Solid objects have both volume and surface area. Formulas for calculations can be expressed using algebra, especially if the measurements of sides are stated by variables. The ratio of volume to surface area is important in applications in chemistry and biology.

The volume of a solid is the amount of space that it contains. For example, the volume of a rectangular prism is measured as its length times width times height. In math language, V=lwh. The volume of a cylinder is π times the radius squared times the height of the cylinder, or V=πr^{2}h. The volume of a cone is 1/3 πr^{2}h, very similar to the formula of the volume of a cylinder. The volume of a pyramid is 1/3 lwh, which is similar to the formula for the volume of a rectangular prism, like a cube or a brick.

The surface area of a solid object is the total area contained in the two-dimensional surfaces of the solid object. (Think of the amount of wrap it would take to completely cover all surfaces without overlapping.) For example, the surface area of a rectangular prism is 2(lw +wh +lh). A cube has 6 faces, a top, bottom, and 4 sides. The surface area of a cylinder has 2 πr^{2}+ 2πrh, because a cylinder has a top circle, a bottom circle, and the circular surface. The surface areas of cones and pyramids are more complex to calculate and depend on the orientation of the object.

Suppose the solid figure has measurements expressed in variables rather than constant numbers. For example, a rectangular prism has a length of c+1, a width of 2c, and a height of c-5. The volume of the rectangular prism would be (c +1)(2c)(c-5). The expression could be rearranged to (2c)[(c +1)(c-5)] using the Commutative and Associative Properties. The expression (c +1)(c-5) could be expanded to c^{2}-4c -5. Multiply each part by 2c for 2c^{3}-8c^{2 }-10c.

The surface area of an object is important to various types of chemical reactions. For example, iron ground into a fine powder has a larger surface area than the same amount of iron in a block. It catches fire easily, while the block of iron is highly stable.

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]]>Trigonometry in the form of triangulation is at the heart of navigation, whether it is by land, sea, or air. GPS systems use triangulation to find and fix positions, extrapolating what is unknown from what is known. Triangulation is also used to measure the distances between earth and distant stars and galaxies, thus aiding future space travel.

The ancient Greeks, Egyptians, and other ancient civilizations developed methods to measure triangles accurately. The Egyptians used principles of trigonometry to build the Pyramids, while the Greeks developed extensive geometric and trigonometric proofs and applied them to many surveying and navigational problems. If the measurement of two angles are known, the third can always be calculated. The tangent of a right triangle can also be used to calculate valuable distances such as the height of a tall tree or of a mountain.

Surveyors use a special type of telescope called a theodolite. It is mounted on a tripod, and measures distances and angles in three-dimensional space. Mounting a telescope on a tripod allows the telescope to be rotated, making very precise measurements possible. Known distances and angles can then be triangulated for accurate mapmaking, road building, and other engineering applications. The procedure for triangulation is similar to measuring the height of a tree. Suppose one knows the accurate measure of baseline AB. The distance to point C can be used by measuring the angles for AC and BC very accurately and using the trigonometric ratios.

Navigation by sea is complicated by large distances without landmarks in open ocean. The principles of trigonometry and triangulation apply. For most of the time humanity has moved through water for long distances, the only landmarks have been the positions of the sun by day and the stars by night. Those angles and distances can be measured accurately by using devices such as the marine sextant (for angles) and the chronometer (for the exact time that measurements are taken). Navigation by sea is based upon spherical trigonometry. The exact position of a ship can be determined by the angle the celestial body makes with the horizon, measured at a precise time. The angle and precise time measurements are compared with tables of known values.

GPS is short for Global Positioning System. It has grown from an original network of 24 satellites to a network of over 30 satellites from the United States. Similar systems are under development from Russia, China, Japan, the European Union, and India, and many satellites are fully operational. Satellites orbit over the same locations every day, and emit signals continuously giving the exact time and their location. Triangulation with particular satellites allows for precise location mapping.

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]]>Students can connect algebra to geometry by expressing geometric inequalities in compound inequality form. Since the sum of the lengths of any two sides of a triangle is greater than the length of the third side, there are three inequality statements. All three inequality statements must be true for three line segments to form a triangle.

The perimeter of any polygon is the sum of the measurements of its sides. Therefore, the perimeter of a triangle is the sum of the measurements of its three sides. Suppose a triangle has 3 sides with measurements a, b, and c. In Euclidean geometry, there are 3 basic inequality statements for any triangle: a +b > c; b +c > a; a +c > b.

In order for a triangle to be a right triangle, one angle must be a 90 degree angle. The sides of the triangle are in special relationship with one another, expressed by the Pythagorean Theorem, such that a^{2}+ b^{2} = c^{2}. Right triangles are a special case of the Triangle Inequality, such that the measure of the hypotenuse c is greater than the measure of either leg a or leg b. However, the measure of c must also be less than the sum of both legs. Suppose an isosceles triangle ABC is constructed so that AB and BC are equal. By definition, the base angles B and C are also equal. If an altitude CD is constructed, it will divide ABC into two right triangles, ACD and CBD. Euclid showed that both AD and CD are shorter than the hypotenuse AC. According to the Triangle Inequality, the measure of AD + CD must be greater than AC.

A figure made up of three line segments is not a triangle if any of the inequalities are untrue. Therefore, all three conditions must be satisfied such that a + b >c; a + c >b, and b + c =a. If any of the elements are untrue, the entire statement is untrue. Students can use the Triangle Inequality as a test for geometric figures, if given the lengths of any three line segments.

One of the most famous of the special triangles is the 3-4-5 triangle. It got its name because one side measures 3 units, one side measures 4 units, and one side measures 5 units. It is also a right triangle, and demonstrates the Pythagorean Theorem, because 3^{2}+ 4^{2}= 5^{2}. It also follows the Triangle Inequality because 3 +4 & gt;5, 3 + 5 >4, and 4 + 5 >3. The sides of the 3-4-5 triangle follow an arithmetic progression. In algebraic terms, sides follow a pattern of a, a + d, and a + 2d. Let a be 3, then 3 + 1 is 4, and 3 + (2·1) = 5. It can be shown that any triangle that follows an arithmetic progression, and fulfills all three inequalities of the Triangle Inequality is similar to the 3-4-5 triangle. Suppose that three line segments follow the arithmetic progression 6, 7, 8. They will form a triangle, because 6 + 7 >8; 7 + 8 >6; 6 + 8 >7. They are also similar to 3, 4, 5 because 6-3 is 3, 7-3 is 4 and 8-3 is 5.

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]]>Any spinning object rotates around a central point called an axis. Tops, dreidels, gyroscopes, and spinning eggs rotate, rise, and seem to defy gravity, as long as they are moving. When they stop moving, they fall.

Angular momentum refers to movement in a circle rather than a straight line. Imagine the movement of one dot on the spinning object as it twirls (such as one dot above the right of the Hebrew letter shin). Its path could be described by angular momentum around the axis through the point of support. The equation that describes angular momentum L takes into account its mass, shape, and rate of speed. In this equation, L = *I*ω, means that angular momentum is the product of an object’s inertia, or resistance to change in its velocity, times its angular (or circular) velocity ω.

Many spinning objects tend to conserve angular momentum. Imagine a figure skater entering a spin. She brings her arms close to her body, spinning faster and faster, and is able to keep balance partly because of angular momentum (and a lot of practice). A rigid object, such as a sphere, will remain upright as it spins on an axis because angular momentum is conserved as long as it is spinning. The system is effectively closed due to Newtonian laws of motion, and can be described in terms of relationships between derivatives.

As a top, planet, or gyroscope spins, its axis does not stay at a fixed point. It tilts and rolls. Precession depends upon whether a force, known as torque, is applied to the spinning object, and how much force is applied. Similar to the equations describing angular momentum, the equations that describe precession also use inertia, rate of speed around the axis, and angle between the symmetry axis and direction of inertia.

Gyroscopes are self-balancing, and have been used since the 18^{th} century in compasses, navigational tools, and in toys. They use general principles of three axes: pitch, roll, and yaw in order to plot precise position in space. Tops have been used as toys in cultures from ancient China and India to Greece and Rome. Spinning eggs and some types of tops actually change axes as they swirl and slow down. All spinning objects rely on similar principles, involving angular momentum, precession, and inertia.

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]]>Trinomial squares are also known as perfect square trinomials, and are the squares of binomial expressions. They factor as (a + b)(a + b) or (a – b)(a – b) where a and b are real numbers. Forms such as (a + b)(a -b) are special products that are also called the difference of squares.

When a squared binomial such as (a + b)(a + b) is multiplied using FOIL, the values of a and b follow a specific pattern. Recall that the first term is a^{2}, the outside and inside terms are ab + ba, and the last term is b^{2}. If a polynomial follows the form ax^{2} + bx + c, a and c are perfect squares, and the b coefficient is twice the sum of ac, it is a perfect trinomial square. Suppose the polynomial is 36x^{2} + 60x + 25. 36x^{2} is a perfect square of 6x, and 25 is a perfect square of 5. The inner and outer terms are 30x + 30x or 60x. That polynomial is (6x + 5)^{2}.

Figure 1: The perfect square trinomial follows a specific pattern.

What if the squared binomial is (a – b)(a – b)? When it is multiplied using FOIL, a^{2} is a perfect square, and so is b^{2}. The sign of 2ab is negative, because it is the sum of two negative products. Suppose the polynomial is 100x^{2} – 80x + 16. The square root of 100x^{2} is 10x, and the square root of 16 is -4. The product of 10 and -4 is -40, and twice -40 is -80. That polynomial is (10x – 4)^{2}.

Figure 2: An example when the middle term is negative.

If the trinomial follows the form –ax^{2} +bx +c or ax^{2} – bx – c or ax^{2} + bx – c, it does not follow the squared trinomial pattern. The coefficient of a squared term cannot be negative even if the term is a perfect square. When a negative is multiplied by another negative, the product is positive. Similarly, if the constant c is not a perfect square, the trinomial does not follow the squared trinomial pattern.

The last special pattern to consider is (a + b)(a – b). Since multiplication is commutative, it is also the same as (a – b)(a + b). It is called the difference of squares. When (a + b)(a – b) is multiplied using FOIL, the first term is a^{2}, and the last term is b^{2}. The outside term is –ab and the inside term is ab, which adds up as zero. They cancel each other out. Suppose a polynomial is 144x^{2} + 81. The square root of 144x^{2} is 12x and the square root of 81 is 9. Following the pattern, the factoring is (12x + 9)(12x – 9).

Figure 3: An example of the form (a + b)(a – b).

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]]>Rational equations have many applications to real-world problems. Some of the types of problems that can be solved include geometric ratios, number problems, motion problems, and work problems.

When solving geometric ratios, it helps to use the problem-solving strategy of making a sketch to check against and see if the answer makes sense. Suppose that Janet and LaToya are building a deck on the back of the house. They know that the width of the deck cannot extend back more than 3 feet, or they will be building into the neighbor’s yard. If the deck will take up 26 square feet of the backyard, what must the length be? We know that the area of the deck is 26 square feet, and since the length times the width is the area, the ratio of the area to the width will give the length. The ratio of 26/3 will give the length of 8.67 feet, or 8 feet 8 inches.

Figure 1: Building a deck uses geometric ratios.

Suppose that one number is 3 times another number, and the sum of their reciprocals is 5/6. What are the numbers? If x is one number, then 3x is the other number. The sum of 1/x + 1/(3x) is 6/5. If every term of the equation is multiplied by 3x to get rid of the denominator then 3 + 1 = (6/5 ∙3x), and 18/5x, or 4 = 3.6x, or x = 0.9.

Suppose that it takes Devon the same amount of time to go downstream 14 miles with a current that moves at 3 miles an hour as it does to go 6 miles upstream. How fast would his canoe go in still water? The time it takes to travel in the canoe can be represented by r for rate of speed. Then r + 3 is rate downstream and r-3 is the rate upstream, against the current. The distance is 14 miles downstream and 6 miles upstream. If time = distance/rate, then the time in still water is 14/(r + 3) and 6/r – 3. Further 14/(r + 3) = 6/(r – 3). Using cross-multiplication 14(r – 3) = 6(r + 3) or 14r – 42 = 6r + 18, or 14r – 6r =18 + 42 or 8r = 60, or r = 7.5. The equations check because 14(7.5) – 42 = 6(7.5) +18

Figure 2: An example of a motion problem.

Suppose that Lori can weed their garden by hand in in 10 hours and Jason can weed their garden in 12 hours. How long will it take them to weed it working together? Let t equal the time worked to weed the garden by hand. Then t/10 + t/12 will equal 1 (the time it takes them to weed the entire garden). Using the LCD, then 6t + 5t = 60, or 11t = 60, 5 hrs, and 5 minutes.

Figure 3: How much time will it take 2 people to care for the garden?

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]]>The relationship of the length of the legs of the right triangle to the hypotenuse is expressed by one of the most famous equations in geometry, the Pythagorean Theorem. Deriving the measurement of the sides of the right triangle was one of the earliest uses of irrational numbers.

Right triangles have one 90-degree angle. The legs of the right triangle form the 90 degree angle, and the side opposite the 90-degree angle is the hypotenuse. The relationship between the lengths of the sides is given by the Pythagorean Theorem, a^{2} + b^{2} = c^{2}. That means that if the length of the hypotenuse and one leg is known, the length of the other leg can be derived. Suppose that the length of the hypotenuse is 5 cm and one leg is 3 cm. Using the Pythagorean Theorem, 3^{2} + b^{2} = 5^{2}, or 9 + b^{2} = 25. So 25 – 9 = b^{2}, or 16 = b^{2}. Taking the square root of both sides, b = 4 cm.

Figure 1: The Pythagorean Theorem.

The followers of Pythagoras soon found that there was a big scar in the perfect geometric world. This was such a shock to them that they took out one Hippasus of Metapontum (who used irrational numbers) and drowned him at sea, or so the story goes. They couldn’t drown the idea so quietly, because the measurement of the side is not always a rational number. Suppose one leg of the triangle measures 10 feet and the other leg measures 15 feet. What is the length of the hypotenuse? 10^{2} or 100 + 15^{2} or 225 equals 325. The square root of 325 (in symbol form √325) is not a perfect square. It is a little more than 18, 18.03 to two decimal places.

Right triangles are found all over, in the angle that a ladder makes with a building, the height of a tree, baseball diamonds and soccer fields. The applications of the Pythagorean Theorem and right triangles can be used to approximate and measure the lengths of the sides of these triangles. (Trigonometry is an entire branch of mathematics that gives more detail.) For example, suppose a ladder is 28 feet long. When the ladder is used against a building, it forms a right triangle. If the ladder is placed so the distance between the wall of the building and the base of the ladder is 2 feet, how high up will the ladder reach? One leg of the triangle is 2 feet, and the hypotenuse is 28 feet long. Using the Pythagorean Theorem, 2^{2} + b^{2} = 28^{2}, or 4 + b^{2} = 784. If b^{2} = 784 – 4 = 780, then b = √780, or 27.93.

Figure 2: A ladder against the side of a burning building forms a real-life right triangle.

Sometimes the right triangle is actually an “air triangle”. The applications of the Pythagorean Theorem and radical expressions still hold. Suppose a plane is at a height of 5000 feet when it approaches an airport. At 5000 feet, the line of sight between the plane and the terminal is 38,000 feet. How far is the horizontal distance between the plane and the airport? This is truly an “air triangle”, as one leg of the triangle is the distance between the plane and the ground, or 5000 feet. Let that be a^{2}. The line of sight between the plane and a point of ground at the terminal is the hypotenuse, or 38,000 feet. So 38000-5000 will be the other leg of the air triangle, or 33000. The square root of 33000 is 181.66.

Figure 3: Right triangles abound in the friendly skies.

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]]>The demonstration of parallax is as close as an observer’s own two eyes. It is the measurement of how the same object appears from two different points of view. Astronomers use the very small angles observed by parallax to estimate distances and relative motion of objects ranging from the sun, moon, and planets to distant stars.

The principles of triangulation are used to measure distances for all celestial objects. Imagine a circle around the earth with 2 observatories roughly at opposite sides of the earth from each other, such as Calgary, Canada and Pretoria, South Africa. That is the baseline of the triangle. The apparent position of the moon in the sky will be at a different angle as seen from one observatory than from another at a different location. The moon is the closest celestial object to the earth, so the angle that can be estimated from its movement is the largest, at nearly one degree. According to the principles of trigonometry, if the baseline of the triangle is known, as well as the top angle (measured in fractions of arcseconds), the length of the long side can be estimated accurately.

The principle of measuring solar parallax, or parallax to any of the other planets or asteroids in the solar system, uses a baseline measurement of the earth at opposite locations of its orbit. For example, apparent motion can be measured in January and in June. Then, similar calculations can be done to estimate the distance to the other celestial object.

The measurement of the parallax of stars outside the solar system uses such small angles that ancient astronomers could not measure them precisely enough. The parsec is a unit of measure that is based upon parallax. It is equal to about 3.26 light years, an unimaginable distance to astronomers in the 1600s and 1700s. Usable measurements of parallax weren’t possible until the middle of the 19^{th} century.

The most accurate measurements of parallax are being made far from the obscuring atmosphere of earth. The Hipparcos satellite was launched in 1989, specifically to measure parallax to distant stars, up to about 1600 light-years away. Its companion Gaia mission measured the distances to over a billion stars. Observations from the Hubble Space Telescope can detect distances to stars over 10,000 light years away.

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]]>One of the biggest controversies in science in the early 18^{th} century was around the development of a new mathematical tool called calculus. In Europe, the mathematician, philosopher, and scientist Gottfried Leibniz held the attention of the scientific community. The most famous scientist of the day, Sir Isaac Newton, was the champion of Great Britain. According to scientific history, both invented calculus by working independently on different aspects.

Many mathematical and geometric ideas were already known before calculus was formulated. Archimedes and other Greek geometers, mathematicians in China and India, and thinkers in the Middle East used methods of calculating area and volume, work with infinite series, and other formulas. However, they did not put all those parts together into a system of thought.

Gottfried Leibniz (1646 -1716) was a German mathematician, philosopher, and scientist who may or may not have been a nobleman. He developed a modern calculating machine, and was an advisor to many political figures in Germany, France, and Austria. One of his many innovations was a version of calculus. He published a paper using the new methods in 1684.

Sir Isaac Newton (1642 – 1727) was one of the most famous scientists and mathematicians of his day. He was President of the Royal Society in Great Britain from 1703 to 1727. Newton’s Three Laws are the basic of classical mechanics and physics, especially gravitation. He developed a mathematical “method of fluxions” which was his form of calculus. He described the geometric background in the Principia Mathematica in 1687, in 1693, and in 1704.

In 1711, some of Newton’s partisans in the Royal Society accused Leibniz of plagiarizing Newton’s system. The controversy escalated between the European scientists and the British scientists, fueled by Newton’s supporters in the Royal Society and the political climate of the time. It wasn’t until the 1800’s that British mathematicians began using the notation that Leibniz developed for calculus concepts such as ∫ for integral, and dx and dy for infinitesimal parts of x and y. Historians of science generally regard both Leibniz and Newton as the inventors of calculus, coming at its mathematical concepts from different directions.

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]]>The AC-method, also known as factoring by grouping, is often used to factor polynomials when the a coefficient is equal to 1. It can also be used when the a coefficient is not equal to 1, similar to the FOIL method.

The first step to defining the polynomial expression, before finding factors or a numerical solution, is to see whether or not it takes the form ax^{2} +bx +c. Many polynomials used in high school math will follow that form, or one that is similar. If it is an expression, the variable x can take a number of different values. If they are in an equation or an inequality, the solution set for the variable will either consist of many solutions, one solution, or no solutions.

Figure 1: Check the form the polynomial takes before finding factors or a numerical solution.

If the polynomial expression has common factors other than 1 or -1, the common factors can be factored out, before looking at the rest of the equation. This is a very important step, because the polynomial that is left may be an ax^{2} +bx +c expression. Suppose the expression is 9x^{3} + 27x^{2} + 24x. The common factor to each element of the expression is 3x. Dividing 9x^{3} by 3xis 3x^{2}; 27x^{2} by 3x is 9x, and 24x by 3x is 8. Therefore one factor is 3x, and the other factor is in the form ax^{2} +bx +c.

Figure 2: An example of reviewing the polynomial for common factors.

The next step is to multiply the leading coefficient a by the constant c. In this example 3∙8 is 24. In the polynomial 14x^{2} + 67x + 81, 14∙81 is 1134. In the polynomial x^{2} -3x + 7, 1∙7 is 7. In symbol terms, find integers p and q such that pq =ac. In symbol terms, integers p and q are such that pq =ac AND p + q =the coefficient b. Then the expression can be factored as (x +p) (x +q). This is also called splitting the middle term, and it is a method of trial and error. Suppose the expression is 3x^{2} -10x -8. The expression is in the form ax^{2} +bx +c. That a coefficient is 3, b is -10, and c is 8, so there are no common factors. In this example ac is equal to -24, so the task is to find two integers p and q, so that pq = -24 and p + q = -10. The factors that are equal to -24 are (-1, 24) with a sum of 23; (1, -24) with a sum of -23, (-2, 12) with a sum of 10, and (2,-12) with a sum of -10; (-3, 8) with a sum of 5; and (3, -8) with a sum of -5; (-4, 6) with a sum of 2, and (4, -6) with a sum of -2. The integers that follow both conditions are 2 and -12.

Figure 3: The process of defining AC and splitting the middle term.

Next, write the expression out and group the factors. The polynomial 3x^{2} -10x – 8 = 3x^{2} – 12x + 2x – 8 = 3x(x – 4), because 3x ∙ x is 3x^{2} and 3x ∙ (-4) is -12x. Also 2x – 8 can be factored as 2(x – 4), because the expression 2x – 8 is equal to 2(x – 4). Putting the expression together 3x(x – 4) +2(x – 4) equals (3x + 2) (x – 4), which can be checked by multiplying using FOIL, so that 3x ∙ x = 3x^{2}, 2x – 12x = -10x, and 2 ∙ 4 is 8.

Figure 4: An example of factoring by grouping.

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