Recall that there are four different base ten blocks: cubes (ones), rods (tens), flats (hundreds), and blocks (thousands). Groups of ten base ten blocks can be regrouped or traded for equivalent amounts of other base ten blocks; for instance, ten cubes can be traded for one rod because both are worth ten. For subtraction, it is useful to know how to trade down rods, flats, and blocks. Trading down means converting larger place value blocks into smaller place value blocks. For instance, one flat can be traded for ten rods since they are both worth 100.

Before describing the subtraction procedure, let’s go over some vocabulary . . .

Minuend – The amount from which you are subtracting.

Subtrahend – The amount that you are subtracting.

Difference – The answer.

In the equation, 234 – 187 = 47, the minuend is 234, the subtrahend is 187, and the difference is 47. Most people don’t bother with the terms minuend and subtrahend, but they are useful in describing the subtraction procedure using base ten blocks.

To begin, represent the minuend with base ten blocks. Try to keep the blocks in order from largest to smallest as this will help to transfer knowledge and skills to paper and pencil methods later on. Remove from the minuend piles, enough blocks to represent the subtrahend. If there aren’t enough blocks available, trade some of the larger place value blocks until there are enough smaller place value blocks to remove. The resulting piles after the subtrahend is removed represents the difference.

In the example, begin by representing 234 with 2 flats, 3 rods, and 4 cubes. The goal is to remove 187 or 1 flat, 8 rods, and 7 cubes from these piles. Removing one flat is simple enough, but 8 rods and 7 cubes are difficult to remove if there are only 3 rods and 4 cubes! To solve this problem, trade in one flat for 10 rods, and one rod for 10 cubes. The result would be 1 flat, 12 rods, and 14 cubes. Removing the subtrahend – 1 flat, 8 rods, and 7 cubes – at this point would leave no flats, 4 rods, and 7 cubes. The difference is whatever is left after removing the subtrahend, so the difference is 47.

For beginners, it would be wise to start with subtraction that does not require trading. For example 1954 – 1831 would require no trading because there are enough blocks in the minuend to remove the subtrahend. For more advanced students, questions that include zeros can present a bit of a challenge. For example, 4000 – 3657 would require several trading steps all starting with four blocks. http://www.math-drills.com has several thousand free math worksheets including subtraction questions with no regrouping (trading). One of the nice features of this website is that answer keys are provided, so students can get feedback on their results.

With enough experience, students learn subtraction on a conceptual level and are better equipped to apply it to pencil and paper methods later on. Students who only learn the paper and pencil method don’t always develop a conceptual understanding of subtraction and are less able to identify errors in their work.

Base ten blocks are not limited to just addition and subtraction of whole numbers. In part III of this series, several other uses of base ten blocks will be explored.

The numbering system that children learn and the one most of us are familiar with is the base ten system. This essentially means that you can only use ten unique digits (0 to 9) in each place of a base ten number. For instance, in the number 345, there is a hundreds place, a tens place and a ones place. The only possible digits that could go in each place are the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, or 9. In this example, the place value of the ones place is 5.

Base ten blocks turn the base ten concept into something children can see and touch.

Base ten blocks consist of cubes, rods, flats, and blocks. Cubes represent the ones place and look exactly like their name suggests – a small cube usually one centimeter by one centimeter by one centimeter. Rods represent the tens place and look like ten cubes placed in a row and fused together. Flats, as you might have guessed, represent hundreds, and blocks represent thousands. A flat looks like one hundred cubes place in a 10 x 10 square and attached together. A block looks like ten flats piled one on top of the other and bonded together.

In order to use base ten blocks to add numbers, students should be familiar with how to represent numbers using base ten blocks. To see what base ten blocks look like, and to try them out, go to the National Library of Virtual Manipulatives:

http://nlvm.usu.edu/en/nav/frames_asid_154_g_1_t_1.html

To represent a number using base ten blocks, make piles of base ten blocks to represent each place value. If your number was 2,784, you would make a pile of 2 blocks, a pile of 7 flats, a pile of 8 rods, and a pile of 4 cubes. It is useful to arrange the piles in a row in the same order that they appear in the number as that will be useful later on when children learn the paper and pencil algorithm.

Another useful skill to practice is trading base ten blocks. Each block can be traded for 10 flats, each flat for 10 rods, and each rod for 10 cubes. Going the other way, 10 cubes can be traded for one rod, 10 rods for one flat, and 10 flats for one block.

One simple use of base ten blocks that translates well to a paper and pencil method of addition is to add by regrouping. To add two or more numbers, start by representing each number with base ten blocks. Put all of the cubes from both numbers in the same pile; do this with the rods, flats, and blocks as well. Next, trade any groups of 10 cubes for a rod. Trade any groups of 10 rods for a flat; then trade any groups of 10 flats for a block. To read the resulting number, count the number of base ten blocks left in each pile and read the number.

To illustrate this procedure, picture the addition question, 568 + 693. After representing both numbers with base ten blocks and combining the piles of like base ten blocks, you should have a pile of 11 cubes, a pile of 15 rods, and a pile of 11 flats. Trading 10 of the cubes for 1 rod means you now have 1 cube, 16 rods and 11 flats. Trading 10 of the rods for one flat results in 1 cube, 6 rods, and 12 flats. Trading 10 of the flats for one block gives you your final piles of 1 cube, 6 rods, 2 flats, and 1 block. The answer to the addition question, therefore, is 1,261.

If you don’t have base ten blocks, you can use the virtual base ten blocks or make paper versions. If you need addition questions (with the answers included), you can access thousands of free math worksheets at http://www.math-drills.com

In future articles, I will describe more uses for base ten blocks including subtraction and multiplication, and I will continue the series with other manipulatives that can help your child or student learn math.

“How much did it cost to fix your car?”

“Six hundred bucks!”

Without any words such as: about, approximately, around, roughly, or nearly, it can be assumed that the second person rounded the actual cost. Before they had their car fixed, they probably received an estimated cost of the repair from the shop. Adults experience rounding and estimation skills in their daily lives. Children need to learn these important skills partly because they often hear estimation and use estimation, but more importantly, it helps to solidify math learning by teaching them the idea of reasonableness.

Even though rounding and estimating are related, there is a significant difference. Rounding involves converting a known number into a number that is easier to use. Estimation is an educated guess of what a number should be without knowing the actual number. In the conversation above, it is unlikely that the second person remembered the actual price of the bill; they likely rounded the number at the time, so they could better remember it.

Children usually learn rounding as an explicit skill, often with the purpose of estimating the answers to math questions. They commonly use estimation to check the reasonableness of an answer by either estimating ahead of time or after they have completed the question. Students run into difficulty when estimating because they don’t have the intuitive sense that adults do to break the rules.

For the uninitiated, the idea of rounding is fairly simple – decide where to round the number (e.g. the hundreds place), either keep the digit at the rounding place the same or round it up, and replace the digits to the right with zeros. The decision to keep the digit the same or to round it up is based on everything that comes after the digit. If it is less than half, the digit remains the same; if it is greater than half, the digit is increased by one; if it is exactly half, the digit remains the same if it is even and increases by one if it is odd. For example, to round 638 to the nearest hundred, you would base your decision on the “38″ portion of the number. Since it is less than half (50), the digit in the hundreds place remains the same, and the 38 is changed to zeros, so the rounded number is 600. If the question is to round 7500 to the nearest thousand, you would round up to 8000. 8500 also rounds to 8000, but 8501 rounds to 9000. Hopefully, this illustrates that rounding follows a strict set of rules that often cause difficulties for children in estimation.

To give you an idea of how following the rounding rules can be problematic in estimation, consider the question 7359 divided by 82. The first difficulty is deciding what place to round to. Let’s say that the student decides to round to the nearest hundred in the first number and the nearest ten in the second number, thus the question is now 7400 divided by 80. At this point some students might resort to a calculator, others to long division, and others might stare confusedly at their paper. An adult with more intuitive sense might look at the numbers and recognize that if she rounded 7359 to 7200, it would be fairly simple to divide by 80 (because 72 divided by 8 is easy).

Many people develop an ability to estimate both by following the rules and by breaking the rules of rounding. Many children need to be taught these skills, so there is a genuine purpose to their estimation rather than just another question to answer. Estimation should be thought of as a tool to quickly determine whether an answer is reasonable or not. One way of teaching estimation for this purpose is by allowing students to break the rounding rules and find an easy question that they can do in their head. In the question 3564 – 2801, rounding to the nearest hundred results in 3600 – 2800, but 3700 – 2700 is much easier to handle, and it is not so far off the real answer. If the purpose of estimating was to get as close to the real answer as possible, you might as well use a calculator to check your answer instead.

Parents can help develop students’ estimation skills by regularly asking real questions. For instance, ask them how long they think it will take to get to hockey practice (time), have them add up the cost of the groceries as you are shopping (money), get them to count the number of people in one area of the mall and have them estimate how many people are in the whole mall (multiplication or addition). Educators should make estimation a regular part of the problem solving process. In a science investigation, students make hypotheses and predictions, so why not make an estimate in a math problem? Students can develop their estimation skills by answering questions on worksheets and comparing their estimated answers to the actual answers. http://www.math-drills.com has thousands of worksheets with answer keys that you could use for this purpose.

Remember these rules for estimation: (i) KISS – keep it simple silly, (ii) break the rounding rules if necessary, (iii) ensure students see a purpose for estimation, (iv) give students a lot of practice and experience with estimation and rounding, (v) include estimation in problem solving and other daily math work. The main rule for parents and teachers: support your students and be flexible!

For the uninitiated, a regular polygon is a closed figure with equal length sides and equal angles. A pentagon with three centimetre sides and 108 degree angles is a regular pentagon. Regular polygons are the figures that are most commonly used to represent each family of polygons.

To experience the most success with this method, it is recommended that you use a full circle protractor. A half circle protractor will work just fine except the procedure changes slightly. The basic procedure for the full circle protractor is to place the protractor on a piece of paper, make a bunch of dots, and join the dots. The trick is dividing the 360 degrees of the circle by the number of vertices in the regular polygon, and making dots at the resulting interval. In a hexagon, for example, there are six vertices, so divide 360 degrees by six to get sixty degrees. Starting at zero degrees, make a mark every sixty degrees around the full circle protractor; there will be dots at 0, 60, 120, 180, 240, and 300 degrees. Join the dots, and voila; you have a perfect regular hexagon. With a half circle protractor, it is necessary to establish a center point first, so when you rotate the protractor to complete the dots on the other side, it can be lined up properly with the zero point and the center point.

The really nice thing about using a 360 degree circle to construct regular polygons is that it works for all of the regular polygons that one would encounter in an elementary or primary school. This is because 360 is divisible by 24 different numbers including 3, 4, 5, 6, 8, 9, 10, and 12. To construct an equilateral triangle, for example, first divide 360 by three to get 120. Make dots at 0, 120, and 240, join the dots, and enjoy a perfectly drawn equilateral triangle. Squares are constructed by marking dots at 90 degree intervals, pentagons at 72 degree intervals, octagons at 45 degree intervals, nonagons at 40 degree intervals, decagons at 36 degree intervals, and dodecagons at 30 degree intervals. “But what about a heptagon?” you may ask. Even numbers that don’t divide evenly into 360 can be approximated using this method. For example, a heptagon (seven sided polygon) can be approximated quite well using 51 degree intervals. It will be hard to tell with the naked eye that you were one or two degrees off.

One limitation of this method is that there is only one size of circle available, so all of the polygons come out quite large. With a little ingenuity, this limitation can be overcome. One simple solution is to cut out a circle of paper and place it on top of the round protractor. Any paper circle smaller than the round protractor can be used. Make the dots around the edge of the paper circle lining them up with the scale on the protractor. The paper circle becomes an intermediate protractor that can be used just as the regular protractor, but it will make a smaller polygon.

Another limitation is that your students might not be at the point where they can divide or find multiples of large numbers. In this case, you could tell your students at which numbers to make the dots, or create paper protractors with just the intervals marked on them for each polygon.

This is the quickest and most efficient method I have seen for constructing regular polygons. It takes little time to teach and little time to learn, and it makes the construction of regular polygons a simple and painless activity for students. And if you need a bit of a challenge, try the 180 sided polygon with two degree intervals. I’ll bet you never guessed you could make one of those so easily!

Then God said, “Let there be light,” and there was light. This ancient description of the creation of the universe found in the Book of Genesis may be accurate after all. The big bang theory describes the beginning of the universe as having been precipitated from an infinitesimally small point. In this small volume, all matter and energy was concentrated until its contents exploded in either a smooth expansion or an incredibly violent energetic explosion that formed the planets, stars and galaxies. Originally this theory had competition from what is called the ‘steady state’ theory whereby the universe is forever expanding and new matter and energy is created spontaneously within the space left by the receding galaxies. However, empirical observations have directed astronomers and scientists into the acceptance of the big bang model. But how did we get to this point in our understanding?

In the early part of the twentieth century the American astronomer Vesto Slipher and the German Carl Wirtz made some important astronomical discoveries. Using spectral analysis, Slipher deciphered the mixtures of gases contained in planetary atmospheres as well as nebulae. What distinguishes his findings is the discovery that most if not all galaxies outside of our own demonstrate what is called a ‘Red Shift.’ This shift is simply a change in the wavelength of the light emitted by those objects under investigation towards a longer wavelength. Wirtz similarly catalogued many red shifts of the nebulae which he chose to study. But it was still to early for them to realize the full potential meaning of their observations. That would wait until Einstein’s General Relativity would be interpreted by other scientists through further mathematical analysis.

His contemporaries demonstrated to Einstein that his new Theory of General Relativity published in 1916 was not compatible with a ‘static’ universe of space time. The theory predicted an expanding or collapsing universe but not a fixed cosmos. Because he personally believed the universe to be an invariable space time continuum, Einstein engaged in a degree of scientific legerdemain. To correct what he perceived to be as ‘flaws’ in his theory he added the contrivance of a cosmological constant known as lambda to force the static universe into reality. Einstein’s view of perfection in an unchanging space time continuum had led him down a blind alley as much as Aristotle’s concept of perfection had brought that great philosopher into the error of believing in a static Earth at the center of the universe.

But even with the addition of the cosmological constant lambda, the universe was still found to be unstable and this whole affair would later be viewed by Einstein as his “greatest blunder.” His cosmological acrobatics behind him, Einstein yielded the stage to others for a clearer understanding of his own theory. It fell to Alexander Alexandrovich Friedmann to consider the consequences of General Relativity without the constant lambda interfering with his study of these relationships. In doing so, the Russian mathematician and cosmologist derived the solution which predicts an ever expanding cosmological structure (1922), a prediction which was disagreeable with Einstein’s concept of universal perfection. A couple of years later, Friedmann published his findings in “About the Possibility of a World with Constant Negative Curvature of Space.” But the entire hypothetical construct still lacked a complete verbalization mathematically and theoretically.

Enter the Reverend Father Georges Lemaitre, a Catholic priest from Belgium. Rev. Fr. Lemaitre provided the equations necessary to formulate the basis of Big Bang theory in his work entitled “Hypothesis of the Primeval Atom.” He postulated that the universe began as a primordial atom of infinitesimal volume and enormous mass energy as well as space and time and everything else comprising the future universe. At some point the universe began with the explosion of this super atom. Lemaitre published his theoretical ideas between the years 1927 and 1933 and speculated that the movement of the nebulae demonstrated the validity of the explosion of his cosmic super atom. Unfortunately, he also wrongly believed that cosmic rays might be an after effect of the super atom’s big bang. These are now known to be generated not from a universal conflagration but from galactic sources unrelated to the big bang.

However, the new theory still lacked a major source of observational support. This would be provided by Edwin Hubble’s observations of the redshift of galaxies. Taking up where Slipher and Wirtz left off, Hubble employed a novel technique to discern the properties of the galactic movements. By choosing to observe stars that are known as Cepheid Variables he could more accurately make measurements. Cepheids are a type of star that brighten and darken and lighten back up in regular periods of time that are well known. Cepheids that have identical cycle times of brightening darkening and brightening again also have identical or nearly identical luminosity. Thus, if one compares the length of the cycle to the amount of light apparent to the observer it is possible to accurately prepare an estimate of the distance to the cepheid.

In this manner, Hubble had found that the nebulae or galaxies exhibited a galactic red shift; in other words, that galaxies were receding away from ours at a speed which is correlated directly with the distance between our vantage point and the galaxy being studied. The further away the galaxies were the faster they appeared to be going in moving away from us. The results of these investigations is now known as Hubble’s Law. Essentially, this law states that universe is in an ever expanding mode whereby the intergalactic distances continue to grow without bound into infinity. Hubble’s Law depends upon the shifting of the wavelength of light and after having been delineated in 1929 has been subsequently proven over and over again. Further, Hubble’s constant has been recalculated to a more ‘perfect’ value and retains a great probability of being ‘recomputed’ in the future based upon new observations.

Thus, it should be clear to the reader that our scientists have a fateful habit of introducing their preconceived notions of beauty into their models. From Aristotle’s static Earth to Einstein’s greatest blunder, the constant which forces a static universe, we proceed only from the wisdom of our weak minds. The more things change the more things stay the same. Man’s hubris knows no limits in our attempts to understand things without the wisdom to comprehend its underlying meaning. Humble we are not. We are making the same mistakes we always have. Back to the future. To be continued…

Left to right addition involves adding the largest place values first. As you move from left to right, you keep a cumulative total, so it is simply a number of smaller addition problems. To give you an idea of how it works and what it sounds like, consider the example, 677 + 938.

Begin by adding the left most place values. In the example this is 600 plus 900 equals 1500. Add the values in the next place, one at a time, to the previous sum, and keep track of the new sum each time. In the example, 1500 + 70 is 1570, 1570 + 30 is 1600. For students who are more proficient at this algorithm, they don’t necessarily think “plus 70″ or “add 30.” Their thought process, if said out loud might sound like, “600, 1500, 1570, 1600, . . .” Continue adding the values in each subsequent place until finished. The final steps in the example are 1600 + 7 is 1607, 1607 plus 8 is 1615. The sum is 1615.

As you can imagine, students need to be proficient at single digit addition and have an understanding of place value before attempting left to right addition. When they are first learning it, they might try repeating sums as they go along (e.g. 1500, 1570, 1570, 1570, 1600, . . .) to help them retain the newest sums. They might also cross out digits as they are adding. There is no rule about having to add in this way mentally. Students could write down the sums as they proceed.

Left to right addition promotes a better understanding of place value than right to left addition. In right to left addition, single digits are carried or regrouped with little emphasis placed on what the value of those carried digits are. In the example, 1246 + 586, students add 6 + 6 to get 12; they write down the 2 and carry the 1 when they should be carrying the ten. In the next step, they add 8 + 4 + 1 to get 13; they write down the 3 and carry the 1 when they should be adding 80 + 40 + 10, writing the 3 in the tens place (i.e. 30) and carrying the hundred. Essentially, right to left addition excludes vocabulary related to place value. Left to right addition, on the other hand, promotes an understanding of place value as each digit is given its correct value. In the example, the one in the thousands place is one thousand, the two in the hundreds place is two hundred, and so on.

Left to right addition is well-suited to mental addition since the sum is cumulative with no steps in between; in other words, there is nothing for the student to keep in mind except for the cumulative sum. In right to left addition, several numbers must be remembered as the student proceeds. To illustrate this, consider the simple example, 64 + 88. In left to right addition, the sum is simple to find: 60, 140, 144, 152. Only one number had to be remembered at any point. In right to left addition, 4 + 8 is 12, so there are already two numbers to remember: the two in the ones place and the regrouped ten. The next step is to add 60 + 80 + 10 to get 150. At this point, the two must be recalled and added to the 150 to get 152. Although this sounds simple, it becomes more complicated with more digits.

Right to left addition does not require numbers to be lined up in a column, but it is often taught that way because the method tends to ignore place value and relies on a student’s ability to line up the place values to compensate. Many errors that students make in right to left addition occur because they don’t have a strong knowledge of place value, and they forget or don’t realize that like place values need to be lined up. They might, for instance, add a digit in the tens place to a digit in the hundreds place. Another scenario is a sloppy recording of numbers where a digit is mistakenly added to the wrong column. In left to right addition, the emphasis is on finding a certain place value in each number rather than relying on the place values being aligned. Students, of course, need to be able to recognize place value before they can be successful at this method. For instance, they should be able to recognize that the ones in the numbers: 514, 1499, and 321 are in the tens, thousands, and ones places respectively. If they can’t, further teaching on place value is required before addition can be taught effectively.

Although left to right addition has several advantages, it isn’t suggested that you scrap everything else. Learning a wide variety of addition methods allows you latitude in problem solving situations. By teaching students this method, you give them another option when they are tackling addition questions.

The Nobel Laureate is a full pull this field having earned a prize for travail on the bulk nuclear pains and he indicated that what is happening today is identical similar to what happened at the 1911 Solvay huddle. Back thence, rontgen rays had recently been discovered and mass energy conservation was unbefitting assault considering of its scandal. Lot theory would serve as needed to solve these problems. Gross further commented that monopoly 1911 ” They were lost something absolutely fundamental, ” now together due to ” we are lost conceivably something being profound seeing they were back ergo. ”

Coming from a scientist ditch establishment credentials this is a curse statement about the state of current conceptual models and most notably string theory. This conceptual model is a means by which physicists come next the bounteous commonly proclaimed particles of particle physics hole up one dimensional objects which are confessed through weight. These bizarre objects were original detected pressure 1968 washed-up the sagacity and muscle of Gabriele Veneziano who was backbreaking to comprehend the muscular nuclear brunt.

Whilst meditating on the sturdy nuclear pow Veneziano detected a rapport between the Euler Beta Function, named for the renowned mathematician Leonhard Euler, and the big conscription. Applying the aforementioned Beta Function to the husky pow he was able to validate a direct conjunction between the two. Interestingly enough, no one knew why Euler ‘ s Beta worked then flourishing pressure mapping the beefy nuclear horsepower data. A proposed solution to this dilemma would come next a few senility later.

Almost two elderliness sequential ( 1970 ), the scientists Nambu, Nielsen and Susskind provided a mathematical description which described the unfeigned phenomena of why Euler ‘ s Beta served seeing a graphical outline for the virile nuclear vigor. By modeling the reinforced nuclear forces seeing one dimensional strings they were able to show why it all seemed to work so well. However, several troubling inconsistencies were immediately seen on the horizon. The new theory had attached to it many implications that were in direct violation of empirical analyses. In other words, routine experimentation did not back up the new theory.

Needless to say, physicists romantic fascination with string theory ended almost as fast as it had begun only to be resuscitated a few years later by another ‘ discovery. ‘ The worker of the miraculous salvation of the sweet dreams of modern physicists was known as the graviton. This elementary particle allegedly communicates gravitational forces throughout the universe.

The graviton is of course a ‘ hypothetical ‘ particle that appears in what are known as quantum gravity systems. Unfortunately, the graviton has never ever been detected; it is as previously indicated a ‘ mythical ‘ particle that fills the mind of the theorist with dreams of golden Nobel Prizes and perhaps his or her name on the periodic table of elements.

But back to the historical record. In 1974, the scientists Schwarz, Scherk and Yoneya reexamined strings so that the textures or patterns of strings and their associated vibrational properties were connected to the aforementioned ‘ graviton. ‘ As a result of these investigations was born what is now called ‘ bosonic string theory ‘ which is the ‘ in vogue ‘ version of this theory. Having both open and closed strings as well as many new important problems which gave rise to unforeseen instabilities.

These problematical instabilities leading to many new difficulties which render the previous thinking as confused as we were when we started this discussion. Of course this all started from undetectable gravitons which arise from other theories equally untenable and inexplicable and so on. Thus was born string theory which was hoped would provide a complete picture of the basic fundamental principles of the universe.

Scientists had believed that once the shortcomings of particle physics had been left behind by the adoption of the exotic string theory, that a grand unified theory of everything would be an easily ascertainable goal. However, what they could not anticipate is that the theory that they hoped would produce a theory of everything would leave them more confused and frustrated than they were before they departed from particle physics.

The end result of string theory is that we know less and less and are becoming more and more confused. Of course, the argument could be made that further investigations will yield more relevant data whereby we will tweak the model to an eventual perfecting of our understanding of it. Or perhaps ‘ We don ‘ t know what we are talking about. ‘