Math Test Template Word
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Print the "worksheet" on the front, then turn the page over and print the "answer key" version on the back. Welcome to Math Assessment! The formative assessment process is a critical element of informed instruction, and making the appropriate next steps in instruction. 8th Grade Math Worksheets Printable with Answers. Don't forget to include additional resources. When you encounter a word problem on the GED Math test (and most are word problems), remember the three Ds — Decipher, Decide, and Do: Decipher — Read the problem and the answers carefully and write. Printable Word Shapes Pages. The result or outcome of the quiz does not really matter. California Standards Test: Algebra II. The questioning on the sheet has helped so many of my students overcome their integers roadblocks. For your benefit, there is a Service Interval Schedule which is being built to enable you to monitor consistent performance of the vehicle. 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Mean, Median, Mode, and More
Today in class we learned about statistics! Statistics are an important part of just about every job and industry. Statistical information is used in the fashion, food and drug, and financial industries, and as a way to track climate, weather, homelessness, and more. Our world is driven by large amounts of data, and data requires the use of statistics to make sense of it all!
In class we learned about the 3 ways we calculate the ‘typical’ value in a set of data. These 3 ways are the mean (also called the “average”; add all the data and divide by the number of data points), the median (the middle value in a set of data points) and the mode (the most common value). We learned about these three measurements of central tendency using a fictional YumYum Bar that students could buy at one of 5 stores. Where did your child choose to buy their YumYum Bar?
There are many ways to illustrate data and today we learned how to create a box-and-whisker plot. This plot is an easy way to see a data set broken up into quartiles (25% pieces) and to illustrate the variability of a given data set. Ask your child to draw you a quick sample of the box-and-whisker plot to see if you can figure out where its name comes from!


Where Binary Heap is Used
A full binary tree (sometimes referred to as a proper binary tree or plane binary tree) is a tree in which every node other than the leaves has two children. In other words, each node has either 0 child or 2 children.
A complete binary tree is a binary tree in which every level, except possibly the last, is completely filled, and all nodes are as far left as possible.
A perfect binary tree is a binary tree with all leaf nodes at the same depth. All internal nodes have degree 2.
A Binary Heap is a binary tree with following 2 properties:


Digital journalism would be impossible without the computer. But before the computer came the digital calculator and before the calculator came a whole host of early mechanical calculating machines. One of the most groundbreaking and beautiful was Charles Babbage’s Difference Engine. Vanessa Edwards has been to London’s Science Museum to discover what made it so special.
The Science Museum is packed with the most astounding scientific and technical achievements.
From early satellites, through the invention of the television and mobile phone, all of them have had a massive impact on the practice and consumption of journalism. But walking through the exhibitions, the power and beauty of the Difference Engine Number Two really caught my eye.
It’s size and complexity are arresting. It is an automatic mechanical calculator designed to tabulate polynomial functions.
In layman’s terms, it could produce the long tables of figures which many Gen X and Baby Boomers used in school and work to calculate important figures. The most common was the logarithm.
The London Science Museum engine
The difference engine on display at the Science Museum is actually the third Babbage designed and was only built and tested after his death.
His difference and the subsequent analytical engine, along with a number of other mechanical calculating machines of different designs provided the core concepts of computing.
Babbage’s own development of the analytical engine from the difference engine saw a genuine move from simple calculation to the more complex general computation.
The video below shows a little bit more of the Science Museum difference engine:
Why the Difference Engine was a computing breakthrough
Although Babbage’s machine worked in decimal (ten digits), he did also consider using the binary system used by modern digital computers.
Perhaps most striking, according to the Computer History Museum website: “The logical structure of the Analytical Engine was essentially the same as that which has dominated computer design in the electronic era – the separation of the memory from the central processor using a ‘fetch-execute cycle’, and facilities for inputting and outputting data and instructions.”
“Calling Babbage ‘the first computer pioneer’ is not a casual tribute.”
Click below to hear my impressions of the Difference Engine Number Two at the Science Museum:


Farkle Dice - nothing like a game of dice. Each player throws the bone, and for this he read out a number of points, depending on the combinations. For a specific combination of symbols will be assigned a specific number of game points. For 3x4 will receive 400 points for the 3x1 has 1000 points, with the 3x5 earn 500 points and so on. Each player on his turn has the ability to roll the dice a few times until the moment when he turns throw a winning combination. If the combinations of non-winning, the course, therefore, goes to the second player.
Download the file:


We study complex functions of two variables through topological studies of their amoebas and coamoebas. The number of holes in the coamoeba to certain polynomials are carefully investigated. Using winding numbers it is proven that a maximum number of holes in the coamoeba cannot be found to every maximally sparse function. The properties of the lopsided and the ordinary coamoeba are compared. In most studied cases the number of holes in the lopsided coamoeba is equal to the number of holes in the ordinary coamoeba. Moreover we numerically nd a set of coecients to the polynomial for which the amoeba is solid. In the study we study the location of the discriminant loci of the polynomial. In all studied cases the discriminant loci are connected to the topological structure of the amoeba.


Our results are twofold. On the one hand, we use the meta-reduction technique to show that the security of Schnorr signatures cannot be proven equivalent to the discrete logarithm problem without programming the random oracle. This result holds under the one-more discrete logarithm assumption and for a large natural class of reductions, we call simple reductions. This class in particular subsumes those used in previous proofs of security in the (programmable) random oracle model.
On the other hand, we show that a stronger result, i.e., an unconditional one, where the meta-reduction also solves the discrete logarithm problem, is most likely not achievable. In fact, we prove that for any randomized signature scheme, finding a meta-reduction that solves a hard non-interactive problem is equivalent to finding an adversary against the strong existential unforgeability of the signature scheme.


Here are some of the routines we will follow this term:
- Homework linked to Literacy and Numeracy will continue to be set on a regular basis. Other pieces of additional topic homework will also be given when necessary. Homework is set on a Friday and due in on Wednesday unless otherwise stated.
- Children should bring water to drink throughout the day in a named, spouted bottle and one pencil case containing a pen, pencil, rubber and ruler.
- Please ensure that your child has an appropriate indoor and outdoor PE kit for this term which is fully labelled. PE is on a Tuesday and Thursday.
- Please ensure your child is reading every day preferably to an adult for some of the time. Guided reading books will be sent home again and should be kept in school bags every day as guided reading sessions are at different times and days each week.
- Please support your child to read and write their spelling words every day.
- Mathletics; TT Rockstars and Hit the Button are great sites for consolidating maths topics and practising times tables, number bonds and division. Please encourage your child to do this for 10 minutes, 3 or 4 times during the week.


There’s a popular NYTimes op’ed titled “Is Algebra Necessary?” which questions the need for kids to learn advanced math. The author cites that failing algebra is a large reason why many students drop out of high school and post-secondary school. Moreover, algebra is not really required for most jobs or liberal arts studies.
This is the wrong idea.
Don’t get me wrong. I hate math with a passion – mainly because I’m absolutely petrified of it. The fear of math is so common, it has a name: number anxiety. In high school calculus, I started panicking before I finished reading equations. Once, my father, a computer programmer, spent hours outside the bathroom door trying to coax me out while I bawled my eyes out after he tried to teach me basic algebra. They enrolled me in Kumon where I learned the art of cheating.
In North America, we are taught to fear math. We are told that math is the sport of geniuses and the rest of us mere mortals should be very afraid. In Asia, kids are taught that anyone can do math with practice and on average, they do better.
But over here, the fear transfers from parents to teachers to kids to popular culture. There’s a similar problem in the African-American community with swimming. Parents who don’t swim are afraid that their kids will drown so they don’t enroll them in swimming lessons. But kids who can’t swim are at the highest risk for drowning. We can’t keep kids from learning advanced math simply because we’re afraid they will fail – doing so discourages them from ever succeeding.
None of those things seemed all the difficult once you figured it out. None of these things seem particularly useful until one day, when it was suddenly crucial. Given the number of doctors, engineers and developers we need, we should be encouraging kids to embrace the difficulty of math like any other subject instead of perpetuating the fear.
P.S. I still suck at math but my Asia-raised parents say that I’m not so bad – I just never gave it a chance.
Here’s more proof that we’re afraid of math via Internet memes (and even more of them on my Pinterest board):


Post multiplication table
The engraving of a multiplication table poster is a great way to teach children to basic mathematics. This multiplication table poster is printed on thick paper, offering excellent image quality and excellent readability. It is available in a variety of sizes and colors to adapt to your needs.
This multiplication table poster is designed to be easy to use and understand. It is divided into sections for each table, with figures that are easy to read. Children can easily learn to make their multiplication tables without having to consult books or abstract paintings. In addition, each section is accompanied by a table of results, which allows children to check their answers.
This multiplication table poster is designed to last. It is printed on superior quality paper which will not fade or will not be damaged easily. In addition, the poster is waterproof and resistant to light, which allows it to be kept in perfect condition for many years.
This multiplication table poster is an excellent addition to your class or your home. It is perfect for children who learn to make their multiplication tables, and is robust enough to be used for many years. So what are you waiting for? Buy your POSTER MULTIGATION TABLE TO today and start teaching basic mathematics for your children.


An Iterative Approach to Inverse Problems using Python's Numpy11:45 AM - 12:10 PM on August 16, 2015, Room 702
- Audience level:
This is a talk about inverse problems and improving on past algorithms that have focused on inverse problems in math, data science, tomography, and image processing. We use a new iterative method coded in Python's Numpy and show it's ability to recover images compared to previous approaches.
This talk will fall in the realm of data, data processing and data science.
Inverse problems commonly occur in image processing, statistical inference, tomography, and any field where one wants to compute some interior properties using exterior meseasurements. Linear regression would be one such problem which solves the least squares problem (where the solution to the least squares problem requires the inversion of a matrix X in order to solve for the optimal beta parameter estimates, b).
In many such problems the data frame or matrix X is ill-conditioned, meaning the matrix iself is close to singular (e.g. a Hilbert matrix that has several close to dependent columns),and the solution, b, is therefore is noisy. A small perturbation in the matrix X will lead to wild adjustements in the solution, b, as is well-known.
One way to deal with such instability is to regularize the problem and solve a "nearby" problem using Lasso methods or the more general Tikhanov reguarlization problem, which add a penalty, lambda, to the optimizaiton problem. However, currently, no efficient method exists for searching for lambda.
In our new innovative method, we have derived an iterative approach to solving the general Tikhanov regularization problem, which converges to the noiseless solution, does not depend strongly on the choice of lambda, and yet still avoids the inversion problem.
This alogirithm is written in a few lines of code using Python's numpy package, and primarily relies on the SVD composition. The solution is applicable to many fields in image processing and data, and the code, as we'll see in the talk, is quite simple.


Become mathematical problem solvers. And I think you're familiar with numbers this large by now. Show the problem in a grid. I don't want to talk down to you. There are 8 candies in one bag. Can you write this as a multiplication expression. Write the problem and use skip counting. I knew it was because he just didn't know those facts.
An analogy is just when you kind of use something as-- well, I won't go too much into it. How can the 3 times some other number still be the same number. From second grade to high school, special needs to gifted, it works and works fast. Solving a word problem, Solving a mental math problem, etc.
Identify the math skills you would like students to work on. The best objects are easy visuals for children to identify and draw, if needed.
As you conclude, ask students to reflect and share: Is this "Two x x". If you are not clear in your instructions or you decide that the problem is too complicated, revise it to suit your students. I could write a huge number, you know, 5, times 0. So this is one 3, two 3's, three 3's, four 3's.
Play a few rounds that hit important multiplication tables.
So I have 1, 2, well, I'm not going to count them because that'll give our answer away. And if you were to add 3 to it, you would move 3 spaces to the right. I hope I'm not confusing you, but I think you'll enjoy this.
So hopefully that gives you a little bit of background, and I really encourage you to try the next exercise, It will give you a lot of practice on being to evaluate expressions like this.
Our system has been used very successfully in many middle schools and high schools Rote memory worked for me. The two students sitting to the right are at Level III for this task.
Can you write a problem about the total number of spots on the ladybugs. But here I'm saying that 2 times 3 is the same thing as 3 plus 3. The real strength of this program is the salvation it offers the academically challenged and at-risk learner.
She needs 1 yard of yellow fabric, 1. Writing & Solving Multiplication Word Problems with One Variable To solve this problem, we first need to write an algebraic expression we can solve from our problem.
Writing & Solving. With a good understanding of the structure of multiplication in the context of a story, word problems can be created and understood. Then I give them simple equations, like 7+4 = 11 and have them do the reverse and write the word problem for that equation.
(It also requires a higher level of Bloom’s Taxonomy when they are creating their. 2 a: a mathematical operation that at its simplest is an abbreviated process of adding an integer to zero a specified number of times and that is extended to other numbers in accordance with laws that are valid for integers.
b: any of various mathematical operations that are analogous in some way to multiplication of the real numbers but are defined for other or larger sets of elements (such. Great question. In algebra, we do indeed avoid using the multiplication sign.
We'll explain it for you here. elleandrblog.com Count Forward From a Given Number (Two Sheets, Download from TpT).
Count Forward by Tens (Harcourt Math). elleandrblog.com Count Forward From a Given Number (3 Pages, Download from TpT). elleandrblog.com Number Formation in Multiple Manners.
elleandrblog.com Games, Auto-Scoring Quizzes, Flash Cards, Worksheets, and tons of resources to teach kids the multiplication facts. Free multiplication, addition, subtraction, and division games.How to write a story problem for multiplication


A little confession from me. I was homeschooled (that's not the confession part), and in 8th grade my algebra textbook had the answers to half the problems in the back. And when I was stumped, I would cheat.
Of course, cheating at math is a terrible way to learn, because the whole point isn't to know the answer to 2x + 2 = 7x - 5, it's to understand the methodology that can solve any like problem.
But what if you could cheat at your homework and learn? That seems to be the premise behind app called Socratic. Or at least that's my takeaway. The app lets you take a picture of a problem (you can also type it in, but that's a little laborious), and it'll not only give you an answer, but the steps necessary to to arrive at that answer — and even detailed explanations of the steps and concepts if you need them.
The app is actually designed to answer any kind of school question — science, history, etc. — but the math thing is the slickest part. For other kinds of questions, Socratic kind of does a bit of Googling, and in my experience can typically find similar word problems on the wide internet, or from its own database of answers. On about half the middle school science problems I tried, the app was able to identify the topic at question and show me additional resources about the concepts involved, but for others it was no more powerful than a simple web search.
But for algebra this thing is sick. I pointed it at 2x + 2 = 7x - 5, which I wrote down at random, and it gave me a 10 step process that results in x = 7/5. It has trouble with word problems, but if you can write down a word problem in math notation it shouldn't be an issue. I also tried it on a weird fraction from an AP algebra exam, which it kind of failed at, but then I swiped over and it was showing me this graph, which included the correct answer:
I love this app, not just because it would've helped 8th grade Paul out of a jam, but because it's such a computery use of computers. You use the tiny computer in your pocket to be basically smarter than you already are. It's technology that augments a human brain, not just a distraction.
The creator of Socratic just open sourced its step-by-step solver, called mathsteps. There are a lot of computer-based algebra solvers out there, but for Socratic they had to do some extra engineering to get at the steps a human would need to solve the same problem.
Also, I'd be remiss not to mention Photomath, which has been doing this since 2014, and actually has step-by-step explanations in the recently released Photomath+ paid version (there's a free trial). I like the Socratic interface and explanations a bit better, but I'm glad to see this is a vibrant market.
Though the site was originally launched with answers written by math tutors and teachers, the plan going forward is to use the peer-to-peer model -- students helping each other on the site. The most useful answers will be rated with stars to distinguish them.
Of course, students have long shared their answers the old fashioned way -- turning to one another for help, sharing their answers and solutions -- whether over the phone or face-to-face, whether transcribed word-for-word from another student's paper or solved thanks to the help and support from a peer. And that will be the model used for Slader: homework answers for students written by students.These are homework answers for students written by students.
Anticipating the criticism, the New York-based startup believes it's a mistake to dismiss this simply as cheating; rather they say the aim is to provide real-time help to students to work through their homework -- an online study hall, if you will. The startup is providing the tools for students to share their work and teach and learn with one another.
That teaching element is important to recognize, and co-founder Scott Kolb says the site is much more of a tutoring resource than simply a place to go look up and jot down the right answer. It's a type of "microtutoring," he says.
That "micro" element doesn't just mean simply that Slader offers help on a specific math problems rather than, say, hiring a math tutor for more generalized help with the subject. The Web site also features "microtransactions." In other words, there's an intellectual and a monetary exchange per answer, handled via points and via a per-answer access. While there is a free version of Slader, there are limitations on the number of answers users can view per day (two).
With a paid subscription, that limitation is still in place: subscription rates range from $2 to $4 per month with the ability to view 5, 15, or 30 solutions per day. Users can also purchase more views (in case of math emergency, I suppose).
It's worth pointing out here that the site is designed with the recognition that most high school students probably don't have credit cards to pay for these sorts of online transactions. As such, one can pay for points via parents' credit cards, but points can also be gifted to another person or offered as "bounties" to answer other questions. And most interestingly, users can also earn points that can actually be "cashed out."
One way to earn points: contributing one's own homework answers back to the Slader community.Users can earn points that can be "cashed out" by contributing one's own homework answers back to the Slader community.
It's an interesting way to encourage students to help one another and to share their homework solutions: doing so allows them to earn royalties of sorts on the work they do. Slader pays points each time a solution is viewed. So ideally the better the solution, the more views, and the more earnings. Users can actually "cash out" too, withdrawing the money they've earned via the site.
The ability for students to earn money from their homework is certainly an interesting twist on "the work" they are doing.
In order to create a platform that can handle math homework (all that mathematical notation and such), the Slader team has created a number of tools, including an equation editor that captures the step-by-step process of moving through a solution. The team has also seeded the site with solutions to the homework problems in most high school level math textbooks. That's no easy task with approximately 100 textbooks in their various versions and editions (so roughly 275 textbooks in all). To create the answers (and it's worth noting too that there are multiple answers to the same question, demonstrating there are multiple ways to solve math problems), Slader enlisted the help of some 2,500 math majors and math teachers.
But the startup's demand for answers to every single homework problem in every single math textbook may be an obstacle that Slader will have to cross if it plans to expand. As is, it's incredibly challenging to keep pace with the ever-changing textbook industry. And right now, much of this work on Slader's part is done by hand. This isn't a mechanized system; this is the Slader team verifying correct answers as well as verifying the pages and exercise numbers in textbooks. Things will be further complicated if, as the startup plans, it expands beyond math to other subject areas.
And the startup also faces competition from a variety of other online homework help sites: Cramster, for example, or Yahoo Answers or Homeworkhelp.com. Googling "I need help with homework" makes evident that there are a lot of questionable Web sites out there, ones with questionable answers and questionable fees. That will make it challenging for Slader to win an SEO game against some of these sites so that the startup actually shows up in searches. But it's also a challenge on the "social" search aspect as well. After all, students want the "right" answer, but they also tend to want to hear it from sources they trust -- and oftentimes that's their friends.
As Slader expands, it will have to win over interest (and pocketbooks) of high school students to convince them to move their homework activities to its online community. It will also have to win over teachers and parents (something that the startup already makes a great effort to do), to help them understand that this isn't about cheating. Rather it's about students teaching and learning and sharing with one another. It's about recognizing that they've always done this. And as such, it's about helping students with the tools and a service so that they can benefit in doing so -- benefit intellectually as well as financially.


Welcome to Tech Rando, a data science blog with complete step-by-step Python tutorials.
Tech Rando began in 2019, with the intent of sharing open source data science solutions with a wide audience across the internet. The ultimate goal of Tech Rando is to make data science and statistics topics accessible, easy to understand, and easy to implement.
About Kirsten Perry, Tech Rando Contributor
I am a data scientist in the renewable energy sector, working on machine learning and artificial intelligence applications in the solar space at the National Renewable Energy Laboratory (NREL) in Golden, CO. Before going into renewables, I worked in the energy industry as an engineer/data scientist at BP America. I have significant experience in database architecture, data engineering, and data visualization. I hold joint undergraduate degrees in mathematics and mechanical engineering from the University of Oklahoma, and a master’s degree in computer science with an emphasis in machine learning from Georgia Institute of Technology.
All thoughts presented in this blog are my own.


March 21 - May 2, 2020
*By completing the enrollment process, the student acknowledges the receipt of the 7EDU Class Policy and agrees to abide by the terms and conditions included therein as defined by 7EDU.
AP Calculus AB
Aims to most effectively prepare students to succeed in the upcoming AP Calculus AB test. The first 4 sessions of the class focus on reviewing important big ideas covered in the AP Calculus AB exam, particularly limits and continuity, derivatives, integrations, and their applications. In the last 4 sessions, students will practice with real or official AP Calculus AB practice tests under strict timing on each section. After each mock test, we will provide a thorough review of the test as well as teach strategies on how to solve problems quickly and accurately.
Students who have taken or are concurrently taking AP Calculus or an equivalent course
Students who need a final boost, and shoot for the full score.
Core topic content development


How to play
The given Sudoku board had 81 cells. The cells can be divided into
- Nine Rows of 9 cells each.
- Nine columns of 9 cells each.
- Nine square boxes (3 x 3 boxes) of 9 cells each.
The puzzle is solved by filling the numbers 1 to 9 in all of the rows, columns and square boxes. No number will be repeated within a row, column or a square box.
Some of the cells are pre-filled with numbers which cannot be changed. The sudoku puzzle has only one solution.


This comprehensive text offers undergraduates a remarkably student-friendly introduction to graph theory. Written by two of the field’s most prominent experts. Read Introduction to Graph Theory book reviews & author details and more at by Gary Chartrand (Author), Ping Zhang (Author). out of 5 stars 1. Introductory Graph Theory presents a nontechnical introduction to this exciting field in a clear, lively, and informative style. Author Gary Chartrand covers the.
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Amazing tour over the basics of graph theory. Jun 05, A. No trivia or quizzes yet.
Jun 11, J. Functional Analysis by Balmohan. The author even represents the Towers of Hanoi problem as a graph.
Introductory Graph Theory
Introduuction theory is used today chartfand the physical sciences, social sciences, computer science, and other areas. As far as I know the content was elementary but covered a wide range of interesting topics Hamiltonian Graphs, Balanced Graphs, Planar Graphs, Orientable Graphs, Chromatic Numbers, to name the immediately memorable.
In all, it is believed that aboutmathematicians have an assigned Erdos number now, and 90 percent of the world’s active mathematicians have an Erdos number lower than 8. Top Reviews Most recent Top Reviews.
Jul intrlduction, Darin marked it as to-read Shelves: If you are a seller for this product, would you like to suggest updates through seller support? Take, for example, the phenomenon of the Erdos number. Graphs as mathematical models Gary Chartrand Snippet view – If you are a software developer, then from time to time you will have to solve an Interesting Problem in optimization, such as finding the best matches on a introduftion site, or the right sharding key for a database cluster.
If you like books and love to build cool products, we may be looking for you.
I would charyrand this book to anyone planning to do much work in graphs. PaperbackAbridgedpages. Simon rated it liked it Jul 04, Write a product review. Elementary Concepts of Graph Theory.
In fact there are some crossovers: Graph theory is used today in the physical sciences, grapb sciences, computer science, and other areas.
Gary ChartrandPing Zhang. Written by two of the field’s most prominent experts, it takes an engaging approach that emphasizes graph theory’s history.
There was a problem filtering reviews right now. Intoduction book is a charming, breezy intro to g If you are a software developer, then from time to time you will have yary solve an Interesting Problem in optimization, vhartrand as finding the best matches on a dating site, or the right sharding key for a database cluster. This book is not yet featured on Listopia.
My library Help Advanced Book Search. Aug 24, Ameer rated it it was amazing Shelves: Diego Stamigni rated it really liked it Nov 07, Want to Read saving…. Selected pages Title Page. To get the free app, enter mobile phone number. It could not be used in a course other than with high school students or undergraduates. The general setup for a chapter is: Page 1 of 1 Start over Page 1 of 1. Introductory Graph Theory by Gary Chartrand.
Avi rated it it was ok Jan 21, In addition, he presents a large variety o Graph theory is used today in the physical sciences, social sciences, computer science, and other areas. So nothing ever gets very gralh, but it does present at least the basic ideas of a lot of things.
This book is a good introduction to intrduction graphs to solve common problems in many different fields and solving various puzzles. The title here is a bit misleading. See all free Kindle reading apps.
See our Returns Policy. Lists with This Book. To ask other readers questions about Gaey Graph Theoryplease sign up. Introductory Graph Theory presents a nontechnical introduction to this exciting field in a clear, lively, and informative style. Visit our Help Pages.
A First Course in Graph Theory – Gary Chartrand, Ping Zhang – Google Books
Cashback will be credited as Amazon Pay balance within 10 days. The book promises to be introductory, and I have now been introduced — the next time I run into a stumper while I’m building software, I feel I have a far better chance of recognizing it as a known problem with a standard solution.
Unique examples and lucid proofs provide a sound yet accessible Lewis Cawthorne rated it really liked it Dec 23, Product details Paperback Publisher: They are not only interesting but they serve as a nice break between sections of purely technical content. Undergraduate mathematics students at every level, puzzlists, and mathematical hobbyists will find well-organized coverage of the fundamentals of graph theory in this highly readable and thoroughly enjoyable book.


Humbugardy: Numb3r5 for 800
This is round 2 of Humbugardy. I’m your host, Alex Scorebard.
The player commonly associated with the number represented here:
Note: In this round, searching the web is allowed.
|Numb3r5||Sudoku||6th Degree Quotes||What and Where||Anagram Lines||Subjective|
|200||200||Bob Timmerman||200||For The Turnstiles||Next…|
|400||For The Turnstiles||400||400||Joe||400|
|1000||1000||1000||Bob Timmerman||For The Turnstiles||1000|
1. Who is Cal Ripken?
2. Jr., that is.
3. Who is Cal Ripken Jr? (just in case - you got me by 5 minutes, TJ!)
4. Even if he was a "Jr." he was the only "Cal Ripken" who was a player, so I hope I'm OK.
This was the first time I've made it to the question before anyone posted anything. Pure luck.
5. "Ahh, I see now"
He says after doing the math.
6. I owe it all to Google (assuming I get the points).
7. I would say type how many points you'd get, but my computer doesn't have the right symbols!
8. T J got it. The board is yours.
For those of you playing at home, the number depicted is 2632, using Mayan numbers.
9. Nice work T J! I like a math-related category a lot better than those anagrams.
10. This was my first real win! I stole the first one.
Anyway, let's try the last untried category. "Subjective" for $200, Alex.
11. So, TJ, if you are comfortable it won't provide aid and comfort to the enemy, what was the google seach that got you this answer?
12. I'm not TJ, but I tried "numerals dots lines" and the answer was on the first page.
13. Ahh . . "numerals" -- "numbers dots lines" on the other hand, does not get you there . . .
14. Be glad to show my work. I was going to do it earlier but I thought it might seem a little... self-absorbed.
Anyway, as best I can recall: I tried "number systems" first. That got me to a site that had a lot of different cultures' number systems. When I saw it was Mayan, I used "mayan numbers" and found a site on Mayan math. Plug and chug after that (base 20! Cool!).
I'm ashamed to say that the number wasn't a dead giveaway to me. I didn't find anyone with that number of career hits at baseball-reference.com, so then I did "2632 baseball." Bingo.


This school year our first cohort of IB students began working its way through the junior year of their two-year course of study. One of the things I’ve found interesting about the experience is teaching with international materials and textbooks published outside of the United States, and seeing how different countries might approach some of the same knowledge. There’s been some slightly new vocabulary to adjust to, and some idiosyncratic decisions when it comes to what is and what isn’t part of the curriculum.
[Tangent—feel free to disregard unless you’re a nerd like me—For example, as far as I can tell, the IB never makes reference to the secant, cosecant, and cotangent trigonometric ratios. My first thought on seeing that was, Good. Those just provide more opportunities for kids to get confused, they’re not present on the TI-84, and they add virtually nothing to the study of trigonometry. But then my calculus-teacher brain kicked in and said, What’s the derivative of tan x then? So I flipped through an IB textbook to see if they continued to avoid the reciprocal functions even then. They do: (d/dx) tan x = 1/cos2x. ]
One of the neat things about this has been that when you ask old questions in new ways, sometimes you stumble across some interesting results. A few weeks back, a test for IB Mathematics SL students which I created using real, released questions from actual IB exams, relied on the students being able to answer the following:
f (2) = _____ f (6) = _____ f (-2) = _____ f (0) = _____ etc.
This wasn’t actually the material being tested—and it’s not directly part of the IB curriculum but rather a prerequisite—but basically, the students would need to know how to answer those questions to answer the questions they were actually being asked.
For perspective, the students in this course have all completed Honors Algebra 2, and some of them have completed Honors Precalculus as well.
Now knowing my readership, at this point half of you are thinking to yourself, That’s a stupidly easy question. Why are you even talking about it? The other half are thinking Ouch! I became a writer because I hate math! Now you made my brain hurt! Either way, relax.
(If you’re very confident with math, you may want to skip the next paragraph.)
If you’ve not looked at math in decades and you don’t remember what the symbolism means, then just take it on faith that this line of questioning should be trivial. If you’re not sure but you’re curious to know what the kids were supposed to do, remember that every point on a coordinate plane has two coordinates, an x-coordinate and a y-coordinate. Listing the coordinates in order gives the “address” of any particular point, kind of like when you played Battleship as a kid. As for the question itself, kids are told from Algebra 1 on (if not sooner) that f (x) is another way of saying y. Even though it contains the letter x, it means y. The reason it contains the letter x is to make it possible to talk about specific values of y that go with specific values of x. So if I ask, “What does f (10) equal?” I’m asking students to find the point where x = 10, and tell me what that point’s y-coordinate equals. So I’ll put the answer to the question here in white, and you can highlight it with your mouse to see if you understood what I meant. f (10) = -9. This represents the lower right dot on the graph, the point (10, -9).
Students in an honors math class ought to know this, but—and this is the important part—this isn’t how we usually ask the question. Usually instead of giving students a graph, we do the more “advanced,” algebraic thing, and we define the function using variables. So we might say that f (x) = (x3 – 10x2 + 8x + 64)/(-16). Then we might ask, “What is f (4)?” The kids, having been trained in the proper algorithm to follow, will “plug in” 4 wherever they see x, and find that:
f (4) = (43 – 10 • 42 + 8 • 4 + 64) ÷ (-16)
= (64 – 160 + 32 + 64) ÷ (-16)
= 0 ÷ (-16)
And we’ll feel good about this as teachers, because they’ll have answered the “harder” question. Wouldn’t you agree that what I just demonstrated was much more work than simply looking up the point on the graph with an x-coordinate of 4, and seeing that its y-coordinate was 0? We’re making the kids work harder, ergo we’re being more rigorous.
Except . . .
Except that when a question like the above came up on a test for my rather advanced students, most of them could not accomplish the relatively trivial task of looking up function values from the graph itself. Then, on a hunch, I asked my Honors Algebra 2 students, and they could not, by and large, either.
What this says to me is that we have succeeded in teaching a rote process, but failed to teach the concept behind the process. Anybody who knows me knows that this sort of thing is a big deal to me. I am constantly stressing in my classes how important it is to focus on why things work a certain way, sometimes to the frustration of those of my students who would just rather I gave them an algorithm to memorize already. Because there’s the rub: memorizing and plug-and-chug may be more tedium, but they’re actually easier than really getting around why things work.
I suspect, deep down, that we educators are often guilty of conspiring with our students to give them the easy way out. Because happy students are students who don’t complain. So we’re all too happy to settle for the appearance of hard work, rather than the reality of wrestling with a concept from all angles.
Another issue is probably our tendency to adopt curricula that is a mile wide and an inch deep–once again, because it gives us the comforting appearance of rigor. When you have three dozen topics to “cover”—and isn’t “cover” a poor synonym for “teach”?—then who has time to approach the same topic from multiple angles? Who has time for high level questioning?
If you’re a math teacher and you happen to stumble across this page while doing a google search for functions or for graphing or something, do me a favor: give the above problem to your kids, say, as a random bellwork problem. Did it seem trivial to you? Were you certain your kids would breeze past it? Did your students surprise you by struggling with it? Please let me know—I’d love to hear some other teachers’ experiences with this.


Some medical imaging procedures are inherently noisy processes. When your target quantities are defined through this data, areas of high noise should produce higher model uncertainty. In this work, we show how we can communicate the model uncertainty to the underlying numerical algorithm, thus yielding more realistic results off the achieved accuracy.
Publication: Soren Hauberg, Michael Schober, Matthew Liptrot, Philipp Hennig, Aasa Feragen: A Random Riemannian Metric for Probabilistic Shortest-Path Tractography . In: Medical Image Computing and Computer-Assisted Intervention (MICCAI), 2015.
We show that some Gaussian process ordinary differential equation solvers have the exact same mean as other well-known ODE solvers. This recovers all theoretical guarantees known for the traditional solvers while putting them in a probabilistic context.
Publication: Michael Schober, David Duvenaud, Philipp Hennig: Probabilistic ODE Solvers with Runge-Kutta Means. In: Advances of Neural Information Processing Systems (NIPS), 2014. Selected for oral presentation.
We use Gaussian process ODE solvers to quantify uncertainty in the approximation quality of ODE solvers. Thus, we can visualize the approximation quality and incorporate the solution quality in downstream computations.
Publication: Michael Schober, Niklas Kasenburg, Aasa Feragen, Philipp Hennig, Soren Hauberg: Probabilistic shortest path tractography in DTI using Gaussian Process ODE solvers. In: Medical Image Computing and Computer-Assisted Intervention (MICCAI), 2014.
There is lots of noise in astronomical images due to long exposure times and low-light settings. I test whether image denoising can be improved by only considering this special case against state-of-the-art algorithms for the general case of any image with i.i.d. Gaussian noise.
Publication: Michael Schober: Camera-specific Image Denoising. Diploma thesis, 2014.
Using training labeled data, I use machine learning algorithms to learn pairwise similarities between instances from a user-specified metric over their labels. Thus, new instances can be compared to labeled instances and predictions of this algorithm are more interprative.
Publication: Michael Schober: Using label metrics for Distance Metric Learning. Research thesis, 2009.
A short collection of tips and tricks to design your first conference poster.
An article I have written for a student magazine outlining basic ideas of causal inference. Unfortunately, only available in German.


In the previous video, you learned how to count from 0 to 9 on the Abacus tool. This video is all about counting from 10th place. You can see small white dots on the beam. The rod immediate left to the dotted rod is called the tens place. Each bead in the tens place rod of the lower deck is valued at 10 and the bead in the upper deck is valued at 50. Isn’t it interesting? Hence, watch the entire video till the end to learn more tricks and tips on Abacus. You can watch these videos as many times as possible for you to get your doubts corrected.


There are two units of work dedicated to each problem-solving strategy.Each unit consists of two pages; an example page, where the problems are worked through step-by-step and explanations and helpful hints are given, and a page of activities that require children to apply the particular strategy. All the activities involve problems that relate to children's everyday experiences, so they can easily visualise such situations.Both sections contain four revision tests to complete along the way, and answers are supplied in a convenient lift-out section.
In this book your child will find:
- An innovative approach to mathematical problem solving
- Self-contained units of work with easy-to-understand explanations
- A wide variety of interesting activities
- revision tests throughout to check their progress
- A lift-out answer section.