideas are creative and
inventive.
Your project contains information presented in a
logical sequence that is easy to follow. Makes
good use of font, color, graphics, effects, etc. to
enhance the presentation.
Several (3-4) different "takes", camera angles,
sound effects, and/or careful of use of zoom
provides variety in the video.
Product shows some original thought. Work shows
new ideas and insights.
Your project is hard to follow because the material
is presented in a manner that jumps around
between unconnected topics. Makes use of font,
color, graphics, effects, etc. but occasionally these
detract from the presentation content.
One or two different "takes", camera angles, sound
effects, and/or careful of use of zoom provides
variety in the video. It uses other people's ideas
(giving them credit), but there is little evidence of
original thinking.
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theoretical probabilities.
The desired outcome to be studied
during the project is erroneous or
irrelevant. Data tables are not
shown or are inaccurate.
Your project is difficult to understand because
there is no sequence of information. Use of font,
color, graphics, effects etc. is evident but these
often distract from the presentation content.
The theoretical probability is missing
or major errors are noted. Does not
compare experimental and
theoretical probabilities.
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Little attempt was made to provide variety in the
video. Uses other people's ideas, but does not
give them credit.
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Unit 12 – Glossary
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Unit 12: Extensions and Applications
Glossary
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combinations
groupings in which the order of members does not matter
compound event
an event with more than one outcome
conclusion
the part of a logical statement that provides the result or
consequences of the hypothesis—In a statement “If x then y”, the
conclusion is y.
conjecture
a statement that attempts to make a conclusion but has not been
proved true or false
counterexample
a situation that provides evidence that a logical statement is false
counting numbers
also called natural numbers, the numbers 1, 2, 3, 4, ...
deductive reasoning
a form of logical thinking that uses generalizations to draw specific
conclusions based on a series of logical steps, deductive reasoning
may use rules, laws, and theories to support or justify a conjecture
dependent events
two or more events for which the occurrence of one affects the
probability of the other(s)
equally likely
having the same likelihood of occurring, such that in a large
number of trials, two equally likely outcomes would happen roughly
the same number of times
event
a collection of possible outcomes, often describable using a
common characteristic, such as rolling an even number with a die
or picking a card from a specific suit
event space
the set of possible outcomes in an event: for example, the event
“rolling an even number” on a die has the event space of 2, 4, and
6
example
a situation that suggests a logical statement may be true
factorial
an abbreviated way of writing a product of all whole numbers from
1 to a given number, indicated by that number followed by an
exclamation point, as in 3! = 3 • 2 • 1
Fundamental Counting
Principle
a way to find the number of outcomes in a sample space by finding
the product of the number of outcomes for each element
generalize
the process of using observations of specific events to make
statements or conjectures about more general situations
hypothesis
the part of a logical statement that provides the premise on which
the conclusion is based—In a statement “If x then y,” the
hypothesis is x.
independent events
two or more events for which the occurrence of one does not affect
the probability of the other(s)
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inductive reasoning
a form of logical thinking that makes general conclusions based on
specific situations, inductive reasoning takes the path of
observation to generalization to conjecture
integers
the numbers !, -3, -2, -1, 0, 1, 2, 3,!
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Unit 12 – Common Core
NROC Algebra 1--An Open Course
Unit 12: Extensions and Applications
Mapped to Common Core State Standards, Mathematics
Unit 12, Lesson 1, Topic 1: Number Sets
Grade: 7 - Adopted 2010
STRAND / DOMAIN
CC.7.NS.
CATEGORY / CLUSTER
STANDARD
7.NS.2.
EXPECTATION
7.NS.2.d.
The Number System
Apply and extend previous
understandings of operations
with fractions to add,
subtract, multiply, and divide
rational numbers.
Apply and extend previous
understandings of
multiplication and division and
of fractions to multiply and
divide rational numbers.
Convert a rational number to
a decimal using long division;
know that the decimal form of
a rational number terminates
in 0s or eventually repeats.
Grade: 8 - Adopted 2010
STRAND / DOMAIN
CC.8.NS.
CATEGORY / CLUSTER
STANDARD
8.NS.1.
The Number System
Know that there are numbers
that are not rational, and
approximate them by rational
numbers.
Understand informally that
every number has a decimal
expansion; the rational
numbers are those with
decimal expansions that
terminate in 0s or eventually
repeat. Know that other
numbers are called irrational.
Unit 12, Lesson 1, Topic 2: Understanding Logical Statements
No Correlations for this Topic.
Unit 12, Lesson 1, Topic 3: Inductive Reasoning
Grade: 7 - Adopted 2010
STRAND / DOMAIN
CC.7.MP.
Mathematical Practices
CATEGORY / CLUSTER
7.MP.2.
CATEGORY / CLUSTER
7.MP.7.
CATEGORY / CLUSTER
7.MP.8.
Reason abstractly and
quantitatively.
Look for and make use of
structure.
Look for and express
regularity in repeated
reasoning.
Grade: 8 - Adopted 2010
!
STRAND / DOMAIN
CC.8.MP.
Mathematical Practices
CATEGORY / CLUSTER
8.MP.2.
Reason abstractly and
quantitatively.
"#$#,!
!
!
CATEGORY / CLUSTER
8.MP.7.
CATEGORY / CLUSTER
8.MP.8.
Algebra 1—An Open Course
Professional Development
Look for and make use of
structure.
Look for and express
regularity in repeated
reasoning.
Grade: 9-12 - Adopted 2010
STRAND / DOMAIN
CC.MP.
Mathematical Practices
CATEGORY / CLUSTER
MP-2.
CATEGORY / CLUSTER
MP-7.
CATEGORY / CLUSTER
MP-8.
STRAND / DOMAIN
CC.F.
Reason abstractly and
quantitatively.
Look for and make use of
structure.
Look for and express
regularity in repeated
reasoning.
Functions
CATEGORY / CLUSTER
F-IF.
Interpreting Functions
STANDARD
EXPECTATION
F-IF.3.
STRAND / DOMAIN
CC.F.
Understand the concept of a
function and use function
notation.
Recognize that sequences are
functions, sometimes defined
recursively, whose domain is a
subset of the integers. For
example, the Fibonacci
sequence is defined
recursively by f(0) = f(1) = 1,
f(n+1) = f(n) + f(n-1) for n
greater than or equal to 1.
Functions
CATEGORY / CLUSTER
F-BF.
Building Functions
STANDARD
EXPECTATION
F-BF.1.
GRADE EXPECTATION
F-BF.1.a.
STRAND / DOMAIN
CC.F.
Build a function that models a
relationship between two
quantities.
Write a function that
describes a relationship
between two quantities.
Determine an explicit
expression, a recursive
process, or steps for
calculation from a context.
Functions
CATEGORY / CLUSTER
F-BF.
Building Functions
STANDARD
EXPECTATION
F-BF.2.
Build a function that models a
relationship between two
quantities.
Write arithmetic and
geometric sequences both
recursively and with an
explicit formula, use them to
model situations, and
translate between the two
forms.
Unit 12, Lesson 1, Topic 4: Deductive Reasoning
Grade: 7 - Adopted 2010
!
STRAND / DOMAIN
CC.7.MP.
Mathematical Practices
CATEGORY / CLUSTER
7.MP.2.
CATEGORY / CLUSTER
7.MP.3.
Reason abstractly and
quantitatively.
Construct viable arguments
and critique the reasoning of
others.
"#$#"!
!
!
CATEGORY / CLUSTER
7.MP.7.
CATEGORY / CLUSTER
7.MP.8.
Algebra 1—An Open Course
Professional Development
Look for and make use of
structure.
Look for and express
regularity in repeated
reasoning.
Grade: 8 - Adopted 2010
STRAND / DOMAIN
CC.8.MP.
Mathematical Practices
CATEGORY / CLUSTER
8.MP.2.
CATEGORY / CLUSTER
8.MP.3.
CATEGORY / CLUSTER
8.MP.7.
CATEGORY / CLUSTER
8.MP.8.
Reason abstractly and
quantitatively.
Construct viable arguments
and critique the reasoning of
others.
Look for and make use of
structure.
Look for and express
regularity in repeated
reasoning.
Grade: 9-12 - Adopted 2010
STRAND / DOMAIN
CC.MP.
Mathematical Practices
CATEGORY / CLUSTER
MP-2.
CATEGORY / CLUSTER
MP-3.
CATEGORY / CLUSTER
MP-7.
CATEGORY / CLUSTER
MP-8.
Reason abstractly and
quantitatively.
Construct viable arguments
and critique the reasoning of
others.
Look for and make use of
structure.
Look for and express
regularity in repeated
reasoning.
Unit 12, Lesson 2, Topic 1: Events and Outcomes (Counting)
Grade: 7 - Adopted 2010
STRAND / DOMAIN
CC.7.SP.
CATEGORY / CLUSTER
STANDARD
7.SP.5.
STRAND / DOMAIN
CC.7.SP.
CATEGORY / CLUSTER
STANDARD
!
7.SP.7.
"#$##!
Statistics and Probability
Investigate chance processes
and develop, use, and
evaluate probability models.
Understand that the
probability of a chance event
is a number between 0 and 1
that expresses the likelihood
of the event occurring. Larger
numbers indicate greater
likelihood. A probability near
0 indicates an unlikely event,
a probability around 1/2
indicates an event that is
neither unlikely nor likely,
and a probability near 1
indicates a likely event.
Statistics and Probability
Investigate chance processes
and develop, use, and
evaluate probability models.
Develop a probability model
and use it to find probabilities
of events. Compare
probabilities from a model to
observed frequencies; if the
agreement is not good,
explain possible sources of the
discrepancy.
!
!
EXPECTATION
7.SP.7.a.
STRAND / DOMAIN
CC.7.SP.
CATEGORY / CLUSTER
STANDARD
7.SP.8.
EXPECTATION
7.SP.8.a.
EXPECTATION
7.SP.8.b.
Algebra 1—An Open Course
Professional Development
Develop a uniform probability
model by assigning equal
probability to all outcomes,
and use the model to
determine probabilities of
events. For example, if a
student is selected at random
from a class, find the
probability that Jane will be
selected and the probability
that a girl will be selected.
Statistics and Probability
Investigate chance processes
and develop, use, and
evaluate probability models.
Find probabilities of
compound events using
organized lists, tables, tree
diagrams, and simulation.
Understand that, just as with
simple events, the probability
of a compound event is the
fraction of outcomes in the
sample space for which the
compound event occurs.
Represent sample spaces for
compound events using
methods such as organized
lists, tables and tree
diagrams. For an event
described in everyday
language (e.g., ''rolling double
sixes''), identify the outcomes
in the sample space which
compose the event.
Grade: 9-12 - Adopted 2010
STRAND / DOMAIN
CC.S.
Statistics and Probability
CATEGORY / CLUSTER
S-CP.
Conditional Probability and
the Rules of Probability
Understand independence and
conditional probability and
use them to interpret data
Understand that two events A
and B are independent if the
probability of A and B
occurring together is the
product of their probabilities,
and use this characterization
to determine if they are
independent.
Statistics and Probability
STANDARD
EXPECTATION
S-CP.2.
STRAND / DOMAIN
CC.S.
CATEGORY / CLUSTER
S-CP.
STANDARD
!
EXPECTATION
S-CP.8.
EXPECTATION
S-CP.9.
"#$#%!
Conditional Probability and
the Rules of Probability
Use the rules of probability to
compute probabilities of
compound events in a uniform
probability model
(+) Apply the general
Multiplication Rule in a
uniform probability model,
P(A and B) = P(A)P(B|A) =
P(B)P(A|B), and interpret the
answer in terms of the model.
(+) Use permutations and
combinations to compute
probabilities of compound
!
!
Algebra 1—An Open Course
Professional Development
events and solve problems.
Unit 12, Lesson 2, Topic 2: Probability of Independent Events
Grade: 7 - Adopted 2010
STRAND / DOMAIN
CC.7.SP.
CATEGORY / CLUSTER
STANDARD
7.SP.5.
STRAND / DOMAIN
CC.7.SP.
CATEGORY / CLUSTER
STANDARD
7.SP.7.
EXPECTATION
7.SP.7.a.
Statistics and Probability
Investigate chance processes
and develop, use, and
evaluate probability models.
Understand that the
probability of a chance event
is a number between 0 and 1
that expresses the likelihood
of the event occurring. Larger
numbers indicate greater
likelihood. A probability near
0 indicates an unlikely event,
a probability around 1/2
indicates an event that is
neither unlikely nor likely,
and a probability near 1
indicates a likely event.
Statistics and Probability
Investigate chance processes
and develop, use, and
evaluate probability models.
Develop a probability model
and use it to find probabilities
of events. Compare
probabilities from a model to
observed frequencies; if the
agreement is not good,
explain possible sources of the
discrepancy.
Develop a uniform probability
model by assigning equal
probability to all outcomes,
and use the model to
determine probabilities of
events. For example, if a
student is selected at random
from a class, find the
probability that Jane will be
selected and the probability
that a girl will be selected.
Grade: 9-12 - Adopted 2010
STRAND / DOMAIN
CC.S.
Statistics and Probability
CATEGORY / CLUSTER
S-CP.
Conditional Probability and
the Rules of Probability
Understand independence and
conditional probability and
use them to interpret data
Understand that two events A
and B are independent if the
probability of A and B
occurring together is the
product of their probabilities,
and use this characterization
to determine if they are
independent.
STANDARD
EXPECTATION
S-CP.2.
Unit 12, Lesson 2, Topic 3: Permutations and Combinations
Grade: 7 - Adopted 2010
!
"#$#&!
!
!
STRAND / DOMAIN
CC.7.SP.
CATEGORY / CLUSTER
STANDARD
7.SP.5.
STRAND / DOMAIN
CC.7.SP.
CATEGORY / CLUSTER
STANDARD
7.SP.7.
EXPECTATION
7.SP.7.a.
Algebra 1—An Open Course
Professional Development
Statistics and Probability
Investigate chance processes
and develop, use, and
evaluate probability models.
Understand that the
probability of a chance event
is a number between 0 and 1
that expresses the likelihood
of the event occurring. Larger
numbers indicate greater
likelihood. A probability near
0 indicates an unlikely event,
a probability around 1/2
indicates an event that is
neither unlikely nor likely,
and a probability near 1
indicates a likely event.
Statistics and Probability
Investigate chance processes
and develop, use, and
evaluate probability models.
Develop a probability model
and use it to find probabilities
of events. Compare
probabilities from a model to
observed frequencies; if the
agreement is not good,
explain possible sources of the
discrepancy.
Develop a uniform probability
model by assigning equal
probability to all outcomes,
and use the model to
determine probabilities of
events. For example, if a
student is selected at random
from a class, find the
probability that Jane will be
selected and the probability
that a girl will be selected.
Grade: 9-12 - Adopted 2010
STRAND / DOMAIN
CC.S.
Statistics and Probability
CATEGORY / CLUSTER
S-CP.
Conditional Probability and
the Rules of Probability
Understand independence and
conditional probability and
use them to interpret data
Understand that two events A
and B are independent if the
probability of A and B
occurring together is the
product of their probabilities,
and use this characterization
to determine if they are
independent.
STANDARD
EXPECTATION
S-CP.2.
Unit 12, Lesson 2, Topic 4: Probability of Dependent Events
Grade: 7 - Adopted 2010
STRAND / DOMAIN
CC.7.SP.
CATEGORY / CLUSTER
!
Statistics and Probability
Investigate chance processes
and develop, use, and
evaluate probability models.
"#$#'!
!
!
STANDARD
7.SP.5.
STRAND / DOMAIN
CC.7.SP.
CATEGORY / CLUSTER
STANDARD
7.SP.7.
EXPECTATION
7.SP.7.a.
Algebra 1—An Open Course
Professional Development
Understand that the
probability of a chance event
is a number between 0 and 1
that expresses the likelihood
of the event occurring. Larger
numbers indicate greater
likelihood. A probability near
0 indicates an unlikely event,
a probability around 1/2
indicates an event that is
neither unlikely nor likely,
and a probability near 1
indicates a likely event.
Statistics and Probability
Investigate chance processes
and develop, use, and
evaluate probability models.
Develop a probability model
and use it to find probabilities
of events. Compare
probabilities from a model to
observed frequencies; if the
agreement is not good,
explain possible sources of the
discrepancy.
Develop a uniform probability
model by assigning equal
probability to all outcomes,
and use the model to
determine probabilities of
events. For example, if a
student is selected at random
from a class, find the
probability that Jane will be
selected and the probability
that a girl will be selected.
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