Unit 12: Extensions and Applications - The NROC Projectcontent.nroc.org/Algebra1/PD12_RESOURCE/Algebra I_PD_U12_InstrGuide_v1... · Algebra 1—An Open Course Professional Development

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Algebra 1—An Open Course Professional Development

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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 12.8

• Tutor Simulation: Probability Game Design


• Puzzle: It’s Bargain Time


• Project: What are the Chances of That? Glossary 12.18 Common Core Standards 12.20

Some rights reserved. Monterey Institute for Technology and Education 2011 V1.1

!Unit 12 – Table of Contents

http://www.montereyinstitute.org/license/pd.html

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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.

Learning Objectives

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Unit 12 – Learning Objectives

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Unit 12

Lesson 1

Topic 1, Presentation – 3.2 minutes

Topic 1, Worked Example 1 – 6.2 minutes











Topic 3, Worked Example 3 – 3 minutes

Topic 4, Presentation – 4 minutes




Unit 12 – Media Run Times

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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.

Unit 12 – Instructor Notes

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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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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:

• 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,

• 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.

Unit 12 – Tutor Simulation

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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.

Unit 12 – Puzzle

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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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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.

1 First problem: Design Experiment

• 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.

• 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:

Unit 12 – Project

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RG RB GR GB BR 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

!

1

6. The

probability would be written P (red, then green).

Hint: If you choose two materials to work with, you will also need

to calculate the probability of the second event.

• 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.

2 Second problem: Calculate Theoretical Probability

1. Now calculate the theoretical probability of the desired outcome of your experiment. Remember to account for replacement and non-

replacement 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

6•1

2=1

12

3 Third problem: Gather Experimental Data

• 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.

• 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.

4 Fourth problem: Compare

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1. 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 Experimental Results

RG heads 4/50 = 8% RG tails 2/50 = 4%

RB heads 5/50 = 10% RB tails 4/50 = 8%

BG heads 6/50 = 12% BG tails 6/50 = 12%

BR heads 3/50 = 6% BR tails 5/50 = 10%

GR heads 4/50 = 8% GR tails 1/50 = 2%

GB heads 2/50 = 4% 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:

• 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 Content Presentation

4 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.

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.

3 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.

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.

2 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

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.

1 The desired outcome to be studied during the project is erroneous or irrelevant. Data tables are not shown or are inaccurate. The theoretical probability is missing or major errors are noted. Does not compare experimental and theoretical probabilities.

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. 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: Extensions and Applications Glossary !

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)

Unit 12 – Glossary

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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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NROC Algebra 1--An Open Course


Mapped to Common Core State Standards, Mathematics

Unit 12, Lesson 1, Topic 1: Number Sets

Grade: 7 - Adopted 2010

STRAND / DOMAIN CC.7.NS. The Number System

CATEGORY / CLUSTER Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

STANDARD 7.NS.2. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

EXPECTATION 7.NS.2.d. 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.


STRAND / DOMAIN CC.8.NS. The Number System

CATEGORY / CLUSTER Know that there are numbers that are not rational, and approximate them by rational numbers.

STANDARD 8.NS.1. 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


STRAND / DOMAIN CC.7.MP. Mathematical Practices

CATEGORY / CLUSTER 7.MP.2. Reason abstractly and quantitatively.

CATEGORY / CLUSTER 7.MP.7. Look for and make use of structure.

CATEGORY / CLUSTER 7.MP.8. Look for and express regularity in repeated reasoning.

Grade: 8 - Adopted 2010 STRAND / DOMAIN CC.8.MP. Mathematical Practices


Unit 12 – Common Core

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Grade: 9-12 - Adopted 2010

STRAND / DOMAIN CC.MP. Mathematical Practices

CATEGORY / CLUSTER MP-2. Reason abstractly and quantitatively.

CATEGORY / CLUSTER MP-7. Look for and make use of structure.

CATEGORY / CLUSTER MP-8. Look for and express regularity in repeated reasoning.

STRAND / DOMAIN CC.F. Functions

CATEGORY / CLUSTER F-IF. Interpreting Functions

STANDARD Understand the concept of a function and use function notation.

EXPECTATION F-IF.3. 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.


CATEGORY / CLUSTER F-BF. Building Functions

STANDARD Build a function that models a relationship between two quantities.

EXPECTATION F-BF.1. Write a function that describes a relationship between two quantities.

GRADE EXPECTATION F-BF.1.a. Determine an explicit expression, a recursive process, or steps for calculation from a context.


CATEGORY / CLUSTER F-BF. Building Functions

STANDARD Build a function that models a relationship between two quantities.

EXPECTATION F-BF.2. 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




CATEGORY / CLUSTER 7.MP.3. Construct viable arguments and critique the reasoning of others.

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CATEGORY / CLUSTER 8.MP.3. Construct viable arguments and critique the reasoning of others.



Grade: 9-12 - Adopted 2010 STRAND / DOMAIN CC.MP. Mathematical Practices

CATEGORY / CLUSTER MP-2. Reason abstractly and quantitatively.

CATEGORY / CLUSTER MP-3. Construct viable arguments and critique the reasoning of others.

CATEGORY / CLUSTER MP-7. Look for and make use of structure.

CATEGORY / CLUSTER MP-8. Look for and express regularity in repeated reasoning.

Unit 12, Lesson 2, Topic 1: Events and Outcomes (Counting)


STRAND / DOMAIN CC.7.SP. Statistics and Probability

CATEGORY / CLUSTER Investigate chance processes and develop, use, and evaluate probability models.

STANDARD 7.SP.5. 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.



STANDARD 7.SP.7. 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.

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EXPECTATION 7.SP.7.a. 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.



STANDARD 7.SP.8. Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.

EXPECTATION 7.SP.8.a. 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.

EXPECTATION 7.SP.8.b. 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

STANDARD Understand independence and conditional probability and use them to interpret data

EXPECTATION S-CP.2. 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.

STRAND / DOMAIN CC.S. Statistics and Probability


STANDARD Use the rules of probability to compute probabilities of compound events in a uniform probability model

EXPECTATION S-CP.8. (+) 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.

EXPECTATION S-CP.9. (+) Use permutations and combinations to compute probabilities of compound

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events and solve problems.

Unit 12, Lesson 2, Topic 2: Probability of Independent Events









Grade: 9-12 - Adopted 2010

STRAND / DOMAIN CC.S. Statistics and Probability




Unit 12, Lesson 2, Topic 3: Permutations and Combinations


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Grade: 9-12 - Adopted 2010 STRAND / DOMAIN CC.S. Statistics and Probability




Unit 12, Lesson 2, Topic 4: Probability of Dependent Events




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