In order to divide rational expressions accurately, special rules for radical expressions can be followed. Some of those rules include the quotient rule, rules for finding the square roots of quotients, and rationalizing the denominator.

Suppose the problem is to evaluate the square root of 36/49. The problem can be written in two different ways, either √ (36)/√ (49) or √ (36/49). The square root of 36/49 solved with the first method is √ (36)/√ (49) or 6/7. The square root of 36/49 solved with the second method is √ (36/49) or 6/7. This leads to the Quotient Rule, that for any non-negative number A and any positive number B, the quotient of two square roots is the same whether the square root is taken from the entire ratio or the square root is taken of the numerator and the denominator separately.

Figure 1: The Quotient Rule in Symbol Form

This rule is commutative, so that it means the same whether the square roots are taken together in a ratio or taken separately. Sometimes it is easier to solve the problem if the square roots of the numerator and the denominator are considered separately. That way, they can be simplified using perfect square roots if they are available. Suppose the problem is to find the square root of 18/50. Neither 18 nor 50 are perfect squares, but the ratio 18/50 can be multiplied by 2/2, to give 36/100. The square root of 36 is 6 and the square root of 100 is 10, to result in a fraction of 6/10, which can be further simplified to 3/5. Alternatively, 18 could be factored as 2 times 9 and 50 as 2 times 25, so the ratio could be divided by 2/2 to be further simplified as 3/5.

One of the ways to move a radical out of the denominator is to multiply the number by a form of 1, so that the denominator becomes a perfect square. Suppose the expression is √ (3/5). While the entire fraction is still under the radical sign, multiply each number by 5/5 so that the new fraction is 15/25 or √ (15)/5.

Figure 2: The process of rationalizing the denominator under the radical sign.

The other way to move a radical out of the denominator is to multiply both the numerator and the denominator for a form of 1 using radicals. Suppose that the ratio is 5/√ (3). In order to move the radical out of the denominator, both the numerator and the denominator can be multiplied by a form of 1 with the radical expression √3/√3 to result in (5√3)/3.

Figure 3: Rationalizing the denominator by using radicals, in symbol form.

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]]>Mathematical patterns underlie everything in the natural world, including those human extensions of the natural world that surround us. Some of those mathematical patterns are expressed in the arts, including visual arts such as painting and sculpture, and the physics and mathematics of music. Other mathematical patterns are expressed in performance arts, such as the choreography of dance.

Renaissance artists represented mathematical patterns and concepts in visual form, using mathematical concepts such as symmetry and perspective. A two-dimensional canvas is used to portray three-dimensional surfaces. Painters used tricks of perspective, optical illusions of geometry such as the vanishing point and the horizon line. They were especially skilled at making objects appear to converge at a single point, just as people might see parallel train tracks in the distance. Those converging points fall along the horizon. Also, objects appear larger the closer they are and smaller the farther away they are, even though the real size of the objects has not changed. Symmetry was also important to create pleasing patterns of objects or shapes in a painting or artwork made of mosaic, fiber, or glass.

From the time of the ancient Greeks, through the Renaissance. The Golden Ratio, also known as the divine proportion, was represented by the Greek letter phi φ. It is an irrational number, about 8/5. Sculptors used it to determine the most pleasing lines in their works. The Golden Rectangle is related to the Golden Ratio. In the Golden Rectangle, the long side of the triangle is φ times as long as the short side. Leonardo da Vinci used the Golden Rectangle in his paintings and sculpture, putting important figures inside one. Painters from Seurat to Mondrian, as well as painters and artists of the present day, use the relationships of the Golden Ratio and the Golden Rectangle in many of their works.

Pythagoras wrote extensively about the geometric foundations of music, the relationships of pitch, modes, and octaves. For example, if a string of a certain length is plucked, dividing it into shorter lengths will produce a higher pitch, as anyone who has ever played a stringed instrument knows. Similarly, a Pan pipe consists of four or five wooden recorders of different lengths that are bound together. Octaves are pleasant because of the relationships between their pitches and frequencies. The Greeks used a 7-note diatonic scale, in various relationships between tones, called modes. Modes predated the 12-tone scale that was well-tempered by Johann Sebastian Bach. Time signatures are standardized, and the length of time each note is played in a time signature can be represented by fractions and patterns.

Dance and other kinetic arts have mathematical relationships of their own, modeled by dynamic equations and the behavior of solid objects in space. Think of the parabolic leaps made by dancers as they seem to fly across the stage and pause in midair, or the relationships between dancers as they cross the stage, from ballroom dancing to hip-hop.

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]]>Mathematical patterns underlie human activity, as far back as we can discover ancient artifacts. Architectural wonders such as the Parthenon, the Pyramids, and other ruins show that civilizations were able to apply geometric and mathematical relationships. Neolithic sites as far away as Newgrange, Stonehenge, and many others around the world, show an ancient grasp of complex mathematical calculations. Artifacts from the smallest decorated cup to the largest mosaics and tessellations show an appreciation of symmetry and balance.

The ancient Greeks built the Parthenon and other complex building as monuments. Not surprisingly, the Parthenon, their most important temple on a high hill overlooking their capital city, is based on their divine proportion phi φ, also called the Golden Ratio. The Golden Ratio is also evident in the Great Pyramids of Egypt. Mayan temples in Central America show complex mathematical relationships that reflect their intense interests in mathematics and astronomy.

Stonehenge is a ring of standing stones as part of the Amesbury complex. It is not known how the Neolithic builders used it, but the rings have relationships to each other, as well to the position of the sun during the summer and winter solstices and the spring and fall equinoxes. Other standing stones exist in many other parts of the world, such as Newgrange in Ireland, Carnac in France, Cueva de Menga in Spain, and other locations throughout Europe.

Many civilizations around the world have developed intricate patterns of mosaics, as well as repeating tile patterns called tessellations. Tiles have been found in many different areas around Mesopotamia, including some intricate mosaic flooring in ancient temples, homes, and other public buildings. Mosaics were used in ancient Greece and Rome, as well as throughout the Byzantine Empire. Tessellations, or repeat patterns, are found all over the world, with some of the most beautiful and intricate patterns in old Seville, Spain, through Moorish architecture.

The earliest examples of weaving show a knowledge of symmetry and design. For example, symmetrical embroidery stitch patterns are echoed in traditional embroidery from East Asia, throughout India, throughout Egypt and the Middle East, to be duplicated in forms throughout Europe. The Hardanger embroidery techniques found in the Scandinavian countries are based upon multiples of threads and complex geometric relationships.

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]]>Numbers in Base 2 are real numbers that can be involved in mathematical operations in the same way as more familiar decimal or Base 10 numbers. The advantage of binary systems — that they contain two numerals 0 and 1 — means that they can be used in situations that have two alternatives.

Base 2, or binary, is the smallest grouping possible. In Base 2, 0 means 0, 1 means 1, and 10_{2 }means 2, 11_{2 }means 3, 100_{2} means 4, 101_{2 }means 5, 110_{2 }means 6, 111_{2 }means 7, 1000_{2 }means 8, 1001_{2 }means 9, and 1010_{2 }means 10 in the decimal (Base 10) system. Notice that place value follows the same pattern in both bases. 10^{0} is 1, and 2^{0} is 1, 10^{1} is 10 and 2^{1} is 2, 10^{2} is 100 and 2^{2 }is 4, represented in Base 2 as 100_{2}; 10^{3} is 1000, and 2^{3 }is 8, represented in Base 2 as 1000_{2}. The difference is that a number such as 10_{2} (which means 2) is read either as “two” or “one-zero”, 100_{2} is read as “eight” or “one-zero- zero”, and so on.

Binary numbers add, subtract, multiply, and divide according to the same rules as numbers in any other base. Addition of two binary numbers involves carrying, similar to adding numbers in Base 10. Suppose the numbers to be added are 110011_{2} and 10001_{2}. Adding from right to left, the sum is 1000100_{2}, because adding 1 and 1 in Base 2 equals 10_{2}. Subtraction uses similar additive inverse rules to other systems, so that subtraction means adding a negative number. More borrowing is used than in Base 10 subtraction.

Multiplication in Base 2 follows a similar repeated addition model as multiplication in decimal systems, and division in Base 2 follows a similar repeated subtraction model as division in decimal systems. Suppose that 110_{2 }is multiplied by 101_{2}. Partial products are used, so that the first row [110_{2 }times 1_{2}] is 110_{2}, the second row is [110_{2 }times 0] moved to the left one space, and the third row is [110_{2 }times 1_{2}] moved to the left 2 spaces. The entire product is 11110_{2}. It really helps to use graph paper or columnar ruled paper to keep the columns in line while adding or subtracting them.

Converting fractions or decimals into the binary Base 2 system presents a challenge, because many fractions and decimals that are exact in the decimal system have approximate values. The decimal 0.10 has an approximation in Base 2, and so do fractions such as ¼, 1/3, ½, and many others. Operations in Base 2 with fractions are approximate, called “floating point arithmetic.”

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]]>The concept of dividing a whole into parts and then dividing the parts into smaller entities is basic to mathematics. That model of division can be illustrated by manipulatives such as fraction bars and Cuisenaire rods. Equality and inequality of fractions can be demonstrated by finding their common denominators and comparing them. By comparing unequal fractions and finding more fractions between them, students can show the density of the number line.

A fraction is a part of a whole. Children can illustrate the concept by taking a whole circle, then cutting the circle in half, then cutting each half into halves to show fourths, then cutting each fourth into eighths, and so forth. Many different nations in ancient times, such as the Egyptians, Hindus, and Babylonians used fractions in different computations. The Arabs separated the numerator and a denominator by a bar between them.

Manipulatives such as fraction bars and Cuisenaire rods are special types of objects that can be used to illustrate the equality or inequality of fractions. Ten white cubes equal the same length as one orange rod, and so do five red rods. It can be shown that one red rod is the same length as two white cubes, in symbol terms 2/10 = 1/5. Similarly, 3 white cubes, or 3/10, is not the same size as 1 purple rod, or 1/3.

The best method to test whether fractions are equal or unequal is to find a common denominator for all the fractions in the group and then compare them. Suppose the fractions are 5/8 and 3/5. The fraction 5/8 is equal to 25/40, but the fraction 3/5 is equal to 24/40. They are not equal. Suppose the fractions are 3/5 and 6/10. Since 3/5 is equal to 6/10, the fractions are equal.

There is always another fraction that can be found between any two fractions on the number line. Take a closer look at the section between 0 and 1 on the number line. At first glance, it appears full. Halfway between 0 and 1 is ½, and halfway between that is ¼, and halfway between that is 1/8, then 1/16, then 1/32, and then there’s a point for ¾, and there’s also a point for 3/8, 5/8, and 7/8, 1/16, 3/16, 5/16, 7/16, 1/32, 3/32, and so on. Fractions are dense along the number line.

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]]>Some special factoring formulas include the difference of two squares, the sum of two cubes, and the difference of two cubes. If there are three terms or more in the polynomial, students can use strategies such as finding common factors and factoring by grouping.

The difference of two squares [(a + b)(a - b)] is a common pattern with binomials involving variables to the second power. However, the concept can also be applied to exponents higher than x^{2}. Any even power (such as x^{2}, x^{4}, x^{6}, and so on) can be factored into squares evenly. For example, 16x^{4} can be rewritten as (4x^{2})^{2} and 49y^{6} can be rewritten as (7y^{3})^{2}. The expression 16x^{4} – 49y^{6} is then factored as (4x^{2} – 7y^{3})(4x^{2} + 7y^{3}).

The product of (a + b)(a^{2} – ab + b^{2}) can be evaluated using FOIL as a^{3} –a^{2}b + ab^{2} +a^{2}b – ab^{2} + b^{3}. That simplifies to a^{3} + b^{3}. Suppose that the binomial that needs to be factored is 27x^{3} + 8. That expression will factor as (3x + 2)(9x^{2} – 6x + 4).

Figure 1: Factoring the sum of two cubes and the difference of two cubes.

The product of (a – b)(a^{2} + ab + b^{2}) can also be evaluated and simplified to a^{3} – b^{3}. The easiest way to remember the direction of the signs when factoring the sum or difference between two cubes is to use the acronym SOAP. The sign between the terms of the binomial factor is in the same direction in both the sum of the cubes and the (a + b) factor. (If the difference of cubes is the issue, the sign in a^{3} – b^{3} and a – b is negative.) The sign is opposite between the a^{2} term and the ab term, such that if it is the sum of cubes the sign between a^{2} and ab is negative, and if it is the difference in cubes, the sign between the a^{2} and ab term is positive. The sign between the ab term and the constant is always positive.

Figure 2: Using the acronym SOAP to remember the direction of the signs.

The first step in factoring a polynomial is always to factor out anything that is common to every term in the polynomial. Suppose that the polynomial to be factored is 3x^{2} + 6x + 9. The first step in factoring would be to remove the common factor of 3 from all the terms as 3(x^{2} + 2x + 3). Next, check to see if it follows any of the special factoring forms. It can be factored by grouping or another method.

Figure 3: Following the general steps to factor a polynomial.

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]]>Space figures exist in three dimensions. Some examples include spheres, cubes, tetrahedra, polyhedra, cylinders, cones, and other types of solids.

Plane figures such as polygons and curved figures are two-dimensional and flat. They have length and width. Space figures exist in three dimensions, adding depth. A sphere has a center point, and the myriad circles that exist around it form the sphere. A square in three dimensions is a cube.

A sphere has one fixed point at the center. When all points on the outside of a sphere are connected with its center, it becomes solid. Many real world objects are spherical or spheroid in shape, such as the earth and other planets, the sun and other stars, and bubbles. A round shape such as an oval can become a three-dimensional ovoid. Real-world examples include eggs or teardrops.

A tetrahedron is a solid with triangular faces. If the triangles that form a solid have any other polygon at the base, that solid is a pyramid. If other polygons are joined into a solid, that solid figure is called a polyhedron. If two faces of the polyhedron are congruent, such as a square at the top, and a congruent square at the bottom, the polyhedron is called a prism. Some polyhedra are regular, and others are irregular. Polyhedra have special properties; and their volume can be measured.

Cylinders are classic solids with two circular bases of the same size. If the bases are directly opposite each other, the figure is called a right cylinder. If the circular bases are not directly opposite, the figure is called an oblique cylinder. Cones are figures that rise from a circular base to a vertex. They may also be right or oblique. Classical solid figures include tetrahedra, cones, cubes, prisms, cylinders, and other polyhedra. However, other solid figures can be easily imagined as combinations of the classic shapes.

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]]>Plane figures in geometry include points, segments, lines, rays, and angles, polygons, curves, and circles. They appear on a flat plane.

Think of a pencil point, and make it smaller, until it is a no-dimensional, undefined, geometric concept that only exists in the minds of mathematicians. Segments are made up of points, and they consist of the beginning, the end and all points between the beginning and the end. They exist in two dimensions. Extend the segment to infinity in both directions, and it is a line. Start from only one point in any direction and extend it to infinity, and it is a ray. Rays are usually designated by the beginning point and the endpoint of a line segment. Two rays that meet at a common point, the vertex, are called an angle, and the angle can be measured with a protractor or duplicated using a drawing compass.

A three-sided polygon is called a triangle, because it consists of line segments arranged in three angles. Four-sided polygons may be squares, if all four sides are equal and each side meets at right angles. Other types of four-sided polygons include the rectangle, the rhombus, the parallelogram, and the trapezoid. Polygons are usually named for the number of sides they have; such as a pentagon with 5 sides, a hexagon with 6 sides, a heptagon with 7 sides, and octagon with 8 sides, and so on.

A set of points connected by a continuous line is called a curve. Curves can be open or closed. A circle is a type of closed curve that has a single center. Each point along the circumference of the circle is at the same distance from the center. The distance from the center to the circle itself is called the radius, and any circle measures 360^{o}.

Many other curved figures exist in geometry and in the real world, such as ellipses, arches, ovals, lenses, and crescents. Technically, polygons are also curves, because they can be formed by a continuous line, even if that line doesn’t appear curved.

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]]>If division were limited to single-digit divisors and math facts, it would be comparatively easy. Long division is more difficult to visualize and tie to real-world manipulative objects. Paradigms exist for long division, and there are tips and tricks for dealing with other problems.

Sometimes it is easier to tell if a dividend is evenly divisible by a divisor before working the problem out on paper. If a number in the units digit is divisible by 2 or 5, the entire number will be divisible by 2 or 5. For example, 385 is divisible by 5. Similarly, if the sum of the digits in a number is divisible by 3 or 9, the number is divisible by 3. If the sum of the last 2 digits is divisible by 4, the number will be divisible by 4. If a number is divisible by both 2 and 3, it will be evenly divisible by 6.

The steps for division include directions for repeated subtraction. This paradigm can be used for one-digit divisors as well as long division. It involves subtracting multiples of the divisor, adding the results up, and continuing until the leftover amount is less than the divisor. Suppose the number is 124/32. Then subtract 124 – 32 = 92 – 32 = 60 – 32 = 28. This method does not use place value, and makes it easier to see the concept.

One of the ways to look more carefully at division is to color-code both the dividend and the divisor using the color-coding that is available for chip-trading, such as red, thousands place, green, hundreds place, blue, tens place, and yellow, units place. Suppose the division problem is 4800/25. That would result in 4 red chips and 8 green chips. The 4 red chips cannot be evenly divided by 25, but they could be traded in for 40 green chips. The 40 green chips can be added to the 8 green chips that are already there to leave 48 green chips. The 48 green chips can then be divided by 25, which will leave 23 green chips. The 1 goes above the green. The 23 green chips become 230 blue chips. 225 of them (25*9) can be eliminated, and the 9 can go above the blue. The remaining 50 is evenly divisible by 25, so the solution is 100 + 90 + 2 = 192.

Normally, any number divided by itself is 1. What happens when zero is divided by itself? The answer is not 1, it is not determined. If any other number is divided by zero, the answer cannot be determined either. Dividing by zero is not allowed, because it doesn’t make any sense.

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]]>Division is the inverse of multiplication. If a times b equals c, and b is not equal to zero, then c divided by b equals a. Also, for any real number a not equal to zero, a times 1/a equals 1.

Division is one of the four basic arithmetic operations. It is usually the last one to be taught, because solving division problems can involve addition, subtraction, and multiplication. Suppose the problem is 36 divided by 4. The answer, 9, is a math fact, because 9 times 4 is 36. It can be solved by using the definition of division, as 36/4 =9. It can be solved by subtraction, as 36 – 4 = 32 – 4 = 28 – 4 = 24 – 4 = 20 – 4 = 16 – 4 = 12 – 4 = 8 – 4 = 4 – 4 = 0. The number 4 was subtracted 9 times from 36. To check division, students can multiply the quotient by the divisor, or add the divisor repeatedly.

Suppose you had 100 movie tickets and you wanted to give 4 free tickets to as many people as possible who were waiting in line outside a theatre. One way that could be done is to measure sets of 4 tickets and give them to the first people in line until there were no more left. In this case, 100 is the dividend, 4 is the divisor, and dividing 100/4, gives 25 groups of 4 tickets each. The first 25 people in line get 4 free movie tickets.

Similarly, suppose the 100 movie tickets were divided into 4 equal parts. How many tickets would be in each set? The quotient would be the same, 25, but this time the divisor 4 represents the number of equal parts. Even though the numbers are the same in both models, the concepts are a little different.

Many hands-on models for division exist, such as dividing beans or cubes into groups, or using the abacus to illustrate division by measurement. When students number off in groups of 4 to take part in activities or register for one of 5 home rooms, they illustrate the partitive model.

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