Math Fluency – Reasoning and understanding


According to the Merriam Webster Dictionary, to be fluent is “to be able to move with ease and grace, effortlessly smooth and flowing, and show mastery subject or skill.” It could also be defined as the ability to use mathematics accurately, efficiently, and effortlessly.

What is math fluency?

Math full means knowing the correct method to correctly solve mathematical problem with ease and in a relatively short time. It is also to be satisfied with different methods to solve Giving problem. The agile man will be in mathematics, the better and smoother, he / she will exactly apply mathematical concepts to real situations. Achieving good fluency in mathematics requires proper, meaningful and consistent practice. Knowing how to apply the formula to solve math problems is not necessarily one is good at math. It takes much more than applying a mathematical formula to be considered good. Research shows that being fluent in math, children should be able to accurately answer math questions within 2 seconds.

Reasoning and understanding the concept of

a student fluent if he / she is able to:

  • Explain solution process step by step and explain the reason for each step.
  • explain the formula and how the formula was.
  • Use alternate methods to solve the problem, if it exists.

Building Blocks

It is very important to fully understand the basic building blocks of mathematics to achieve mastery of the material. One should have mastered the concepts add, drag, multiplication and division before moving on to higher math scores. While understands how to perform calculations and has some fluency in math facts, he / she still needs constant work. Understanding is essential for math fluency, but not enough; practice and repetition is what creates perfection.

If you are able to establish a link between mathematical operations and build on them as you go, you should be able to master the material with ease.


Police Exam Sample Math Question – Get Police Practice Test math questions


There are 8 main components of the civil service police exam. One of the eight seems to be a problem for most applicants math section. Do not get me wrong some newcomers navigate through this section without a hitch, but many fail miserably. If you are not strong in math, I recommend sites like to help you brush up.

The good thing is you can find various sites on the internet to improve math. Now if you decide to invest in police exam study guide you will get the job done the math questions as well as a thorough review of other 7 part of the police test.

Here is a sample police exam math problem, Sample Math Question :.

Members traffic enforcement department issued the following quotes in a recent six-day period

Officer Millett out 33 citations

Officer Rodgers out 108 citations.

Officer Smith issued 40 citations.

Officer Fixing issued 24 citations.

How many citations were issued two traffic enforcement officers write the most citations for this season?

(A) 143

(B) 75

(C) 134

(D) 148

answer D. is 108 + 40 = 148

The mathematics part of the police officer exam is not very difficult, but it can be if you do not follow instructions. The real reason why some candidates not this section is not because they can not add or subtract. Applicants fail because of not following the instructions properly. That is why it is so important to read the instructions carefully before choosing your response.


Algebra for Beginners – Tips To Become An Algebra Star


Many people are kind of weak in the area of ​​algebra mathematics. A lot of themself read books, attend lessons and research the web to find information, interactive lessons and websites That could help Them Improve Their algebra skills. Interactive learning (eg online courses) is far more fun than reading and Entertaining algebra books. But Can interactive learning alone help you become an algebra star?

In order to master mathematics you have to master the algebra concepts. Learn the basics of algebra efficiently, move on to more complicated issues Lear theme. Then move forward to, even more complex issues. And so on.

If you do These steps over and over, your brain will start to adjustability and you Will like it. In fact, if you successfully Learn a few algebra concept then you’ll want to have more.

Practise is really importand in algebra. You Can not expect to Learn everything from online courses or turorials. You need to go through the concepts and the principles of algebra again and again in order to Learn Them.

An Effective method is to take your algebra books and write down all the Equations and concepts in a sheet of paper. Keep Each concept on a differentiable sheet. Take notes and write down your opinion for the every equation or algebra concept. Keep this notebook in your pocket and continue filling it with notes. You may not like it at first, but the more you Practise and use this personal algebra notebook the better for you.

Do not be hasty. Do not rush. Relax. Learning Algebra requires time. Sit down, try to figure out the concepts. You are not alone. We’ve all been there. Well Done! You’ve now created your own Algebra Notebook. In case you did not know this is really importand.

Here are some more tips:

– There are thousands of books out there Algebra That break down algebra to Easiest its components. Not all books are good but some are really exceptional. (Tip: Go to college book stores. They definitely have algebra texts or books That You Can Borrow or takeaway). Do not stick with one book or lesson. Many books explain some algebra concepts better than other. Choose a variety of books or online courses.

– You Can get a live tutor. Eveeybody needs a tutor from time to time! It Will Be Much Easier to Learn from someone who ‘s Already familiar with the concepts of algebra. Some schools offer no cost Tutors to help you through the basics of algebra. Just make sure you get a Skilled one.

– Join a website and go through some online courses. You will love it! There are online communities where people discussants Their algebra problems and help eachother. There are a lot of websites Offering free online math lessons. Join Them. Have some fun.

Even if you are a member of one the Many popular interactive algebra communities you still need to study and Practise a lot. You Can combine studying and practicing with interactive lessons-which is fun. Interactive courses, labs Math, Algebra Tutorials and Books are great but what’s really importand and the effective is to do some homework.

Never forget that the secret is Practise. It’s about understanding the reason behind the concepts and the steps. By time you Will find it a lot Easier to follow the procedure. I wish you the best of luck and hope you’ll become the next star algebra.


Cool Math Tricks


Super 11 rule

There is a strong likelihood that many people know 10 rule. That is, when you multiply the number by 10, you add 0 at the end of the number. However, rule 11 may be known to many. It is so easy to understand

-. As you multiply any two-digit number of 11

– Take 36 in this example (3-6) and take a gap between them.

– The sum of two numbers, 3 and 6 together

-. Place the result, 9 of the gap between

– It is simply done 39 x 11 = 3 96

However, there is one tricky thing you should remember, if the result after you have added two places is more than9, put only numbers representing the “ones” in the gap. For example:

… 74×11 7_4 …. 4 + 7 = 11 …

Add one gap, and add 1 to 7 to obtain 8, the answer to 74 x 11 = 814

How to multiply by 999, 99 or 9

Multiply the number of 9 is just like multiplying it by 10-1.

As such, multiply 9 × 9 is9 (10-1) will determine the outcome of 90 -9 = 81

Another example:

9 × 68 = 68 … 01/10)

= 680-68

= 612

To multiply the number by 99 is the same as the (100-1)

47 multiplied by 99 …. 47 (100-1)

= 4700-47 = 4653 …

Multiply the number 1000 means the same as multiplying the same by (1000- 1)

Therefore 55 multiplied by 1000 is the same as…..55(1000-1)



How squ9are to two digit figures

Take, for example, you want to square 48. The first step is squaring each of the letters to get partial results.

4 × 4 and 8 × 16 8 = 64. Then put two numbers together, 1664.

From there, you should multiply the numbers the number you want to square …. 4 × 8 = 32

Double product to get 64

Add 64 to 1664 to get the answer

Math can be terrifying idea for many but with this cool mental math tricks and a little practice to perfect them, general knowledge and speed in mathematics will be considerably improved.


Graph Theory – More on tree


Last time we showed the basic characteristics of wood; it is connected to a minimum and maximum serial port. We are now going to investigate the number of edges of the tree in relation to the number of vertices.

First, I want to show that every tree has “leaves.” A leaf is a vertex of degree 1. To do this, consider the longest path in the tree and mark vertices on this path 1, 2, …, k. Now vertex 1 is not able to connect to the 2, 3, …, k, as this would create a cycle. Vertex 1 is not able to connect with other vertex in the graph, it would be a longer path in the tree. The degree of this vertex will be 1. In fact, for the same reasons, vertex K also has a grade 1.

This little result is very useful when it comes to theorizing about the tree. It is useful for a wonderful mathematical tool that is stimulated. I’ll give you an example.

I’ll show a tree with n vertices has n-1 edges. We shall do this by induction. Consider Singleton graph at one vertex; This has no edges and one vertex. Now consider the graph of n vertices. Pick a leaf on this graph and remove it. Now you have a graph on n-1 vertices exposed to n-2 edges with the induction hypothesis. But you have only removed one edge of the original graph, which must have had n-1 edges.

Now we have shown that trees with n vertices will have N-1 edges.


Brahmagupta Private zero


Zero is not just a word or number but a symbol of the pride of the Indian nation and the fact that research in Eastern culture and development in science and mathematics is the very foundation of discoveries medieval world was made and even more future ones anyway be forgotten zeroes in the east.

In this ever changing dynamic world nobody really cares about this cute little zero, but imagine if this is taken out of our lives then all of science will come to a chaotic end and 07:56 Photo salary will be one digit trauma. NASA must stop trips to Mars and the Moon, Hubble will just fall to the ground. Furthermore, our modern mathematics will be shaken and will find new suddenly face to explain complex sentence her. In short, the whole world will be a garden and a bear in the absence of this single number called “Zero” is written as “0” and sometimes flashy and just O.

Zero was not the brainchild of Greek, Arab or other Western world than the product of the Indian mathematician called Brahmagupta in 598 AD. Brahmagupta was born in Gujarat in the city Bhinmal which is now located in North West Rajasthan. Brahmagupta was head of the department of mathematics and astronomy at the University set up by another great Vedic Indian mathematician Aryabhatta the city of Ujjain who was at that time a great center for learning Sanskrit literature Science and art forecast called astronomy.


Music and Mathematics – there are many Connections


If you thought music was not mathematical language, then think again. In fact, music and mathematics are very much intertwined, so much so that I think you might say could not live without the other. Here we explore the relationship clearly shows the strength of this tie. Let the music begin.

For those with a rudimentary knowledge of music, diatonic scale is something quite familiar. To understand why certain pairs of notes sound good together and others do not, you need to look into the points pattern wave physics and frequencies. The sine wave is one of the basic pattern surge in mathematics and is displayed well alternately Crest-lowest extent. Many physical phenomena and real-world can be explained by this basic wave pattern, including many of the fundamental tonic properties of music. Certain musical notes sound good together (musically this is called reconciliation or consonance ) due dots pattern waves strengthen each other at select intervals.

If you play the piano, then how each of the different notes sound to you depends on how your instrument is configured. There are different ways to tune instruments and these methods depend on mathematical principles. These artefacts are based on the whole frequency applied to a particular mind, and as such, this product whether groups of notes sound well together, but we say such comments are in harmony, or incompatible, but we say such remarks are out of harmony or dissonant.

When this product coming from the launch of criteria instruments manufacturer and today there are certain criteria that these fabricators follow. Still criteria Despite a multiple nature of mathematics. For example, in more advanced mathematics, students learn some numbers. A series is simply the pattern numbers are determined by some rule. One famous series is harmonic series . This includes the reciprocals of integers, it is 1/1, 1/2, 1/3, 1/4 … The harmonic series serves as one set of criteria for certain tunings, one in particular called Pythagorean intonation .

In Pythagorean intonation, comments are set according to “ rule that perfect fifth .” A perfect fifth consists of “ musical distance ” between the two tones, such as C and G. Again without trying to turn this article into a thesis on musical theory, are notes between C and GC #, D, D #, E, F, F #, and G. The “ from ” between each of these notes is called a half-step. Thus, the perfect fifth 7 half-step, CC #, C #, D, DD #, D # -E, EF, FF # and F # -G. When we consider the notes in a musical harmonic series, a number considered a C note and attributed to G mind will always be in the ratio of 2: 3 So the frequency of these notes will be adjusted to their ratios in accordance with the 2: 3 There is C- note the frequency will be 2/3 G-note frequency, or vice versa, G note frequency will be 3/2 C note frequency, the frequency is measured in cycles per second or Hertz.

Now, continue to adjust according to the perfect fifth, fifth above G is D. Use perfect fifth rate , D note will be tuned to the frequency of 3: 2 G frequency or watched from below, G note is 2/3 frequency D note. We can continue the same way until we complete what is called Ring of the fifth , bringing us back to the C note by applying a series of ratios 3/2 in the previous note in the cycle. This includes the Twelve Steps and when complete, the frequency of the second C or higher octave C note should be exactly twice frequency lower C note. This is a requirement of all octaves. But this does not happen by applying this ratio to 3/2.

Musicians have corrected this problem by resorting to none other than the field irrational number . Recall that the numbers are such that they can not give up the offense, that is, a decimal representation of them, like the number pi or the square root of two, providing not and do not repeat. Thus failure of a Pythagorean tuning method for producing perfect octaves, tuning methods have been developed to prevent this situation. Is called “ equal temperament ” Tuning, and this is standard procedure for most practical applications. Believe it or not, this tuning method involves the rational powers of the number two. That’s right: folded chose number two . So if you thought you were studying rational exponents for nothing in algebra class, here is one example of where such a debate is used in real life

Route equal temperament tuning works as follows :. Each note through its octave has a frequency multiplier repeated twelfth root of two to get to the next higher note. That is, if we start with the traditional note, which vibrates at 440 Hertz, let’s say, getting to # 440 multiplied this by 2 ^ (1/12). Because the twelfth root of two is equal to 1.05946 to five decimal places, A # was tuned to 440 * 1.05946 or 464.18 Hertz. And so continues tuning with the next Note B by taking 2 ^ (2/12) * 440. Note that with the increase of twelfth power 2 by 1 every time the power 2 of 1 / 12, 2/12, 3/12, etc.

What is nice about this process is exactness, unlike inexactness of Pythagorean intonation method discussed earlier. Thus when we arrived at the octave note, almost on top of the standard A, which should vibrate twice the original frequency of 440 Hertz A, we get A octave = 440 * 2 ^ (12/12) 440 * 2 = 880 Hertz, as it should be — exactly. As we noted earlier, when the set of the Pythagorean method, this does not happen because of repeated use rate 3/2, and the accommodation must be made to bring into line the inexactness of this approach. These great cause noticeable dissonance between certain notes in certain keys.

This tuning exercise shows that math and music are closely intertwined, and indeed it can be said that these two disciplines are inseparable. Music truly is mathematics and mathematics is well, yes music. Since many people think of musical talent coming from “ creative” species and math capabilities come from “ nerdy ” or non-creative types, this article is in some part help disabuse these same people of this idea. Yet the question is: If two ostensibly different fields like music and mathematics are happily married, how many other areas out there that at first seem to have nothing to do with mathematics, are just as intricately linked to this most fascinating material. Consider that for a while.


Math Games – the number of rows (Part 3 of 3)


In the previous two articles we looked at the types of numbers the number sequences are often used as the basis of “What comes next?” questions IQ test. In this key findings article we will look at two more types of number sequences that are used in ‘What comes next? “Questions. Geometric progressions and special or” one-off “number sequences are the least numbers progressions, but it is a different reason for this in each case. Geometric progressions are mathematically complex than numbers progressions, making only the simplest geometric number sequences suitable for use the IQ test questions. The goal IQ tests measure intelligence, based on the combination of speed and accuracy of the answers of the other party, meaning questions that require long calculation is not suitable for inclusion. However, individual rows speak rarely used in IQ tests for they tend to favor those with more extensive education and pattern recognition skills, rather than those with innate intelligence.

Geometric Number Sequences

A geometric progression is a sequence of numbers each number in the series, but the first is found by multiplying the previous one by a certain number, which is known as a common rate. The corporate rate can be either positive or negative number.

Positive Common ratios

For example, the order of 2, 8, 32, 128 … is a geometric progression with a common ratio of 4, 40, 10, 2.5, … is a geometric series with common ratio of 0.25.

To check whether the sequence of numbers is geometric, one simply checks whether the subsequent entries in the series all have the same ratio.

If the common ratio is greater than 1, the series will grow inexorably to infinity. However, the common rate between zero and 1 will produce a series of decays toward zero. A common ratio 1 produces dullest geometric series, stable series where each number in the sequence is the same example, 5, 5, 5, 5, 5, 5 …

Negative Common ratios

Negative common ratios produce alternating series, where the legs numbers change from positive to negative and back again.

For example, the sequence 1, 2, 4, -8, 16, -32 … is a geometric progression with normal rated -2.

If compared to the common rate is between zero and -1 it will produce alternating order decays toward zero. A common ratio of less than -1 leads alternating series showing exponential growth towards infinity. A common ratio of -1 produces alternating continuous series such as -5, 5, -5, 5, …

final chance to be aware of the common rate of 0, which produces a range comprising the first number and infinite series zero eg 5, 0, 0, 0, 0, 0, …

Special Number Series

Special or “one-off” series of statistical series that have mathematical basis, but recognizable by the model rather than the underlying mathematics. The best-known series, which might seem a question of IQ test, prime numbers. A key is a number that has exactly two integer share, and the number itself and 1. Series starts primes 2, 3, 5, 7, 11, 13, 17, 19, 23, …

Another, sophisticated, special series is a series of different numbers. A perfect number is a positive integer that is the sum of its proper positive Divisors. The first perfect number is 6 (the sum of its proper positive share ,, 1 + 2 + 3 = 6) and second 28 (1 + 2 + 4 + 7 + 14). In the next two numbers in the series are 496 and 8128, you will appreciate that we are entering a state of higher mathematics.

Returning to normal plane, we will do by looking at the ‘What comes next? “The question left unanswered in the second article


The problem is this question raises is that, because it seems on the surface to be in alphabetical order, many seek for an alphabetic solution. Side thinkers who consider the possibility that it is a statistical series are quickly rewarded with a correct answer. The series is the first letters of the written number sequences







make the missing letters S and E for seven and eight.

The question may seem either delightfully simple, or simply not fair, depending on whether you got the answer right or wrong. It does, however, show why train you, or your children, to perform better in the ‘what comes next’ questions should be regarded as a fun activity, rather than a sure-fire way to improve your IQ test scores.


Get Lucky Lottery Numbers með því að nota einfalda stærðfræði


Margir leikmenn segjast hafa sumir efst aðferðir um hvernig á að fá heppinn happdrætti tölur. En ef það væri sannarlega svo einfalt að vinna í lottóinu, þá hver maður þarna úti myndi fara fyrir það. Svo sem happdrætti aðferðir raunverulega vinna?

Fólk heldur að tína af handahófi tölur eða hafa það sem kallað er “fljótur velja” einhvern veginn að gefa þeim tækifæri til að vinna. Einnig, sumir myndu stöðugt standa með uppáhalds númer samsetningu þeirra. Sannleikurinn er, allar þessar aðferðir einfaldlega virka ekki, að minnsta kosti af þeim tíma. Hvers vegna? Þar sem happdrætti er ekki allt heppni. Hluti af því er byggt á líkum og stærðfræði.

Svo hvaða hjartarskinn stærðfræði að gera með heppinn happdrætti tölur? Svarið er mikið, því að happdrætti felur kenningar líkindum COMBIN aðgerð og sjálfstætt auk háð atburðum. Fyrst og fremst, kenningin um líkur er í tengslum við lögum meðaltöl og er hugmyndin að á lengri tíma, tölur dregin af nákvæmlega sama hátt er líklegt að meðaltali út á fjölda skipta sem þeir eru valdir.

Til dæmis, þegar þú flettir upp peningi, það eru 2 líklegar niðurstöður, sem eru annaðhvort höfuð eða hala. Ef þú flettir peningnum nokkrum sinnum, getur þú byrjað að sjá mynstur. Miðað við að það eru bara 2 líklegan árangur, og við fáum einhvers konar sögu síðustu niðurstöðum með að fá eitthvað eins og 18 höfuð og 12 hala, getum við gert ráð fyrir að líkurnar á að fá höfuð í eftirfarandi Flip er meira en að hafa hala. Þegar þú sækir um sömu hugmyndina að happdrætti, það er ekki mikill munur. Happdrætti hafa verið í áratugi núna, þannig að við höfum meira en nóg sögulegum vinnur að byggja númer samsetningar okkar á. Lítilsháttar munur er að fella við stigi handahófi sem er eðlislæg í öllum happdrætti.

Á hinn bóginn er Combin Hlutverk ráðstafanir sem tala af lifnaðarhættir sem ákveðin sett af tölum má vera á tilteknu happdrætti atburðarás. Til dæmis, að nýta Combin Virka getum við strax meta að í happdrætti 49 bolta, það eru 13,983,816 leiðir þróa ákveðna setja af 6 boltum og svona, líkurnar á hitting gullpottinn (ef þú kaupir einn miða) er 1 13,983,816.

Loks hafa óháðir atburðir engin áhrif á atburðum framtíðarinnar, né eru þeir fyrir áhrifum af niðurstöðum sem gerst áður. Teikningar eru tilvalin dæmi af sjálfstæðum atburðum, þar sem hvert jafntefli er aðskilin frá öðrum í vissum skilningi að tölurnar, sem valdir eru algjörlega ekkert að gera með tölurnar valdir í fyrra teikningu. Nokkrir leikmenn gera mistök að trúa að því lengur sem einkum koma af tölum eru ekki valið, mun meiri líkur á því sem sett er að vera valinn í síðari dregur.

Using Kerfi

A happdrætti kerfi er fær um að hækka líkurnar á að fá þá heppinn happdrætti tölur vegna þess að þeir bera nú flókið formúlu. Þetta er byggt á stærðfræðilegum útreikningum sem hafa verið þróaðar af fyrri verðlaunahafa sjálfir sem hafa notað eigin reynst sínum aðferðum til að vinna.


Mathematician and Poets – Two of a Kind?


“A mathematician WHO is not also something of a poet will never be a complete mathematician.” – Karl Weierstrass

C’mon a mathematician Who’s a poet! Give me a break. Is not that as compatible as a snake and a mongoose? You mean there really is some truth in the quote by Weierstrass, one of the most famous mathematician of all time, WHO is probably Responsible for the rigors of calculus? Indeed mathematics has a rhythmic structure Which, When probed, Reveals its poetic and musical beauty. And Any person who masters this discipline rightfully then shouldnt be regarded as something of a poet.

When my Spanish II professor Suggested That I go to Salamanca, Spain to study Castilian, I was pleasantly surprised. She had recommended participate in go Because of my outstanding performance in her class and Because I had shown Such intense interest in the subject. To this recommendation, I replied That I was a mathematics major WHO also loved-Besides foreign languages-the classics in literature as well as the sciences. To this, the professor replied, “So you’re a modern day renaissance man.”

Many people think That Those WHO excel in mathematics igniting to shun things like art and literature. Mathematics, They think, is too rigorous a subject, and Any WHO Embraces Such, could not Possibly have the softest emotional side to embrace Such flowery subjects like art and literature. Yet this is not true at all. Because mathematics has an inherent rhythmicity to its structure, mathematician are really quite sensitive to the Humanities like art and music. Even in the structure of poem, with its varied meters and rhyming schemes, Can be found the essence of mathematics.

Given the above considerations, a person should not feel awkward at being both matches mathematician and poet. Nor Should one feel Any wonder at how easilyNavigation math portability translates Into poetic portability: how one Can compose poems of varying length, meters, and rhyming schemes Because of a predispositions toward math. You see, this creative portability all comes naturally to the mathematically inclined person. And WHO knows how many of the world’s greatest poets were great mathematician? So parents, if your son our daughter shows a flair for poetry, bear in mind That you might have a budding mathematician on your hands. And Those That have children WHO are great at math, Who Knows? -Your Might very well have on your hands a world class poet.