text: Unit 3 Quadratic Tools and Apps. The first abstraction, which is shared by many animals, was probably that of numbers: the realization that a collection of two apples and a collection of two oranges (for example) have something in common, namely the quantity of their members. x n = a number. When and where does your problem occur? What are the causes of your problem? Think about all the possible causes. Have fun learning about numbers and math with these free number games for kids. This Solver (Compound Interest Solver) was created by by jim_thompson5910(35256) : View Source, Show, Put on YOUR site About jim_thompson5910: If you need more math help, then you can email me. Math Solver From adding up numbers, to solve even the complex mathematical problems, Math Solver is a fantastic program for your assistance. Simply named Microsoft Maths Solver, the app's functionalities include solving basic equations to more complex integrations and limits. Beast Academy is published by the Art of Problem Solving® team, which has developed resources for outstanding math students since 1993. “Help me solve this Math problem!” Let us know what kind of task you face problems with, and a dedicated specialist will do it in around 3 hours. Besides, rest assured that you will get custom-tailored assistance. It work 100% real time. Many students find solving algebra word problems difficult. Award winning personalized learning Math program with unlimited practice on any device, anywhere, anytime. You encounter and solve basic math word problems on a daily basis without thinking about it. In this math lesson we will learn how to solve problems involving simple interest. Desmos offers best-in-class calculators, digital math activities, and curriculum to help every student love math and love learning math. Venn Chart is a helpful tool to solve math problems which requires logical thinking and deductive reasoning. This is not just another website that answers math problems. K-12 Math Puzzles – Links to math word problems and puzzles. Solving Inequality Word Questions. Trigonometry of angles and sides can be used on a daily basis in the. There are a number of strategies used in solving math word problems; if you don't have a favorite, try the Math-Drills. Take a photo of your homework question and get answers, math solvers, explanations, and videos. The hardest thing about doing word problems is using the part where you need to take the English words and translate them into mathematics. Word problems on mixed fractrions. This website offers teachers and students authentic mathematics problems based upon NASA press releases, mission science results, and other sources. Weekly workbooks for K-8. Choose from the topics below to create a variety of custom worksheets. To get students to stop relying on key words and think of situations instead, I do an introductory lesson involving four word problems (shown above). In elementary algebra, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. Owl Hat is a tutor and next-level math solver at your fingertips. Use plenty of math operators and keep it as simple as possible. Note: This intervention relies on the teaching of certain common structures of math word/story problems, the identification of the structure by the student, and the visual representation of the story problem before solving. As they reread the word problems, fluency and comprehension increase. Pythagorean theorem word problems. Amy and Beth each had a mix of fifty-cent and ten-cent coins. Solve Absolute Value Inequalities. In this lesson we will talk about simple interest. Interest Rate Problems 2. Free student math practice. These problems will include all four operations (addition, subtraction, multiplication, and division). Check your division skills and your ability to solve division word problems. I don't have a specific problem, but often, I have been in this situation often. All he has to do is point the iPhone’s camera at a math problem, and the app shows the result along with detailed step-by-step instructions. Word Problem Mnemonics. With every upgrade, Owl Hat takes another step toward being a true virtual tutor, and in the meantime, it's the coolest calculator and math s…. 9 Topics 46 Posts Last post by Math Tutor Tue Apr 21, 2020 6:17 am. These simple everyday actions will help your child make the necessary connections to solve 3rd grade math word problems. But figuring out the actual equation can seem nearly impossible. Solving Equations Containing x 3, x 4, etc. In addition to open-ended problems, I have found Math Talks particularly effective with word/story problems and assigned problems that were difficult or confusing to several students' multi-step problem solving. Select problem types to focus on Result Unknown, Change Unknown, and Start Unknown. Epic is the leading digital reading platform—built on a collection of 40,000+ popular, high-quality books from 250+ of the world’s best publishers—that safely fuels curiosity and reading confidence for kids 12 and under. With Photomath Plus, users get access to upgraded features including custom-made solutions and explanations for all problems in specific math textbooks. Mathematics resources for children,parents and teachers to enrich learning. Statistics problems are something that most of the student don't enjoy solving. Over 1000 free skill testing apps and games - tablet and chromebook friendly. Thank you for visiting Figure This! Math Challenges for Families. Time Useful lessons, activities, and tips. corestandards. For instance, Brain Blast explores and encourages divergent thinking, necessary for Step 2, Determining Possibilities and Choices. Get help on the web or with our math app. The concepts which seem boring are more fun and maths is no longer a struggle to solve through pages in order to reach the conclusions. 99) This adventure app asks students to complete different races using math facts. And yes, by all we mean word problems and equations too!. PhotoMath only works with printed text, so unfortunately it won't be solving your scribbles anytime. Choose from the topics below to create a variety of custom worksheets. Problem Solving Games These free maths problems activities are great for teaching and learning the skills needed to solve mathematical problems as they are engaging for young children. Among the reasons why students prefer our service to those of our competitors is because we pay attention to even the minutest deals. After the. USE OWL HAT. Another of the apps that solve math word problems. Build confidence with hands-on learning. Aug 29, 2016. 1 Define Your Problem. Time and work word problems. Supports arithmetic, algebra, calculus, trigonometry. Population Problems 4. This app is designed to help 1st-grade school students understand and solve basic math word problems. At times, you will have problems involving equilibrium expressions with the variable "x" raised to a power "n". So it is better for you to ask for help on math problems once you receive this challenging assignment. Using APKPure App to upgrade 600+ Math Word Problems (LITE), fast, free and save your internet data. You can use this tool for solving quadratic equations or calculating the angles of a triangle. References to complexity and mode refer to the overall difficulty of the problems as they appear in the main program. Identify the graph. How I Teach Word Problems by Andre Toom (PDF). Practise maths online with unlimited questions in more than 200 year 7 maths skills. I charge $2 for steps, or$1 for answers only. And yes, by all we mean word problems and equations too!. If playback doesn't begin shortly, try restarting your device. net and master equation, complex fractions and plenty of other algebra topics. The experts on our team possess all of the needed abilities. Word Problem Mnemonics. Here are some articles that will help you to get more detail about the Solve Problems With Technology so just go through the link. Your elementary grade students will love this Logic and Math (middle school) Word Problems. However, word problems stumped him on a regular basis. com, a math practice program for schools and individual families. Like many kids, my son has little problem with his math facts. Math games online that practice math skills using fun interactive content. Memorizing math facts and mastering basic calculations will get you only so far. Supports arithmetic, algebra, calculus, trigonometry. Solving word problems can get confusing for students. Solving Word Problems Using Tape Diagrams. Basic Math Solver offers you solving online fraction problems, metric conversions, power and radical problems. To study maths you have to roll up your sleeves and actually solve some problems. The description of Math word problems solver game Free math word problems for 3rd grade and math worksheets include various elementary adding and subtraction games for 1st grade. Photomath Plus is a premium addition to the Photomath experience. Stress less, learn better. Classkick is a free app that shows teachers in real-time exactly what students are doing and who needs help so they can provide instant feedback. In this lesson we will talk about simple interest. Math Vs Zombies; Reminiscent of the popular app, Plants vs Zombies, Math vs Zombies allows students to fight virtual zombies from taking over the world by using simple math skills. I highly recommend the program. With Photomath Plus, users get access to upgraded features including custom-made solutions and explanations for all problems in specific math textbooks. Photomath is a new app that not only solves complex math equations but teaches you how to solve them yourself. ’ 5- FX Math Solver ‘FX Math Solver is a comprehensive math software, based on an automatic mathematical problem solving engine, and ideal for students preparing term math exams, ACT, SAT, and GRE…’. Solving Math Word Problems. Come to Pocketmath. Change answer; Math. Solving word problems can be both a challenging and rewarding activity (like many things that are challenging!). Over 1000 free skill testing apps and games - tablet and chromebook friendly. How Mathway works? Type your math problem (or simply scan it using camera). Ratio and proportion word problems. They were chatting about math, and this lad whipped out his iPhone and showed my friend an app that he employs to do his math problems. Download this app from Microsoft Store for Windows 10, Windows 8. Using Bar Models for Multiplication and Division in One-Step Multiplication Word Problems. Moved Permanently. » 2 Print this page. Word math problems and solved exercises count 231. Questioning - Tools for Questioning (From Now On) The Questioning Toolkit provides different kinds of questions, along with tools to answer those questions. Her explanation might help you discover if she needs help with number skills, such as addition, subtraction, multiplication and division, or with the concepts involved in solving the problem. Assists teachers and improves students standardized test performance. Multiplication Word Problems. UMS is the most powerful step-by-step math. An explanation of the common word problem structures can be found on the attached sheets. If you like this Site about Solving Math Problems, please let Google know by clicking the +1 button. MyScript gives the feeling of writing on the paper and solving it over the app. How Mathway works? Type your math problem (or simply scan it using camera). Statistics word problem solver online. While doing your task, we always follow the required stages of general problem-solving methods until we reach the correct conclusion. Percentage word problems Profit and loss word problems Markup and markdown word problems Decimal word problems. Venn Chart is a helpful tool to solve math problems which requires logical thinking and deductive reasoning. You enter your own problems to find the answers. 1 Define Your Problem. Grade 3 Module 1: Properties of Multiplication and Division and Solving Problems with Units of 2–5 and 10 This 25-day module begins the year by building on students’ fluency with addition and knowledge of arrays. Photomath is free and works without wi-fi or data. This app has a child-friendly environment which makes it easy for children. In addition to open-ended problems, I have found Math Talks particularly effective with word/story problems and assigned problems that were difficult or confusing to several students' multi-step problem solving. Each week I give students a different “answer” and they need to write a. Your elementary grade students will love this Logic and Math (middle school) Word Problems. 99) This adventure app asks students to complete different races using math facts. Consider your own. At the moment, the app can't recognise handwritten problems, but does a good job identifying printed ones. How Mathway works? Type your math problem (or simply scan it using camera). By teaching students how to solve the kinds of problems they haven’t seen before, our materials have helped enthusiastic math students prepare for —and win!—the world’s hardest math competitions, then go on to succeed at the most prestigious colleges. $$\frac{a}{b}=\frac{x}{100}$$. You can actually jump difficult hoops of math problems with their help at all times. The students who attended youcubed summer camp describe the power of mindset math in their own words!. Math Tutor DVD provides math help online and on DVD in Basic Math, all levels of Algebra, Trig, Calculus, Probability, and Physics. Forgot Password Log in with Clever. 127 Quadratic Word Problems quadratic word problems, quadratic equation word problems, quadratic function word problems, quadratic word problems projectile motion, quadratic application word problems, solving quadratic word problems, quadratic application problems, quadratics, quadratic equations, projectile motion, consecutive integers. This is a fun game that also helps students master elementary math skills. * Increasing levels of difficulty Easy > Medium > Hard > Expert. Let’s take down the last challenging question together! PSLE challenging math problems: 2015 Paper 2 Question 17 Three girls, Amy, Beth and Cindy had the same number of coins. Use it to complete an hour of physics in 5 minutes!. Basic Math Video Tutorials How to Study for a Math Test Free Basic Math Quiz Improve your skills in topics like algebra, math word problems, arithmetic, and decimals. 3rd Grade Place Value Test When taking this test, students will identify place value of whole numbers through thousands and round numbers to the nearest ten or hundred. Snap a pic of your math problem With our mobile app, you can take a photo of your equation and get started, stat. Besides, rest assured that you will get custom-tailored assistance. As special educators, there is much we can and need to do in this area that calls for so much greater attention than we have typically provided. Solving Equations Containing x 3, x 4, etc. This website offers teachers and students authentic mathematics problems based upon NASA press releases, mission science results, and other sources. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring (direct factoring, grouping, AC method), completing the square, graphing and others. Mathway is the most usable and Best Math Problem Solver App. Designed to target the kids of grades 1st, 2nd, 3rd, 4th, 5th and 6th along with basic review and practice, this app is geared towards highlighting fun math word problems. Access Lesson Plan Resources for Solve proportions: word problems. Note: This intervention relies on the teaching of certain common structures of math word/story problems, the identification of the structure by the student, and the visual representation of the story problem before solving. Math homework isn't the easiest thing to get through quickly, no matter how prepared you are. It's perfect for elementary school. References to complexity and mode refer to the overall difficulty of the problems as they appear in the main program. By teaching students how to solve the kinds of problems they haven’t seen before, our materials have helped enthusiastic math students prepare for —and win!—the world’s hardest math competitions, then go on to succeed at the most prestigious colleges. Photomath is a new app that not only solves complex math equations but teaches you how to solve them yourself. com/itzmeflorenz/ Use my code ITZMEFLORENZ for 12-15% off all YesStyle orders  www. Velocity: The velocity of moving object is modeled by a vector whose direction is the direction of motion and whose magnitude is the speed. PhotoMath also provides a step-by-step guide of how each problem is solved, a feature that some consumers have lauded as a potentially. 127 Quadratic Word Problems quadratic word problems, quadratic equation word problems, quadratic function word problems, quadratic word problems projectile motion, quadratic application word problems, solving quadratic word problems, quadratic application problems, quadratics, quadratic equations, projectile motion, consecutive integers. You enter your own problems to find the answers. How far above the ground is the mark at this point? Solution: This problem requires that we apply much of what we've. Coloring Lettering Worksheets For Educational Kids Kindergarten Free Lowercase Printable With Extraordinary Word Problems In Math Solver Algebra Exam Study Guide One Step Educational Worksheets For Kids School Worksheets preschool tracing worksheets best educational websites for kids learning apps for 4 year olds preschool counting worksheets free educational websites for preschoolers Why. of rational problem-solving and help students to practice specific skills required to successfully solve problems, in a fun and non-threatening manner. By (date), after given real-life word problems and pictures of the word problems, (name) will listen to and read along with the problem 2-3 times, underlining key language that signals use of mathematical operations (e. The do's and don'ts of teaching problem solving in math General advice on how you can teach problem solving in elementary, middle, and high school math. 1 Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e. While doing your task, we always follow the required stages of general problem-solving methods until we reach the correct conclusion. Here’s a heated chat between the editors who stopped doing any. Problem Solving- (Process Standard) • A problem solver understands what has been done, knows why the process was appropriate, and can support it with reasons and evidence. Photomath is a new app that not only solves complex math equations but teaches you how to solve them yourself. Suitable for grades 1 - 8, Math Car Racing lets you choose the expression that will yield the highest number to win. Find out why our students win so many awards. They were chatting about math, and this lad whipped out his iPhone and showed my friend an app that he employs to do his math problems. Once you answer a word problem the answer is shown in a proof. Multiplication Word Problems. Solving word problems can be both a challenging and rewarding activity (like many things that are challenging!). Problem-solving definition: the act or process of finding solutions to problems, esp by using a scientific or | Meaning, pronunciation, translations and examples. The best way to approach word problems is to “divide and conquer”. Get help with your Math Word Problems homework. 100's of new math brain teasers! We have recently added 400 new printable math IQ puzzles for you to solve. If you’re looking to sharpen your competition skills, or just want to engage in good old-fashioned mathematical battle with friends or strangers, then this is the arena for you. By teaching students how to solve the kinds of problems they haven’t seen before, our materials have helped enthusiastic math students prepare for —and win!—the world’s hardest math competitions, then go on to succeed at the most prestigious colleges. That is, most of the times, you’ll be provided with a question, and you’ll be required to select the correct answer. A word problem is a few sentences describing a 'real-life' scenario where a problem needs to be solved by way of a mathematical calculation. It's perfect for elementary school. If you aren't familiar with these games, mark the checkbox below, in order to get instructions, or to set game options, for each Daily game before you play. It is a truly a great piece of math software. When a student looks at a word problem from a situation standpoint, they are reading for meaning and really understanding what operation is required to solve the problem. Choose from the topics below to create a variety of custom worksheets. Math is omnipresent in our everyday world. Inputs Simply enter your linear programming problem as follows 1) Select if the problem is maximization or minimization 2) Enter the cost vector in the space provided, ie in boxes labeled with the Ci. Teacher resources for elementary students. x n = a number. Their share their answers and justify their thought processes with the class. When and where does your problem occur? What are the causes of your problem? Think about all the possible causes. $$\frac{a}{b}=\frac{x}{100}$$. The key to solving 3rd grade math word problems is to find out what, exactly, you are trying to solve. With Photomath, learn how to approach math problems through animated steps and detailed instructions or check your homework for any printed or handwritten problem. Photomath is free and works without wi-fi or data. An app-based recommendation framework for investor adoption of crypto assets. I love how their basic math problem solver lets you try it first to see how it makes math easy before you buy. Aug 29, 2016. Work Word Problems - Sample Math Practice Problems The math problems below can be generated by MathScore. The “Four-Step Problem Solving” plan helps elementary math students to employ sound reasoning and to develop mathematical language while they complete a four-step problem-solving process. You can find area and volume of rectangles, circles, triangles, trapezoids, boxes, cylinders, cones, pyramids, spheres. Daily Puzzles and Games These are the dailies for January 11, 2021. Assists teachers and improves students standardized test performance. Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e. The problem is, the hallway turns and you have to fit your sofa around a corner. Take a picture of your math problems and get answers. Our site offers online addition, subtraction, multiplication and division math word problems, students can practice and learn about measurement and solve equation. In these games, the child is presented with a math problem and must find the creature that's holding the correct answer and smack it on the head with the hammer. As special educators, there is much we can and need to do in this area that calls for so much greater attention than we have typically provided. With every upgrade, Owl Hat takes another step toward being a true virtual tutor, and in the meantime, it's the coolest calculator and math s…. Take a photo of your homework question and get answers, math solvers, explanations, and videos. Learn math, check homework and study for upcoming tests and ACTs/SATs with the most used math learning app in the world! Got tricky homework or class assignments? Get unstuck ASAP with our step-by-step explanations and animations. com, the web equivalent of Mathematica, to solve them. Select the range of sums/minuends for the prob…. In this problem, the $100 is the present value (PV), N is 5, and i is 10%. Download the SuDoku Templates (both 4x4 and 6x6 grids) to use as a reusable overhead or place the templates in sheet protectors so that students can use dry erase markers to solve the posted SuDoku problems and then reuse the templates for future problems. I want to solve a set of equations, linear, or sometimes quadratic. Go to our Lesson Plans index. In order to make their meerkat avatar run and jump fast enough to be a winner, they have to answer each question quickly and. Developer Description: Online math solver with free step by step solutions to algebra, calculus, and other math problems. Before working through the worksheets, discuss with your children any phrases or vocabulary that they may be unsure of. Snap a pic of your math problem With our mobile app, you can take a photo of your equation and get started, stat. Microsoft Math solver app provides help with a variety of problems including arithmetic, algebra, trigonometry, calculus, statistics, and other topics using an advanced AI powered math solver. Dig deeper into specific steps Our solver does what a calculator won't: breaking down key steps into smaller sub-steps to show you every part of the solution. Art of Problem Solving's Premier Game FTW has it all: an unparalleled array of MATHCOUNTS-style problems, elite competition, customizable games, and player rankings. These apps are all timed, helping students to be quicker in solving math problems. Take a photo of your homework question and get answers, math solvers, explanations, and videos. , by using visual fraction models or equations to represent the problem. How Mathway works? Type your math problem (or simply scan it using camera). Solving word problems can be both a challenging and rewarding activity (like many things that are challenging!). We use it a lot for warm ups. Geometry problems test a student’s understanding of the geometric concepts learned in the past, and require students to apply that understanding to solve the problem presented. Problem solving - use acquired knowledge to solve simultaneous equations practice problems System of 3 Equations Word Problem Examples 7:43 Download the app. Problem Solving Games These free maths problems activities are great for teaching and learning the skills needed to solve mathematical problems as they are engaging for young children. A versatile cross-platform mind mapping tool. Go to our Lesson Plans index. Have fun learning about numbers and math with these free number games for kids. You enter your own problems to find the answers. The concepts which seem boring are more fun and maths is no longer a struggle to solve through pages in order to reach the conclusions. common app short essay prompts sythesis com obama michelle thesis best problem solving writer for hire us best paper ghostwriter site ca 800 word essay sample. Shop for Homeschool, Subjects, Math, Math Subjects, Problem Solving & Word Problems and much more. Once you download it, you can take a picture of a math problem with your phone… and the app not only solves the. Problem Solving lesson plans are in "Multidisciplinary" in the "Lessons and Curricula" of the Awesome Library. Three pages of complex math problems draw on critical thinking skills and logic in order to find the correct solution. com/itzmeflorenz/ Use my code ITZMEFLORENZ for 12-15% off all YesStyle orders  www. Mega-Mathematics- lesson ideas on interesting subjects such as coloring and graphing games. Free interactive printable math activities for kids - reinforcement of the curriculum. Used by over 11 million students, IXL provides personalized learning in more than 8,500 topics, covering math, language arts, science, social studies, and Spanish. For me personally, STEM includes an indepth, integrated focus on science and math, and on using the engineering design process to solve real-world problems. Photomath is free and works without wi-fi or data. The history of mathematics can be seen as an ever-increasing series of abstractions. Understand the Problem 2. $$\frac{a}{b}=\frac{x}{100}$$. Time and work word problems. Download Problem Solving Math Challenge and enjoy it on your iPhone, iPad, and iPod touch. Solve calculus and algebra problems online with Cymath math problem solver with steps to show your work. It is a truly a great piece of math software. ‘Solve your algebra problems step-by-step with MathPapa!MathPapa can solve your equations (and show the work!) and help you when you're stuck on your math homework. Welcome to IXL's primary 3 maths page. 31 Mixture Word Problems Mixture Problems, Solving Mixture Problems, solving mixture problem, solve mixture problems, mixture problem, math mixture problems, mixture word problems, algebra mixture problem, solve mixture problem, algebra mixture problems, Linear equations & inequalities, Mixture Word Problems. PBL in mathematics, particularly when completed in teams, helps learners "model with mathematics" as they "apply the mathematics they know to solve problems arising in everyday life, society, and the workplace," "use tools strategically," and "construct viable arguments and critique the reasoning of others," as noted in the Common Core. Problem Solving- (Process Standard) • A problem solver understands what has been done, knows why the process was appropriate, and can support it with reasons and evidence. com gives both interesting and useful tips on solve my math word problem for free, matrix algebra and logarithmic and other algebra subject areas. In this entertaining app, students also widen their understanding of whole numbers, learn about problem solving, study geometry, graphs, and other math essentials. Have fun learning about numbers and math with these free number games for kids. Each week I give students a different “answer” and they need to write a. Learn to Identify the Question. 6 Math Problems That You Can Solve To Earn Thousands Of Dollars In Prize Money. Solve real-world and mathematical problems involving area, surface area, and volume. Keep in mind that some tasks may require more time than others. It indicates the ability to send an email. 99) This adventure app asks students to complete different races using math facts. Constants arise in many areas of mathematics, with constants such as e and π occurring in such diverse contexts as geometry, number theory, and calculus. MathShot! uses advanced character recognition to convert math problems into symbolic expressions in a formal system. Physics Problem Solver online that shows work and takes care of units. In this lesson, the goal is to show you detailed worked solutions of some problems with varying levels of difficulty. Check your division skills and your ability to solve division word problems. How Mathway works? Type your math problem (or simply scan it using camera). A small mark is painted at the very top of the tire, and then the tire is rolled forward slightly so that the mark rotates through an angle of pi/4 radians. Download the SuDoku Templates (both 4x4 and 6x6 grids) to use as a reusable overhead or place the templates in sheet protectors so that students can use dry erase markers to solve the posted SuDoku problems and then reuse the templates for future problems. Already in the first stage of problem solving, understanding stage jobs, it is useful to try to answer the question: “is it possible to satisfy the condition?” is not always able to answer this question, but sometimes it can be done. 2 Solve Problems Using. Problem solving - use acquired knowledge to solve simultaneous equations practice problems System of 3 Equations Word Problem Examples 7:43 Download the app. This problem-solving plan consists of four steps: details, main idea, strategy, and how. [+] Word Problems Video Playlist Word Problems Calculators: (41) lessons. But when faced with a fraction such as 3/8, students. This app is efficiently composed of more than a hundred Symbolab most delegate and powerful calculators. A screencasting app gives children the ability to create a video that shows them writing on their iPad screen and captures their voice as they solve a word problem. January 23, 2018 by Guest Post In an age when attention spans of student is diminishing rapidly and gadgets play a major role in their lives, forward-thinking teachers are trying to find new ways to help their students learn better – and keep them interested in their studies. Get step-by-step answer instantly. Use plenty of math operators and keep it as simple as possible. Once you download it, you can take a picture of a math problem with your phone… and the app not only solves the. Otherwise, solving a math problem becomes a "mere calculation" and the student is viewed as not having true understanding. Most bearing word problems involving trigonometry and angles can be reduced to finding relationships between angles and the measurements of the sides of a triangle. Walt Hickey. Billions of people use this application. Welcome to IXL's year 5 maths page. com/itzmeflorenz/ Use my code ITZMEFLORENZ for 12-15% off all YesStyle orders  www. Skill development steps are incremental and focus on attaining success. Welcome to Big Ideas Math! Let's get you registered. Word math problems and solved exercises count 231. And with Math Word Problems Made Easy: Grade 6, they just might enjoy themselves while doing so. Choose from the topics below to create a variety of custom worksheets. Find and fix missing skills with personalized learning, pinpoint assessments, and motivating rewards. Learn the math behind your money. We will offer you special pricing for purchases of 20 apps or more. colors worksheets for preschoolers free printable. PayForMathHomework is a professional website that solves word problems for you. Practice the Rules of Exponents (JavaScript) New. Our word scramble solver is a tool that helps players solve word scramble puzzles and mobile phone games such as Scrabble and Words With Friends. Another of the apps that solve math word problems. Grade 5 » Number & Operations—Fractions » Use equivalent fractions as a strategy to add and subtract fractions. Learning a consistent problem-solving approach, becoming familiar with and practicing effective problem-solving strategies, and applying these ideas in word-problem contexts help students become more effective problem solvers and mathematicians. 1 Applying Trigonometry with 2. So, from equations to word problems, for a math operation to be worth solving, the need for a solution must have real-life implications to which students can relate. In Class Labs - Students work through a range of problem solving strategies. Statistics word problem solver online. To get students to stop relying on key words and think of situations instead, I do an introductory lesson involving four word problems (shown above). I love how their basic math problem solver lets you try it first to see how it makes math easy before you buy. PhotoMath also provides a step-by-step guide of how each problem is solved, a feature that some consumers have lauded as a potentially. It indicates the ability to send an email. Three Types of Math Word Problems. Welcome to IXL's year 6 maths page. common app short essay prompts sythesis com obama michelle thesis best problem solving writer for hire us best paper ghostwriter site ca 800 word essay sample. Time and work word problems. Use mental math and estimation strategies (such. Enter your problem statement (as algebra, not words!) where it says "Enter a problem" near the bottom of the solver. When a student looks at a word problem from a situation standpoint, they are reading for meaning and really understanding what operation is required to solve the problem. Basic Math Plan. Epic is the leading digital reading platform—built on a collection of 40,000+ popular, high-quality books from 250+ of the world’s best publishers—that safely fuels curiosity and reading confidence for kids 12 and under. Middle school teachers will love the math worksheets that will excite, enrich, challenge, and instruct kids in your class. Usually, once you get the math equation, you're fine; the actual math involved is often fairly simple. Already in the first stage of problem solving, understanding stage jobs, it is useful to try to answer the question: “is it possible to satisfy the condition?” is not always able to answer this question, but sometimes it can be done. Math in Everyday Life. Solving problems with percent. To get students to stop relying on key words and think of situations instead, I do an introductory lesson involving four word problems (shown above). Small technologies can solve big problems. To solve word problems we need to write a set of equations that represent the problem mathematically. What follows is a list of hints and helps. However, word problems can present a real challenge if you don't know how to break them down and find the numbers underneath the story. The key to solving 3rd grade math word problems is to find out what, exactly, you are trying to solve. com Best Math Apps for 2020 : https://ww. Solve word problems involving two-variable equations Z. Free interactive printable math activities for kids - reinforcement of the curriculum. Click on the one that you want to review: 1. Just because you found an answer to your equation does not necessarily mean you are finished with the problem. I don't have a specific problem, but often, I have been in this situation often. Statistics Problem Solver software can solve statistical problems and generate step-by-step solution for it. Whether you're on the bus, in a cave, or just offline, core Photomath is always available. Math Solver Free for Windows 8 is a handy tool for performing frequently used operations used for solving math problems. As you enter your math problems, the solver will show you the Math Format automatically to make sure you have effectively entered the math problem you really want it to solve. As special educators, there is much we can and need to do in this area that calls for so much greater attention than we have typically provided. All he has to do is point the iPhone’s camera at a math problem, and the app shows the result along with detailed step-by-step instructions. This app is designed to help 1st-grade school students understand and solve basic math word problems. It's perfect for elementary school. Using these strategies can also help you to improve your math skills overall. Already in the first stage of problem solving, understanding stage jobs, it is useful to try to answer the question: “is it possible to satisfy the condition?” is not always able to answer this question, but sometimes it can be done. You also get private 1:1 tutoring sessions, plus a world-class curriculum with critical thinking, logical reasoning, and word problems. The practice worksheets are based on Math Essentials textbook and math common core standards. Velocity: The velocity of moving object is modeled by a vector whose direction is the direction of motion and whose magnitude is the speed. Download this app from Microsoft Store for Windows 10, Windows 8. Later when we solve quadratic word problems, my students can choose to solve by factoring or with the Quadratic Formula. After the. PhotoMath, from PhotoPay Limited, will solve all of your complicated mathematical problems for you. Let's solve them together. Welcome to IXL's year 7 maths page. — but the math thing is the slickest part. Suitable for grades 1 - 8, Math Car Racing lets you choose the expression that will yield the highest number to win. A math word problem solver is a tool that is used to help individuals who want to solve any math problem. No need to even type your math problem. com, a math practice program for schools and individual families. doc) Answer the question in the problem. Questioning - Tools for Questioning (From Now On) The Questioning Toolkit provides different kinds of questions, along with tools to answer those questions. For example, students may need a way to figure out what 7 × 8 is or have previously memorized the answer before you give them a word problem that involves finding the answer to 7 × 8. Includes chapters on: Sequences, Problem-solving, Money, Percents, Algebraic Thinking, Negative Numbers, Logic, Ratios, Probability, Measurements, Fractions, Division. Math and Logic Problems for Teenagers. All answers are 20 or less on the iPad and 10 or less on the iPod. Math learning difficulties are common, significant, and worthy of serious instructional attention in both regular and special education classes. This means you need to find the question in the problem. Math Stars Newsletters: problem solving for talented math students Math Worksheets for K-8 students Grades 7-8. An explanation of the common word problem structures can be found on the attached sheets. Unit 4 Exponential Functions Trigonometry Lesson on Word Problems 2. com, the math site with 1000's of free and teacher-made math worksheets, workbooks, math puzzles and brain teasers for kids in primary math education. The Math App For Students. Grade 3 Module 1: Properties of Multiplication and Division and Solving Problems with Units of 2–5 and 10 This 25-day module begins the year by building on students’ fluency with addition and knowledge of arrays. Check your division skills and your ability to solve division word problems. Mathinenglish. Percent problems, straightforward Finding percent change Markup, discount, and tax (easy, hard) Proportions Proportion word problems Similar figures Similar figure word problems Simple and compound interest. Statistics being indeed one hardest branch of mathematics it requires more time and exposure to perform well in the homework. Math Word Problems – Help the bees with honey word problems. UMS is the most powerful step-by-step math. 10 Free Math Apps for Students that Teachers Swear By. We offer cheap help for all academic levels. Model your word problems, draw a picture, and organize information!. 6th Grade Math Worksheets Order Of Operations. 6m members in the AskReddit community. Model your word problems, draw a picture, and organize information!. Interactive questions, awards, and certificates keep kids motivated as they master skills. How many books does she need to sell to make at least$120. You love challenging math problems. Later when we solve quadratic word problems, my students can choose to solve by factoring or with the Quadratic Formula. Before working through the worksheets, discuss with your children any phrases or vocabulary that they may be unsure of. This calculators will solve three types of 'work' word problems. From famine to poverty, water scarcity to business management, or healthcare to education, technology has all the answers…just ask any question! Recommended Articles. Students will solve problems involving multiple steps (for example, students may be asked to first add, then subtract). Most bearing word problems involving trigonometry and angles can be reduced to finding relationships between angles and the measurements of the sides of a triangle. Linear Equations Solving word problems related to exponential and logarithmic functions. Trigonometry of angles and sides can be used on a daily basis in the. There's no avoiding math when it comes to money. Our powerful anagram engine will automatically rearrange the letters into potential words with one click. Primary Grade Challenge Math by Edward Zaccaro. Take a picture of your math problems and get answers. Simply take a picture of the math problem you want to solve. During the class discussion phase of the math task, students can post their solutions on the wall using text and images. PhotoMath does just that, utilizing your smartphone's camera to quickly solve math problems. This Logic and Math (middle school) Word Problems is perfect to practice problem solving skills. Let’s look closely at what it means to be a problem solver, to communicate mathematically and to demonstrate mathematical reasoning ability. There are several maths apps that can solve any problem easily along with steps. Math Stars Newsletters: problem solving for talented math students Math Worksheets for K-8 students Grades 7-8. Help Solving Math Word Problems Solve This Math Problem. Word problems on mixed fractrions. , by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. » 2 Print this page. Also, it will provide a detailed explanation. Simplex Algorithm Calculator is an online application on the simplex algorithm and two phase method. Math Stars Newsletters: problem solving for talented math students Math Worksheets for K-8 students Grades 7-8. Free Algebra 1 worksheets created with Infinite Algebra 1. Keep in mind that some tasks may require more time than others. Online math solver with free step by step solutions to algebra, calculus, and other math problems. Problem solving skills are highly sought after by employers as many companies rely on their employees to identify and solve problems. Select the range of sums/minuends for the prob…. This application solves every problem like Basic math, Pre-Algebra, Trigonometry, Pre-calculus, calculus and so more. While doing your task, we always follow the required stages of general problem-solving methods until we reach the correct conclusion. Story Problems Build student's problems solving skills. Before working through the worksheets, discuss with your children any phrases or vocabulary that they may be unsure of. Math concepts can be explored in a variety of ways using a flexible set of tools to sketch, write, and build equations. By using this website, you agree to our Cookie Policy. Here, students can review multi-digit multiplication, long division and study equations with the help of easy-to-understand diagrams. Problem-solving definition: the act or process of finding solutions to problems, esp by using a scientific or | Meaning, pronunciation, translations and examples. The mnemonic reminds students to work step-by-step through a word problem. Stress less, learn better. r/AskReddit is the place to ask and answer thought-provoking questions. Webmath is a math-help web site that generates answers to specific math questions and problems, as entered by a user, at any particular moment. They use the same models and problem-solving methods from grade to grade, so math concepts stay with them, year after year. Epic is the leading digital reading platform—built on a collection of 40,000+ popular, high-quality books from 250+ of the world’s best publishers—that safely fuels curiosity and reading confidence for kids 12 and under. As they reread the word problems, fluency and comprehension increase. Solving Equations Containing x 3, x 4, etc. 58 MB) Math Word Problems for 2nd Grade focuses on understanding, planning and solving a word problem. In this lesson we will talk about simple interest. For K-12 kids, teachers and parents. Worksheets, learning resources, and math practice sheets for teachers to print. Tool helps minority students improve math word problem solving by boosting reading skills. Equation of a line / linear equation. This calculators will solve three types of 'work' word problems. Why are math word problems SO difficult for elementary school children? Hint: it has to do with a "recipe" that many math lessons follow. From math word problem solver to math review, we have every part included. Get free kid-tested teacher worksheets, workbooks, puzzles, and resources that will help develop each child's knowledge and love of school. Learning a consistent problem-solving approach, becoming familiar with and practicing effective problem-solving strategies, and applying these ideas in word-problem contexts help students become more effective problem solvers and mathematicians. Math Word Problems is targeted to children in Grades 2-5 who struggle with understanding math word problems and how to do them. There is a "trick" to doing work problems: you have to think of the problem in terms of how much each person / machine / whatever does in a given unit of time. OTHER TOPICS. r/AskReddit is the place to ask and answer thought-provoking questions. There are a number of strategies used in solving math word problems; if you don't have a favorite, try the Math-Drills. I hated them as a kid and I hate them even more now that I have to help my kids solve them! However, now I have a powerful tool to help my kids break the cycle of Word. Simply take a picture of the math problem you want to solve. Understand the Problem 2. The probability (chance) is a value from the interval 0;1> or in percentage (0% to 100%) expressing the occurrence of some event. Problem solving - use acquired knowledge to solve simultaneous equations practice problems System of 3 Equations Word Problem Examples 7:43 Download the app. com brings you a greater love of mathematics, both for its beauty and its power to help solve everyday problems. Online math classes, advanced math textbooks, math games, and more for high school and middle school students. PBL in mathematics, particularly when completed in teams, helps learners "model with mathematics" as they "apply the mathematics they know to solve problems arising in everyday life, society, and the workplace," "use tools strategically," and "construct viable arguments and critique the reasoning of others," as noted in the Common Core. Writing out your problem will help to organize information, see it from new angles, and identify the most important issues. Multiplication story problems are multiplication problems presented through a story. Math / Physics Problem Solver This program solves simple math and physics problems stated in English. You can find area and volume of rectangles, circles, triangles, trapezoids, boxes, cylinders, cones, pyramids, spheres. The man makes $10 with each sale, therefore he earns a total of$20. Select problem types to focus on Result Unknown, Change Unknown, and Start Unknown. Printable Math Worksheets Use our Math Worksheet Generators to whip out some printable math worksheets in a snap. This can be challenging for some but introducing your child to different types of word problems can be mentally stimulating for him. Amy and Beth each had a mix of fifty-cent and ten-cent coins. Dyslexics will appreciate the ST Math app, which visually represents and animates math concepts without words. This approach not only complicates the simplest of math problems; it also. com will be the excellent place to explore!. In this blog post, we illustrate some of the common types of word problems involving Multiplication and Division and how to solve them using this powerful heuristic. An app-based recommendation framework for investor adoption of crypto assets. This type of technology gives children a space to draw pictures, annotate an image and narrate a multi-step procedure. How Mathway works? Type your math problem (or simply scan it using camera). There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring (direct factoring, grouping, AC method), completing the square, graphing and others. Velocity: The velocity of moving object is modeled by a vector whose direction is the direction of motion and whose magnitude is the speed. Select the range of sums/minuends for the prob…. Linear Equations Solving word problems related to exponential and logarithmic functions. If you’re looking to sharpen your competition skills, or just want to engage in good old-fashioned mathematical battle with friends or strangers, then this is the arena for you. Solving Word Problems in Chemistry/ 2 cannot solve a problem about nothing. The description of 600+ Math Word Problems (LITE) Over 600 MATH WORD PROBLEMS to solve for elementary and middle school students (Grades 1, 2, 3, 4, 5, 6, 7, 8). of rational problem-solving and help students to practice specific skills required to successfully solve problems, in a fun and non-threatening manner. So for example supposed we have this as our system. With Photomath, learn how to approach math problems through animated steps and detailed instructions or check your homework for any printed or handwritten problem. To solve the problem, one must read the story and figure out what math equation will help find the answer to the question asked at the end of the story. To solve problems with percent we use the percent proportion shown in "Proportions and percent". Epic is the leading digital reading platform—built on a collection of 40,000+ popular, high-quality books from 250+ of the world’s best publishers—that safely fuels curiosity and reading confidence for kids 12 and under. Investigating Exponents challenges students to identify the different patterns found in the units digits of numbers raised to different powers. Practise maths online with unlimited questions in more than 200 primary 3 maths skills. As you enter your math problems, the solver will show you the Math Format automatically to make sure you have effectively entered the math problem you really want it to solve. Math Games provides a suite of free, customizable games and a digital textbook with hundreds of math exercises to suit different students’ abilities and grade levels. Problems,children's solutions,interactivities,games,articles. The concepts which seem boring are more fun and maths is no longer a struggle to solve through pages in order to reach the conclusions. Wolfram Alpha (\$3 on iOS / Android / Windows Phone, free on web): Enter a math problem or word problem to solve and get step-by-step solutions. https://www. I highly recommend the program. Supports arithmetic, algebra, calculus, trigonometry. Word problems build critical problem-solving, higher-order thinking, and reasoning. Okay where we can spread 5y on both sides then we divide. Once proficient, these skills are combined as your child works through complete problems. It’s easy to set up a free Padlet account and create a math problem solving “wall” for each rich math task. However, word problems can present a real challenge if you don't know how to break them down and find the numbers underneath the story. Unit 1 covers the following concepts: add and subtract numbers up to 20; add three or four numbers; find partners and totals; write equations for math mountains; tell if numbers are odd or even (why?) understand doubles facts; solve one-step word problems. Word problems on constant speed. Students work with a partner to come up a with word problems, involving time, money, and simple fractions.


text:
## Meeting new people at ICFP

This afternoon I’ll be getting on a plane to Vancouver for ICFP. I’m looking forward to seeing many friends, of course, but I also enjoy meeting new people—whether or not they are “famous”, whether or not I think they can “advance my career”. So I’ll just throw this out there: if you will be in Vancouver this week and would like to meet me, just leave a comment and I will make a point of trying to find you to chat! I’ll be attending the Haskell Implementor’s Workshop, the Ally Skills Tutorial, ICFP itself, the Haskell Symposium, and FARM, but there’s also plenty of time to chat in the hallway or over a meal.

Posted in haskell | 1 Comment

## Catsters guide is complete!

About a year and a half ago I announced that I had started creating a guide to the excellent series of category theory YouTube videos by the Catsters (aka Eugenia Cheng and Simon Willerton). I am happy to report that as of today, the guide is finally complete!

As far as possible, I have tried to arrange the order so that each video only depends on concepts from earlier ones. (If you have any suggestions for improving the ordering, I would love to hear them!) Along with each video you can also find my cryptic notes; I make no guarantee that they will be useful to anyone (even me!), but hopefully they will at least give you an idea of what is in each video.

If and when they post any new videos (pretty please?) I will try to keep it updated.

Posted in haskell, math, teaching | Tagged | 1 Comment

## The Species of Bracelets

Summary: The species package now has support for bracelets, i.e. equivalence classes of lists up to rotation and reversal. I show some examples of their use and then explain the (very interesting!) mathematics behind their implementation.

I recently released a new version of my species package which adds support for the species of bracelets. A bracelet is a (nonempty) sequence of items which is considered equivalent up to rotation and reversal. For example, the two structures illustrated below are considered equivalent as bracelets, since you can transform one into the other by a rotation and a flip:

In other words, a bracelet has the same symmetry as a regular polygon—that is, its symmetry group is the dihedral group $D_{2n}$. (Actually, this is only true for $n \geq 3$—I’ll say more about this later.)

Bracelets came up for me recently in relation to a fun side project (more on that soon), and I am told they also show up in applications in biology and chemistry (for example, bracelet symmetry shows up in molecules with cycles, which are common in organic chemistry). There was no way to derive the species of bracelets from what was already in the library, so I added them as a new primitive.

Let’s see some examples (later I discuss how they work). First, we set some options and imports.

ghci> :set -XNoImplicitPrelude
ghci> :m +NumericPrelude
ghci> :m +Math.Combinatorics.Species

Unlabelled bracelets, by themselves, are completely uninteresting: there is only a single unlabelled bracelet shape of any positive size. (Unlabelled species built using bracelets can be interesting, however; we’ll see an example in just a bit). We can ask the library to tell us how many distinct size-$n$ unlabelled bracelets there are for $n \in \{0, \dots, 9\}$:

ghci> take 10 $unlabelled bracelets [0,1,1,1,1,1,1,1,1,1]  Labelled bracelets are a bit more interesting. For $n \geq 3$ there are $(n-1)!/2$ labelled bracelets of size $n$: there are $(n-1)!$ cycles of size $n$ (there are $n!$ lists, which counts each cycle $n$ times, once for each rotation), and counting cycles exactly double counts bracelets, since each bracelet can be flipped in one of two ways. For example, there are $(5-1)!/2 = 24/2 = 12$ labelled bracelets of size $5$. ghci> take 10$ labelled bracelets
[0,1,1,1,3,12,60,360,2520,20160]

In addition to counting these, we can exhaustively generate them (this is a bit annoying with the current API; I hope to improve it):

ghci> enumerate bracelets [0,1] :: [Bracelet Int]
[<<0,1>>]

ghci> enumerate bracelets [0..2] :: [Bracelet Int]
[<<0,1,2>>]

ghci> enumerate bracelets [0..3] :: [Bracelet Int]
[<<0,1,2,3>>,<<0,1,3,2>>,<<0,2,1,3>>]

And here are all $12$ of the size-$5$ bracelets, where I’ve used a different color to represent each label (see here for the code used to generate them):

As a final example, consider the species $B \times E^2$, the Cartesian product of bracelets with ordered pairs of sets. That is, given a set of labels, we simultaneously give the labels a bracelet structure and also partition them into two (distinguishable) sets. Considering unlabelled structures of this species—that is, equivalence classes of labelled structures under relabelling—means that we can’t tell the labels apart, other than the fact that we can still tell which are in the first set and which are in the second. So, if we call the first set “purple” and the second “green”, we are counting the number of bracelets made from (otherwise indistinguishable) purple and green beads. Let’s call these binary bracelets. Here’s how many there are of sizes $0$ through $14$:

ghci> let biBracelets = bracelet >< (set * set)
ghci> take 15 $unlabelled biBracelets [0,2,3,4,6,8,13,18,30,46,78,126,224,380,687]  Let’s use the OEIS to check that we’re on the right track: ghci> :m +Math.OEIS ghci> let res = lookupSequence (drop 1 . take 10$ unlabelled biBracelets)
ghci> fmap description res
Just "Number of necklaces with n beads of 2 colors, allowing turning over."

Unfortunately the species library can’t currently enumerate unlabelled structures of species involving Cartesian product, though I hope to fix that. But for now we can draw these purple-green bracelets with some custom enumeration code. You can see the numbers $2, 3, 4, 6, 8, 13$ show up here, and it’s not too hard to convince yourself that each row contains all possible binary bracelets of a given size.

If you’re just interested in what you can do with bracelets, you can stop reading now. If you’re interested in the mathematical and algorithmic details of how they are implemented, read on!

## Exponential generating functions

The exponential generating function (egf) associated to a combinatorial species $F$ is defined by

$\displaystyle F(x) = \sum_{n \geq 0} |F[n]| \frac{x^n}{n!}.$

That is, the egf is an (infinite) formal power series where the coefficient of $x^n/n!$ is the number of distinct labelled $F$-structures on $n$ labels. We saw above that for $n \geq 3$ there are $(n-1)!/2$ labelled bracelets of size $n$, and there is one bracelet each of sizes $1$ and $2$. The egf for bracelets is thus:

$\displaystyle B(x) = x + \frac{x^2}{2} + \sum_{n \geq 3} \frac{(n-1)!}{2} \frac{x^n}{n!} = x + \frac{x^2}{2} + \sum_{n \geq 3} \frac{x^n}{2n}.$

(Challenge: show this is also equivalent to $\frac{1}{2}(x + x^2/2 - \ln(1-x))$.) This egf is directly encoded in the species library, and this is what is being used to evaluate labelled bracelets in the example above.

Incidentally, the reason $(n-1)!/2$ only works for $n \geq 3$ is in some sense due to the fact that the dihedral groups $D_2 = Z_2$ and $D_4 = K_4$ are a bit weird: every dihedral group $D_{2n}$ is a subgroup of the symmetric group $\mathcal{S}_n$ except for $D_2$ and $D_4$. The problem is that for $n < 3$, “flips” actually have no effect, as you can see below:

So, for example, $D_4$ has $4$ elements, corresponding to the identity, a 180 degree rotation, a flip, and a rotation + flip; but the symmetric group $\mathcal{S}_2$ only has two elements, in this case corresponding to the identity and a 180 degree rotation. The reason $(n-1)!/2$ doesn’t work, then, is that the division by two is superfluous: for $n < 3$, counting cycles doesn’t actually overcount bracelets, because every cycle is already a flipped version of itself. So it would also be correct (if rather baroque) to say that for $n < 3$ there are actually $(n-1)!$ bracelets.

I find this fascinating; it’s almost as if for bigger $n$ the dihedral symmetry has “enough room to breathe” whereas for small $n$ it doesn’t have enough space and gets crushed and folded in on itself, causing weird things to happen. It makes me wonder whether there are other sorts of symmetry with a transition from irregularity to regularity at even bigger $n$. Probably this is an easy question for a group theorist to answer but I’ve never thought about it before.

## Ordinary generating functions

The ordinary generating function (ogf) associated to a species $F$ is defined by

$\displaystyle \tilde F(x) = \sum_{n \geq 0} |F[n]/\mathord{\sim}| x^n$

where $\sim$ is the equivalence relation on $F$-structures induced by permuting the labels. That is, the coefficient of $x^n$ is the number of equivalence classes of $F$-structures on $n$ labels up to relabelling. There is only one unlabelled bracelet of any size $n \geq 1$, that is, any bracelet of size $n$ can be transformed into any other just by switching labels around. The unique unlabelled bracelet of a given size can be visualized as a bracelet of uniform beads:

though it’s occasionally important to keep in mind the more formal definition as an equivalence class of labelled bracelets. Since there’s just one unlabelled bracelet of each size, the ogf for bracelets is rather boring:

$\displaystyle \tilde B(x) = x + x^2 + x^3 + \dots = \frac{x}{x - 1}$.

This is encoded in the species library too, and was used to compute unlabelled bracelets above.

## egfs, ogfs, and homomorphisms

egfs are quite natural (in fact, species can be seen as a categorification of egfs), and the mapping from species to their associated egf is a homomorphism that preserves many operations such as sum, product, Cartesian product, composition, and derivative. ogfs, however, are not as nice. The mapping from species to ogfs preserves sum and product but does not, in general, preserve other operations like Cartesian product, composition or derivative. In some sense ogfs throw away too much information. Here’s a simple example to illustrate this: although the ogfs for bracelets and cycles are the same, namely, $x/(1-x)$ (there is only one unlabelled bracelet or cycle of each size), the ogfs for binary bracelets and binary cycles are different:

ghci> -- recall biBracelets = bracelet >< (set * set)
ghci> let biCycles = cycles >< (set * set)
ghci> take 15 $unlabelled biBracelets [0,2,3,4,6,8,13,18,30,46,78,126,224,380,687] ghci> take 15$ unlabelled biCycles
[0,2,3,4,6,8,14,20,36,60,108,188,352,632,1182]

(Puzzle: why are these the same up through $n=5$? Find the unique pair of distinct binary $6$-cycles which are equivalent as bracelets.)

Clearly, there is no way to take equal ogfs, apply the same operation to both, and get different results out. So the species library cannot be working directly with ogfs in the example above—something else must be going on. That something else is cycle index series, which generalize both egfs and ogfs, and retain enough information that they once again preserve many of the operations we care about.

## Cycle index series

Let $\mathcal{S}_n$ denote the symmetric group of order $n$, that is, the group of permutations on $\{1, \dots, n\}$ under composition. It is well-known that every permutation $\sigma \in \mathcal{S}_n$ can be uniquely decomposed as a product of disjoint cycles. The cycle type of $\sigma$ is the sequence of natural numbers $\sigma_1, \sigma_2, \sigma_3, \dots$ where $\sigma_i$ is the number of $i$-cycles in the cycle decomposition of $\sigma$. For example, the permutation $(132)(45)(78)(6)$ has cycle type $1,2,1,0,0,0,\dots$ since it has one $1$-cycle, two $2$-cycles, and one $3$-cycle.

For a species $F$ and a permutation $\sigma \in \mathcal{S}_n$, let $\mathrm{fix}\ F[\sigma]$ denote the number of $F$-structures that are fixed by the action of $\sigma$, that is,

$\displaystyle \mathrm{fix}\ F[\sigma] = \#\{ f \in F[n] \mid F[\sigma] f = f \}.$

The cycle index series of a combinatorial species $F$ is a formal power series in an infinite set of variables $x_1, x_2, x_3, \dots$ defined by

$\displaystyle Z_F(x_1, x_2, x_3, \dots) = \sum_{n \geq 0} \frac{1}{n!} \sum_{\sigma \in \mathcal{S}_n} \mathrm{fix}\ F[\sigma] x_1^{\sigma_1} x_2^{\sigma_2} x_3^{\sigma_3} \dots$

We also sometimes write $x^\sigma$ as an abbreviation for $x_1^{\sigma_1} x_2^{\sigma_2} x_3^{\sigma_3} \dots$. As a simple example, consider the species of lists, i.e. linear orderings. For each $n$, the identity permutation (with cycle type $n,0,0,\dots$) fixes all $n!$ lists of length $n$, whereas all other permutations do not fix any lists. Therefore

$\displaystyle Z_L(x_1, x_2, x_3, \dots) = \sum_{n \geq 0} \frac{1}{n!} n! x_1^n = \sum_{n \geq 0} x_1^n = \frac{1}{1 - x_1}.$

(This is not really that great of an example, though—since lists are regular species, that is, they have no nontrivial symmetry, their cycle index series, egf, and ogf are all essentially the same.)

Cycle index series are linked to both egfs and ogfs by the identities

$\displaystyle \begin{array}{rcl}F(x) &=& Z_F(x,0,0,\dots) \\ \tilde F(x) &=& Z_F(x,x^2,x^3, \dots)\end{array}$

To show the first, note that setting all $x_i$ to $0$ other than $x_1$ means that the only terms that survive are terms with only $x_1$ raised to some power. These correspond to permutations with only $1$-cycles, that is, identity permutations. Identity permutations fix all $F$-structures of a given size, so we have

$\begin{array}{rcl} Z_F(x,0,0,\dots) &=& \displaystyle \sum_{n \geq 0} \frac{1}{n!} \mathrm{fix}\ F[\mathit{id}] x^n \\ &=& \displaystyle \sum_{n \geq 0} |F[n]| \frac{x^n}{n!}. \end{array}$

To prove the link to ogfs, note first that for any permutation $\sigma \in \mathcal{S}_n$ with cycle type $\sigma_1,\sigma_2,\sigma_3,\dots$ we have $\sigma_1 + 2\sigma_2 + 3\sigma_3 + \dots = n$. Thus:

$\begin{array}{rcl} Z_F(x,x^2,x^3,\dots) &=& \displaystyle \sum_{n \geq 0} \frac{1}{n!} \sum_{\sigma \in \mathcal{S}_n} \mathrm{fix}\ F[\sigma] x^{\sigma_1}x^{2\sigma_2}x^{3\sigma_3}\dots \\ &=& \displaystyle \sum_{n \geq 0} \frac{1}{n!} \sum_{\sigma \in \mathcal{S}_n} \mathrm{fix}\ F[\sigma] x^n \\ &=& \displaystyle \sum_{n \geq 0} |F[n]/\mathord{\sim}| x^n \end{array}$

where the final step is an application of Burnside’s Lemma.

The important point is that the mapping from species to cycle index series is again a homomorphism for many of the operations we care about, including Cartesian product and composition. So in order to compute an ogf for some species defined in terms of operations that are not compatible with ogfs, one can start out computing with cycle index series and then project down to an ogf at the end.

## Cycle index series for bracelets

Let’s now see how to work out the cycle index series for the species of bracelets. For $n=1$, the single bracelet is fixed by the only element of $\mathcal{S}_1$, giving a term of $x_1$. For $n=2$, the single bracelet is fixed by both elements of $\mathcal{S}_2$, one of which has cycle type $2,0,0,\dots$ and the other $0,1,0,\dots$. Bracelets of size $n \geq 3$, as discussed previously, have the dihedral group $D_{2n}$ as their symmetry group. That is, every one of the $(n-1)!/2$ size-$n$ bracelets is fixed by the action of each element of $D_{2n}$, and no bracelets are fixed by the action of any other permutation. Putting this all together, we obtain

$\begin{array}{rcl} Z_B(x_1, x_2, x_3, \dots) &=& \displaystyle \sum_{n \geq 0} \frac{1}{n!} \sum_{\sigma \in \mathcal{S}_n} \mathrm{fix}\ B[\sigma] x_1^{\sigma_1}x_2^{\sigma_2}x_3^{\sigma_3}\dots \\ &=& \displaystyle x_1 + \frac{1}{2}(x_1^2 + x_2) + \sum_{n \geq 3} \frac{1}{n!} \sum_{\sigma \in D_{2n}} \frac{(n-1)!}{2} x^\sigma \\ &=& \displaystyle x_1 + \frac{1}{2}(x_1^2 + x_2) + \sum_{n \geq 3} \frac{1}{2n} \sum_{\sigma \in D_{2n}} x^\sigma. \end{array}$

Our remaining task is thus to compute $\sum_{\sigma \in D_{2n}} x^\sigma$, that is, to compute the cycle types of elements of $D_{2n}$ for $n \geq 3$. I don’t know whether there’s a nice closed form for $Z_B$, but for our purposes it doesn’t matter: it suffices to come up with a finite algorithm to generate all its terms with their coefficients. A closed form might be important if we want to compute with $Z_B$ symbolically, but if we just want to generate coefficients, an algorithm is good enough.

In general, $D_{2n}$ has $n$ elements corresponding to rotations (including the identity element, which we think of as a rotation by $0$ degrees) and $n$ elements corresponding to reflections across some axis. Below I’ve drawn illustrations showing the symmetries of bracelets of size $5$ and $6$; each symmetry corresponds to an element of $D_{2n}$.

The lines indicate reflections. You can see that in general there are $n$ lines of reflection. The curved arrows indicate clockwise rotations; taking any number of consecutive arrows from $0$ to $n-1$ gives a distinct rotational symmetry. Let’s label the rotations $R_k$ (for $k \in \{0, \dots, n-1\}$), where $R_k$ indicates a rotation by $k/n$ of a turn (so $R_0$ is the identity element). We won’t bother labelling the reflections since it’s not clear how we would choose canonical names for them, and in any case (as we’ll see) we don’t have as much of a need to give them names as we do for the rotations. The only thing we will note is that for even $n$ there are two distinct types of reflections, as illustrated by the dark and light blue lines on the right: the dark blue lines pass through two vertices, and the light blue ones pass through two edges. In the odd case, on the other hand, every line of reflection passes through one vertex and one edge. If you haven’t studied dihedral groups before, you might want to take a minute to convince yourself that this covers all the possible symmetries. It’s clear that a rotation followed by a rotation is again a rotation; what may be less intuitively clear is that a reflection followed by a reflection is a rotation, and that a rotation followed by a reflection is a reflection.

So the name of the game is to consider each group element as a permutation of the labels, and compute the cycle type of the permutation. Let’s tackle the reflections first; we have to separately consider the cases when $n$ is odd and even. We saw above that when $n = 2m+1$ is odd, each line of reflection passes through exactly one vertex. As a permutation, that means the reflection will fix the label at the vertex it passes through, and swap the labels on other vertices in pairs, as shown in the leftmost diagram below:

So the permutation has cycle type $1,m,0,0,\dots$. There is one $1$-cycle, and the remaining $n-1 = 2m$ elements are paired off in $2$-cycles. There are $n$ of these reflections in total, yielding a term of $nx_1x_2^m$ (where $m = \lfloor n/2 \rfloor$).

When $n = 2m$ is even, half of the reflections (the light blue ones) have no fixed points, as in the middle diagram above; they put everything in $2$-cycles. The other half of the even reflections fix two vertices, with the rest in $2$-cycles, as in the rightmost diagram above. In all, this yields terms $mx_1^2x_2^{m-1} + mx_2^m$.

Now let’s tackle the rotations. One could be forgiven for initially assuming that each rotation will just yield one big $n$-cycle… a rotation is just cycling the vertices, right? But it is a bit more subtle than that. Let’s look at some examples. In each example below, the green curved arrow indicates a rotation $R_k$ applied to the bracelet. As you can check, the other arrows show the resulting permutation on the labels, that is, each arrow points from one node to the node where it ends up under the action of the rotation.

Do you see the pattern? In the case when $k=1$ (the first example above), or more generally when $n$ and $k$ are relatively prime (the second example above, with $n=5$ and $k=2$), $R_k$ indeed generates a single $n$-cycle. But when $n$ and $k$ are not relatively prime, it generates multiple cycles. By symmetry the cycles must all be the same size; in general, the rotation $R_k$ generates $(n,k)$ cycles of size $n/(n,k)$ (where $(n,k)$ denotes the greatest common divisor of $n$ and $k$). So, for example, $2$ cycles are generated when $n=6$ and $k=2$ or $k=4$ (the next two examples above). The last example shows $n=12$ and $k=3$; we can see that three $4$-cycles are generated. Note this even works when $k=0$: we have $(n,0) = n$, so we get $n$ cycles of size $n/n = 1$, i.e. the identity permutation.

So $R_k$ contributes a term $x_{n/(n,k)}^{(n,k)}$. However, we can say something a bit more concise than this. Note, for example, when $n=12$, as the contribution of all the $R_k$ we get

$x_1^{12} + x_{12} + x_6^2 + x_4^3 + x_3^4 + x_{12} + x_2^6 + x_{12} + x_3^4 + x_4^3 + x_6^2 + x_{12}$

but we can collect like terms to get

$x_1^{12} + x_2^6 + 2x_3^4 + 2x_4^3 + 2x_6^2 + 4x_{12}$

For a given divisor $d \mid n$, the coefficient of $x_{n/d}^d$ is the number of nonnegative integers less than $n$ whose $\gcd$ with $n$ is equal to $d$. For example, the coefficient of $x_{6}^2$ is $2$, since there are two values of $k$ for which $(n,k) = 12/6 = 2$ and hence generate a six-cycle, namely, $2$ and $10$. So as the contribution of the $R_k$ we could write something like

$\displaystyle \sum_{d \mid n} \left( \#\{ 1 \leq k \leq n \mid (k,n) = d \} \right) x_{n/d}^d$

but there is a better way. Note that

$\displaystyle \#\{ 1 \leq k \leq n \mid (k,n) = d \} = \#\{ 1 \leq j \leq n/d \mid (j,n/d) = 1 \}$

since multiplying and dividing by $d$ establishes a bijection between the two sets. For example, we saw that $2$ and $10$ are the two numbers whose $\gcd$ with $12$ is $2$; this corresponds to the fact that $2/2 = 1$ and $10/2 = 5$ are relatively prime to $12/2 = 6$.

But counting relatively prime numbers is precisely what Euler’s totient function (usually written $\varphi$) does. So we can rewrite the coefficient of $x_{n/d}^d$ as

$\displaystyle \varphi(n/d) x_{n/d}^d$.

Finally, since we are adding up these terms for all divisors $d \mid n$, we can swap $d$ and $n/d$ (divisors of $n$ always come in pairs whose product is $n$), and rewrite this as

$\displaystyle \varphi(d) x_d^{n/d}$.

To sum up, then, we have for each $n \geq 3$:

1. When $n$ is odd, $n x_1 x_2^{\lfloor n/2 \rfloor}$.
2. When $n$ is even, $(n/2) x_1^2 x_2^{n/2 - 1} + (n/2)x_2^{n/2}$.
3. For each $d \mid n$, we have $\varphi(d) x_d^{n/d}$.

The only overlap is between (2) and (3): when $d = 2$ both generate $x_2^{n/2}$ terms. Using Iverson brackets (the notation $[P]$ is equal to $1$ if the predicate $P$ is true, and $0$ if it is false), we can thus write the sum of the above for a particular $n$ as

$\displaystyle [n\text{ odd}](n x_1 x_2^{\lfloor n/2 \rfloor}) + [n\text{ even}]\left(\frac n 2 x_1^2 x_2^{n/2 - 1}\right) + \sum_{d \mid n} \left(\varphi(d) + [d = 2]\frac n 2\right) x_d^{n/d}$.

Substituting this for $\displaystyle \sum_{\sigma \in D_{2n}} x^\sigma$ yields a full definition of $Z_B$. You can see the result encoded in the species library here. Here’s the beginning of the full expanded series:

ghci> :m +Math.Combinatorics.Species.Types
ghci> take 107 $show (bracelets :: CycleIndex) "CI x1 + 1 % 2 x2 + 1 % 2 x1^2 + 1 % 3 x3 + 1 % 2 x1 x2 + 1 % 6 x1^3 + 1 % 4 x4 + 3 % 8 x2^2 + 1 % 4 x1^2 x2"  This, then, is how unlabelled biBracelets (for example) is calculated, where biBracelets = bracelet >< (set * set). The cycle index series for bracelet and set are combined according to the operations on cycle index series corresponding to * and ><, and then the resulting cycle index series is mapped down to an ogf by substituting $x^k$ for each $x_k$. ## Bracelet generation The final thing to mention is how bracelet generation works. Of course we can’t really generate actual bracelets, but only lists. Since bracelets can be thought of as equivalence classes of lists (under rotation and reversal), the idea is to pick a canonical representative element of each equivalence class, and generate those. A natural candidate is the lexicographically smallest among all rotations and reversals (assuming the labels have an ordering; if they don’t we can pick an ordering arbitrarily). One easy solution would be to generate all possible lists and throw out the redundant ones, but that would be rather inefficient. It is surprisingly tricky to do this efficiently. Fortunately, there is a series of papers by Joe Sawada (Generating bracelets with fixed content; A fast algorithm to generate necklaces with fixed content; Generating bracelets in constant amortized time) describing (and proving correct) some efficient algorithms for generating things like cycles and bracelets. In fact, they are as efficient as possible, theoretically speaking: they do only $O(1)$ work per cycle or bracelet generated. One problem is that the algorithms are very imperative, so they cannot be directly transcribed into Haskell. But I played around with benchmarking various formulations in Haskell and got it as fast as I could. (Interestingly, using STUArray was a lot slower in practice than a simple functional implementation, even though the imperative solution is asymptotically faster in theory—my functional implementation is at least $O(\lg n)$ per bracelet, and quite possibly $O(n)$, though since $n$ is typically quite small it doesn’t really matter very much. Of course it’s also quite possible that there are tricks to make the array version go faster that I don’t know about.) The result is released in the multiset-comb package; you can see the bracelet generation code here. Posted in combinatorics, haskell, math, species | | 5 Comments ## Ally Skills Tutorial at ICFP I just signed up for the Ally Skills Tutorial at ICFP, and if you are (1) a man who (2) will be at ICFP in Vancouver, I encourage you to sign up, too! From the website: The Ally Skills Tutorial teaches men simple, everyday ways to support women in their workplaces and communities. Participants learn techniques that work at the office, in classrooms, at conferences, and online. The skills we teach are relevant everywhere, including skills particularly relevant to open technology and culture communities. At the end of the tutorial, participants will feel more confident in speaking up to support women, be more aware of the challenges facing women in their workplaces and communities, and have closer relationships with the other participants. This sounds super helpful—I suspect there is often a large gap between the extent to which I want to support women and the extent to which I actually know, practically, how to do so. The workshop will be taught by Valerie Aurora, Linux filesystem developer and Ada Initiative co-founder; I expect it will be high quality! Posted in haskell | Tagged , , , , | 2 Comments ## Crystal Ball Connection Patterns via Species and Generating Functions A couple weeks ago, Denise posted Puzzle: Crystal Ball Connection Patterns on her blog, Let’s Play Math. I had fun playing with it and thought I would demonstrate how to apply some high-powered combinatorial techniques to it (probably not what Denise had in mind!). The setup is that there are $n$ (distinct) friends who can talk to each other on the phone. Only two people can talk at a time (no conference calls). The question is to determine how many different “configurations” there are. Not everyone has to talk, so a configuration consists of some subset of the friends arranged in (unordered) conversational pairs. Warning: spoilers ahead! If you’d like to play around with this yourself (and it is indeed a nice, accessible combinatorics problem to play with), stop reading now. My goal in this post is to have fun applying some advanced tools to this (relatively) simple problem. # Telephone numbers Let’s start by visualizing some configurations. In her post, Denise illustrated the complete set of configurations for $n = 4$, which I will visualize like this: Notice how I’ve arranged them: in the first row is the unique configuration where no one is talking (yes, that counts). In the second row are the six possible configurations with just a single conversation. The last row has the three possible configurations with two conversations. One good approach at this point would be to derive some recurrences. This problem does indeed admit a nice recurrence, but I will let you ponder it. Instead, let’s see if we can just “brute-force” our way to a general formula, using our combinatorial wits. Later I will demonstrate a much more principled, mechanical way to derive a general formula. Let’s start by coming up with a formula for $T_{n,k}$, the number of configurations with $n$ people and $k$ conversations. The number of ways of choosing $k$ pairs out of a total of $n$ is the multinomial coefficient $\displaystyle \binom{n}{2,2,\dots,2,n-2k} = \frac{n!}{(2!)^k(n-2k)!}$. However, that overcounts things: it actually distinguishes the first pair, second pair, and so on, but we don’t want to have any ordering on the pairs. So we have to divide by $k!$, the number of distinct orderings of the pairs. Thus, $\displaystyle T_{n,k} = \frac{n!}{2^k (n-2k)! k!}.$ Let’s do a few sanity checks. First, when $k=0$, we have $T_{n,0} = \frac{n!}{n!} = 1$. We can also try some other small numbers we’ve already enumerated by hand: for example, $T_{4,1} = \frac{4!}{2 \cdot 2 \cdot 1} = 6$, and $T_{4,2} = \frac{4!}{4 \cdot 1 \cdot 2} = 3$. So this seems to work. For $n$ people, there can be at most $\lfloor n/2 \rfloor$ conversations. So, the total number of configurations is going to be $\displaystyle T_n = \sum_{k=0}^{\lfloor n/2 \rfloor} T_{n,k}$. We can use this to compute $T_n$ for the first few values of $n$: $\begin{array}{rcl}T_0 &=& 1\\T_1 &=& 1 \\ T_2 &=& 1 + 1 = 2 \\ T_3 &=& 1 + 3!/2 = 4 \\ T_4 &=& 1 + 6 + 3 = 10 \\ T_5 &=& 1 + 5!/(2 \cdot 3!) + 5!/(4 \cdot 2) = 1 + 10 + 15 = 26 \\ T_6 &=& 1 + 6!/(2 \cdot 4!) + 6!/(4 \cdot 2 \cdot 2) + 6!/(8 \cdot 3!) = 1 + 15 + 45 + 15 = 76 \end{array}$ At this point we could look up the sequence 1,1,2,4,10,26,76 on the OEIS and find out all sorts of fun things: e.g. that we are also counting self-inverse permutations, i.e. involutions, that these numbers are also called “restricted Stirling numbers of the second kind”, some recurrence relations, etc., as well as enough references to keep us busy reading for a whole year. # Species We can describe configurations as elements of the combinatorial species $C = E \circ (E_2 + X)$. That is, a configuration is an unordered set ($E$) of ($\circ$) things ($E_2 + X$), where each thing can either be an unordered pair ($E_2$) of people talking on the phone, or ($+$) a single person ($X$) who is not talking. We can now use the Haskell species library to automatically generate some counts and see whether they agree with our manual enumerations. First, some boilerplate setup: ghci> :set -XNoImplicitPrelude ghci> :m +NumericPrelude ghci> :m +Math.Combinatorics.Species Now we define the species of configurations: ghci> let configurations = set o (set ofSizeExactly 2 + singleton) We can ask the library to count the number of configurations for different $n$: ghci> take 10 (labelled configurations) [1,1,2,4,10,26,76,232,764,2620]  Oh good, those numbers look familiar! Now, I wonder how many configurations there are for $n = 100$? ghci> labelled configurations !! 100 24053347438333478953622433243028232812964119825419485684849162710512551427284402176  Yikes! We can also use the library to generate exhaustive lists of configurations, and draw them using diagrams. For example, here are all $76$ configurations for $n=6$. (If you want to see the code used to generate this diagram, you can find it here.) And just for fun, let’s draw all $232$ configurations for $n = 7$: Whee! # A general formula, via generating functions Finally, I want to show how to use the species definition given above and the theory of generating functions to (somewhat) mechanically derive a general formula for the number of configurations. (Hopefully it will end up being equivalent to the formula we came up with near the beginning of the post!) Of course, this is also what the species library is doing, but only numerically—we will do things symbolically. First, note that we are counting labelled configurations (the friends are all distinct), so we want to consider exponential generating functions (egfs). Recall that the egf for a species $F$ is given by $\displaystyle F(x) = \sum_{n \geq 0} |F[n]| \frac{x^n}{n!}$, that is, a (possibly infinite) formal power series where the coefficient of $x^n/n!$ is the number of distinct labelled $F$-structures of size $n$. In our case, we need $\displaystyle E(x) = \sum_{n \geq 0} 1 \cdot \frac{x^n}{n!} = e^x$, since there is exactly one set structure of any size, and $\displaystyle E_2(x) = \frac{x^2}{2}$, which is just the restriction of $E(x)$ to only the $x^2$ term. Of course, we also have $X(x) = x$. Putting this together, we calculate $\begin{array}{rcl}\displaystyle (E \circ (E_2 + X))(x) &=& e^{(x^2/2 + x)} \\[1em] &=& \displaystyle \sum_{n \geq 0} \frac{(x^2/2 + x)^n}{n!} \\[1em] &=& \displaystyle \sum_{n \geq 0} \sum_{k = 0}^n \frac{1}{n!} \binom{n}{k} \left(\frac{x^2}{2}\right)^k x^{n-k} \\[1em] &=& \displaystyle \sum_{n \geq 0} \sum_{k=0}^n \frac{1}{n!} \binom{n}{k} \frac{x^{n+k}}{2^k} \end{array}$ Ultimately, we want something of the form $\displaystyle \sum_{m \geq 0} f_m \frac{x^m}{m!}$, so we’ll need to collect up like powers of $x$. To do that, we can do a bit of reindexing. Right now, the double sum is adding up a bunch of terms that can be thought of as making a triangle: Each ordered pair in the triangle corresponds to a single term being added. Each column corresponds to a particular value of $n$, with $n$ increasing to the right. Within each column, $k$ goes from $0$ up to $n$. The powers of $x$ in our double sum are given by $n+k$. If we draw in lines showing terms that have the same power of $x$, it looks like this: So let’s choose a new variable $m$, defined by $m = n + k$. We can see that we will have terms for every $m \geq 0$. We will also keep the variable $k$ for our other index, and substitute $n = m - k$ to get rid of $n$. In other words, instead of adding up the triangle by columns, we are going to add it up by diagonals. Previously we had $k \leq n$; substituting for $n$ that now turns into $k \leq m - k$. Adding $k$ to both sides and dividing by $2$ yields $k \leq \lfloor m/2 \rfloor$ (we can round down since $k$ is an integer). Looking at the diagram above, this makes sense: the height of each diagonal line is indeed half its index. Rewriting our indices of summation and substituting $m - k$ for $n$, we now have: $\begin{array}{rcl}\displaystyle \sum_{n \geq 0} \sum_{k=0}^n \frac{1}{n!} \binom{n}{k} \frac{x^{n+k}}{2^k} &=& \displaystyle \sum_{m \geq 0} \sum_{k=0}^{\lfloor m/2 \rfloor} \frac{1}{(m-k)!} \binom{m-k}{k} \frac{x^m}{2^k} \\[1em] &=& \displaystyle \sum_{m \geq 0} \sum_{k=0}^{\lfloor m/2 \rfloor} \frac{1}{k!(m-2k)!} \frac{x^m}{2^k} \\[1em] &=& \displaystyle \sum_{m \geq 0} \frac{x^m}{m!} \sum_{k=0}^{\lfloor m/2 \rfloor} \frac{m!}{k!(m-2k)!2^k} \end{array}$ And hey, look at that! The coefficient of $x^m/m!$ is exactly what we previously came up with for $T_m$. Math works! Posted in combinatorics, haskell, math, species | | 1 Comment ## BlogLiterately 0.8.1, now with HTTPS! I’ve just released version 0.8.1 of BlogLiterately, a tool for formatting and posting stuff (especially Haskelly stuff) to blogs. This is in conjunction with the release of haxr-3000.11. After much blood, sweat, and tears, I was able to rip the HTTP package out of the guts of haxr, replace it with http-streams, and carefully sew everything back together around the edges. The result is that haxr now finally supports making XML-RPC calls via HTTPS, which in turn means that BlogLiterately once again works with WordPress, which no longer supports XML-RPC over HTTP. Happy blogging! Posted in haskell | Tagged , , , , | 1 Comment ## Thoughts on grade inflation, part I: is grade inflation bad? Grade inflation. It’s terrible, horrible, no good, very bad, and ruining everything. …right? Well… I’m not so sure. What I do know is that the typical conversation around grade inflation frustrates me. At best, it often leaves many important assumptions unstated and unquestioned. Is grade inflation really bad? If so, why? What are the underlying assumptions and values that drive us to think of it in one way or another? At worst, the conversation is completely at the wrong level. Grade inflation is actually a symptom pointing at a much deeper question, one that gets at the heart of education and pedagogy: what do grades mean? Or, put another way, what do grades measure? This will be a two-part series. In this first post, I consider the first question: is grade inflation bad? In most conversations I have been a part of, this is taken as given, but I think it deserves more careful thought. I don’t know of any reasons to think that grade inflation is good, but I also don’t buy many of the common arguments (often implied rather than explicitly stated) as to why it is bad; in this post I consider three common ones. ## What is grade inflation? Just to make sure everyone is on the same page: by grade inflation I mean the phenomenon where average student grades are increasing over time, that is, the average student now receives higher grades than the average student of n years ago. (You could also think of it as the average value of a given grade going down over time.) This phenomenon is widespread in the US. I am only really familiar with the educational system in the US, so this post will of necessity be rather US-centric; I would be interested to hear about similarities and differences with other countries. Let’s now consider some common arguments as to why grade inflation is bad. ## The “Back in my day…” argument This is not so much an “argument” as an attitude, and it goes something like this: “Back in MY day, a C really meant a C! These ungrateful, entitled young whippersnappers don’t understand the true value of grades…” This is a caricature, of course, but I have definitely encountered variants of this attitude. This makes about as much sense to me as complaining “back in MY day, a dollar was really worth a dollar! And now my daughter is asking me for twenty dollars to go to a movie with her friends. TWENTY DOLLARS! These ungrateful, entitled young whippersnappers don’t understand the true value of money…” Nonsense, of course they do. It just costs$20 to go to a movie these days. A dollar is worth what it is now worth; a C is worth what it is now worth. Get over it.  It’s not that students don’t understand “the true value of grades”, it’s just that the value of grades is different than it used to be.

If I may hazard a guess, I think what this often boils down to is that people blame grade inflation on student attitudes of entitlement. As a potential contributing factor to grade inflation (and insofar as we would like to teach students different attitudes), that is certainly worth thinking about. But grade inflation potentially being caused by something one dislikes is not an argument that grade inflation itself is bad.

## The compression argument

Of course, there’s one important difference between money and grades: amounts of money have no upper limit, whereas grades are capped at A+.  This brings us to what I often hear put forth as the biggest argument against grade inflation, that it compresses grades into a narrower and narrower band, squeezed from above by that highest possible A+.  The problem with this, some argue, is that grade compression causes information to be lost.  The “signal” of grades becomes noisier, and it becomes harder for, say, employers and grad schools to be able to distinguish between different students.

My first, more cynical reaction is this: well, cry me a river for those poor, poor employers and grad schools, who will now have to assess students on real accomplishments, skills, and personal qualities, or (more likely) find some other arbitrary measurement to use. Do we really think grades are such a high-quality signal in the first place? Do they really measure something important and intrinsic about a student? (More on this in my next post.)  If the signal is noisy or arbitrary in the first place then compressing it really doesn’t matter that much.

Less cynically, let’s suppose the grade-signal really is that high-quality and important, and we are actually worried about the possibility of losing information. Consider the extreme situation, where grade inflation has progressed to such a degree that professors only give one of two possible grades: A (“outstandingly excellent”) or A+ (“superlatively superb”). An A- is so insultingly low that professors never give it (for fear of lawsuits, perhaps); for simplicity’s sake let’s suppose that no one ever fails, either. In this hypothetical scenario, at an institution like Williams where students take 32 courses, there are only 33 possible GPAs: you could get 32 A+’s, or one A and 31 A+’s, or two A’s and 30 A+’s… all the way down to getting all A’s (“straight-A student” means something rather different in this imaginary universe!).

But here’s the thing: I think 33 different GPAs would still be enough! I honestly don’t think companies or grad schools can meaningfully care about distinctions finer than having 33 different buckets of students. (If you think differently, I’d love to hear your argument.) If student GPAs are normally distributed, this even means that the top few buckets have much less than 1/33 of all the students. So if the top grad schools and companies want to only consider the top 1% of all students (or whatever), they can just look at the top bucket or two. You might say this is unfair for the students, but really, I can’t see how this would be any more or less fair than the current system.

Of course, under this hypothetical two-grade system, GPAs might not be normally distributed. For one thing, if grade inflation kept going, the distribution might become more and more skewed to the right, until, for example, half of all students were getting straight A+’s, or, in the theoretical limit, all students get only A+’s. But I really don’t think this would actually happen; I think you would see some regulating effects kick in far before this theoretical limit was reached. Professors would not actually be willing to give all A+’s (or even, for that matter, all A’s and A+’s).

The GPAs could also be very bimodal, if, for example, students are extremely consistent: a student who consistently scores in the top 40% of every class would get the same grades (all A+’s) as a student who consistently scores in the top 10%. However, I doubt this is how it would work (as any professor knows, “consistent” and “student” are a rare pairing). It would be interesting to actually work out what GPA distributions would result from various assumptions about student behavior.

## The moving target argument

The final argument against grade inflation that I sometimes hear goes like this: the problem is not so much that the average GPA is going up but simply that it is moving at all, which makes it harder for grad schools and employers to know how to calibrate their interpretations. But I don’t really buy this one either. The value of money is moving too, and yes, in some grand sense I suppose that makes it slightly harder for people to figure out how much things are worth. But somehow, everyone seems to manage just fine. I think employers and grad schools will manage just fine too. I don’t think GPAs are changing anywhere near fast enough for it to make much difference. And in any case, most of the time, the only thing employers and grad schools really care about is comparing the GPAs of students who graduated around the same time, in which case the absolute average GPA doesn’t matter at all. (One can make an argument about the difficulties caused by different schools having different average GPAs, but that is always going to be an issue, grade inflation or no.)

In the end, then, I am not so sure that grade inflation per se is such a terrible thing. However, it is well worth pondering the causes of grade inflation, and the deeper questions it leads to: what are grades? Why do we give them? What purposes do they serve, and what do they measure? I’ll take up these questions in a subsequent post.

Posted in teaching | Tagged , , , | 3 Comments

format: plain