Mathematics is the study of order, relation and pattern. From its origins in counting and measuring it has evolved in highly sophisticated and elegant ways to become the language now used to describe many aspects of the world in the twenty-first century.
Mathematics is a core subject for Years 7 to 12 with some electives in Years 11 and 12.
Tutorials (and Mathematics teachers) are available throughout the year for students from all year levels.
Tuesday/Wednesday from 2:45–4:00pm in room MTE1.
Senior Subject Book (PDF, 4.1MB) provides information on subjects offered in Year 11 and 12 for 2020, and describes the required prerequisites.
Year 9 and 10 Subject Book (PDF, 3.5MB) provides information on subjects offered in Year 9 and 10.
|Years 7 and 8|
|Years 9 and 10|
|Years 11 and 12|
- General Mathematics
- Mathematical Methods
- Specialist Mathematics
|Mathematics||Year 07 A (PDF, 608.19 KB)|
Year 07 B (PDF, 627.04 KB)
Year 08 A (PDF, 537.88 KB)
Year 08 B (PDF, 946.92 KB)
Year 09 (PDF, 820.92 KB)
Year 10 (PDF, 900.66 KB)
|Year 07 (PDF, 734.66 KB)|
Year 08 (PDF, 363.56 KB)
Year 09 (PDF, 831.83 KB)
Year 10 (PDF, 831.83 KB)
|General Mathematics||Year 11 Unit 1 (PDF, 652.82 KB)|
Year 12 Unit 3 (PDF, 686.03 KB)
|Mathematical Methods||Year 11 Unit 1 (PDF, 742.69 KB)||Year 11 Unit 1 (PDF, 742.69 KB)|||||
|Specialist Mathematics||Year 11 Unit 1 (PDF, 889.17 KB)||Year 11 Unit 1 (PDF, 889.17 KB)|||||
Activities and competitions
Students are offered the opportunity to participate in the following activities / competitions:
- JCU MEOW (Maths Enhances Our World) – Year 8 students invited to apply.
- QUT STEM Camp – high achieving Year 11 Maths/Science students invited to apply.
- IAES – Indigenous Australian Engineering School – Indigenous students in Years 10, 11 and 12 interested in Engineering as a career are invited to apply.
- Australian Mathematics Competition – offered to students achieving at an A level in Semester 1.
- Year 7 and 8 Mathematics Quiz – held during Term 3.
More information can be found in school newsletters.
Why study mathematics?
Learning mathematics creates opportunities for and enriches the lives of all Australians. The Australian Curriculum: Mathematics provides students with essential mathematical skills and knowledge in number and algebra, measurement and geometry, and statistics and probability. It develops the numeracy capabilities that all students need in their personal, work and civic life, and provides the fundamentals on which mathematical specialties and professional applications of mathematics are built.
Mathematics has its own value and beauty and the Australian Curriculum: Mathematics aims to instil in students an appreciation of the elegance and power of mathematical reasoning. Mathematical ideas have evolved across all cultures over thousands of years, and are constantly developing. Digital technologies are facilitating this expansion of ideas and providing access to new tools for continuing mathematical exploration and invention. The curriculum focuses on developing increasingly sophisticated and refined mathematical understanding, fluency, reasoning, and problem-solving skills. These proficiencies enable students to respond to familiar and unfamiliar situations by employing mathematical strategies to make informed decisions and solve problems efficiently.
The Australian Curriculum: Mathematics ensures that the links between the various components of mathematics, as well as the relationship between mathematics and other disciplines, are made clear. Mathematics is composed of multiple but interrelated and interdependent concepts and systems which students apply beyond the mathematics classroom. In science, for example, understanding sources of error and their impact on the confidence of conclusions is vital, as is the use of mathematical models in other disciplines. In geography, interpretation of data underpins the study of human populations and their physical environments; in history, students need to be able to imagine timelines and time frames to reconcile related events; and in English, deriving quantitative and spatial information is an important aspect of making meaning of texts.
The curriculum anticipates that schools will ensure all students benefit from access to the power of mathematical reasoning and learn to apply their mathematical understanding creatively and efficiently. The Mathematics curriculum provides students with carefully paced, in-depth study of critical skills and concepts. It encourages teachers to help students become self-motivated, confident learners through inquiry and active participation in challenging and engaging experiences.
Contact details for the Heads of Departments are listed under the contact us page.


!! THIS SITE HAS MOVED !!
Please visit the new Palo Verde PTA Website: http://paloverde.paloaltopta.org
New standards, called Common Core State Standards, are on the way for all students in California, beginning in the 2014-2015 school year. These standards set high expectations for all students. The aim of this state-led initiative is to develop standards that ensure all students are held to consistent expectations that will prepare them for college and career.
Find out what your child will be learning, at each grade level, in Mathematics and English Language Arts once the Common Core State Standards are in place. You’ll also learn more about how you can support your child’s learning and other helpful information.


James R. Gray, 83, of the Rooster Valley in rural Spring Grove, died Monday, September 11, 2023, at Gundersen Medical Center in La Crosse, Wis.
James was born September 30, 1939, in Panama City, Fla., to Randall and Nathalie (Waller) Gray. He graduated from Bay High School, Panama City. He received his Bachelor of Science and Master of Science degrees in mathematics from Florida State University. On June 9, 1962, he married Julia Hobbs in Port Royal, Va. He was employed by the Institute for Instrumental Mathematics of the University of Bonn, Bonn, West Germany, in 1963 and 1964. From November 1964 to September 1965 he worked for IBM in Poughkeepsie, N. Y. From September 1965 through June 1969 he was a PhD student in mathematics at the University of Wisconsin, Madison, Wis. He was a professor of mathematics at California State University – Los Angeles. In 1971 James and Julia purchased a farm in Rooster Valley, Black Hammer Township, Houston County, Minn. From 1977 to 1984 they had a small dairy herd. In 1984, the family moved back to Madison and he continued to work at UW – Madison until retiring back to the farm in 1999. He enjoyed mathematics, classical music, especially Richard Wagner, art, and foreign movies.
James is survived by his wife Julia and two sons, Andreas of Spring Grove, and Alexander (Lisa) of Champaign, Ill.; two grandchildren, Sierra and Maia Gray of Cleveland, Ohio; and a sister, Julia Elizabeth Gray of Evans, Ga.
He was preceded in death by his parents and a brother, Clifford Timothy Gray of Tallahassee, Fla.
A Celebration of Life for James will be at 12 noon on September 30 at their farm, 13878 Rooster Valley Rd, Spring Grove, Minn. The Mengis Funeral Home in Mabel is assisting with arrangements.


Large-scale calculations of nuclear-structure data for simulation data basesP. Möller1, R. Bengtsson2, K.L. Kratz3 and H. Sagawa4
1 Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA
2 Department of Mathematical Physics, Lund Institute of Technology, 22100 Lund, Sweden
3 Max-Planck Institut für Chemie, Otto-Hahn Institut, 55128 Mainz, Germany
4 Center for Mathematical Sciences, University of Aizu, Aizu-Wakamatsu, Fukushima 965-8580, Japan
Published online: 21 May 2008
In the macroscopic-microscopic model we have (1) enhanced our FRDM (1992) model of nuclear ground-state masses and shapes, (2) performed a global calculation of nuclear shape isomers, in which we characterize the ground-state and isomer minima in terms of their relative energies and shapes, and furthermore provide the saddle heights between all pairs of minima, and (3) performed a calculation of fission potential-energy surfaces for more than 5 million different shapes for each of 5254 nuclei from A = 170 to A = 330. We use an immersion technique borrowed from geography to determine saddle points and minima in these surfaces. We also use this technique to establish if structures such as deep valleys separated by high ridges are present. These would then give rise to different modes of fission.
© CEA 2008


24-square cabled entrelac cube with connectionlines, originally uploaded by fuzzyjay.
This is one of the simpler ways of showing the connections between squares in the 24-square cabled entrelac cube. In this type of cube, each square is attached to 6 others: the 4 that are adjacent on each side, and the 2 that are on the lower right and upper left corners of the square as knitted. Each square has a cable that runs from lower right to upper left, and joins cables on two other squares. This is shown with thicker lines


The Member of Parliament (MP) for Assin South, Rev John Ntim Fordjour, has charged the government to ensure that Science, Technology, Engineering and Mathematics (STEM) is incorporated well into the education policy of Ghana.
According to him, it is the fulcrum for growth and development, and what has brought the difference between developing and developed countries.
Mr Ntim Fordjour made the call in a statement issued on Sunday, March 14, 2021, to commemorate International Mathematics Day.
“The world today has become a technology-driven society. The status quo of complexities of global health challenges, environmental sustainability concerns, and the fact that commerce today is rapidly being driven by technology, would demand pragmatic integration of Science Technology Engineering and Mathematics (STEM) into the fabric of education policy. Undoubtedly, principal to addressing the gaps in technical and professional roles in key engineering and technology related sectors is STEM,” he said in the statement.
According to him, STEM is instrumental today and will certainly become the niche and direction in education for future generations.
He stressed that considering the fast spate with which the world is tilting towards technology, societies get better when they embrace STEM and prioritize the learning of Mathematics today.
He noted that many misconceptions had been associated with the study of mathematics and is therefore a reserve of a few exceptionally brilliant people.
“Another myth relating to Maths is the school of thought that Mathematics is not meant for women. I have seen many females and a good number of males avoid Maths like a plague in the course of their study. Against all these myths, Maths is equally for females as it is for males. Over the years, I have come across smart females who are astute in Mathematics and demystify the study of Mathematics,” he said.
Mr Ntim Fordjour, a professionally trained and practicing engineer, said his experience over the years indicates that “Mathematics had been sine qua non and woven into the fabric of learning.”
He further called on Ghanaians to celebrate men and women of honour and repute “who have excelled in various facets of Mathematics and to recognize those in authority who continue to promote Mathematics, making it attractive for learning.”


0.3.5 - Refined
Other Available Downloads
- Instance operations are now protected internal and no-longer public. All operations are now performed with mathematical operators. This may cause problems for people using languages other than C# or VB.NET. Let me know if this broke anything for you.
- Trigonometric functions improved: in all earlier releases digits beyond the 10th decimal place were unreliable and useless. In 3.5 these digits are now truncated.
- Large number accuracy: 0.3 introduced a bug where significant digits would be ignored if the number was larger than an arbitrary threshhold. This behaviour has been fixed.
- Major functions now moved to the BigMath class to match the design of the native math library in .NET
- Furthermore, the mathematical definitions of more useful functions such as Natural Log and the Exponential function have been defined in BigMath.cs but not implemented because I don't have the time or inclination. Anyone is free to implement them (shouldn't take more than a few hours). Just uncomment the #define at the top of BigMath.cs
- FxCop compliance increased again
Thank you dude for your library! I've made cool app based on it:
A calculator similar to the one from the Dos Navigator - old and lovely file manager.
Three number representations, no silly buttons and other unnecessary stuff.
I discovered some bugs in your release, could point them if you want.
on Dec 8, 2009 at 8:19 AM
Opera does not support ClickOnce X
To install this application, save it and then open it. Opening it directly from Opera will not work correctly.


Advantages of Playing On Video Roulette Machines
Are you searching for a Roulette Machine Strategy? If that’s the case, then you are in the right place. In this article, I will share with you a Roulette Machine Strategy that I use to consistently earn money from my machines. In addition, you will learn a simple and easy way to “break” the big payoff pots through the use of bonus signs.
The initial strategy is actually quite simple. In any casino that offers Roulette, there exists a Roulette Wheel. In fact, the majority of casinos have a spinning wheel that spins around. On every spin, lots is randomly chosen. You can bet on that number and if it is correct, then you win how much the bet – IF your original bet was add up to or more than the designated amount on the roulette wheel.
But this is not the only Roulette Machine Strategy that works! You’ll be able to beat the video roulette with some simple mathematics skills. For example, if the number on the video roulette is 5, then it is very likely that someone (maybe even you) has bet on that number before and won. What you must do is figure out how many people have bet on this number before to be able to bet on a single number, multiply this by five and you have a new number that you can try and win.
The next Roulette Machine Strategy that I will show you relates to the specific chip denomination that’s being used in video roulette. Whenever a bet is positioned on a roulette game, it will be paying out in another of three different chip denominations. It may be either the exact same chip denomination, a different denomination or a combination of both. What you want to do is figure out which chip denomination you can be playing on and then utilize this information to try and figure out what odds are accessible to you.
You can find two types of roulette that folks can play; table roulette and video roulette. When you are playing a table game, you then are basically doing a game of chance when it comes to figuring the odds. The exact same table, the same spin, exactly the same numbers will always bring about the same outcome whether you’re playing a video or table game.
In table roulette players are betting the number of the pot, which is equivalent to the quantity of the bet. The bets bypass until someone 온라인 카지노 gets lucky and hits a “match” which results in spending the exact level of the bet. These matches are called strikes and there are usually up to 10 strikes for each bet made. The odds on video roulette machines aren’t nearly as good as a real wheel. The reason for the reason being there are fewer chips on these machines and the random number generators do not take the time to create a match for every possible outcome.
One of the major differences between the video roulette machine and the original roulette table is the house edge. The home edge is the sum of money that a player would have to keep paying to the overall game if they lost the entire amount of the bet. In roulette, a player may end up losing even more money than they placed into the game. The reason behind this is that there are many more spins each hour on video machines than there are on a normal roulette table. If a player is playing the roulette game for pure entertainment and contains no other intentions of winning, they are not going to have that large of an investment to lose.
If you’re going to be playing at video roulette machines, you need to know about the several benefits of playing on these machines. Firstly, video roulette offers you the opportunity to play for smaller stakes and far lower minimum bets. There’s usually no minimum for online roulette games aswell, so you can win as much or more than you would if you were playing in a real live casino. If you are looking for more excitement, then roulette video machines might be the best way to go. Just ensure that you know all the rules for these games in order that you don’t end up receiving disqualified due to betting an excessive amount of when you aren’t actually thinking about the outcome of the overall game.


Negative– When the correlation produces a downward slope that means the relationship between the variables is inversely proportional. If Variable 1 increases, variable 2 will decrease – and vice versa. This means if you have more than two columns of variable, the matrix will contain the correlation coefficient for all combinations of variables. Supply the Input Range for the correlation calculation. This should be a range with numerical values organized into columns or rows.
This compensation may impact how and where listings appear. Investopedia does not include all offers available in the marketplace. Correlation combines several important and related statistical concepts, namely, variance and standard deviation. Variance is the dispersion of a variable around the mean, and standard deviation is the square root of variance. Mastering Excel is also highly beneficial for students who want an efficient and high data storage workbook.
Using CORREL Formula
According to the Bureau of Labor Statistics, the job outlook for operations research analysts is projected to grow by 25 percent between 2020 to 2030. Statistical analysis and correlation coefficient calculations are widely used in these occupations. Excel offers a simple metrics breakdown that offers logical values for positive, negative, and no correlation. Let’s see an example to know how to calculate the correlation coefficient in excel.
In our example, we’ll display the output in another workbook, so we will select New Worksheet Ply, and then click OK. The correlation coefficient measures the association between two variables. Correlations are shown as values between -1.0 and 1.0, from no correlation to positive correlation.
Select the Labels in first row check box if the selected range contains column headers. Plot a correlation graph to get the visual representation of the data relationship. In statistics, it is the most popular correlation type, and if you are dealing with a “correlation coefficient” without further qualification, it’s most likely to be the Pearson. The numerical measure of the degree of association between two continuous variables is called the correlation coefficient .
In the below-mentioned example, the table contains two variables, one in column X & the other in column Y. Where both the datasets contain positive values. With the help of the Correl function, I need to find out the correlation coefficient between two datasets or variables. How are the sales of coffee related to football games?
Let’s use the CORREL function to get the correlation coefficient. In our example, that value is a little over .5, which indicates that there is a fairly strong positive correlation. Now go to the Add-Ins option and in the Manage select Excel Add-ins from the drop down. It is entered as a second cell reference or range of values.
Find the Pearson correlation coefficient with the CORREL function. In statistics, they measure several types of correlation depending on type of the data you are working with. In this tutorial, we will focus on the most common one. A coefficient of 0 means no relationship between two variables – the data points are scattered all over the graph.
I hope this was explanatory enough to explain the correlation coefficient. You can now create your own correlation coefficient calculator in excel. A positive correlation how to find correlation coefficient in excel coefficient tells us that if one variable’s value increases, other variable’s value also increases. This means that if x increases by 1 unit, y also increases by 1 unit.
On the main screen, click on the File tab, if you are using the latest versions of Excel. If otherwise, click on the menu button found on the top left side of the screen. Then, click another empty cell within the same screen and name this cell as “Correlation results.”
This can be quickly done by using the Excel RANK.AVG function. Depending on whether there are or there are no ties in the ranking , the Spearman correlation coefficient can be calculated with one of the following formulas. Open the menu for selecting the input values and output parameters . In the fields for the specify range of the input data, which describes the options and influence the factor . In Microsoft Excel Calculating correlation is one of the simplest tasks to do. A correlation graph shows the relation between two or more variables.
Excel calculate Covariance, Coefficient of Correlation
Where a – are the regression coefficients, x – the influencing variables, k – the number of factors. Consider the example to the construction of a regression model in Excel and the interpretation of the results. Merged cells are one of the most popular options used by beginner spreadsheet… This will open up the Data Analysis menu and you can then select Correlation from the options and press the OK button. Excel comes with a powerful statistical tools add-in, but you need to enable it to use it first and it’s quite hidden. These are required inputs and must be a single column or single row array of numbers.
Put the cursor in any cell and click the fx button. To find the paired coefficients applied CORREL function. And based on the main factors you may to predict, to plan the development of priorities areas, to make to the management decisions. Lover on the specific practical examples, we consider these two are very popular analysis among economists. And give an example of the receiving the results when they are combined.
- To use the Analysis Toolpak add-in in Excel to quickly generate correlation coefficients between multiple variables, execute the following steps.
- For example, when an independent variable increases, the dependent variable decreases, and vice versa.
- The correlation analysis helps to the analyst to determine whether it is possible for the value of one indicator to predict the potential value of the other one.
- Since we are trying to calculate a numeric relationship, we need to have numeric data points.
- The null hypothesis is that sociability is not correlated with job satisfaction.
It is important to note that both functions REQUIRE numeric values! Since we are trying to calculate a numeric relationship, we need to have numeric data points. This is why we converted wins and losses to 1s and 0s. An industrial psychologist rated the sociability of 20 employees of the FarDrate Timepiece Corporation. Each FarDrate employee also rated his or her job satisfaction on a scale of 1 to 80 . The null hypothesis is that sociability is not correlated with job satisfaction.
In this case, you’d be wise to use the Spearman rank correlation instead. For our sample data set, the correlation graphs look like shown in the image below. Additionally, we displayed https://1investing.in/ R-squared value, also called the Coefficient of Determination. This value indicates how well the trendline corresponds to the data – the closer R2 to 1, the better the fit.
How to Calculate the Correlation Coefficient in Excel
Excel is a powerful tool that has some amazing functions and functionalities when working with statistics. For this example, we’ll be using a similar data set with the one above with the addition of ‘Z Variables’. You can’t use the ‘CORREL’ function to determine the relationship of more than 2 variables. Also, it is useful in data science to find the relationship between 2 variables. Optimal PortfolioPortfolio optimization refers to the strategic process of making improvements or positive changes to the investor’s portfolio containing various assets. The Sharpe ratio helps to analyze the returns from an optimal portfolio.
But what if you want to create a correlation matrix across a range of data sets? To do this, you need to use Excel’s Data Analysis plugin. The plugin can be found in the Data tab, under Analyze. The importance of learning how to use Excel sheets is beneficial across a wide range of professional careers. Whether you want to become a project manager, a mechanical engineer, or a business analyst, working with Excel sheets is the norm. Excel sheets offer solutions for easy data analysis, correlation calculations, and employee data collection.
Then you can choose to output on the same sheet or on a new sheet. Learning to find the correlation coefficient in Excel is useful for enhancing your business productivity. There are several ways technology helps business professionals. For example, Excel’s correlation coefficient formula allows you to seamlessly measure the correlation between work productivity and behaviors for higher profits. In this tutorial, I will show you two really easy ways to calculate correlation coefficient in Excel.
The alternative hypothesis is that these two variables are correlated. Spearman Rank Correlation evaluates the monotonic relationship between the ranked values. In a monotonic relationship, the variables also tend to change together, but not necessarily at a constant rate. Consider how with helping Excel tools to find the correlation coefficient. The classification of correlations for different areas will be different. If the value of the coefficient is 0 linear dependence between samples does not exist.
As the ‘X Variables’ increase, the ‘Y Variables’ increases also. The correlation coefficient (a value between -1 and +1) tells you how strongly two variables are related to each other. We can use the CORREL function or the Analysis Toolpak add-in in Excel to find the correlation coefficient between two variables. Correlation Vs CovarianceCovariance and Correlation are two terms which are exactly opposite to each other; both are used for statistics and regression analysis. Covariance reflects how two variables vary from each other, whereas correlation depicts the relationship between two variables.
The Pearson correlation is not able to distinguish dependent and independent variables. However, you could switch around the variables and get the same result. So, someone may conclude that higher heater sales cause temperature to fall, which obviously makes no sense.
To compute a correlation coefficient by hand, you’d have to use this lengthy formula. To find correlation coefficient in Excel, leverage the CORREL or PEARSON function and get the result in a fraction of a second. A coefficient of 1 means a perfect positive relationship – as one variable increases, the other increases proportionally. A coefficient of 0 indicates no linear relationship between the variables. This is what you are likely to get with two sets of random numbers.


Roulette is really a well-known casino game started in Italy and in addition probably named after the French term meaning wheel; the wheel can be an old-fashioned term used in Italian, and could also be referred to as the ‘lion’ or ‘wheel of fortune’. The wheel has been used since ancient times in many countries. One of these brilliant countries is France. Roulette can be called, in English, ‘the game of luck’, and is popular in the United States. Roulette also has a second name, which is, needless to say, ‘the game of chance’.
You can find two major factors to consider when playing roulette: the home edge and the number of bets. The house edge is the expected amount of money a player would have to win a single round of playing for the house to pay off the complete pot; the number of bets is referred to as the number of spins. The ball player can adjust the quantity of bets they make, though according to the house edge, it might be difficult. That will help you with calculating the best bets, we have included some tips and techniques. In the same way in other types of gambling, roulette relies on strategy and probability.
In roulette, there are many bets which might be made. The bets can be made on either the winning hand or losing one, though losing one will always incur a penalty in roulette. However, the strategies used are what make the difference in winning and losing. One great way of making the most out of every bet is to understand the mathematics of probability.
You can find 3 basic ways of winning in roulette; all of them are based on probability. First, the player needs to know how many cards come up as the outcome of the initial few spins. There are many of ways to do that. A few of these are by guessing, where the player simply counts the number of cards that come up being an outcome. Different ways are to count the total amount of cards in the deck.
There are also casino games where in fact the player is given the group of cards and will then spin them in order to get a number from them. That is called the fan method. And in casino games that rely on outside bets, there are lots of methods used. The bets are placed on lots or numbers that have recently been picked out.
This is a simple game for those who want to play with numbers. The benefit of playing 에볼루션 카지노 roulette with numbers is that it offers the player an idea of how much they can win, in comparison to betting on other numbers. Most bets in roulette involve inside bets, but since there is no ceiling on the sum of money that can be wagered on numbers, the options of getting a high payout is high. The disadvantage of playing roulette with numbers is that the bigger bet wins aren’t that large when compared to small bets, which may not be that profitable. This means that the players may become influenced by getting inside bets, which increases the risk factor. People playing roulette with numbers usually prefer to play larger bets so they won’t be vunerable to losing more money when times get rough.
People have used the Fibonacci system in other areas of life to predict future winnings or losses. For instance, the gambler Pat Coscarelli used the Fibonacci system to carefully analyze the odds of the blackjack table he was betting on. He discovered that the best time to place his bets is once the odds are lower, since the odds of losing is lowest at this time. This is much like what the roulette system is wanting to do, which is to increase profits while minimizing the total amount that is lost. However, there are limitations to this technique.
In roulette along with other casino games, the casinos don’t offer any sort of wagering methods, since the odds can virtually be figured out by considering the past results. However, you should use a method known as inside betting to dramatically increase your odds of winning. The inside bet takes place once you bet on the initial number drawn rather than betting on the first number on the flop or after the flop, and contains been used for decades to improve the profitability of many games. Since roulette and most casino games are based on chance, inside bets have already been around for a long period and are a technique used by professional gamblers worldwide.


Yesterday Bethany participated in the Whatcom County Math Championships with 3 of her classmates. Her teacher gave them a few extra math lessons after school on Mondays for a few weeks and took the group down to the community college to compete in individual, mental math, and team tests.
That’s right. Three girls. One boy. Kind of goes against the typical stereotype there. Ed went along as a parent chaperone and he says that one boy is super-smart. They were doing math problems on the way and Ed says he got every one right. But still. Three girls. One boy. Niiiiiiice!
Many, many, moons ago I also went to the Math Olympiad, as it was called back then. I, unlike Bethany, was not good at Math. But, my brother was going and I think my math teacher thought maybe if she pretended I was good in math and treated me like I was good in math, I might just be good in math. And it worked. For the two years I had that teacher, I got the best grades in math of my life, until I got to college. (Although my getting a straight A in “Teaching Elementary Math” might not count to some people). Anyway, all that to say, teachers have a big impact in their students’ lives. And so far, Mrs. T. is having a great impact on my Bethany.
Since Ed was on the field trip, the pictures are minimal and this is all I’ve got of Mrs. T, but at least she’s being very teacher-ly:
Anyway, Bethany truly is good at math. In the individual tests, she scored the highest out of her group of 4, which earned her a ribbon!
Congratulations, Bethany! We are so proud of you!


A top Chicago area company is looking to hire a Statistical Analyst to join it’s Insights Team.
The Statistical Analyst must have an advanced knowledge of regression models, customer analytic techniques, and SAS. Ideal candidates will have extensive hands on modeling experience using SAS and an ability to interpret results into business decisions. This is a great opportunity for someone who has a strong modeling background and and would love to be a internal consultant to a large organization.
Master's degree in Statistics, Applied Economics, Mathematics, or related quantitative discipline Ability to translate technical material to non-technical audiences 3+ years of experience working in analytics and statistical modeling Hands on experience with SAS required


Arithmetic Concepts in GMAT Quant
GMAT ratio and proportion problems
One type of question on the GMAT quantitative section that can appear tricky is a question involving ratios and proportions. However, with the right strategy you can make these easy for yourself.
Consider the following question from GMAT Tutor:
A bottle contains a certain solution. In the bottled solution, the ratio of water to soap is 3:2, and the ratio of soap to salt is three times this ratio. The solution is poured into an open container, and after some time, the ratio of water to soap in the open container is halved by water evaporation. At that time, what is the ratio of water to salt in the solution?
How to solve the problem:
The first step is to translate the numbers in the question and map it out as best you can.
The obvious place to start is with a ratio of water to soap of 3:2. Draw it out:
The ratio of soap to salt is three times this ratio. To calculate this, we take the ratio, convert it to a fraction, and then perform the arithmetic:
3:2 = 3/2
3/2 * 3 = 9/2
which converts back to 9:2.
Note: three times the ratio 3:2 is NOT simply multiplying 3 to both parts of the ratio.
Add this to your ratio table:
Now, there is another condition we must consider:
The solution is poured into an open container, and after some time, the ratio of water to soap in the open container is halved by water evaporation.
Ah, so the ratio of water to soap must be halved. Again, to do this, let’s convert to fractions and perform the arithmetic:
3:2 = 3/2
3/2 * 1/2 = 3/4
3/4 = 3:4
So, now we have:
...and our soap to salt remains:
At this point, we notice two different ratio relationships for the three entities. Wouldn’t it be great if we can get them all on one line as a single, combined ratio? Let’s do just that.
Notice that the soap is common to both ratios, so if the soap had the same ratio value for both, we can write the ratio on one line.
So, to do this, make the soap ‘common’. One number is 4 and the other is 9, so we can convert them up to a common value of 36 (or any multiple of 4 and 9). We carry out the arithmetic as follows:
|x 9 =||x 9 =|
|x 4 =||x 4 =|
With soap now carrying a common value of 36, we can write the three-part ratio as follows:
And now, for the final question we were being asked:
At that time, what is the ratio of water to salt in the solution?
We can see that the answer to this would be 27 : 8.
Let’s summarise the method:
- Organise the initial ratios.
- Carry out any arithmetic operations related to expansion or reduction.
- Produce a combined ratio of the three components.
- Answer the question based on the final ratios in the table.
A step-by-step methodical approach to such questions will lead you to right answer more quickly.
This was a sample of the in-depth instruction that GMAT Tutor offers about solving ratio and proportion questions in the GMAT Quant section. For complete and interactive lessons and online tutor support, subscribe to one of GMAT Tutor's top-rated GMAT prep plans. Commitment-free trials are available for seven days.


Surface Area (BSA) = ( [Height (cm) x Weight (kg) ] / 3600 )1/2
Index (CI) = Cardiac Output (CO) / Body Surface
Arterial Pressure (MAP) = (2 x DBP) + SBP / 3
Vasular Resistance (PVR) = PAM - PCWP / CO X 80
Index (SI) = Stroke Volume (SV) / Body Surface Area (BSA)
Volume (SV) = is the amount of blood ejected by the left ventricle into
the vasculature in one heart beat.
Vascular Resistance (SVR) = (MAP - CVP) / CO X 80


There was great excitement last week when the three senior infant classes got a visit from two newborn lambs. They were very small so we tried to be very quiet and gentle when we were petting them. Thank you to Ms. Egan and Ms. De Paor for organising the visit!
We have been doing lots of active maths in the last few weeks and it is great to see the children playing maths games and practising their adding at the same time. Thank you to all the helpers who come in on Mondays!
We are also learning lots about the dentist in our Aistear groups this month. We have been checking our teeth, practising our brushing, making appointments and getting checkups.


Having recently bought a few Treefolk kickers, I thought that it would be a good idea to learn how they work before putting them into battle against my arch-rival and co-contributor to this blog, Nathan Miller.
As a lover of Spreadsheets, I set about producing one to carry out the calculations for the basic actions of each unit. I calculated the expected value for each action for each type of unit, and the corresponding Standard Deviation for each, both under normal circumstances and, in the case of Manoeuvres, I also looked at how the Rapid Growth racial ability assists in terrains with the Earth element.
Being the generous person that I am, I have decided to share the fruits of my travails with the readers of this blog, in the hope that they might find them useful.
Using the Tables
Using the tables to get useful results it easy. To get the expected value for an action for an army, simply add together the numbers in the ‘mean’ column.
e.g. I have a small 9 health army on a coastland terrain. Units are:-
1 x Eldar Dryad
1 x Dryad
1 x Pine
2 x Pineling
If I want to know the expected result for, say saves, for this army, I add together 1.0 + 0.833 + 0.833 + 0.333 + 0.333 = 3.333.
If I want the expected number of Manoeuvre results, I add 0.5 + 0.333 + 1.0 + 0.333 + 0.333 = 2.5
On a terrain containing the earth element (yellow), and we are counter-manoeuvring, the manoeuvres are read from the ‘Rapid Growth’ row, so our army would now get 0.917 + 0.611 + 1.5 + 0.556 + 0.556 = 4.14
As you are aware, the expected result is only one weapon that we have in our armoury. This tells us the ‘middle’ of all the values that we would get if we rolled these dice indefinitely, but it tells us nothing about the spread of our results. The standard deviation column tells us this.
For the non-statisticians amongst us, if there are any, one standard deviation either side of the mean gives us the boundaries for where 66% of our results will lie, and 95% of our results will fall between the mean minus two standard deviations and the mean plus two standard deviations.
e.g. I have an army of Darkwombles that have a melee expected value of 6 hits with a standard deviation of 1.5 – this tells me that I should be expecting 66% of my results to lie between 6-1.5 and 6+1.5 , in other words between 4.5 and 7.5 . Furthermore, I know that 95% of my results will be between 6 – 2 x 1.5 and 6 + 2 x 1.5 , i.e. 95% of the time I will score somewhere between 3 and 9 melee results for this particular army of Darkwombles.
Unfortunately, standard deviations do not simply add together like means (expected values) but, if you think of our old Greek friend Pythagoras, he gives us the answer.
To combine standard deviations, you simply square the numbers, add them together, and then take the square root. Exactly as we do to find the length of hypotenuse from the other sides of a right angled triangle.
The Standard Deviation for saves in the above army, therefore is:-
S.D. = SQRT ( 1.414 ^ 2 + 1.213 ^ 2 + 1.213 ^ 2 + 0.471 ^ 2 + 0.471 ^ 2 )
= SQRT ( 2 + 1.471 + 1.471 + 0.222+ 0.222 ) = SQRT( 5.386 ) = 2.32
This means that my 9 health army will produce between 3.333 – 3.32 and 3.333 + 3.32 ( between 0.013 and 6.653) two thirds of the time.
Taking this army on a terrain containing the earth element and looking at manoeuvres:-
S.D. = SQRT ( 1.382 ^ 2 + 0.921 ^ 2 + 0.866 ^ 2 + 0.497 ^ 2 + 0.497 ^ 2 )
= SQRT ( 1.910 + 0.848 + 0.75 + 0.247 + 0.247 ) = SQRT ( 4.002 ) = 2
So, this army will score between 2.14 and 6.14 manoeuvres, using Rapid Growth ability during counter-manoeuvre, 66% of the time.
If these tables are useful, please let me know and I can publish other races.
My Treefolk calculator spreadsheet, which does all the above calculations for you, can be downloaded from here:-


Read Monadologie by Gottfried Wilhelm Leibniz Free Online
Book Title: Monadologie|
The author of the book: Gottfried Wilhelm Leibniz
Edition: Philipp Reclam jun. GmbH & Co. KG, Stuttgart
Date of issue: January 1st 1998
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ISBN 13: 9783150078532
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In den 90 Paragraphen der sogenannten Monadologie gibt Leibniz eine begriffliche Fassung und perspektivische Auslegung der Grundbegriffe seiner Metaphysik, die durch ihre Stringenz und deduktive Geschlossenheit die Position des Autors in der Philosophiegeschichte wesentlich bestimmt haben. Hier findet sich nicht nur eine ausführliche Diskussion seines "Monade"-Begriffs, sondern auch die Verbindung zur "prästabilierten Harmonie", in der die Gesetze des Körpers und der Seele zur Übereinstimmung kommen. Die neue, zweisprachige Ausgabe ist ausführlich kommentiert.
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Read information about the authorGottfried Wilhelm von Leibniz (/ˈlaɪbnɪts/; German: [ˈɡɔtfʁiːt ˈvɪlhɛlm fɔn ˈlaɪbnɪts] or [ˈlaɪpnɪts]; July 1, 1646 – November 14, 1716) was a German polymath and philosopher.
He occupies a prominent place in the history of mathematics and the history of philosophy. Most scholars believe Leibniz developed calculus independently of Isaac Newton, and Leibniz's notation has been widely used ever since it was published. It was only in the 20th century that his Law of Continuity and Transcendental Law of Homogeneity found mathematical implementation (by means of non-standard analysis). He became one of the most prolific inventors in the field of mechanical calculators. While working on adding automatic multiplication and division to Pascal's calculator, he was the first to describe a pinwheel calculator in 1685 and invented the Leibniz wheel, used in the arithmometer, the first mass-produced mechanical calculator. He also refined the binary number system, which is the foundation of virtually all digital computers.
In philosophy, Leibniz is most noted for his optimism, i.e., his conclusion that our Universe is, in a restricted sense, the best possible one that God could have created, an idea that was often lampooned by others such as Voltaire. Leibniz, along with René Descartes and Baruch Spinoza, was one of the three great 17th century advocates of rationalism. The work of Leibniz anticipated modern logic and analytic philosophy, but his philosophy also looks back to the scholastic tradition, in which conclusions are produced by applying reason of first principles or prior definitions rather than to empirical evidence.
Leibniz made major contributions to physics and technology, and anticipated notions that surfaced much later in philosophy, probability theory, biology, medicine, geology, psychology, linguistics, and computer science. He wrote works on philosophy, politics, law, ethics, theology, history, and philology. Leibniz's contributions to this vast array of subjects were scattered in various learned journals, in tens of thousands of letters, and in unpublished manuscripts. He wrote in several languages, but primarily in Latin, French, and German. There is no complete gathering of the writings of Leibniz.
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In this self-paces course you may study all about decimals, fractions, powers of ten algebraic pondering, and even properties of shapes. Learn fifth grade math—fractions, decimals, shapes, and more! This Basic geometry and measurement course is a refresher of length, area, perimeter, quantity, angle measure, and transformations of 2D and 3D figures.
Use your KS3 maths abilities to answer questions on fractions, decimals, percentages and ratios and reunite the islands. Learn third grade math—fractions, space, arithmetic, and a lot https://www.topschoolreviews.com/mathsisfun-review extra. Learn Algebra 1 aligned to the Eureka Math/EngageNY curriculum —linear capabilities and equations, exponential progress and decay, quadratics, and extra.
Abstract as it might be, quantity theory may also be one of the more familiar types of math to many of us. It encompasses the multiplication, subtraction, division, and addition of integers, or entire numbers and their adverse counterparts. Learning maths is fun with these on-line maths video games from BBC Bitesize. All the games are free to play and are designed to help develop maths ideas and skills. The algebra worksheets present practice in translating, evaluating and simplifying algebraic expressions.
- But nature is a wonderful realm in which to look at beauty born out of mathematical relationships.
- Join a quick game to play on-line, or set up a personal game to play with your folks.
- Khan Academy’s 100,000+ free practice questions give instant suggestions, don’t need to be graded, and don’t require a printer.
- Learn pre-algebra—all of the basic arithmetic and geometry skills needed for algebra.
What you do next is multiply and divide from left to proper. The number sense worksheets familiarize kids with primary math operations and assist them comprehend place values and kinds of numbers like odd, even, prime, composite and more. Inspire a lifetime of learning and discovery with our free, enjoyable instructional program for kids ages two to eight. K5 Learning provides free worksheets, flashcardsand inexpensiveworkbooksfor children in kindergarten to grade 5.
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Kids use visual area models to compare fractions with widespread denominators on this intergalactic math sport. Kids can also study independently in the Khan Academy Kids Library—a curated assortment of actions, books, videos, and coloring pages. Our guide reader allows kids to follow together with recorded audio narration or learn on their own across our fiction, non-fiction and fiction leveled books.
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Know more about derivatives primarily based on power, product and quotient guidelines, exponents, definite integrals and more. Equip youngsters with this set of measurement worksheets to assist them determine the attributes of size, time, weight and capacity. Learn to check sizes, read clocks, and calculate money as nicely.
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With Zearn, kids be taught the same concept twice—once with the trainer and once of their digital lessons—so they achieve a deep understanding of math. Their digital math lessons cover every K–6 math goal, and all supplies are research-backed and crafted to represent all youngsters. Khan Academy presents apply exercises, tutorial movies, and a personalized studying dashboard that empower learners to review at their own tempo in and outdoors of the classroom.


A ﬂow based pruning scheme for enumerative equitable coloring algorithms
|Collection Location||Koleksi E-book & E-Journal Perpustakaan Pusat Unila|
|Author(s)||Koster, A. M. C. A.
Tieves, · M.
|Subject(s)||Business and Management
|Abstract/Notes||Abstract An equitable graph coloring is a proper vertex coloring of a graph G where the sizes of the color classes differ by at most one. The equitable chromatic number, denoted by χeq(G), is the smallest number k such that G admits such equitable k-coloring. We focus on enumerative algorithms for the computation of χeq(G) and propose a general scheme to derive pruning rules for them: We show how the extendability of a partial coloring into an equitable coloring can be modeled via network ﬂows. Thus, we obtain pruning rules which can be checked via ﬂow algorithms. Computational experiments show that the search tree of enumerative algorithms can be signiﬁcantly reduced in size by these rules and, in most instances, such naive approach even yields a faster algorithm. Moreover, the stability, i.e., the number of solved instances within a given time limit, is greatly improved. Since the execution of ﬂow algorithms at each node of a search tree is time consuming, we derive arithmeticpruningrules(generalizedHall-conditions)fromthenetworkmodel.Addingthese rules to an enumerative algorithm yields an even larger runtime improvement.|
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Towards Proof-of-Work Cryptocurrency Valuation: Mining Games, Network Effects and the Social Value of Blockchain
32 PagesPosted: 5 Apr 2019
Date Written: March 8, 2019
In this study, a mathematical model of proof-of-work cryptocurrency valuation is developed based on the concepts of simultaneous equilibria in two mining games, the purchasing power parity in the system of equations of exchange and the network effects of transaction cost optimisation in the economy where agents utilise both conventional and digital currencies to process payments.
The model lays the foundation for rigorous long-term proof-of-work cryptocurrency valuation superior to existing approaches that are solely based on variations of Metcalfe’s law or costs of mining.
A Skeptic's View of Crypto (from the Point of View of Monetary Economics)
Based on the model, the long-term Bitcoin equilibrium price conditional on current block size limit is $106, while if the block size limit is abandoned, the conditional equilibrium price is $6725. Network hashrate is consistent with the no-limit scenario, while the transaction fees are consistent with the status quo, revealing an important source of uncertainty on the market.
The developed framework is also applied to the measure the net social value of proof-of-work cryptocurrencies, and Bitcoin long-term social value of equilibrium is shown to be positive at $126,200 per block.
The comparison of equilibrium and optimal cases shows that external intervention is not necessary to guarantee socially beneficial outcomes.
Keywords: cryptocurrency, valuation, proof-of-work, bitcoin, network effects, transaction costs, alternative investments, simultaneous equilibria, social value of blockchain
JEL Classification: C62, D53, D61, E42, G12, G19
Suggested Citation:Suggested Citation
Shanaev, Savva and Shuraeva, Arina and Vasenin, Mikhail and Kuznetsov, Maksim, Towards Proof-of-Work Cryptocurrency Valuation: Mining Games, Network Effects and the Social Value of Blockchain (March 8, 2019).
Available at SSRN: https://ssrn.com/abstract=3352098 or http://dx.doi.org/10.2139/ssrn.3352098


My son and I like math and numbers. We enjoy repetitive, consecutive, or palindrome numbers. 3:33 is one of our favorite times of day, as is 12:34. Pi Day has been a favorite unofficial holiday for us every year. We usually celebrate with pie – pizza for dinner, and pie for dessert. It’s a silly but fun little (pun) tradition that we enjoy.
Today is Super Pi Day, because the date (American style) is 3/14/15, and the beginning of Pi is 3.1415. This morning at 9:26:53 is even better – 3.141592653. If you don’t want to use military time, it will happen again this evening.
My son has a “Pi chart” in his room that we got him for his birthday one year. He memorized the first 67 decimals, or as he put it “enough for a stranger to deduce that I have no life” – he’s funny!


A semi-rigid connection must have the required strength, stiffness and ductility. Some typical moment versus rotation (M-θ) curves and various analytical and mathematical models representing the M-θ relationships can be found in previous chapters. Conveniently, a joint can be considered in an analysis as dimensionless with location at the intersection of the element centre lines. Further, a rotational spring element satisfying the M-θ relationship is inserted into each end of beam element to model the connection behaviour. The joint equilibrium condition can be expressed as,
in which Me and Mi are the moments at the two ends of a connection (see Figure1). The corresponding external node is connected to the global node and the internal node is joined to the beam element.
(a) Equilibrium at a joint (b) External and internal rotations
Figure1 Modeling of Semi-Rigid Jointed Member
The stiffness of the connection, S, can be related to relative rotations at the two ends of the connection spring as,
where θe and θi are the conjugate rotations for the moments Meand Mi (see figure above). Rewriting Equation (2) in matrix form, the stiffness matrix of a connection spring can be written as,
A typical beam element bending stiffness matrix can be expressed as,
in which kij are the stiffness coefficients of a prismatic beam. Here, the imperfect PEP element proposed by Chan and Zhou (1995) is adopted and more details can be referred to the original reference. Therefore, a hybrid element can be obtained by directly adding the two ends connection stiffness to the PEP element bending stiffness matrix as,
In which the first subscript refers to node 1 or node 2. The internal degrees of freedom of the stiffness expression can be eliminated by a standard static condensation procedure. The stiffness expression of a beam element with both ends connected to a pair of springs can be finally written as,
and β is given by,
where S1 and S2 are the connection stiffness at nodes 1 and 2, respectively. When the spring stiffness Si is zero, it means that the corresponding end is pinned end; when the spring stiffness Si is infinite, it means that the corresponding end is rigid end. For semi-rigid case, the spring stiffness Si can be determined by the given M-θ function.


By Anton Deitmar
A primer in harmonic research at the undergraduate level
Gives a lean and streamlined creation to the relevant ideas of this gorgeous and utile theory.
Entirely in line with the Riemann necessary and metric areas rather than the extra not easy Lebesgue indispensable and summary topology.
Almost all proofs are given in complete and all vital strategies are awarded clearly.
Provides an creation to Fourier research, top as much as the Poisson Summation Formula.
Make the reader conscious of the truth that either central incarnations of Fourier thought, the Fourier sequence and the Fourier rework, are particular situations of a extra common conception bobbing up within the context of in the community compact abelian groups.
Introduces the reader to the thoughts utilized in harmonic research of noncommutative teams. those options are defined within the context of matrix teams as a imperative instance
Read Online or Download A First Course in Harmonic Analysis (Universitext) PDF
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Extra info for A First Course in Harmonic Analysis (Universitext)
Ek ) = Span(a1 , . . , ak ). Then put k ek+1 = ak+1 − ak+1 , ej ej . j=1 For j = 1, . . , k then ek+1 , ej = 0. Further, the linear independence implies that ek+1 cannot be zero, so put ek+1 ek+1 = ek+1 . Then e1 , . . , ek+1 are orthonormal. If H is ﬁnite-dimensional, this procedure will produce a basis (ej ) in ﬁnitely many steps and then stop. If H is inﬁnite-dimensional, it will not stop and will thus produce a sequence (ej )j∈N . By construction we have span(ej )j = span(aj )j , which is dense in H.
The fact that it is a pre-Hilbert space is then immediate. Let us ﬁrst prove the convergence of the scalar product. Since |f (s)g(s)| = |f (s)||g(s)|, it suﬃces to prove the claim for real-valued nonnegative functions f and g. Let F be a ﬁnite subset of S. There are no convergence problems for 2 (F ); hence the latter is a Hilbert space and the Cauchy -Schwarz inequality holds for elements of 2 (F ). Let f, g ∈ 2 (S) be real-valued and nonnegative and let fF and gF be their restrictions to F , which lie in 2 (F ).
Show that 2 2 ||v + w|| + ||v − w|| 2 2 = 2 ||v|| + 2 ||w|| . This equality is known as the parallelogram law. 17 Let H be a Hilbert space and let T : H → H be a map. , there is a map T ∗ on H such that T v, w = v, T ∗ w for all v, w ∈ H. Show that T and T ∗ are both linear. 18 Let V be a ﬁnite-dimensional Hilbert space. A linear operator A : V → V is called self-adjoint if for any two vectors v, w ∈ V we have Av, w = v, Aw . , that V has a basis consisting of eigenvectors of A. (Hint: Show that if A leaves stable a subspace W of V , then it also leaves stable its orthogonal space W ⊥ .
A First Course in Harmonic Analysis (Universitext) by Anton Deitmar


Our Book-making workshop involves learning several methods of bookbinding (such as stick-bound, pamphlet, accordion and Japanese stab-binding). Book-making also involves some math, such as measurement, fractions, geometry and graph theory. Participants will make books from a variety of fun reclaimed materials, such as pages made from graph paper or covers made from wallpaper. Books are great blank canvases for many learning outcomes, such as making cookbooks as part of a food unit, or students writing their own stories to be shared in the classroom library. There is so much to explore when making books!
Please complete our Program Request Form if you’re interested in scheduling a Book-Making ReMaker Playshop!


Thanks everyone for your kind comments yesterday. I decided not to respond individually to everyone and just do a group thanks.
I put all of the Kaffe strip pieced blocks up on a design wall to get an idea what I'm doing. I have used up all of the 7" scraps which were trimmed to 6.5" blocks when sewn.
I have a pile of 11" strips left over from this quilt. I was thinking about cutting them down to 7" but then I played around with math. The blocks I have made will finish at 6". If I sew the 11" strips I can cut blocks at 10.5" which will finish at 10" If I made the center of the quilt with nine 10" blocks (3 rows of 3 blocks, 30" on a side) I can add rows of five 6" blocks around it (also 30"). It will take 24 six inch blocks to go around the center 10" blocks once. I have 37 blocks made. So............the plan is: make nine 10" blocks and then cut the rest of the strips down to 7" to make more of the smaller blocks.


Bezier and B-spline Techniques with Matlab (Paperback)
The aim of this book is to teach students the essential of Bezier and B-spline techniques with the aid of examples. Computer codes, which give an easy interface of Bezier and B-spline techniques to the users, are implemented as Matlab programs. The reason to choose Matlab is that it is easy to use and has a good graphical user interface. This book focuses on curves and surfaces using Bezier and B-spline techniques. It is based on the theory "Bezier and B-spline Techniques" which are known in mathematics. Interpolation and approximation methods have been illustrated intensively. Some of algorithms are represented using practical cases for example Casteljau algorithm. Students and researchers can use this book to succeed good understanding of Bezier and B-spline techniques for reliable and efficient studies in accordance within scientific applications.


Grandparents Teach, Too
MARQUETTE – Fine motor is the coordination of hand muscles so children can print and do other tasks correctly. There are some, simple fun activities that can help teach, practice, and refine these skills.
For more help developing your young children’s skills to help them succeed in school and have a lifetime of learning see the authors’ book “Learning Through the Seasons” at grandparents teachtoo.org, museums, and bookstores, available in English and Spanish.
Materials You Will Need:
A pair of dice, pencil, paper, and ruler
What To Do:
Help your children investigate a pair of dice. What do they see? What numbers are represented by the dots? What shape is a dice? How many sides are there?
Explain a guessing game you can play. With your help, young children draw a chart using paper, pencil, and ruler.
Across the top should be six squares. Help your children write the numerals 1 – 6 and draw a vertical line from the top to the bottom of the paper from each numbered square.
Explain that tally marks keep track of how many times a number is rolled. Discuss that tally marks are another way of representing numbers and a way of recording an outcome. Ask your children which number they think will be rolled the most.
Have your children roll one of the dice. Count the number of dots on the side facing up. Show how to make a tally and have your child make a tally mark under the correct number.
Continue doing this for ten rolls. Show how you record five tallies by placing a slanted line over four of them. Talk about how it is easier to see and count which number has the most.
After ten rolls, ask children if their guess was correct. Which number did come up the most? WhichWoodlandnumberMeadowscameOpenhouse:up the least?Layout
How This Will Help My Child?
The simple act of rolling dice helps children gain strength in the shoulder, wrist, hands, and fingers. This control leads to correctly using tools like pencils, crayons, scissors, and other objects. Tally marks are a useful tool for children to keep track as they count and collect data quickly in science and math. Grouping tally marks also reinforces counting by fives.
What Else Can We Do?
You may want to use the pattern of the dots in the tally squares instead of the numeral itself. This helps your children visually understand a number. For an older child, you may use the numbers up to 12.
A child will need to add the faces of both dice to record the correct number. This helps practice addition skills. Count with tallies often. When your children understand five tallies, add on more tally marks2 for6/counting4/2012 practice.2:24 PM “TallyPage O’Malley” by Stuart J. Murphy and “Tally Cat Keeps Track” by Trudy Harris are great read together books.


iMode Top : CLDC 16::16 FixedPoint Library
This is an implementation of a simple fixed point math library for the IAppli (CLDC) Java platform.
Note, I am not a numerical methods or computer arithmetic hacker. Not having easy access to my computer arithmetic texts from college (I'm in Japan at the moment) I scrounged the algorithms from various sources on the net (all references are in the source code). In each case I have very little clue as to how "good" these algorithms are in terms of maximum absolute error, or performance. Most of the functions are documented with some clues as to what kind of performance you can expect.
The package works on fixed-point 32 bit ints, with 16 bits of integer and 16 bits of fraction.
The logarithm function is still under development, and I haven't implemented an exponent function that works yet. If anyone knows a good implementation for fixed point exponential, please email me.
I tested the Sin, Cosine, Tangent and ArcTangent functions pretty well, but not some of the others. In particular watch out for invalid ranges of inputs. The Sin and Cosine should handle inputs from 0 to 2 PI (radians).
Below is a graphic demo that runs on the 503i IAppli phones. I have tested this on the F503i.
For higher precision and a more versatile approach, I recommend looking into the CORDIC algorithms, as described in this article.
Line intersection, Sin and Cosine tests. Hey, that circle looks a little flat on the tops and bottoms!


In my article today, I will give a quick introduction to OpenBOM calculations and formulas. It is a powerful mechanism, which allows you to perform flat and structured calculations. Examples of calculations – to calculate a total cost of inventory, calculation total mass or total cost of the BOM. Another example of the calculation is to calculate the unit cost based on the purchased material length. There are other examples. Learn more about formulas and cost in our online training library.
Formula builder allows you to create a formula. You can type the names of the processes (instead of cell numbers like in Excel or Google Sheet).
All OpenBOM formulas are applied to a specific line in a BOM or Catalog or to all lines. The formula includes names of properties and operations. The scope of the line formula is in the single line.
Structure Calculations (or Rollups)
The calculation with the scope goes between lines is called rollup. In such a case OpenBOM applies all calculations in a specific level (eg. sub-assembly) and the total cost of the sub-assemblies applies to the upper “unit cost” level. In such a case, OpenBOM can calculate different values (eg. cost, mass, etc.) and make totals for the BOM.
This is the simplest calculation available in OpenBOM – Totals. It can be done in any table and it includes automatically sum of all line values.
In the following two videos I will demonstrate the use of both formulas. I will use line calculation and flat totals in the catalog. Here is the video.
The following video shows how formulas can be applied in the structure. In the video I demo how you can create a cost rollup and apply it to all BOM levels.
OpenBOM calculations and formulas is a super powerful tool that can simplify your life when making calculating in BOMs, Catalogs, Order BOMs for many engineering and procurement use cases. Learn more about it in our Online Training Library.
Try OpenBOM today. Register for FREE here.
Best, Oleg @ openbom dot com.


Computational Mathematical Sciences
We are interested in developing and analyzing computational algorithms that are working in solving mathematical modeling problems including (fractional) partial differential equations and integral equations describing physical, biological, chemical, and engineering phenomena. We want to see the world beyond the experimental world with mathematical theories and computer simulation.
Numerical analysis methods, Nonlinear Dynamics, Mathematical Data Analysis
Fractional and Partial Differential Equations, Biodiversity, Networks, Text mining, Reservoir computing
Research Keywords and Topics
Numerical analysis methods: Fractional, Partial Differential & Integral equations
Nonlinear Dynamics: Biodiversity, fractional dynamics, networks
Mathematical Data Analysis: Text mining, Deep learning & Reservoir computing
Kim, Hyunju; Kim, Keon Ho; Lee, Seyeon; Jang, Bongsoo*, " New explicit and accelerated techniques for solving fractional order differential equations", Appl. Math. Comput. 379 (2020), 125228, 21 pp.
H.A.A, El-Saka, Seyeon Lee, Bongsoo Jang*, "Dynamic analysis of fractional-order predator-prey biological economic system with Holling type II functional response", Nonlinear Dyn. 96, 407–416 (2019)
Junpyo Park, Younghae Do, Bongsoo Jang*, and Ying-Cheng Lai, "Emergence of unusual coexistence states in cyclic game systems", Scientific Reports 7 (2017) 7465.
System and Method for Transferring Style of Recognition Area: Application No, 10-2019-0059322, Registered No, 10-2057713, Kyunghoon Kim, Bongsoo Jang
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Korean Text Clustering System and Method: Application No, 10-2016-0036552, Registration No/date, 10-1769035/Kyunghoon Kim, Bongsoo Jang
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Maths Games Competition
In January, the school took part in a really exciting Maths week where we had the opportunity to design our own Maths board game. Each class shortlisted their games to be put forward for a competition. Mr Horne found it too tricky to choose as they were all so fantastic and so enlisted the help of the school council to judge.
We had to choose a winner for each year group to win a prize and an overall winner of the school.
It was so much fun playing all the children's games!


Mathematics as Problem-Solving
The ability to apply mathematical skills confidently and meaningfully to unfamiliar situations is the goal of mathematics instruction.
Mathematics as Communication
Reading, writing, modeling, drawing and discussing are all important tools in helping children to explore, convey and clarify mathematical concepts and ideas.
Mathematics as Reasoning
Giving children opportunities to analyze, draw conclusions, and justify their thinking helps them gain a sense of self-reliance and confidence in their mathematical abilities and helps them see that mathematics makes sense.
Mathematics as Connections
Helping children see how concepts and ideas in one area of math relate to other areas of math, other subject areas, and their everyday lives foster an appreciation of the usefulness of mathematics.


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Karnoven Heart Rate | Body Mass Index | RMR and Calorie Needs
Karnoven Heart Rate Formula (HR)
Formula: THR = ((HRmax - HRrest) × %Intensity) + HRrest
||Training (or Target) Heart Rate
||Maximum Heart Rate
||Resting Heart Rate Males = 220 - AGE Females = 226 - AGE
||The percentage of your HRmax
The Karvonen method calculates your training heart rate zones based on your fitness level.
As your fitness level increases, your resting (minimum) heart rate drops. The heart rate reserve is your maximum heart rate minus your resting heart rate. The resting heart rate is determined by testing -- put your heart rate monitor on just after you wake up and keep it on for about fifteen minutes. Your heart rate at its lowest during this period is your resting heart rate.
Body Mass Index (BMI)
BMI is the measurement of choice for many physicians and researchers studying obesity. BMI uses a mathematical formula that takes into account both a person's height and weight. BMI equals a person's weight divided by their height. (BMI= weight/height). The following table further describes the risk level associated with BMI and waist size.
Risk of Associated Disease According to BMI and Waist Size
||Waist less than or equal to 40 in. (men)
or 35 in.(women)
|Waist greater than 40 in. (men)
or 35 in. (women)
|18.5 or less
|18.5 - 24.9
|25.0 - 29.9
|30.0 - 34.9
|35.0 - 39.9
|40 or greater
Caloric Expenditure using Harris-Benedict Equation
The Harris Benedict Equation uses your Resting Metabolic Rate (RMR) and then applies an activity factor to determine your total daily energy expenditure (or calorie needs). The only factor omitted by the Harris Benedict Equation is lean body mass. Remember, leaner bodies need more calories than less leaner ones. Therefore, this equation will be very accurate in all but the very muscular (will under-estimate calorie needs) and the very fat (will over-estimate calorie needs).
Harris- Benedict Formulas:
- Males: RMR = 66 + (13.7 × W in kg*) + (5 × H in cm**) - (6.8 × Age in yrs.)
- Females: RMR = 655 + (9.6 × W in kg) + (1.8 × H in cm) - (4.7 × Age in yrs.)
- *2.2 Pounds = 1 Kilograms
- **1 inch = 2.54 centimeters
Calorie Needs are determined by multiplying your RMR with an activity level. The following table describes each Activity multiplier:
Calorie requirements based on activity factors
||aerobic exercise 3 times a week
||aerobic exercise 5 times a week
||aerobic exercise 7 times a week


Canon of Insolation and the Ice-Age Problem
The original edition of "The Canon" was published in German (1941). It was translated into Serbian (1997), into English in Jerusalem (1969), and finally into English in Belgrade (1998). The book has 57 illustrations, 28 tables and 1338 equations. "The Canon" was an interdisciplinary, work between mathematics, astronomy, geophysics and geology. With astronomical methods, Milankovich solved the problems of the secular perturbations in the movements of the Earth; with geophysical methods he solved the law of the Earth s insolation: while with geological methods he studied succession and the history of Quaternary glaciations.
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It all adds up to a thorough education for New Rickstones Academy students.
Year 7 and 8 students spent a whole week focusing on numbers, whether the lesson was science, music or business studies.
This is the first time the school, in Conrad Road, Witham, has held a Numeracy Week, with plans to now hold one each term.
Each member of staff wore a maths question around their neck for students to work out during the event and the Happy Puzzle Company took along a selection of puzzles, games and challenges to get the students thinking.
Teachers from all subjects worked numeracy into their lessons to keep students on topic for the five days and tutor periods were spent on group maths activities.
Oli Bamigbele, who organised the event, said: “The students spent a lot of time working on shapes, building them and calculating their area, plus looking at the perimeter of 3D shapes and working out capacities.
“We also invited parents to come in to work with their children and the Happy Puzzle Company to give them an insight into what their children are working on and to improve communication between them.
“The Numeracy Week was about raising attainment in numeracy by showing the students numeracy is used in everything we do.”
Penny Tuffey, Year 8, said: “The week was really fun. It gave us a chance to think about puzzles and to try new things


The period of graph concept started with Euler within the yr 1735 to resolve the well-known downside of the Königsberg Bridge. Within the trendy age, graph concept is an integral part of pc science, synthetic engineering, machine studying, deep studying, knowledge science, and social networks. Trendy Purposes of Graph Idea discusses many cutting-edge purposes of graph concept, akin to site visitors networks, navigable networks and optimum routing for emergency response, and graph-theoretic approaches to molecular epidemiology.
What’s Graph Idea?
A graph G(V, E) is a non-linear knowledge construction, which consists of pair of units (V, E) the place V is the non-empty set of vertices (factors or nodes). E is the set of edges (traces or branches) such that there’s a mapping f: E →V i.e., from the set E to the set of ordered or unordered pairs of parts of V. The variety of known as the order of the graphs and the variety of edges known as the scale of graph G (V, E).
Graphs are of three varieties Undirected Graphs, Directed graphs, and weighted graphs.
- Undirected graphs: In Undirected Graphs, the perimeters are related to an unordered pair of vertices. A graph G (V, E) and not using a loop and parallel edges known as a easy graph. A graph that has multiple edge between any pair of vertices known as a multigraph. Once more if any multigraph incorporates loops then the graph is a Pseudo graph. In keeping with construction, there are various kinds of undirected graphs, akin to Null graphs, full graphs, Common graphs, bipartite graphs, Cycles, Wheels, Eulerian graphs, and Hamiltonian graphs.
- Directed Graph: A directed or digraph graph G consists of a set V of vertices and a set E of edges such that eϵE is related to an ordered pair of vertices i.e., every edge has a course. There are various kinds of directed graphs. Symmetric directed graphs, easy directed graphs, full directed graphs, quasi-transitive digraphs, and oriented graphs.
- Weighted Graphs: Many graphs can have edges containing a weight related to characterize real-world implications akin to value, distance, and amount. Weighted graphs might be directed or undirected graphs.
- Timber are some of the generally used sub-categories of graphs. In computing, bushes are helpful for organizing and storing knowledge in a database. A tree is a linked acyclic graphic with no cycle. A tree T with n vertices has n-1 edges. A subgraph T a linked graph G (V, E) known as a spanning tree if T is a tree and if contains each vertex of G. There are two algorithms a) BFS (Breadth-first search) and b) DFS (Depth-first Search) for setting up the spanning bushes of a given undirected graph G. For weighted graphs one can assemble the minimal spanning tree utilizing Prim’s and Kruskal’s algorithm. The Binary bushes having one vertex of diploma two and the opposite vertices of diploma one or diploma three, are used to characterize an algebraic expression and storage illustration. Storage Illustration of Binary tree has two methods a) Sequential illustration and b) Hyperlink illustration.
Ex. Use a binary tree to characterize the expression ((a + b)* c) + (d/e)
How does Graph Idea Work?
Graph concept is in the end about finding out the relationships between totally different nodes (vertices) and connections (edges). The research of graphs throughout a construction offers solutions to quite a few issues in structure, networking, optimization, matching, and operation.
Graph Colouring Issues
Graph coloring is without doubt one of the most helpful strategies wherein adjoining vertices get hold of totally different colours. The minimal variety of colours used for the proper coloring of the graph is our aim which is an optimization downside.
The issue of graph coloring has many purposes, akin to Making a Schedule or Time Desk, Cell Radio Frequency Project, Sudoku, Register Allocation, and Map Coloring.
Time Scheduling Downside
Take into consideration a particular semester; there are college students taking every of the next mixtures of matters. On this downside, our intention is to search out the minimal variety of examination days for scheduling the examination within the 8 topics in order that college students taking any of the given mixtures of the topic haven’t any battle.
As well as, discover an out there schedule utilizing a minimal variety of days.
Desk: Combos of Topics
The result of the issue
Some Classical Issues of graph concept
- An outdated downside is to attach 4 homes H1, H2, H3, and H4 to a few utilities every – water (W), fuel (G), electrical energy (E), and TV cable line (C). Can every service be linked to every of the 4 homes with out having two cross-connections between them?
- Travelling Salesman Downside:
Suppose that the territory of a vendor contains a number of cities with highways linking some pairs of those cities. He ought to go to each metropolis as soon as. Graph concept could be helpful in fixing this transport system. The issue could be represented graphically by a graph G whose vertices correspond to the cities. The 2 vertices are joined by an edge if and provided that a freeway connects the corresponding cities. Beginning at vertex a, the salesperson can go to by taking the perimeters e1,e2, e3, e4, e5, and e6 and again to vertex a.
Algorithm for Trendy Actual-life Utility
Google maps use graphs for development and transport methods. The intersection of two (or extra) roads is taken into account a vertex, and the highway connecting two vertices is taken into account an edge. Their navigation system then makes use of the algorithm to calculate the shortest path between two vertices. In GPS we additionally use totally different shortest path algorithms akin to DFS (Depth first search) and BFS (Breath first search) algorithm. By the Dijkstra algorithm, one can discover the shortest route between a given node (supply node) and all different nodes (vacation spot node) in a graph. This algorithm makes use of edge weights to discover a option to scale back the full distance (weight) between the supply node and all different nodes.
Fb and LinkedIn
Ever marvel how Fb is aware of how an individual is your mutual buddy or how LinkedIn is aware of if a connection is a second or third one? Fb and LinkedIn mannequin their customers as a graph wherein every vertex is a consumer profile. The sting between two individuals is the truth that they’re associates amongst themselves or comply with each other. Fb and LinkedIn Buddy suggestion algorithm makes use of graph concept. Fb is one instance of an undirected graph.
World Broad Internet
On the World Broad Internet, net pages are thought of vertices. There may be an edge between web page ‘u’ and one other web page ‘v’ if there’s a hyperlink from web page ‘v’ to web page ‘u’. That’s an instance of a directed graph. That’s the fundamental idea behind Google Web page Rank Algorithm.
On social networking websites, we use graphs to trace consumer info. Appreciated exhibiting most well-liked put up strategies, suggestions, and so on. Thus, the event of algorithms to handle graphs is of nice curiosity within the area of data expertise.
Graph concept is utilized in Netflix and different OTT platforms to reinforce advice methods. By representing customers and content material as nodes and their relationships as edges, graph concept helps determine patterns and connections. It permits customized suggestions by analyzing the consumer’s viewing historical past, rankings, and comparable preferences of different customers with comparable viewing patterns. By setting up a graph of interconnected content material, OTT platforms can recommend related reveals or films based mostly on the consumer’s pursuits and the preferences of comparable customers, resulting in a extra participating and customized streaming expertise.
Attributable to rising the applying of Synthetic Intelligence, Machine Studying, Deep Studying, Knowledge Science, and Cryptography in numerous fields like Well being Science, Social Science, Manufacturing Business, Defence providers, and totally different authorities actions, the graph theoretical method, and its software is a really demanding topic for the researcher. After ending the research of graph concept, college students might be able to apply their information of graph concept in numerous fields of recent science.


Ever since I opened my Etsy shop, I’ve gotten obsessed with researching other possibilities for it. And I have to say, I LOVE reading other people’s blogs and seeing projects and getting to know the community more.
I will share other links soon, but I thought these math creations deserved a post all unto themselves. I really love every single project posted here, and the math side of it is fascinating. The fact that Op art is one of my absolute favorite movements, and so many of these pieces look like knitted Op art projects is just simply the cherry on top. Enjoy!
By the way, just a fun little family fact – my husband can solve the Rubik’s cube in a minute or somewhere thereabouts. He taught himself, and says it is all about the math patterns too. 🙂


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This unique approach makes for a meaningful and memorable learning experience. Everything at Gizmos & Gadgets is hands on, cross-disciplinary, and has real world applications.
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In a world that is defined by rapid change – particularly in technology – there are concerns that students today won’t be prepared for the challenges of the future. Of all U.S. high school students who graduated in 2011, only 45 percent were ready for college-level math and 30 percent for science, according to ACT, a college-entrance testing agency. As jobs increasingly require proficiency or expertise in STEM subjects (science, technology, engineering, and math), it’s more important than ever that children are inspired to explore these subjects and understand their real impact on our everyday lives.
STEM drives an incredible number of the innovations we see today, from the fastest jet planes to the cars in our driveways to the televisions in our living rooms. Even the video and computer games that children and teenagers turn to for entertainment are based on STEM.
However, interest among students in these important subjects is lagging. And with the extraordinary number of careers and opportunities for growth in STEM fields, many organizations in both the private and public sectors are taking action to bolster student interest and enthusiasm in this area. Samsung, for example, is working to make STEM fun and exciting with their STEM education program called Solve for Tomorrow. It’s a national contest that encourages teachers and students to creatively use STEM to explore and improve their local environment and community.
“Our goal is to raise awareness and interest in STEM subjects, while providing teachers and students with the resources they need to innovate and grow,” says David Steel, executive vice president of corporate strategy for Samsung Electronics North America.
“We as a nation need to make STEM education a top priority,” says Betsy Landers, President of the National Parent Teacher Association (PTA), who joined as a program partner this year. “According to the U.S. Department of Labor, 15 of the 20 fastest growing occupations projected for 2014 require significant mathematics or science preparation. It is clear that making STEM education a priority is important, for our nation’s short and long-term future. We commend Samsung for their efforts in making a difference in the education and lives of children.”
To learn more about the contest, please visit Samsung.com/solvefortomorrow.
As the world continues to change in unexpected ways, the demand for technological developments will only increase. By equipping students of today with knowledge and interest in STEM subjects, we’re fostering the innovators of tomorrow.


Sorry it’s been awhile. I love being a dad and giving an attempt at that while going to graduate school is not an easy combination. But I’m still a heavy consumer of all the great Mathematics posts I see on Twitter. Usually when I see a post that catches my eye, I’m able to contemplate how I respond… but then immediately get sucked into preparing food or trying to set up a new gadget or just making faces at the kiddo.
But this one really got into my brain. I thought: what the hell kind of mutant number always contains the digit 1? I’m pretty good at simply multiplying numbers in my head, so I thought I could solve this fairly quickly.
First I tried some simple ones. 11? Nah, we can see that it has no 1 when multiplied by… like anything. 21… 31… 41… all lose their ‘oneness’ when multiplied by anything. Time to think bigger…
How about 379? Dude that doesn’t even have a 1 right now.
Ok, let’s get crazy. How about the number 123,456,789? Surely when you multiply it by anything a magical one pops into view… Gotta bust out a calculator for that one. Ok, type it in, multiply by 2 and I get 246,913,578. Aha, a 1! Clearly I’ve solved this problem, let’s just check once more to prove to myself how AWESOME I am at solving math puzzles. So we type in 123,456,789, multiply by 3 and… 370,370,367… No 1 in sight.
Clearly this is my cup of tea: simple-on-the-face, but can’t be solved without effort. What I want to show here is another way to think about problems like this. How can computers, and more specifically Python can help solve this.
I’ve been studying programming in the business sense for the last semester (see the zero posts I’ve made in last 6 months). But I want to show you how easy it is to get started with a language like Python – a free, open source programming language that is a breeze to install and use. Python has tutorials all over the place if you want to try on your own, but here’s a starter package.
First, we need a function to test whether a number actually has a 1 in it or not. That might seem unnecessary at first blush, after all can’t we just look at the number? Well… I guess it’s possible, but then it can’t be automated. The purpose of setting up programming is to let the computer do all the heavy lifting.
This program can be pretty simple, we can even get it a badass name like “The MF’ing One Extractor”, or maybe “one_tester” for short. Now Python functions are just like algebraic functions – you pass some stuff in and get something out. In this case, we’re going to pass a number in and get either True or False (stored as ‘flag’) out based on whether it contains a 1 (and yes the capitalization matters, Python is a little finicky about punctuation). Here’s our script:
if ‘1’ in str(word):
flag = True
flag = False
Now the main thing I’ve learned is that when we test programs like this, we need to try some weird inputs to see if our script holds up. Let’s try a few inputs: 435909; 000000001111100000000, and just for kicks, let’s throw in ‘brandon1rules’.
Perfect, we could keep testing, but seems like that part is working. Now that we have a function for testing for ones, we can do something a little more advanced. For a given candidate number to the solution of the puzzle, here is what we would like to do:
- Test if the candidate contains a ‘1’
- If it does not, stop the script and send a message to the user
- If it does, try multiplying by 2
- Go back to the first step and continue the process
Essentially we’re trying to do the opposite of the puzzle: find a multiple of the user’s chosen number that doesn’t have a ‘1’.
flag = False
multiplier = 1
while (flag == False):
if (one_tester(multiplier * num) == True):
multiplier = multiplier + 1
print str(num) + ” does not have a 1 when multiplied by ” + str(multiplier)
flag = True
Now, is this script the best we can do? Absolutely not. But it really gives the user the power to do something really simple and easy, rather than manually trying to find a multiple of a candidate that doesn’t contain a 1. In fact this program is so poor that if we actually found a candidate whose multiples always contain a 1, it would be caught in an infinite loop. But this is the kind of beginning that takes the burden of getting started off of the puzzle solver. I can start plugging in numbers quickly and try to find patterns.
Say you have a hunch that the number 574,381 is the one to solve this problem. Instead of trying to multiply over by one and manually checking for a ‘1’, you can simply execute the script:
Now we’ve been freed up from manually checking numbers and can move our brainpower to looking for patterns of numbers.
Over the last couple of years, I’ve really been converted. I think that we have to prioritize computer literacy, not how to format generic powerpoints about solar systems, but rather being able to have computers take the burden of certain tasks . We have super computers in our pockets and on our laps, and they can be used so much more powerfully than glorified word processors.