In the 1960s we had New Math. In the 1980s and 1990s we had a reiteration called Reform Math, which is still with us. These are all inferior curricula and have the counterintuitive result of making children unable to do much math or science.
Reform Math consists of about a dozen separate textbooks. The most widely used of the lot is Everyday Mathematics (also called EM and Chicago Math). Two things you can safely say about Everyday Math: it’s rarely loved and often hated. Millions of parents have wasted billions of hours trying to explain to their kids why they have to learn math in such a stupid way.
Children forced to learn with any variety of Reform Math bounce from topic to topic without mastering any of it. Most end up discouraged, and rarely go on to calculus, physics, chemistry, and the sciences generally. It’s a kid-killer and a civilization-killer.
To give people watching from the sidelines a sense of what Everyday Math is like, I collected war stories and first-person accounts. The material is sad, revealing, and somewhat repetitive. You can skip along any way you like but please read the concluding remarks. (Quotes are lightly edited for smoother reading; links to originals are provided.)
PARENT: “My son was actually crying at the table because he couldn't figure out how to do his division. He worked on the problem for over a half an hour, and could not get the answer. My child is not a slouch. He is very good at school, very good. I taught him how to divide the way that we were all taught in school. He had every problem on the page done in less than five minutes....What this is going to boil down to in my eyes is possibly when my child hits college, it's going to be more money out of my pocket and I'm going to have to pay for remedial math courses for my kid to get into college or to maintain a college grade point average....Every parent that I talked to hates this 'Everyday Math' and wishes it was gone."
PARENT: “My 9 year old is an honor student in the 4th grade at his school. He is a good athlete and generally well-rounded in writing, grammar, music, art and science. He gets good grades in math only because his parents are teaching him old time arithmetic, so he can understand what the ‘strategies’ are for. The school systems are puking all over our kids.”
WIKIPEDIA: “Almost as soon as the first edition was released, it became part of a nationwide controversy over reform mathematics. In October 1999, US Department of Education issued a report labeling Everyday Mathematics one of five ‘promising’ new math programs. The perceived endorsement of Everyday Mathematics and a number of other textbooks by an agency of the US government caused such outrage among practicing mathematicians and scientists that a group of them drafted an open letter to then Secretary of Education Richard Riley urging him to withdraw the report. The letter appeared in the November 18, 1999 edition of the Post and was eventually signed by over two hundred prominent mathematicians and scientists including four Nobel Laureates (one of whom, Steven Chu, has since become Secretary of Energy), three Fields Medalists, a National Medal of Science winner from the University of Chicago, and the some chairs of math departments.”
WHAT WORKS CLEARINGHOUSE: WWC reviewed the evidence in support of the Everyday Mathematics program. Of the 61 pieces of evidence submitted by the publisher, 57 did not meet the WWC minimum standards for scientific evidence, four met evidence standards with reservations, and one of those four showed a statistically significant positive effect.
READINGTON SCHOOL PARENTS: “Today our district is said to be reviewing the program, along with many others, to determine what course our math curriculum should take in the coming years. Some critics believe that this review process is a sham and that the de-facto decision has already been made to forge ahead with an updated version of Everyday Mathematics, a program frequently despised by parents and other community members...Critics say that, whatever the goals of Everyday Math, in practice it is a curriculum which is a mile wide and an inch deep, it is confusing to students who seem to chase their tails finding different computational methods instead of finding the answers, it doesn’t provide enough practice for students to master fact recall, it fails totally in areas like fractions and pre-algebra, and it forces parents or the private tutors they hire to supplement with more traditional methods at home.”
PARENTS: "Everyday Mathematics--popularly known as ‘Chicago Math’ and published by Wright Group/McGraw-Hill...is one of the most pervasive of the original fuzzy, constructivist math programs. It was developed starting in 1985 by a group called the University of Chicago School Mathematics Project (UCSMP). It is legendary for its problems without solutions, incredibly frustrating ‘games,’ shallow interest in effective algorithms, heavy use of the demoralizing practice of spiraling, and oddball methods such as ‘lattice multiplication.’”
TEACHER: “We’d gone through six straight wrong answers, and now the kids were tired of feeling lost. It was only October, and already my fourth-grade public school class in the South Bronx was demoralized. Day after day of going over strange, seemingly disconnected math lessons had squelched my students’ interest in the subject....Then, quietly, 10-year-old David spoke up. ‘Mr. Clavel, no one understands this stuff.’ He looked up at me with a defeated expression....If school officials knew how far my lessons would deviate from the school district-mandated math program in the months ahead, they probably would have fired me on the spot.....But boy, did my kids need a fresh approach.... Since kindergarten, most of them had been taught math using this same dreadful curriculum, called Everyday Mathematics....Unfortunately, a student in a Fuzzy Math program—including Everyday Mathematics—is unlikely to master much of anything. The hours of logically linked lessons that old-style math classes spent on practicing operations so that they became second nature to students just are not there. As one local paper, complaining about Fuzzy Math, put it, ‘Rote learning and the memorization of traditional algorithms appear to have been completely thrown out the window.’
Teachers frustrated by this incoherent approach got little sympathy from school administrators. District officials told us that we should just keep going—even if not a single child in our rooms understood what we were talking about.”
PARENTS: “Where Everyday Math will do real damage is in the communities who don’t have the knowledge or the resources to overcome the shortcomings. And, sadly, who really gets shortchanged here? The kid who might be mathematically gifted but who has a language disability. All kids should have the opportunity to be good at something; these kids can’t even have that.”
PARENT: “If your school uses Everyday Math, you should be extremely watchful. Everyday Math is problematic because it is a language-intensive-based program that stresses the use of calculators; wants the kids to come up with their own ways to solve the problems; doesn't teach the traditional algorithms (the multiplication and division methods that they teach break down when using large numbers, but there is absolutely no reason to be able to compute large numbers nowadays, is there?); and does not advocate drill in any form. Now, this means that some kids lose out: Kids who might have a language problem but would be really good at mathematics, kids who need the ‘rules’ first and then they can come to the concepts (think phonics versus whole language), and kids who need drill in order to retain concepts.”
PARENT: “At first blush my instincts told me that it was not rigorous enough, nor content rich. However, I decided at the time to be deferential to the educators and accepted the program assuming that educators knew more about mathematics education than I. I rue the day that I accepted it without question and that I did not investigate further. My children's days were to be filled with counting on a scroll to 1,000,000 and playing games. Addition and subtraction facts were not emphasized properly. Fractions in the 5th grade were glossed over. These should have been major red flags to me as these skills are all very critical to success in algebra. It should have occurred to me that whoever wrote this program had a significant deficit in understanding or remembering advanced mathematics and in knowing what it takes for children to be successful in advanced mathematics.....Children who are unfortunate enough to receive this curriculum who do not have their education supplemented will be shut out of many careers that depend on mastering the sequential elements of the hierarchy that is math.
PARENT: “I live in an Everyday Math district in Pennsylvania, and while scores on our state test, the PSSA, are rising, our SAT scores are dropping.
....A child could be getting straight As in Everyday Math and still be counting on his fingers. I have heard Everyday Math referred to as ‘Every Night Math’ because every night parents are teaching their children the math they should be learning in school or driving them to tutoring centers to supplement their education.”
PARENT: “Dear Faculty and Staff: I am writing to express my concern about Chicago Math, which is currently being taught in the Troy elementary school system. This program has failed my two daughters, and is not the best choice for our students....I pursued advanced mathematics in high school, and then in college, obtaining a degree from Michigan State University, and another degree from the University of California Berkeley....I believe the awkward algorithms promoted by Chicago Math are inappropriate for most of our students. A small percentage of gifted children may grasp the deeper meaning, the detailed construction of the car's engine, but most will not. Many children are left behind, and cannot perform the simplest arithmetic operations that we take for granted. This is especially true for the kids who are already struggling.”
Reform Math basically drives most of its victims away from any interest in math, and prevents skill at math.
To be successful in math, children need to believe they are in control, that they understand what they’re doing. When children feel that they are winners, they’ll want to go on to the next step. If they don’t have that feeling of victory, they fall by the wayside. Reform Math virtually guarantees that children will not have that confidence.
If you had children who hated Everyday Math, some expert might try to tell you that a different school or better trained teachers could make it work. That’s the big lie. These stories make clear this bogus pedagogy is designed not to work.
The perversity in Reform Math is so great and so glaring. You cannot trust the integrity of the people who come up with such dreck. These professors are from the field of education, not math. They have ideological goals. They think that girls and minorities can’t even learn math, so they needed to create a debased form of math. Having created that debased form of math, they then think they are entitled to force it on everyone, even the students who supposedly can learn regular math.
Reform Math is basically a psycho but one that clearly identifies itself as such (a psycho in psycho’s clothing, we might say). One of the oddball features of New Math/Reform Math was that parents could not understand them and could not help their little third-graders with arithmetic homework. That tells the whole story.
It’s helpful to contrast Everyday Math with Saxon Math, the highly successful books written and published by John Saxon circa 1985. Saxon believed in teaching things slowly, lovingly, with lots of practice and lots of review. The whole point was to make the child feel successful, and be successful. Saxon believed in incremental learning with mastery all along the way. His ideas were the mirror opposite of what Reform Math preaches. Saxon Math took ordinary kids and made them into, if not whizzes, then at least highly accomplished problem solvers. Reform Math destroys all but the most gifted math students.
The American tragedy is that we have ideological extremists who insist on designing flawed methods.
Concerned about STEM? Get rid of Reform Math.
RELATED ARTICLE: "36: The Assault on Math"
Bruce Deitrick Price is an author, artist, and education reformer. He founded Improve-Education.org in 2005. This site has 65 unique articles. Many of them deconstruct the theories and methods often found in the public schools.
For example, there are articles on whole word, reform math, cooperative learning, learning styles, constructivism, prior knowledge, etc. There are also many articles on intellectual subjects such as sophistry, Latin, design, poetry, and grammar.
The site is also concerned with the optimal ways to teach subjects, as discussed for example in "26: How To Teach History, Etc. and the article above.


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The Justice League have had a great start to the year at Caldew Lea. All the children have settled in well and have been working hard since their first day back. Recently we have written Tanka poetry, formal letters (which we plan on sending to the Prime Minister) and have begun work on adventure stories during the past week. In Maths lessons the children have worked very well on both place value and addition and subtraction. Our topic for this term is Ancient Egypt. The children have written their names in hieroglyphics on papyrus which they made themselves and made their own cartouches. We have also learnt about Howard Carter and begun to investigate the Rosetta Stone.
Our PE lessons are on Tuesdays and Fridays this year so please ensure you child has their full PE kit with them on these two days.


Member Since May 2020
Dr. Rothberg has served in senior leadership positions in optimization software companies for more than twenty years. Prior to his role as Gurobi CEO, Dr. Rothberg held the Gurobi COO position since co-founding Gurobi in 2008, and prior to that he led the ILOG CPLEX team. Dr. Edward Rothberg has a BS in Mathematical and Computational Science from Stanford University, and an MS and PhD in Computer Science, also from Stanford University. Dr. Rothberg has published numerous papers in the fields of linear algebra, parallel computing, and mathematical programming. He is one of the world’s leading experts in sparse Cholesky factorization and computational linear, integer, and quadratic programming. He is particularly well known for his work in parallel sparse matrix factorization, and in heuristics for mixed integer programming.
Operations researchers use mathematics to solve business problems.
Aug 9, 2021
Numerous high-profile incidents have shone a spotlight on some of the inherent shortcomings and unintended biases of AI technologies, shaking our confidence in the accuracy of these solutions.
Mathematical optimization and machine learning are two tools that, at first glance, may seem to have a lot in common.
Many businesses today are scrambling to become quantum-ready and looking to lay the foundation for quantum computing in their organizations.
Gurobi produces the world’s fastest and most powerful mathematical optimization solver – the Gurobi Optimizer – which is used by leading global companies across more than 40 different industries to rapidly solve their complex, real-world problems and make automated decisions that optimize their efficiency and profitability. As the market leader in mathematical optimization software, we aim to deliver not only the best solver, but also the best support – so that companies can fully leverage the power of mathematical optimization (on its own or in combination with other AI techniques such as machine learning) to drive optimal business decisions and outcomes. Founded in 2008, Gurobi has operations across the USA, Europe, and Asia and more than 2,400 customers globally including SAP, Air France, Uber, and the National Football League (NFL).


Suppose you have a document with a date and time column. Calculating time differences is easy, you can simply subtract an earlier time from a later one1. In the screenshot below, column C shows the difference between the time in consecutive rows (column B). However, you can see that this simple approach fails just after midnight:
The time difference in row 4 is negative and also wrong. This becomes very clear when you switch to a 24h time format and enable negative times2:
Solution: Include the date in the calculation
Internally, dates are represented as whole numbers and times as fractional numbers smaller than 1 (1 would be 24h = 1 whole day). This means you can simply add date and time! Therefore, a better formula to calculate time differences is:
By including the date, this formula works fine after midnight.
What if you don’t have a date column?
You can use a clever approach I found here:
A3-A2 is the simple time difference calculation we used in the beginning. (A3<A2) returns TRUE only when the next day starts and the time is “smaller” than in the row above (see row 5, column C). What makes this work for our purposes it that TRUE is evaluated as 1 while FALSE is 0 (see column D). As mentioned above, a date/time value of 1 corresponds to 1 day (24h).
In row 5, Excel is therefore calculating the difference between 24h and 23h 59min 59s, which is 1 second.
1 This will work if Excel correctly recognized the value as a time (not a text). You can test this with the ISNUMBER function which should return TRUE.
2 The easiest way to get Excel to show negative times is by enabling the 1904 date system in the advanced options.
3 If you want to see time values of 24h or more, use an elapsed time format with square brackets, e.g. [h]:mm:ss instead of h:mm:ss.


Blume et al. (2017) posted a paper to the ArXiv about a Second generation p-value.
The proposal is essentially as follows.
- Define an interval [a, b] defining a range of estimates that would be considered support for the null. E.g., perhaps a d statistic would have a null interval of -.1, .1, with the explanation that if the true d statistic falls within that interval, the effect is useless/incompatible with substantive hypothesis/etc.
- Compute the statistic and some interval representing the range of values compatible with the data. In frequentist terms, this may be a CI that is constructed such that any values outside of the interval would be rejected IF the null hypothesis were the point estimate. In Bayesian terms, this may be some highest density or quantile interval of the posterior for the estimate.
- Compute the proportion of the statistic-interval that is contained within the null-interval. If the statistic-interval is “too large”, then apply a correction to make the proportion max out at .5.
- The second generation p-value is then the proportion of the “data-consistent interval” that overlaps with the “null-consistent interval”.
Some properties include:
- If H0 is true, such that the true parameter value is within the null interval, then as information increases, the p-value will increase to 1.
- If H1 is true, such that the true parameter is outside of the null interval, then as information increases, the p-value will decrease to 0.
- If an estimate is directly on the null interval border, the p-value will be .5.
- The authors argue that it outperforms in other ways, such as controlling t1 error rates, even compared to FDR/FWER correction procedures.
I like the idea, though it sounds so much like a frequentist has rediscovered the Bayesian ROPE concept — Define a Region of practical equivalence, and use the posterior densities to evaluate the probability of the parameter falling within the ROPE. In fact, it’s pretty much the same thing, with the only difference being that the second-generation p-value uses overlapping intervals, rather than probability mass contained within an interval. The Bayesian ROPE can have a probabilistic interpretation, while the authors argue that the second generation p-value is just a descriptive, a proportion, not a posterior probability or anything similar to that. I respect them for admitting that, rather than trying to shoehorn in a probabilistic interpretation.
One thing I’m curious about, is whether another X-generation p-value could essentially be a marginalized p-value.
This make already exist, in which case I am rediscovering it much like they may have rediscovered the ROPE.
The p-value (for a nil-null hypothesis) is defined as: $p(t > T(y,\theta)|\theta = 0)$. What if instead of defining an interval, we defined the null hypothesis as a density.
E.g., instead of saying $H0: \theta = [-.1,1]$ we said $H0: \theta \sim N(0,.05)$, or something similar.
Then computed: $\int p(t > T(y,\theta)|\theta)p(\theta)d\theta$.
Before any frequentists jump on me for putting a GASP parameter on a distribution: just imagine this is a probabilistic weight, for penalization, rather than some philosophical statement about a-priori plausibility of parameters.
But I think that, too, would have interesting and similar properties.
I may explore that at some point. It’d be conceptually similar to assurance, which is just marginal power when an effect size is uncertain.
The stickiest points, for me, are the use of CIs as the data-consistent intervals — Only certain CIs can be built with the property needed for this to work, and even then they can be problematic. So that leaves one with credible intervals as an alternative, but then you may as well just use the posterior for inference anyway; you’re in Bayes land, just enjoy your time there.
The other issue is in the case where the true hypothesis is, somehow, EXACTLY equal to a null interval boundary, the second generation p-value would asymptotically be .5. I’m not sure that’s bad or unintended, given that it would imply you can’t really choose between the null or the alternative, but it’s a bit annoying to have a singularity of indecision.
Anyway, it may be an interesting metric. I haven’t decided yet whether I like it more than a p-value (but still less than a Bayesian treatment, of course); but I see it being a potentially useful metric nevertheless.


Constraint programming is a relatively new research area that studies methods for solving problems by exploring constraints, which must be satisfied by a solution. In our research, we use a constraint programming framework over finite domains. It uses methods from operational research, mathematics, computational geometry, graph theory, and artificial intelligence to build efficient methods for combinatorial problems. We apply constraint programming to both optimization and satisfiability problems in embedded systems design problems.
Over the last years we have developed our own constraint solver, called JaCoP - Java Constraint Programming library (http://www.jacop.eu). The solver implements state of the art constraints and has been used in many projects, such as identification and selection of processor accelerators, or scheduling and assignment of operations.
Our research is also focused on parallel and distributed constraint solving. Constraint programming is declarative, hence, there are good opportunities for automatic parallelization. Our work so far has dealt with communication schemes, load-balancing methods, and task parallelism in constraint programming. Parallelizing the solving process is one of the latest trends in the constraint programming community.


Includes bibliographical references (p. 211-212).
|LC Classifications||QA107 .S5384 2000|
|The Physical Object|
|Pagination||viii, 212 p. ;|
|Number of Pages||212|
|LC Control Number||00032709|
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Some friends, one of whom is a high school student, asked me to explain the phenomenon that the ratio of successive Fibonacci numbers tends toward the golden ratio. Some of the explanations I have seen involve fairly heavy use of linear algebra; others involve generating functions and the use of some analysis. I wanted to give an explanation that I would have had a prayer of understanding when I was a high school student. I’m not sure whether I have succeeded—there are a great many things I found hard to understand as a high school student (and now as well)—so I would be happy to hear about alternatives or other suggestions. As I see it, there are five main steps in the reasoning. This proof is the one in the Wikipedia article on the Fibonacci numbers, but I have expanded the explanation quite a bit. I highly recommend Peter Cameron’s series of posts about the Fibonacci numbers (see I, II, III, IV, V, and VI). Part 3 of that series is closely related to the topic of this post, but at a somewhat higher level.
Step 1: As is often the case in mathematics, in order to understand a phenomenon, it is helpful to consider a more general phenomenon. The sequence of Fibonacci numbers is just one of infinitely many “Fibonacci-like” sequences that obey the Fibonacci rule . The Fibonacci numbers themselves are generated under the assumptions but if you choose and to be any other random numbers that you like, and then apply the Fibonacci rule to generate and so on, you get another Fibonacci-like sequence.
An example of such a sequence is the Lucas numbers, which are generated from the starting pair . The resulting sequence is
You can try generating some random Fibonacci-like sequences, and you will find, unless you are extremely unlucky (or perhaps I should say lucky), that the ratio of successive terms tends toward the golden ratio, just as ratios of Fibonacci numbers themselves do. What I mean by “extremely unlucky” is that there do exist choices of starting pair for which you get a ratio different from the golden ratio, but these choices are very rare, and you are unlikely to pick one of them by chance.
Step 2: If you multiply every number in a Fibonacci-like sequence by the same value, the resulting sequence is also Fibonacci-like. For example, if you multiply every Fibonacci number by 5, you get the sequence which is Fibonacci-like since it obeys the Fibonacci rule, . Furthermore, if you add two Fibonacci-like sequences together, term-by-term, the resulting sequence is Fibonacci-like. For example, adding the Fibonacci and Lucas sequences gives
which is Fibonacci-like.
Make sure you understand why both of these facts are true. You can, of course, combine the two operations together. For example, you could multiply the Fibonacci numbers by 5 and the Lucas numbers by 12, and then add the resulting sequences term-by-term, and the result will be another Fibonacci-like sequence. In this example you would get
Step 3: Any Fibonacci-like sequence can be expressed as a combination of the Fibonacci sequence and the Lucas sequence. If we let be the Fibonacci sequence and we let be the Lucas sequence, then any other Fibonacci-like sequence equals for some suitable choice of the numbers and
For example, consider the Fibonacci-like sequence with starting pair 4, 9. The sequence this pair produces is How can we express this in terms of Fibonacci and Lucas sequences as The sequence is and the sequence is The sequence is therefore To find and we require that the terms in this sequence match our sequence So we have to solve the simultaneous linear equations
In fact, we can ignore all the equations from the third on, since if the first two equations are satisfied, all the higher ones will automatically hold. (Make sure you understand why.) So we carry out the algebra to find the and that will make the first two equations hold true. You should try this. The answer is You can then verify that if you multiply the Fibonacci numbers by 14, multiply the Lucas numbers by −5, and add the resulting sequences term-by-term, you do indeed get our sequence
Since the same algebraic procedure can be carried out for any Fibonacci-like sequence whatsoever, any Fibonacci-like sequence is a combination of the Fibonacci and Lucas sequences.
There is nothing sacred about using the Fibonacci and Lucas numbers as a basis, however. Any two Fibonacci-like sequences—as long as they are not multiples of each other—can be used as a basis for all other Fibonacci-like sequences. So if and are Fibonacci-like sequences, and is not simply equal to for some number then any Fibonacci-like sequence can be written as
Step 4: To understand why the ratio of successive terms in a Fibonacci-like sequence (usually) tends toward the golden ratio, I will choose the basis sequences and in a very special way. Since we are interested in ratios of successive terms in Fibonacci-like sequences, we ask ourselves, is there a Fibonacci-like sequence for which the ratio of successive terms doesn’t merely tend toward some fixed ratio, but exactly equals the same ratio for all pairs of successive terms in the sequence?
A sequence in which the ratio of successive terms is always the same is called a geometric sequence. If the initial term is and the ratio between successive terms is then the geometric sequence is For example, if the intial term is and the ratio between successive terms is , then the geometric sequence is
Could a Fibonacci-like sequence be a geometric sequence? For this to be the case, we need for the Fibonacci rule, to hold for the terms of the geometric sequence. This means that a system of simultaneous equations must be satisfied:
If the first of these equations holds, then all the others do to, since they are obtained by multiplying both sides of the first equation by or or or a higher power of
So can we find a ratio such that We can get rid of the factor of so we need to solve or The quadratic formula gives two solutions which we will call and
You will recognize as the golden ratio. Since and are the roots of you can write
from which you can conclude that and You can also check these directly, the second one by writing and then rationalizing the denominator.
What this means is that for a Fibonacci-like sequence to be a geometric sequence, the ratio between successive terms has to be either or Choosing the initial term to be 1, the two geometric, Fibonacci-like sequences are
Of course, neither if these is the Fibonacci or Lucas sequence, or any other familiar sequence. The numbers in these sequences are not whole numbers, after all. Rounded to three decimal places, these sequences are
Among all Fibonacci-like sequences, these two sequences are very special: they are geometric. If you multiply either of them by a number, the resulting sequence will still be geometric. But as soon as you start forming other combinations, by adding sequences together, the result will no longer be geometric (although, as discussed in Step 2, it will remain Fibonacci-like).
Step 5: Just as in Step 3, where we expressed Fibonacci-like sequences using the Fibonacci and Lucas sequences as a basis, we can express any Fibonacci-like sequence using the geometric sequences and as a basis, that is, we can write any Fibonacci-like sequence as
As an example, let’s try to express the Lucas sequence in terms of and We need to find numbers and so that
So we need to solve the simultaneous equations
I’ll leave it for you to carry out the algebra, but you should find that This means that the nth Lucas number, which we call equals (Verify this. You should find that and so on.) This is a remarkable formula. The Lucas numbers are whole numbers, whereas and are irrational numbers. The formula says that if you add the nth powers of and together, you always get a whole number.
Now here’s the important point. The nth power of is huge once reaches appreciable size. This is because is a number bigger than 1, and so grows exponentially. On the other hand, the nth power of is tiny since is smaller than 1 in magnitude, which means that decays exponentially. Since is a whole number, but is tiny, we see that will be very close to a whole number once reaches appreciable size.
In other words, although the powers of the golden ratio are irrational numbers, these powers will be very nearly whole numbers once the power gets large. Since the tiny term is alternately positive and negative, the dominant term will be alternately smaller than and bigger than the whole number it approximates.
At last the Fibonacci numbers! We will write the sequence of Fibonacci numbers using the geometric sequences and as a basis. The algebra is similar to the algebra for the Lucas sequence. We require
If you solve for and and rationalize the denominators you should find that
You can also divide numerators and denominators by and you will find that
Hence the nth Fibonacci number, which we call can be written as
Let’s now compute the ratio of the nth and (n−1)st Fibonacci numbers:
Observe that is smaller than 1 in magnitude, so is tiny when gets large. Hence the ratio of successive Fibonacci numbers, which equals times a fraction whose numerator and denominator both get closer and closer to 1 as gets large, approaches as gets large.
Some final remarks. I said that, with some rare exceptions, the ratio of successive terms in any Fibonacci-like sequence—call it —will be close to Let me explain this. Using the sequences and as a basis, so that our sequence is expressed as for some numbers and we see that the nth term in our Fibonacci-like sequence is Take the ratio of the (n+1)st and nth terms:
If is not zero, then numerator and denominator can be divided by to give
which tends to as gets large, since becomes tiny. On the other hand, if is zero, we cannot divide numerator and denominator by In fact, the ratio is simply rather than So this is the special case where the limiting ratio is not the golden ratio. In this situation, the limiting ratio instead is −1 divided by the golden ratio.
We could say that any Fibonacci-like sequence is a mixture of the sequence of powers of and the sequence of powers of If there is no in the mixture, then the ratio of successive terms is simply If there is no in the mixture, then the ratio of successive terms is simply If the mixture contains both and then since gets huge while gets tiny, the contribution can be ignored for large, and so the ratio of successive terms tends toward An “unlucky choice” of initial terms would be or more generally which would generate a Fibonacci-like sequence that is a multiple of and therefore contains no in the mixture.
If you have read this far, you may have noticed that linear algebra did make an appearance after all. The use of the word “basis” is a clue that the idea of a vector space is lurking. More specifically, the Fibonacci-like sequences form a two-dimensional vector space. The focus on Fibonacci-like sequences that are also geometric sequences is a way of sneaking in the matrix diagonalization proof described in Peter Cameron’s third post.


with Carol Essig
Saturday, October 14th 1-3 pm
44 Edgemont Rd., Katonah
Sacred Geometry is the study of symbols and geometric patterns that have been used by the ancients and incorporated in nature since the beginning of time. Each pattern carries a mathematical sequence that aligns it to an energetic frequency. Join Carol as she leads you in the exploration of a very small portion of this vast world. She will discuss the history, meaning, and how we can implement several of these patterns into our daily lives and environment to raise the frequency of the energy around us. This workshop is an introduction to a whole new world for your consciousness.”
REGISTER ONLINE : $40
To find out more about Carol, please feel free to visit her website at : http://www.magentaway.com An ordained interfaith minister of the Order of Melchizedek through the Sanctuary of the Beloved since 1999, Carol is also a Master of Shamballah Multidimensional Healing since 2000 and a Certified Sound Healer.


Office: 345 Gordon Palmer Hall
Each student entering the University takes a mathematics placement examination. Students placed in MATH 005 must complete MATH 005 as a prerequisite for MATH 100. Students placed in MATH 100 must complete MATH 100 as a prerequisite for MATH 110, MATH 112, MATH 115, or any other MA-designated course. A grade of “C-” or higher is required in all prerequisite mathematics courses.
MATH 005 Remedial Mathematics. No credit awarded.
Prerequisite: One unit of high-school mathematics.
Brief review of arithmetic operations followed by intensive drill in basic algebraic concepts: factoring, operations with polynomials and rational expressions, linear equations and word problems, graphing linear equations, simplification of expressions involving radicals or negative exponents, and elementary work with quadratic equations. Grades are reported as pass/fail.
MATH 100 Intermediate Algebra. 3 hours.
Prerequisites: Placement and two units of college-preparatory mathematics; if a student has previously been placed in MATH 005, a grade of “C-” or higher in MATH 005 is required.
Intermediate-level course including work on functions, graphs, linear equations and inequalities, quadratic equations, systems of equations, and operations with exponents and radicals. The solution of word problems is stressed. NOT APPLICABLE to UA Core Curriculum mathematics requirement. Grades are reported as “A,” “B,” “C,” or “NC” (No credit).
MATH 110 Finite Mathematics. 3 hours.
Prerequisites: Placement and two units of college-preparatory mathematics; if a student has previously been placed in MATH 100, a grade of “C-” or higher in MATH 100 is required.
Sets and counting, permutations and combinations, basic probability, conditional probability, matrices and their application to Markov chains, and a brief introduction to statistics. Grades are reported as “A,” “B,” “C,” or “NC” (No credit).
MATH 112 Precalculus Algebra. 3 hours.
Prerequisites: Placement and three units of college-preparatory mathematics; if a student has previously been placed in MATH 100, a grade of “C-” or higher in MATH 100 is required.
A higher-level course emphasizing functions including polynomial functions, rational functions, and the exponential and logarithmic functions. Graphs of these functions are stressed. The course also includes work on equations, inequalities, systems of equations, the binomial theorem, and the complex and rational roots of polynomials. Applications are stressed. Grades are reported as “A,” “B,” “C,” or “NC” (No credit).
Continuation of MATH 112. The course includes study of trigonometric functions, inverse trigonometric functions, trigonometric identities, and trigonometric equations. Complex numbers, De Moivre’s Theorem, polar coordinates, vectors, and other topics in algebra are also addressed, including conic sections, sequences, and series. Grades are reported as “A,” “B,” “C,” or “NC” (No credit).
MATH 115 Precalculus Algebra and Trigonometry. 3 hours.
Prerequisite: Placement and a strong background in college-preparatory mathematics, including one-half unit in trigonometry.
Properties and graphs of exponential, logarithmic, and trigonometric functions are emphasized. Also includes trigonometric identities, polynomial and rational functions, inequalities, systems of equations, vectors, and polar coordinates. Grades are reported as “A,” “B,” “C,” or “NC” (No credit). Degree credit will not be granted for both MATH 115 and MATH 112 or MATH 113.
A brief overview of calculus primarily for students in the Culverhouse College of Commerce and Business Administration. Warning: This course is not satisfactory preparation for curricula requiring standard calculus or higher mathematics, and it is not a prerequisite to calculus or higher mathematics. Includes differentiation and integration of algebraic, exponential, and logarithmic functions, and applications in business and economics. Some work on functions of several variables and Lagrange multipliers is done. L’Hopital’s Rule and multiple integration are included. Only business-related applications are covered. Degree credit will not be granted for both MATH 121 and MATH 125.
This is the first of three courses in the basic calculus sequence. Topics include the limit of a function; the derivative of algebraic, trigonometric, exponential, and logarithmic functions; and the definite integral. Applications of the derivative are covered in detail, including approximations of error using differentials, maxima and minima problems, and curve sketching using calculus. There is also a brief review of selected precalculus topics at the beginning of the course. Degree credit will not be granted for both MATH 121 and MATH 125.
This is the second of three courses in the basic calculus sequence. Topics include vectors and the geometry of space, applications of integration, integration techniques, L’Hopital’s Rule, improper integrals, parametric equations, polar coordinates, conic sections, and infinite series.
Honors sections of MATH 125.
Honors sections of MATH 126.
MATH 208 Mathematics for Elementary School Teachers: Numbers and Operations. 3 hours.
Prerequisites: Elementary education or special education major and grade of “C-” or higher in MATH 100.
Arithmetic of whole numbers and integers, fractions, proportion and ratio, and place value. Class activities initiate investigations underlying mathematical structure in arithmetic processes and include hands-on manipulatives for modeling solutions. Emphasis is on the explanation of the mathematical thought process. Students are required to verbalize explanations and thought processes and to write reflections on assigned readings on the teaching and learning of mathematics.
MATH 209 Mathematics for Elementary School Teachers: Geometry and Measurement. 3 hours.
Prerequisites: Elementary education or special education major and grade of “C-” or higher in MATH 208.
Properties of two- and three-dimensional shapes, rigid motion transformations, similarity, spatial reasoning, and the process and techniques of measurement. Class activities initiate investigations of underlying mathematical structure in the exploration of shape and space. Emphasis is on the explanation of the mathematical thought process. Technology specifically designed to facilitate geometric explorations is integrated throughout the course.
MATH 210 Mathematics for Elementary School Teachers: Data Analysis, Statistics, and Probability. 3 hours.
Prerequisites: Elementary education or special education major and grade of “C-” or higher in MATH 209.
Data analysis, statistics, and probability, including collecting, displaying/representing, exploring, and interpreting data, probability models, and applications. Focus is on statistics for problem solving and decision making, rather than calculation. Class activities deepen the understanding of fundamental issues in learning to work with data Technology specifically designed for data-driven investigations and statistical analysis is integrated throughout the course.
This is the third of three courses in the basic calculus sequence. Topics include vector functions and motion in space, functions of two or more variables and their partial derivatives, applications of partial derivatives (including Lagrange multipliers), quadric surfaces, multiple integration (including Jacobian), line integrals, Green’s Theorem, vector analysis, surface integrals, and Stokes’ Theorem.
Fundamentals of matrices and vectors in Euclidean space. Topics include solving linear systems of equations, matrix algebra, inverses, determinants, eigenvalues and eigenvectors. Also covers the basic notions of span, subspace, linear independence, basis, dimension, linear transformation, range, and null-space. Use of mathematics software is an integral part of the course.
MATH 238 Applied Differential Equations I. 3 hours.
Prerequisite: MATH 227.
Introduction to analytic and numerical methods for solving differential equations. Topics include numerical methods and qualitative behavior of first order equations, analytic techniques for separable and linear equations, applications to population models and motion problems; techniques for solving higher-order linear differential equations with constant coefficients (including undetermined coefficients, reduction of order, and variation of parameters), applications to physical models; the Laplace transform (including initial value problems with discontinuous forcing functions). Use of mathematics software is an integral part of the course.
Honors sections of MATH 227.
A theory-oriented course in which students are expected to understand and prove theorems. Topics include vector spaces and subspaces, linear independence, bases and dimension of vector spaces, solving systems of linear equations, matrices, determinants, linear transformations, eigenvalues, eigenvectors, and diagonalization.
MATH 300 Introduction to Numerical Analysis. 3 hours.
Prerequisites: MATH 227, CS 114 or GES 126, and ability to program in a high-level programming language.
Credit will not be granted for both MATH 300 and MATH 411. A beginning course in numerical analysis. Topics include number representation in various bases, error analysis, location of roots of equations, numerical integration, interpolation and numerical differentiation, systems of linear equations, approximations by spline functions, and approximation methods for first-order ordinary differential equations and for systems of such equations.
An introductory course that primarily covers logic, recursion, induction, modeling, algorithmic thinking, counting techniques, combinatorics, and graph theory.
MATH 303 Contemporary Applied Mathematics. 3 hours.
Prerequisites: CS 110 or CS 114, and MATH 125.
The course is primarily concerned with mathematical models of real-world situations in the physical and social sciences and the professions. It provides excellent background material for middle-school and secondary-school mathematics teachers. Usually offered in the fall semester.
MATH 307 Introduction to the Theory of Numbers. 3 hours.
Prerequisite: MATH 227.
Divisibility theory in the integers, the theory of congruencies, Diophantine equations, Fermat’s theorem and generalizations, and other topics. Usually offered in the spring semester.
Provides background material for middle school and secondary school mathematics teachers. Topics include logic and proof, set theory, mathematical induction, Cartesian products, relations, functions, cardinality, basic concepts of higher algebra, and field properties of real numbers. Usually offered in the fall semester.
MATH 343 Applied Differential Equations II. 3 hours.
Prerequisite: MATH 238.
Continuation of MATH 238. Topics include Laplace Transform methods, series solutions of second-order differential equations, the method of Frobenius, Bessel equations and functions, Fourier series, separation of variable method, elementary boundary valve problem for the Laplace, heat and wave equations, an introduction to Sturm-Liouville boundary valve problems, and phase plane analysis. Usually offered in the fall semester.
The foundations of the theory of probability, laws governing random phenomena, and their practical applications in other fields. Topics include probability spaces, properties of probability set functions, conditional probability, an introduction to combinatorics, discrete random variables, expectation of discrete random variables, Chebyshev’s Inequality, continuous variables and their distribution functions, and special densities.
Topics include inner product spaces, norms, self adjoint and normal operators, orthogonal and unitary operators, orthogonal projections and the spectral theorem, bilinear and quadratic forms, generalized eigenvectors, and Jordan canonical form. Usually offered in the spring semester.
Further study of calculus with emphasis on theory. Topics include limits and continuity of functions of several variables; partial derivatives; transformations and mappings; vector functions and fields; vector differential operators; the derivation of a function of several variables as a linear transformation; Jacobians; orthogonal curvilinear coordinates; multiple integrals; change of variables; line integrals; and Green’s, Stokes’, and Divergence Theorems.
MATH 402 History of Mathematics. 3 hours.
Prerequisite: Permission of the department; background in traditional high-school geometry, algebra, or calculus is recommended.
Survey of the development of some of the central ideas of modern mathematics, with emphasis on the cultural context.
Topics include advanced Euclidean and analytic geometry. The course provides excellent background material for middle school and secondary school mathematics teachers. Usually offered in the fall semester.
MATH 406 Curriculum in Secondary Mathematics. 3 hours.
Prerequisites: Admission to the teacher education program in secondary mathematics, BCT 300, and MATH 227; or permission of the instructor.
Future secondary mathematics teachers examine advanced concepts, structures, and procedures that comprise secondary mathematics.
Further study of matrix theory, emphasizing computational aspects. Topics include direct solution of linear systems, analysis of errors in numerical methods for solving linear systems, least-squares problems, orthogonal and unitary transformations, eigenvalues and eigenvectors, and singular value decomposition. Usually offered in the spring semester.
MATH 411 Introduction to Numerical Analysis (previously MATH 311). 3 hours.
Prerequisites: MATH 238; MATH 237 or MATH 257; CS 114 or GES 126; and ability to program in a high-level programming language.
Credit will not be granted for both MATH 411 and MATH 300. A rigorous introduction to numerical methods, formal definition of algorithms, and error analysis and their implementation on a digital computer. Topics include interpolation, roots, linear equations, integration and differential equations, and orthogonal function approximation. Usually offered in the fall semester.
Quadratic functionals on finite dimensional vector spaces, variational formulation of boundary value problems, the Ritz Galerkin method, the finite-element method, and direct and iterative methods for solving finite-element equations.
A one-semester introduction to both linear and nonlinear programming for undergraduate students and non-math graduate students. Emphasis is on basic concepts and algorithms and the mathematical ideas behind them. Major topics in linear programming include the simplex method, duality, sensitivity, and network problems; major topics in nonlinear programming include optimality conditions, several search algorithms for unconstrained problems, and a brief discussion of constrained problems. In-depth theoretical development and analysis are not included.
In-depth theoretical development and analysis of linear programming. Topics include formulation of linear programs, various simplex methods, duality, sensitivity analysis, transportation and networks, and various geometric concepts.
In-depth theoretical development and analysis of nonlinear programming with emphasis on traditional constrained and unconstrained nonlinear programming methods and an introduction to modern search algorithms.
Topics include the basic no-arbitrage principle, binomial model, time value of money, money market, risky assets such as stocks, portfolio management, forward and future contracts, and interest rates.
MATH 428 Introduction to Optimal Control. 3 hours.
Prerequisite: MATH 238.
Corequisite: MATH 410 or permission of the instructor.
Introduction to the theory and applications of deterministic systems and their controls. Major topics include calculus of variations, the Pontryagin’s maximum principle, dynamic programming, stability, controllability, and numerical aspects of control problems. Usually offered in the fall semester.
Survey of several of the main ideas of general theory with applications to network theory. Topics include oriented and nonoriented linear graphs, spanning trees, branching and connectivity, accessibility, planar graphs, networks and flows, matching, and applications. Usually offered in the fall semester.
Methods of solving the classical second-order linear partial differential equations: Laplace’s equation, the heat equation, and the wave equation, together with appropriate boundary or initial conditions. Usually offered in the fall semester.
MATH 442 Integral Transforms and Asymptotics. 3 hours.
Prerequisite: MATH 441.
Complex variable methods, integral transforms, asymptotic expansions, WBK method, Airy’s equation, matched asymptotics, and boundary layers. Usually offered in the spring semester.
MATH 445 Theoretical Foundations of Fluid Dynamics I. 3 hours.
Prerequisite: MATH 343, AEM 264 or equivalent, or permission of the department.
Introduction to continuum mechanics and tensors. Local fluid motion. Equations governing fluid flow and boundary conditions. Some exact solutions of the Navier-Stokes equations. Vortex motion. Potential flow and aerofoil theory.
MATH 451 Mathematical Statistics with Applications I. 3 hours.
Prerequisites: MATH 237, or MATH 257 and MATH 355.
Introduction to mathematical statistics. Topics include bivariate and multivariate probability distributions, functions of random variables, sampling distributions and the central limit theorem, concepts and properties of point estimators, various methods of point estimation, interval estimation, tests of hypotheses, and Neyman-Pearson Lemma, with some applications. Usually offered in the fall semester.
Further applications of the Neyman-Pearson Lemma, Likelihood Ratio tests, Chi-square test for goodness of fit, estimation and test of hypotheses for linear statistical models, analysis of variance, analysis of enumerative data, and some topics in nonparametric statistics. Usually offered in the spring semester.
MATH 457 Stochastic Processes with Applications I. 3 hours.
Prerequisite: MATH 355 or equivalent.
Introduction to the basic concepts and applications of stochastic processes. Markov chains, continuous-time Markov processes, Poisson and renewal processes, and Brownian motion. Applications of stochastic processes including queueing theory and probabilistic analysis of computational algorithms.
MATH 459 Stochastic Processes with Applications II. 3 hours.
Prerequisite: MATH 457 or equivalent.
Continuation of MATH 457. Advanced topics of stochastic processes including Martingales, Brownian motion and diffusion processes, advanced queueing theory, stochastic simulation, and probabilistic search algorithms (simulated annealing). Usually offered in the fall semester.
MATH 460 Introduction to Differential Geometry. 3 hours.
Prerequisite: MATH 486, or MATH 382 and permission of the department.
Introduction to basic classical notions in differential geometry: curvature, torsion, geodesic curves, geodesic parallelism, differential manifold, tangent space, vector field, Lie derivative, Lie algebra, Lie group, exponential map, and representation of a Lie group. Usually offered in the spring semester.
MATH 465 Introduction to General Topology. 3 hours.
Prerequisite: MATH 486.
Basic notions in topology that can be used in other disciplines in mathematics. Topics include topological spaces, open sets, closed sets, basis for a topology, continuous functions, separation axioms, compactness, connectedness, product spaces, quotient spaces, and metric spaces. Usually offered in the spring semester.
MATH 466 Introduction to Algebraic Topology. 3 hours.
Prerequisites: MATH 465 and MATH 470.
Homotopy, fundamental groups, covering spaces, covering maps, and basic homology theory, including the Eilenberg Steenrod axioms. Usually offered in the fall semester.
This is a second course in axiomatic geometry. Topics include Euclidean and non-Euclidean geometry, studied from an analytic point of view and from the point of view of transformation geometry. Some topics in projective geometry may also be treated. Usually offered in the spring semester.
A first course in abstract algebra. Topics include groups, permutation groups, Cayley’s theorem, finite abelian groups, isomorphism theorems, rings, polynomial rings, ideals, integral domains, and unique factorization domains. Usually offered in the fall semester.
MATH 471 Principles of Modern Algebra II. 3 hours.
Prerequisite: MATH 470.
Introduction to the basic principles of Galois Theory. Topics include rings, polynomial rings, fields, algebraic extensions, normal extensions, and the fundamental theorem of Galois Theory. Usually offered in the spring semester.
Introduction to the rapidly growing area of cryptography, an application of algebra, especially number theory. Usually offered in the fall semester.
MATH 485 Introduction to Complex Calculus. 3 hours.
Prerequisite: MATH 227.
Some basic notions in complex analysis. Topics include analytic functions, complex integration, infinite series, contour integration, and conformal mappings. Usually offered in the spring semester.
Rigorous development of the calculus of real variables. Topics include topology of the real line, sequences, limits, continuity, and differentiation. Usually offered in the fall semester.
Riemann integration, introduction to Reimann-Stieltjes integration, series of constants and convergence tests, sequences and series of functions, uniform convergence, power series, Taylor series, and the Weierstrass Approximation Theorem. Usually offered in the spring semester.
Offered as needed.


There are some great lists out there to follow for Data Science and Technology – Check out a few I have been honored to be listed on – Please feel free to check out everyone on these lists, some really great folks out there!
50 Data Science Gurus (and 10 Organizations) You Must Follow On Twitter ->
PRESENTING: The 100 Most Influential Tech Women On Twitter ->
10 Big Data Pros to Follow on Twitter ->
4 Women Leading the Way in Business Intelligence ->
Top 200 Thought Leaders in Big Data & Analytics
Top 5 Data Science Gals
Big Data: Experts to Follow on Twitter
I appreciate all the mentions and inclusions to these lists and WOW, I’m in great company!
Lastly, this is not a list but some advice -> “An understanding of math is important,” he says, “but equally important is understanding the research. Understanding why you are using a particular type of math is more important than understanding the math itself.”
Have fun with DATA 🙂


Institute Name: Science Maths Classes.
Telephone Number: 7891931771
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Location: Address - Chitrakoot Scheme Akshardham Chauraha , Vidhyut Nagar , Chitrakut, District/City - Jaipur, State - None, Region - Not known.
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|Name of Institute||:||Science Maths Classes|
|Institution type 1||:||FM Cable Education|
|Institution type 2||:||Not known|
|Status||:||Government Colleges & Universities|
|Address||:||Chitrakoot Scheme Akshardham Chauraha , Vidhyut Nagar , Chitrakut|
|Affiliation Type||:||Not known|
Science Maths Classes Email and Website and Phone Number
The email, website, and phone number of Science Maths Classes are
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- Phone: 7891931771
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The School is the largest School and it comprises six departments, namely: Department of Biology, Department of Chemistry, Department of Computer Science, Department of Integrated Science, Department of Mathematics, and Department of Physics.
Students are exposed to the scientific principles and techniques during classroom teaching, practical exercises and field trips as means of demonstrating academic and professional excellence in teaching science.
Philosophy of the School
The philosophy of the School is to:
- produce competent, professional and effective teachers of science education who will be able to develop in students an appreciation and understanding of Mathematics and Science.
- impart the intrinsic values as well as the utility values of Mathematics and Science in all spheres of human activities.
- develop in the students the ability to formulate problems, identify variables and design experiments, interpret results, recognize patterns, generate hypothesis, draw conclusions and develop theoretical models.
- develop academic and professional competency which will enable individual earn a living.
- help students become intellectually informed in Mathematical ideas, notations and skills for logical reasoning.
To realise the stated philosophy above, the School has the following objectives:
(i) to equip the students to be able to demonstrate practical skills in handling scientific apparatus.
(ii) to equip the student to be able to demonstrate excellence and professional competence in the teaching of Mathematics and Science subjects.
(iii) to develop an individual who will make a successful carrier in Mathematics and Science teaching.


The Indian Certificate of Secondary Education or ICSE is a private board for school education in India. 7th is considered to be the foundational class which consists of basic core topics. It covers a range of topics that are essential for students to master before moving on to higher-level mathematics. These topics are important for future classes as many of these topics in the ICSE maths syllabus for class 7 are repeated in detail in higher classes.
The syllabus is divided into six major parts: Number System, Commercial Mathematics, Algebra, Geometry, Mensuration, and Data Handling.
Detailed Syllabus of Class 7 Maths ICSE
Chapter 1: Number System
Numbers are the basic concept in mathematics. There are various types of numbers used by humans to cater to their various needs and applications. The different types of numbers that you will learn in this chapter of the ICSE maths syllabus for class 7 are natural numbers, even & odd numbers, whole numbers, integers, and rational & irrational numbers.
- Natural Numbers
- Even & Odd Numbers
- Whole Numbers
- Prime & Composite Numbers
- Rational Numbers
- Irrational Numbers
- Number Line
- Face Value & Place Value of a Number
- Fundamental Operation of Numbers – Addition
- Fundamental Operation of Numbers – Subtraction
- Fundamental Operation of Numbers – Multiplication
- Fundamental Operation of Numbers – Division
Chapter 2: Power and Roots
There are many quantities or measurements that need very large numbers such as the mass of the Earth and the Sun. In such cases, we use a special type of number called exponents. This chapter introduces you to exponents and roots.
- Properties of Exponent
- Square of a Number
- Cube of a Number
- Square Root – Prime Factor Method
- Square Root – Prime Division Method
- Perfect Square
- Cube Root – Prime Factor Method
- Cube Root – Using Index Form
Chapter 3: HCF and LCM
There are situations where we want to split or group a number of items into smaller groups or form bigger groups and want to know the number of new groups formed. To solve such types of problems, we use the concept of HCF and LCM. In this chapter of the ICSE maths syllabus for class 7, you will learn about HCF and LCM and different methods of finding them.
- Factors of a Number
- Common Factors
- Prime Factors
- Finding HCF – Common Factor Method
- Finding HCF – Prime Factor Method
- Finding HCF – Division Method
- Multiples of a Number
- Common Multiples
- Finding LCM – Prime Factors Method
- Finding LCM – Common Division Method
- Product of HCF and LCM
Chapter 4: Fraction
Fractions are special types of numbers that represent a part of a whole. This chapter explains the various types of fractions and the basic operations that can be performed on them.
- Classification of Fractions – Vulgar Fractions
- Classification of Fractions – Decimal Fractions
- Classification of Fractions – Proper Fractions
- Classification of Fractions – Improper Fractions
- Classification of Fractions – Mixed Fractions
- Reducing Fractions to Lowest Form
- Equivalent Fractions
- Like and Unlike Fractions
- Comparing Fractions
- Inserting Fractions Between Two Fractions
- Operations on Fractions – Addition
- Operations on Fractions – Subtraction
- Operations on Fractions – Multiplication
- Operations on Fractions – Division
- Using “Of”
- Using BODMAS
- Using Brackets – Removal & Insertion of Brackets
Chapter 5: Decimal Fractions
Decimals are also used to represent part of a whole and are similar to fractions. In this chapter, you will learn about decimals, their similarities with fractions, and the basic operations that are performed on decimals.
- Reading Decimal Numbers
- Converting Decimal Numbers to Fractions
- Decimal Places
- Addition of Decimal Numbers
- Subtraction of Decimal Numbers
- Multiplication of Decimal Numbers
- Division of Decimal Numbers
- Types of Decimal Numbers – Terminating, Non-Terminating But Recurring & Non-Terminating and Recurring
- Rounding Off Decimal Numbers
- Significant Digits
Chapter 6: Unitary Method
The unitary method is a very simple and easy way of solving problems on variations. In this chapter of the ICSE maths syllabus for class 7, you will understand the use of the unitary method in solving problems on direct and inverse variations.
- Unitary Method
- Direct Variation
- Inverse Variation
- Time and Work
Chapter 7: Speed, Distance, and Time
Speed is a term associated with an object in motion. You know that a moving car has a speed. This chapter explains the concept and relation between speed, distance covered, and time taken by a moving object. In this chapter, you will also learn two types of speeds – uniform speed and variable speed.
- Uniform Speed and Variable Speed
- Converting Speed From One Unit to Other
Chapter 8: Ratio and Proportion
Ratios and proportions are used when we distribute items among many people. This chapter explains the terms ratios and proportions and the relation between these two. You will also learn about a type of proportion called continued proportion that involves three or more quantities.
Chapter 9: Percent and Percentage
A percentage is a term used in many areas. The mark sheets of students show the performance in terms of percentage. In this chapter of the ICSE maths syllabus for class 7, you will learn the meaning of percentages and how to solve problems based on percentages.
- Expressing Percentage
- Finding Percentage of a Quantity
- Percent Change
Chapter 10: Profit, Loss, and Discount
When you visit a mall or a shop, you come across the term called ‘discount’. In this chapter, you will understand the meaning of discount and how it is calculated along with the two other important and related terms – profit and loss.
Chapter 11: Simple Interest
Whenever someone deposits money in a bank, the bank pays back an extra amount along with the money deposited. This extra amount received by a person is called interest. In this chapter, you will learn the meaning of simple interest and how it is computed.
Chapter 12: Averages
An average of a list of data is the expression of the central value of a set of data. It is also called the mean of the data set. In this chapter, you will learn the process of calculating an average of a list of numbers.
- Finding Average
Chapter 13: Fundamental Concepts
The expressions formed using variables and constants are known as algebraic expressions. In this chapter, you will learn about algebraic expressions, their types, and the four basic operations that can be performed on algebraic expressions.
- Algebraic Expressions and Related Terms
- Types of Algebraic Expressions
- Addition of Algebraic Expressions
- Subtraction of Algebraic Expressions
- Multiplication of Algebraic Expressions
- Division of Algebraic Expressions
Chapter 14: Products
Algebraic identities are expressions that are used to solve algebraic expressions quickly and easily. There are many algebraic identities that are used to make our computations easy and simple. Learn about these identities in this chapter.
- Algebraic Identities
Chapter 15: Simplifications
While solving algebraic expressions, we generally group and regroup the component terms and expressions to make our calculations easy. This chapter explains the ways of simplifying algebraic expressions by using brackets.
- Simplifying Algebraic Expressions
- Using Brackets
- Inserting and Removing Brackets
Chapter 16: Equations and Inequations
Two or more algebraic expressions separated by an equal sign are called equations. When this equal-to sign is replaced by some other sign such as less than or greater than it becomes an inequation. In this chapter, you will understand the difference between equations and inequations and how they are solved.
- Solving Equations – Transpose Method
- Solving Equations With Variables on Both Sides
- Equations Involving Fractions
- Replacement Set and Solution Set
- Properties of Inequations
- Graphical Representation of Inequations
Chapter 17: Formula: Change of Subject of a Formula
In mathematics, we use formulas to compute the value of many things like speed, area, volume, etc. In this chapter of the ICSE maths syllabus for class 7, you will learn how to formulate formulas for a given situation and how to change them depending on the requirement of a problem.
- Changing the Subject of a Formula
Chapter 18: Problems Based on Simple Equations
Many real-world problems such as the ones based on numbers, ages, and geometry are solved using simple equations. In this chapter, you will learn how to formulate and solve an equation for a given problem.
- Problems Involving Numbers
- Problems Involving Ages
- Miscellaneous Problems
Chapter 19: Factorization
You know the terms factors and multiples of a number. Similarly, algebraic expressions can also be decomposed into smaller expressions known as factors. The process of finding the factors of an algebraic expression is called factorization. Learn about the different methods of factorizing algebraic expressions.
- Factors and Factorization
- Types of Factorization – Taking Out a Common Factor
- Types of Factorization – Grouping
- Types of Factorization – Difference of Two Squares
Chapter 20: Graphs
Coordinate geometry is used to represent a point in a plane using two values – x and y. In this chapter, you will learn how to represent a point in a plane using x and y coordinates and also to draw linear graphs in a Cartesian plane.
- Representation of a Point in a Plane
- Draw a Linear Graph in Two Variables
Chapter 21: Relations and Mapping
Mapping is any prescribed way of assigning to each object in one set a particular object in another (or the same) set. Mapping applies to any set: a collection of objects, such as all whole numbers, all the points on a line, or all those inside a circle. In this chapter, you learn about the meaning of relations and mapping and how they are used.
- Ordered Pair
- Representing Relation
- Domain and Range of a Relation
Chapter 22: Indices
An index, or power, is a small floating number that appears after a number or letter. The plural of an index is indices. Indices show how many times a number or letter has been multiplied by itself. This chapter explains the meaning of indices and the laws associated with it which are used to solve problems based on indices.
- Indices or Exponent
- Laws of Indices
Chapter 23: Lines and Angles
This chapter of the ICSE maths syllabus for class 7, deals with basic terms, types of lines, types of angles, and the construction of angles. In the solutions, students are provided with the steps to be followed in the construction of various structures.
- Basic Terms
- Measurement of an Angle
- Types of Angles
- Pairs of Angles
- Parallel Lines and Transversal
- Construction of Angles
Chapter 24: Triangles
This chapter of the ICSE Maths Syllabus for Class 7 helps you understand the types of triangles based on the angles and lengths of sides. You will also learn about the important properties of triangles and how to construct triangles based on given measurements.
- Interior and Exterior Angles
- Classification of Triangles Based on Angles
- Classification of Triangles Based on Sides
- Properties of Triangles
- Construction of Triangles
Chapter 25: Symmetry
A geometric figure is symmetric if the line of symmetry divides it into two identical parts. In this chapter, you will learn about symmetric figures and the line of symmetry required for reflection and rotation.
- Line Symmetry
- Rotational Symmetry
Chapter 26: Congruent Triangles
The word “congruent” means “equal”. In this chapter, you will learn the meaning of congruency with respect to triangles and the different rules or criteria for proving the congruency of triangles.
- Meaning of Congruency
- Congruency of Triangles – Corresponding Sides and Angles
- Conditions of Congruency – SSS
- Conditions of Congruency – SAS
- Conditions of Congruency – ASA
- Conditions of Congruency – RHS
Chapter 27: Polygons
A polygon is a closed plane figure bounded by at least three line segments. This chapter of the ICSE Maths Syllabus for Class 7 explains the various types of polygons and the important property of polygons related to their interior angles.
- Types of Polygons
- Sum of Interior Angles of a Polygon
Chapter 28: Construction of Quadrilaterals
Quadrilaterals are a type of polygon with four sides, four vertices, and four angles. This chapter explains the step-by-step process of constructing different types of quadrilaterals.
- Construction of Quadrilaterals
- Construction of Parallelograms
- Construction of Rectangles
- Construction of Squares
- Construction of Rhombus
Chapter 29: Introduction to Theorem
In mathematics, a theorem is a statement that has been proved or can be proved. In this chapter of the ICSE maths syllabus for class 7, you will understand the meaning of a theorem and how it is proved. You will also understand the difference between two related terms – axiom and postulate.
- Axiom, Postulate, and Theorem
- Proving Theorems
Chapter 30: Circles
A circle is a basic 2D shape that is measured in terms of its radius. The circles divide the plane into two regions – interior and exterior. In this chapter of the ICSE Maths Syllabus for Class 7, you will learn about a circle and the terms related to it.
- Describing a Circle – Related Terms
Chapter 31: Perimeter and Area
Perimeter is the total length of all the sides of a polygon taken together, whereas area is the region occupied by a polygon in a 2D plane. Learn about finding the perimeter and area of two basic 2D figures – rectangle and square.
Chapter 32: Volume and Surface Area
Surface area is the sum of areas of all the faces of a solid shape and volume is the space occupied by a 3D object in space. This chapter of the ICSE maths syllabus for class 7, explains the methods of finding volume and surface areas of two basic 3D shapes – cuboid and cube.
- Surface Area of Cuboid
- Surface Area of Cube
- Volume of Cuboid
- Volume of Cube
- Area of Four Walls of a Room
Chapter 33: Collection and Tabulation of Data
Data is a collection of values used to convey some meaning. Learn about data and its different types and about frequency distribution. In this chapter of the ICSE Maths Syllabus for Class 7, you will also learn how to represent data graphically using a pie graph, bar graph, or a line graph/
- Data and its Types
- Frequency Distribution
- Graphical Representation of Data – Pie Graph
- Graphical Representation of Data – Bar Graph
- Graphical Representation of Data – Line Graph
- Arithmetic Mean
Chapter 34: Set Concepts
A collection of related objects is called a set. For example, the collection of books in your school library can be called a set of books. In this chapter, you will learn the meaning of a set and how it is represented. You will also learn about the different types of sets.
- Sets and Related Terms
- Representation of a Set
- Types of Sets
Chapter 35: Subsets and Universal Set
The smaller sets that can be formed from large sets are known as subsets. In his chapter of the ICSE maths syllabus for class 7, you will learn about sets and related terms. You will also learn the different operations that are performed on sets.
- Universal Set
- Complement of a Set
- Operations on Sets – Union
- Operations on Sets – Intersection
- Operations on Sets – Difference
Chapter 36: Venn Diagrams
Venn diagrams are the visual representation of sets. In this chapter, you will learn how to represent sets using Venn diagrams.
- Venn Diagrams
Best Maths Reference Books For Class 7 ICSE
If you are looking for the best Maths reference books for Class 7 ICSE, then this list is for you. Here we have listed some of the best Maths books that will help you score high marks in your exams. These books cover the entire maths 7 class maths syllabus and are a great resource for students who want to excel in this subject. In this list, we will recommend three of the best math reference books for 7th grade students who are following the ICSE curriculum.
- The first book on our list is “Concise Mathematics Middle School for Class 7 – Examination 2023-24” by R.K. Bansal. This is very popular among students and teachers alike. If you’re a student of class 7 ICSE, it’s an essential book to practice maths. Check Price
- “Essential ICSE Mathematics for Class 7” by A.D. Gupta & A. Kumar of Bharati Bhavan publications is also one of the best books for students of ICSE class 7. The subject contained in this book is explained in easy language and covers many examples from real-life situations. Check Price
- Another good reference book for ICSE class 7 students is “ICSE Guided Mathematics Class 7” by Abhijit Mukherjee of Oxford Publications. It is also one of the best books for students of ICSE class 7 as it covers all the topics in detail and also provides solved examples to help students understand the concepts better. Check Price
In conclusion, the ICSE Class 7 Maths Syllabus is designed to challenge and improve students’ understanding of the subject. The coursework is demanding but ultimately rewarding, and students who complete it will be well-prepared for further study in mathematics.
What are the topics in the class 7 ICSE maths syllabus?
The class 7 ICSE maths syllabus is divided into six major parts: Number System, Commercial Mathematics, Algebra, Geometry, Mensuration, and Data Handling.
Is ICSE harder than CBSE?
ICSE Syllabus is considered to be more difficult than CBSE. Because the ICSE syllabus has more internal assessments and concepts as compared to CBSE.


A couple of weeks ago, I noted that there was a distinct risk that Celtic would win the league at the next Old Firm match at Ibrox in a couple of weeks. Under the circumstances, I felt that this was extremely worrying, and that it would be better if they could win the league absolutely anywhere else. (That they will win the league isn't really in question - it's just a question of where and when.)
Fortunately, Celtic dropped points at the weekend, and although Rangers lost their match the mathematics has changed.
If Celtic win the Old Firm match, that will now place them on 78 points. If Rangers win all of their other matches, they could (in theory) also reach 78 points. Furthermore, if Motherwell win all of their remaining matches, they too can reach 78 points.
In order for Celtic to win the league at Ibrox, then, they need Rangers to first drop points at Dundee United, and for Motherwell to drop points against either Aberdeen or Kilmarnock. (Plus, of course, they would need to win the match at Ibrox.) Granted, that's not a hugely unlikely sequence of results, but it seems much less likely than the previous "win all your games, and expect Rangers won't win all of theirs" - this now depends on two teams dropping points rather than one, and is almost entirely out of Celtic's hands.
Of course, there's still virtually no doubt that Celtic will win the league - any two wins and a draw in their remaining games will guarantee that. It's just now more likely that they'll do so at Celtic park on 1st April (against St Johnstone) than at Ibrox. And that's much less of a powder keg.
All in all, it's a bit of a relief.


The latlongdistance( function calculates the distance between two points on the earth (or any other sphere).
This function has five parameters:
latitude1 – the latitude of the first point, expressed in decimal degrees (for example 34.1614).
longitude1 – the longitude of the first point, expressed in decimal degrees (for example –118.0054).
latitude2 – the latitude of the second point.
longitude2 – the longitude of the second point.
units/radius – the units for this calculation. This parameter is optional – if omitted, the default is miles. The distance can be calulated in miles, kilometers or nautical miles (and the abbreviations mile, kilo, km and nm are also accepted). This parameter can also be used to spedify the radius of the earth or any other sphere.
This function calculates the distance between two points on the earth (or any other sphere). The points
35,-118 are about 69 miles apart.
latlongdistance(34,-118,35,-118) ☞ 69.097585
The distance is normally calculated in miles, but you can also specify kilometers or nautical miles.
latlongdistance(34,-118,35,-118,"miles") ☞ 69.097585 latlongdistance(34,-118,35,-118,"kilometers") ☞ 111.194927 latlongdistance(34,-118,35,-118,"nm") ☞ 60.039326.
The final parameter can also be used to specify the radius of the earch (or other sphere). In that case, the units can be whatever you want – the results will be in the same units. This example calculates a distance between two objects on the earth in meters (as noted below, this calculation isn’t really accurate to the meter, much less a fraction of a millimeter).
latlongdistance(34,-118,35,-118,6378100) ☞ 111318.845021
This example calculates a distance on the moon in kilometers (the radius of the moon is 1,737 km).
latlongdistance(-118,34,-118,35,1737) ☞ 30.316369
This function uses the Haversine formula to calculate distances, which is well conditioned for all distances except for those close to 180° (see Great Circle Distance). The function does assume that the Earth is a perfect sphere.
|10.0||New||New in this version.|


Arithmate Pro Calculator Alternatives
Arithmate Pro Calculator is described as 'A modern, scientific calculator that is easy to use' and is an app in the Education & Reference category. There are more than 50 alternatives to Arithmate Pro Calculator for a variety of platforms, including Android, Windows, Mac, iPhone and iPad. The best alternative is SpeedCrunch, which is both free and Open Source. Other great apps like Arithmate Pro Calculator are Qalculate! (Free, Open Source), Windows Calculator (Free, Open Source), Soulver (Paid) and Calculator (Free).
- 72 alternatives
- RealCalc Scientific CalculatorAndroid's #1 Scientific Calculator. A fully featured scientific calculator which looks and operates like the real thing. View 100 alternatives to RealCalc Scientific Calculator
- NumWorks designed an intuitive and evolutive graphing calculator to make learning maths easier. View 109 alternatives to NumWorks Graphing Calculator
- Free • Open Source


Image puzzles often go viral on social media to boggle our minds.
Today we are here with one such picture puzzle that apparently features a panda toy but we tell you that it also has a real panda that you have to figure out.
So have you found the real panda in the picture? If not, we will let you know.
The correct answer is in the picture below.
Can you solve this 150 year old math puzzle?
A question raised on social media has left people confused, which is a simple math problem.
This question is almost 150 years old, recently presented by a YouTube channel in its video, this question was also part of the MIT entrance exam in 1876.
It is true that this question has been solved by many brilliant students but you don’t need much math skills to solve it.
Also read: Which of these four cups will be filled with coffee first?
Question was A father said to his son, 2 years ago I was 3 times older than you, but after 14 years I am twice as old as you, then how old will both of them be?
Now you try to think what the answer might be.
If you can’t solve it, you can see in this video that it can be solved using algebra.
So, you must have seen the answer, but here we are writing it as, father is 50 years old and son is 18 years old.
2 years ago father is 48 and son is 16, so father is 3 times older than son.
Similarly, after 14 years father will be 64 and son will be 32, then this difference will double.


By N. L. Carothers
This brief path on classical Banach house idea is a traditional follow-up to a primary path on sensible research. the themes coated have confirmed important in lots of modern learn arenas, similar to harmonic research, the idea of frames and wavelets, sign processing, economics, and physics. The e-book is meant to be used in a sophisticated issues path or seminar, or for self sustaining examine. It deals a extra hassle-free advent than are available within the present literature and comprises references to expository articles and proposals for additional examining.
Read or Download A Short Course on Banach Space Theory PDF
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Extra resources for A Short Course on Banach Space Theory
That is, T maps bounded sets into compact sets. 8 actually shows something more: T fails to be an isomorphism on any in nite dimensional subspace of `r . " Or, in still other words, each in nite 54 CHAPTER 5. BASES IN BANACH SPACES III dimensional subspace of `r contains an isomorphic copy of `r and the restriction of T to this copy of `r cannot be an isomorphism. A bounded linear map T : X ! Y which fails to be an isomorphism on any subspace of X is said to be strictly singular . ) Thus, every map T : `r !
Show that (X Y )1 = (X Y )1, isometrically. 8. Prove that N can be written as the union of in nitely many pairwise disjoint in nite subsets. 9. Find a \natural" copy of (`p `p )p in Lp(R). 10. If (Xn ) is a sequence of Banach spaces, prove that (X1 X2 )p is a Banach space for any 1 p 1. 11. 10 requires that ((Y Z ) (Y Z ) )p (Z Z )p (Y Y )p. Verify this claim. 12. 10 in the case X = c0. Chapter 6 Special Properties of c0, `1, and `1 The spaces c0, `1 , `2, and `1 play very special roles in Banach space theory.
Show \by hand" that (xn) is not equivalent to (en) in `p, for 1 p < 1, but that (xn) is equivalent to (en) in c0. What can you say about the block basis yn = xn=n? 2. 6. 3. 7. 4. If T : c0 ! `p, 1 p < 1, is bounded and linear, show that kTenkp ! 0. 5. Show that every bounded linear map T : `r ! `p , 1 p < r < 1, or T : c0 ! `p, 1 p < 1, is compact. ] 6. Show that every bounded linear map T : c0 ! `r or T : `r ! c0, 1 r < 1, is strictly singular. 7. Show that (X Y )1 = (X Y )1, isometrically. 8.
A Short Course on Banach Space Theory by N. L. Carothers


LA&S Mathematics & Science AS62-64
This curriculum is designed to serve the interests of those students with goals and strengths in the mathematics and science fields while broadening their knowledge in allied disciplines and clarifying career objectives. In collaboration with a faculty advisor, students can plan a program of study that will prepare them to transfer to a baccalaureate program. Those areas of study available: Biology & Health Science: Life Science Department Chemistry: Engineering, Computer & Physical Sciences Department Geology: Life Science Department Mathematics: Mathematics Department Physical Education/Science: Life Science Department Sports Medicine: Physical Education Department Physics: Engineering, Computer & Physical Sciences Department Pre-Environmental Science & Forestry: Life Science Department
- Graduates of the program transfer to a 4-year institution in mathematics or related field
- Students will demonstrate awareness of mathematical career opportunities.
- Graduates have completed at least 8 of the 10 SUNY silos of General Education.
- Students of the program will demonstrate the ability to interpret and communicate mathematics orally.
- Students of the program will demonstrate the ability to interpret and communicate mathematics in writing.
- Students will communicate results of scientific inquiry.
- Students will communicate information and ideas clearly in written form using correct structure, grammar, spelling and organization.
- Students will demonstrate an ability to write proofs using rigorous mathematical reasoning.
- Students will demonstrate an ability to solve word problems using rigorous mathematical reasoning.
- Students will be able to work effectively within a group by demonstrating openness toward diverse points of view, drawing upon knowledge and experience of others to function as a group member, demonstrating skill in negotiating differences and working toward solutions.
- Graduates who entered into the program beginning fall 2008 will complete the Diversity and Global View requirement.
- Students will be able to state a problem correctly, reason analytically to a solution and interpret the results.
- Students will demonstrate the ability to solve equations using rigorous mathematical reasoning.
- Students will be able to solve application problems from numerical, graphical and/or analytical perspectives.
- Students will be able to demonstrate an understanding of how mathematics can be used to analyze real world situations.
- Students will be able to use technology appropriately to solve problems.
- Students will be able to collect and/or analyze data using the appropriate technology.
- Students will use traditional and contemporary information technology.
- Students will identify, access, and appropriately use authoritative sources of information.


1st Grade STEMInstructor: Mrs. Althoen
Welcome to STEM! Together we will discover more about Science, Technology, Engineering and Mathematics. Not only will you learn more about these subjects and how they are interconnected, STEM is designed to challenge you! It helps you develop critical thinking and problem solving skills, logical reasoning, and requires collaborating with others to apply appropriate technology.
STEM is a fun and messy world!
- Mrs. Althoen
"The future belongs to the curious. The ones who are not afraid to try it, explore it, poke at it, question it and turn it inside out."
Class Web Pages
- Mrs. Althoen's Fairy Tales Symbaloo
- Mrs. Althoen's Fall Fun Symbaloo
- Mrs. Althoen's Math Symbaloo
- Mrs. Althoen's Read Aloud Books Symbaloo
- Mrs. Althoen's Reading & Writing Symbaloo
- Mrs. Althoen's STEM Hurricanes Symbaloo
- Mrs. Althoen's Spring Symbaloo
- Mrs. Althoen's Winter Symbaloo
- Mrs.Althoen's Presidents' Day Symbaloo