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Introduction to Estimation Theory
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This module introduces estimation theory and its terminology, including bias, consistency, and ...

This module introduces estimation theory and its terminology, including bias, consistency, and efficiency. In searching for methods of extracting information from noisy observations, this chapter describes estimation theory, which has the goal of extracting from noise-corrupted observations the values of disturbance parameters (noise variance, for example), signal parameters (amplitude or propagation direction), or signal waveforms. Estimation theory assumes that the observations contain an information-bearing quantity, thereby tacitly assuming that detection-based preprocessing has been performed (in other words, do I have something in the observations worth estimating?). Conversely, detection theory often requires estimation of unknown parameters: Signal presence is assumed, parameter estimates are incorporated into the detection statistic, and consistency of observations and assumptions tested. Consequently, detection and estimation theory form a symbiotic relationship, each requiring the other to yield high-quality signal processing algorithms.

Material Type:
Readings
Syllabi
Provider:
Rice University
Provider Set:
Connexions
Author:
Don Johnson
Investigating the Dynamic Martian Polar Caps
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In this activity, students download NASA Hubble Space Telescope (HST) images of ...

In this activity, students download NASA Hubble Space Telescope (HST) images of the Martian polar ice caps in summer and winter. Using image processing techniques, students measure and compare various images of the changing Martian and Earth polar ice caps.

Subject:
Algebra
Functions
Material Type:
Activities and Labs
Assessments
Lesson Plans
Provider:
Montana State University
NASA
Provider Set:
NASA/MSU Center for Educational Resources (CERES)
Math, Grade 6, Unit 8
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Distributions and Variability Type of Unit: Project Prior Knowledge Students should be ...

Distributions and Variability

Type of Unit: Project

Prior Knowledge

Students should be able to:

Represent and interpret data using a line plot.
Understand other visual representations of data.

Lesson Flow

Students begin the unit by discussing what constitutes a statistical question. In order to answer statistical questions, data must be gathered in a consistent and accurate manner and then analyzed using appropriate tools.

Students learn different tools for analyzing data, including:

Measures of center: mean (average), median, mode
Measures of spread: mean absolute deviation, lower and upper extremes, lower and upper quartile, interquartile range
Visual representations: line plot, box plot, histogram

These tools are compared and contrasted to better understand the benefits and limitations of each. Analyzing different data sets using these tools will develop an understanding for which ones are the most appropriate to interpret the given data.

To demonstrate their understanding of the concepts, students will work on a project for the duration of the unit. The project will involve identifying an appropriate statistical question, collecting data, analyzing data, and presenting the results. It will serve as the final assessment.

Subject:
Mathematics
Mathematics
Mathematics
Mathematics
Mathematics
Mathematics
Mathematics
Mathematics
Mathematics
Mathematics
Statistics and Probability
Material Type:
Unit of Study
Provider:
Pearson
Math, Grade 7, Unit 2
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Proportional Relationships Type of Unit: Concept Prior Knowledge Students should be able ...

Proportional Relationships

Type of Unit: Concept

Prior Knowledge

Students should be able to:

Understand what a rate and ratio are.
Make a ratio table.
Make a graph using values from a ratio table.

Lesson Flow

Students start the unit by predicting what will happen in certain situations. They intuitively discover they can predict the situations that are proportional and might have a hard time predicting the ones that are not. In Lessons 2–4, students use the same three situations to explore proportional relationships. Two of the relationships are proportional and one is not. They look at these situations in tables, equations, and graphs. After Lesson 4, students realize a proportional relationship is represented on a graph as a straight line that passes through the origin. In Lesson 5, they look at straight lines that do not represent a proportional relationship. Lesson 6 focuses on the idea of how a proportion that they solved in sixth grade relates to a proportional relationship. They follow that by looking at rates expressed as fractions, finding the unit rate (the constant of proportionality), and then using the constant of proportionality to solve a problem. In Lesson 8, students fine-tune their definition of proportional relationship by looking at situations and determining if they represent proportional relationships and justifying their reasoning. They then apply what they have learned to a situation about flags and stars and extend that thinking to comparing two prices—examining the equations and the graphs. The Putting It Together lesson has them solve two problems and then critique other student work.

Gallery 1 provides students with additional proportional relationship problems.

The second part of the unit works with percents. First, percents are tied to proportional relationships, and then students examine percent situations as formulas, graphs, and tables. They then move to a new context—salary increase—and see the similarities with sales taxes. Next, students explore percent decrease, and then they analyze inaccurate statements involving percents, explaining why the statements are incorrect. Students end this sequence of lessons with a formative assessment that focuses on percent increase and percent decrease and ties it to decimals.

Students have ample opportunities to check, deepen, and apply their understanding of proportional relationships, including percents, with the selection of problems in Gallery 2.

Subject:
Mathematics
Mathematics
Mathematics
Mathematics
Mathematics
Mathematics
Mathematics
Mathematics
Mathematics
Mathematics
Ratios and Proportions
Material Type:
Unit of Study
Provider:
Pearson
Math, Grade 7, Unit 2, Lesson 1
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Students are asked whether they can determine the number of books in ...

Students are asked whether they can determine the number of books in a stack by measuring the height of the stack, or the number of marbles in a collection of marbles by weighing the collection.

Students are asked to identify for which situations they can determine the number of books in a stack of books by measuring the height of the stack or the number of marbles in a collection of marbles by weighing the collection.

Key Concepts

As students examine different numerical relationships, they come to understand that they can find the number of books or the number of marbles in situations in which the books are all the same thickness and the marbles are all the same weight. This “constant” is equal to the value BA for a ratio A:B; students begin to develop an intuitive understanding of proportional relationships.

Goals and Learning Objectives

Explore numerical relationships

SWD:
Some students with disabilities will benefit from a preview of the goals in each lesson. Have students highlight the critical features or concepts to help them pay close attention to salient information.

Subject:
Numbers and Operations
Material Type:
Lesson Plans
Provider:
Pearson
Math, Grade 7, Unit 5
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Algebraic Reasoning Type of Unit: Concept Prior Knowledge Students should be able ...

Algebraic Reasoning

Type of Unit: Concept

Prior Knowledge

Students should be able to:

Add, subtract, multiply, and divide rational numbers.
Evaluate expressions for a value of a variable.
Use the distributive property to generate equivalent expressions including combining like terms.
Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true?
Write and solve equations of the form x+p=q and px=q for cases in which p, q, and x are non-negative rational numbers.
Understand and graph solutions to inequalities x<c or x>c.
Use equations, tables, and graphs to represent the relationship between two variables.
Relate fractions, decimals, and percents.
Solve percent problems included those involving percent of increase or percent of decrease.

Lesson Flow

This unit covers all of the Common Core State Standards for Expressions and Equations in Grade 7. Students extend what they learned in Grade 6 about evaluating expressions and using properties to write equivalent expressions. They write, evaluate, and simplify expressions that now contain both positive and negative rational numbers. They write algebraic expressions for problem situations and discuss how different equivalent expressions can be used to represent different ways of solving the same problem. They make connections between various forms of rational numbers. Students apply what they learned in Grade 6 about solving equations such as x+2=6 or 3x=12 to solving equations such as 3x+6=12 and 3(x−2)=12. Students solve these equations using formal algebraic methods. The numbers in these equations can now be rational numbers. They use estimation and mental math to estimate solutions. They learn how solving linear inequalities differs from solving linear equations and then they solve and graph linear inequalities such as −3x+4<12. Students use inequalities to solve real-world problems, solving the problem first by arithmetic and then by writing and solving an inequality. They see that the solution of the algebraic inequality may differ from the solution to the problem.

Subject:
Mathematics
Mathematics
Mathematics
Mathematics
Mathematics
Mathematics
Mathematics
Mathematics
Mathematics
Mathematics
Algebra
Material Type:
Unit of Study
Provider:
Pearson
Math, Grade 7, Unit 5, Lesson 10
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Gallery Overview Allow students who have a clear understanding of the content ...

Gallery Overview

Allow students who have a clear understanding of the content thus far in the unit to work on Gallery problems of their choosing. You can then use this time to provide additional help to students who need review of the unit's concepts or to assist students who may have fallen behind on work.

Gallery Descriptions

Flight of the Albatross

Students use estimation to learn about the distances albatrosses fly.

Sales

Students use equivalent expressions to solve two-step percent of increase or percent of decrease problems in one step.

Carpet Cleaning

Students write and solve equations to model the relationship between hours worked and the cost of carpet cleaning.

Equations About Numbers

Students write an equation to solve a problem about the sum of three consecutive numbers. Then they write and solve their own problem about the sum of three consecutive numbers.

Equations About Sports

Students write and solve equations to solve problems about a basketball game and about walking versus bicycling.

Equations About Figures

Students write and solve equations to model the side lengths and areas of rectangles.

A Number Trick

Students use what they know about simplifying algebraic expressions to do number tricks.

Subject:
Mathematics
Material Type:
Lesson Plans
Provider:
Pearson
Math, Grade 7, Unit 6
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Samples and ProbabilityType of Unit: ConceptualPrior KnowledgeStudents should be able to:Understand the ...

Samples and ProbabilityType of Unit: ConceptualPrior KnowledgeStudents should be able to:Understand the concept of a ratio.Write ratios as percents.Describe data using measures of center.Display and interpret data in dot plots, histograms, and box plots.Lesson FlowStudents begin to think about probability by considering the relative likelihood of familiar events on the continuum between impossible and certain. Students begin to formalize this understanding of probability. They are introduced to the concept of probability as a measure of likelihood, and how to calculate probability of equally likely events using a ratio. The terms (impossible, certain, etc.) are given numerical values. Next, students compare expected results to actual results by calculating the probability of an event and conducting an experiment. Students explore the probability of outcomes that are not equally likely. They collect data to estimate the experimental probabilities. They use ratio and proportion to predict results for a large number of trials. Students learn about compound events. They use tree diagrams, tables, and systematic lists as tools to find the sample space. They determine the theoretical probability of first independent, and then dependent events. In Lesson 10 students identify a question to investigate for a unit project and submit a proposal. They then complete a Self Check. In Lesson 11, students review the results of the Self Check, solve a related problem, and take a Quiz.Students are introduced to the concept of sampling as a method of determining characteristics of a population. They consider how a sample can be random or biased, and think about methods for randomly sampling a population to ensure that it is representative. In Lesson 13, students collect and analyze data for their unit project. Students begin to apply their knowledge of statistics learned in sixth grade. They determine the typical class score from a sample of the population, and reason about the representativeness of the sample. Then, students begin to develop intuition about appropriate sample size by conducting an experiment. They compare different sample sizes, and decide whether increasing the sample size improves the results. In Lesson 16 and Lesson 17, students compare two data sets using any tools they wish. Students will be reminded of Mean Average Deviation (MAD), which will be a useful tool in this situation. Students complete another Self Check, review the results of their Self Check, and solve additional problems. The unit ends with three days for students to work on Gallery problems, possibly using one of the days to complete their project or get help on their project if needed, two days for students to present their unit projects to the class, and one day for the End of Unit Assessment.

Subject:
Mathematics
Mathematics
Mathematics
Mathematics
Mathematics
Mathematics
Mathematics
Mathematics
Mathematics
Mathematics
Statistics and Probability
Material Type:
Unit of Study
Provider:
Pearson
Math, Grade 7, Unit 6, Lesson 16
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Lesson Overview Students estimate the length of 20 seconds by starting an ...

Lesson Overview

Students estimate the length of 20 seconds by starting an unseen timer and stopping it when they think 20 seconds has elapsed. They are shown the results and repeat the process two more times. The first and third times are recorded and compiled, producing two data sets to be compared. Students analyze the data to conclude whether or not their ability to estimate 20 seconds improves with practice.

Key Concepts

Measures of center and spread
Line plots, box plots, and histograms
Mean absolute deviation (MAD)

Goals and Learning Objectives

Apply knowledge of statistics to compare sets of data.
Use measures of center and spread to analyze data.
Decide which graph is appropriate for a given situation.

Subject:
Statistics and Probability
Material Type:
Lesson Plans
Provider:
Pearson
Modeling Hot and Cold Planets: Activity C Approximating the Average Surface Temperature of the Earth
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In this activity, students explore the importance of adequate sampling strategies when ...

In this activity, students explore the importance of adequate sampling strategies when conducting a scientific investigation. They are tasked with determining the average temperature of the Earth, using data sets easily found on the Internet, and determine the kind and size of sample necessary to calculate a representative average. The resource includes a student data sheet and an authentic assessment for the module, where students discuss the establishment of a habitation site on Mars. This is Activity C in module 2, titled "Modeling Hot and Cold Planets," of the resource, Earth Climate Course: What Determines a Planet's Climate? The course aims to help students to develop an understanding of our environment as a system of human and natural processes that result in changes that occur over various space and time scales.

Subject:
Life Science
Mathematics
Atmospheric Science
Material Type:
Activities and Labs
Data
Instructional Material
Provider:
NASA
Provider Set:
NASA Wavelength
Normal Estimate Experiment
Conditions of Use:
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This resource consists of a Java applet and expository text. The applet ...

This resource consists of a Java applet and expository text. The applet simulates a random sample from a normal distribution, and computes standard point estimates of the distribution mean and standard deviation. The bias and mean square error are also computed.

Subject:
Education
Material Type:
Activities and Labs
Instructional Material
Interactive
Simulations
Provider:
University of Alabama in Huntsville
Provider Set:
Virtual Laboratories in Probability and Statistics
Author:
Kyle Siegrist
One Grain of Rice
Conditions of Use:
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Beginning with the famous story of the village girl trying to feed ...

Beginning with the famous story of the village girl trying to feed her people, the lesson involves students in the mathematics of exponential growth. Students work collaboratively to come up with a bargaining plan to trick a raja into feeding the village using algebra and estimation. The complete activity includes the development of an exponential equation, but just following the growth of the number of rice grains throughout the story gives a good introduction to exponential growth. Questions for students and ideas for assessment are provided.

Subject:
Algebra
Material Type:
Interactive
Lesson Plans
Provider:
National Council of Teachers of Mathematics
Provider Set:
Illuminations
PhET Simulation: Estimation
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This interactive Flash animation allows students to explore size estimation in one, ...

This interactive Flash animation allows students to explore size estimation in one, two and three dimensions. Multiple levels of difficulty allow for progressive skill improvement. In the simplest level, users estimate the number of small line segments that can fit into a larger line segment. Intermediate and advanced levels offer feature games that explore area of rectangles and circles, and volume of spheres and cubes. Related lesson plans and student guides are available for middle school and high school classroom instruction. <i>Editor's Note: When the linear dimensions of an object change by some factor, its area and volume change disproportionately: area in proportion to the square of the factor and volume in proportion to its cube. This concept is the subject of entrenched misconception among many adults. This game-like simulation allows kids to use spatial reasoning, rather than formulas, to construct geometric sense of area and volume.</i> This is part of a larger collection developed by the Physics Education Technology project (PhET).

Subject:
Education
Mathematics
Physics
Material Type:
Activities and Labs
Instructional Material
Interactive
Provider:
ComPADRE Digital Library
Florida Center for Research in Science, Technology, Engineering, and Mathematics
Science and Math Informal Learning Educators (SMILE)
University of Colorado Boulder
Provider Set:
ComPADRE Classroom-Ready Resources
ComPADRE: Resources for Physics and Astronomy Education
iCPALMS: A Standards-based K-12 Resources and Tools Pathway
SMILE Pathway: Science and Math Activities in One Search
PhET Interactive Simulations
Author:
Excellence Center of Science and Mathematics Education at King Saud University
Michael Dubson
Mindy Gratny
National Science Foundation
O'Donnell Foundation
PhET
PhET at University of Colorado
PhET Interactive Simulations
Physics Education Technology Project
The William and Flora Hewlett Foundation
Wireless Generation
Planning a Project
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Gantt charts, critical path analysis, SMART objectives and estimation skills are just ...

Gantt charts, critical path analysis, SMART objectives and estimation skills are just some of the topics covered in this unit to help you understand how to plan for a project. You will gain an appreciation of the range of planning techniques available and the situations in which it is appropriate to use them.

Subject:
Business and Communication
Material Type:
Activities and Labs
Readings
Syllabi
Provider:
The Open University
Provider Set:
Open University OpenLearn
Quantitative Research in Political Science and Public Policy, Spring 2004
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Introduction to the application of elementary statistics to political analysis. A basic ...

Introduction to the application of elementary statistics to political analysis. A basic literacy subject, teaching the student how to read and interpret the quantitative literature in various subfields of political science and public policy. Students develop elementary statistical computation skills and learn to use a statistical computing package. From the course home page: This course provides students with a rigorous introduction to Statistics for Political Science. Topics include basic mathematical tools used in social science modeling and statistics, probability theory, theory of estimation and inference, and statistical methods, especially differences of means and regression. The course is often taken by students outside of political science, especially those in business, urban studies, and various fields of public policy, such as public health. Examples draw heavily from political science, but some problems come from other areas, such as labor economics.

Subject:
Statistics and Probability
Political Science
Material Type:
Full Course
Textbooks
Provider:
M.I.T.
Provider Set:
M.I.T. OpenCourseWare
Author:
Ansolabehere, Stephen
Randomized Algorithms, Fall 2002
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Studies how randomization can be used to make algorithms simpler and more ...

Studies how randomization can be used to make algorithms simpler and more efficient via random sampling, random selection of witnesses, symmetry breaking, and Markov chains. Models of randomized computation. Data structures: hash tables, and skip lists. Graph algorithms: minimum spanning trees, shortest paths, and minimum cuts. Geometric algorithms: convex hulls, linear programming in fixed or arbitrary dimension. Approximate counting; parallel algorithms; online algorithms; derandomization techniques; and tools for probabilistic analysis of algorithms.

Subject:
Computer Science
Geometry
Material Type:
Full Course
Textbooks
Provider:
M.I.T.
Provider Set:
M.I.T. OpenCourseWare
Author:
Karger, David
Rational Number Project Fraction Operations and Initial Decimal Ideas
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Twenty-eight lesson plans make up this research-based fourth- through sixth-grade fraction and ...

Twenty-eight lesson plans make up this research-based fourth- through sixth-grade fraction and decimal curriculum, covering through addition and subtraction of decimals, and multiplication and division of fractions. It includes a teacher guide with theoretical framework and instructions and templates for making manipulatives, as well as suggestions for using the lessons with English-language learners (ELL). The Rational Number Project is an NSF-funded multi-university research project.

Subject:
Education
Mathematics
Material Type:
Instructional Material
Lesson Plans
Provider:
NSDL Staff
Provider Set:
NSDL Math Common Core
Author:
Kathleen Cramer
Seth Leavitt
Terry Wyberg