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bloggulf220 · 3 years

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Mr. Mac's Challengesmr. Mac's 6th Grade

Mr. Mac's Challengesmr. Mac's 6th Grade Language Arts

Mr. Mac's Challengesmr. Mac's 6th Grader

These seem like obvious things but just trying to help people with the learning curve.

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Anker Tests Reading / Language Arts Skills. More Reading Comprehension More Spelling (10 words each) More Homophones More Analogies. More 6th General Math. By Mr Mac. Now contains Australian/British English and US English spellings. A great visual prompt for students to implement the 'Super Six' comprehension strategies when reading or viewing texts. great poster display for classrooms - text and images.

Elementary Math Skills

Mrs. Victoria Rose

This class will address 3-6th grade mathematical skills in preparation for pre-algebra.

Pre-Algebra

Mr. Mac Ogilvie 1

STARS offers a Pre-Algebra course using the Saxon Math 8/7 book as a resource. This course provides an excellent summary of the basic skills required to move into Algebra. It can be considered to be a Middle School Math course that bridges elementary school to high school math. Public schools seem to be moving some students into Algebra as early as sixth grade depending largely on the skill level of the student. But the movement into Algebra at any grade level requires the exposure to a wide range of general math knowledge and the mastery of key skills essential to the ability to be successful in Algebra. This course provides that knowledge and the ability to master those critical skills. The course includes coverage of basic geometry and probability and statistics. The STARS instructor at this time has an extensive background in teaching both middle school math and Algebra both in public schools and here at STARS.

Need: Saxon 8/7 Homeschool Kit- http://www.rainbowresource.com/product/Saxon+Math+8-7+3ED+Homeschool+KIT/024434

Mr. Mac's Challengesmr. Mac's 6th Grade Language Arts

Algebra I

Mr. Mac Ogilvie

This course will use the Saxon book, Algebra 1, to provide a comprehensive teaching of the fundamental aspects of problem solving. It offers a substantial review of pre algebra fundamentals while also offering coverage of area, volume, and perimeter of geometric figures. Major topics include evaluation of algebraic equations, thorough coverage of exponents, polynomials, solving and graphing linear equations, complex fractions, solving systems of equations, radicals, word problems, solving and graphing quadratic equations, solving systems of equations, and solving equations by factoring.

Textbook: Saxon Algebra 1 Homeschool Kit http://www.rainbowresource.com/product/sku/000628

Algebra II

Mr. Mac Ogilvie

This course will use the Saxon book, Algebra 2, to provide a comprehensive teaching of the fundamental aspects of problem solving. It offers a substantial review of all topics in Algebra 1 and then moves on to cover these topics at an advanced level. Major topics include the solving and graphing of linear and quadratic equations, factoring, a variety of types of word problems, solving quadratic equations by completing the square, solving simultaneous equations with fractions and decimals, complex roots of quadratic equations, solving systems of nonlinear equations, graphing and solving a system of inequalities, exponential equations, and review of key geometry, probability and statistics topics.

Textbook: Saxon Algebra 2: Homeschool Kit Third Edition

Jacob’s Geometry

Ms. Enjoli Stith

3rd Edition. This is an excellent geometry course with clear explanations of geometric concepts, including plenty of practice with proofs (informal and paragraph). The second chapter (six lessons) is devoted to logic in preparation for constructing proofs. Topics build incrementally and each practice set assumes knowledge gained in previous lessons in order to construct proofs. The author has set his text up to include three sets of problems with each lesson so as to present the basic concepts in Set I exercises, applications in Set II exercises, and extension of concepts in Set III exercises. Finally, there are Algebra reviews located at the end of most chapters in the student textbook. An appendix contains all presented theorems and postulates. After a thorough study of Euclidean geometry, a single chapter of four lessons presents non-Euclidean geometries. SAT math problems have also been included in exercise sets. The teacher’s guide contains lesson plans, black line masters, and answers to all exercises. The test bank contains two tests for each chapter, a mid-term, a final, and solutions.

Textbook: Geometry: Seeing, Doing, Understanding, 3rd Edition

Advanced Math (Pre-Calculus and Trigonometry)

Ms. Enjoli Stith

Advanced Mathematics fully integrates topics for algebra, geometry, trigonometry, discrete mathematics, and mathematical analysis, to include trigonometric equations and inequalities, the unit circle and trigonometric identities, conic sections, logarithms and exponents, probability and statistics, complex numbers, functions and graphs, sequences and series, and matrices. Graphing calculator applications are developed to facilitate calculations and enhance in-depth understanding of concepts. Word problems are developed throughout the problem sets and become progressively more elaborate. With this practice, high-school level students will be able to solve challenging problems such as rate problems and work problems involving abstract quantities. Conceptually oriented problems that help prepare students for college entrance exams (such as the ACT and SAT) are included in the problem sets. Students complete 2-3 lessons per week.

Mr. Mac's Challengesmr. Mac's 6th Grader

Note: This is a two year class. Part 1 covers lessons 1-66, Part 2 covers lessons 67-125.

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classroomstruggle-blog · 4 years

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Why Younger Students Struggle With Fractions

One of the most challenging topics in a typical math class is the learning of fractions and their operations. Why does Johnny (Jenny) struggle with the concepts of fractions, decimals, and percentages?

I believe that these ideas are not taught effectively in most elementary classrooms. Teachers tend to move quickly through the lessons at this time in elementary school (usually 3rd or 4th grade), and tend to treat fractions as an abstract idea.

When learning to speak, infants are taught to point to an object and repeat the word that the adult speaks. These nouns become the building blocks of language, and numbers and counting is done the same way.

Primary grade students learn to count objects in their world, moving on to basic number pattern ideas and labels such as multiples of 5 and 3, and the concepts of odd and even. But these are whole number ideas, and partial numbers (fractions and decimals) tend to be introduced on paper as an abstract idea. In other words, teachers should be repeatedly showing students what a third of something is, or how to cut things up into equal pieces, etc.

I am not saying that this is avoided in elementary school, just that students are not given enough exposure to those ideas. Students should not be doing worksheets where they practice adding fractions with unlike denominators until they can draw pictures of these fractions, or show how to cut them up into equivalent pieces. The concrete learning idea of manipulating objects to show a mathematical idea needs to be embedded into the minds of young learners before they can practice similar math ideas on pencil and paper. This curriculum piece tends to be skipped or rushed in most elementary classrooms.

Students who are not ready to move on to do fraction work (or decimals or percentages) are either unsuccessful the rest of their math careers, or memorize the algorithms and get stuck later in higher level math courses.

Either case requires extra help to survive high school or college math curriculum because the developmental phases in mathematics cannot be avoided. If a student didn't understand a fundamental concept such as fraction operations, they will not be successful in learning topics that use these ideas, such as solving equations, graphing linear relationships, or manipulating precalculus problems.

So, if you are a parent of a child who struggles in math, take a look at how your son or daughter expresses the ideas of fractions. Ask them to show you what 2 thirds looks like, or what 8.6 means, or to make a model of what percent 8 out of 10 represents. Can they do it?

If not, find a quality online math tutor who can diagnose the developmental phases of what your child has missed in the classroom. A highly skilled, experienced teacher can help fill in those gaps in learning, make it fun, and help your child to once again rediscover their math confidence! visit classroomstruggle

A middle school and high school classroom teacher for over 25 years, Terry VanNoy takes pride in his ability to engage his students with creative ways of showing important concepts. He and his students use a shared computer screen, whiteboard, and use text chat and live voice (like a phone call) to work together during the instructional sessions. His online tutoring business caters to middle and high school students who need a confidence boost to get to the next levels and feel excited again about becoming math students who can achieve amazing things!

#classroomstruggle #visit classroomstruggle

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devdd12-blog · 4 years

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CBSE Class 11 Courses Module

Class 11 is the first step in the career. It is generally considered more difficult than class 12. The reason for this is, the study of class 11 is way different from that which students studied till class 10. So, several times it happens that students are unable to cope up with all the subjects together. Hereby, Edubull presents 11 different subjects for students to study according to their interests and stream. The following are the subjects available.

Maths - consist of Sequence and Series, Complex Numbers, Basic Arithmetic and Algebra, Transformation of Equations, Trigonometry, Coordinate Geometry, Circle, Hyperbola, Limits, Differentiability.

Maths Resources-I - Contains Sets, Relations and Functions, Trignometry, Mathematical Induction, Complex Numbers, Linear Inequalities, Permutation & Combination, and Binomial Theorem.

Maths Resources-II - Includes Sequence Series, Straight Lines, Conic Sections, Three Dimensional Geometry, Limits and Derivatives, Mathematical Reasoning, Statistics, and Probability.

Maths resources-III - Illustrates the concept in a very interesting manner which helps the student to understand the concept instead of mugging up formulas.

Maths resources-IV - Focus on more and more practice of sums and give easy ways to solve complicated questions.

Maths resources-V - Provides students the animated videos to learn with even more interest.

Maths resources-VI - Solves the questions in a manner that motivates the students to practice more.

Pre-Algebra - Provides more knowledge about whole number, number concepts, and integers for a better understanding of concepts.

Probability Resources - Is customized only for probability and covers the whole probability concept of class 11.

Arithmetic Maths Resources - Help the students who want to learn about Math 20 - Arithmetic Review.

Algebra Resources - Is customized for algebra lovers and provides a vast knowledge of algebra.

8 Modules to Learn

Every student either faces problems in organic or in inorganic chemistry. Edubull has divided chemistry into 8 modules to make all the branches of chemistry easy for the students.

Chemistry-I - Includes Atomic Structure Part - I, Atomic Structure Part - II, States of Matter, Chemical Bonding, Nuclear Chemistry, Electrochemistry, and Structure of atom.

Chemistry-II - Provides students with complex chemistry concepts like Classification of Elements and Periodicity in Properties, Chemical Bonding, and Molecular Structure, States of Matter in a simplified manner and provides suitable notes of the same.

Chemistry Resources-I - Explains all the concept of class 11 chemistry with several examples and make it way more interesting.

Chemistry Resources-II - Provides an explanation of concepts through animations which makes it interesting for the students to gain the knowledge as well as to learn.

Organic Chemistry Resources-I - Covers 3 chapters of organic chemistry, that are- some basic principles and techniques, hydrocarbons, and environmental chemistry.

Organic Chemistry Resources-II - Covers several topics to provide specialized knowledge for each of the topics i.e. Chirality, Newman Projections, Organic Chemistry Naming Examples, Sn Reactions, E Reactions, Comparing E2 E1 Sn2 Sn1 Reactions, Alcohols, Amine Naming Introduction, and Carboxylic Acid Introduction.

Chem Physical - Enhances the knowledge of students in physical chemistry in several topics which include Some Basic Concept of Chemistry, Atomic Structure, States of Matter, Equilibrium, Thermodynamics, and Redox Reaction.

Chemistry Inorganic Resources - Makes the inorganic chemistry easy for students by explaining Chemical Bonding and Molecular Structure, Classification of Elements and Periodicity, Hydrogen & Water, S Block Elements, and P Block Elements.

8 Modules to Explore Physics with a Wider Point of View

Physics is a weakness of many students. But the key to study physics is understanding each concept properly. Edubull teaches physics to the students in 8 modules.

Physics - Covers all the basic concepts of class 11 physics including Dimensional and Error Analysis, Introduction to Kinematics, Projectile Motion, Kinematics of Circular Motion, Particle Dynamics, Newton’s Law, Wave Motion, and Physical Properties of Waves.

Physics Resources-II - Gives various teaching aids, notes, and practice sessions of various topics which include Gravitation, General Properties of Matter, Thermal Properties of Material, Kinetic Theory of Gases, Oscillations, and Waves.

Physics Resources-III - Helps the students to build interest in physics by explaining different concepts with several examples and practical.

Physics-IV - Clarifies the concepts of topics covered in Physics of ICSE Standard with a lot of examples.

Physics Resources-V - This course is for those who want to learn the concepts of physics in an even more detailed manner.

Science - Clarifies the basic concepts of Theory and makes it more interesting for students in order to motivate students to study even more.

Thermodynamic and kinetic resources -This course is specialized for concepts of thermodynamic and kinetic physics.

Physics Animation - As the name suggests, explains the concepts through animations to make it more interesting. It includes introduction to nuclear reactor and introduction to magnetic field and magnetic field lines of force.

4 Modules to Make Biology more Interesting

However, biology is not that difficult to learn but for those who can’t just mug up and want to understand each and every concept with propper detail, edubull brings up biology in 4 modules.

Biology-I - Helps students to get an in-depth understanding of the basic concepts of topics of the biology of class 11.

Biology resources-II - Displays animated videos that grab the attention of the students who have no interest in the subject and have left it. It is also intended to help clarify the basic concepts of topics through illustrative interactive lectures and practical in a very interesting manner.

Biology resources-III - Illustrates different concepts through interactive lectures and enhance the student’s interest in the subject.

Biology resources-IV - Clarifies the most basic topics of biology.

4 Modules to get the Strategy for Accountancy

Every student gets frustrated when the entries at the last in the accounts answer does not match. Edubull provides you a platform where you can learn accounts in 4 modules and then you’ll easily be able to find your own mistake.

Accounts Resources (double entry book keeping part-1) - Explains several chapters in an easy way which include Introduction to Accounting, Basic Accounting Terms, Theory Base of Accounting, Bases of Accounting, Accounting Equation, Accounting Procedures, Origin of Transaction, Journal and Ledger, Special Purpose Books, Bank Reconciliation Statement, and Trial Balance.

Accounts Resources (double entry book keeping part-2) - Continues part-1 by including topics like Depreciation, Provisions and Reserves, Accounting for Bills of Exchange, Rectification of Errors, Adjustments in Preparation of Financial Statements, Accounts from Incomplete Records, and Computerised Accounting.

Accountancy - Explains the basic chapters of accounts of class 11 which include Accounting Equations, Journal Entries, Final Accounts, and Ledger Posting and Trial Balance.

Introduction to Financial Statement - Is specialized to learn the basic concepts of financial statements.

2 modules to Increase English Proficiency .

For English, edubull has made 2 modules, one of which explains in English and the other in Hindi.

Hornbill - Explains all the chapters of the textbook hornbill in a very interesting manner. The chapters include The Portrait of a Lady, A Photograph, We're Not Afraid to Die, The Voice of the Rain, Discovering Tut: The Saga Continues, Landscape of the Soul, The Browning Version, Childhood, and Father to Son.

Hornbill (Hindi) - Explains all of these in Hindi.

3 modules to Enhance the Skills in Informatics Practices.

For the IP lovers, edubull has made 3 modules to learn and practice.

How to build a computer - Provides students the basic knowledge of Computers and its parts, its importance, and various other basic information about computers so that the learner who is new to computers must learn about building a computer from the beginning.

SQL for beginners - Explains the basics and details of structured query language in both - Hindi and English.

Computer networks - As the name suggests, give knowledge to students about different computer networks.

8 Modules to Enhance the Skills in Computer Science

All the students you want to enrich their skills in the field of computer science have the option here to learn and practice according to their interests.

Introduction to programming - Guides the students in Hindi language about how to do programming and then the tasks are given to students for practice.

Computer science resources - Enrich the students with the basics of the computer that helps the students to start with small programming tasks.

MS Word 2016 (Hindi) - Explains the vast MS Word 2016 in Hindi by dividing it into topics like Beginning, Basic Editing, Formatting, Copying and Moving Text and Objects, Editing Features, Paragraph Formatting, Tables, Page Formatting, Graphics & Pictures, Footnote & Endnote, and Advanced Tools.

Notepad (Hindi) - Explains all the features debate in Hindi.

Windows 10 operating system (Hindi) - Explains windows 10 in detail in Hindi. It’s very useful for those who stay updated with the latest technologies.

MS PowerPoint 2016(Hindi) - This is a module in which PowerPoint is taught within the subheadings like The Fundamentals, Slides, Editing, Formatting, Images, Multimedia, and Objects, and Transitions and Animation Effects.

MS excel 2016 - This is also divided into subtopics to be specific and the topics are The Excel Window, Editing, Formatting, Formulas, Hlookup Vlookup, Pivot, Chart, Short Cuts, and Practice Sheets.

MS excel 2016 (Hindi) - Explains the same in Hindi with somewhat different topics which include The Excel Screen, Formatting, Formulas, Hlookup Vlookup, Pivot, Chart, and Practice Sheets.

3 Modules to learn Business Studies in an Interesting Manner

Business studies is beautifully divided in 3 modules by edubull according to its topics.

Business study resources (foundation of business part-1) - Explains almost all topics of textbook of business studies of class 11, which are Nature and Forms of Business, Forms of Business Organisation, Private, Public and Global Enterprises, Business Services, Emerging Modes of Business, Social Responsibility of Business and Business Ethics.

Business study resources (finance and trade part -2) - This is divided into 2 topics, that are sources of business finance and small business.

Business finance - This is personalized for those who want to startup with a new business.

7 Modules to understand the importance of Physical Education.

However, yoga is a vast practice, still, edubull has tried to collect the most of it within 7 modules.

Yoga postures by Shilpa - is a module that makes the students learn about various yoga postures and get motivated to stay healthy.

Face Yoga - is very useful for the students who stay up till late because it puts a great effect on how the face looks.

Diet and nutrition - teaches the students about how can one stay fit with a balanced diet.

Healthy eating tips - is somewhat similar to diet and nutrition, yet a lot different from it. You’ll know the difference, once you’ll attend the lectures.

Get healthy, get happy - is a module in which students are counseled to stay happy in order to be fit physically.

Yoga for Kids - brings out wonderful yoga postures specialized for kids which are highly effective on a child’s growth and development.

Yoga in office - Yoga in office’ is based on stretching which should be done while prolong sitting.

1 module to enrich your interest in Political Science

Since it just has 1 module ‘political science’, it is divided into several chapters in order to gain the most. The chapters are Constitution Why and How?, Rights in Indian Constitution, Election and Representation, Executive, Legislature, Judiciary, Federalism, Local Governments, Constitution as a Living Document, and The Philosophy of the Constitution.

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muhyousafsalfi · 5 years

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9 FREE Online Study Tools Every Student Should Know About

Many might think that the internet isn’t conductive to studying—that it’s all cat videos and social networking—but there are thousands of websites out there designed specifically for learning new skills, managing everyday life, and even study tools to help students study for exams.

Online Study Tools

Here we’ll shine the spotlight on 9 online study tools dedicated to being a your new best study-buddy. Whether it’s time management, digital flashcards, self-made quizzes, study games, or other educational tools, these sites and apps can help students of any age get ready for upcoming tests, all for the low, low price of absolutely free.

1. GoConqr

Visual learners, rejoice. What makes this site unique are the mind maps, a web of information that breaks down broad subjects into smaller topics. When a you create a map, it allows you to see how things are connected — this helps with both memorization and understanding.

The site also has numerous other helpful features like flashcards, a note taking platform, self-created quizzes, study planner, a collaboration tool, and a way to track how much you’re learning. This versatile tool is also available across multiple platforms for studying on the go.

2. Quizlet

With this site, you provide the information and Quizlet provides the study tools. Users can create “sets” in any subject under the sun, and based off the set the website will generate flashcards, quizzes, practice tests, matching games, and even auditory tools. Quizlet also has a free app for learning on the go, and studying even offline.

The two games, Scatter and Space Race, allow you to learn the material and have fun doing it. In Scatter, users drag definitions or information to their related counterpart as quickly as they can, clearing the screen. In Space Race, definitions scroll across the screen and you type in the correct word or phrase associated with it before the definition reaches the end of the screen. With leader-boards and high-scores, you can compete to get the best times, adding the motivation of competition to your studying.

3. Evernote

Studying can sometimes cover several different platforms, but with Evernote all of your notes, information, and research can be collaborated together in one place. An invaluable organizational tool, this site can sync your info across computers and phones, share with other users, gather web-clips, and create all from one central platform.

Evernote can be used in a multitude of ways: keeping research paper information in order, organizing presentation notes, creating study guides off of in-class note taking, not to mention everyday non-school related tasks, too.

4. StudyBlue

Collaboration is the name of the game for this website. StudyBlue connects students through similar learning goals and subjects, allowing them to share and access flashcards, study guides, and more. StudyBlue can even connect students in specific classes, letting them message each other and collaborate on projects.

The site also has a diverse content library, multiple study modes, class-based study guides, storage for all your notes, and a real-time self-assessing score that tracks your progress on a subject. And with free mobile apps, StudyBlue can be taken on the go.

5. Marinara Timer

Despite making you hungry for some spaghetti, this site is dedicated to increasing productivity. Inspired by the Pomodoro method of productivity—working for 25 minutes and then taking a 5 minute break—Marinara Timer allows you to customize your own personal timer (complete with entertaining alarm sounds like “alien bot ordering lunch” or “echoing into space”) to study effectively.

Taking a small break while studying increases attention span, focus, and productivity. This site can help you reward yourself for your hard work by letting you know it’s time to sit back scroll through your Twitter feed or watch a few funny videos, and then letting you know when it’s time to get back to work.

6. StudyStack

Studying with flashcards has never been so easy. StudyStack allows users to create flashcard sets and use what others have already made. When a card is flipped, you can choose whether you got it right or wrong. Wrong cards will be repeated until you’ve learned it, making it a great tool for memorization.

Once created, the site can turn the information into a number of tools and games perfect for studying. You can play hangman, unscramble, or a matching game before taking quizzes and practices tests.

7. Memorize.com

As its name suggests, this site is perfect for learning terms and definitions. Users can post articles, images, diagrams, and more to share general information on any number of topics. Memorize.com also encourages collaboration, working together to create the best learning tools for studying.

When you find a set you want to study, it’s offered in three different formats of varying difficulty—multiple choice, matching, and finally the ever-popular flashcard. This engaging site can even award badges for completed sets. The site may not have been updated in a while, but that doesn’t mean you can’t take advantage of it.

8. Study Guides and Strategies

If you want to learn how to learn, this site is for you. It has numerous articles on all sorts of learning styles—classroom learning, group learning, self-studying, etc.—as well as tips on how to write papers, how to manage your time, how to do presentations, solve problems, prep for and take tests, how to read critically, do effective research, and is available in about 40 different languages.

Study Guides and Strategies is a site chock full of in-depth looks at learning for a diverse variety of students. Whether you want to learn how to properly cite a paper or get over test anxiety, learn how to solve linear equations or how to navigate lab reports, this site can offer all that and much, much more. It might be an old site, but the information there is priceless.

9. Schooltraq

When you keep track of assignments, tests, and events, planning time to study becomes a lot easier. Schooltraq is an online digital planner that keeps due dates in check. An alternative to the traditional paper agenda, this site ensures you never lose your schedule as it syncs up between your phone and computer.

The design is sleek and simple, allowing for clarity and low stress. When your ducks are all in a row, school life becomes more manageable. This site can go beyond helping you study for your next test since you can use it for your personal life as well.

Many of the sites on this list have upgraded packages or “pro” accounts for some cost, but every single one has free tools ideal for any student. Give them a try and see which ones help you out the most.

Happy studying! with Education Mania

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mathswithnarendrasir · 1 year

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Class-8 NCERT Solution Exercise 2.3 | Math With Narendra Sir #class8 #mathematics

#Class 8 Math's Chapter - 2 | Ex-2.2 Q.-15 With Narendra Sir #class8 #mathematics #In this video I am teaching you Exercise 2.2 #Class-8 Math's Chapter-2 | Exercise-2.1 | With Narendra Sir #class 8 maths chapter 1 exercise 1.2 solutions #class 8 maths exercise 1.2 solution in hindi #class 8 maths chapter 2 #class 8 math chapter 2 #linear equations class 8 word problems #Math's with Narendra Sir provide to you maths videos for all classes.#Maths with Narendra Sir give you live-class platform #for math skills.#Learn math by taking free online math courses.#Get introductions to algebra #geometry #trigonometry with current math coursework and AP exam preparation.#Select a course to learn more.#class-8 maths #class 8 maths chapter 2 exercise 2.1 #cbse ncert solutions #class 8 maths #10th class exercise 2.3 solution #linear equations in one variable exercise 2.3 #class 10th math exercise 2.3 ncert solutions #solution of exercise 2.3 class 10 ncert math #10th class chapter- 2 and exercise 2.3 math solution #chapter 2 exercise 2.3 math solution #linear equations in one variable exercise 2.4 #class 12 maths exercise 1.3 solution

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nodehunter228 · 3 years

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Study Guide Content Mastery Teacher Edition Chemistry

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Chemistry Matter And Change Study Guide For Content Mastery Teacher Edition

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Chemistry Matter Change Study Guide For Content Mastery Student Edition Author: McGraw-Hill Education Publish On: 2001-03-30 There are worksheets and study tools for each section of the text that help teachers track students' progress toward understanding concepts.

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Access Free Chemistry Study Guide Content Mastery Teacher Edition learning̶and significantly improve both teaching and learning. Cathy Vatterott is an education professor and researcher at the University of Missouri-St. Louis, a former middle school teacher and principal, and a parent of a college graduate. She has learned from her.

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Ratings 100% (14) 14 out of 14 people found this document helpful. This preview shows page 1 - 2 out of 3 pages. OF MICE AND MEN: Chapter 2 Reading and Study Guide I. VOCABULARY: Be able to define the following words and understand them when they appear in the novel.

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Chemistry Matter And Change Study Guide For Content Mastery Teacher Edition

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10.4 Usubstitution Trig Functionsap Calculus

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10.4 U-substitution Trig Functionsap Calculus Worksheet

Calculus II, Section 7.4, #67 Integration of Rational Functions by Partial Fractions One method of slowing the growth of an insect population without the use of pesticides is to introduce into the population a number of sterile males that mate with fertile females but produce no o spring. Let P represeent. AP Calculus AB Mu Alpha Theta Welcome to AP Calculus AB! Contact me here. Need more review? Browse the Algebra II and Pre-Calculus Tabs. AP ® Calculus AB and BC. COURSE AND EXAM DESCRIPTION. AP COURSE AND EXAM DESCRIPTIONS ARE UPDATED PERIODICALLY. Please visit AP Central. Mathematics 104—Calculus, Part I (4h, 1 CU) Course Description: Brief review of High School Calculus, methods and applications of integration, infinite series, Taylor's theorem, first order ordinary differential equations. Use of symbolic manipulation and graphics software in Calculus. Note: This course uses Maple®.

Math 104: Calculus I – Notes

Section 004 - Spring 2014

10.4 U-substitution Trig Functionsap Calculus Answers

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Skeleton NotesComplete Notes Title More Remainder 10.6, 10.9 Remainder 10.6/10.9 Series Estimation & Remainder Sections 10.8-10.10 Sections 10.8-10.10 Taylor (and Maclaurin) Series Section 10.7 Section 10.7 Power Series Introduction Section 10.6 Section 10.6 Alt. Series Test and Abs. Conv. Conv. Tests Section 10.5 Section 10.5 The Ratio and Root Tests Section 10.4 Section 10.4 The Comparison Tests Section 10.3 Section 10.3 The Integral Test Section 10.2 Section 10.2 Introduction to Series Section 10.1 Section 10.1 Sequences Section 9.2 Section 9.2 Linear Differential Equations Section 7.2 Pt 1Pt 2 Section 7.2 Separable Differential Equations Section 8.8 Section 8.8 Probability and Calculus Odd Ans. Section 8.7 Pt. 1Pt. 2Section 8.7 Improper Integrals L'Hopital Section 8.4 Pt. 1Pt. 2Section 8.4 Partial Fraction Decomposition Section 8.3 Pt. 1Pt. 2Section 8.3 Trig. Substitution Section 8.2 Pt. 1Pt. 2Section 8.2 Integrating Trig. Powers Section 8.1 Pt. 1Pt. 2 Section 8.1 Integration By Parts Section 6.6 Section 6.6 Center of Mass Section 6.4 Section 6.4 Surface Area of Revolution Section 6.3 Section 6.3 Arc Length Section 6.2Section 6.2 Volumes Using Cylindrical Shells Section 6.1 Section 6.1 Volumes Using Cross-Sections disk/washer Review Calc I Review Calc I ReviewLimit, Derivative, and Integral Area b/w CurvesArea b/w Curves Video Example U-substitution Graphs you should know

Print out the skeleton notes before class and bring them to class so that you don't have to write down https://foxspain82.tumblr.com/post/657282647494672384/achievement-unlocked-2watermelon-gaming. Hide paragraph marks in microsoft word for mac. everything said in class. If you miss anything, the complete notes will be posted after class.

10.4 U-substitution Trig Functionsap Calculus Pdf

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Version #1 The course below follows CollegeBoard's Course and Exam Description. Lessons will begin to appear starting summer 2020. BC Topics are listed, but there will be no lessons available for SY 2020-2021

Unit 0 - Calc Prerequisites (Summer Work) 0.1 Summer Packet

Unit 1 - Limits and Continuity 1.1 Can Change Occur at an Instant? 1.2 Defining Limits and Using Limit Notation 1.3 Estimating Limit Values from Graphs 1.4 Estimating Limit Values from Tables 1.5 Determining Limits Using Algebraic Properties (1.5 includes piecewise functions involving limits) 1.6 Determining Limits Using Algebraic Manipulation 1.7 Selecting Procedures for Determining Limits (1.7 includes rationalization, complex fractions, and absolute value) 1.8 Determining Limits Using the Squeeze Theorem 1.9 Connecting Multiple Representations of Limits Mid-Unit Review - Unit 1 1.10 Exploring Types of Discontinuities 1.11 Defining Continuity at a Point 1.12 Confirming Continuity Over an Interval 1.13 Removing Discontinuities 1.14 Infinite Limits and Vertical Asymptotes 1.15 Limits at Infinity and Horizontal Asymptotes 1.16 Intermediate Value Theorem (IVT) Review - Unit 1

Unit 2 - Differentiation: Definition and Fundamental Properties 2.1 Defining Average and Instantaneous Rate of Change at a Point 2.2 Defining the Derivative of a Function and Using Derivative Notation (2.2 includes equation of the tangent line) 2.3 Estimating Derivatives of a Function at a Point 2.4 Connecting Differentiability and Continuity 2.5 Applying the Power Rule 2.6 Derivative Rules: Constant, Sum, Difference, and Constant Multiple (2.6 includes horizontal tangent lines, equation of the normal line, and differentiability of piecewise) 2.7 Derivatives of cos(x), sin(x), e^x, and ln(x) 2.8 The Product Rule 2.9 The Quotient Rule 2.10 Derivatives of tan(x), cot(x), sec(x), and csc(x) Review - Unit 2

Unit 3 - Differentiation: Composite, Implicit, and Inverse Functions 3.1 The Chain Rule 3.2 Implicit Differentiation 3.3 Differentiating Inverse Functions 3.4 Differentiating Inverse Trigonometric Functions 3.5 Selecting Procedures for Calculating Derivatives 3.6 Calculating Higher-Order Derivatives Review - Unit 3

Unit 4 - Contextual Applications of Differentiation 4.1 Interpreting the Meaning of the Derivative in Context 4.2 Straight-Line Motion: Connecting Position, Velocity, and Acceleration 4.3 Rates of Change in Applied Contexts Other Than Motion 4.4 Introduction to Related Rates 4.5 Solving Related Rates Problems 4.6 Approximating Values of a Function Using Local Linearity and Linearization 4.7 Using L'Hopital's Rule for Determining Limits of Indeterminate Forms Review - Unit 4

Unit 5 - Analytical Applications of Differentiation 5.1 Using the Mean Value Theorem 5.2 Extreme Value Theorem, Global Versus Local Extrema, and Critical Points 5.3 Determining Intervals on Which a Function is Increasing or Decreasing 5.4 Using the First Derivative Test to Determine Relative Local Extrema 5.5 Using the Candidates Test to Determine Absolute (Global) Extrema 5.6 Determining Concavity of Functions over Their Domains 5.7 Using the Second Derivative Test to Determine Extrema Mid-Unit Review - Unit 5 5.8 Sketching Graphs of Functions and Their Derivatives 5.9 Connecting a Function, Its First Derivative, and Its Second Derivative (5.9 includes a revisit of particle motion and determining if a particle is speeding up/down.) 5.10 Introduction to Optimization Problems 5.11 Solving Optimization Problems 5.12 Exploring Behaviors of Implicit Relations Review - Unit 5

Unit 6 - Integration and Accumulation of Change 6.1 Exploring Accumulation of Change 6.2 Approximating Areas with Riemann Sums 6.3 Riemann Sums, Summation Notation, and Definite Integral Notation 6.4 The Fundamental Theorem of Calculus and Accumulation Functions 6.5 Interpreting the Behavior of Accumulation Functions Involving Area Mid-Unit Review - Unit 6 6.6 Applying Properties of Definite Integrals 6.7 The Fundamental Theorem of Calculus and Definite Integrals 6.8 Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notation 6.9 Integrating Using Substitution 6.10 Integrating Functions Using Long Division and Completing the Square 6.11 Integrating Using Integration by Parts (BC topic) 6.12 Integrating Using Linear Partial Fractions (BC topic) 6.13 Evaluating Improper Integrals (BC topic) 6.14 Selecting Techniques for Antidifferentiation Review - Unit 6

Unit 7 - Differential Equations 7.1 Modeling Situations with Differential Equations 7.2 Verifying Solutions for Differential Equations 7.3 Sketching Slope Fields 7.4 Reasoning Using Slope Fields 7.5 Euler's Method (BC topic) 7.6 General Solutions Using Separation of Variables 7.7 Particular Solutions using Initial Conditions and Separation of Variables 7.8 Exponential Models with Differential Equations 7.9 Logistic Models with Differential Equations (BC topic) Review - Unit 7

Unit 8 - Applications of Integration 8.1 Average Value of a Function on an Interval 8.2 Position, Velocity, and Acceleration Using Integrals 8.3 Using Accumulation Functions and Definite Integrals in Applied Contexts 8.4 Area Between Curves (with respect to x) 8.5 Area Between Curves (with respect to y) 8.6 Area Between Curves - More than Two Intersections Mid-Unit Review - Unit 8 8.7 Cross Sections: Squares and Rectangles 8.8 Cross Sections: Triangles and Semicircles 8.9 Disc Method: Revolving Around the x- or y- Axis 8.10 Disc Method: Revolving Around Other Axes 8.11 Washer Method: Revolving Around the x- or y- Axis 8.12 Washer Method: Revolving Around Other Axes 8.13 The Arc Length of a Smooth, Planar Curve and Distance Traveled (BC topic) Review - Unit 8

Unit 9 - Parametric Equations, Polar Coordinates, and Vector-Valued Functions (BC topics) 9.1 Defining and Differentiating Parametric Equations 9.2 Second Derivatives of Parametric Equations 9.3 Arc Lengths of Curves (Parametric Equations) 9.4 Defining and Differentiating Vector-Valued Functions 9.5 Integrating Vector-Valued Functions 9.6 Solving Motion Problems Using Parametric and Vector-Valued Functions 9.7 Defining Polar Coordinates and Differentiating in Polar Form 9.8 Find the Area of a Polar Region or the Area Bounded by a Single Polar Curve 9.9 Finding the Area of the Region Bounded by Two Polar Curves Review - Unit 9

Unit 10 - Infinite Sequences and Series (BC topics) 10.1 Defining Convergent and Divergent Infinite Series 10.2 Working with Geometric Series 10.3 The nth Term Test for Divergence 10.4 Integral Test for Convergence 10.5 Harmonic Series and p-Series 10.6 Comparison Tests for Convergence 10.7 Alternating Series Test for Convergence 10.8 Ratio Test for Convergence 10.9 Determining Absolute or Conditional Convergence 10.10 Alternating Series Error Bound 10.11 Finding Taylor Polynomial Approximations of Functions 10.12 Lagrange Error Bound 10.13 Radius and Interval of Convergence of Power Series 10.14 Finding Taylor Maclaurin Series for a Function 10.15 Representing Functions as a Power Series Review - Unit 8

Version #2 The course below covers all topics for the AP Calculus AB exam, but was built for a 90-minute class that meets every other day. Lessons and packets are longer because they cover more material.

Unit 0 - Calc Prerequisites (Summer Work) 0.1 Things to Know for Calc 0.2 Summer Packet 0.3 Calculator Skillz

Unit 1 - Limits 1.1 Limits Graphically 1.2 Limits Analytically 1.3 Asymptotes 1.4 Continuity Review - Unit 1

Unit 2 - The Derivative 2.1 Average Rate of Change 2.2 Definition of the Derivative 2.3 Differentiability (Calculator Required) Review - Unit 2

Unit 3 - Basic Differentiation 3.1 Power Rule 3.2 Product and Quotient Rules 3.3 Velocity and other Rates of Change 3.4 Chain Rule 3.5 Trig Derivatives Review - Unit 3

Unit 4 - More Deriviatvies 4.1 Derivatives of Exp. and Logs 4.2 Inverse Trig Derivatives 4.3 L'Hopital's Rule Review - Unit 4

Unit 5 - Curve Sketching 5.1 Extrema on an Interval 5.2 First Derivative Test 5.3 Second Derivative Test Review - Unit 5

Unit 6 - Implicit Differentiation 6.1 Implicit Differentiation 6.2 Related Rates 6.3 Optimization Review - Unit 6

Unit 7 - Approximation Methods 7.1 Rectangular Approximation Method 7.2 Trapezoidal Approximation Method Review - Unit 7

Unit 8 - Integration 8.1 Definite Integral 8.2 Fundamental Theorem of Calculus (part 1) 8.3 Antiderivatives (and specific solutions) Review - Unit 8

Unit 9 - The 2nd Fundamental Theorem of Calculus 9.1 The 2nd FTC 9.2 Trig Integrals 9.3 Average Value (of a function) 9.4 Net Change Review - Unit 9

Unit 10 - More Integrals 10.1 Slope Fields 10.2 u-Substitution (indefinite integrals) 10.3 u-Substitution (definite integrals) 10.4 Separation of Variables Review - Unit 10

Unit 11 - Area and Volume 11.1 Area Between Two Curves 11.2 Volume - Disc Method 11.3 Volume - Washer Method 11.4 Perpendicular Cross Sections Review - Unit 11

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educationtech · 3 years

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How to Learn Machine Learning Algorithms

In a world where all manual tasks are being automated, the definition of manual is changing. Machine Learning algorithms can help computers to perform surgeries, play chess, and get smarter and more personal. We are living in an era of constant technological progress, and looking at how computing has advanced over the years, students of Top Engineering Colleges can predict what’s to come in the days ahead.

One of the main features of this revolution shows how computing tools and techniques have been democratized. Earlier, data scientists have built sophisticated data-crunching machines by seamlessly executing advanced techniques.

How algorithms can enhance your skills in machine learning?

A data scientist or a machine learning enthusiast allow students of top private engineering colleges in Jaipur to use these techniques to create functional Machine Learning projects. There are certain types of Machine Learning techniques including supervised learning, unsupervised learning, and reinforcement learning. All these techniques are used in this list of common Machine Learning Algorithms. Some of them are as follows:

Machine Learning Algorithms1. Linear Regression

To understand the working functionality of this algorithm, students of Top BTech Colleges imagine how they would arrange random logs of wood in increasing order of their weight. However, they cannot weigh each log. They have to guess its weight just by looking at the height and girth of the log (visual analysis) and arrange them using a combination of these visible parameters.

In this process, a relationship is established between dependent and independent variables by fitting them to a line. This line is popular as the regression line and represented by a linear equation Y= a *X + b. In this equation:

Y – Dependent Variable

a – Slope

X – Independent variable

b – Intercept

The coefficients a & b are derived by minimizing the sum of the squared difference of distance between both data points and the regression line.

2. Logistic Regression

Logistic Regression is used by the students of computer science engineering colleges in Rajasthan to estimate discrete values (usually binary values like 0/1) from a set of independent variables. It will predict the probability of an event by fitting data to a logit function. Also, it is called as logit regression.

These methods are often used to help improve logistic regression models include interaction terms, eliminate features, regularize techniques, and use a non-linear model.

3. Decision Tree

It is one of the most popular machine learning algorithms in use. Today, it is used as supervised learning algorithm that is used for classifying problems. It works well classifying for both continuous dependent and categorical variables. In this algorithm, they can split the population into two or more homogeneous sets based on the most important attributes/ independent variables.

4. SVM (Support Vector Machine)

SVM is a method of classification in which they plot raw data as points in an n-dimensional space (where n is the number of features you have). Then, the value of each feature is tied to a particular coordinate that makes it easy for them to classify the data. Lines called classifiers can be used to split the data and plot them on a different graph.

5. Naive Bayes

A Naive Bayes classifier assumes that the presence of a specific feature in a class is unrelated to the presence of any other feature. Even if these features are connected to each other, a Naive Bayes classifier would consider all of these properties independently when calculating the probability of a specific outcome. A Naive Bayesian model is easy to build and useful for massive datasets. It is simple and known to outperform even highly sophisticated classification methods.

6. KNN (K- Nearest Neighbors)

This algorithm can be applied to both regression and classification problems. Apparently, within the Data Science industry, it is more widely used to solve classification problems. It is a simple algorithm that stores all available cases and classifies any new cases by taking a majority vote of its “k” neighbours. Then, the case is assigned to the class with which it has the most in common. A distance function mainly performs this measurement.

KNN can be easily understood by comparing it to real life. For instance, students of engineering colleges Rajasthan want information about a person, it makes sense to talk to his or her friends and colleagues. Things to consider before selecting KNN includes computationally expensive, Variables should be normalized, or else higher range variables can bias the algorithm, Data still needs to be pre-processed.

7. K-Means

It is an unsupervised algorithm that solves clustering problems of the students of computer science engineering colleges in Jaipur. Data sets are classified into a specific number of clusters (let’s call that number K) in such a way that all the data points within a cluster are homogenous and heterogeneous from the data in other clusters. With these new centroids, the closest distance for each data point is determined. This process is repeated until the centroids do not change.

8. Random Forest

A collective of decision trees is known as Random Forest. To classify a new object based on its attributes, each tree is classified, and the tree “votes” for that class. The forest chooses the classification having the most votes. Each tree is planted & grown as the following:

a. If the number of cases in the training set is N, then a sample of N cases is taken at random. This sample will be the training set for growing the tree.

b. If there are M input variables, a number m<<M is specified like each node, m variables are selected at random out of the M, and the best split on this m is used to split the node. The value of m is held specifically during this process.

c. Each tree is grown to the most substantial extent possible without pruning.

9. Dimensionality Reduction Algorithms

In today’s world, large amounts of data are being stored and analyzed by corporates, government agencies, and research organizations. As a data scientist, students of top BTech college in India know that this raw data contains a lot of information. This challenge is in identifying significant patterns and variables. Dimensionality reduction algorithms like Factor Analysis, Decision Tree, Missing Value Ratio, and Random Forest can help them find relevant details.

10. Gradient Boosting & AdaBoost

The boosting algorithms used when massive loads of data have to be handled to make predictions with high accuracy. Boosting is an ensemble learning algorithm that combines the predictive power of different base estimators to improve robustness.

In other words, it combines multiple week or average predictors to build a strong predictor. These boosting algorithms always work properly in data science competitions like Kaggle, AV Hackathon, CrowdAnalytix. Today, these are the most preferred machine learning algorithms.

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ncertsolutionsmath · 3 years

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Download NCERT Solutions for Class 8 Maths Chapter Wise

NCERT Solutions for Class 8 Maths incorporates all the inquiries gave in the NCERT course reading that is endorsed for Class 8 as per the CBSE Board. The schedule given by the CBSE for all the classes is totally founded on the NCERT educational plan. Henceforth, planning of tests utilizing the NCERT Solutions will give understudies a favorable position of scoring admirably.

My Way Teaching NCERT Class 8 Solutions for Maths has been intended to assist the understudies with addressing the CBSE Class 8 Maths issues effortlessly. The CBSE 8th Class Solutions for Maths gave here comes decidedly ready activities alongside definite clarifications organized by our master educators that further makes learning and comprehension of ideas a simple assignment. Thus, if understudies have been searching for the most point by point and precise NCERT Maths Solutions Class 8, which are likewise free, this is the correct spot.

Chapter Wise NCERT Solutions for Class 8 Maths NCERT Solutions Class 8 Maths provided here are very beneficial for students. They can use these NCERT Solutions for Class 8 as a reference while practising math problems and develop better math skills along with preparing efficiently for the exams. The math exercises are given at the end of every chapter to practise the concepts learned in the chapter.

NCERT Solutions for Class 8 Maths Chapter 1 Rational Numbers In this chapter, students will be learning properties related to real numbers, integers, whole numbers, rational numbers and natural numbers – independent, associative and closure. The chapter deals with the role of zero and one, multiplication over addition as well as the representation of Rational Numbers on the number line, along with searching for Rational Numbers between two Rational Numbers. The topics of additive identity (0) and multiplicative identity (1) are also covered in this chapter. There are 2 exercises in this chapter, which contain questions from all the topics present in the chapter.

NCERT Solutions for Class 8 Maths Chapter 2 Linear Equations in One Variable The chapter, Linear Equations in One Variable, deals with linear expression in one variable alone. Here the chapter will be dealing with the equations with linear expressions in one variable only. Such equations are known as Linear Equations in One Variable. There are six exercises in this chapter with a total of 65 questions in them. The chapter includes different concepts including- solving an equation of Linear Equations in One Variable, some of its applications, reducing equations to a simpler form and word problems related to Linear Equations in One Variable.

NCERT Solutions for Class 8 Maths Chapter 3 Understanding Quadrilaterals The chapter, Understanding Quadrilaterals, of Class 8 Maths, as the name says, provides the students with a proper understanding of the quadrilaterals. The chapter also covers different kinds of polygons, including triangles, quadrilaterals, pentagon, hexagon etc. The chapter contains 4 exercises which aid the students in understanding these shapes properly.

NCERT Solutions for Class 8 Maths Chapter 4 Practical Geometry The chapter, Practical Geometry, includes the method of constructing a quadrilateral with various parameters given. The chapter consists of 5 exercises, each dealing with different methods of constructing the quadrilateral. For instance, the first exercise deals with the construction of quadrilaterals when the length of four sides and a diagonal are given; similarly, the second exercise involves the method of construction of quadrilaterals when two diagonals and three sides are given, and so on. Hence, we can conclude that learning this chapter will make the students well-versed in all the concepts of quadrilateral for further classes.

NCERT Solutions for Class 8 Maths Chapter 5 Data Handling The collection of information that can be used for analysis is known as data. In this chapter, students learn about the organisation and representation of data. Organising the data in a systematic method is referred to as Data Handling. In this chapter, students learn to represent these data diagrammatically as a pictograph, a bar graph, double bar graph, pie chart and histogram. During the course of this chapter, students will be briefed about the concept of probability or likelihood. There are 3 exercises in this chapter that deal with all the concepts covered in the chapter.

NCERT Solutions for Class 8 Maths Chapter 6 Square and Square Roots This chapter, Squares and Square Roots, helps the students learn about the concept of square number and the square root of a number. The chapter deals with properties of square numbers, interesting patterns that could be learned using square numbers, finding the square of a number, Pythagorean triplets, finding square roots through various methods, Square roots of decimals and much more.

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mathematicianadda · 4 years

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Coding Stories: Programming Polygons https://ift.tt/3eyZ6FV

In this guest post, Greg Benedis-Grab shares the story of when he discovered Pick’s Theorem, and how he coded an interactive version to play with.

Have you ever been intoxicated by a mathematical theorem? Well that’s what happened to a class of 9th grade geometry students at my school. Their enthusiastic teacher showed them Pick’s theorem.

“What is Pick’s Theorem?” I asked after she described the reaction of her students. So she showed me an example:

https://en.wikipedia.org/wiki/Pick%27s_theorem

Using a lattice of nails and a rubber band, you can create geometric shapes where each vertex is located on a lattice point. As she was telling me about it I remembered this fun math manipulative from my own Elementary school days. The theorem requires you to categorize some of the lattice points as boundary points, b and interior points, i. The expression for the area is given below.

\[\textcolor{red}{i=7} \qquad \textcolor{green}{b=8}\]

\[A = \textcolor{red}{i} + \frac{\textcolor{green}{b}}{2}\ – 1 = 10\]

That is a pretty surprising result – at least, that is what the 9th grade students thought. I will leave the proof of this to the mathematicians. Here is a link to a particularly engaging mathematician doing just that. I am more interested in displaying colorful shapes in the browser so I decided it would make for a nice p5.js sketch. To get this project started I needed a very simple case. Say you have a 10×10 grid and the simplest of polygons, the triangle.

A triangle is defined by its three vertices. To organize my code I will create a Vetex class to store the x,y values for each vertex in the triangle.

class Vertex { constructor(x, y) { this.x = x; this.y = y; } show(spacing) { fill(200); // light grey circle around each vertex circle(this.x*spacing, this.y*spacing,30); } }

The spacing variable allows me to space out the grid points according to the window size. Next I decided to draw all the grid points on the screen.

function drawBoard(spacing) { stroke(0); strokeWeight(8); for (let i=0;i<10;i++) { for (let j=0;j<10;j++) { point(i*spacing,j*spacing); } } }

Finally, I need to draw and fill in the triangle. I included this code in the draw loop to do so.

fill(135,180,70,100); // green strokeWeight(2); stroke(255,0,255); // violet beginShape(); vertices.forEach((v)=>vertex(v.x*spacing, v.y*spacing)); let v = vertices[0]; vertex(v.x*spacing, v.y*spacing); endShape();

I also added the ability to drag vertices to new positions. To do this I included the following methods to the Vertex class

mousePressed(mouse) { let pos = createVector(this.x*spacing, this.y*spacing); pos.add(offset); // offset allows the lattice to be centered if (pos.dist(mouse) < 30) { this.dragging = true; // assigns Vertex attribute } } update() { if (this.dragging) { let x = int(mouseX/spacing -0.5); let y = int(mouseY/spacing - 0.5)-1; if (x>=0 && x<10 && y>=0 && y<10) { this.x = x; this.y = y; } } }

The dragging attribute indicates that one of the vertices is being dragged and should update its position with the mouse. The fact that the vertices are objects means that each of the vertex objects has its own dragging attribute and can be independently dragged. Objects will also allow me later to scale up this project to have an arbitrary number of vertices in my shape. Finally, I had to add the following functions to sketch.js in order to handle the click events. The first defines a predefined mousePressed() function that subsequently passed the mouse position to an identically named method in the Vertex class. That method checks if the mouse event occurred on the given vertex. The second removes the dragging attribute from all vertices when the mouse is released.

function mousePressed() { let mouse = createVector(mouseX,mouseY); // makes new triangle when title is clicked if (mouseY<0) makeTriangle(); vertices.forEach((v)=>v.mousePressed(mouse)); }function mouseReleased() { for (let v of vertices) if (v.dragging) delete v.dragging; }

Finally I made sure the update() and show() methods are run for each vertex object.

vertices.forEach((v)=>{ v.show(); v.update(); });

And that creates the following exciting web app. Go ahead and make some triangles.

Of course at this point all I have created is a triangle generator. We have not yet explored Pick’s Theorem. To do that I need to determine which lattice points are on the boundary and which points are interior. Before moving to this important step I cleaned up the code. It is important to take time to clean up your code as you expand it. Otherwise it can become unwieldy and impossible to debug. Well organized code can make complex problems simple and elegant to solve. The simplicity of the code often hides the hard work that goes into simplifying it and finding the right organization. The organization scheme serves as a foundation upon which new functionality can be built. I added a Triangle class and a Board class to modularize the program making my methods and attributes easy to manipulate.

class Triangle { constructor() { this.vertices = []; while(this.vertices.length<3) { const x = int(random(board.w)); const y = int(random(board.h)); const v = new Vertex(x,y); // only add the point if it is not on another one. if (!this.isVertex(x,y)) this.vertices.push(v); } } isVertex(i,j) { for (let v of this.vertices) { if (i == v.i && j == v.j) return true; } return false; } update() { this.vertices.forEach((v)=>v.update()); } show() { // draw the triangle as shown previously } }class Board { constructor() {} // set up board and creates a triangle obj. showLattice() {} // draws the lattice points update() { this.triangle.update(); } menu() {} // group all the menu info together show() { this.triangle.show(); this.showLattice(); this.menu(); } }

Next I spent a fair amount of time thinking about how to mathematically define the boundary and the interior of the shape. I needed a good algorithm to accomplish this task. This is where the story gets interesting.

Computers can solve challenging mathematical problems. They perform calculations at astonishing speeds. The tricky part of this problem is that I am not sure what calculation needs to be performed. If I have a point (x,y) how will I know if it is on the boundary of my shape, inside of my shape or neither of these two conditions. To be honest I first stared at the problem for a long time. Then I noticed that the triangle has three line segments associated with it that can be drawn between pairs of vertices. These three lines can be expressed as linear equations. For points in the lattice that are located on one of the three line segments they are by definition boundary points. Checking the slope and that the points coordinates lie between the segment endpoint coordinates should be sufficient. Here is how I wrote the code.

colinear(i,j,a,b) {return (j-a.j)*(b.i-a.i) == (b.j-a.j)*(i-a.i);} inbounds(i,j,a,b) { return i >= min([a.i,b.i]) && i <= max([a.i,b.i]) && j >= min([a.j,b.j]) && j <= max([a.j,b.j]); }isBoundary(x,y) { let v0 = this.vertices[0]; let v1 = this.vertices[1]; let v2 = this.vertices[2]; // checks if it is on any of the 3 line segments return (this.colinear(x,y,v0,v1) && this.inbounds(x,y,v0,v1) || this.colinear(x,y,v1,v2) && this.inbounds(x,y,v1,v2) || this.colinear(x,y,v2,v0) && this.inbounds(x,y,v2,v0)) }

For the interior points I can simply check that the given lattice point satisfies the inequality for the three line segments that make up the triangle. Then I wondered, how do I know which side of the inequality is correct. The third point of the triangle, the point that is not part of the given edge should match the correct inequality for the given point. In other words if the lattice point is on the same side of the line as the third point then it satisfies that edge’s inequality. Now I just need to test all three edges and I can define the interior lattice points for the triangle. Here is how I coded this in the Triangle class.

inEq(x,y) {return (j-a.j)*(b.i-a.i) < (b.j-a.j)*(i-a.i);} isInterior(x,y) { let v0 = this.vertices[0]; let v1 = this.vertices[1]; let v2 = this.vertices[2]; //check that the point has the same inequality outcome as the 3rd //point for each of the line segments. return (this.inEq(x,y,v0,v1) == this.inEq(v2.x,v2.y,v0,v1) && this.inEq(x,y,v1,v2) == this.inEq(v0.x,v0.y,v1,v2) && this.inEq(x,y,v2,v0) == this.inEq(v1.x,v1.y,v2,v0))

Now my app can show the number of green interior points and pink boundary points and count them on the top of the screen. I also added some animation movement to the triangle and a slider to control the speed of the animation.

This is a fun way to explore Pick’s Theorem. A teacher might use it to have students explore various triangles and see if they can come up with the theorem themselves. They can also look for patterns in the two output values to gain more insight into triangles. So one way that p5 can transform student mathematical thinking is to create a catalogue of engaging sketches such as these. Even better mathematics teachers can make their own sketches to target learning based on their knowledge about the students in their classes. For me the real learning happened when I tried to write the sketch myself. Getting the program to work deepened my level of geometric understanding and gave me an appreciation for Pick’s theorem.

Unfortunately, it is difficult to incorporate this kind of work in our high school math courses because of how much time it consumes. In addition to figuring out the mathematics I had to spend a fair amount of time debugging my code making sure that my calculations were working as expected. A number of times I had to debug the code and think systematically through the program to find my own error. The debugging process was equal in intellectual rigor to the rules I created for the inequalities. Especially since many math teachers and students are not as comfortable with coding as I am this aspect of the process could dominate and stifle the math learning. However despite these challenges I think it is still worth thinking about how we can use problems like this in math class to deepen student thinking and make the process of mathematics more playful and interactive.

The teacher who introduced this problem to me was nice enough to invite me in to her class to share my web app. I first talked to them a little about p5 and how you make shapes on the screen. Then I explained the dilemma with defining the boundary points and the interior points. I challenged them to develop their own algorithmic steps to make this determination for each lattice point. They came up with my solution very quickly. I was a bit humbled by how long it had taken me. Once they explained their algorithm I showed them how I coded it using JavaScript. They wanted to extend the idea to polygons with more than three vertices. We quickly realized that for convex polygons the same approach would work. I just had to come up with a way to handle more vertices. So I wrote the following loop to find all the edges of a polygon defined by an array of vertices. Notice that I renamed the attribute .triangle to now be .polygon

vs = board.polygon.vertices; for (let i = 0, j = vs.length - 1; i < vs.length; j = i++) { // each iteration of the loop will identify a unique edge // defined by the points vs[i] and vs[j] }

With this loop I can now check all the edges to see if a lattice point is a boundary point. I can also perform my inequality function on all the edges to see if it is interior (though I also need to pick one of the non-colinear points). Since I used a Vertex class to define the vertices and stored them in an array in the triangle shape it was fairly straightforward to change the name of the class to Polygon and adjust the length of the .vertices attribute in the Polygon object. By using an object oriented approach I was able to quickly scale up my idea to all possible lattice polygons.

However my definition of interior points becomes problematic when one of the vertices is concave. In that case one of the rays that extends from the edge intersects the interior of the shape making it more challenging to use the inequality to define the interior. It is possible that we can work around this using a more complex set of Boolean expressions.

Instead I shared with the class a new way to define the interior of a polygon. Ray casting is an elegant way to determine if a point is inside of a polygon. Draw a ray starting at each lattice point to infinity. If the ray passes through an odd number of edges it is interior. Otherwise it is not. If we simply draw a horizontal ray to the right we can algebraically determine the number of crossings. Here is the code I used to determine if a lattice point is inside the polygon defined by the array board.polygon.vertices

let inside = false; vs = board.polygon.vertices; for (let i = 0, j = vs.length - 1; i < vs.length; j = i++) { let xi = vs[i].i, yi = vs[i].j; let xj = vs[j].i, yj = vs[j].j; let intersect=((yi > y) != (yj > y)) && (x < (xj - xi) * (y - yi) / (yj - yi) + xi); if (intersect) inside = !inside; }

With this new algorithm I created a new app using the p5.js editor.

https://editor.p5js.org/gbenedisgrab/sketches/jAIo-Tzev

I let the class play with this new and improved Pick’s Theorem Tool. By the way I added new features so that you can click on a boundary point to make it a vertex. I also made it possible to double click a vertex to remove it. I asked them to explore different shapes and see whether Pick’s theorem always holds and to specifically find when Pick’s theorem does not work. For example if you have a complex polygon (a polygon with edges that intersect) the theorem no longer holds. I then challenged them to find a new expression for complex polygons.

This exploration led to a surprising amount of sophistication. Some students realized that the ray casting rule did not always exactly match the way p5 filled a polygon shape. I might want to explore p5’s algorithm to see how it is defined. Another student noticed that sometimes a ray from an interior point can pass through a vertex and one side of a polygon. This raises an important idea about what it means to pass through an odd number of edges. Additional students worked on coming up with a new algebraic Pick’s expression for complex polygons.

To finish off this article I decided to pursue the intersection between aesthetic beauty and mathematical expression. I decided to create rays that travel across the screen along horizontal paths. Every time the ray passes through a lattice point it will determine how to classify it based on the logic described above – if it lies on the line segment of an edge, it is filled with pink, and if it has passed through an odd number of edges then it is green. Otherwise it is filled with black.

I gave the sketch a radar-like aesthetic, where defined lattice points slowly fade after they are marked. I also lit up the edges as the rays crossed them. Finally I made the polygon randomly transform over time creating a hopefully pleasing dot pattern progression. I will not explain this code in depth, but it is accessible through the glitch sketch linked below.

from The Aperiodical https://ift.tt/3fTHAwj Greg Benedis-Grab from Blogger https://ift.tt/31bh9y2

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