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The 12 Days of Christmas Math Problem: A Festive Exploration of Sequences, Series, and Combinatorics
Author: Dr. Evelyn Reed, PhD in Mathematics Education, Professor of Mathematics at the University of California, Berkeley. Dr. Reed has published extensively on the use of popular culture in mathematics education and is a leading expert in developing engaging mathematical problems for students of all ages.
Keyword: 12 days of christmas math problem
Introduction:
The seemingly simple carol, "The Twelve Days of Christmas," offers a surprisingly rich tapestry of mathematical concepts, making it a perfect vehicle for exploring various mathematical principles. The "12 days of Christmas math problem" centers around calculating the cumulative gifts received throughout the twelve days, prompting investigations into arithmetic sequences, geometric series, and combinatorial analysis. This article delves into the intricacies of this problem, highlighting its educational significance and exploring its diverse applications in mathematics education. We'll explore different approaches to solving the problem, discuss the underlying mathematical concepts, and demonstrate how this festive challenge can be adapted for various age groups and learning levels.
Understanding the 12 Days of Christmas Math Problem:
The core of the "12 days of Christmas math problem" lies in the cumulative nature of the gift-giving described in the carol. Each day, the recipient receives a new set of gifts in addition to the gifts received on previous days. This isn't simply a matter of multiplying the number of gifts received on a single day by twelve. The calculation requires understanding and applying principles of arithmetic sequences and series.
The problem typically involves calculating the total number of gifts received over the twelve days. Let's break down the gifts received each day:
Day 1: 1 Partridge in a Pear Tree
Day 2: 2 Turtle Doves + 1 Partridge in a Pear Tree
Day 3: 3 French Hens + 2 Turtle Doves + 1 Partridge in a Pear Tree
...and so on.
To find the total, one must sum the arithmetic series that unfolds throughout the twelve days. A straightforward approach involves calculating the number of each gift received over the twelve days and then summing those totals. A more elegant solution involves using the formula for the sum of an arithmetic series, which significantly reduces calculation time.
Mathematical Concepts Involved:
The "12 days of Christmas math problem" elegantly integrates several key mathematical concepts:
Arithmetic Sequences: The number of each type of gift received each day forms an arithmetic sequence. For example, the number of partridges received forms the sequence 1, 2, 3,... 12.
Arithmetic Series: The total number of each type of gift received over all twelve days is the sum of an arithmetic series. This can be calculated using the formula: S = n/2 [2a + (n-1)d], where 'n' is the number of terms (days), 'a' is the first term, and 'd' is the common difference.
Geometric Series: While not immediately apparent, a variation of the problem can involve considering the value of each gift, introducing the concept of a geometric series if the value increases exponentially.
Combinatorics: Advanced variations of the problem might explore combinatorial possibilities, such as the number of ways the gifts could be arranged or the probability of receiving specific combinations of gifts.
Solving the 12 Days of Christmas Math Problem:
Several methods can solve the problem. The simplest involves manually summing the gifts received each day. However, this is time-consuming and prone to errors. A more efficient method uses the formula for the sum of an arithmetic series for each gift type separately and then adds the totals.
Let's illustrate this using the formula for partridges:
n = 12 (number of days)
a = 1 (partridges on day 1)
d = 1 (common difference)
S = 12/2 [2(1) + (12-1)(1)] = 78 partridges
We repeat this for each gift type (turtle doves, French hens, etc.) and sum the results to find the total number of gifts.
A further extension could involve assigning monetary values to each gift, transforming the problem into one involving geometric series if the values increase exponentially, or using other mathematical approaches depending on the assumptions made.
Educational Significance of the 12 Days of Christmas Math Problem:
The "12 days of Christmas math problem" is invaluable in mathematics education because it:
Engages Students: The festive context makes the problem more appealing and memorable than traditional mathematical exercises.
Connects Abstract Concepts to Reality: It provides a tangible example of abstract concepts such as arithmetic sequences and series.
Encourages Critical Thinking: It promotes problem-solving skills and encourages students to explore different solution strategies.
Develops Mathematical Fluency: It reinforces the application of mathematical formulas and procedures.
Facilitates Collaboration: It provides opportunities for students to work together, discuss strategies, and learn from each other.
Adapting the Problem for Different Age Groups:
The "12 days of Christmas math problem" can be adapted to suit different age groups and learning levels:
Elementary School: Focus on simple addition and counting, perhaps limiting the number of days or gift types.
Middle School: Introduce the concept of arithmetic sequences and series, guiding students towards using the formula for the sum of a series.
High School: Explore more advanced concepts like geometric series, combinatorial analysis, and potentially creating variations with differing conditions.
Conclusion:
The "12 days of Christmas math problem" is far more than a festive curiosity. It's a powerful tool for engaging students with core mathematical concepts, promoting critical thinking, and fostering mathematical fluency. Its adaptability makes it suitable for diverse learning environments and age groups, showcasing the practical application of abstract mathematical principles within a familiar and engaging context. By exploring this problem, students not only improve their mathematical skills but also develop a deeper appreciation for the interconnectedness of mathematics and everyday life.
FAQs:
1. What is the total number of gifts in the "12 Days of Christmas" song? The total number of gifts is 780.
2. How do I calculate the total number of gifts without using the formula for the sum of an arithmetic series? You can manually add the number of gifts received each day.
3. Can this problem be adapted for younger children? Absolutely! Simplify it by focusing on smaller numbers of days or fewer types of gifts.
4. What are some advanced variations of the "12 Days of Christmas math problem"? Introduce varying gift values (geometric series), consider permutations and combinations of gifts, or add probabilistic elements.
5. How can this problem be used in a classroom setting? As a collaborative activity, a problem-solving challenge, or an assessment tool.
6. Are there online resources to help solve the "12 Days of Christmas math problem"? Yes, many websites and educational platforms offer resources and explanations.
7. What mathematical software can be used to solve this problem? Spreadsheet software like Excel or Google Sheets can be used to automate the calculations.
8. Can this problem be used to teach other subjects besides mathematics? It can be used to teach language arts (analyzing the lyrics) and music (understanding the melody).
9. What is the historical significance of the "12 Days of Christmas" carol? The carol's origins are uncertain, but it’s likely a mnemonic device to teach children the Christian feast days of the twelve days of Christmas.
Related Articles:
1. "Arithmetic Sequences and Series: A Comprehensive Guide": A detailed explanation of the mathematical concepts underlying the 12 Days of Christmas problem.
2. "Geometric Series: Applications and Solutions": Explores geometric series, relevant for variations of the problem involving increasing gift values.
3. "Combinatorics: Counting and Probability": Discusses combinatorial analysis, which can be applied to more advanced versions of the problem.
4. "Problem-Solving Strategies in Mathematics": Provides a general overview of problem-solving techniques applicable to the 12 Days of Christmas problem.
5. "Using Popular Culture in Mathematics Education": Explores the use of engaging examples, such as this problem, to make math learning more relevant.
6. "Engaging Math Activities for Middle School Students": Provides ideas for using the problem in a middle school classroom.
7. "Creative Math Projects for High School Students": Offers advanced variations and extensions of the problem suitable for high schoolers.
8. "The Mathematics of Christmas": A collection of various math problems and activities related to Christmas.
9. "Spreadsheet Applications in Mathematics Education": Demonstrates how to use spreadsheet software to solve the "12 Days of Christmas Math Problem" efficiently.
| 12 days of christmas math problem: Engaging Maths Catherine Attard, 2013-04-19 |
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| 12 days of christmas math problem: The True Gift Patricia MacLachlan, 2009-10-06 Experience the magic of authentic giving in this holiday classic from the Newbery Award–winning author of Sarah, Plain and Tall. All year long Lily and Liam look forward to the holidays at their grandparents’ farm. It’s always the perfect trip: walking to the lilac library, trimming the tree, giving gifts. But this year, thanks to a white cow alone in the meadow, things will be different. This holiday, Lily and Liam will find out the meaning of a special gift. This holiday classic from a beloved author rings in the season by celebrating the joys of family, community, and true giving. |
| 12 days of christmas math problem: Humble Math - 100 Days of Timed Tests Humble Math, 2020-04-05 100 days of multiplication problems. Discover the difference a couple minutes of practice can make. These are reproducible practice sheets to help students learn their multiplication facts and recall them with fluidity. This book focuses on digits 0-12. An answer key is included in the back of this book, so students can easily check their own work. |
| 12 days of christmas math problem: Engaging Maths Catherine Attard, 2015-04-28 |
| 12 days of christmas math problem: Emma's Christmas Irene Trivas, 2012-10-01 When Emma, the farmer's daughter, politely declines the proposal of the local prince, he begins to send his royal messengers with gifts. . . . In nontraditional style she declines castle life, inviting him to be her Farmer Prince instead. . . . Refreshing, fanciful, and great for holiday storyhours.--Booklist, starred review. Full-color illustrations. |
| 12 days of christmas math problem: Math with Bad Drawings Ben Orlin, 2018-09-18 A hilarious reeducation in mathematics-full of joy, jokes, and stick figures-that sheds light on the countless practical and wonderful ways that math structures and shapes our world. In Math With Bad Drawings, Ben Orlin reveals to us what math actually is; its myriad uses, its strange symbols, and the wild leaps of logic and faith that define the usually impenetrable work of the mathematician. Truth and knowledge come in multiple forms: colorful drawings, encouraging jokes, and the stories and insights of an empathetic teacher who believes that math should belong to everyone. Orlin shows us how to think like a mathematician by teaching us a brand-new game of tic-tac-toe, how to understand an economic crises by rolling a pair of dice, and the mathematical headache that ensues when attempting to build a spherical Death Star. Every discussion in the book is illustrated with Orlin's trademark bad drawings, which convey his message and insights with perfect pitch and clarity. With 24 chapters covering topics from the electoral college to human genetics to the reasons not to trust statistics, Math with Bad Drawings is a life-changing book for the math-estranged and math-enamored alike. |
| 12 days of christmas math problem: The 12 Days of Valentine's Jenna Lettice, 2017-12-26 Count the 12 days leading up to Valentine’s Day with this fun-filled picture book inspired by “The 12 Days of Christmas”—perfect for fans of Natasha Wing’s “The Night Before” series. The first day of Valentine’s starts with ONE warm, fuzzy hug. On the second day, the crafts begin with TWO cups of sparkles. On the third day, let’s make our cards with THREE pink pens! Each of the 12 busy days leading up to Valentine’s Day are celebrated in this cumulative rhyming storybook based on “The 12 Days of Christmas.” Kids will love spotting all the fun ways a family gets ready for Valentine’s Day! Also available: The 12 Days of Kindergarten, The 12 Days of Halloween, and The 12 Days of Preschool. will be a fast favorite with the preschool and kindergarten sets.—ReadBrightly.com |
| 12 days of christmas math problem: The Twelve Days of Christmas Jane Ray, 2011 A truly perfect version of much-loved festive song, The Twelve Days of Christmas |
| 12 days of christmas math problem: Math Wise! Over 100 Hands-On Activities that Promote Real Math Understanding, Grades K-8 James L. Overholt, Laurie Kincheloe, 2010-02-12 A fun, easy-to-implement collection of activities that give elementary and middle-school students a real understanding of key math concepts Math is a difficult and abstract subject for many students, yet teachers need to make sure their students comprehend basic math concepts. This engaging activity book is a resource teachers can use to give students concrete understanding of the math behind the questions on most standardized tests, and includes information that will give students a firm grounding to work with more advanced math concepts. Contains over 100 activities that address topics like number sense, geometry, computation, problem solving, and logical thinking. Includes projects and activities that are correlated to National Math Education Standards Activities are presented in order of difficulty and address different learning styles Math Wise! is a key resource for teachers who want to teach their students the fundamentals that drive math problems. |
| 12 days of christmas math problem: The 12 Days of Halloween Jenna Lettice, 2017-07-25 Count the 12 days leading up to Halloween with this fun-filled picture book inspired by “The 12 Days of Christmas”—perfect for fans of Natasha Wing’s “The Night Before . . .” series! On the first day of Halloween, ONE very eager kid starts trick-or-treating a little too early! On the second day, he and his brother create TWO scarecrows for their front porch. The third day? Let’s carve THREE pumpkins! Each of the 12 busy days leading up to Halloween are celebrated in this cumulative rhyming storybook based on “The 12 Days of Christmas.” Excited trick-or-treaters will love spotting all the fun ways a family gets ready for Halloween. Also available in the series: The 12 Days of Kindergarten. |
| 12 days of christmas math problem: The Problem with Math Is English Concepcion Molina, 2012-09-06 Teaching K-12 math becomes an easier task when everyone understands the language, symbolism, and representation of math concepts Published in partnership with SEDL, The Problem with Math Is English illustrates how students often understand fundamental mathematical concepts at a superficial level. Written to inspire ?aha? moments, this book enables teachers to help students identify and comprehend the nuances and true meaning of math concepts by exploring them through the lenses of language and symbolism, delving into such essential topics as multiplication, division, fractions, place value, proportional reasoning, graphs, slope, order of operations, and the distributive property. Offers a new way to approach teaching math content in a way that will improve how all students, and especially English language learners, understand math Emphasizes major attributes of conceptual understanding in mathematics, including simple yet deep definitions of key terms, connections among key topics, and insightful interpretation This important new book fills a gap in math education by illustrating how a deeper knowledge of math concepts can be developed in all students through a focus on language and symbolism. |
| 12 days of christmas math problem: Math Before Bed Jonathan Orr, 2017-12-05 The benefits of reading stories to our children at nighttime have been shared countless times over, and for good reason. Reading promotes literacy. Why is it that we don't do math with our children before bed? This book is a collection of prompts that can inspire mathematical discussions that you and your children can have before bed, at dinner, or at anytime. |
| 12 days of christmas math problem: Lower Elementary Math Task Cards: Multiplication ETC Montessori Digital, The third in the series of The Lower Elementary Math Task Cards for the Montessori classroom. This compilation of cards deals with both Operations and Facts for MULTIPLICATION. With 386 pages, the following concepts are included: Operations: Golden Beads Stamp Game Small Bead Frame Large Bead Frame Checker Board Golden Bead Frame Bank Game Facts: Level 1 Bead Games Multiplication Chart Missing Factors Level 2 Bead Games Snake Game |
| 12 days of christmas math problem: The Magic of Maths Arthur Benjamin, 2015-09-08 The world's greatest mental mathematical magician takes us on a spellbinding journey through the wonders of numbers (and more) Arthur Benjamin ... joyfully shows you how to make nature's numbers dance.--Bill Nye (the science guy) The Magic of Math is the math book you wish you had in school. Using a delightful assortment of examples-from ice-cream scoops and poker hands to measuring mountains and making magic squares-this book revels in key mathematical fields including arithmetic, algebra, geometry, and calculus, plus Fibonacci numbers, infinity, and, of course, mathematical magic tricks. Known throughout the world as the mathemagician, Arthur Benjamin mixes mathematics and magic to make the subject fun, attractive, and easy to understand for math fan and math-phobic alike. A positively joyful exploration of mathematics. -Publishers Weekly, starred review Each [trick] is more dazzling than the last. -Physics World |
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| 12 days of christmas math problem: Algebra: Themes, Tools, Concepts -- Teachers' Edition Henri Picciotto, Anita Wah, 1994 |
| 12 days of christmas math problem: Super Tricky Brain Teasers Richard Malena-Webber, 2022-05-17 Give your brain a workout with perplexing puzzles and games Solving brain teasers is a fun and effective way to exercise your mental acuity at any age. This book is filled with 120 mind-bending puzzles and riddles that boost your brainpower and provide hours of stimulating entertainment. What sets this book apart from other puzzle books for adults: It's not just for adults—Explore challenging brain games that adults, teens, and preteens will enjoy solving. Progressive difficulty—Warm up with some easier exercises, and then dive into brain puzzles that grow more complex as you progress. A variety of challenges—Solve an engaging mix of logic puzzles, ciphers, sequences, anagrams, crosswords, word chains, train puzzles, and more. Fascinating trivia—Discover intriguing factoids sprinkled throughout the book. For example, did you know that the longest palindromic word in English comes from the author James Joyce? Flex your mind muscles and have a blast with Super Tricky Brain Teasers. |
| 12 days of christmas math problem: The Twelve Days of Christmas Jan Brett, 1989-10-13 A centuries-old favorite holiday carol, The Twelve Days of Christmas was one of the first to celebrate the secular tradition of gift-giving. This sumptuously illustrated version features not only the gifts of a suitor to his true love but a loving family's Christmas preparations, how to say Merry Christmas in 12 different language, and a vast array of charming woodland animals. |
| 12 days of christmas math problem: Ask a Manager Alison Green, 2018-05-01 'I'm a HUGE fan of Alison Green's Ask a Manager column. This book is even better' Robert Sutton, author of The No Asshole Rule and The Asshole Survival Guide 'Ask A Manager is the book I wish I'd had in my desk drawer when I was starting out (or even, let's be honest, fifteen years in)' - Sarah Knight, New York Times bestselling author of The Life-Changing Magic of Not Giving a F*ck A witty, practical guide to navigating 200 difficult professional conversations Ten years as a workplace advice columnist has taught Alison Green that people avoid awkward conversations in the office because they don't know what to say. Thankfully, Alison does. In this incredibly helpful book, she takes on the tough discussions you may need to have during your career. You'll learn what to say when: · colleagues push their work on you - then take credit for it · you accidentally trash-talk someone in an email and hit 'reply all' · you're being micromanaged - or not being managed at all · your boss seems unhappy with your work · you got too drunk at the Christmas party With sharp, sage advice and candid letters from real-life readers, Ask a Manager will help you successfully navigate the stormy seas of office life. |
| 12 days of christmas math problem: The Virginia Mathematics Teacher , 1992 |
| 12 days of christmas math problem: Arnold's Problems Vladimir I. Arnold, 2004-06-24 Vladimir Arnold is one of the most outstanding mathematicians of our time Many of these problems are at the front line of current research |
| 12 days of christmas math problem: The Girl who Played with Fire Stieg Larsson, 2010 When the reporters to a sex-trafficking exposé are murdered and computer hacker Lisbeth Salander is targeted as the killer, Mikael Blomkvist, the publisher of the exposé, investigates to clear Lisbeth's name. |
| 12 days of christmas math problem: The Singapore Mathematics Calendar 2020 Yan Kow Cheong, The Singapore Mathematics Calendar is a three-book series that provides an informal yet creative way for both parents and homeschoolers to support their child succeed in math. This calendar is designed in such a way that the answer to the problem on each day is the date on which the problem appears. Hints and solutions are also provided for nonroutine or brain-unfriendly questions. Besides, each month begins with some elements of enrichment or recreational math. The Singapore Mathematics Calendar Series aims to convey the message that mathematics needn’t be drill-and-kill exercises—it can be fun, yet challenging for students to be exposed to the beauty and joy of mathematics. |
| 12 days of christmas math problem: Behave Robert M. Sapolsky, 2017-05-02 Why do we do the things we do? Over a decade in the making, this game-changing book is Robert Sapolsky's genre-shattering attempt to answer that question as fully as perhaps only he could, looking at it from every angle. Sapolsky's storytelling concept is delightful but it also has a powerful intrinsic logic: he starts by looking at the factors that bear on a person's reaction in the precise moment a behavior occurs, and then hops back in time from there, in stages, ultimately ending up at the deep history of our species and its genetic inheritance. And so the first category of explanation is the neurobiological one. What goes on in a person's brain a second before the behavior happens? Then he pulls out to a slightly larger field of vision, a little earlier in time: What sight, sound, or smell triggers the nervous system to produce that behavior? And then, what hormones act hours to days earlier to change how responsive that individual is to the stimuli which trigger the nervous system? By now, he has increased our field of vision so that we are thinking about neurobiology and the sensory world of our environment and endocrinology in trying to explain what happened. Sapolsky keeps going--next to what features of the environment affected that person's brain, and then back to the childhood of the individual, and then to their genetic makeup. Finally, he expands the view to encompass factors larger than that one individual. How culture has shaped that individual's group, what ecological factors helped shape that culture, and on and on, back to evolutionary factors thousands and even millions of years old. The result is one of the most dazzling tours de horizon of the science of human behavior ever attempted, a majestic synthesis that harvests cutting-edge research across a range of disciplines to provide a subtle and nuanced perspective on why we ultimately do the things we do...for good and for ill. Sapolsky builds on this understanding to wrestle with some of our deepest and thorniest questions relating to tribalism and xenophobia, hierarchy and competition, morality and free will, and war and peace. Wise, humane, often very funny, Behave is a towering achievement, powerfully humanizing, and downright heroic in its own right. |
| 12 days of christmas math problem: Lessons for Algebraic Thinking Ann Lawrence, Charlie Hennessy, 2002 These lessons show how to maximize instruction that prepares students for formal algebra. Through a series of investigations, students build their proficiency with key algebraic concepts. Connections between arithmetic and algebra are made through the use of drawings, tables, graphs, words, and symbols. Lessons include a technology component with suggestions for teaching with graphing calculators. |
| 12 days of christmas math problem: Algebra Anita Wah, Creative Publications, Inc, 1994 |
| 12 days of christmas math problem: Fostering Children's Mathematical Power Arthur J. Baroody, Ronald T. Coslick, 1998-09-01 Teachers have the responsibility of helping all of their students construct the disposition and knowledge needed to live successfully in a complex and rapidly changing world. To meet the challenges of the 21st century, students will especially need mathematical power: a positive disposition toward mathematics (curiosity and self confidence), facility with the processes of mathematical inquiry (problem solving, reasoning and communicating), and well connected mathematical knowledge (an understanding of mathematical concepts, procedures and formulas). This guide seeks to help teachers achieve the capability to foster children's mathematical power - the ability to excite them about mathematics, help them see that it makes sense, and enable them to harness its might for solving everyday and extraordinary problems. The investigative approach attempts to foster mathematical power by making mathematics instruction process-based, understandable or relevant to the everyday life of students. Past efforts to reform mathematics instruction have focused on only one or two of these aims, whereas the investigative approach accomplishes all three. By teaching content in a purposeful context, an inquiry-based fashion, and a meaningful manner, this approach promotes chilren's mathematical learning in an interesting, thought-provoking and comprehensible way. This teaching guide is designed to help teachers appreciate the need for the investigative approach and to provide practical advice on how to make this approach happen in the classroom. It not only dispenses information, but also serves as a catalyst for exploring, conjecturing about, discussing and contemplating the teaching and learning of mathematics. |
| 12 days of christmas math problem: Mathematics for Elementary Teachers Albert B. Bennett, Leonard T. Nelson, 2004 Emphasizing conceptual understanding through the use of models and visuals, this text helps students connect ideas and concepts while providing them with useful methods for teaching math to elementary school children. Mathematics for Elementary Teachers: A Conceptual Approach addresses the NCTM standards by encouraging active student participation through features such as Math Activities and Math Investigations. The Math Investigations now appear on the companion website. |
| 12 days of christmas math problem: Math Word Problems Sullivan Associates Staff, 1972 |
| 12 days of christmas math problem: Pete the Cat's 12 Groovy Days of Christmas James Dean, Kimberly Dean, 2018-09-18 This New York Times bestselling Pete the Cat hardcover picture book is a perfect Christmas gift for Pete the Cat fans who are eagerly awaiting Santa’s arrival! Pete and his friends are rockin’ and groovin’ while counting down the days to Christmas! Who needs five golden rings, when you can have five onion rings? Join Pete and the gang for one-of-a-kind holiday cheer. He adds a cool-cat spin on a well-known Christmas carol, bringing a hip and energetic spirit to the season. Plus, don't miss Pete’s other spin on a holiday classic, Pete the Cat Saves Christmas. And Pete’s other groovy guides, Pete the Cat’s Groovy Guide to Life, Pete the Cat’s Groovy Guide to Love, and Pete the Cat’s Groovy Guide to Kindness! |
| 12 days of christmas math problem: Which One Doesn't Belong? Christopher Danielson, 2019-02-12 Talking math with your child is simple and even entertaining with this better approach to shapes! Written by a celebrated math educator, this innovative inquiry encourages critical thinking and sparks memorable mathematical conversations. Children and their parents answer the same question about each set of four shapes: Which one doesn't belong? There's no one right answer--the important thing is to have a reason why. Kids might describe the shapes as squished, smooshed, dented, or even goofy. But when they justify their thinking, they're talking math! Winner of the Mathical Book Prize for books that inspire children to see math all around them. This is one shape book that will both challenge readers' thinking and encourage them to think outside the box.--Kirkus Reviews, STARRED review |
| 12 days of christmas math problem: The Great Mental Models, Volume 1 Shane Parrish, Rhiannon Beaubien, 2024-10-15 Discover the essential thinking tools you’ve been missing with The Great Mental Models series by Shane Parrish, New York Times bestselling author and the mind behind the acclaimed Farnam Street blog and “The Knowledge Project” podcast. This first book in the series is your guide to learning the crucial thinking tools nobody ever taught you. Time and time again, great thinkers such as Charlie Munger and Warren Buffett have credited their success to mental models–representations of how something works that can scale onto other fields. Mastering a small number of mental models enables you to rapidly grasp new information, identify patterns others miss, and avoid the common mistakes that hold people back. The Great Mental Models: Volume 1, General Thinking Concepts shows you how making a few tiny changes in the way you think can deliver big results. Drawing on examples from history, business, art, and science, this book details nine of the most versatile, all-purpose mental models you can use right away to improve your decision making and productivity. This book will teach you how to: Avoid blind spots when looking at problems. Find non-obvious solutions. Anticipate and achieve desired outcomes. Play to your strengths, avoid your weaknesses, … and more. The Great Mental Models series demystifies once elusive concepts and illuminates rich knowledge that traditional education overlooks. This series is the most comprehensive and accessible guide on using mental models to better understand our world, solve problems, and gain an advantage. |
| 12 days of christmas math problem: 100 Days of Money, Fractions, & Telling the Time Humble Math, 2020-09-02 100 days of practice counting money, learning fractions, and telling the time. Three core areas of maths that students often need additional help to master. Start by counting groups of coins that have a value of a pound or less. Slowly learn to count greater amounts of money that include small and large notes too. Add and subtract values, and even learn to make change. Learn fractions one step at a time. Start with simple fractions and eventually learn to add and subtract mixed numbers. Telling the time is one of the more difficult concepts for children to learn. Gradually progress from learning to tell hours, to quarter hours. Eventually tell time in five minute and minute increments. This is a great workbook to start out with. Most students will grasp the concepts within each of the three sections. If a student still needs more practice, Humble Math has workbooks that provide additional practice for these concepts too. |
这12个视频解析下载地址，网上视频均可下载，视频素材多到手软
Nov 11, 2022 · 这个网站视频（包括vip视频）解析之后支持在线观看，但是不能下载。支持解析百度网盘、优酷、乐视、芒果、搜狐、快手、土豆、pptv、华数tv、爱奇艺、腾讯、优酷、acfun …
想知道住房公积金5%，8%,12%都是怎么算钱的？具体是多少钱？ …
按照12%的比较缴纳就是3500x0.12=420元，加上公司为你缴纳的420元，一共是840元。 另外，这些问题大家也可以看看： 公积金里面有多少钱才能进行贷款。? 未给员工足额缴存公积金 …
都说13代、14代酷睿处理器缩肛，具体是什么情况? - 知乎
13、14代可以粗糙的认为是12代超频的产品。默认频率高了，自然容易烧坏，烧坏么就缩缸，缩了性能就降低，最多损失达30%，且无法恢复。 缩缸不是一定发生，长时间高负载更容易导致 …
为什么DeepSeek获取数据停留在23年10月12号，无法准确回答后 …
为什么DeepSeek获取数据停留在23年10月12号，无法准确回答后续时间的相关问题？ 刚开始问了他25年春节档哪部电影值得一看，他给予的回答模棱两可，然后又问他唐探1900怎么样，最 …
逆天｜详细说说苹果M4、M4 Pro和M4 Max - 知乎
从多核跑分上来看也是不得了，10核的M4达到了12核M2 Max的水平。 这里简单说下单核跑分和多核跑分的意义。 单核跑分反映其中的一个CPU核心的性能。
正在组装电脑中，14600KF到底容易爆雷或缩肛吗？有没有必要多 …
RT，求各位大神不吝赐教，谢谢！ 13 14缩肛是因为夏天高温蓝屏，主板以为电压不够就加压—蓝屏—加压—蓝屏—缩肛，很多不锁电压的直接干到1.5 1.6v了
2025年国产各品牌平板电脑推荐（618更新）618平板电脑选购指南
May 29, 2025 · 3.联想平板PadPro 12.7 如果想要性价比高一些的大屏平板，可以考虑联想平板PadPro 12.7，性能也不错。 联想这边的手写笔不要买错，目前共三款手写笔，每个平板兼容不 …
2025年 618 电脑配置推荐（配置单可以直接照抄） - 知乎
May 30, 2025 · 2025年618台式机电脑diy配置推荐（这篇文章每月都会更新，可以收藏） 本文内配置单无任何利益相关，配置推荐以性价比为主，每月月初会及时更新，希望大家可以点赞支持 …
2025年618 CPU选购指南丨CPU性能天梯图（R23 单核/多核性能跑 …
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集成显卡：Intel® Iris® Xe Graphics到底在显卡中算什么水平？
小结一下. 这次英特尔的Iris Xe 96EU核显进步是真的大，在搭配LPDDR4X-4266Mhz内存时，其跑分性能甚至能力压NVDIA GeForece MX350——这直接导致英伟达提前结束MX350显卡的生 …
这12个视频解析下载地址，网上视频均可下载，视频素材多到手软
Nov 11, 2022 · 这个网站视频（包括vip视频）解析之后支持在线观看，但是不能下载。支持解析百度网盘、优酷、乐视、芒果、搜狐、快手、土豆、pptv、华数tv、爱奇艺、腾讯、优酷、acfun …
想知道住房公积金5%，8%,12%都是怎么算钱的？具体是多少钱？ …
按照12%的比较缴纳就是3500x0.12=420元，加上公司为你缴纳的420元，一共是840元。 另外，这些问题大家也可以看看： 公积金里面有多少钱才能进行贷款。? 未给员工足额缴存公积金 …
都说13代、14代酷睿处理器缩肛，具体是什么情况? - 知乎
13、14代可以粗糙的认为是12代超频的产品。默认频率高了，自然容易烧坏，烧坏么就缩缸，缩了性能就降低，最多损失达30%，且无法恢复。 缩缸不是一定发生，长时间高负载更容易导致 …
为什么DeepSeek获取数据停留在23年10月12号，无法准确回答后 …
为什么DeepSeek获取数据停留在23年10月12号，无法准确回答后续时间的相关问题？ 刚开始问了他25年春节档哪部电影值得一看，他给予的回答模棱两可，然后又问他唐探1900怎么样，最 …
逆天｜详细说说苹果M4、M4 Pro和M4 Max - 知乎
从多核跑分上来看也是不得了，10核的M4达到了12核M2 Max的水平。 这里简单说下单核跑分和多核跑分的意义。 单核跑分反映其中的一个CPU核心的性能。
正在组装电脑中，14600KF到底容易爆雷或缩肛吗？有没有必要多 …
RT，求各位大神不吝赐教，谢谢！ 13 14缩肛是因为夏天高温蓝屏，主板以为电压不够就加压—蓝屏—加压—蓝屏—缩肛，很多不锁电压的直接干到1.5 1.6v了
2025年国产各品牌平板电脑推荐（618更新）618平板电脑选购指南
May 29, 2025 · 3.联想平板PadPro 12.7 如果想要性价比高一些的大屏平板，可以考虑联想平板PadPro 12.7，性能也不错。 联想这边的手写笔不要买错，目前共三款手写笔，每个平板兼容不 …
2025年 618 电脑配置推荐（配置单可以直接照抄） - 知乎
May 30, 2025 · 2025年618台式机电脑diy配置推荐（这篇文章每月都会更新，可以收藏） 本文内配置单无任何利益相关，配置推荐以性价比为主，每月月初会及时更新，希望大家可以点赞支持 …
2025年618 CPU选购指南丨CPU性能天梯图（R23 单核/多核性能跑 …
May 4, 2025 · cpu型号名称小知识 amd. 无后缀 ：普通型号; 后缀 g ：有高性能核显型号（5000系及之前系列 除了后缀有g的其他均为 无核显，7000除了后缀f，都有核显）
集成显卡：Intel® Iris® Xe Graphics到底在显卡中算什么水平？
小结一下. 这次英特尔的Iris Xe 96EU核显进步是真的大，在搭配LPDDR4X-4266Mhz内存时，其跑分性能甚至能力压NVDIA GeForece MX350——这直接导致英伟达提前结束MX350显卡的生 …
