**Overview: **

Angles and the lines that form them are an essential part of geometry. Understanding the relationship between parallel lines, lines that are not parallel, and the different types of angles within figures is important to determining their measurement.

**What Are the Relationships Between Angles on a Straight Line?**

Suppose that three points A, B, and C are all on the same line. Then ∠ABC measures 180^{o} by definition. If a ray goes between point B through point D, two new angles are formed, ∠DBC and ∠DBA. Those angles are called a linear pair because they have a common side, ray DB. The measurements of a linear pair, because the angles are on a straight line, add up to the same measurement as a straight line, or 180^{o}. If ∠DBC measures 70^{o}, then ∠DBA measures 180-70, or 110^{o}. Similarly, if ∠DBC measures 135^{o}, then ∠DBA measures 180-135, or 45°.

**What Are Vertical Angles?**

Suppose that ray DB were extended through point B and point E into a line. There would then be two new angles, ∠EBC and ∠EBA. There are now two new linear pairs, ∠DBA and ∠EBA and ∠DBC and ∠EBC. Suppose that ∠DBC measures 70°, then ∠EBC measures 110°, or 180-70. Since ∠DBC and ∠DBA are a linear pair, ∠DBA also measures 110°, the same as ∠EBC. The angles ∠DBA and ∠EBA are also a linear pair, and since ∠DBA measures 110°, ∠EBA measures 180-110, or 70°, the same as ∠DBC. The angles ∠DBA and ∠EBC are opposite each other, as are the angles ∠DBC and ∠EBA. They are called vertical angles, and vertical angles have the same measurement.

**What Are Parallel Lines?**

Parallel lines are on the same plane, but do not have any points in common. They never meet at any point on the plane. Suppose a parallel line to the line containing points A, B, and C were drawn through point D. That line contained points F and G, as well as D, and none of those points are on line ABC. The line BDE that intersects both parallel lines is called a transversal. The only point it has in common with line ABC is point B, and the only point it has in common with line FDG is point D.

**What Is the Measurement of Angles Created by Parallel Lines?**

Parallel lines go in the same direction, so the measurements of the new angles created by the intersection of the transversal with the new parallel line are equal to the measurements of the intersection of the transversal with the original line. Those angles are called corresponding angles.

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]]>In the United States, two separate measurement systems are used, the conventional or English system and the metric system. The conventional system with inches, pounds, cups, and quarts, is in use in many aspects of everyday life, in carpentry, and in many trades, while the metric system is used in science, medicine, and many other fields. Some manufacturing is done using the metric system, while other manufacturing is done with the conventional system.

**Why Convert Between Systems?**

Many countries throughout the world use the metric system in all areas of life. It is important to know the relationship between kilometers and miles when traveling in Canada, or the relationship of liters and quarts, kilograms and pounds. When importing or exporting goods throughout the world, conversions between the conventional and metric systems are essential.

**Converting Between Length Measurements**

One length measurement is an exact conversion. One inch is exactly 2.54 centimeters. However, the other length measurements are approximate, so that 1 meter is approximately 39.37 inches, and 1 kilometer is approximately 0.62 miles. (The mathematical sign ≈ means approximately equal to.) Many highway signs in Canada state a speed limit of 90 km/hr or 100 km/hr. A quick estimation without a calculator is that 1 km is about 3/5 of a mile, so that 90 km/hr is about 55 miles per hour, and 100 km/hr is about 60 miles per hour. That’s good to know to avoid a speeding ticket.

**Converting Between Weight Measurements**

Weight measurements between the conventional and metric systems are approximations, so that 1 kilogram (kg) is approximately 2.2 pounds. A kilogram block of cheese is going to weigh more than a two-pound block. Suppose a weight is 5 kilograms. On the other end of the scale one will need a weight of (5 times 2.2) or about 11 conventional pounds to balance it. Similarly, if a weight is 15 pounds, its weight in kilograms will be (15\2.2), about 6.82 kilograms.

**Converting Between Volume Measurements**

A liter is approximately 1.06 quarts, so that a 2 liter bottle of soda is about 2.12 quarts of soda. Conversions become difficult between conventional volumes and metric volumes, because we use other measurements than quarts. For example, gasoline is measured in gallons, and there are 4 quarts in a gallon, or about 4.24 liters. Suppose a tank holds 20 gallons. That would be 80 quarts or about (80 times 1.06) 84.8 liters.

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Pedigrees are visual representations of a family’s genetic history. Each pedigree focuses on one specific gene that is studied by geneticists. There are several benefits of studying a family’s genetic history. By reviewing past generations, scientists can deduce the mode of inheritance for a specific gene. A person can study their family’s pedigree to determine if they have the trait or are a carrier of the gene for that trait. Parents can also determine if their children will inherit the trait or become carriers. Pedigrees play a significant role in the studies of genetic diseases and illnesses.

Autosomal dominant genes are found on autosomal chromosomes, and therefore are unaffected by gender. Affected individuals are common, and have at least one parent with the trait. The trait often appears in each generation. Parents who are unaffected will only produce children without the phenotype.

Autosomal recessive genes are unaffected by gender, since they are located on autosomal chromosomes. Parents do not need the phenotype to produce a child with the trait, but parents who both have the trait will only have affected children. The phenotype can sometimes skip generations and appear in younger family members.

The diagrams below show how autosomal domninant and autosomal recessive traits are passed on through generations.

Figure 1: Offspring of homozygous dominant and heterzygous dominant parent.

In figure 1, we can see that a parent that is homozygous dominant and a parent that is heterzygous dominant (only one allele carries the dominant trait), will produce offspring that will all phenotypically show the dominant trait. Each child has a 50% chance of presenting the dominant trait on both alleles and 50% chance of carrying the dominant trait on one allele.

Figure 2: Offspring of homozygous dominant and homozygous recessive parents.

In figure 2, we can see that a parent that is homozygous dominant and a parent that is homozygous recessive (does not carry the trait at all), will produce offspring that only carry the trait on one allele. Phenotypically, this means that all offspring will show the dominant trait.

Figure 3: Offspring of heterzygous dominant parents

In figure 3, we can see that if both parents are heterzygous dominant (only carry the dominant trait on one allele), then their offspring may have 3 possible genotypes. Each offpspring will have a 25% chance of being homozygous dominant for the trait (carries dominant trait on both alleles), a 50% chance of being heterzygous dominant for the trait (carries dominant trait on one allele) and a 25% chance of being homozygous recessive for the trait (does not carry dominant trait at all). Phenotypically, there is a 75% chance that each offspring will show the dominant trait and a 25% chance of not showing the trait.

Y-linked genes are found only on the male Y chromosome. As a result, only males in a given family are affected. A father will pass this trait onto all of his sons, as each son inherits an X chromosome from his mother and his father’s Y chromosomes. Daughters will never inherit the trait since the Y chromosome is not inherited.

Figure 4: Offspring of Y – linked trait

Dominant genes found on the X chromosome affect both males and females of a pedigree. The gene can be inherited from either the mother or father, and passed onto the children. Both genders will always express the phenotype if the gene is inherited. A father with the trait will always pass it onto his daughters

Genes that are both recessive and found on the X chromosome can also affect both genders. However, the majority of affected family members are male; the phenotype is not always expressed if the gene is inherited. A female must inherit two X chromosomes with the gene in order to express the trait, while a male must inherit only one affected X chromosome. A mother who expresses the phenotype will pass it onto all her sons, and parents who both have the trait will have only affected children.

The diagrams below show how X-linked dominant and recessive traits are passed on through generations.

Figure 5: Offspring where both parents are affected.

Figure 6: Offspring where only the father is affected.

Figure 7: Offspring where only the mother is affected.

The patterns and characteristics within pedigrees allow geneticists to determine individual genotypes. Modes of inheritance can be determined through careful analysis of a family’s genetic history and pedigree. With the results of these studies, scientists can determine causes of hereditary diseases and illnesses. Parents have the ability to determine the probability of their children inheriting genes associated with health risks. As medical technologies advance, traditional pedigrees continue to shed light onto the study of genetics.

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**Overview**

Rational exponents are an extension of the rules of exponents. They can be solved by applying those rules and often take more than one step.

**What Is an Exponent?**

An exponent such as 3^{2} is a direction to multiply 3 by itself twice, such as 3∙3, or 9. Similarly, an exponent such as 2^{3} is a direction to multiply 2 by itself 3 times, as 2∙2∙2. It is not the same as 2∙3, which is 6. Therefore, a number such as a^{n} means to multiply a by itself n times. The rules of exponents have been extended so that x^{0} equals 1, so that 5^{0} equals 1, 7^{0} equals 1, and so forth. By definition, x^{1}is x by itself, so that 2^{1} equals 2, 5^{1} equals 5, and 100,521,362^{1} equals 100,521,362.

**What about Roots of Numbers?**

By definition, the root of a number is the inverse operation of multiplying by an exponent. If 3^{2} equals 9, the square root of 9 (√9) is 3. Similarly, if 5^{3} means 5∙5∙5, then the cube root of 125 is 5. Also, if 2^{4} equals 16, then the fourth root of 16 is 2. In addition, (√2)(√2) is the same thing as saying (√2)^{2} or 2. That is also the same thing as 2^{1}.

**What about Rational Exponents?**

By the rules of adding exponents, 2^{2∙}∙2^{4} is the same thing as 2^{6}, or 2^{2+4}. Therefore, 2^{½} ∙ 2^{½}would be the same as 2^{½ + ½} or 2^{1}. Now 2^{1} equals 2, and so does (√2)^{2}. By definition, the square root of any positive number n, when n>0, equals n^{1/2}. That definition can also be extended to roots other than square roots, such that n^{1/3} means the cube root of n. Therefore, 1728^{1/3}is equal to the cube root of 1728, which is 12, because 12^{3} is 1728.

**What about Other Fractional Exponents?**

The rules of defining fractional exponents such as 25^{1/2 }mean that 25^{1/2} is the same thing as the square root of 25, or 5. A fractional exponent as 25^{3/2}, however, means (25 ^{1/2}) ^{3} or (√25)^{3}. It is a good rule to take the square root first, so the square root of 25 is 5, and 5^{3}is 125. Suppose the number were –(16)^{1/2}. The square root of 16 is 4, and taking the negative would be -4. The square root of -16 is not a real number.

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]]>When in a chemistry lab or just writing a test, many students struggle with how to create a solution of known concentration. Students find it difficult because they try to calculate the solution in terms of moles, which is a unit that cannot be measured in the lab. There are some easy to follow steps on how to prepare a proper solution with a known concentration.

The first step is to determine what concentration of solution you need to make along with the volume you are looking to end up with. In this example we are going to make 1234 mL of a 1.54 molar solution of NaCl (Sodium chloride).

First, we will determine the number of mols of NaCl that are contained in 1234mL of a 1.54 molar solution. To do this, we will use the following formula:

Where:

n = number of moles of solution in mol

C = concentration of solution in mol/L

V = volume of solution in L

**NOTE:** Always ensure that your units are consistent.

Note that the unit of volume is mL while the unit for C is mol/L. This means that we need to convert mL to L before solving the equation. 1234 mL is the same at 1.234 L. Now that all of our units are consistent, we can substitute our known values for concentration and volume:

We now know that we need to add 1.90 mol of NaCl to our solution; however, mol is not a unit we can measure out with common lab instruments. We must first convert it to a unit that is easily measured in the lab. This requires a new equation:

Where:

n = number of mole in mol

m = grams of compound in g

M = molar mass of compound in g/ mol ( This is taken directly from the periodic table)

To determine M, we need to take a closer look at our compound:

NaCl = 1 Na + 1 Cl

= (1 * 22.99 g/mol) + (1 * 35.45 g/mol)

= 58.44 g/ mol

We have determined our molar mass (M) and calculated n in Step 1. We are now ready to use substitution to find m. Since all of our units are consistent, no further adjustments need to be made.

m = n * M

We now know that we must measure out 111.0 grams of NaCl on a scale for it to be added to solution.

**NOTE:** The key when creating any solution is to slowly add the compound to water while stirring. This will start off with a very low concentration of the compound and slowly increase to the desired concentration. If water is added to the compound instead, this will start off with a very high concentration of the solution, potentially resulting in a hazardous compound.

Fill a beaker about ¾ full with deionized water, then add the 111.0 g of NaCl. After the NaCl is added, fill the container the rest of the way to the 1.234L mark. You now have 1.234 L of a 1.54 molar solution of NaCl.

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**Overview:**

Estimation is an important skill in mathematics, because sometimes an approximate figure is needed in everyday life. Suppose a person is deciding if they have enough money to buy all the items on a shopping list or how many materials will be needed to finish a project. Types of estimation include rounding up, rounding down, and rounding to the nearest.

**Why Estimate?**

Estimates are often used when an exact value is not available yet. Suppose that a new road is being built. Before companies can bid on the job, they will need to make an estimate of how much materials and labor will cost to build it. Similarly, if people are going on a journey, they will need to estimate how much money they will need for the trip, as well as anything they might want to pack. Also, sometimes estimates are easier to work with than exact figures.

**What Is Rounding Up?**

Many estimates involve “rounding up”. Suppose the calculation for how many hours an individual works monthly shows that they will work 94.6 hours in the next month. As this is only an estimate of how many hours they will work, it is usually rounded up to the next whole number. They will work about 95 hours in the next month. Many small objects, such as pens, pencils, and paper clips are sold in boxes rather than individually. If pens are sold is packages of 25, there will need to be at least 2 packages for a class of 42 students.

**What Is Rounding Down?**

Rounding down is a form of estimation where part of the amount is not taken into consideration. For example, if the interest on a savings account is rounded down to the nearest cent, and the calculated interest is $15.242, only $15.24 would be paid out in interest. Similarly, many calculators truncate any numbers past the 10^{th} decimal place, so that a decimal such as .1236789452 would be read as .123678945. The directions that come with calculators tell how many places they will show.

**What Is Rounding to the Nearest?**

Usually when rounding numbers either up or down, more directions are given. The distance between the earth and the sun is 92,955, 807 miles. It is often rounded to the nearest million, or 93 million miles. Suppose the distance from the earth and the sun were given to the nearest 10 miles. It would be rounded to 92,955, 810 miles. Suppose it were given to the nearest 100 miles. Then the estimate would be 92, 955, 800 miles.

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]]>Sir Isaac Newton is well-known for his work on creating the three laws of motion based on the theories of previous scientists as well as his own speculations. These 3 simple, intuitive laws lay the foundation for the basics of classical, Newtonian mechanics. The following are Newton’s three laws in more detail:

The first law of motion states that objects will remain in their current state of motion unless acted upon by an external, unbalanced force. This means that objects that are at rest, will remain at rest unless acted upon by an external, unbalanced force while objects that are moving, will continue in a constant state of movement unless acted upon by an external, unbalanced force.

In order to understand the law of inertia, we can imagine that a hockey puck is on ice (which in this case is frictionless). At first, the hockey puck is not moving, and we can assume that if it is not moving, it will stay that way. However, an unbalanced, external force such as a hockey player hitting the puck with their stick could cause the hockey puck to come out of rest and start to move.

In another scenario, we can imagine that a hockey puck is moving at a constant speed on frictionless ice. It will continue to move in constant motion unless something gets in its way and stops the motion, such as a hockey player stopping the puck.

The second law is based on the following equation:

F = m * a

In this equation, “F” represents a force measured in joules, “m” represents mass in kg and “a” represents acceleration in m/s^{2}.

This law states that the acceleration of the object is dependent upon the magnitude of the force, and the size of the mass. The acceleration of any object is directly proportional to the force applied to the object. That is, a stronger force applied to the object will result in a larger acceleration. The acceleration of any object is also inversely proportional to the mass of the object. That is, the larger the mass of the object, the lower the acceleration will be.

Newton’s third law of motion states that “every action has an equal and opposite reaction.”

This law states that if object A applies a force to object B, object B will, in turn, apply a force of equal magnitude on object A. is applied from one body to another, no matter how large the force is, there will always be an equal force applied right back in the opposite direction but with the same amount of force. For example, when an individual punches a wall, they will feel the force of that punch hit them right back with the same magnitude, causing pain.

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A hyperbola is a conic section that is formed when a plane intersects a double cone, so that it forms two identical curves that go in opposite directions. It has two separate foci and the two branches are mirror images of each other.

**What Is a Hyperbola?**

The two branches of the hyperbola have a transverse axis that bisects them along a line of symmetry and a conjugate axis that is perpendicular to them. The branches mirror each other exactly. Suppose that one branch of the hyperbola has a vertex at (0, 4). The other branch of the hyperbola will have a vertex at (0, -4). Another feature of both branches of the hyperbola is that the sides of each branch come very close to extended diagonal lines, but never quite touch them.

**What Is The Equation for a Hyperbola?**

Using the Distance Formula, if the foci are at (0, 5) for the “positive” hyperbola and (0, -5) for its “negative” mirror image, the equation that describes its location on the plane is 9y^{2} – 16x^{2} = 144. The equation for a hyperbola can also be written as y^{2}/a^{2 –} x^{2}/b^{2 }when the foci are at coordinates (0, c), and (0, -c). This is a special type of second-degree equation that has 2 separate squared variables.

**What Is a Rectangular Hyperbola?**

A rectangular hyperbola is a special type of hyperbola where the asymptotes are the x and y axes. It is defined by an equation of the form xy = k, where k is a constant. (This is also the form for inverse variation, but only one of the branches is usually meaningful.) If k is greater than 0, then one branch of the hyperbola will have positive coordinates and the other branch will have negative coordinates, but if k is less than 0 then the values will be reversed.

**What Are Real-World Examples of Hyperbolas?**

One of the earliest real-world examples of hyperbolas was the path of light traced on a sundial. It was studied in ancient Greece, who first noticed the relationship between the shape of the hyperbola and the angle of the sun’s rays at a given time of year. Because hyperbolas have constant differences, they are often used to solve problems in navigation and with LORAN and GPS transmitters. High-energy atomic particles also follow a hyperbolic path.

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]]>A parabola is a conic section that forms an open curve. Graphs of quadratic functions have the shape of a parabola.

**What Is A Parabola?**

A parabola is the curve that is formed on either side of an axis of symmetry. If the parabola is vertical, a line to one side of the parabola (called the directrix) is vertical. If it is horizontal, the directrix will be horizontal. The axis of symmetry contains two named points, the focus of the parabola, which is usually inside the parabola, and the vertex, which is the point at the tip of the parabola, where it changes direction.

**Why Is the Shape of a Quadratic Function Parabolic?**

The equation of a parabolic function is a quadratic function. In general terms, the equation is ax^{2} + bx + c. If the vertex is at (h,k), the Distance Formula can be used can be derive a general equation, so that y = a(x-h)^{2} + k. Suppose the focus of a parabola is at (3, 5), and the directrix is y = 1. The axis of symmetry is at x =3. Using the equation for the vertex, the parabola will change direction at (3, 3). Its equation is (x-3)^{2}+ 24 = 8y or x^{2 } – 6x + 9 + 24 = 8y or (x^{2} – 6x + 33)/8 = y.

**What Are Some Parabolic Shapes that Occur in Nature?**

Some of the parabolic shapes that occur in nature include the trajectory of a moving body under a uniform gravitational field. For example, a baseball being thrown into the air at a low speed, or the center of gravity for a diver jumping off a diving board, without air resistance form a parabolic shape. The orbits of comets as they pass close to the Sun are also nearly parabolic.

**How Is the Parabola Used in Calculation?**

The parabola has many applications in calculation. For example, many bridges are constructed with parabolic arches, and it is a common architectural form. The focus points are also used to concentrate waves of sound in a parabolic antenna or in a parabolic mirror. Vertical curves in roads also form a parabolic design.

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]]>Motion problems are special types of rational expressions that are based on the relationship between distance, rate, and time (d=rt). In order to solve them, it is important to read the story problem, recognize the essential information, and organize the facts in an equation.

**Reading the Story Problem**

Story problems usually contain all the information that is necessary, and sometimes additional information that is not needed. If the problem involves the speed of something traveling, people or vehicles either going in opposite directions or overtaking each other, and a rate of speed that may change or differ, it will follow the distance = rate X time pattern. Suppose two drivers start out at the same time travelling in opposite directions. One driver was driving 6 miles slower than the other driver. After 10 hours they were 940 miles apart. Find the speed each driver was driving.

**Recognizing the Essential Information**

In the sample story problem, there are several things that are known. First, the total distance, 940 miles, is known. Second, the time for each driver is known (10 hours). If Driver A is going r miles per hour, Driver B is going r-6 miles per hour. We know the rate is miles per hour rather than feet per second or kilometers per hour, because the time traveled is measured in hours and the distance measured is measured in miles.

**Organizing the Data**

The data can then be organized in an equation. If d=rt, the value for d, 940 miles, is already known. That is the same value for both Driver A and Driver B. In addition, we also know that the time for both drivers is the same (10 hours). The rate for Driver A is r and the rate for Driver B is r-6. In order to set up the equation so that both Driver A and Driver B are included, the equation would then be 10r (the rate of Driver A) + 10(r-6) (the rate of Driver B) = 940.

**Solving and Checking the Equation**

The equation, 10r + 10(r-6) = 940, can be expanded to 10r +10r -60 =940. Then like terms can be combined as 20r – 60 = 940. Then 60 can be added to each side so that 20r = 940 + 60 or 20r = 1000. Dividing both sides by 20, r equals 50 and r – 6 = 44. Also, 20(50) – 60 = 940.

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