Homework Maths Year 6 Worksheets
Free Math Worksheets for Grade 6
This is a comprehensive collection of free printable math worksheets for sixth grade, organized by topics such as multiplication, division, exponents, place value, algebraic thinking, decimals, measurement units, ratio, percent, prime factorization, GCF, LCM, fractions, integers, and geometry. They are randomly generated, printable from your browser, and include the answer key. The worksheets support any grade 6 math program, but have been specifically designed to match Math Mammoth grade 6 curriculum.
The worksheets are randomly generated each time you click on the links below. You can also get a new, different one just by refreshing the page in your browser (press F5).
You can print them directly from your browser window, but first check how it looks like in the "Print Preview". If the worksheet does not fit the page, adjust the margins, header, and footer in the Page Setup settings of your browser. Another option is to adjust the "scale" to 95% or 90% in the Print Preview. Some browsers and printers have "Print to fit" option, which will automatically scale the worksheet to fit the printable area.
All worksheets come with an answer key placed on the 2nd page of the file.
In sixth grade, students will start the study of beginning algebra (order of operations, expressions, and equations). They learn about ratios & percent and start using integers. Students also review long division, factoring, fraction arithmetic, and decimal arithmetic.In geometry, the focus is on the area of triangles and polygons and the volume of rectangular prisms. Other topics include rounding, exponents, GCF, LCM, and measuring units. Please note that these free worksheets do not cover all 6th grade topics; most notably, they do not include problem solving.
Multiplication and Division and Some Review
- 1-digit divisor, 5-digit dividend, no remainder
- 1-digit divisor, 5-digit dividend, with remainder
- 1-digit divisor, 6-digit dividend, no remainder
- 1-digit divisor, 6-digit dividend, with remainder
- 1-digit divisor, 7-digit dividend, no remainder
- 1-digit divisor, 7-digit dividend, with remainder
- 2-digit divisor, 5-digit dividend, no remainder
- 2-digit divisor, 5-digit dividend, with remainder
- 2-digit divisor, 6-digit dividend, no remainder
- 2-digit divisor, 6-digit dividend, with remainder
- 2-digit divisor, 7-digit dividend, no remainder
- 2-digit divisor, 7-digit dividend, with remainder
- 3-digit divisor, 6-digit dividend, no remainder
- 3-digit divisor, 6-digit dividend, with remainder
- 3-digit divisor, 7-digit dividend, no remainder
- 3-digit divisor, 7-digit dividend, with remainder
- Multiply decimals, writing the numbers under each other (0-2 decimal digits)
- Divide a whole number or a decimal by a whole number, need to add zeros to the dividend
- Convert a fraction to a decimal using long division, rounding the answers to three decimals
Convert measuring units using long division & multiplication
Primary Grade Challenge Math by Edward Zaccaro
A good book on problem solving with very varied word problems and strategies on how to solve problems. Includes chapters on: Sequences, Problem-solving, Money, Percents, Algebraic Thinking, Negative Numbers, Logic, Ratios, Probability, Measurements, Fractions, Division. Each chapter’s questions are broken down into four levels: easy, somewhat challenging, challenging, and very challenging.
Order of operations
- Three operations, uses ÷ for division, no exponents
- Four operations, uses ÷ for division, no exponents
- Two or three operations, uses the fraction line for division, no exponents
- Two or three operations, uses the fraction line for division, exponents included
- Two, three, or four operations, uses the fraction line, exponents included
Key to Algebra Workbooks
Key to Algebra offers a unique, proven way to introduce algebra to your students. New concepts are explained in simple language, and examples are easy to follow. Word problems relate algebra to familiar situations, helping students to understand abstract concepts. Students develop understanding by solving equations and inequalities intuitively before formal solutions are introduced. Students begin their study of algebra in Books 1-4 using only integers. Books 5-7 introduce rational numbers and expressions. Books 8-10 extend coverage to the real number system.
=> Learn more
Fractions vs. Decimals
- Decimals to fractions or mixed numbers (tenths/hundredths/thousandths)
- Decimals to fractions or mixed numbers (up to millionths)
- Mixed numbers to decimals (denominators 10, 100, and 1000)
- Proper and improper fractions to decimals (denominators 10, 100, or 1000)
- Proper fractions to decimals (denominators powers of ten till 1,000,000)
- Mixed numbers to decimals (denominators powers of ten, till 1,000,000)
- Fractions or mixed numbers to decimals (easy, varied denominators)
- Fraction to decimals - need long division
- Fractions to decimals - mixed practice
Decimal Addition and Subtraction
Key to Decimals Workbooks
This is a workbook series by Key Curriculum Press that begins with basic concepts and operations on decimals. Then the books cover real-world uses of decimals in pricing, sports, metrics, calculators, and science.
The set includes books 1-4.
=> Learn more
Multiply in columns
- Easy decimal division (dividend has 1-2 decimal digits, a whole number divisor)
- Like above, but missing dividend or divisor
- Divide decimals by decimals (Think of how many times the divisor fits into the quotient.)
- Mixed multiplication and division problems 1 (1 decimal digit)
- Divide whole numbers and decimals by 10, 100, or 1000
- Same as above, missing dividend or divisor
- Multiply or divide decimals & whole numbers by 10, 100, and 1000
- Divide whole numbers and decimals by 10, 100, 1000, or 10,000
- Divide whole numbers and decimals by 10, 100, 1000, or 10,000 - missing dividend or divisor
Convert measuring units using long division & multiplication (paper & pencil) or mental math
- Convert between inches & feet
- Convert between inches, feet, and yards
- Convert between ounces & pounds
- Convert between tons & pounds
- Convert between cups, pints, and quarts
- Convert between cups, pints, quarts, and gallons
- Convert between ounces, cups, and quarts
- All customary units except miles - mixed practice
- All customary units except miles - mixed practice - a challenge
Convert using a calculator, with decimals
- Convert between mm, cm and m - using decimals
- Convert between mm, cm, m, and km - using decimals
- Convert between ml & l and g & kg - using decimals
- All metric units mentioned above - mixed practice - using decimals
- Metric system: convert between the units of length (mm, cm, dm, m, dam, hm, km)
- Metric system: convert between the units of weight (mg, cg, dg, g, dag, hg, kg)
- Metric system: convert between the units of volume (ml, cl, dl, L, dal, hl, kl)
- Metric system: convert between the units of length, weight, and volume
Prime factorization, GCF, and LCM
Fraction addition and subtraction
In all fraction multiplication and division problems, it helps to simplify before you multiply.
Convert fractions to mixed numbers and vv
Simplify or equivalent fractions
Fractions vs. Decimals
Addition & subtraction
Addition and subtraction of integers are beyond the Common Core Standards for grade 6 but some curricula or standards may include them in 6th grade.
Multiplication & Division
Multiplication and division of integers are beyond the Common Core Standards for grade 6 but the worksheets links are included here for completeness sake, as some curricula or standards may include them in 6th grade.
Area - these worksheets are done in the coordinate grid.
Volume & surface area
Since these worksheets below contain images of variable sizes, please first check how the worksheet looks like in print preview before printing. If it doesn't fit, you can either print it scaled (such as at 90%), or make another one by refreshing the worksheet page (F5) until you get one that fits.
If you wish to have more control on the options such as number of problems or font size or spacing of problems, or range of numbers, just click on these links to use the worksheet generators yourself:
Free printable math worksheets
Home > Printables > Worksheets
HelpingWithMath.com offers a wide range of math worksheets that are ideal for parents of children in Kindergarten through 8th Grade who want extra homework help in mathematics.
Free math worksheets
The pre-made worksheets above are categorized by both subject and by grade level. Clicking the links will list these worksheets. The worksheets include arithmetic operations, (addition, subtraction, multiplication and division) fractions, decimals, percentages, geometry, place value, integers, and more. Practicing math with the help of these worksheets will be a valuable homework activity.
As mentioned above, you will also find many free math worksheet generators here and they will provide limitless questions along with answers.
How to use the printable worksheets
The worksheets are set up for easy printing. Just click the blue text links to open the document containing the worksheet. Either use the print function on your web browser or use the Print Worksheet option at the top of each worksheet to send as many copies as you wish to your printer. Each worksheet also has an option to show and print the answers for each questions.


Air in An Insulated Piston Cylinder Mass
Air in an insulated piston cylinder is initially at 300K and 100 kPa. The air is allowed to expand isentropically until the air’s specific volume reaches 2.0 m
/kg. How much work per mass (kJ/kg) is produced during the expansion? Use the IG model (variable specific heat) to calculate the answer.
a) 156.6 kJ/kg
b) 61.4 kJ/kg
c) 32.3 kJ/kg
d) -32.3 kJ/kg


How verbal items are stored in long term affect how they are retrieved. This in turn affects how we play a fair number of games, as they require retrieval of items from long term memory. In particular, I consider such games as Codenames, Scattergories, and Taboo.
For those who want to know more about the mathematics and computation behind memory retrieval:
Raaijmakers, J.G.W. (2008). Mathematical models of human memory. In H. L. Roediger, III (Ed.), Cognitive Psychology of Memory. Vol. 2 of Learning and Memory: A Comprehensive Reference, 445-466. Oxford: Elsevier.


Thirty one students from Noble Middle School were among the 120,000 students worldwide to participate in this year’s Math Olympiad program. They participated in a series of monthly contests of five problems each, with weekly practice sessions under the supervision and coaching of MSAD 60 Excel program teacher, Emily MacKinnon. She helped students to solve unusual and difficult problems through creative thinking in math. Mrs. MacKinnon has been teaching at the Berwick school for 16 years.
Most of the students that participate in Math Olympiad contests are among the best mathematics students at their schools, therefore, to earn an individual award, a student ranks among the best of the best internationally. Twenty one Noble Middle School students earned national awards for excellence.
*The gold pin is awarded to those in at least the 98th percentile (top 2%). At NMS, Rain Bugado, with a score of 22 of 25 points was in this extraordinary group.
*The silver pin is awarded to those in the 90th-97th percentiles, including the following students: Ethan Bent, Lucas Bent, Anthony Prak, Ian Tasker, Jamie George, and Wallace Morton.
* The embroidered felt patch is awarded to the top 50% of all participants, including the following students: John Kodzis, Liam Sullivan, Luke Safford, Colin Kitchen, Breenah Williams, Emma Badger, Stuart Powers, Lily Smith, Isaac Hayes, Cormac Mallett, Nicholas Sirois, Celia Momonee, Melissa Pass, and Noah Spinney-Hernandez.


is the Highest Level of Z?
Inside my very last essay , I clarified what derived that this concept out of a verse from the Bible and are the highest amount of mathematics. Now, I want to provide a conclusion of this concept to you. It is easy to know very well what may be the lab report item principle of math. The Product Rule in Math can be found in many math textbooks and is well known in the industry. The rule states that when working with numbers, and particularly facts about number values, facts should be computed in order of magnitude and not the letter values of these numbers. Here's an intriguing fact about this particular rule: what's the maximum amount of math? The number also the period , or three the percent eleven or five, or one's series plus the series of zero and one. All these numbers are figured in order of size rather than the correspondence values. The principle of several is referred to as the product principle, Because you can see. However, the highest level of this rule can be calculated by multiplying the numbers all together. The truth is that this might be the easiest way to figure the amount of this particular rule. The reason that this may be so easy to figure out is because all of the facts about one plus are already figured in order of magnitude. All that the reader needs to do is multiply them all together. Of course, it is necessary to use some type of multiple of the factor by which you are working with in order to get the right answer. But, the factors are already figured in order of magnitude, so all that needs to be done is to multiply them all together. But, what is the product rule in math? The answer may be hidden in the book of Numbers. In the book of Numbers, it says that two plus two equal five. However, the rule in the book of Numbers is that two plus two equal four and not five. The question would be: What is the highest degree of mathematics if fact loved ones in mathematics is always to be utilized? In actuality, while the reader has got the principle in mind, it needs to be read, then he or she can apply it. To check the answer for the question above, the reader will need to look at the fact family in math and compare the number five to the series of the five facts in the first fact family. This is the level of math. It merely takes one very fact to be able to reach the clear answer, in fact, before reaching this level, far much more specifics will have to get understood. The 5 in fact family is actually a collection of 5. Utilizing this fact in fact family in math, the reader gets got the answer for the problem previously. Thus, the answer to this question above, what's the highest degree of math, can be that a string of 5 facts from fact family in mathematics. The reader has got the response to the problem"what's the highest level of mathematics" with no to make use of any additional provisions. Does that mean that the answer isn't the obvious"truth?" Nothis does not mean the clear answer is a fact, since there isn't any gap amongst fact family in math. The reader understands the reply does not will need to refer to almost any other conditions, and to the question over is that a series of 5 facts in actuality household in mathematics. Should involve some knowledge of this fact residing in mathematics. Teachers who do not know about the fact residing in mathematics isn't going to know that they must ask the question,"what's the highest level of mathematics " Before using a word that does not implement.


Some of you love it, some of you hate it and I bet your parents feel the same. My goal is for you to ENJOY maths and understand why on earth we learn what we do. Below are some links for games and helpful information for your parents and family. If you have some other website suggestions please let me know.
Our first area of maths in this year I like to call ‘Maths Market’.We will be exploring how maths can help us at the market or when we go shopping and traveling. Would you like to know some cool short cuts? Would you like to improve your mental maths work so that you can beat your parents in a shop?


The solution that fits you best!
The 7 SunnyPower-solar panels you would like to place on your roof have a rated power of 350 Watt. In the Netherlands such a solar panel would yield approximately 300 kWh per year.
Maximum yield = 2100 kWh
(7 solar panels X 300 kWh = 2100 kWh)
Expected yield = 1680 kWh
(2100 kWh X 0,8 = 1680 kWh)
The yield in euro naturally depends on the price of energy. Using a price of 20 cent per kWh (prijspeil bron) this leads to the yearly savings shown here.
Expected savings: €336,-
(1680 kWh X €0.20,- = €336,-)


Number Time Trio Mini Stamp Set
Sale Price: $4.25
Product ID : 2627
A simple collection of three number styles, sized to coordinate with the largest clock face in the Hands of Time stamp set. However, don't let your ideas be limited to that pairing. These numbers have lots of possibilities for scrapbook layouts, birthday cards, school themed projects and so much more!
*This stamp set does not include a CD case.
This set comes on a single 3x4 acetate sheet. All of the numbers average about 3/8" tall.
Sale Price: $9.00 Sale Price: $4.50
Sale Price: $1.00 Sale Price: $0.85


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Item Dimensions: 936634174100
Languages: EnglishPublishedEnglishOriginal LanguageGermanUnknown
4 black & white tables, biography
Number Of Items:
Number Of Pages:
May 29, 2007
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Editorial Review:Product Description:
This textbook presents mathematical models in bioinformatics and describes biological problems that inspire the computer science tools used to manage the enormous data sets involved. The first part of the book covers mathematical and computational methods, with practical applications presented in the second part. The mathematical presentation avoids unnecessary formalism, while remaining clear and precise. The book closes with a thorough bibliography, reaching from classic research results to very recent findings. This volume is suited for a senior undergraduate or graduate course on bioinformatics, with a strong focus on mathematical and computer science background.
Average Rating: none


- Kresge 3-443
I am interested primarily in the philosophy of logic and mathematics, with associated interests in early analytic philosophy and Kant. Ages ago, I worked on a dissertation on the normativity of logic. My published work has been on the logic of neo-Fregean abstraction. But my interests in logic are not limited to that, and my interests in general are not limited to logic.
One moment, publications loading ... (or view them here)
- "Deductive Cardinality Results and Nuisance-like Principles". Forthcoming in The Review of Symbolic Logic.
Abstract. The injective version of Cantor’s theorem appears in full second-order logic as the inconsistency of the abstraction principle, Frege’s Basic Law V (BLV), an inconsistency easily shown using Russell’s paradox. This incompatibility is akin to others—most notably that of a (Dedekind) infinite universe with the Nuisance Principle (NP) discussed by neo-Fregean philosophers of mathematics. This paper uses the Burali-Forti paradox to demonstrate this incompatibility, and another closely related, without appeal to principles related to the axiom of choice—a result hitherto unestablished. It discusses both the general interest of this result, its interest to neo-Fregean philosophy of mathematics, and the potential significance of the Burali-Fortian method of proof.
- "Abstraction Principles and the Classification of Second-Order Equivalence Relations", Notre Dame Journal of Formal Logic, 60:1, 2019.
This paper improves two existing theorems of interest to neo-logicist philosophers of mathematics. The first is a classification theorem due to Fine for equivalence relations between concepts definable in a well-behaved second-order logic. The improved theorem states that if an equivalence relation is defined without non-logical vocabulary, then the bicardinal slice of any equivalence class—those equinumerous elements of the equivalence class with equinumerous complements—can have one of only three profiles. The improvements to Fine's theorem allow for an analysis of the well-behaved models had by an abstraction principle, and this in turn leads to an improvement of Walsh and Ebels-Duggan's relative categoricity theorem.
- "Relative Categoricity and Abstraction Principles” (joint work with Sean Walsh, UC Irvine). The Review of Symbolic Logic, 8:3 (572–606), 2015.
Many recent writers in the philosophy of mathematics have put great weight on the relative categoricity of the traditional axiomatizations of our foundational theories of arithmetic and set theory (Parsons, 1990; Parsons, 2008, sec. 49; McGee, 1997; Lavine, 1999; Väänänen & Wang, 2014). Another great enterprise in contemporary philosophy of mathematics has been Wright’s and Hale’s project of founding mathematics on abstraction principles (Hale & Wright, 2001; Cook, 2007). In Walsh (2012), it was noted that one traditional abstraction principle, namely Hume’s Principle, had a certain relative categoricity property, which here we term natural relative categoricity. In this paper, we show that most other abstraction principles are not naturally relatively categorical, so that there is in fact a large amount of incompatibility between these two recent trends in contemporary philosophy of mathematics. To better understand the precise demands of relative categoricity in the context of abstraction principles, we compare and contrast these constraints to (i) stability-like acceptability criteria on abstraction principles (cf. Cook, 2012), (ii) the Tarski-Sher logicality requirements on abstraction principles studied by Antonelli (2010b) and Fine (2002), and (iii) supervaluational ideas coming out of the work of Hodes (1984, 1990, 1991).
- “The Nuisance Principle in Infinite Settings”. Thought: A Journal of Philosophy, 4:4 (263–268), December 2015.
Neo-Fregeans have been troubled by the Nuisance Principle (NP), an abstraction principle that is consistent but not jointly (second-order) satisfiable with the favored abstraction principle HP. We show that logically this situation persists if one looks at joint (second-order) consistency rather than satisfiability: under a modest assumption about infinite concepts, NP is also inconsistent with HP.


The program used in this work:
equil-time-series.c You will also need
nrutil.h for a few definitions.
Examples of use:
equil-time-series 1326 2 ftse1326.dat
(Input time series ftse1326.dat; number of values to be read, 1326, must be equal to, or less than, the number of values in the file; and the value of 2, number of categories used in the coding, should be fixed always at 2.)
equil-time-series 1326 2 r
(Use 1326 uniformly distributed random values.)


Exhibits, Demos & Posters
A Hybrid Peer-to-Peer and Grid Job Scheduling System for Teaming Up Desktop Resources with Computer Clusters to Perform Turbulence Simulations
- Kay Dörnemann, Department of Mathematics and Computer Science, University of Marburg, Germany
- Tim Dörnemann, Department of Mathematics and Computer Science, University of Marburg, Germany
- Bernd Freisleben, Department of Mathematics and Computer Science, University of Marburg, Germany
- Tobias M. Schneider, Department of Physics, University of Marburg, Germany
- Bruno Eckhardt, Department of Physics, University of Marburg, Germany
Simulating turbulence in fluids is a fascinating part of physics which requires a high amount of computational power. Since transitional Reynolds numbers for each simulation run can be performed on a single contemporary CPU, turbulence studies are ideally suited for distributed computing where each node performs a simulation for a single initial condition.
The approach presented in this paper makes use of unused computational power by integrating a dynamically changing set of possibly unreliable desktop PCs into a grid infrastructure of attentively administered dedicated cluster resources. The basic idea is to use peer-to-peer (P2P) technology for managing the set of computers and develop a "bridge" to interface the P2P network with a grid meta-scheduler which in turn interfaces with the grid middleware.
This eliminates the need for central administration and continuous resource availability. It provides distributed scheduling, replicated storage, and system monitoring capabilities. Experimental results obtained from an evaluation of our implementation show that our approach is both scalable and resilient in the presence of node failures and network churn.


Michael L. Roukes
Frank J. Roshek Professor of Physics, Applied Physics, and Bioengineering
Caltech Physics, Mathematics and Astronomy
B.A., University of California (Santa Cruz), 1978; Ph.D., Cornell University, 1985. Associate Professor, Caltech, 1992-96; Professor of Physics, 1996-2002; Professor of Physics, Applied Physics, and Bioengineering, 2002-11; Abbey Professor, 2011-17; Roshek Professor, 2018-; Director, Kavli Nanoscience Institute, 2004-06; Co-Director, 2008-2013.
Pasadena CA 91125 USA
Office Phone: 626-395-2916 work


Week 1: “Genetic Optimization Visualization”
For our first session of Games & Art, we looked at Rock Paper Scissors (RPS) as well as at strategies for winning this time-honored game.
In the standard game, one point is awarded for rock beating scissors, paper beating rock or scissors beating paper. With this point assignment, there is no numerical strategy that will create a distinct advantage. Basically, the best way to win is to try to psyche the other player out or, to go a bit further, to make your rock, paper or scissors choices be as random as possible. In other words, R, P and S should each have a 33.3% chance of being chosen in each round. This makes sense given that RPS is generally thought of being more along the lines of a coin toss than a game.
If we change the point distributions awarded to a winning R, P or S, however, the possibility of numerical strategies for winning begins to emerge.
One variation we considered in class was the following set of point assignments: R:+10, P:+3, S:+1
In this variation, the reward for winning with an S is only +1, which is much less than R or P. To make matters worse, if the other player happens to choose R, then the effective cost of choosing S can be thought of as -11 since the other player has gained 10 while you have also lost the opportunity to gain 1. Therefore, we can see that an optimal strategy for winning at this variation of the game would not be an even 33%/33%/33% distribution between R, P and S. Instead, we should favor our choices towards R or P. But to what extent?
To find this out, I looked towards genetic algorithms. With the help of Dan Shiffman’s excellent genetic algorithm text, I devised the following model for an algorithm that could be used to determine the optimal choice-probability distribution for any particular point distribution:
First, I defined a Player class of objects, which contains four data points: a R, P, and S choice-probability value as well as the number of points the player has scored. Next, an initial population of players is generated. The choice-probabilities for each player are initially random, but hover somewhere close to an even 33%/33%/33% distribution. Once the population has been generated, the players are put into matches against one another. In each match, each player chooses an R, P or S in accordance with its choice-probability model. The winner of each match is awarded with the appropriate number of points. After the players have played a sufficient number of games against one another, their fitness for mating is determined by the number of points they have scored. So if one player has scored 10 points, the probability of its being selected for mating is ten times that of a player who has scored only 1 point.
The mating process consists of pairing two players together and using some of each of their “genetic” traits to form a child player. I had to experiment with the best way to go about this mating process. Since the R, P and S choice probabilities for each player must add up to a total of 100%, we cannot disassociate these values from one another. My first thought was to simply average the two parents’ choice-probability values to form the child’s values, but this turned out to produce somewhat muddy results and does not always come out to 100% probability total. The approach I eventually implemented was to take a number line between 0 and 1 and insert two barriers into it: one barrier separating R from P and one separating P from S. These two barriers can be moved anywhere in the number line and so long as they don’t cross one another or go above 1 or below 0, we will end up with positive R, P and S probability values that sum to 1.0 (or 100%). Using this way of the dividing the probability space works well for the mating process. Once two parents have been chosen for mating, the algorithm randomly decides whether to take one barrier value from each parent or both from one. Finally, to ensure that the gene pool does not get too incestuous, each child is given a 10% chance of mutating. If a child is chosen for mutation, it’s barrier values are moved randomly in either direction by a small amount.
With this genetic algorithm, the program is able to automatically find an optimal choice-probability model for a given RPS point distribution by using a survival of the fittest approach where fitness is determined by the player’s accumulated winnings over many games. The size of the population, the number opponents each player plays against, the number of matches that each pair of plays against one another and the number of generations over which the algorithm is run all have an effect on the accuracy of the algorithm’s determination of what constitutes an “optimal” choice-probability model.
Next, I set about the task of creating a visualization that would present the optimal choice-probability distribution for a group of RPS point distribution variations. For visualization purposes, I had the good fortune of RPS being a three-dimensional system, which maps well to both 3D coordinates and RGB values.
I assigned Rock to the X axis, Paper to the Y axis and Scissors to the Z axis. As we move along the X from 0 to 9, for instance, we add one to the value of winning with a rock and so forth for each of the axises and its respective R, P or S point value.
At each coordinate in the grid, I represent the choice-probability distribution with a color. Rock is associated with the Red color dimension, Paper with Green and Scissors with Blue. So, if for instance, Rock was worth 9 and Paper and Scissors were both worth 0, the color of that grid coordinate should in theory be bright Red as an optimal strategy for this point distribution would presumably bias towards the choice of Rock.
To show how the number of players, opponents, matches and generations affect the successfulness of the optimization process, I ran my visualization using varied parameters for these variables and recorded the results. Here are a few samples:
Notice that with a small number of players, opponents and matches, the arrangement of color values is relatively unordered. With a larger number of players, opponents and matches, the color arrangements take on a linear gradient quality. This shows that greater optimization (or accuracy if you prefer) has been achieved with the larger population. If the smaller population were allowed to run for a much larger number of generations, it’s accuracy would begin to approach that of the larger population.
I wrote the genetic algorithm in C++ and the visualization uses openFrameworks.
WEEK 2: “Genetic RPS Newsfeed”
For last week’s assignment, I was interested in exploring the aesthetics of RPS as a type of zero-player game. In the book, Evolution and the Theory of Games, John Maynard Smith considers the ways in which game theory or the investigation of strategic advantage can be applied to our understanding of individual fitness and evolution within a population. For instance, with the Hawk-Dove game, he demonstrates how the aggressiveness of an individual’s strategy in competing for food affects population balance and, in turn, how evolutionarily stable populations can emerge from the proper mix of strategies utilized by individuals as part of an ecosystem. To call RPS a zero-player game is in some respects counter intuitive. There are, of course, two players. In a statistical sense, though, these two individuals are merely playing out the probabilities. Though each run of the aforementioned genetic algorithm is likely to produce slightly different optimal probability distributions for a given point distribution, the results vary only slightly and therefore indicate that to a large extent, there can be little individual differentiation in the numerical aspect of choosing a winning strategy.
What remains, then, is a sort of mating dance. If no individual is truly more fit than any other, competition must be about the extent to which an individual is able to purport him or herself as more fit than another. That is to say, RPS is about psyching the other player out. The best way to win seems to be to focus on executing the steps of your own dance while trying remain undistracted by that of your opponent. It is hard to conjure random or statistically-weighted choices while in the midst of a mating dance. So for this week, I was interested in testing whether strategic advantage could be achieved by enhancing a player’s ability to do so.
Using my genetic algorithm, I precompiled a list of R,P or S choices for a particular point distribution (R:10 P:3 S:1) and converted this list to an audio feed. Before entering a game of RPS with a human opponent, I donned a TV news anchor-style earpiece, which would be used to feed me a stream of R, P or S’s during game play. I then played a number of games against opponents who were not given any technological aid. After playing several opponents over a number of games, it was clear that this aid was indeed giving me an advantage over my opponents. Of course, RPS remains a game of probability and so I found that this device created an advantage but did not lead to a complete domination of all opponents in all cases.
Is RPS a zero-player game? Understanding, of course, that I am using this term metaphorically and not by the technical definition (which is intended to describe dynamics more like Conway’s Game of Life, etc), I submit that RPS is a game of narrow margins. But so too is the mating game. If RPS falls somewhat short in meeting the criteria required by Smith’s comprehension of evolutionary advantage, perhaps it forms a better metaphor for the evolutionary state of humankind. That is to say, the availability of technology and the complexity of our society is such that the fitness criteria for an individual human cannot be clearly named. Rather, we live and die by something far more ephemeral, with all the same technologies, medicines, etc at our disposal, we have only the slight bias of our mating dance, our swagger, to put us ahead. In the end, this seems to make all the difference or possibly none at all.


Repeated measures and longitudinal data offer rich possibilities for business insights and research advances. Whether you have experimental data, need to monitor simple time trends, or have complex data with unknown time trends, Elkhart Group Ltd. is ready to use modern analytics to help you understand your data and answer your questions with confidence. Continuous or categorical variables, missing data, or nonrandom study dropout, our statistical consultants are ready to tackle even the most complex questions and give you simple answers and graphs to understand and take action on your results.
Experiments often collected repeated measures throughout the study, and require special analyses that account for non independence of each assessment. Our statistical consultants can help you take advantage of all the measures in an experiment by using repeated measures ANOVA or more modern mixed models for your data. If you are collecting physiological or biological data, we can use area under the curve (AUC) analyses to compute the total amount of a particular analyte output during the course of the experiment.
Whether you are monitoring trends in attitudes toward your company's products, exploring health behaviors after a heart attack, or looking at fluctuations in employment and marriage, we can help you analyze and interpret your longitudinal data. Our statistical consultants can use mixed effects models or latent growth models to capture individual differences in starting points and growth or change over time. We can help detect and explore nonlinear time trends, and account for residual autocorrelation or dependence in your data.
For complex longitudinal categorical and count data, we can help you setup mixed models to test for significant effects, and also help you convert back to interpretable metrics, like changes in the probability of an event occurring. For rare events or models too complex for classical statistics, we can code hierarchical Bayesian models.
Not all individuals follow the same trajectories. Our statistical consultants can use latent class analysis and growth mixture models to answer hypotheses that that there are distinct classes of people with their own trajectories over time, whether you have discrete time measures or a truly continuous time assessment. We will partner with you to compare results from models specifying different numbers of distinct classes and explain the assumptions of each model so that you find a solution that both makes theoretical sense and fits the data well.
Suppose a researcher studied depressive symptoms and measured them at four time points in cancer patients after diagnosis. Traditional latent growth models and mixed effects models for longitudinal and repeated measures data assume that everyone has the same general trajectory. Our analysts can use growth mixture models or latent class analyses to explore whether there are distinct trends in your data. The results of such an analysis on a sample dataset finds four classes with the means of each class at each time point shown below. The pattern is that patients in Class 4 are stable and high in depressive symptoms, Class 2 are stable and low in depressive symptoms, and Classes 1 & 3 start low and high, respectively, and then switch later on, representing a late reactivity and a recovery trajectory.
Plots of the raw data, separated by group after the growth mixture model identified unique trajectories are shown here.
Intensive data collection such as ecological momentary assessment (EMA) where there are many observations for each participant often may not follow simple time trends. We can use piecewise or spline models for time to fit complex time trends. We also know that data may be cyclical or exhibit unknown time trends. Our statistical consultants can use generalized additive mixed models (GAMMs) to estimate models that flexibly fit time trends without having to specify the functional form.
Although most analyses focus on one or two outcome variables, some questions require assessing the trajectories of multiple outcomes simultaneously. For example, do individuals' slopes of reaction times to a short puzzle task predict cognitive performance and decline over a five year period? In newly cohabiting couples, do brand preferences change together, or does one partner influence the other? Should advertising be targeted at both members of the couple or is targeting one member sufficient to eventually change the other? At Elkhart Group, we are committed to answering your questions with straightforward answers. Our statistical consultants can design parallel growth models to examine whether trajectories predict each other or move in tandem. We can analyze your data to answer whether multiple processes move in synchrony or co-regulate each other.
In longitudinal studies, participants are lost to follow up and missing data is the rule, not the exception. If you have intermittent missing data, our statistical consultants can use multilevel multiple imputation or full information maximum likelihood. If your data may be missing not at random, we can implement pattern mixture models, and shared parameter or joint models where longitudinal trajectories and time until study dropout is simultaneously estimated to account for non random dropout.
Premier: Our most bespoke offering. Let our consultants create solutions and reports that turn your data into strategic insight.
Academic: For accredited, not-for-profit universities. From methods write up assistance to advanced model analysis, we are experienced researchers who enjoy scientific discovery.
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Blackjack is a card game that has no dearth of geniuses. From rags to riches to the best of the best, the game has given us moments and names that we will never forget. The blackjack professor, Edward O Thorp is a name who has his class. Thorp was born in 1932 and went on to become a mathematics professor, writer, hedge fund manager, and also one of the best blackjack researchers. He’s best known for his work in correlations in the financial gains, and application of the probability theory. He later applied this knowledge in the stock market as well as in blackjack that gained him quite a fortune. Apart from that, his work continues to gain him the respect that he deserves. He is also the author of the popular book on the casino games Beat the Dealer that gives insights into how the house advantage can be beaten by counting cards. In addition to that, he has a variety of skills in the financial market and offers insights into that as well.
Thorp is a Ph.D. holder from the University of California from the class of 1958. In addition to that, he has also worked at MIT, remained as professor of mathematics in the New Mexico State University (1961-65). He was also working as a professor of Mathematics at the University of California from 65 to 77. He also worked as a professor of math and finance from 1977-1982.
His initial finding in the game was inspired by a paper that was published in 1956 by Kelly. He based his theory on the Kelly criterion to ascertain the probabilities of beating the house edge and winning the game. In addition to that, he analysed the game, and then went on to develop the card counting schemes that could help you beat the edge. This could help you beat the odds, and increase your chances of winning the game.
For the theory to work in the field, it had to be implemented. So, he decided to make sure that he was put to work. With an investment of $10,000 at a local blackjack table, he along with many Kimmel managed to win $11000 in a week. This proved the potency of the theory, and he went on to play at various tables. In addition to that, the casinos soon discovered his techniques and he had to play in disguise. In a matter of months, his name became so popular that everybody was inspired by the new techniques. Thorp wrote the book Beat the Dealer in 1966. The book took the casino world by storm and sold 700,000 copies. The book made it to the New York best-selling list.
He has written a couple of books in this regard. These include A Man for all Markets, Elementary probability, A winning Strategy for the Game of Twenty-One, the Mathematics of Gambling, and Beat the Market. They offer unique insights into the game and allow the player to know different aspects of the card game.


In this test, the beating heart simulation is performed with a rotationally symmetric left ventricle model. The same time profile of calcium ion concentration as in Isometric Contraction is given for all places in the muscle. The half cross sectional area is divided into quadratic elements, and your sarcomere model is imbedded into each of elements. The momentum equation of the heart wall is solved under the equilibrium condition with the intracavity pressure P, where the equilibrium conditions with the circuit models are also imposed. The basal portion of the model is fixed in the simulation.
The geometry in the unloaded condition of the half sectional model is given by ellipsoidal curves. At each point in the heart wall model, the normal frame composed of so called the cardiac vectors c, l and r is given. The longitudinal unit vector l is a vector directed from the apex to the base and parallel to the ellipsoidal curve. The circumferential unit vector c is a vector perpendicular to the sectional plane and directed to the back of the plane. The radial vector is given by r=c×l. In the heart beat test, we assume that the fiber orientation f is perpendicular to the radial direction r and given by the rotational angle α from the circumferential vector c. You can specify the angles α in the following ways:


Reading a new opinion columnist in the NYT—Stanley Strogatz. His column is called “From Fish to Infinity” (which I can only guess is a paean to cosmologist George Gamow's One, Two, Three...Infinity) and explores what he calls a second chance at math. It is a new way (to me anyway) for subjects to be explored in the Times because it takes a slightly more academic bent in its treatment of sources. It cites them. Who knew? That is a ‘new’ model. Better yet it has introduced me to a book and an article that I might never have seen as a math lover from afar.
- “E. Wigner, “The unreasonable effectiveness of mathematics in the natural sciences,” Communications in Pure and Applied Mathematics, vol. 13, No. I (February 1960), pp. 1-14. A pdf version is here. I put this one in my iPod touch and use the app, Papers, to read it. My skim of it can only come up with one word--mordant.
- P. Lockhart, “A Mathematician’s Lament: How School Cheats Us Out of Our Most Fascinating and Imaginative Art Form” (Bellevue Literary Press, 2009). This one goes on my wishlist at Amazon. Please, anyone feeling generous? Otherwise I will have to get it via Interlibrary Loan (which the Firefox extension Book Burro found for me quite handily).


Let D(K) be the positively-clasped untwisted Whitehead double of a knot K, and T(p,q) be the (p,q) torus knot. We show that D(T(2,2m+1)) and D2(T(2,2m+1)) differ in the smooth concordance group of knots for each m>1. In fact, they generate a Z2 summand in the subgroup generated by topologically slice knots. Our main tool is the Heegaard Floer correction term for the double cover of S3 branched along a knot. We also present some sufficient conditions for general knots to have this independence property. Interestingly, these results are not easily shown using other concordance invariants such as tau-invariants of the knot Floer theory and s-invariants of Khovanov homology.


2D projection of a 3D Phasefield simulation computed on a GPU (deeper layers appear bluer, areas of higher concentration of component A appear lighter). The simulation code was developed using OpenCL. Besides the usual Cahn-Hilliard formalism (shown here for an immiscible alloy) the simulation adds plastic deformation, modeled using a simple plane shifting algorithm. Both post processing for visualization as well as characterization using spherically averaged structure factors are computed on the GPU.
The animation on the left shows a 3D random walk originating in the center of a sphere, shown until it first crosses the surface of the sphere. This is called a first passage event. The system is rotated around the x axis to keep the particle in a plane parallel to the screen and going through the center of the sphere. That way it always hits the surface of the sphere at the outer perimeter line, making the first passage event more obvious in the 2D representation.


To reduce the cost of storing, processing and visualizing a large-scale point cloud, we propose a randomized resampling strategy that selects a representative subset of points while preserving application-dependent features. The strategy is based on graphs, which can represent underlying surfaces and lend themselves well to efficient computation. We use a general feature-extraction operator to represent application-dependent features and propose a general reconstruction error to evaluate the quality of resampling; by minimizing the error, we obtain a general form of optimal resampling distribution. The proposed resampling distribution is guaranteed to be shift-, rotation- and scale-invariant in the 3D space. We then specify the featureextraction operator to be a graph filter and study specific resampling strategies based on allpass, lowpass, highpass graph filtering and graph filter banks. We validate the proposed methods on three applications: large-scale visualization, accurate registration and robust shape modeling demonstrating the effectiveness and efficiency of the proposed resampling methods.


MATHEMATICAL ANALYSIS II
The Euclidean space R^. Functions between Euclidean spaces, limit and continuity of functions. Differentiation of vector-valued functions of a single variable, application in mechanics and differential geometry, polar, cylindrical and spherical coordinates. Differentiable functions (partical derivatives, directional derivatives, differential). Vector fields. Gradient-divergence-curl. Fundamental theorems of differentiable functions (mean value theorem, Taylor). Inverse function theorem. Implicit function theorems. Functional dependence. Local and conditional extremes. Double and triple integrals: definitions, integrability criteria, properties. Change of variables, applications. Contour integrals: Contour integral of the first and second kind, contour integrals independent of path, Green’s Theorem, simply and multiply connected domains of R^2 and R^3. Elements of surface theory. Surface integrals of the first and second kind. Fundamental theorems of vector analysis (Stokes and Gauss) applications.
Introduction to differential equations (definitions). First order differential equations (separable variables, total differential and Euler multiplier, linear, Bernoulli, homogeneous Riccati, Clairaut, Lagrange). Qualitative theory of differential equations (general). Higher order linear differential equations (general theory). Linear differential equations with constant coefficients (solution of linear equations, variation of parameter method, method of undetermined coefficients, Euler’s differential equations, applications).


Strolling through most math classes today, many would be astounded to see that understudies are as yet utilizing calculators with working frameworks from 1999.
It’s hard to believe, but it’s true. The TI-83, which hasn’t been refreshed since 1999, it is still the standard in many schools. Nearby it is the TI-84, which, despite the fact that has a marginally further developed working framework, has a goal that feels old.
Texas Instruments has turned out with some extraordinary more up to date forms of charting calculators, however schools are reluctant to embrace them.
The TI-Nspire models, for instance, nearly capacity like little workstations. They have screens that can be explored by touchpads, the capacity to spare reports, and have such a high goals, that clients can transfer pictures.
My graphing calculator audit is that numerous schools feel that these calculators are excessively best in class. A portion of these calculators can even associate with the Internet.
Thus, schools keep on making understudies to buy the old TI-83 and TI-84’s – calculators that are increasingly costly, and thirteen years more established than a portion of the further developed, contemporary calculators.
While composing this graphing calculator audit, it struck me that these models (counting the TI-83 and 84) are extraordinary learning instruments that can make math simple and fun.
Be that as it may, if there are less expensive, further developed models out there, for what reason would we say we are making understudies buy the more costly ones?
In American schools, we center such a great amount around “shutting the accomplishment hole,” now and then overlooking that purchasing a less propelled calculator for more cash doesn’t bode well for certain families that are monetarily hindered nor does it bode well for schools on a spending limit.
Another motivation behind why schools don’t move up to the further developed calculators is that reading material and exercise plans are now composed obliged the TI-83 and TI-84.
Buying another class set of TI-Nspires would mean advising distributors to refresh their course books to have another graphing calculator survey.
Aside from graphing calculators, we also have Velocity Calculator. Visit us to find more variety of calculators online. …


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Amazing NEW Board Game!
Math Ops (Ages 9-11)
MATH OPS© is poised to become the world’s most exciting MATH board game! It was conceived as a way of overcoming the real or perceived fear of Mathematics especially by primary and middle-schoolers.
Math-Ops is an easy to follow and understand board game designed to help children learn, love and have fun with Math. After a few rounds of exciting games you will conquer your Math fears and be ready to excel!
Math-Ops is a fun and exciting game that children can have fun with and enjoy with friends.
Math-Ops is also a useful educational tool to learn and perfect a range of math concepts, from arithmetic to probability, and beyond.
As players compete to win, they make use of and develop skills critical to their understanding of Math.


Here's a table summarising the equations you need to calculate the lift force on most of a rocket's lifting surfaces (nosecone, fins, body, transitions).
Haven't seen them all in one place before, so figure it might save another poor soul from wading through all of the Barrowman equations.
Only use the fin-body interference factor when calculating fin lift.
Can generally neglect body lift for regular looking rockets.
Discussions about techniques for rocket building. Leave your tips and tricks to making the perfect rocket.
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(Note: The author, Brian Boley, has a bachelor's degree in Physics and has taught physics labs at the college level. He has also tutored algebra, calculus, physics, chemistry, and biology at both the high school and college level. He has 4 children, all of whom are home-schooled.)
Perhaps the hardest class for most home school parents and professional teachers to teach properly is Mathematics. This is because few among us have truly learned Math -- most of us struggled through the courses. You cannot let most students simply follow a textbook, since this only provides half of the information needed to properly solve Math problems.
Math is very different from English or History -- or even Science -- because Math is the only subject that shares two characteristics:
In other classes such as History, one can start at any point and study just that particular part of History. In English, there are various degrees of correctness and opinions may differ as to what is correct. But not in Math: Math is precise and requires exact execution. Math answers are either right or they are wrong. ("Partial credit" is merely a device used by teachers from the 1970's who were too concerned with the short-term self-esteem of their pupils.)
In this article, we will develop several Concepts, which will enable us to teach Math correctly. As we learn a Concept, we will see how to apply it in the classroom. We will also see the effect that the Concept has on the student.
This is the first Concept: Math answers are either right or they are wrong. There is no opinion -- only proof.
By the time most students get into serious Math -- roughly the 4th grade -- they have been taught that the answers that they are expected to come up with are either correct, partially correct, or wrong. They have also been taught that the question of "correctness" depends upon opinion and that their teacher's opinion of the correctness of an answer is most critical. Thus, you'll see many students look at the eyes of the teacher when answering a question. This is so the student can make an instant determination of correctness based upon the teacher's opinion. If the student can make a strong case, they can improve their grade -- even if the answer is objectively wrong.
This "opinion-based" teaching has developed since the 1970's with the change from strict memorization of facts to the more discussion-based approach to teaching. Whole Language has further emphasized the "opinion-based" view of learning among the students -- "Your opinion is just as good as anyone else's opinion." Self-esteem is preferred to objective answers.
But now we come to Math. 2 x 2 = 4 and it doesn't matter what the students or teachers say or argue. This is the single most important issue that elementary math students (and teachers) must learn!
The Concept seems so obvious, yet is not. The reason it is not is because most of us studied at elementary schools that encouraged fact memorization. The most important things to learn in History about the Civil War when we were in school were the dates and places of battles -- today's History spends the most time in a debate over the causes and outcomes of the war. When we were young, right and wrong were obvious. Today, they are not.
If your student is doing excellent in History, English, Reading, and Science --- yet poor in Math -- this Concept -- the Concept of Right and Wrong -- is what needs explaining.
Explain to the student repeatedly -- the methods of Math determine the answer. What the teacher says does not matter -- the methods determine the answer. While you are going over a problem, explain that the method determines the answer. "What if I said the answer were 8? I'd be wrong, because the Math determines the answer to be 5." Do this repeatedly for months and the student(s) will get the message. This is a particularly important concept to emphasize in the classroom to normal 2nd to 4th graders.
Students that look at your face when you ask, "What is 5 x 5?" are students that believe they need to guess your opinion of the right answer. (Students that get the message will either look away while thinking, or will reply confidently "25!")
Avoid the "partial credit" trap when teaching middle school and high school students. Someday you may drive over a bridge which one of your students designed. Do you expect him to have calculated the loads correctly or should he get "partial credit" for getting a close answer? And all because you taught him that using the right equation was worth 90% of the problem -- and adding 2 + 5 = 8 was only 10% off.
In late 1999, a Mars probe, which cost $125 million, was lost because one team involved used the English unit of pounds for force calculations and the other team used the metric unit of Newtons. They made the "simple mistake" of failing to convert the units. The probe, after traveling about 100 million miles, was 60 miles low when it came time to enter Mars orbit -- and the probe crashed. By "partial credit" standards, the flight team would get a 99% grade. But the probe crashed -- and the taxpayers lost $125 million for nothing. In reality, the team got the answer 100% wrong.
Unlike other areas of study, one can't just pick up a Math book at the college level and begin unless the proper sequence of preliminary courses have been taken. This is because the study of Math is the study of a procedure and of a methodology -- not a set of facts or opinions. A body of facts, such as History or Geography, can be entered at many points. A methodology must be taught from the beginning.
Learning Math is similar to learning how to cook. A beginning cook cannot simply walk into a kitchen and bake a cake from scratch. A terminology must be known first: What does "blend" mean? What is a "tablespoon"?
In Math, simple routines and terminology must be learned before advanced routines and terminology can be learned. (I say "advanced" rather than "complex" because higher Math is not complicated -- provided you understand how to do simple Math.) Any student that is not capable of 99% accuracy on basic addition, subtraction, and multiplication is going to have tremendous problems with long division -- simply because multiplication and subtraction are the basis of division. Similarly, any student that has trouble with fractions will have trouble with algebra -- fractions are a vital part of algebraic manipulation.
Hence, it is vital that students understand completely each building block as they learn Math. If you encounter a student that has trouble with division -- throw in a few multiplication and subtraction problems to diagnose potential mis-understandings in those areas. When starting algebra, be sure to thoroughly test your students of their knowledge of fractions and basic area/volume formulas. If a student shows a weakness in these areas, a couple of weeks of review are definitely in order. In some cases, that student may have to spend a couple of months reviewing the missed material.
Many students, taught by the system to learn facts and then forget them to make way for more facts, will forget Math methods after a test. We suggest using the Saxon Publishing Math series as a way to cure this problem. The Saxon books simply do not let a student forget a procedure -- it comes back again and again and again
A simple concept that constantly caused problems with my daughter was her tendency to look at a problem that contained a zero and treat the zero as though it wasn't there. For example, as part of a long division problem of 500/20 = ?, she would look at the zeros and say "how many times does 20 go into zero? -- It doesn't go into zero." She would then go onto the next digit, skipping the place value. Instead, she should have said,” 20 goes into zero exactly zero times" and put a 0 into the place. Clearly, she had not understood that zero was just as important as every other numeral. We had to take a week and do many problems using zero.
Another subtle effect of advanced Math problems is this. Imagine that your student can do single-digit multiplication with 90% accuracy. In a long division problem such as 46535 divided by 15, there will be at least 5 double-digit multiplication problems, or a total of 10 single digit multiplications. If the 90% accuracy holds, then probability says that the student has a 35% chance of getting the correct answer -- assuming that the student makes no mistakes on the many subtraction steps. However, if the student has 99% accuracy in single digit multiplication, the student has a 90% chance of getting the problem correct. This is why it is important to drill and drill the basic functions until they are 99% accurate or better.
Many students that develop Math problems in the middle grades do so because of this effect. The multi-step problems magnify minor inaccuracies into major, test-failing problems.
To correct this problem, give the student a complex division problem. Have the student go slowly through the problem and write down each step separately, spread out across the paper or chalkboard for effect. Now point out the number of steps involved and the fact that every step must be correct for the problem to be correct. And identify which basic building blocks the student is having difficulty solving. Work with the student on those issues.
Student Math difficulties fall into three categories -- Conceptual, Algorithmic, and Process. Let's see what I mean by these categories.
My daughter's problem with zero was a Conceptual problem. She didn't clearly understand the idea of zero. Similarly, students -- particularly those who miss class -- can miss a critical concept that can cause problems for years to come.
Other concepts that are commonly missed include basic geometrical concepts such as area, volume, diameter, etc; negative numbers and their manipulation; the fraction as division; exponents; logs; trig functions; and other items that are covered quickly.
Fixing Conceptual problems is simplest -- explain the concept to the student until they understand it.
Algorithmic problems are problems with the method of solving the problem. For example, a student may simply not understand the step-by-step procedure of solving a division problem. Since we tend to spend the most time on these issues, this is not generally a problem with most students -- it's what we are already focusing on.
The general procedure to solve an algorithmic problem is to first teach the student that an algorithm is a step-by-step procedure, similar to using a recipe to bake a cake. The recipe must be followed, or you will not get a cake. Then, the task is to teach the student the recipe clearly. Feel free to write down step-by-step instructions just as you would for baking a cake. (Warning -- don't try to wing this. Few people can write down an algorithm correctly the first time. You will need to make a couple of revisions, which means that you should do this during class preparation time -- not during teaching time.) Then walk the students through the step-by-step process. Have them take notes and keep the algorithms in a notebook for reference. After they refer to it several times, they will have the procedure memorized.
Process problems are general problems with the manner in which the student solves Math problems -- any Math problems. As such, they are really personal habits that need to be corrected. Process problems cause most of the difficulties Math students have. Yet, we spend the least time fixing these problems, since they are not covered in most textbooks. That is why the next 6 Concepts are devoted to the most common process problems.
Process problems cause the most Math difficulties and lead to the most problems and furthermore, process problems are the cause of long-term Math difficulties. What are some sample process problems?
Sloppiness -- If your student doesn't write neatly, he will consistently line up columns incorrectly, resulting in mistaken addition and subtraction. This problem is particularly a problem with bright students who "get it" in the early grades. But as the problems get more and more complex, their sloppiness will tend to prevent them from getting correct answers.
Taking Shortcuts -- Students who can solve simple problems easily will often have trouble with complex problems because they are used to taking shortcuts rather using a robust methodology.
No Checking Methods -- Students who get good grades rarely check their answers. Once again, this comes back later to haunt them.
All of these Process problems are related -- they are most commonly found in students who succeed in Math for years -- and then start having problems. Process problems can be identified when a student starts making "dumb mistakes" or "simple mistakes" such as not carrying a number properly, or adding 8+7=17. This commonly occurs in the middle school grades, but can sometimes not surface until high school. A few students will experience the problems first in long division.
The root cause of all Process problems is a lack of self-discipline. A student with a spotless room rarely will have Process problems for long, while a student slob will have a difficult time correcting Process problems -- yet can benefit from it in many ways. Students without Process problems will often be described as Methodical. Students with Process problems will often be described as Brilliant but having difficulties. We will explore the most common Process problems, their effects, and cures in the next few Concepts.
There are 3 reasons why the military emphasizes cleanliness and neatness:
· It saves time since things don't get lost.
· It leads to self-discipline and planning.
· It leads to greater self-confidence and pride.
These three benefits also apply in the area of Math.
In Math, Neatness makes a big difference once 4-digit addition is introduced. It is vital that the student keep the columns in alignment in order to get the right answer consistently. A sloppy student will accidentally misalign columns of numbers and add them incorrectly.
There are several points where neatness must be upheld in Math:
· Columns must be neatly aligned vertically -- particularly in long division.
· Decimal points must be aligned vertically in arithmetic problems.
· Numerals and variable letters must be precisely and neatly written.
· If unclear, the numeral "7" should be written with a dash across the midpoint to clearly distinguish from "1".
· In the same manner, "Z" should be written with a slash to distinguish from "2".
· When solving equations, each step should be written on a separate line.
· "=" signs should be lined up vertically.
· After the introduction of variables, the "x" sign should be replaced by the "*" for multiplication.
· After the introduction of fractions, the fraction bar should replace the simple division sign.
· Solve equations 1 step at a time. Avoid trying to do multiple steps at once because you will get confused.
· Only 1 equal sign should exist in any equation.
After equation work has begun in Pre-algebra, be sure that students always start with plenty of room. Don't let them zigzag across the page. Instead, insist upon a vertical column of precisely aligned equations.
To get the point across to the students, one day without previous notice, pass out a handwritten assignment page, which is SLOPPY! Violate as many of the neatness points above as you can, and have the students work on the page as a typical homework assignment. Be sure to pass out the assignment late in class so that the students will have to sweat over the assignment at home and be unable to ask you questions easily. Make it really hard to tell your 1 and 7 apart and the difference between your 2 and Z. Misalign your columns on purpose and make it hard to tell whether you meant x as a variable or x as a multiplication sign.
The next day, be prepared for comments. Encourage a gripe session about the difficulty of reading the problems. After the gripes are going strong, explain to the students that it was to illustrate a point. (If you have time, you might even grade the homework in class -- be merciless in your grading!) Then begin your lecture on neatness.
For weeks afterward, grade papers on neatness. Get your students in the habit of being precise. That precision will begin to show in their work, also, as neatness-related Process mistakes gradually disappear.
One of the common problems that develop both in the early middle school years and again in the high school years is the Process problem where the student approaches each problem as a totally different problem. This leads to confusion and a tendency to get the process right once, but then to drift off course. The solution is DITS-WET - "Do It The Same Way Each Time".
Encourage your student to look at categories of problems. Each type of problem has one method to solve that problem. Emphasize that once a student finds a way to solve a category of problem, he or she can use that method to solve ALL problems that are in the category. Then, use that method over and over and over again. If the student understands and uses this Concept, then they will concentrate on the higher-level question of "which category is this problem a member of?"
DITS-WET is particularly good for students that have difficulty with multi-step problems (such as long division or basic algebra), and with word problems. Imitate a machine or a computer as you do a sample problem (such as an 8-digit addition problem). Emphasize DITS-WET each cycle of the problem.
Another manifestation of this problem is my daughter's previously mentioned difficulty with zero. In 3rd or 4th grade, many students transition from a memorization-based arithmetic (multiplication tables) to an algorithm-based approach. They have a bit of difficulty understanding that the method doesn't depend upon the numbers involved. "0" and "1" appear to be exceptions to the methods to many of the students. So they are still caught in this middle ground where they are trying to learn a method, yet they know that the numbers make a difference to the outcome, but don't understand that following the method will handle the difference in numbers.
We used the DITS-WET Concept with our daughter to emphasize that it did not matter WHICH numbers were in the problem, the METHOD was the same. We emphasized the difference between Method and Data, explaining them as concepts. Then we drilled over and over again that you don't treat "0" different from the other digits. DITS-WET.
Later, DITS-WET was invoked with basic geometric questions of area of a circle and volume of a cube. I remember using the procedure in college to emphasize the step-by-step procedure for solving a word problem. (Details below).
Many students of Math fall into a common trap -- A simple problem is presented as an introduction to a class of problems. The brighter students quickly realize that the problem can be solved by mental shortcuts and don't really study the method rigorously. A few weeks or months later, a much more complicated instance of the problem appears and the shortcuts don't work because the complexity is too great.
In effect, avoiding shortcuts is a variation on DITS-WET. It is important for students to learn the methods that always work. Teaching shortcuts is something that should only be done with students who have demonstrated a mastery of the complete method. Worst is a method of teaching which I've seen occasionally, where a simple method is used to solve a simple problem. But the method doesn't work with more complex problems. In effect, the student must learn several methods to solve problems of increasing complexity, when one moderately complex method would always solve any of the problems. In the long term, it is to the student's benefit to have one thoroughly understood method that always works, than be faced with the added complexity of having several methods for different situations.
To break a student of using shortcuts, toss in complex problems in a sea of simple problems. Or better yet, quickly progress the student to the level of solving the complex problems regularly. With more mature students, the benefits of learning DITS-WET can be taught and the shortcuts kept in their proper place.
Many students never check their answers. It's particularly important as multi-step problems appear. And reversing the process can check almost any problem.
The best way to teach checking is to require it. Require a student to solve the problem and put the reverse check beside the problem.
Many students don't work with units for several years after they are introduced. Units tend to be added at the end of the problem as an afterthought rather than pulled through the method with the numbers. The result is that many mistakes, which could be caught by paying attention to units, are not caught.
For example, if a student divides 2 hours by 150 miles to get miles per hour, he will catch this mistake (Oops, that's hours per mile -- obviously wrong) if he uses his units, but may not catch it if he simply adds "mph" at the end.
Require units be pulled through the entire problem and count the problem wrong if they are not.
No student can develop the step-by-step concentration that Math requires in an environment surrounded by rock music and television. That is because modern music and TV are designed to grab and hold the listener or viewer's attention.
Mathematicians almost without exception have noted that the best environment for doing difficult Math problems is either a quiet place (Newton invented calculus sitting outside under the trees.) or refer to baroque music such as that of Bach, Vivaldi, Telemann, or Handel. This is because the baroque music is very orderly and is without attention-grabbing climaxes. Certain quiet "New Age" and Jazz music has the same quality.
To encourage your students to study with this music, consider using Bach as a background to seatwork. Exposure will gradually convince students of the merits of the music -- Bach's "Jesu, Joy of Man's Desire" became a top 5 hit in the 1970's after a Boston radio station gave it heavy air play as part of an experiment.
As problems grow more complex, luckily they begin to have more and more relevance to the world. I encourage each of my students to look at their answers and give it a reality check -- "If I drove to Chicago from New York at 60 miles per hour, does it make sense that I would get there in 2 hours?"
The best way to begin to encourage this is first to explain the technique. The next day, tease the students when they have obviously impossible answers. Do this a couple of times and emphasize the reality check Concept.
Word Problems cause many difficulties for many students. Yet, there is a straightforward Process that allows all Word Problems to be methodically solved. Most Math students are not taught this process and are not held to the process. Thus, they have considerable problems with this class of Math problem. The following method works beginning with Pre-Algebra.
Make a drawing or chart of the information contained in the problem.
Step 2 -- Write down as a vertical list a set of equations expressing known facts. For example:
Total Area = Area1 + Area2 + Area3
Height = 5 cm.
Area = Length x Width
(Advanced students can use variables to represent words i.e. A=LW).
Length1 = 10 cm
Length2 = Length1
Width1 = 5 cm
Be sure to list ALL the known facts -- either values or relationships -- as equations.
Step 3 -- Write down all the unknown variables under the know facts.
Length3 = ?
W = ?
Important Note: Keep it neat and organized!
Step 4 -- Solve any equations that you can and substitute values into other equations to solve them.
Gradually put numbers where variables or words used to be until you have numbers for everything.
A good workman needs tools. And he needs the right tools for the job. In any profession, better tools are invented as time progresses -- tools that allow difficult tasks to be handled quickly and easily. For example, today’s modern carpenter begins work with a hammer, nails, and a handsaw. Yet he has available an electric skill saw, various specialty forms of handsaw, an electric table saw, a band saw, and even more advanced tools. Yet Jesus was a carpenter and had no electricity in His shop (other than in His words!). He was able to accomplish his carpentry tasks using very basic tools -- it just took longer and took more effort to cut a board than it would have if He had owned an electric table saw.
Is the table saw more difficult to use than the handsaw? Even though carpenters begin with the handsaw, the handsaw probably requires more skill and practice to use correctly. So the handsaw (the simple tool) is more difficult to use than the table saw (the advanced tool).
In the same way, mathematicians started with addition and subtraction. Multiplication was simply a tool that sped up addition. Algebra is a wonderful tool for solving many different types of problems and trigonometry saves considerable time and effort in calculating heights and elevations for surveyors. Calculus was fully developed to make solving the motion of the planets much easier, but is invaluable to the civil engineer and economist. And complex mathematics considerably simplifies the development of complex electronic circuits.
There is a tendency among Math students to say, "______ (division, algebra, geometry, trig, calculus, etc) is supposed to be really hard." (Invariably the level of math in question is a couple of years ahead of the student.) This frightens the student and adds to their Math anxiety.
Point out the Toolbox Concept. Point out that the various tools actually make solving problems easier. For example, using a handy pattern of tiles (linoleum or ceiling tiles), have the students determine the area of a large area. If there are several students, have half the students determine the area with nothing but counting. Let the other students use multiplication. See which ones finish first and are more accurate. (If there is one student, have him perform the counting task first, timing his speed. Next, have him multiply for the result, once again timing him.) Which method is easier and quicker? Now explain that this is what the higher Math tools do -- they make problem solving easier. Help the students understand that they are filling a Toolbox with Tools.
As you can see, the keys to success in Math are not Brilliance and Intuition -- rather, a step-by-step approach and mentality is required. The Brilliance and Intuition in Math is required for new Math discoveries -- Newton with Calculus, Godel with Incompleteness, and Einstein with Tensor Calculus. But the task of Math up until at least the level of Differential Equations is the task of the good workman.
Math success is driven by Persistence. Ideally, solving a complex algebra problem should have the same feel as painting a house -- gradually the outlines of the beautiful ending appear as the step-by-step transformation occurs. Mathematics has a Concept of Elegance, which is defined as a step-by-step solution to a problem which is clear, easy to follow, as short as possible, and comes quickly to the correct answer. As you paint a house, you cover the walls step by step. When you solve a Math problem, you go step by step.
In the same way that a brilliant artist is not the type of person who you'd want to paint a house (imagine the odd colors!), a brilliant student can have difficulties with complex Math problems -- not because he can't develop an approach to the problem, but because he won't stick to the mundane Processes required to solve the problem step-by-step. The brilliant student needs to be reminded to stay on course and continue step-by-step neatly and precisely until the problem is completed.
As a workman, the Math student needs to keep his toolbox organized. Know what tools are available and what types of problems they solve. This isn't a function of brilliance -- only of organization. Make a list if necessary -- formulas for area, volume, perimeter; methods of algebraic transformation; definitions of trig functions; a card catalog of algorithms; a listing of integration techniques, etc.
Copyright Ó 1999 - Brian L. Boley. All Rights Reserved.


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John Urschel speaks about graduate studies in mathematics, professional football, and career aspirations
Urschel: ‘In going from football to math, I think football really helped me have perspective.’
Professor John Urschel of the Mathematics Department was appointed as an assistant professor in the fall of 2023. Urschel completed his PhD at MIT in 2021 and was a former NFL player for the Baltimore Ravens for three years.
The Tech spoke with Urschel as he reflected on his time in the NFL and as a PhD student. This interview has been edited for length and clarity.
TT: Who or what inspired you to play football professionally/pursue math at a high level?
Urschel: My dad inspired me to play football. When I was a kid, I would visit my dad in Canada and he had this office and his office had a portrait of himself next to the door, and it was him in his football uniform. I always looked at that when I was a little kid. My dad played linebacker at the University of Alberta.
As far as math goes, as long as I can remember when I was a little kid, I loved puzzles and quantitative games. I can’t actually ever really pinpoint a moment where it's not like there was a moment where I really started to love math. I think that math is loosely defined. I've loved it ever since I can remember.
TT: How did you balance Division 1 football and academics at Penn State?
Urschel: When I first got to college, it was quite overwhelming. But eventually you get into a groove into a routine. You sort of learn. You learn when to do what. I really had a good habit in college of making sure I do the most important things first. I always made sure that I scheduled my classes as early in the day as possible to the extent that I could. I scheduled my personal football training as early as possible when there were choices about which time slot to do lifting or do conditioning.
TT: What was your journey to entering graduate school at MIT?
Urschel: Before I got to the league, I finished my undergrad in three years. I got a Master’s because I really wanted to keep playing football. At the same time, I didn't want to do my PhD at Penn State because I did my undergrad there.
I'm listed on, you know, [NFL] draft rankings, and I decided to put the PhD on hold. I'm gonna play the league for a couple years, retire and go for my PhD. I play my first year in the league. Things went quite well and I started a bunch of games, but I felt unsatisfied like in my personal life I realized that I really missed the academic environment and being around other people who want to learn.
After my first year [in the NFL] I decided I needed to apply to a PhD program and I got into MIT. And I thought it was a really good fit for me because the sort of math I do is a little bit more on the applied side. I felt like I belonged here, so I accepted the offer and the rest is history.
TT: Was balancing graduate studies and professional football different from doing both in your undergrad? How did you strike the balance during your PhD?
Urschel: Very poorly. When I was a math major at Penn State, PhD classes there felt manageable. MIT’s PhD program doesn't include all part-time students. I can tell you being a professional football player simultaneously was just too much. I was constantly stressed at work. There are only so many hours in the day. I can say in hindsight it makes for a funny story, but it was not the most pleasant thing to be doing both of these things at the same time.
TT: How do you spend your typical day?
Urschel: The role of a math professor is a pretty nice one. There's no lab, so there's no physical thing I'm responsible for. It's just math problems that I'm trying to solve and people I am trying to solve them with. I'm also trying to teach how to solve math problems.
My typical day starts with me and my son, and we get to MIT via the T and by foot. Every morning my day starts with a long two mile walk before I settle down here. Depending on the morning, if it isn't teaching, I hold office hours and meet with collaborators, but my day mostly consists of, or at least my favorite days, consist of thinking about math.
TT: What did you find to be the most unexpected intersection between your math research and your career in the NFL?
Urschel: I definitely think as a kid, being quantitatively minded definitely helped me in a lot of the more analytic aspects of football. In particular, understanding how these responsibilities of different people fit together in a larger sort of theory or scheme while playing football was helpful. Being able to take information about what I'm seeing helped me more than maybe other people.
In going from football to math, I think football really helped me have perspective. I think a lot of times, math can feel very intense if you let it because you're trying to solve some math problem, and it's a little different than doing an experiment or testing some hypothesis. You can spend a very long time thinking about something and not really make any progress either way, so in that way, it can be a little frustrating.
I always kind of think and pretend to myself that I've already retired. Right now this is just this is my post retirement career. And so, I find that's a good way for me to look at it. My day as an MIT professor is much easier than my day as a professional football player.
TT: How did you respond to challenges in football versus in math (and how these overlap)?
Urschel: You're told these things that really get beat into you that you're constantly chasing this perfection that you're never actually going to achieve consistently in terms of performance. You're constantly trying to do better no matter how well you've done and when things are going wrong or when things are performing poorly. The key thing is to always lean into your training, lean into your preparation, and stay the course. The idea that often shows up in football is the idea of purpose. Perfect practice makes perfect.
The way you are setting yourself up for success is something that has carried over into math, especially when I feel really stuck and feel like I'm not making progress. It's often helpful for me to take a step back and think about whether I am actually setting myself up in the best way to be successful at this? Do I need to go read some different resources to try to help me, you know, be better prepared to solve this problem? Should I talk to some other people to get some perspective?
TT: Which area do you find to be more competitive: football or academia?
Urschel: I will say football is very much more directly competitive, whereas in math it is much more indirect. As a math community, we really do usually feel like we're all part of the same community, with some exceptions. This is how it should be. In general, one shouldn't think of math as like some zero sum game. We're working together to try to solve problems of course, but how much you share with other people varies because you also need the job. There can be a little bit of competition because multiple people will be applying for the same position. There aren’t many academic jobs compared to how many talented and qualified mutations there are, so it can be quite tough. But the good news for mathematicians, both undergraduates and PhDs, is that there are rewarding jobs in industry whether it be tech, finance or other fields.
TT: How do you visualize or conceptualize the math problems you work on? And what do you love most about the field of math?
Oftentimes, I do think very conceptually about things. I'll be walking home and decompressing from the day and I’m slowly letting the things I've been thinking about all day sort of seep in, and really let them sit. This happens a lot before I go to bed. I find that I've been thinking a lot during the day about different things I've been picking very actively. Oftentimes, I have an insight that I text to myself and I put my phone down and in the morning like often I find that I actually had some good insight.
Taking a step back and thinking about that work with separation is often a very productive way of sort of understanding how things work at a high level. You can do rough calculations in your head or thinking through steps in your head. But I find that sometimes you just get the answer or you sort of figure out the right technique by thinking about it directly. I find that to be really productive.
The thing I love most about math is learning. There are so many rich and beautiful results in mathematics that connect to each other. There's so much that I don't know. Every time I learn something new or slightly surprising in math, it's a really great feeling. The act of learning is just a really enjoyable thing. That's one of the great things about being a professor is that you get to just keep doing that the rest of your life.
TT: What are your thoughts regarding the general public perception of math and math education?
Urschel: That's a great question. I think there's often a perception in this country that being bad at math or not liking math is something that is joked about often, even by smart people. And I think that numeracy in the same way that literacy is important is a crucial skill in our society. I think these are skills that are useful for many jobs, but more generally, it's just useful for everyone in your day to day life, some fundamental level of quantitative reasoning.
I think that this is something that we really need to value as a society and it doesn't always get sort of the level of value that I think it does. And I also think that top careers in STEM also should be glamorized a little bit more than they are.
I recognize that I'm a former pro football player. I've had a much larger platform than most mathematicians because of what I did outside of math. I don't think that's fair because I know so many amazing mathematicians with a lot to contribute to the world and a lot to share with other people, but that's the way our society is right now.
TT: What do you believe is your biggest accomplishment in math? And, are there any future problems you’d really like to solve?
Urschel: I’m too young—I’m 32, I just got here. Let’s save the greatest accomplishment five years from now, but I would say that my best results certainly lie in matrix analysis and matrix computations. Whether it’s fundamental results about the nature of Gaussian elimination, with techniques that you learned in school to solve a system of equations or results about graphs and networks through looking at algebraic properties. I think that’s probably the area of biggest contributions.
There are a couple of concrete problems for which I say “this is a problem I would like to see solved in the next however many years” and it's important to set those goals so that you are moving towards something concrete. But the joy of it, at least for me, is not the moment I solve that problem. The joy of it is the progress of getting to solving that problem, the joy of being at odds with a problem like struggling against the problem.
When you finally solve this problem, this is like the cherry on top. I feel like I understand everything or at least I understand enough to really say something powerful. I find this process of getting to solving the problem to be the most rewarding.
TT: What would you say your greatest accomplishment in football is, or your greatest accomplishment, in general?
Urschel: I’m quite proud of my time in Penn State, my alma mater—especially in my later years. This was a really tough time for a lot of people in that community. And in hindsight, when I think about my football career, that’s the time period I really look back on very fondly, and that I’m quite proud of.
TT: How do you define progress in math?
Urschel: Who knows? Oftentimes it can be helpful when you have some problem and you can try to break it up into smaller pieces that you think might be a useful ingredient for the bigger problem. Or you can simplify your problem in a number of ways. You can try to prove some lemma or some complimentary thing that maybe helps you understand the nature of the problem. You can also try approaching the problem with some technique and understand the limits of that technique.
Progress can be hard to track. For instance, I'm working on a research project with a former undergraduate, and we do not know how to solve the problem. We just don't and that's normal. But I had an idea for an insight that might give us a way in that other people haven't thought about. And so you try this insight. You try to prove some things in the direction that you're trying to go at the end. Maybe all the things we proved aren't useful because this technique doesn't work. It's hard to know that until you get to the end and so the more experience you have doing math often the better you are at roughly guessing what your progress looks like. It's a feeling you have.
TT: What are your future career goals and aspirations?
Urschel: At this point in my career, I'm really happy to be back at MIT. I'm here as an assistant professor, so I'll be here for you know, probably the next six years or so. My big career aspirations are to keep doing math, solve really interesting problems and I am eventually looking forward to being in a tenured position at some point, settling down, buying a permanent house. That's a milestone.