Probability Using Dice

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "Probability Using Dice"

Transcription

1 Using Dice One Page Overview By Robert B. Brown, The Ohio State University Topics: Levels:, Statistics Grades 5 8 Problem: What are the probabilities of rolling various sums with two dice? How can you analyze this situation? Getting Started: Ask what sums are possible if two dice are rolled. Do all the different sums that you came up with have the same chance of happening? Is it easier to think about these questions if the two dice are of different colors or of the same color, or does it even matter? 1. Number, Number Sense and Ohio Academic Content Standards, 2002 NCTM Principles and Standards, x 1. Number, Number Sense and 1. Number, Number Sense and 1. Number and x 1. Number and 2. Measurement 2. Measurement 2. Measurement 2. Algebra 2. Algebra 3. Geometry and Spatial Sense 4. Patterns, Functions and Algebra Mathematical Processes Reasoning X 3. Geometry and Spatial Sense 4. Patterns, Functions and Algebra Mathematical Processes Reasoning X 3. Geometry and Spatial Sense 4. Patterns, Functions and Algebra 5. Data Analysis and Mathematical Processes 3. Geometry 3. Geometry 4. Measurement 4. Measurement X 6. Problem Solving 6. Problem Solving 7. Reasoning and Proof x 7. Reasoning and Proof 8. Communication 8. Communication 9. Connections 9. Connections 10. Representation 10. Representation Note: Capital X denotes major emphasis; lower case x denotes minor emphasis. Using Dice p. 1

2 Using Dice By Robert B. Brown, The Ohio State University Topics:, Statistics Levels: Grades 5 8 Materials: A pair of dice of the same color for each pupil Timing: One hour Prerequisites: Knowledge of fractions Problem: What are the probabilities of rolling various sums with two dice? How can you analyze this situation? Goals: Introduce the notion of a set of outcomes and the probability of a particular outcome Provide a theoretical analysis of dice rolling using a simple example which can be used later to illustrate many simple notions in probability and statistics Experience the difference between a set of outcomes that are not equally likely and a set which are equally likely Big Ideas: Equally likely Statistics Using Dice p. 2

3 Procedure: 1. Give each pupil a pair of dice of the same color. Ask them what sums are possible when they roll the dice. After a little bit of thought they will agree that all sums from 2 to 12 and no others are possible. 2. Ask them if they think that all of these sums have the same chance of being rolled? They will probably agree that the answer is no. They may reason that 2 and 12 have the smallest chance of being rolled because there is only one way to get a 2 and only one way to get a 12, but other sums can be gotten in more than one way. 3. Tell them that they are now going to analyze the chances that any particular sum is rolled. To help with this give each pupil a pair of dice of different colors, let s just say that the colors are red and green. 4. Have everyone take their red die. Ask what could they get if they roll the red die by itself? The numbers 1 6, of course. Are the chances the same for each of these? They will agree the answer is: yes, if the die is a good one. Is it the same for the green die by itself? Sure. 5. If you roll the two dice together, the red one and the green one, and you write down what comes up on each of them, how many possibilities are there? Some will say at first 12 possibilities, 6 for the red and 6 for the green. Have them do some actual rolls and write down all the different outcomes. In short order they will see that the number of different outcomes is 36, which is 6 6, rather than Ask them if they think that all of the 36 different outcomes have the same chance of occurring? (This is a good opportunity to introduce the terminology equally likely outcomes if you want to.) They will probably agree that all 36 outcomes are equally likely. 7. Now you are ready to calculate the probabilities of the different sums. Ask for the probability of getting a 2. Of the 36 equally likely outcomes only one yields a sum of 2, namely, red 1 & green 1. So the chance of getting a sum of 2 is 1 out of 36. What about a sum of 3? That you can get in two ways, red 1 & green 2 and red 2 & green 1. So the chances of getting a sum of 3 are 2 out of 36. This is an opportunity to introduce more terminology if you want to. You can explain that the technical way of expressing the chances of getting a 3 is, The probability of getting a sum of 3 is 2/36. When you do that, they may observe that 2/36 is the same as 1/18. But writing it as 1/18 obscures the fact that these probabilities were calculated using 36 equally likely outcomes, and it makes more difficult the comparison of the probabilities of different sums. Using Dice p. 3

4 Extensions: Tie probability to statistics by comparing theoretical results with experimental results. It is quick and easy to collect data by rolling dice. If you break the students into small groups and let each group present data on the sum of two dice for, say, 36 rolls, they will see that rolling only 36 times can miss the predicted distribution of outcomes by quite a bit due to randomness alone. In fact, the data from different groups of pupils is likely to miss the predicted distribution in different ways. Then you can have the students pool their data to see how much closer the data get to the predicted distribution with a larger number of trials. The Mathematics: Replacing two dice of the same color with two dice of different colors makes it easier to count the 36 different equally likely outcomes for each roll. Sum Number of ways (red 1 & green 2 and red 2 & green 1) As a check, add up the column number of ways, and you will get 36. To see that these probabilities for sums do not change when the differently colored dice are replaced by dice of the same color, you can ask the pupils to imagine that all of the color on each die gets concentrated at one tiny dot on each die. This concentration does not change the probabilities of the different sums, but now it looks like they are just rolling two dice of the same color. If you give them dice of the same color to work with, the following situation may come up. They will analyze all the different ways of getting sums with dice of the same color and come up with the following: Using Dice p. 4

5 Sum Number of ways & & & 3, 2 & & 4, 2 & & 5, 2 & 4, 3 & & 6, 2 & 5, 3 & & 6, 3 & 5, 4 & & 6, 4 & & 6, 5 & & & 6 If you add up the number of ways you get 21. Therefore one might think that the probabilities have denominator 21. It might take some experimenting to see that these 21 legitimately counted outcomes are not equally likely. For example, look at the sum of 4. Using a red and a green die there is only one way to get 2 & 2. But you can get 1 & 3 in two ways, red 1 & green 3 and red 3 & green 1. So getting 1 & 3 is twice as likely as getting 2 & 2. Using two dice of the same color, it would become apparent after a lot of rolls that 1 & 3 is more likely than 2 & 2. Using Dice p. 5

6 Relationships to the Ohio Academic Content Standards, 2002: Grades 5 7: Data Analysis and Standard Find all possible outcomes of simple experiments or problem situations, using methods such as lists, arrays and tree diagrams. Describe the probability of an event using ratios, including fractional notation. Make and justify predictions based on experimental and theoretical probabilities. Mathematical Processes Standard Use inductive thinking to generalize a pattern of observations for particular cases, make conjectures, and provide supporting arguments for conjectures. Grades 8 10: Data Analysis and Standard Compute probabilities of compound events, independent events, and simple dependent events. Make predictions based on theoretical probabilities and experimental results. Mathematical Processes Standard Apply reasoning processes and skills to construct logical verifications or counter examples to test conjectures and to justify and defend algorithms and solutions. Relationships to the NCTM Principles and Standards, 2000: Grades 6 8: Data Analysis and Standard Instructional programs from pre kindergarten through grade 12 should enable all students to Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them. Understand and apply basic concepts of probability. Reasoning and Proof Standard Instructional programs from pre kindergarten through grade 12 should enable all students to Make and investigate mathematical conjectures. Using Dice p. 6

Statistics and Probability (Data Analysis)

Statistics and Probability (Data Analysis) Statistics and Probability (Data Analysis) Kindergarten Grade 1 Grade 2 Grade 3 Grade 4 Specific Learning Outcomes Specific Learning Outcomes Specific Learning Outcomes 2.SP.1. Gather and record data about

More information

NEW MEXICO Grade 6 MATHEMATICS STANDARDS

NEW MEXICO Grade 6 MATHEMATICS STANDARDS PROCESS STANDARDS To help New Mexico students achieve the Content Standards enumerated below, teachers are encouraged to base instruction on the following Process Standards: Problem Solving Build new mathematical

More information

What is the Probability of Pigging Out

What is the Probability of Pigging Out What is the Probability of Pigging Out Mary Richardson Susan Haller Grand Valley State University St. Cloud State University richamar@gvsu.edu skhaller@stcloudstate.edu Published: April 2012 Overview of

More information

Mathematics Assessment Collaborative Tool Kits for Teachers Looking and Learning from Student Work 2009 Grade Eight

Mathematics Assessment Collaborative Tool Kits for Teachers Looking and Learning from Student Work 2009 Grade Eight Mathematics Assessment Collaborative Tool Kits for Teachers Looking and Learning from Student Work 2009 Grade Eight Contents by Grade Level: Overview of Exam Grade Level Results Cut Score and Grade History

More information

Probability and Statistics

Probability and Statistics Probability and Statistics Activity: TEKS: (8.11) Probability and statistics. The student applies concepts of theoretical and experimental probability to make predictions. The student is expected to: (C)

More information

Probability, statistics and football Franka Miriam Bru ckler Paris, 2015.

Probability, statistics and football Franka Miriam Bru ckler Paris, 2015. Probability, statistics and football Franka Miriam Bru ckler Paris, 2015 Please read this before starting! Although each activity can be performed by one person only, it is suggested that you work in groups

More information

PA Common Core Standards Standards for Mathematical Practice Grade Level Emphasis*

PA Common Core Standards Standards for Mathematical Practice Grade Level Emphasis* Habits of Mind of a Productive Thinker Make sense of problems and persevere in solving them. Attend to precision. PA Common Core Standards The Pennsylvania Common Core Standards cannot be viewed and addressed

More information

PA Academic Standards. 2.4. Mathematical Reasoning and Connections 2.5. Mathematical Problem Solving and Communication

PA Academic Standards. 2.4. Mathematical Reasoning and Connections 2.5. Mathematical Problem Solving and Communication Curriculum Category Shift Standard Areas Grades K-8 and High School PA Academic Standards 2.4. Mathematical Reasoning and Connections 2.5. Mathematical Problem Solving and Communication PA Common Core

More information

This document contains Chapter 2: Statistics, Data Analysis, and Probability strand from the 2008 California High School Exit Examination (CAHSEE):

This document contains Chapter 2: Statistics, Data Analysis, and Probability strand from the 2008 California High School Exit Examination (CAHSEE): This document contains Chapter 2:, Data Analysis, and strand from the 28 California High School Exit Examination (CAHSEE): Mathematics Study Guide published by the California Department of Education. The

More information

7.S.8 Interpret data to provide the basis for predictions and to establish

7.S.8 Interpret data to provide the basis for predictions and to establish 7 th Grade Probability Unit 7.S.8 Interpret data to provide the basis for predictions and to establish experimental probabilities. 7.S.10 Predict the outcome of experiment 7.S.11 Design and conduct an

More information

Grade 5 Math Content 1

Grade 5 Math Content 1 Grade 5 Math Content 1 Number and Operations: Whole Numbers Multiplication and Division In Grade 5, students consolidate their understanding of the computational strategies they use for multiplication.

More information

The Comparisons. Grade Levels Comparisons. Focal PSSM K-8. Points PSSM CCSS 9-12 PSSM CCSS. Color Coding Legend. Not Identified in the Grade Band

The Comparisons. Grade Levels Comparisons. Focal PSSM K-8. Points PSSM CCSS 9-12 PSSM CCSS. Color Coding Legend. Not Identified in the Grade Band Comparison of NCTM to Dr. Jim Bohan, Ed.D Intelligent Education, LLC Intel.educ@gmail.com The Comparisons Grade Levels Comparisons Focal K-8 Points 9-12 pre-k through 12 Instructional programs from prekindergarten

More information

Probability. A random sample is selected in such a way that every different sample of size n has an equal chance of selection.

Probability. A random sample is selected in such a way that every different sample of size n has an equal chance of selection. 1 3.1 Sample Spaces and Tree Diagrams Probability This section introduces terminology and some techniques which will eventually lead us to the basic concept of the probability of an event. The Rare Event

More information

Glencoe. correlated to SOUTH CAROLINA MATH CURRICULUM STANDARDS GRADE 6 3-3, 5-8 8-4, 8-7 1-6, 4-9

Glencoe. correlated to SOUTH CAROLINA MATH CURRICULUM STANDARDS GRADE 6 3-3, 5-8 8-4, 8-7 1-6, 4-9 Glencoe correlated to SOUTH CAROLINA MATH CURRICULUM STANDARDS GRADE 6 STANDARDS 6-8 Number and Operations (NO) Standard I. Understand numbers, ways of representing numbers, relationships among numbers,

More information

Session 8 Probability

Session 8 Probability Key Terms for This Session Session 8 Probability Previously Introduced frequency New in This Session binomial experiment binomial probability model experimental probability mathematical probability outcome

More information

Performance Level Descriptors Grade 6 Mathematics

Performance Level Descriptors Grade 6 Mathematics Performance Level Descriptors Grade 6 Mathematics Multiplying and Dividing with Fractions 6.NS.1-2 Grade 6 Math : Sub-Claim A The student solves problems involving the Major Content for grade/course with

More information

Basic Probability. Probability: The part of Mathematics devoted to quantify uncertainty

Basic Probability. Probability: The part of Mathematics devoted to quantify uncertainty AMS 5 PROBABILITY Basic Probability Probability: The part of Mathematics devoted to quantify uncertainty Frequency Theory Bayesian Theory Game: Playing Backgammon. The chance of getting (6,6) is 1/36.

More information

A STATISTICS COURSE FOR ELEMENTARY AND MIDDLE SCHOOL TEACHERS. Gary Kader and Mike Perry Appalachian State University USA

A STATISTICS COURSE FOR ELEMENTARY AND MIDDLE SCHOOL TEACHERS. Gary Kader and Mike Perry Appalachian State University USA A STATISTICS COURSE FOR ELEMENTARY AND MIDDLE SCHOOL TEACHERS Gary Kader and Mike Perry Appalachian State University USA This paper will describe a content-pedagogy course designed to prepare elementary

More information

Mathematics Cognitive Domains Framework: TIMSS 2003 Developmental Project Fourth and Eighth Grades

Mathematics Cognitive Domains Framework: TIMSS 2003 Developmental Project Fourth and Eighth Grades Appendix A Mathematics Cognitive Domains Framework: TIMSS 2003 Developmental Project Fourth and Eighth Grades To respond correctly to TIMSS test items, students need to be familiar with the mathematics

More information

K-12 Mathematics Framework

K-12 Mathematics Framework DRAFT IN PROCESS San Diego City Schools Institute for Learning MATHEMATICS DEPARTMENT K-12 Mathematics Framework Introduction Mathematics Content Mathematics Processes Note: The content in this document

More information

Chapter 111. Texas Essential Knowledge and Skills for Mathematics. Subchapter B. Middle School

Chapter 111. Texas Essential Knowledge and Skills for Mathematics. Subchapter B. Middle School Middle School 111.B. Chapter 111. Texas Essential Knowledge and Skills for Mathematics Subchapter B. Middle School Statutory Authority: The provisions of this Subchapter B issued under the Texas Education

More information

Drawing 3-D Objects in Perspective

Drawing 3-D Objects in Perspective Mathematics Instructional Materials SAS#.1 (one per pair of students) SAS#.2 (one per pair of students) TIS#.1 (transparency) TIS#.2 (transparency) TIS#.3 (Journal prompt) Isometric Dot Paper Isometric

More information

Conducting Probability Experiments

Conducting Probability Experiments CHAPTE Conducting Probability Experiments oal Compare probabilities in two experiments. ame. Place a shuffled deck of cards face down.. Turn over the top card.. If the card is an ace, you get points. A

More information

Grade 6 Mathematics Assessment. Eligible Texas Essential Knowledge and Skills

Grade 6 Mathematics Assessment. Eligible Texas Essential Knowledge and Skills Grade 6 Mathematics Assessment Eligible Texas Essential Knowledge and Skills STAAR Grade 6 Mathematics Assessment Mathematical Process Standards These student expectations will not be listed under a separate

More information

The Normal Approximation to Probability Histograms. Dice: Throw a single die twice. The Probability Histogram: Area = Probability. Where are we going?

The Normal Approximation to Probability Histograms. Dice: Throw a single die twice. The Probability Histogram: Area = Probability. Where are we going? The Normal Approximation to Probability Histograms Where are we going? Probability histograms The normal approximation to binomial histograms The normal approximation to probability histograms of sums

More information

MAT 1000. Mathematics in Today's World

MAT 1000. Mathematics in Today's World MAT 1000 Mathematics in Today's World We talked about Cryptography Last Time We will talk about probability. Today There are four rules that govern probabilities. One good way to analyze simple probabilities

More information

Math Content by Strand 1

Math Content by Strand 1 Math Content by Strand 1 Number and Operations with Whole Numbers Multiplication and Division Grade 3 In Grade 3, students investigate the properties of multiplication and division, including the inverse

More information

Mathematics Success Grade 8

Mathematics Success Grade 8 T92 Mathematics Success Grade 8 [OBJECTIVE] The student will create rational approximations of irrational numbers in order to compare and order them on a number line. [PREREQUISITE SKILLS] rational numbers,

More information

Number and Numeracy SE/TE: 43, 49, 140-145, 367-369, 457, 459, 479

Number and Numeracy SE/TE: 43, 49, 140-145, 367-369, 457, 459, 479 Ohio Proficiency Test for Mathematics, New Graduation Test, (Grade 10) Mathematics Competencies Competency in mathematics includes understanding of mathematical concepts, facility with mathematical skills,

More information

AMS 5 CHANCE VARIABILITY

AMS 5 CHANCE VARIABILITY AMS 5 CHANCE VARIABILITY The Law of Averages When tossing a fair coin the chances of tails and heads are the same: 50% and 50%. So if the coin is tossed a large number of times, the number of heads and

More information

Mathematical goals. Starting points. Materials required. Time needed

Mathematical goals. Starting points. Materials required. Time needed Level SS4 of challenge: B SS4 Evaluating statements about length and length areand area Mathematical goals Starting points Materials required Time needed To help learners to: understand concepts of length

More information

Prentice Hall: Middle School Math, Course 1 2002 Correlated to: New York Mathematics Learning Standards (Intermediate)

Prentice Hall: Middle School Math, Course 1 2002 Correlated to: New York Mathematics Learning Standards (Intermediate) New York Mathematics Learning Standards (Intermediate) Mathematical Reasoning Key Idea: Students use MATHEMATICAL REASONING to analyze mathematical situations, make conjectures, gather evidence, and construct

More information

Chapter 3: The basic concepts of probability

Chapter 3: The basic concepts of probability Chapter 3: The basic concepts of probability Experiment: a measurement process that produces quantifiable results (e.g. throwing two dice, dealing cards, at poker, measuring heights of people, recording

More information

Section 6.2 Definition of Probability

Section 6.2 Definition of Probability Section 6.2 Definition of Probability Probability is a measure of the likelihood that an event occurs. For example, if there is a 20% chance of rain tomorrow, that means that the probability that it will

More information

3. Data Analysis, Statistics, and Probability

3. Data Analysis, Statistics, and Probability 3. Data Analysis, Statistics, and Probability Data and probability sense provides students with tools to understand information and uncertainty. Students ask questions and gather and use data to answer

More information

Number System Properties Grade Nine

Number System Properties Grade Nine Ohio Standards Connection: Number, Number Sense and Operations Benchmark C Apply properties of operations and the real number system, and justify when they hold for a set of numbers. Indicator 1 Identify

More information

Chapter 11 Number Theory

Chapter 11 Number Theory Chapter 11 Number Theory Number theory is one of the oldest branches of mathematics. For many years people who studied number theory delighted in its pure nature because there were few practical applications

More information

Hands-On Math Data Analysis and Probability

Hands-On Math Data Analysis and Probability Hands-On Math Data Analysis and Probability by Robert H. Jenkins illustrated by Lois Leonard Stock Contents To the Teacher... v 1. Spinnering... 1 Activities focus on students ability to construct sample

More information

Greater Nanticoke Area School District Math Standards: Grade 6

Greater Nanticoke Area School District Math Standards: Grade 6 Greater Nanticoke Area School District Math Standards: Grade 6 Standard 2.1 Numbers, Number Systems and Number Relationships CS2.1.8A. Represent and use numbers in equivalent forms 43. Recognize place

More information

Tasks to Move Students On

Tasks to Move Students On Maths for Learning Inclusion Tasks to Move Students On Part 1 Maths for Learning Inclusion (M4LI) Tasks to Move Students On Numbers 1 10 Structuring Number Maths for Learning Inclusion (M4LI) Tasks to

More information

Current Standard: Mathematical Concepts and Applications Shape, Space, and Measurement- Primary

Current Standard: Mathematical Concepts and Applications Shape, Space, and Measurement- Primary Shape, Space, and Measurement- Primary A student shall apply concepts of shape, space, and measurement to solve problems involving two- and three-dimensional shapes by demonstrating an understanding of:

More information

MA 1125 Lecture 14 - Expected Values. Friday, February 28, 2014. Objectives: Introduce expected values.

MA 1125 Lecture 14 - Expected Values. Friday, February 28, 2014. Objectives: Introduce expected values. MA 5 Lecture 4 - Expected Values Friday, February 2, 24. Objectives: Introduce expected values.. Means, Variances, and Standard Deviations of Probability Distributions Two classes ago, we computed the

More information

6.3 Conditional Probability and Independence

6.3 Conditional Probability and Independence 222 CHAPTER 6. PROBABILITY 6.3 Conditional Probability and Independence Conditional Probability Two cubical dice each have a triangle painted on one side, a circle painted on two sides and a square painted

More information

Third Grade Math Games

Third Grade Math Games Third Grade Math Games Unit 1 Lesson Less than You! 1.3 Addition Top-It 1.4 Name That Number 1.6 Beat the Calculator (Addition) 1.8 Buyer & Vendor Game 1.9 Tic-Tac-Toe Addition 1.11 Unit 2 What s My Rule?

More information

A Correlation of Pearson Texas Geometry Digital, 2015

A Correlation of Pearson Texas Geometry Digital, 2015 A Correlation of Pearson Texas Geometry Digital, 2015 To the Texas Essential Knowledge and Skills (TEKS) for Geometry, High School, and the Texas English Language Proficiency Standards (ELPS) Correlations

More information

6.042/18.062J Mathematics for Computer Science. Expected Value I

6.042/18.062J Mathematics for Computer Science. Expected Value I 6.42/8.62J Mathematics for Computer Science Srini Devadas and Eric Lehman May 3, 25 Lecture otes Expected Value I The expectation or expected value of a random variable is a single number that tells you

More information

Discrete Math in Computer Science Homework 7 Solutions (Max Points: 80)

Discrete Math in Computer Science Homework 7 Solutions (Max Points: 80) Discrete Math in Computer Science Homework 7 Solutions (Max Points: 80) CS 30, Winter 2016 by Prasad Jayanti 1. (10 points) Here is the famous Monty Hall Puzzle. Suppose you are on a game show, and you

More information

Prentice Hall Mathematics: Course 1 2008 Correlated to: Arizona Academic Standards for Mathematics (Grades 6)

Prentice Hall Mathematics: Course 1 2008 Correlated to: Arizona Academic Standards for Mathematics (Grades 6) PO 1. Express fractions as ratios, comparing two whole numbers (e.g., ¾ is equivalent to 3:4 and 3 to 4). Strand 1: Number Sense and Operations Every student should understand and use all concepts and

More information

STRAND: Number and Operations Algebra Geometry Measurement Data Analysis and Probability STANDARD:

STRAND: Number and Operations Algebra Geometry Measurement Data Analysis and Probability STANDARD: how August/September Demonstrate an understanding of the place-value structure of the base-ten number system by; (a) counting with understanding and recognizing how many in sets of objects up to 50, (b)

More information

Mathematics Interim Assessment Blocks Blueprint V

Mathematics Interim Assessment Blocks Blueprint V 6-7 Blueprint V.5.7.6 The Smarter Balanced Interim Assessment Blocks (IABs) are one of two distinct types of interim assessments being made available by the Consortium; the other type is the Interim Comprehensive

More information

That s Not Fair! ASSESSMENT #HSMA20. Benchmark Grades: 9-12

That s Not Fair! ASSESSMENT #HSMA20. Benchmark Grades: 9-12 That s Not Fair! ASSESSMENT # Benchmark Grades: 9-12 Summary: Students consider the difference between fair and unfair games, using probability to analyze games. The probability will be used to find ways

More information

South Carolina College- and Career-Ready (SCCCR) Probability and Statistics

South Carolina College- and Career-Ready (SCCCR) Probability and Statistics South Carolina College- and Career-Ready (SCCCR) Probability and Statistics South Carolina College- and Career-Ready Mathematical Process Standards The South Carolina College- and Career-Ready (SCCCR)

More information

Problem of the Month Through the Grapevine

Problem of the Month Through the Grapevine The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards: Make sense of problems

More information

Probability and Expected Value

Probability and Expected Value Probability and Expected Value This handout provides an introduction to probability and expected value. Some of you may already be familiar with some of these topics. Probability and expected value are

More information

Article from: ARCH 2014.1 Proceedings

Article from: ARCH 2014.1 Proceedings Article from: ARCH 20141 Proceedings July 31-August 3, 2013 1 Comparison of the Standard Rating Methods and the New General Rating Formula Muhamed Borogovac Boston Mutual Life Insurance Company*; muhamed_borogovac@bostonmutualcom

More information

Open-Ended Problem-Solving Projections

Open-Ended Problem-Solving Projections MATHEMATICS Open-Ended Problem-Solving Projections Organized by TEKS Categories TEKSING TOWARD STAAR 2014 GRADE 7 PROJECTION MASTERS for PROBLEM-SOLVING OVERVIEW The Projection Masters for Problem-Solving

More information

Grade 6 Grade-Level Goals. Equivalent names for fractions, decimals, and percents. Comparing and ordering numbers

Grade 6 Grade-Level Goals. Equivalent names for fractions, decimals, and percents. Comparing and ordering numbers Content Strand: Number and Numeration Understand the Meanings, Uses, and Representations of Numbers Understand Equivalent Names for Numbers Understand Common Numerical Relations Place value and notation

More information

Big Ideas in Mathematics

Big Ideas in Mathematics Big Ideas in Mathematics which are important to all mathematics learning. (Adapted from the NCTM Curriculum Focal Points, 2006) The Mathematics Big Ideas are organized using the PA Mathematics Standards

More information

Chapter 4: Probability and Counting Rules

Chapter 4: Probability and Counting Rules Chapter 4: Probability and Counting Rules Learning Objectives Upon successful completion of Chapter 4, you will be able to: Determine sample spaces and find the probability of an event using classical

More information

Why Teach Probability in the Elementary Classroom?

Why Teach Probability in the Elementary Classroom? Why Teach Probability in the Elementary Classroom? Felicia M. Taylor Abstract While the curriculum in the United States continues to focus on basic skills well past the fourth grade, classrooms in Japan

More information

Understanding Place Value of Whole Numbers and Decimals Including Rounding

Understanding Place Value of Whole Numbers and Decimals Including Rounding Grade 5 Mathematics, Quarter 1, Unit 1.1 Understanding Place Value of Whole Numbers and Decimals Including Rounding Overview Number of instructional days: 14 (1 day = 45 60 minutes) Content to be learned

More information

Problem of the Month: Fair Games

Problem of the Month: Fair Games Problem of the Month: The Problems of the Month (POM) are used in a variety of ways to promote problem solving and to foster the first standard of mathematical practice from the Common Core State Standards:

More information

Algebra I. In this technological age, mathematics is more important than ever. When students

Algebra I. In this technological age, mathematics is more important than ever. When students In this technological age, mathematics is more important than ever. When students leave school, they are more and more likely to use mathematics in their work and everyday lives operating computer equipment,

More information

Statistics and Probability

Statistics and Probability Statistics and Probability TABLE OF CONTENTS 1 Posing Questions and Gathering Data. 2 2 Representing Data. 7 3 Interpreting and Evaluating Data 13 4 Exploring Probability..17 5 Games of Chance 20 6 Ideas

More information

Math/Stats 425 Introduction to Probability. 1. Uncertainty and the axioms of probability

Math/Stats 425 Introduction to Probability. 1. Uncertainty and the axioms of probability Math/Stats 425 Introduction to Probability 1. Uncertainty and the axioms of probability Processes in the real world are random if outcomes cannot be predicted with certainty. Example: coin tossing, stock

More information

TIMSS 2011 Mathematics Framework. Chapter 1

TIMSS 2011 Mathematics Framework. Chapter 1 TIMSS 2011 Mathematics Framework Chapter 1 Chapter 1 TIMSS 2011 Mathematics Framework Overview Students should be educated to recognize mathematics as an immense achievement of humanity, and to appreciate

More information

Prentice Hall Connected Mathematics 2, 7th Grade Units 2009

Prentice Hall Connected Mathematics 2, 7th Grade Units 2009 Prentice Hall Connected Mathematics 2, 7th Grade Units 2009 Grade 7 C O R R E L A T E D T O from March 2009 Grade 7 Problem Solving Build new mathematical knowledge through problem solving. Solve problems

More information

+ Section 6.2 and 6.3

+ Section 6.2 and 6.3 Section 6.2 and 6.3 Learning Objectives After this section, you should be able to DEFINE and APPLY basic rules of probability CONSTRUCT Venn diagrams and DETERMINE probabilities DETERMINE probabilities

More information

5 th Grade Common Core State Standards. Flip Book

5 th Grade Common Core State Standards. Flip Book 5 th Grade Common Core State Standards Flip Book This document is intended to show the connections to the Standards of Mathematical Practices for the content standards and to get detailed information at

More information

THE WHE TO PLAY. Teacher s Guide Getting Started. Shereen Khan & Fayad Ali Trinidad and Tobago

THE WHE TO PLAY. Teacher s Guide Getting Started. Shereen Khan & Fayad Ali Trinidad and Tobago Teacher s Guide Getting Started Shereen Khan & Fayad Ali Trinidad and Tobago Purpose In this two-day lesson, students develop different strategies to play a game in order to win. In particular, they will

More information

NF5-12 Flexibility with Equivalent Fractions and Pages 110 112

NF5-12 Flexibility with Equivalent Fractions and Pages 110 112 NF5- Flexibility with Equivalent Fractions and Pages 0 Lowest Terms STANDARDS preparation for 5.NF.A., 5.NF.A. Goals Students will equivalent fractions using division and reduce fractions to lowest terms.

More information

Executive Summary Principles and Standards for School Mathematics

Executive Summary Principles and Standards for School Mathematics Executive Summary Principles and Standards for School Mathematics Overview We live in a time of extraordinary and accelerating change. New knowledge, tools, and ways of doing and communicating mathematics

More information

7 Probability. Copyright Cengage Learning. All rights reserved.

7 Probability. Copyright Cengage Learning. All rights reserved. 7 Probability Copyright Cengage Learning. All rights reserved. 7.2 Relative Frequency Copyright Cengage Learning. All rights reserved. Suppose you have a coin that you think is not fair and you would like

More information

RUTHERFORD HIGH SCHOOL Rutherford, New Jersey COURSE OUTLINE STATISTICS AND PROBABILITY

RUTHERFORD HIGH SCHOOL Rutherford, New Jersey COURSE OUTLINE STATISTICS AND PROBABILITY RUTHERFORD HIGH SCHOOL Rutherford, New Jersey COURSE OUTLINE STATISTICS AND PROBABILITY I. INTRODUCTION According to the Common Core Standards (2010), Decisions or predictions are often based on data numbers

More information

Unit 1: Family Letter

Unit 1: Family Letter Name Date Time HOME LINK Unit 1: Family Letter Introduction to Third Grade Everyday Mathematics Welcome to Third Grade Everyday Mathematics. It is part of an elementary school mathematics curriculum developed

More information

Are You Game? Explorations in Probability. Math

Are You Game? Explorations in Probability. Math Are You Game? Explorations in Probability Supplemental Math Materials for Grades 3 6 Math Academy A Thank You to Our Corporate Sponsors The Actuarial Foundation would like to thank the following sponsors

More information

Law of Large Numbers. Alexandra Barbato and Craig O Connell. Honors 391A Mathematical Gems Jenia Tevelev

Law of Large Numbers. Alexandra Barbato and Craig O Connell. Honors 391A Mathematical Gems Jenia Tevelev Law of Large Numbers Alexandra Barbato and Craig O Connell Honors 391A Mathematical Gems Jenia Tevelev Jacob Bernoulli Life of Jacob Bernoulli Born into a family of important citizens in Basel, Switzerland

More information

Intro to Data Analysis, Economic Statistics and Econometrics

Intro to Data Analysis, Economic Statistics and Econometrics Intro to Data Analysis, Economic Statistics and Econometrics Statistics deals with the techniques for collecting and analyzing data that arise in many different contexts. Econometrics involves the development

More information

Curriculum Map by Block Geometry Mapping for Math Block Testing 2007-2008. August 20 to August 24 Review concepts from previous grades.

Curriculum Map by Block Geometry Mapping for Math Block Testing 2007-2008. August 20 to August 24 Review concepts from previous grades. Curriculum Map by Geometry Mapping for Math Testing 2007-2008 Pre- s 1 August 20 to August 24 Review concepts from previous grades. August 27 to September 28 (Assessment to be completed by September 28)

More information

Indiana State Core Curriculum Standards updated 2009 Algebra I

Indiana State Core Curriculum Standards updated 2009 Algebra I Indiana State Core Curriculum Standards updated 2009 Algebra I Strand Description Boardworks High School Algebra presentations Operations With Real Numbers Linear Equations and A1.1 Students simplify and

More information

Probability: Terminology and Examples Class 2, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom

Probability: Terminology and Examples Class 2, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom Probability: Terminology and Examples Class 2, 18.05, Spring 2014 Jeremy Orloff and Jonathan Bloom 1 Learning Goals 1. Know the definitions of sample space, event and probability function. 2. Be able to

More information

Maths Progressions Number and algebra

Maths Progressions Number and algebra This document was created by Clevedon School staff using the NZC, Maths Standards and Numeracy Framework, with support from Cognition Education consultants. It is indicative of the maths knowledge and

More information

DELAWARE MATHEMATICS CONTENT STANDARDS GRADES 9-10. PAGE(S) WHERE TAUGHT (If submission is not a book, cite appropriate location(s))

DELAWARE MATHEMATICS CONTENT STANDARDS GRADES 9-10. PAGE(S) WHERE TAUGHT (If submission is not a book, cite appropriate location(s)) Prentice Hall University of Chicago School Mathematics Project: Advanced Algebra 2002 Delaware Mathematics Content Standards (Grades 9-10) STANDARD #1 Students will develop their ability to SOLVE PROBLEMS

More information

1. The Fly In The Ointment

1. The Fly In The Ointment Arithmetic Revisited Lesson 5: Decimal Fractions or Place Value Extended Part 5: Dividing Decimal Fractions, Part 2. The Fly In The Ointment The meaning of, say, ƒ 2 doesn't depend on whether we represent

More information

Grade 7 Mathematics. Unit 7. Data Analysis. Estimated Time: 18 Hours

Grade 7 Mathematics. Unit 7. Data Analysis. Estimated Time: 18 Hours Grade 7 Mathematics Data Analysis Estimated Time: 18 Hours [C] Communication [CN] Connections [ME] Mental Mathematics and Estimation [PS] Problem Solving [R] Reasoning [T] Technology [V] Visualization

More information

Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit?

Question: What is the probability that a five-card poker hand contains a flush, that is, five cards of the same suit? ECS20 Discrete Mathematics Quarter: Spring 2007 Instructor: John Steinberger Assistant: Sophie Engle (prepared by Sophie Engle) Homework 8 Hints Due Wednesday June 6 th 2007 Section 6.1 #16 What is the

More information

Mathematics in hair and beauty studies principal learning

Mathematics in hair and beauty studies principal learning Mathematics in hair and beauty studies principal learning There are opportunities to develop mathematics to support and enhance the business and scientific aspects of the hair and beauty studies principal

More information

First Grade Exploring Two-Digit Numbers

First Grade Exploring Two-Digit Numbers First Grade Exploring Two-Digit Numbers http://focusonmath.files.wordpress.com/2011/02/screen-shot-2011-02-17-at-3-10-19-pm.png North Carolina Department of Public Instruction www.ncdpi.wikispaces.net

More information

PROBABILITY. Thabisa Tikolo STATISTICS SOUTH AFRICA

PROBABILITY. Thabisa Tikolo STATISTICS SOUTH AFRICA PROBABILITY Thabisa Tikolo STATISTICS SOUTH AFRICA Probability is a topic that some educators tend to struggle with and thus avoid teaching it to learners. This is an indication that teachers are not yet

More information

LAKE ELSINORE UNIFIED SCHOOL DISTRICT

LAKE ELSINORE UNIFIED SCHOOL DISTRICT LAKE ELSINORE UNIFIED SCHOOL DISTRICT Title: PLATO Algebra 1-Semester 2 Grade Level: 10-12 Department: Mathematics Credit: 5 Prerequisite: Letter grade of F and/or N/C in Algebra 1, Semester 2 Course Description:

More information

MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab

MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab MATH 0110 Developmental Math Skills Review, 1 Credit, 3 hours lab MATH 0110 is established to accommodate students desiring non-course based remediation in developmental mathematics. This structure will

More information

If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C?

If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Problem 3 If A is divided by B the result is 2/3. If B is divided by C the result is 4/7. What is the result if A is divided by C? Suggested Questions to ask students about Problem 3 The key to this question

More information

Basic Probability Theory I

Basic Probability Theory I A Probability puzzler!! Basic Probability Theory I Dr. Tom Ilvento FREC 408 Our Strategy with Probability Generally, we want to get to an inference from a sample to a population. In this case the population

More information

NCTM Curriculum Focal Points for Grade 5. Everyday Mathematics, Grade 5

NCTM Curriculum Focal Points for Grade 5. Everyday Mathematics, Grade 5 NCTM Curriculum Focal Points and, Grade 5 NCTM Curriculum Focal Points for Grade 5 Number and Operations and Algebra: Developing an understanding of and fluency with division of whole numbers Students

More information

Gaming the Law of Large Numbers

Gaming the Law of Large Numbers Gaming the Law of Large Numbers Thomas Hoffman and Bart Snapp July 3, 2012 Many of us view mathematics as a rich and wonderfully elaborate game. In turn, games can be used to illustrate mathematical ideas.

More information

Assessment For The California Mathematics Standards Grade 6

Assessment For The California Mathematics Standards Grade 6 Introduction: Summary of Goals GRADE SIX By the end of grade six, students have mastered the four arithmetic operations with whole numbers, positive fractions, positive decimals, and positive and negative

More information

Introduction and Overview

Introduction and Overview Introduction and Overview Probability and Statistics is a topic that is quickly growing, has become a major part of our educational program, and has a substantial role in the NCTM Standards. While covering

More information

Project Maths. Mathematics Resources for Students. Junior Certificate Strand 1. Statistics and Probability

Project Maths. Mathematics Resources for Students. Junior Certificate Strand 1. Statistics and Probability Project Maths Mathematics Resources for Students Junior Certificate Strand 1 Statistics and Probability NCCA 2009 PROJECT MATHS - Mathematics Resources for Students Introduction This material is designed

More information

Such As Statements, Kindergarten Grade 8

Such As Statements, Kindergarten Grade 8 Such As Statements, Kindergarten Grade 8 This document contains the such as statements that were included in the review committees final recommendations for revisions to the mathematics Texas Essential

More information

In the situations that we will encounter, we may generally calculate the probability of an event

In the situations that we will encounter, we may generally calculate the probability of an event What does it mean for something to be random? An event is called random if the process which produces the outcome is sufficiently complicated that we are unable to predict the precise result and are instead

More information