Adults use rounding and estimation in their everyday lives. They approximate the temperature, the cost of items, the time, and even their age. Consider this conversation:

“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. .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!

In part one of this article, you read about representing and adding numbers using base ten blocks. Once these two skills are mastered, it is time to move onto many a child’s nightmare: subtraction. Subtraction, as you may have heard, is essentially addition in reverse. It can be an arduous task on paper, but it can be quite easy with base ten blocks.

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. .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.

Last December (’05), physicists held the 23rd Solvay Conference in Brussels, Belgium. Amongst the many topics covered in the conference was the subject matter of string theory. This theory combines the apparently irreconcilable domains of quantum physics and relativity. David Gross a Nobel Laureate made some startling statements about the state of physics including: “We don’t know what we are talking about” whilst referring to string theory as well as “The state of physics today is like it was when we were mystified by radioactivity.”

The Nobel Laureate is a heavyweight in this field having earned a prize for work on the strong nuclear force and he indicated that what is happening today is very similar to what happened at the 1911 Solvay meeting. Back then, radioactivity had recently been discovered and mass energy conservation was under assault because of its discovery. Quantum theory would be needed to solve these problems. Gross further commented that in 1911 “They were missing something absolutely fundamental,” as well as “we are missing perhaps something as profound as they were back then.”

Coming from a scientist with establishment credentials this is a damning statement about the state of current theoretical models and most notably string theory. This theoretical model is a means by which physicists replace the more commonly known particles of particle physics with one dimensional objects which are known as strings. These bizarre objects were first detected in 1968 through the insight and work of Gabriele Veneziano who was trying to comprehend the strong nuclear force.

Whilst meditating on the strong nuclear force Veneziano detected a similarity between the Euler Beta Function, named for the famed mathematician Leonhard Euler, and the strong force. Applying the aforementioned Beta Function to the strong force he was able to validate a direct correlation between the two. Interestingly enough, no one knew why Euler’s Beta worked so well in mapping the strong nuclear force data. A proposed solution to this dilemma would follow a few years later.

Almost two years later (1970), the scientists Nambu, Nielsen and Susskind provided a mathematical description which described the physical phenomena of why Euler’s Beta served as a graphical outline for the strong nuclear force. By modeling the strong nuclear forces as 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.’

It may not sound like a difficult task, but constructing hexagons and other polygons can be a frustrating and daunting task for children and adults. A sketch of a square is fairly simple to make as the corners are familiar right angles that most people have no trouble creating. Every other regular polygon from equilateral triangles to dodecagons and beyond can be a challenge without a highly developed ability to recognize and construct a variety of angles. Thankfully, there is a slick technique for constructing all sorts of regular polygons based on the fact that all regular polygons fit neatly inside of a circle.

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!

Theoretical cosmologists spend much of their time perfecting what is now known as the ‘Big Bang’ theory. This concept originates from ideas percolating in the minds of scientists, theologians and astronomers down through the ages. However, much of what they consider as proof for the ‘Big Bang’ is dependent upon uncontrolled experimentation that is molded to meet their expectations.

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…

In the first two parts, representing, adding, and subtracting numbers using base ten blocks were explained. The use of base ten blocks gives students an effective tool that they can touch and manipulate to solve math questions. Not only are base ten blocks effective at solving math questions, they teach students important steps and skills that translate directly into paper and pencil methods of solving math questions. Students who first use base ten blocks develop a stronger conceptual understanding of place value, addition, subtraction, and other math skills. Because of their benefit to the math development of young people, educators have looked for other applications involving base ten blocks. In this article, a variety of other applications will be explained.

Multiplying One- and Two-Digit Numbers

One common way of teaching multiplication is to create a rectangle where the two factors become the two dimensions of a rectangle. This is easily accomplished using graph paper. Imagine the question 7 x 6. Students colour or shade a rectangle seven squares wide and six squares long; then they count the number of squares in their rectangle to find the product of 7 x 6. With base ten blocks, the process is essentially the same except students are able to touch and manipulate real objects which many educators say has a greater effect on a student’s ability to understand the concept. In the example, 5 x 8, students create a rectangle 5 cubes wide by 8 cubes long, and they count the number of cubes in the rectangle to find the product.

Multiplying two-digit numbers is slightly more complicated, but it can be learned fairly quickly. If both factors in the multiplication question are two-digit numbers, the flats, the rods, and the cubes might all be used. In the case of two-digit multiplication, the flats and the rods just quicken the procedure; the multiplication could be accomplished with just cubes. The procedure is the same as for one-digit multiplication - the student creates a rectangle using the two factors as the dimensions of the rectangle. Once they have built the rectangle, they count the number of units in the rectangle to find the product. Consider the multiplication, 54 x 25. The student needs to create a rectangle 54 cubes wide by 25 cubes long. Since that might take a while, the student can use a shortcut. A flat is simply 100 cubes, and a rod is simply 10 cubes, so the student builds the rectangle filling in the large areas with flats and rods. In its most efficient form, the rectangle for 54 x 25 is 5 flats and four rods in width (the rods are arranged vertically), and 2 flats and five rods in length (with the rods arranged horizontally). The rectangle is filled in with flats, rods, and cubes. In the whole rectangle, there are 10 flats, 33 rods, and 20 cubes. Using the values for each base ten block, there is a total of (10 x 100) + (33 x 10) + (20 x 1) = 1350 cubes in the rectangle. Students can count each type of base ten block separately and add them up.

Division

Base ten blocks are so flexible, they can even be used to divide! There are three methods for division that I will describe: grouping, distributing, and modified multiplying.

To divide by grouping, first represent the dividend (the number you are dividing) with base ten blocks. Arrange the base ten blocks into groups the size of the divisor. Count the number of groups to find the quotient. For example, 348 divided by 58 is represented by 3 flats, 4 rods, and 8 cubes. To arrange 348 into groups of 58, trade the flats for rods, and some of the rods for cubes. The result is six piles of 58, so the quotient is six.

Dividing by distributing is the old “one for you and one for me” trick. Distribute the dividend into the same number of piles as the divisor. At the end, count how many piles are left. Students will probably pick up the analogy of sharing quite easily - i.e. We need to give everyone an equal number of base ten blocks. To illustrate, consider 192 divided by 8. Students represent 192 with one flat, 9 rods and 2 cubes. They can distribute the rods into eight groups easily, but the flat has to be traded for rods, and some rods for cubes to accomplish the distribution. In the end, they should find that there are 24 units in each pile, so the quotient is 24.

To multiply, students create a rectangle using the two factors as the length and width. In division, the size of the rectangle and one of the factors is known. Students begin by building one dimension of the rectangle using the divisor. They continue to build the rectangle until they reach the desired dividend. The resulting length (the other dimension) is the quotient. If a student is asked to solve 1369 divided by 37, they begin by laying down three rods and seven cubes to create one dimension of the rectangle. Next, they lay down another 37, continuing the rectangle, and check to see if they have the required 1369 yet. Students who have experience with estimating might begin by laying down three flats and seven rods in a row (rods vertically arranged) since they know that the quotient is going to be larger than ten. As students continue, they may recognize that they can replace groups of ten rods with a flat to make counting easier. They continue until the desired dividend is reached. In this example, students find the quotient is 37.

Changing the Values of Base Ten Blocks

Up until now, the value of the cube has been one unit. For older students, there is no reason why the cube couldn’t represent one tenth, one hundredth, or one million. If the value of the cube is redefined, the other base ten blocks, of course, have to follow. For example, redefining the cube as one tenth means the rod represents one, the flat represents ten, and the block represents one hundred. This redefinition is useful for a decimal question such as 54.2 + 27.6. A common way to redefine base ten blocks is to make the cube one thousandth. This makes the rod one hundredth, the flat one tenth, and the block one whole. Besides the traditional definition, this one makes the most sense, since a block can be divided into 1000 cubes, so it follows logically that one cube is one thousandth of the cube.

Representing and Working With Large Numbers

Numbers don’t stop at 9,999 which is the maximum you can represent with a traditional set of base ten blocks. Fortunately, base ten blocks come in a variety of colors. In math, the ones, tens, and hundreds are called a period. The thousands, ten thousands, and hundred thousands are another period. The millions, ten millions and hundred millions are the third period. This continues where every three place values is called a period. You may have figured out by now that each period can be represented by a different colour of place value block. If you do this, you eliminate the large blocks and just use the cubes, rods, and flats. Let us say that we have three sets of base ten blocks in yellow, green, and blue. We’ll call the yellow base ten blocks the first period (ones, tens, hundreds), the green blocks the second period, and the blue blocks the third period. To represent the number, 56,784,325, use 5 blue rods, 6 blue cubes, 7 green flats, 8 green rods, 4 green cubes, 3 yellow flats, 2 yellow rods, and 5 yellow cubes. When adding and subtracting, trading is accomplished by recognizing that 10 yellow flats can be traded for one green cube, 10 green flats can be traded for one blue cube, and vice-versa.

Integers

Base ten blocks can be used to add and subtract integers. To accomplish this, two colours of base ten blocks are required - one colour for negative numbers and one colour for positive numbers. The zero principle states that an equal number of negatives and an equal number of positives add up to zero. To add using base ten blocks, represent both numbers using base ten blocks, apply the zero principle and read the result. For example (-51) + (+42) could be represented with 5 red rods, 1 red cube, 4 blue rods, and 2 blue cubes. Immediately, the student applies the zero principle to four red and four blue rods and one red and one blue cube. To finish the problem, they trade the remaining red rod for 10 red cubes and apply the zero principle to the remaining blue cube and one of the red cubes. The end result is (-9).

Subtracting means taking away. For instance, (-5) - (-2) is represented by taking two red cubes from a pile of five red cubes. If you can’t take away, the zero principle can be applied in reverse. You can’t take away six blue cubes in (-7) - (+6) because there aren’t six blue cubes. Since a blue cube and a red cube is just zero, and adding zero to a number doesn’t change it, simply include six blue cubes and six red cubes with the pile of seven red cubes. When six blue cubes are taken from the pile, 13 red cubes remain, so the answer to (-7) - (+6) is (-13). This procedure can, of course, be applied to larger numbers, and the process might involve trading.

Other Uses

By no means have I explained all of the uses of base ten blocks, but I have covered most of the major uses. The rest is up to your imagination. Can you think of a use for base ten blocks when teaching powers of ten? How about using base ten blocks for fractions? So many math skills can be learned using base ten blocks simply because they represent our numbering system - the base ten system. Base ten blocks are just one of many excellent manipulatives available to teachers and parents that give students a strong conceptual background in math.

The base ten blocks skills described above can be applied using worksheets from .math-drills.com. The worksheets come with answer keys, so students can get feedback on their ability to correctly use base ten blocks.

Heralding a new age in the cosmos, Norwegian Kristian Birkeland predicted that the universe likely consisted of an exotic component that would later be called dark matter. His comments about this subject matter appeared in a description of the Norwegian Aurora Polaris Expedition (1902-1903). Birkeland’s ideas about the Expedition were published in the fateful year of 1913 which would see the rise of the socialist Federal Reserve System and the Income Tax in the United States of America, two key components of the communist manifesto. Evolutionary processes were in motion throughout all fields of endeavor. Economics, politics, science and the hearts and minds of men and women were in the balance whilst relativism not truth held sway over the modern imagination. Cosmology would suffer from the same ‘evolutionary’ mindset and Birkeland wrote as much:

“We have assumed that each stellar system in evolutions throws off electric corpuscles into space. It does not seem unreasonable therefore to think that the greater part of the material masses in the universe is found, not in the solar systems or nebulae, but in “empty” space.”

In this fashion, Birkeland predicted that because of the ‘evolutions’ present within the cosmos most of the matter in the universe must be found in ‘empty’ space rather than that which is observable in stellar objects. It is currently believed that only four percent of the universe is of this ordinary visible stellar type. Further, about a quarter of the universe is made up of the ubiquitous dark matter with the rest of the cosmos being filled with the even more bizarre dark energy. It was Fritz Zwicky, a swiss astrophysicist working for Caltech, who would further the concept of dark matter through the aegis of the Virial Theorem.

This mathematical relation is a formula which bounds the energy of a set of particles. In another dark year in the steady evolution to slavery since 1933 saw the removal of gold from the accounts of american citizenry, Zwicky used the Virial Theorem in an attempt to ascertain the validity of the dark matter hypothesis. He focussed his attention on the Coma galactic cluster and his analysis provided prima facie confirmation for the existence of dark matter. By evaluating the amount of movement of those galaxies at the periphery of the cluster he was able to approximately surmise the aggregate of all the matter therein.

He was astonished to learn that this sum total of mass is different from a separately computed estimate. This other value was obtained by analyzing the sum total of galaxies and the brightness of the Coma cluster. Juxtaposing this value with the periphery computation he observed that there was a discrepancy of at a minimum four hundredfold. Since the galaxies were insufficiently massive to cause the computed orbital velocities there must be some other mechanism to explain this phenomena. This conundrum became in the scientific lexicon the missing mass problem. Zwicky had established the need for the existence of an invisible source of mass hitherto unknown which must provide the necessary gravitational effect for the cluster.

Thus, it is a fact of the current state of cosmology that the greatest set of evidence for dark matter comes from this galactic gravitational data. Scientists have even made galactic curves describing the rotational properties of stars versus the distance from the galactic center. When the gravitational data is plotted it can be shown that only a small portion of the observed speeds are explicable by classical computations. In other words, there is a scarcity of visible mass in the observed galaxies to attribute the sum total of gravitational effects to visibly observable stars planets and galaxies. Thus, the simplest way to explain this galactic mystery of insufficient mass is to hypothesize a non-detectable type of mass known as dark matter which can be the cause for the gravitational effects.

As more and more data is collected on these and other aspects of the universe, formulae and cosmological postulates are generated describing the results so obtained. Fulfilling the requirements of the aforementioned aspects leads some scientists to propose several different types of dark matter. The four main types of dark matter are called 1- baryonic dark matter; 2- warm dark matter; 3- cold dark matter and 4- hot dark matter. Dark matter ranges from the known to the predicted, from black holes to brown dwarfs to the massive compact halo objects (MACHOs), the neutrino, axions, WIMPS or weakly interacting massive particles and the esoteric neutralino. However, there is an alternative explanation for the gravitational effects which originally created the dark matter concept.

If an incomplete understanding of gravitation is factored into the picture, then it can be asserted that the dark matter interpretation is incorrect because some other cause is generating these phenomena. Several different contending theories have been developed to describe the observed galactic data. In particular, one of the main competing explanations is given by scalar tensor theories which try to combine the teachings of quantum mechanics with gravity. Amplifying these ideas leads to a variety of exotic ideas which challenge our most fundamental notions of physics and astronomy. Other concepts go even further and have been the subject of interest for astronomers like Dr. Riccardo Scarpa since these allow for a cosmology without the inclusion of the enigmatic dark matter.

Dr. Scarpa works at the European Southern Observatory in Santiago Chile using the Very Large Telescope Array at Paranal. With all of his experience in this field, it is interesting to note some of his most recent comments on the superfluous dark matter:

“Dark matter is the craziest idea we’ve ever had in astronomy. It can appear when you need it, it can do what you like, be distributed in any way you like. It is the fairy tale of astronomy.”

In view of these comments one should ask if another scientific idea might be on the verge of collapsing. Indeed, astronomers are routinely using these other theoretical principles on a daily basis in infrared observatories around the world. Thus, it is very likely that we are simply wrong about all of this dark matter. It is within all probability that the only dark matter that we will ever find is that ignorant dark matter between our ears.

Base ten blocks are an excellent tool for teaching children the concept of addition because they allow children to touch and manipulate something real while learning important skills that translate well into paper and pencil addition. In this article, I will describe base ten blocks and how to use them to represent and add numbers.

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:

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 .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.

The concept of the invisible ether or ‘aether’ is an old concept dating to the time of the ancient Greeks. They considered the ether as that medium which permeated all of the universe and even believed the ether to be another element. Along with Earth, Wind, Fire and Water Aristotle proposed that the ether should be treated as the fifth element or quintessence; this term which literally means ‘fifth element’ has even survived down to the present day to explain an exotic form of ‘dark energy’ which is crucial in some cosmological models. These ideas spread throughout the world until the advent of a new springtime in scientific thought. The first person in the modern era to conceive of the idea of an underlying ether to support the movement of light waves was seventeenth century dutch scientist Christiaan Huygens.

Many others followed in expressing their opinions on the ether concept. Whilst Isaac Newton disagreed with Huygens wave theory he also wrote about the ‘aethereal medium’ although he expressed his consternation in not knowing what the aether was. Newton later renounced the ether theory because in his mind the infinite stationary ether would interrupt the motions of the enormous masses (the stars and planets) as they moved in space. This rejection was reinforced by some other problematical wave properties which were not explicable at the time; most notably, the production of a double image when light passes through certain translucent materials. This property of matter known as ‘birefringence’ was an important hurdle to be overcome for a proper understanding of the wave nature of light.

Some time later (1720) whilst working on other astronomical issues related to light and the cosmos, English scientist James Bradley made observations in hopes of quantifying a parallax. This effect is an apparent motion of foreground objects in comparison to those in the background. Whilst he was unable to discern this parallax effect he happened to reveal another effect which is prevalent in cosmological observations; this other effect is known as stellar aberration. Bradley was able to easily describe this aberration in terms of Newton’s particle theory of light. However, to do so in light of the wave or undulatory theory was difficult at best since to do so would have required a ‘motionless’ medium; the static nature of this ether concept was of course the property which had originally caused Newton’s denial of the idea.

But Newton’s acolytes would find themselves in a difficult position when it was shown that birefringence could be explained through another interpretation of the nature of light. If light was treated as being in a side to side action or ‘transverse motion’ then birefringence could be attributed to a light wave rather than the particle or corpuscular theory of Newton. This along with the detection of an interference effect for light by Thomas Young in 1801 renewed the ascendancy of the wave theory of light. These findings however carried with them all of the preconceived notions prevalent in the scientific mind. Since it was assumed that waves like water and sound waves required a medium of propagation, it was similarly assumed that light still needed a medium or ether for its waves to be transmitted across the universe.

However, further problems would afflict the ether theory. Because of the unique properties of a transverse wave it became apparent that this hypothetical explanation required the ether to be a solid. In response, Cauchy, Green and Stokes contributed theoretical and mathematical observations to an ‘entrainment’ hypothesis which later came to be known as the ‘ether drag’ concept. But nothing would give more impetus to these ideas than when James Clerk Maxwell’s equations (1870s) required the constancy of the speed of light (c). When the implications of Maxwell’s equations are worked out by physicists, it was understood that as a result of the need for a constant speed of light only one reference frame could meet this requirement under the teachings of Galilean Newtonian relativity. Therefore, scientists expected that there existed a unique absolute reference frame which would comply with this need; as a result, the ether would again be stationary.

As a consequence, by the late nineteenth century the aether was assumed to be an immovable rigid medium. However, earlier previous theories existed as to the nature of the aether. One of the most famous of these is known as the ‘aether drag’ hypothesis. In this concept, the aether is a special environment within which light moves. Also, this aether would be connected to all material objects and would move along with them. Measuring the speed of light in such a system would render a constant velocity for light no matter where one tested for light’s speed. This ‘aether drag’ idea originated in the aftermath of Francois Arago’s experiment which appeared to show the constancy of the speed of light. Arago believed that refractive indexes would change when measured at different times of the day or year as a result of stellar and earthly motion. In spite of his efforts, he did not notice any change in the refractive indexes so measured.

Many other experiments would follow; these were performed in order to find evidence of the aether in its many different abstractions. However the most important of these was conducted by american scientists Michelson and Morley. Their experiment considered another alleged effect of a different aether theory which came to be known as the aether wind. Since the aether permeated the entire universe, the earth would move within the ether as it spun on its axis and moved within the solar system about the sun. This movement of the earth with respect to the aether gave rise rise to the idea that it would be possible to detect an ‘ether wind’ which would be sensed because of the aforementioned movement. Thus, their experiment was essentially an attempt to detect the so-called ether wind. This mysterious zephyr would be nearly impossible to detect because the aether only infinitesimally affected the surrounding material world. Michelson first experimented in 1881 with a primitive version of his interferometer; a mechanism designed to measure the wave like properties of light. He would follow this by combining forces with Morley in the most famous ‘null’ experiment of physics.

In this investigation, Michelson utilized an improved version of his interferometer device. Michelson’s apparatus would help him win the Nobel prize for his optical precision instruments and the investigations carried out with them. His most important study being what became known as the Michelson Morley experiment of 1887. Michelson and Morley used a beam splitter made of a partially transparent mirror and two other mirrors arranged horizontally and vertically from a light source. When a beam of light traveled from a source of coherent light to the half-silvered mirror (the semitransparent mirror) it is transmitted to either of the horizontal or vertical mirrors. When the light returned to the eyepiece of an observer the separately returning light waves would combine destructively or constructively. This phenomenon is known as the interference effect for light. It was hoped that a shifting of the interference fringes from that which was normally predicted would be able to ascertain the existence of the aether wind.

To detect this effect, the Michelson interferometer was prepared in such a manner as to minimize any and all extraneous sources of experimental error. It was located in a lower level of a stone edifice to eliminate heat and oscillatory effects which might comprise the experimental results. Additionally, the interferometer was mounted atop a marble slab that was floated in a basin of mercury. This was so that the apparatus could be moved through a variety of positions with respect to the invisible ether. But despite their many preparations the experiment did not yield the expected fringe patterns. Thus, Michelson and Morley concluded that there was no evidence for the existence of the ether. Others would replicate the experiment in different incarnations which modified the premise of the experiment. Each and every one returning a similar negative result. Modern theorists have taken these results and those of many other experiments as being indicative of the non-existence of the aether. However, even the negative result of Michelson Morley has come in to question as far back as 1933.

In that year, Dayton Miller demonstrated the fact that even though the duo’s experiment had not specifically found the expected range of interference patterns, they had found an interesting little noticed effect. Miller then went on to suggest that Michelson Morley had found an experimental sine wave like set of data that correlated well with the predicted pattern of data. He also described how thermal and directional assumptions inherent in the experimental arrangement may have impacted badly on the fringe interference data. Thus, the test may have been performed in an imperfectly conceived experimental setup and with a built in mathematical bias against the detection of an appropriate outcome. Thus, in the future the aether theory in some form or another may still be sustainable as a foundational theory of physics.

Perhaps it is best to leave with these ideas as expressed in 1920 by Einstein who stated that he believed the ether theory to still be relevant to his ideas on space and time:

“More careful reflection teaches us, however, that the special theory of relativity does not compel us to deny ether. We may assume the existence of an ether”

he continued:

“Recapitulating, we may say that according to the general theory of relativity space is endowed with physical qualities; in this sense, therefore, there exists an ether”

and finally:

“According to the general theory of relativity space without ether is unthinkable; for in such space there not only would be no propagation of light, but also no possibility of existence for standards of space and time (measuring-rods and clocks), nor therefore any space-time intervals in the physical sense. But this ether may not be thought of as endowed with the quality characteristic of ponderable media, as consisting of parts which may be tracked through time. The idea of motion may not be applied to it.”

More than likely, when you learned how to add, you started on the right and moved to the left. If you were adding whole numbers, you added the ones, “carried” if necessary, and repeated for the tens, hundreds and so on. This works well on paper, and it is the most efficient paper and pencil method; however, adding in the other direction has several desirable advantages: the left to right method promotes a better understanding of place value, it can be done mentally with much greater ease, and it does not require that numbers be lined up in a column. Students can learn left to right addition, so they have another method to choose from when presented with addition problems.

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.