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NWN: 0075 Algebra Mind Map 6: 3.5.1.3.3 (Tanith): Sequences can be based on an infinite number of relationships between numbers. Two common ones are arithmetic and geometric sequences, where numbers are added or multiplied together to get the next number in the sequence. In formulas, a is the number you get when the sequence has gone through its function n times (written a sub n). Neverwinter Nights: Zanzi meets the last courtesan, Tanith. Although she's curious to know what sequence of events led him to work at the Moonstone Mask, her curiosity ebbs almost as soon as he begins to explain. Tanith will at least engage in some sort of conversation with Zanzi, after she gives him the Pass Coin, but when she asks him anything about the plague he's quick to change the subject. "Oh well, that's 200GP wasted," she thinks to herself and then somewhat embarrassed by the awkward situation says her farewells: "Excuse me, Mr Tanith, it's rather warm in here." She's not tempted to avail herself of this courtesan's services today, despite Tanith's undoubted appeal in this regard.
NWN: 0074: Algebra Mind Map 6: 3.5.1.3.1 (Tamora): To graph rational expressions you need up to four basic bits of information (depending on how many variables there are in the equation) and to plot a few points to confirm where on the graph the lines will be positioned: First you need to find the x intercept by making the numerator of the rational expression zero, therefore set a value for x which makes the numerator zero (because the only way to set y to zero is to set the numerator of the rational expression to zero). Next find the y intercept by making x zero, what you get is the y intercept. Then find the vertical asymptote by choosing a value for x in the denominator that makes the denominator zero. Finally find the horizontal asymptote by making x very large and you'll see that for polynomial fractions with the same degree x variables, the horizontal asymptote becomes the ratio of leading coefficients for those x variables. If you need to you can plot about four points just to confirm which quadrants on the coordinate plane the lines of the graph will be located. All first degree rational expressions (e.g. y =(3x-2) / (3x+5)) will produce similar shaped graphs. So when you look at those rational expressions that have x variables of the same degree in the numerator and denominator you should see that the numerator will provide the x intercept, the denominator will provide the vertical asymptote, the ratio of the leading coefficients provides the horizontal asymptote and letting x = 0 provides the y intercept. Neverwinter Nights: Here Zanzi meets Tamora in her sumptuous bedroom in the hope of asking her a few plague-related questions. However, Tamora looks immodestly at Zanzi and says, "Sorry little lady, I'm not your type. You'll have to go talk to Tanith. He's a big sweety." She then winks at Tomi and makes him an obscene gesture. "Forget it, Tomi," says Zanzi, "she's got the whole rationale for our being here wrong." So alas for Tomi, Zanzi makes it clear to him that Tamora is like an asymptote: he can look but not touch. 3.5.1.3.2 (Luce): Graphs based on simple exponential and logarithmic equations have two common features: points plotted by raising a base to any positive or negative exponent will always be positive for the second (y) coordinate ordered pair, and all these graphs will have a point that intersects 1 either on the y axis (i.e. (0,1) for exponential graphs) or x axis (i.e. (1,0) for logarithmic graphs). They all show the same graph characteristic, with log graphs showing that logs are inverse functions of exponential functions, as indicated by the two types of graphs being reflections of each other through the line y = x. Note the that tipping point for these graphs occurs when y = 1^x = 1, because if the base is >1 the graph curve for log and exponent plots will increase from left to right and if the base >0 and <1 then the graph curve for log and exponential plots will decrease from left to right. Neverwinter Nights: Zanzi meets Luce in the next sumptuous bedroom but like Tamora, Luce has no useful information to offer and Zanzi's hopes that the money she spent on getting access to the back rooms of the Moonstone Mask will prove fruitful in her quest to combat the Wailing Death are now in exponential decline. Unlike Tamora, Luce has an honest face and an amiable attitude. So Tamora's basic character is quite diminutive (<1), if not zero, in relation to Luce's (>1). Evidently, these two people are like chalk and cheese despite their identical professions.
NWN: 0073: Algebra MM6 3.5.1.3 (Moonstone Mask (second floor corridor)): Graphing Radical Equations using a Table: The graphs obtained when plotting radicals are very similar, except that even roots (square root, fourth root, etc.) come to an end at (0,0) as points for such radicals they can never be negative. Odd roots (cubed roots, fifth roots, etc.) can go into the negative x and y quadrant, showing symmetry about the line y = -x. As the number for the root increases the steepness of the graphs for both even and odd roots becomes shallower. Neverwinter Nights: As Zanzi somewhat apprehensively enters the second floor corridor of the Moonstone Mask she muses on the radically different lifestyle she leads from the high class courtesans working in these rooms, and the negativity of the situation isn't lost on Tomi, who thinks it odd that Zanzi would want to visit these courtesans. Zanzi is however very thorough, and even if nothing positive results from speaking to these people she likes to leave no stone unturned.
NWN: 0072 Algebra MM6 3.5.1.2 (Bedroom two): Introduction to Conic Sections: The study of circles, ellipses and hyperbolas is actually the study of the perimeters of conic sections that result when a plane cuts a double cone. A circle is a type of ellipse formed when a plane cuts parallel to either base of the double cone, a parabola is a perimeter formed when a plane cuts parallel to the side of either cone, an ellipse is a perimeter formed when a plane is angled so that it intersects one half of the double cone and a hyperbola is a perimeter formed when a plane is angled so that it intersects both halves of the double cone. The standard equation for an ellipse is x^2/a^2 + y^2/b^2 =1 (so the unit circle is an ellipse where a and b =1); the standard equation for a hyperbola is x^2/a^2 - y^2/b^2 = 1, which is the same except for the negative sign. (Note that as subtraction is non-commutative another equation for a hyperbola is y^2/a^2 - x^2/b^2 = 1). When we have these equations in standard form the resulting graph always shows the ellipse or hyperbolas centred at the origin of the x and y axes. Also note that with ellipses you will always be able to find the x and y intercepts (using the standard method of making first the x term zero then the y term zero), but with hyperbolas only the positive term will give you the intercepts. If you take the negative term of a hyperbola equation and treat it as a positive (as if you were about to graph an ellipse) the point you get will enable you to complete a rectangle on the graph, and lines drawn from the origin through the corners of this rectangle are equivalent to asymptotes of the two hyperbolic graphs. (This becomes much clearer when you see how to graph hyperbolas). Neverwinter Nights: Zanzi has a definite sense of deja vu when she enters this second bedroom - the bed, the chest the armoire are exactly like those she was so familiar with when she studied at the academy. But again she feels the room's welcoming familiarity is entirely illusory, as the Moonstone Mask has nothing in common with the academy. Elves, like humans, are often guilty of hyperbole, but it is no exaggeration to say that Zanzi's normally centred emotions have been somewhat elliptically distorted by this place in this setting, and now she truly aches to be back in the academy, where life was much simpler and her role and duty were clearly defined.
NWN: 0071: Algebra Mind Map 6: 3.5.1.1 (Bedroom one): There's a lot of new stuff here. There are two equations to remember, the first deals with finding the distance between two points on a coordinate plane and the second deals with graphing circles. To find the distance between two points we use the distance formula: d = [(x2-x1)^2 + (y2-y1)^2]^0.5, which means that for two points on a coordinate system if you take the square root of the sum of the quantity difference of the x coordinates squared and the quantity difference of the y coordinates squared, the result you get is the distance between those two points. The equation for the circle is derived from the distance equation and is r^2 = (x-a)^2 + (y-b)^2, where r is the radius of the circle and a and b are the coordinates of the centre of the circle. The OPPOSITE values of a and b in the equation give the point on the coordinate plane at the centre of the graphed circle. Also r will always be positive, so when working with the circle equation you can disregard any negative values resulting from the use of radicals when solving for r. When calculating a and b for given values of r, x and y it's often necessary to complete the square (i.e. (a=1 and 1/2b)^2) for either the x coordinate or y coordinate or both to get your results for a and b in the circle equation. Note than when a and b are zero the centre of the circle will be at the origin of the x and y axes. Circles will always have a coefficient of 1 for the x and y coordinates of the circle equation. Neverwinter Nights: The first of two bedrooms in the reception area, and Zanzi looks at the familiar objects in this room but in the context of the Moonstone Mask she sees them in a totally new light. Paradoxically, in some ways she has come full circle (with her familiarity of the bed and desk) yet there is a massive distance from the feelings originally evoked by these objects.
NWN: 0070: Algebra MM6 3.5.1 (Moonstone Mask (backroom reception area)): The Discriminant. We learn three new things here, firstly we note that if the coefficient (a) of the x^2 term in a quadratic is negative the parabola will lead up to a vertex then go symmetrically downwards from there, we also learn that the vertex at x for any parabola is found using x = -b/2a. Finally we learn that the term b^2-4ac is known as the discriminant (of a quadratic equation) and that if this value is negative we will have two complex coordinates (because the discriminant lies within square root of the quadratic equation, so -b+(discriminant)^0.5 and -b-(discriminant)^0.5), meaning that the parabola will not intercept the x axis when graphed (i.e if the vertex is above the x axis then the arms of the parabola will plot up away from the x axis and if the vertex is below the x axis then the arms of the parabola will plot down away from the x axis). Neverwinter Nights: Here in the backroom reception area the courtesans get to meet their clients. These girls, though certainly very pretty, are not too well read so cannot engage in anything other that small talk and physically pleasuring their clients. So the more discriminating client will not cross the line with these courtesans.
NWN: 0069: Algebra Mind Map 6: 3.4 (Judge Oleff): Neverwinter Nights: In this lesson we review much of what we learned about graphing using algebra when in the West Corridor, Auditorium and South Corridor (minor) of the Academy, and in the same way we revisit Judge Oleff for him to explain himself and to come clean about his visits to the Moonstone Mask, or else - Zanzi gets 500GP hush money from Oleff here. Oleff also gave 500GP for Jacob's Quill obtained from Gilles and her alignment also shifted 3 points towards Good. 3.5: (Moonstone Mask (backdoor corridor): Here we learn that the vertex of a parabola is the lowest (or highest) point of the parabolic curved line. We also note that for parabolas derived from x^2 values, numbers added or subtracted to x after the x value has been squared move graphs higher or lower and that values added or subtracted to x before the x value has been squared move graphs left or right. Neverwinter Nights: With the hush money from Oleff we purchase a pass coin from Torgo and are now in the back corridor - Zanzi has reached a new low (vertex) with this tactic.
NWN: 0068: Algebra Mind Map 6: 3.1 (Gilles): Neverwinter Nights At this point we use what we've learned when meeting Clancy on entering the Moonstone Mask back entrance by doing some application problems. During this time she meets a servant of the Church of Talona named Gilles. (Talona is a lesser deity of chaotic evil alignment who goes by the title of "Lady of Poison" or "Mistress of Death". Zanzi suspects that she maybe responsible for the Wailing Death as Talona thrives on extreme suffering caused by chronic pain and disease and slow-acting poisons). Gilles offers Zanzi more money than Judge Oleff for the Tyrran artefacts found in the tomb quest, but then she must break her promise to Oleff and side with the evil forces at work in Neverwinter. (This is a practicality verses clear conscience dilemma - like applying maths to everyday situations, which generally provides greater rewards but at the expense of denying the true spirit of mathematics, which is philosophical and imaginary). Zanzi could do with as much money as she can procure, and for a while she seriously considers handing over the Tyrran artefacts to Gilles, but when he mentioned that he had stolen an item from a dying man (Jacob's quill) at the tomb excavation in the Peninsula District she becomes enraged: "Give me the quill now, you evil bastard!" and grumbling bitterly Gilles complies before scampering away in fear on hearing Zanzi's threatening tone.
NWN: 0067: Algebra Mind Map 6: 3.0 (Moonstone Mask): The matrix method relies on an "augmented matrix" made up of coefficients from the variables of a system of equations. This method works well when there are more that two variables in a system of two or more equations. So at the start we have our system of equations and we take the coefficients of the variables and write the coefficients as numbers in an augmented matrix. Then our goal is to set the lower leftmost number in the matrix to zero for a 2 x 2 system of two equations of x and y variables, or to set the three leftmost numbers to zero for a 3 x 3 system of three equations of x, y and z variables. The rules by which we achieve these zero values are determined by what are called elementary row operations, namely: 1: We can interchange any two rows. 2: We can multiply any row by a non-zero constant. 3: We can add to any row a constant multiple of another row. After we've used the elementary row operations to obtain our zero or zeros in the right places in the augmented matrix, we then take those numbers in the matrix and place them back as coefficients in a system of equations which will be equivalent to our original system of equations but which will be much more easy to solve. Neverwinter Nights: After meeting Clancy she enters a building by the name of the "Moonstone Mask" by the back entrance, which is not the most obvious way to gain entry to these premises. (Likewise, solving equations by this augmented matrix method is not the most obvious way to go about solving systems of equations). Tomi asks Zanzi why she has entered these premises by the back entrance. "In normal circumstances I wouldn't be seen dead entering this place," she replies, "It's nothing more than a glorified whorehouse - and not that glorified either." 3.2 (Ophala Cheldastorn): During this lesson we're introduced to determinants, which, for the general procedure, uses two arrays: a number (coefficient) array, called the determinant array, and a sign array. In addition to the general method for working with determinants there are two other methods which are specific to a 2x2 determinant array and a 3x3 determinant array. The procedure for whichever method chosen is somewhat bizarre and appears perplexing at first, but will be of value when solving equations that have many variables and coefficients. Method specific for the 2x2 determinant array: Multiply down the diagonal (top left number to bottom right number) and then subtract from that product what you get when you multiply up the diagonal (bottom left number to top right number). Method specific for the 3x3 determinant array: Copy the first two columns and place them to the right of the determinant array, then multiply down the three diagonals and then subtract from that product what you get when you multiply up the three diagonals. General method for any size determinant array (2x2, 3x3, 4x4, 5x5, etc.): To the right of the determinant array write an alternating sign array, always beginning with a + sign in the top left (e.g. + - +), the number of signs and how many columns and rows corresponding to the columns and rows of the numbers in the determinant array. Then "expand" across one of the rows or down one of the columns of the determinant array by writing below the determinant array the signs for the corresponding row or column in the sign array; then place those numbers for that row or column of the determinant to the right of the signs you've taken from the sign array. You then have to create a smaller array to the right of each of those numbers by taking numbers remaining from determinant array after mentally removing the row you are expanding across and the column down (or column you are expanding down and the row across) for each number in that row (or column). Then, as in the 2x2 array method, multiply down the diagonals and subtract from that product what you get when you multiply up the diagonals. Finally multiply the result
NWN: 0066: Algebra MM6 2.0: Solving Systems of Equations using Substitution: 2.0 (Clancy): In this lesson we actually learn two methods for solving systems of equations (or simultaneous equations). This first, called the addition or elimination method, utilises the addition property of equality. Firstly we arrange and or multiply one of two equations so that one of its terms has the same coefficient as one of the terms in the other equation, we then add or subtract one equation from the other so that the term with the same coefficient adds to zero (and is therefore eliminated) and we can solve for the other variable. We then plug in the solved variable into either equation to find the value of the other variable. The second method, called substitution, requires expressing one equation of at least two equations in terms of one of the variables in that equation and then substituting that expression for same variable in any of the other equations. It is a requirement to show the solved variables x and y as an ordered pair (x, y) as a final step when completing these sorts of calculations for two linear equations in two variables. There are a couple of special cases that occasionally result when solving systems of equations. In both cases the variables will be eliminated from the final calculated equation and that final equation will either be a true statement or a false statement. So you might end up with the true statement 0 = 0. In algebra we describe this situation by saying that the true statement implies that there will be an infinite number of ordered pairs that if present in one equation in a system of equations will also be present in the other equation, and this can be written as, for example, {(x,y)| -2x + 6y = 8}. Geometrically, we describe this situation by saying that the two lines coincide (i.e. the graph for one equation lies exactly on top of the graph for the other equation). In the situation where you end up with a false statement, the final statement will say something like 0 = 1 or 19 = -17, 24 = 20, etc. In algebra we describe this situation by saying that the false statement implies that there are no solutions to the system of equations. Geometrically, we describe this situation by saying that the two lines are parallel. A note on the addition property of equality. Expressed algebraically this means that if A and B are equal and C and D are equal then A + C and B + D will also be equal. So adding expressions that are equal to each other to both sides of an equation won't change the solution set. Neverwinter Nights: Clancy is a common member of the townsfolk that Zanzi just happened to meet after speaking to Erevian. It is obvious that Clancy is a strong and burly fellow who clearly has not succumbed to the Wailing Death: his strong constitution has eliminated any chance of the disease catching hold in his system . Zanzi looks at him and wonders if there's a way to somehow magically substitute Clancy's constitution for her own.
NWN: 0065 Algebra Mind Map 6: 1.0 (Erevian): Solving a system of equations by graphing means finding the ordered pair common to two or more graphs; thus, this ordered pair will be at the point where those graphs intersect. A note about fractional gradients. In slope intercept form y = mx + c, if m is a fraction the numerator moves a point on the graph up if positive (often described as the "rise") or down if negative and the denominator moves a point right if positive (often described as the "run") and left if negative. The variable c is the y intercept value. A note about graphing linear equations with just one variable. In this situation we graph either a straight vertical or horizontal line. E.g. x = 3 will result in a vertical line through x at 3 on the horizontal axis, and y = 3 will result in a horizontal line at y = 3 on the vertical axis. Question: In x = 3 how is it that there are an infinite number of y values that can be plotted on the x = 3 graph but y is not showing in the equation? Answer: In x = 3 for every y value in any ordered pair, x always equals 3. We can see what's happening to y when we consider the slope intercept equation: y = mx + c. x = 3, so y = m3 + c. Now c in this equation represents the y intercept, but because x = 3 produces a vertical line that will never cross the y axis there is no value for c, therefore c = 0 to represent a zero intercept for y. This leaves us with y = m3. The slope, m, is a calculation of the ratio of the vertical rise of a graph to the horizontal run, but there is no horizontal component to a vertical line, only a vertical component. This means that m must be undefined. Therefore y = m3 must also be undefined, and must always be no matter what value we assign for x. So although y values exist in the ordered pairs for the graph x = 3, there is no value for the slope of the equation x = 3 and because of this the y variable cannot be shown as a term in this equation. In y = 3, same question but in relation to x: How is it that there are an infinite number of x values that can be plotted on the y = 3 graph but x is not showing in the equation? Answer: In y = 3 for every x value in any ordered pair, y always equals 3. We can see what's happening to x when we consider the slope intercept equation again: y = mx + c. c represents the y intercept, which will occur when x = 0, giving us y = mx + 3. As y = 3 produces a horizontal line, the slope m has no vertical component. Since we know that m is a ratio of the vertical rise of a graph to the horizontal run, m must equal zero. So y = 0x + 3 will always produce y = 3 for all values of x, and x itself cannot be shown as term in the y = 3 equation since it is being multiplied by a zero coefficient. Neverwinter Nights: Having decided to keep Tomi as her henchman after all, Zanzi exits the Trade of Blades. She then meets another one of the city guards. His name is Erevian and is a run-of-the-mill city guard - he will help maintain order, but that's all he will do - like graphing to solve a system of equations (or simultaneous equations), this is a fairly basic modus operandi.
NWN: 0064 Algebra Mind Map 5: 5.6 (Bodyknock Glinckle): Solving equations using logarithms means taking the log for the values on either side of the equation and then manipulating the equation so that you get the variable on one side. You can find the log of any base by representing the expression in exponential form and then taking the logs of both sides. Change of Base: The reason we can take the log of any base of any number and represent it as a ratio of a base 10 log of that number to a base 10 log of the base is because ratios of logs always produce the same answer provided the bases used in the ratio calculation remain the same. E.g. 1) log base 2 of 256 is 8 and 2) log base 2 of 16 is 4; if you divide equation 1) by equation 2) the ratio is 8/4 = 2. Also 3) log base 10 of 256 is 2.4082... and 4) log base 10 of 16 is 1.2041...; if you divide equation 3) by equation 4) the ratio is also 2. In other words the numerator and denominator of the fraction change proportionally with the change in base. You could just as well take log base 3, 4, 5 etc. ratios for any base of a number and you'd end up with the same ratio value. So equations 3) and 4) could log base 3 or log base 4 or log base 5, etc., just so long as the bases remain the same when obtaining the ratio value. We use base 10 because that's what many calculators have a Log for base 10 button. Neverwinter Nights: Boddyknock Glinckle is a gnome sorcerer who has a fancy for novelties and newness, so he will be delighted on how we wield our new skill of finding the value of exponents using logarithms.
NWN: 0063 Algebra MM5: 5.5 (Linu La'neral): Solving Logarithmic Equations. When solving equations containing logs we need to make use of the definition for logs (that the exponent used to raise a base to a certain number is the log of that number) and we always remember that that "certain number" can never be less than or equal to zero. Neverwinter Nights: Next among the potential henchmen is Linu La'neral, a cheerful if clumsy elf, and being of the same race she would appear to be a good companion for Zanzi. However, she will be prone to forget to disregard extraneous values of x (i.e. non-positive values of x) obtained from quadratic equations that sometimes occur when solving log equations - so be careful with her.
NWN: 0062 Algebra Mind Map 5: 5.3 (Sharwyn): In this mini lecture we revise much of what we learned when meeting Daelan. We also learn that to solve an equation involving logs after we've simplified the equation as much as we can using the three properties of logs, we can re-write the equation in exponent form and consequently take out the logarithmic expressions. The equation can then be solved as such and such value(s) of various degrees equals a constant. E.g. log base 3 of x(x-2) = 1 becomes 3^1 = x(x-2). Be careful to identify extraneous values when solving log equations because y = log x means that the domain, x, cannot be less than or equal to zero. Neverwinter Nights: Sharwyn is a hypnotically beautiful human bard who is motivated to resolve two main issues in her life: She needs to obtain an elixir to cure her ailing mother and to obtain an item that can break a spell cast by a queen on Sharwyn's ex-lover. These seem complicated problems to sort out, but once the elixir and item have been obtained these issues will be simple to resolve - like changing log format into exponent format, solving what looks complicated becomes easy. Sharwyn would not be a good henchman for Zanzi in this game because Sharwyn, being also a bard, will have similar strengths and weaknesses to Zanzi. So Sharwyn is to logarithms as Zanzi is to exponents - they are virtually the same bar the fact that Zanzi is an elf. 5.4 (Grimgnaw): In this mini lecture we learn how to use a calculator to find common and natural logs for a given number, and also the inverse function to find the number given the log (exponent) value. We note that the log button is the common log with base 10 implied and that ln is the natural logarithm with implied base e. The respective inverse functions for these logs are the 10^x and e^x functions. Note to use a calculator to find the value of e use the e^x function and enter 1 as the value for x. You will then see that ln(e) or ln(ANS) = 1, which makes sense because logs are exponents and ln is the inverse function of e^x. Neverwinter Nights: Grimgnaw is an evil dwarven monk who detests the undead as this state of non-death escapes natural death or common death. As a henchman Grimgnaw's skills as a fighter, spy and assassin would prove very complimentary to Zanzi's range of arcane spells. His lawful evil alignment might cause problems, but Zanzi is of chaotic neutral alignment so she is quite nonchalant about such matters, just so long as Grimgnaw's abysmal charisma rating doesn't distract others from Zanzi's elven charms.
NWN: 0061 The properties of logarithms, and firstly we note that the logarithm of a product is the sum of the logs; so log base b of x.y is equal to log base b of x + log base b of y . Secondly we note that the logarithm of a quotient is the difference of the logs; so log base b of x / y is equal to log base b of x - log base b of y. Thirdly we note that logarithms will take exponents and turn them into coefficients; so log base b of x^y is equal to y.log base b of x. So what logarithms can do is change higher difficulty operations to lower difficulty operations, turning quotients into differences, products into sums and exponents into coefficients. And this procedure of changing from higher difficulty operations to lower in log calculations is called expanding the logs. The reverse procedure is called simplifying or condensing the logs. Note that when simplifying logs, it is apparently okay to end up with a log that has radical showing in the denominator of a fraction. Neverwinter Nights: After speaking with Graxx she spies a large half-orc fighter-barbarian by the name of Daelan Red Tiger. Daelan's mo in this game is pretty unsophisticated, he uses simple brute force to overcome spells, enemies and obstacles to expand Zanzi's chances of making it through the game. Daelan informs Zanzi that he is looking for a talisman from his mother that will let him communicate with his God, Uthgar.
NWN: 0060 Algebra Mind Map 5: 5.1 (Graxx): In this mini lecture we just go through what we've already learned when entering the Trade of Blades, except we learn a neat trick to graphing logarithms - start by expressing the logarithm in exponential form, then graph the inverse of the exponential form and then obtain the log graph by reflecting the inverse graph about y = x. When graphing inverses we simply transpose the x and y coordinates. Thus for y = log base 2 of x, show this in exponent form, 2^y = x and then transpose x and y to get the inverse, 2^x = y. Graph this and then reflect that graph about y = x to obtain y = log base 2 of x. Neverwinter Nights: The first person Zanzi speaks to in the Trade of Blades is a diminutive fellow by the name of Graxx. Although Graxx isn't a henchman he isn't quite just another member of the townsfolk. For a fairly small fee he can provide Zanzi the means to undertake a sub-quest: The Gauntlet is a fighting arena located at the Board Laid Bare tavern in the Blacklake District. Graxx has a Gauntlet pass to give to the bartender at the Board Laid Bare tavern so Zanzi can participate in fighting in the Gauntlet arena. This is really a fighter, barbarian quest but because Zanzi has such strong persuasive skills, she can eventually invert Graxx's initial refusal to sell her the pass.
NWN: 0059 Algebra Mind Map 5: 5.0 (Trade of Blades): A logarithm is a function that enables us to write an equation that was in exponential form (i.e. showing a product as equal to a base raised to a power) and change it to logarithmic form (i.e. showing an exponent in terms of its base and the product). In logarithmic form we say that log base b of x is equal to y if and only if b^y=x. What this means is that logarithms are exponents. Note that if a function has a one-to-one domain-to-range relationship then it will have an associated inverse function. So because all exponential functions are one-to-one functions (think of their graphs) each exponential function will have an associated inverse function. The inverse function of an exponential function is called the logarithmic function. The "if and only if" phrase has the symbol <=> and means that if one mathematical statement is true then the other statement referred to must be true also; if one is false then the other referred to must also be false. There are a couple of restrictions that need to be placed on the base in logarithms: b>0, so the base must be a positive number, and b not=1. ++++++ Questions: If the base were negative then ...? E.g. (-2)^2 = 4, (-2)^3 = -8, (-2)^4 = 16, etc. The exponential graph would fluctuate between positive and negative range values (step function?) Calculator produces an error when (-2)^2.5 or (-2)^(5/2). What does this mean? Negative bases when raised to a fractional power require imaginary number i? ++++++ Neverwinter Nights: After talking to Cabbage she enters a building called the Trade of Blades, at which point she has reached a critical part of Neverwinter Nights as this is the place where most of the henchmen in the game hang out. It is vital that Zanzi makes the right choice of henchman to compensate for her deficiencies (she can't be good at everything). Henchmen will exponentially raise Zanzi's ability to defeat foes in the game. In logarithmic terms, whoever Zanzi chooses to be her henchman, that henchman is the power to which she (the base) will be raised. So henchmen are analogous to logarithms in this game.
NWN: 0058 Algebra Mind Map 5: 3.2.2 (Girl's bedroom): At this point we are revising a lot of what we've previously learned about functions. We remember that to find the inverse of a function y = x^z, or whatever (must be y in terms of x), we simply reverse the values of the ordered pairs on the original x y chart, or we could draw a graph which is symmetrical about the line y = x. We also remember that if the inverse graph of a function has duplicate values in the first coordinate in its ordered pairs then this means there is more than one y value for a given x value, which means that the inverse can't be a function. 4.0 (Cabbage): Exponential functions work in the same way as regular functions in that when you want to find the value of a function you replace x with that value and simplify. So f(x)=2^x and if f(2) is required then f(2)=2^2=4. Neverwinter Nights: From the mother's bedroom she enters the girl's bedroom, and here Zanzi has a sense of deja vu as it reminds her of her bedroom when she was a child. At this point she and Tomi decide to leave this house, and once outside Zanzi sees a burly looking man by the name of Cabbage, but he's just another member of the townsfolk. Zanzi has a look at her map and sees her shorthand notes, as she did when she was in the west passageway (south) in the academy. She added something in superscript above one of the notes on her map before walking away from the house.
NWN: 0057 Algebra Mind Map 5: 2.3.1 (Mother's bedroom): The formula relating distance to rate (or speed) and time is D=RT. As with other word problems, drawing a table headed for rate, time and distance will help organise our thoughts. During the following calculations you can either use values that show some relationship of speed to distance and time or two values (or even three), x and y (z) and solve by substitution into an equation of x expressed in terms of y (and/or z) or vice versa. Either way you'll end up with (usually) two equations which when solved will provide the solutions to the problem. Check your solutions for correctness, pay attention to the units, and then write out the solutions as a sentence answering the question posed in the problem. Neverwinter Nights: At this point Zanzi enters the mother's bedroom but she could just as well be in the child's bedroom - both rooms perform the same function, namely they are rooms that house a bed to sleep in. Likewise it doesn't matter which method you use to solve these word problems the end result will be the same.
NWN: 0056 Algebra Mind Map 5: 3.2 (Home (upper floor): Age word problems, and these will typically involve two people of different ages and involve a relationship between their ages at some time in the past or in some time in the future. Drawing a table often helps to organise these relationships. Write out an equation or equations with the information provided. After working out the solution(s), check for correctness and then write out a sentence answering the question posed in the problem. Neverwinter Nights: After spending what seems like an age in the living area of the house, Zanzi then walks up to the upper floor of the house, where she sees two bedrooms, one for the adults and the other is a child's bedroom.
NWN: 0055 Algebra Mind Map 5: 3.1.1 (Chest of Drawers): When solving coin word problems draw a table with N, V and T headings (for number, value and total) for each type of coin, fill in as much information into the table as given in the word problem. If there are two unknowns to be found you can either fill in the table letting one unknown be equal to x and the other in relation to x, which will leave you with one linear equation to solve, or you could let one unknown be equal to x and the other to y, which will leave you with a system of equations to solve. Whatever denomination is used to represent the value of the coins, the same denomination must be used when working out the solution to the equation(s). Finally check the solutions are correct and then write out the solutions as a sentence answering the question posed in the word problem. For mixture quantities, remember amt.price + amt.price = amt of mix.price. Neverwinter Nights: In the living area she sees a chest of drawers, and in the first drawer she opened she saw a number of bits and bobs, in the second draw she opened she found a gem and thinks, "This gem looks valuable." The third drawer she opened was totally empty.
NWN: 0054 Algebra Mind Map 5: 3.1 (Home (living area)): When solving word problems involving geometric shapes draw a picture of the shape, mark on it the known values and the relationship of the variables to each other. Then proceed to calculate the solution from your own general knowledge of the shape (e.g. the area of a triangle = 1/2bh, perimeter of a rectangle =2l.2w, etc.), remember to use appropriate units as supplied in the problem. Finally write out the solution as a sentence. Neverwinter Nights: Zanzi walks into the main living area of the house and surveys the geometry of the living room - she looks at how far the walls are from each other, how rectangular the room looks and what the angles are that make up the room.
NWN: 0053 Algebra Mind Map 5: 3.0 (Home (entrance area)): When dealing with number and work word problems you need to look carefully at the wording of the problem and try and figure out what the relationship is between known values and unknown values, then write an expression or equation that represents this relationship. Neverwinter Nights: After speaking with Issabon, she enters an unlocked private residence. Most people who live together in such residences are in some way family-related, but Zanzi and Tomi are not at all related to these people and they should not be in this stranger's house - they are intruders so they must carefully word their explanation for this unsolicited entry.
NWN: 0052 Algebra Mind Map 5: 1.0 (Mulland): Solving for quadratic inequalities means following exactly the same procedure of factoring to products once the quadratic is in standard form, but then you make a sign chart by drawing a number line for each factor and indicate with positive or negative signs values that satisfy the inequality statement for each factor, using a vertical dividing line between positive and negative values. Finally obtain the product of all the factors by stacking the positives and negatives (e.g. two stacked negatives or two stacked positives gives a positive value, one negative stacked with one positive gives a negative value). You can translate the results either by graphing the number line to show the range of values that satisfy the original inequality statement or you can also use interval notation. Neverwinter Nights: After talking to Winthorp she now approaches Castle Never where Lord Nasher, the ruler of Neverwinter, resides. Outside the main door to Castle Never is an administrator by the name of Mulland, and he is adamant that no one should be let in to see Lord Nasher. Zanzi rolls her eyes to the heavens on hearing this and thinks to herself, "This is the second time my authority as an agent for Aribeth has been ignored in the City Core. Maybe I should chart on my map where, if anywhere, there are signs Aribeth has any kind of clout with the people in this place." 2.0 (Issabon): With absolute values we are thinking in terms of magnitude only (not sign), which, for solving inequalities, means considering the distance from zero the inequality expression is on the number line. You must also isolate the absolute value expression so it is on one side of the inequality sign and the constants (or whatever) are on the other side. Neverwinter Nights: As she walked away from Castle Never, Zanzi met Issabon, who is a very snooty lady with an overtly disdainful attitude towards all persons of lower social standing to herself - she is the absolute epitome of inequality, an isolationist, yet Zanzi remains un-phased and absolutely ignores this.
NWN: 0051 Before leaving this mind map, some things you should know before progressing to the next one. Neverwinter Nights: Zanzi takes a breather. (Vocabulary of Quadratic Polynomials, Set Operation: Union, Set Operation: Intersection).
NWN: 0050 An imaginary number is what we get when we take the square root of -1, and whatever number -1^0.5 is, it isn't real, meaning it can't be placed anywhere on the number line. The square root of -1 is represented by the lower case letter i and is used to enable us to undertake calculations that involve negative numbers under a radical. The result of such a calculation produces two numbers, one being a real number and the other imaginary. E.g. (-9)^0.5 = [9(-1)]^0.5 = 3i. Also, because i = (-1)^0.5 it follows that i^2 = -1. The term "complex number" is used to describe any number that has both a real and imaginary part, e.g. 3i. Neverwinter Nights: Zanzi Naburu is the main protagonist in this Game of Neverwinter Nights. She is a young elf (a spring chicken at just 73) and like many of her race has a penchant for the finer things in life, such as poetry, song and magical arts. By elvan standards she is unusually tall at 5.5 feet, but apart from this aspect of her physique she is very much a typical elf with a slim, lithe body, very smooth pallid skin, deep green eyes and long dark brown hair which she generally wears in a ponytail. Her captivating natural beauty only adds to her charisma, which at the beginning of the Game is set at a very respectable 15. She is quintessentially a chaotic neutral character: an individualist who sees social restrictions and traditions as irrelevancies to be ignored or challenged. She seems to be pragmatist whose motivation to confront the dangers facing Neverwinter appears tempered by an almost forensic examination of what she considers realistically achievable and by a belief that "good" cannot exist without "evil". Zanzi evidently takes every character she meets in this Game at face value and she will not make friends or enemies easily and will not judge herself or others by reputation. Belief in herself is central to who she is, in that she considers herself to be the author of her own destiny. Her vital Neverwinter Nights statistics at the start of the Game: Strength: 12, Dexterity: 16, Constitution: 12, Intelligence: 12, Wisdom: 10, Charisma: 15. Given that she has always been prone to live her life immersed in the fantasies of the songs she hears and novels she reads, it is inevitable that Zanzi would progress towards a bardic way of life, and what this means for this Game is that her strongest magic will emphasise spells that incorporate charm and illusion. This is a mercurial, almost disturbing, character that everyone who meets would like to get to know but can never really figure out. What, if anything, is truly motivating Zanzi is a mystery.
NWN: 0049 Algebra Mind Map 4: 5.0 (Winthrop): Solving inequalities is very similar to solving equations, in that the addition and multiplication properties of equality and inequality are almost equivalent (meaning that you can add or subtract equal values to both sides of the equals or inequality sign without affecting the calculation) EXCEPT for one very important proviso for the multiplication property of equality: when multiplying or dividing a term in an inequality by a negative number you must change the direction of the inequality sign, and you must do this at the same time as multiplying or dividing by that number. This is because if a>0 then -a<0 (e.g. as 4>0 then -4<0, also as 4>2 then -4<-2). Neverwinter Nights: Having been hastily moved along by Ibarum with a flea in her ear, she walks with some purpose away from main city gate exit from the City Core. Then she accidentally bumps into an NPC called Winthorp, who, like the rest of the townsfolk in this godforsaken city, is just another victim of the cruel inequalities of life. She looks into his eyes full of fear and thinks, "He needs a drop of Potion of Bless." (See MM1 4.1 (Dining room)).
NWN: 0048 Algebra Mind Map 4: 4.0 (Ibarum): A formula is a concise way of expressing information symbolically. Formulas always have an equal sign and at least one variable, e.g. E=mc^2. Neverwinter Nights After obtaining the Blacklake Zoo quest from Nyatar, Zanzi sees a guard standing by the main gates out of the city core. His name is Ibarum and he told Zanzi in as concise a manner as any formula that she could not leave the quarantined city of Neverwinter by this exit.
NWN: 0047 Algebra Mind Map 4: 3.4 (Lion): Use of the quadratic equation, x = (-b +-[b^2-4ac]^0.5)÷2a, is employed to find x whenever a quadratic equation in standard form (ax^2+bx+c=0) is not factorable (or not obviously factorable). Quadratic in form equations are equations were a variable (e.g. "y") is substituted into a non-quadratic equation to make it quadratic in form; in this form the equation can either be factored to the two binomial factors, each equal to zero (as we need to have the quadratic in standard form), or we can use the quadratic equation formula for the solutions. Once values for y have been obtained, we back substitute y into the original expression and find the value for the original variable (conventionally expressed as "x"). Neverwinter Nights: Finally Nyatar tells her about a lion that is to be freed from the Blacklake Zoo, and like the lion who is considered to be the "king of the jungle", the quadratic equation formula is the ultimate solution method to solving quadratic equations, in that it always works whether the equations are factorable or not.
NWN: 0046 Algebra Mind Map 4: 3.3 (Wolf): There are three main steps to completing the square. Firstly add or subtract as necessary to remove the constant term from the standard form quadratic equation on the left of the equals sign and then add or subtract it to the right of the equals sign. Secondly obtain a 1 coefficient for the x^2 term of the quadratic by dividing that coefficient by itself (and of course dividing every other term in the equation by the same value). Thirdly, look at the x term coefficient, half it and then square it and then make that number the new constant term of the quadratic expression (which will now be a perfect square expression) and also to the constant term to the other side of the equals sign. From then on use factoring and the square root property of quadratics to solve for x. Neverwinter Nights: Also to be freed from the Blacklake Zoo is a wolf, and once you get the hang of this completing the square method, it's very straightforward, and like a wolf's cry, is a howling success in solving quadratics and always works whether the initial equation is factorable or not.
NWN: 0045 Algebra Mind Map 4: 3.2 (Black panther): The square root property of perfect square quadratics means that taking the square root of squared numbers results in two values, one positive and the other negative. We can then manipulate the equation so that the variable is by itself on one side of the equation. Neverwinter Nights: Nyatar also wants a black panther freed from the Blacklake Zoo, and like a black panther in the dark, the point of using the square root property of quadratics may seem hard to see, given that the equation must be factorable and can therefore be foiled, but in the daylight we can see this square root property of quadratics is a path to derivation of the quadratic equation formula by the "completing the square" method. The square root property method of finding solutions to a perfect square quadratic equation is also faster than factoring by FOILing.
NWN: 0044 Algebra Mind Map 4: 3.1 (Bear): Factoring quadratic equations relies on the zero factor property (or the zero product property), the fact that the only way to get a zero product is to multiply by zero. So after factoring a quadratic in standard form you're left with two factors, either of which equates to zero, giving you a value for x for each factor. So for solving quadratic equations we rely on the BEAR-EST number, which is zero. Neverwinter Nights: The first of the captive animals in Blacklake Zoo Nyatar is worried about is a bear. Zanzi will have to use the scented fetish to initiate dialog with this animal and guide it out of the zoo and to safety.
NWN: 0043 Algebra Mind Map 4: 3.0 (Nyatar): Equations containing radicals, and we use the squaring, cubing, 4th power, 5th power, etc. property of equality to find a real number solution to the equation, if there is one. E.g. Use the squaring property of equality to solve (x+19)^0.5 = x+7. First square both sides: [(x+19)^0.5]^2 = (x+7)^2 to get x+19 = x^2+14x+49. Then put the equation in standard form: 0 = x^2+13x+30. Factor: (x+10)(x+3) = 0 to obtain solutions x=-10 or x=-3. Finally check each solution: (-10+19)^0.5 = -10+7 which equals 9^0.5=-3, and because a negative number can never be the result of taking the square root of any real number we can say that this is a false statement or extraneous solution. Check next solution: (-3+19)^0.5=-3+7 which equals 16^0.5=4, which is a true statement. Neverwinter Nights: After exiting the Cloak Tower, she approaches a man standing underneath the mother of all oak trees. He is Nyatar, a druid from the divinity school of magic. Everything Nyatar believes in centres around the balance and equality of nature. When talking to him she learns of his concern for the welfare of four captive animals in the Blacklake District Zoo. She wants to help but he initially refuses, seeing her as too much of a radical, like an unfathomable ratio, to be of assistance. (See Basic Maths MM1 6.1.1 (Elynwyd) - remember that Zanzi has studied some diametrically incompatible magic to divinity, especially in relation to her rogue-ish skills). However, she is a bard of great charisma and eventually persuades Nyatar to task her with the quest of freeing the four animals from the Blacklake Zoo.
NWN: 0042 Algebra Mind Map 4: 2.1.1.4 (Kindling wood): The process of eliminating radicals from the denominator is called rationalising the denominator, and it works by multiplying the offending fraction by the number one in the form of root-over-root, where the root is the same as the radical denominator. E.g. (1/2)^0.5 simplifies to 1^0.5/2^0.5. We need to rationalise the 2^0.5 denominator thusly: (1^0.5/2^0.5)(2^0.5/2^.05) = 2^0.5/4^0.5 = 2^0.5/2. Neverwinter Nights The fourth and final item is kindling wood, which Eltoora can use to synthesise a fireball and spells of mass destruction. Sanctioning the production of such spells can only be rationalised in times of war, and Neverwinter is at war.
NWN: 0041 Algebra Mind Map 4: 2.1.1.3 (Puff of fog): Multiplication of radicals, and these sorts of problems are often presented as roots within two or three binomial expressions multiplied together, in which case use the commutative and distributive properties of multiplication to group numbers and roots, then multiply accordingly. Neverwinter Nights: The third item is a puff of fog, the multiplicity water droplets of which Eltoora can use to synthesise spells that can radically obscure the vision of adversaries.
NWN: 0040 Algebra Mind Map 4: 2.1.1.2 (Flask of Water): When adding or subtracting radicals (which is also described as "combining" roots) we often have to simplify the roots of each term and then see if we can find a common radical (or "radicand"). And the procedure for simplifying the roots requires that we look at the number under the root and see if it will factor into a perfect square (for square roots) or perfect cube (for cube roots). So 75 has the factors 3 and 25 and 24 has the factors 3 and 8. Then take the perfect square or cube out from under the radical, i.e. 5.3^0.5 and 2.3^(1/3). Finally add or subtract expressions of the same radicand, e.g. 5.3^0.5 + 6.3^0.5 = (5+6).3^0.5 = 11.3^0.5. The radical parts of each expression are like variables, and we can use the distributive property of multiplication to allow us to add the numbers multiplying the radicals as if they were coefficients multiplying variables . Neverwinter Nights: The second item is a flask of water, and the flask and water are completely different states of matter (in the same way, radicals are not variables), yet only in this combined form can Eltoora make use of their potential to synthesise magic potions to eradicate enemies.
NWN: 0039 Algebra Mind Map 4: 2.1.1.1 (Rare earth clay): When simplifying radicals we want to take as much out from under the radical as we can, which means factoring numbers and variables to obtain square, cube, etc. (as required by the square, cube, etc. root that encompasses the numbers and variables we are dealing with) values that when taken out from under the radical will show a non-zero integer and or a variable with an integer exponent multiplied by the root encompassing the remaining factored number and or variables. Where a variable is involved it's important to realise that for any even root, i.e. square root, fourth root, sixth root, etc., the number for the variable under the root must be positive, because there is no real number that when multiplied or divided by itself an even number of times will produce negative value. E.g. Simplifying (x^2)^0.5 means that x, if positive, will equal x after the calculation, which is true because we end up with the same value for x after we square it and then take the square root of the square of x. If x is negative then we need to obtain the opposite value of x in order to make it positive under the square root. So if x = -4, we must have ([-1(-4)]^2)^0.5. Now the answer is 4, which is true because [-1(-4)] is what we originally squared. So when we see a problem that asks us to simplify a radical like (49x^2)^0.5 we must specify whether x >=0. And what we are doing when x is negative and we go through the process of making it positive so we can simplify the root is taking its absolute value. Thus, the absolute value of x when x is negative is the same as -1(x) or -x, which is always positive. So |x| = -x when x is negative. Neverwinter Nights: The first of these four items is some rare earth clay, which is in the wizard lab in the Peninsula District, and what Eltoora will do with this is magically mould this seemingly useless clay and synthesize it into a spell that can be of positive use in defeating enemies.
NWN: 0038 Algebra Mind Map 4: 2.1.1 (Eltoora Sarptyl): Synthetic division is a simplified long division algorithm used to calculate the quotient of a polynomial divided by a rational expression of the form x+k or x-k, where k is a constant term. With synthetic division, the polynomial (dividend) coefficients of the second and subsequent terms (including, if present, the last constant term) are multiplied by the opposite sign constant term of the rational expression (divisor), the coefficient of the first term of the polynomial having first been "dropped down" unchanged to enable the first multiplication operation. If the final product in this procedure is a non-zero number then this number becomes part of the fractional remainder and is the numerator of the fraction with the rational expression as the denominator. Neverwinter Nights: On exiting the hallway she meets Eltoora Sarptyl, leader of the Many-Starred Cloak guild. As Zanzi is a bard, Eltoora can offer her a quest which, if successful, will enable her to join the guild: Four items are required by Eltoora to enable her to synthesise various arcane magic potions and spells which will simply and quickly eradicate and divide many enemies. Each item is located in a wizard laboratory, one lab for each of the four districts outside Neverwinter's City Core.
NWN: 0037 Algebra Mind Map 4: 2.1 (Cloak Tower hallway): Any graph of the form y=a/x, where a is a constant integer and x is a variable, will produce a graph with two curved lines each with a horizontal and vertical asymptote. Neverwinter Nights: She entered this hallway through the front door (like an X intercept). She exited this hallway (like a Y intercept). And she never touched either side of the hallway (like vertical and horizontal asymptotes). To confirm her path to and from Eltoora she noted a few of her steps along the way (like checking the graph with a few easy co-ordinate points).
NWN: 0036 Algebra Mind Map 4: 2.0 (Outside the Cloak Tower): Equations containing rational expressions, and often the easiest way to solve these equations is by eliminating the fractions altogether by multiplying both sides of the equation by the LCD and then dividing out any common variables. It's important that check for extraneous results (or excluded values) by plugging in your calculated solution into the original equation to see if any denominators produce a zero value. Neverwinter Nights: Feeling cleansed by the priest's blessings, she now stands outside the Cloak Tower, equating how the insecurity she felt when breaking into the small crate in food store 2 (the money found in that crate reminded her of (the lowest common) Dendy's extraneous parrying techniques) with the confidence she now feels when entering the Cloak Tower - she has sorcerer-related powers and can legitimately enter the tower.
NWN: 0035 Algebra Mind Map 4: 1.0 (Priest of Helm): Sometimes rational expressions take the form of complex fractions and when this is the case we can employ two strategies to simplify the expression. We can multiply the dividend rational expression by the reciprocal of the divisor rational expression and we can use the LCD. Neverwinter Nights: Both Zanzi and Tomi are glad to have made it out of the City Core's food storage facility without getting caught for stealing, and from there they walk towards a man wearing a scarlet uniform. On speaking with this person Zanzi learns that he is a Priest of Helm and he asks her if she would like to receive a blessing from him. It's as if he can sense the complex guilt Zanzi is harbouring (she hates stealing and her sense of purpose to allow it is fractional). She acquiesces, and receives two blessings from this priest, one to multiply and reciprocate her sense of purpose and the other to reduce her enemies to the lowest common denominator.
NWN: 0034 Algebra Mind Map 3: 5.2.3 (Small crate): Just like fractions, calculations for addition and subtraction of rational expressions can become a lot more convoluted than those for multiplication and division, because if the denominators are not the same you will need to factor them out and then use the lowest common denominator method to obtain the same denominator for both rational expressions. Only when you've done that can you go ahead and add or subtract the numerators. Neverwinter Nights: The third and final cube she delves into is another crate, slightly smaller than the huge crate she'd just looked in, but still pretty big all the same. In here she finds three gold pieces, and she thinks to herself, "Actually, I've found quite a lot of money so far. I'd better spend it on some decent protective items." At the back of her mind is Dendy, who taught her about parrying. Trouble is, parrying has never worked in Neverwinter Nights and hence the need to compensate by purchasing lots of defensive protection, be it physical or magical.
NWN: 0033 Algebra Mind Map 3: 5.2.2 (Crate): Multiplying and dividing two rational expressions, and what we are relying heavily on here is the associative and commutative property of multiplication (remembering that division is multiplication by the reciprocal). The thing to bear in mind is that the aim here is not actually to multiply the two rational expressions but rather to obtain the solution in lowest terms. So first combine the two rational expressions into a single fraction and then, if necessary, rearrange the terms before dividing out any common factors. It's easy to get too involved with the calculation and carry out the multiplication without dividing out common factors, in which case you'd only end up having to factor the product of the numbers you've just multiplied - inefficient and a waste of time. Neverwinter Nights: The next cube she looks in is a pretty big crate. In here she finds seven bullets. She becomes quite emotional when looking at these bullets as they remind her that there's no going back (like how she felt when Pavel told her of the death of his brother in the academy library) and also of the futility of trying to learn anything from the books she saw on the first bookshelf. The point being that it doesn't matter in which order she uses these bullets, they are all equally potent weapons if used correctly and in this respect are associative and commutative weapons.
NWN: 0032 Algebra Mind Map 3: 5.2.1 (Box): Reducing rational expressions, and first we need a definition for the term "rational expression". A rational expression is a fraction that has a polynomial in both the numerator and denominator. If either polynomial has a variable, x, say, then the exponent of this variable will be a positive whole number. So (3x^5 - 9) / (x-3) would be an example of a typical rational expression. Another term for "rational expression" would therefore be "polynomial fraction". The procedure for reducing rational expressions is exactly the same as that for reducing fractions, namely you factor the numerator and denominator and divide out any common factors. For rational expressions that have variables in the denominator it's critical to consider their domains because you can't have a value for variables that results in a denominator of zero. A numerator of zero is okay though. Neverwinter Nights: The first cube she looks in is a fairly large box, and she knows that items found in this game vary on each re-load. At this point Zanzi breaks the fourth wall and speaks directly to the Game Player saying, "Look if I don't like what's in this box, do me a favour and re-load the game." To which the Game Player replies, "That's cheating!" To which she scoffs, "Trust you to be an ethical gamer! Just factor the variables will you and don't leave me with something that reduces my chance of success or is of zero use to me, like an armoire."
NWN: 0031 Algebra Mind Map 3: 5.2 (Food store 2): The equations for the sum or difference of two cubes just have to be memorised. The sum of two cubes: a^3 +b^3=(a+b)(a^2 -ab+b^2). The difference of two cubes: a^3 -b^3=(a-b)(a^2 +ab+b^2). Note that the trinomial factors in both cases are never factorable. Neverwinter Nights: From the first food store she entered the second food store. All the containers in here are cubes, the sum of the contents of which could make a big difference on how well she progresses through this chapter of the game. She thinks, "Everything about this place is square." (What she means is that the trinomials factored from the sum or difference of cubes can never themselves be factored, which is a bit of a bummer).
NWN: 0030 Algebra Mind Map 3: 5.1 (Food store 1): The difference between two squares takes the form of a^2 -b^2, which will always factor into the product of the sum of a+b and the difference a-b. Problems that involve the difference of two squares crop up everywhere in algebra, so learn the squared values up to at least 15^2 in order to instantly recognise squared differences. Neverwinter Nights: On entering the first food store she sees lots of different kinds of poultry and meat products hanging from hooks, whereupon she suddenly feels pangs of hunger. She hasn't eaten in a while and could do with a good square meal right now. 5.1.1 (Chest): Perfect square trinomials, and if you have a situation where you recognise that the a and c numbers of ax^2 +bx+c or ax^2 -bx+c are squares then there's a good chance that the trinomial resulted from the expansion of ([a^0.5]x +c^0.5)^2 or ([a^0.5]x-c^0.5)^2 respectively. So to shave a few seconds off estimating numbers to go into binomials, look at the a and c numbers and plug their square roots into a binomial, square that binomial, multiply it out and see if the resulting trinomial matches the one you're trying to find factors for. If it does then you have found the factors of a perfect square trinomial. For example, 36x^2 -84x+49 might well be a perfect square trinomial resulting from the expansion of (6x-7)^2. Test this by expanding (6x-7)(6x-7) and 36x-84x+49 results, confirming that (6x-7)^2 is the correct factorisation of 36x^2 -84x+49. Neverwinter Nights: Whilst in food store one she sees a chest that is quite similar to the one she saw in her bedroom in the academy and also the one she saw in the other (GCF) store room. In this chest she finds three gold pieces. Everything about this chest reminds her of three. She thinks, "If I find one more of these chests that would make a perfect square."
NWN: 0029 Algebra Mind Map 3: 4.1 (Borlan): Factoring a trinomial will result in two binomials, and likewise multiplying two binomials will result in a trinomial. Consider two binomials (x+y)(x+z). Multiplying using the FOIL method these we get x^2 +xz+yx+yz, which equals x^2 +(z+y)x+yz. Usually when presented with factoring trinomials the x variable is the unknown term and the (z+y) coefficient of x and constant yz term will be supplied. So whatever number is provided for the constant term, it will be the product of y and z, and the number for the coefficient of x will be the sum of y plus z. Neverwinter Nights: After speaking to Phoebe she speaks to Borlan, who also comes from Beggar's Nest and can therefore also not return home because of the zombie infestation in that district. So in Zanzi's mind Phoebe and Borlan are two known people who she believes she can help. The unknown x factor is the zombies, the undead people whom she knows nothing about but is still confident she can work out a way to kill them all. 5.0 (Food store corridor): More on factoring trinomials, and if the factors of the constant term, c, are not common to both a and b in the trinomial ax^2 +bx+c then neither of its binomial factors will share a common factor. For example the trinomial 6x^2 +7x+2 could have factors (3x+?)(2x+?) because 3x.2x = 6x^2. For the constant term 2 we only have two possible factors, 1 and 2. When choosing which number 1 or 2 to place in which of the two binomials be guided by the fact that the 2 can't go into the 2x+? binomial because then we would have (2x+2), which has a common factor of 2, as in 2(x+1). This can't be correct because although 6 has 2 as a factor, 7 does not. Therefore the factors for 6x^2 +7x+2 must be (3x+2)(2x+1). This can be seen if we let x = 2, say, meaning that 6x^2 +7x+2 = 40. If 2x+2 were a factor then 6 would be a factor of 40, which is not the case. And it turns out that whatever value we choose for x, 2x+2 will never be a factor for this trinomial. So if x=1, we get 15 and 4 is not a factor, if x=3, we get 77 and 8 is not a factor, if x=4 we get 126 and 10 is not a factor, and so on. Neverwinter Nights: After speaking with Borlan she makes her way to the city core food storage facility with a mind to take more loot for herself, and once again she relies on Tomi to unlock the door so she can enter the store corridor. Zanzi does admire Tomi's skill at picking locks and as she see's him unlock this door with apparent ease she's half a mind to ask him to teach her how he does it (this lock had three bolts to overcome after all). But then she dismisses the idea - lock-picking is best practiced by those with a thieving disposition and though she can make use of some rogue skills, she favours "lawful" pursuits and tries to convince herself she has nothing in common with the likes of Tomi. (Zanzi can be very hypocritical at times!)
NWN: 0028 Algebra Mind Map 3: 4.0 (Phoebe): Whenever you have four terms in an expression that you want to simplify you'll want to look for any factors common to all those terms, however you'll most likely not be able to find common factors for all the terms in the expression. By grouping those terms that do have common factors you'll at least be able to simplify the expression to some extent. This procedure is called "factoring by grouping" and it works like this: Consider the four term expression ax+bx+ay+by. There is no common factor for all four of these terms but there are common factors for the two pairs of terms: ax+bx, namely x, and ay + by, namely y. So factoring out the x and y we have x(a+b)+y(a+b). Suppose we let a+b=2, we can then see that the expression becomes 2x+2y and 2 is clearly a common factor for x and y, so we can now factor this to get 2(x+y). Likewise x(a+b)+y(a+b) can also be factored further to the simpler expression (x+y)(a+b) because whatever the value of (a+b) is, it's just a number. The same is of course true for (x+y), it too is just a number. Neverwinter Nights: Having exited the store she now approaches one of two wretched people sleeping rough on the street. This person's name is Phoebe and she hails from the Beggar's Nest district of Neverwinter. Problem is, she can't get home because a group of zombies is infesting that district. Despite Phoebe's somewhat dismissive response when Zanzi suggests she may be able to help her return home, Zanzi is more confident as knows that zombies have common frailties which, when factored together, can be used to kill the entire group of them.
NWN: 0027 Algebra Mind Map 3: 3.1.3 (Chest): The greatest common factor of a polynomial (GCF) is found by factoring out the largest number that divides the coefficients and then factoring out the largest common variable. E.g. 6x^6 +24x^2 will have a GCF of 6x^2 because 6 is the largest number that divides 6 and 24 leaving no remainder and x^2 is the largest common variable as x^2 divides into itself once and x^6 three times. Note that we don't have to worry about not having remainders when considering the variable for the GCF because exponents on variables represent repeated multiplication, so if we consider x.x.x.x.x.x / x.x.x.x.x = x^6 / x^5 = x. So the GCF for 6x^6 +24x^5 would be 6x^5. Neverwinter Nights: Next she sees a chest which is quite similar to the one in her bedroom in the academy. Inside the bedroom chest were items that defined why she was in the academy, namely to learn how to fraction the enemies of Neverwinter. In this chest she finds a throwing axe, which she will also use to fraction the enemy (into pieces). There are factors of great commonality between this chest and the one in her academy bedroom.
NWN: 0026 Algebra Mind Map 3: 3.1.2 (Crate): Scientific notation is used to represent very large or very small numbers and takes the form of a whole number or decimal multiplied by ten to whatever power required to make the number equivalent to the standard notation form. E.g. 1234 in scientific notation is written as 1.234 x10^3. There should be no trailing zeros in the decimal, so 12340 would be 1.234 x10^4. Writing 1.2340 x10^4 would not be correct. In general, a number in scientific notation is written as "a x 10^n", where the absolute value of the a variable (absolute value because negative numbers can also be written in scientific notation) has values such that 1<=a<10. If the value of the number in standard form is between 0 and 1 then the exponent, n, on the 10 in scientific notation will be negative. So 0.1234 in scientific notation would be 1.234 x10^-1 and 0.01234 would be 1.234 x10^-2, and so on. Neverwinter Nights: In the crate she finds two gold pieces or 2 x10^0 in scientific notation.
NWN: 0025 Algebra Mind Map 3: 3.1 (Store): When dividing a polynomial by a monomial we need to see if the numerator is factorable and then see if we can divide out common factors. E.g. (10x^3 - 15x^2) / 5x = 5x^2(2x-3) / 5x = x(2x-3). So the largest common factor we could extract for the numerator is 5x^2 and then we can divide 5x^2 by 5x to leave x. We could if we want go one step further and multiply out the bracket for x(2x-3) to give 2x^2 -3x. Neverwinter Nights: At the end of the corridor she enters the store. Whatever she can find here she will keep most (poly) of the good stuff for herself and give just the odd titbit (mono) to Tomi. Not that she's selfish, mind, but after all it's she and not Tomi who's tasked with saving Neverwinter. 3.1.1 (Box): Dividing a polynomial with a polynomial first we need to see if the numerator is factorable and then see if we can divide out any common factors. In the following example the numerator isn't factorable and in this situation we need to use the long division algorithm. E.g. (x^2 -5x+8) / (x-3). So x-3 divides into x^2 x times to get x^2 -3x. Then subtract x^2 -3x from x^2 -5x to get 0 -2x to which we bring down the 8 to give -2x+8. So x-3 divides into -2x+8 negative two times to give -2x+6. Subtract -2x+6 from -2x+8 to get 0 +2. So the resulting quotient is x-2 with a remainder of 2 / (x-3). If the dividend polynomial has "missing powers" then we must supply them as placeholders in the form of 0x to whatever power is required. E.g. Divide (2x^3 -3x+2) / (x-5). Again the numerator is not factorable so we have to take the long division route. First we need to write the numerator as 2x^3 +0x^2 -3x+2. We've added the 0x^2 term purely in order facilitate the long division algorithm. So x-5 divides into 2x^3 twice x^2 to get 2x^3 -10x^2. Then subtract 2x^3 -10x^2 from 2x^3 +0x^2 to get 0 +10x^2. We then bring down the -3x to get 10x^2 -3x. So x-5 divides into 10x^2 -3x ten x times to get 10x^2-50x. Subtract 10x^2 -50x from 10x^2 -3x to get 0 +47x. Bring down the +2 to get 47x+2. So x-5 divides into 47x+2 forty seven times to get 47x-235. Subtract 47x-235 from 47x+2 to get 0 +237. The quotient is therefore 2x^2 +10x+47 with a remainder of 237 / (x-5). Neverwinter Nights: The first thing she looks into is a box and after a long time delving through the numerous (poly) pieces of rubbish in this box she finally finds one gold piece. She thinks, "I'll need many (poly) more of these GP's if I'm going to buy the kit necessary to divide and conquer Neverwinter's enemies."
NWN: 0024 Algebra Mind Map 3: 3.0 (Store corridor): Multiplication of polynomials usually entails multiplying a bracketed polynomial expression by a monomial or two or more bracketed polynomial expressions together to obtain an expression that has no brackets. The phrase "multiply out bracket" is often used in this regard, and when we do this we change the calculation from one whose primary operation was multiplication into one whose primary operation is addition and/or subtraction. Multiply a monomial by a polynomial: 3x^3(2x^2 +4x+5) = 3x^3.2x^2 +3x^3.4x+3x^3.5 = 6x^5 +12x^4 +15x^3. So we distribute the monomial to each term within the bracket in order to eliminate the bracket. We then multiply those terms together to end up with a string of terms that can be added together whenever we plug in a value for x. Multiply a binomial by a binomial: (3x-5)(2x-1) = 6x^2 -3x-10x+5 = 6x^2 -13x+5. Each term is multiplied together in a systematic way: First term (3x.2x), Outside term (3x.(-1)), Inside term ((-5)(2x)), Last term ((-5)(-1)), or FOIL. The expression resulting from the multiplication of two binomials that have a single x (or whatever is assigned) variable in each binomal is called a quadratic. Beyond multiplying two binomials (i.e. multiplying a binomial by a trinomial, multiplying a trinomial by a trinomial, etc.) the procedure of multiplying each term in sequence to every other term becomes very tedious. This is where study of Pascal's Triangle and augmented matrices will become applicable. Neverwinter Nights: Somewhat sombre after speaking with Halion, she makes her way to the first of two store buildings in the city core. Zanzi is soon glad she has chosen Tomi as a henchman as his multiple lock-picking skills have given her access to the store and she is now able to open the door and walk through the store corridor.
NWN: 0023 Algebra Mind Map 3: 2.0 (Halion): Adding and subtracting polynomials is really just a matter of adding and subtracting like terms. Add: (2x^2 -5x+3)+(4x^2 + 7x-8) = 2x^2 -5x+3 +4x^2 +7x -8 = 2x^2 +4x^2 -5x+7x+3 -8 = 6x^2 +2x -5. Subtract: (3x^2 +x+4) - (x^2 +2x+3) = 3x^2 +x+4 -x^2 -2x-3 = 3x^2 -x^2 +x-2x+4 -3 = 2x^2 -x+1. Neverwinter Nights: After speaking to Bethany she talks to one of the city guards. His name is Halion and gives her additional information on the numerous plague related issues facing Neverwinter city core. The city has been quarantined and none of the many (poly) people within the city core can enter (add to city population) or leave (subtract from city population) without authorisation (only like terms).
NWN: 0022 Algebra Mind Map 2: 2.4 (Oleff): Operations with monomials. Multiplication and division of monomials - these operations are carried out on each like term, i.e. the constants and the like variables. E.g. multiply (-3x^2)(4x^3) = -12x^5 and divide 15x^3 / 3x^2 = 5x. Note that when multiplying the like variables we add their exponents and when dividing the like variables we subtract their exponents. Addition and subtraction of monomials - these operations can only be carried out if the variable part of each term is the same. E.g. -3x^2 +15x^2 = 12x^2 and 9x^2y - 20x^2y = -11x^2y. Note that the resulting sum or difference will have the variable part of the expression unaltered and that the only thing that changes is the value of the coefficient. You can't combine monomials (and therefore polynomials, as monomials are a type of polynomial) that have different variables for addition and subtraction. E.g. 5x^2 +8y^2 cannot be added to a single total and as there are no common factors for the terms the expression is already in its simplest form, namely a binomial. Neverwinter Nights: She accepts Judge Oleff's quest to search for Lord Never's tomb. There is ONE tomb in each of the four outlying sectors of the city, so this is a MONOmial quest for each sector. Oleff gives Zanzi a letter signifying that Zanzi acts with his sanction. She is to give the letter to guard Briley in the Peninsula district. Briley is guarding the one tomb already found. Algebra Mind Map 3: 1.0 (Bethany): Neverwinter Nights: She then leaves the Hall of Justice and her attention is immediately focused on a young girl begging for help in the street. On speaking to this girl, whose name is Bethany, Zanzi learns that after the prison break in the Peninsula district the prisoners are on the rampage killing everyone. Bethany will not be able to return home until this district has been made safe, so Zanzi promises to help if she can, as a charitable act for Bethany - there are no common factors between this quest and Judge Oleff's, so Zanzi will have to undertake each one separately. She cannot combine unlike MONOmial quests.
NWN: 0021 Algebra Mind Map 2: 2.3 (Sergol): At this stage we're only really interested in learning the terminology of polynomials - what's a monomial, binomial, trinomial and polynomial. Take particular note that the procedure for determining the degree of a monomial is different from determining it for all other types of polynomial. Neverwinter Nights: In the Temple of Tyr, Sergol explains to her that his magic is centred around teleportation and use of the Stone of Recall. There are three situations when this magic is useful: firstly to unload goods (monomial), secondly to heal (binomial) and thirdly to return to return to your original location for a small fee (trinomial). Further, this teleportation magic works at almost any time throughout this part of the game (polynomial).
NWN: 0020 Algebra Mind Map 2: 1.0 (Sanatorium): Just as coefficients can represent multiple addition operations on a number, exponents represent mutliple multiplication operations on a number. So a+a+a = 3a, where 3 is the coefficient which denotes how many times the same variable undergoes multiple addition, and a.a.a = a^3, where 3 is now the exponent to which a, the variable, is "raised". In the case of the coefficients, the variable's status remains unchanged but when the variable has an exponent it is also referred to as a "base" and the exponent as a "power". So we would describe "a^3" as "a raised to the third power". Considering a^2, note that this is a.a, and a^1 is a, and a^0 is a/a=1, and a^-1 is a/(a.a) = 1/a, and a^-2 is a/(a.a.a)=1/(a.a), etc. So the number for the exponent denotes how my times a base undergoes repeated multiplication operations. All other definitions for exponents are derived from this basic definition. Neverwinter Nights: Zanzi begins Chapter One of Neverwinter Nights by meeting Fenthick and Desther in the city core sanatorium. Fenthick explains to her the grave situation facing Neverwinter. All around her are desperately ill sufferers of the Wailing Death. This disease is spreading EXPONENTIALLY throughout the city core. 2.1 (Aribeth): One of the properties for exponents of bases that undergo division is that if the bases are the same then you subtract the exponent of the divisor base (or base in the denominator) from the exponent of the dividend base (or base in the numerator). So a^x / a^y = a^(x-y). E.g. a^7 / a^2 = a^5 and a^7 / a^-2 = a^7 -(-2) = a^9. Another property is that exponents outside bracketed fractional expressions are distributed to the bases when the brackets are removed. So (a / b)^x = a^x / b^x. E.g (2 / 3)^3 = 2^3 / 3^3 = 8 / 27, and you can see this is true because (2 / 3)^3 is (2 / 3)(2 / 3)(2 / 3) = (2.2.2) / (3.3.3) = 8 / 27. Neverwinter Nights: In the Temple of Tyr, Neverwinter's Hall of Justice, she gets to speak in depth with Lady Aribeth, and it soon becomes clear to her that Aribeth is unsure of whom to trust and even unsure of herself - she is very divided and this subtracts from her confidence. Aribeth instructs Zanzi to find the four Waterdhavian creates and bring them back to her. As a secondary quest, she also instructs Zanzi to find clues about what has caused the plague. 2.2 (Tomi): Here we learn many examples of multiplication of exponents and there are a couple of rules we learn here, namely (a^x)^y=a^(xy) and (a^x.b^y)^z=a^(xz).b^(yz). When learning these examples the word "power" is often used for the exponents. E.g.(5^3)^2 is described as "five cubed raised to the second power". Neverwinter Nights: Also in the Temple of Tyr she meets Tomi Undergallows, who is wanted for a number of crimes, including "deflowering" a pasha's daughter and pickpocketing. As Tomi has thieving skills that Zanzi is lacking, she (the base) pays him to be her henchman (the power) to exponentially raise her ability to progress through the game. Tomi provides Zanzi with information regarding the strange goings on in the Peninsula district prison - the warden's let out all the prisoners and appears to be deranged. Tomi is a halfling of the underworld in Calimport. He, like Zanzi, is of chaotic neutral alignment. He is a very skilled rogue who tells Zanzi that he was employed by a powerful medusa. His last job for this medusa was to infiltrate the house of the famous vizier Sabbalan Viheyed and kill him, but he refuses to provide any more information about this to Zanzi at this time.
NWN: 0019 Algebra Mind Map 1: 11.0 (Stables): Inverse functions can be explained in relation to ordered pair coordinates, such that if and only if the ordered pair (x,y) belong to a function then the ordered pair (y,x) will belong to the inverse of that same function. Inverse functions can also be explained in relation to each other, such that f composed of f inverse of x will equal x and f inverse composed of f of x will also equal x. So what this means that whatever f does to x to get y, then f inverse will do the opposite to y to get x and vice versa. So if f(x) = 2x then f inverse (x) = x/2. Then f(f inverse(x)) = 2(x/2) = x. Inverse functions can also be explained by the fact that graphs have symmetry about the line y = x. If a function is graphed and you reflect that graph about the line y = x the resulting graph will be the inverse of that function. It might well be for non linear functions that this reflected graph is not itself a function. If it has duplication of first coordinates for different ordered pairs then it won't be a function. This is easy to see graphically because a non-function graph will cut a vertical line in more than one place - the "vertical line test" - indicating duplication of x coordinates. Inverse function notation is written as f^-1(x), g^-1(x), h^-1(x), etc. The superscript ^-1 is just a way to indicate these functions are inverse functions and does not mean 1/f(x), 1/g(x), 1/h(x), etc. The reciprocal of a function is not the same as its inverse function. (From the above example f(x) = 2x and f inverse (x) = x/2, then 1/(2x) is not equal to x/2 for all values of x except x=1 and x=-1). Neverwinter Nights: In the stables she sees the loss of the Waterdhavian creatures. She also says goodbye to Pavel and meets Fenthick Moss and Desther Indelayne. Fenthick and Desther are apparently working together, yet they are totally at odds as regards the method of solving the crisis facing Neverwinter. Fenthick is young, attractive, gentle, soft spoken and very pro-Lady Aribeth and Desther is old, ugly, aggressive, coarse and definitely very anti-Lady Aribeth, and they sort of like INVERSES of each other. Having defeated an elite goblin during the fight for the Waterdhavian creatures, Zanzi gained enough experience points to level up. She is now a third level bard; most notably her Bard Song has been enhanced with the Extra Music feat and she has learned the Mage Armor spell from Conjuration.
NWN: 0018 Algebra Mind Map 1: 10.0 (Common room): Composition of functions is a non-commutative operation, symbol o (ring operator). This function notation is written (f o g)(x) and is read ‘f composed with g of x’. If f(x) = 2x-1 and g(x) = x^2 -4, f composed with g of x will give us f(x^2 -4)=2(x^2 -4)-1 = 2x^2 -8 -1 = 2x^2-9. And g composed of f of x will give us (g o f)(x) = g(2x-1) = (2x-1)^2-4 = 4x^2 -4x+1 -4 = 4x^2 -4x-3. So for (f o g)(x) we do f to what we get when we do g of x, and for (g o f)(x) we do g to what we get when we do f of x. Note that (f o g)(x) is the same as f(g(x)) and (g o f)(x) is the same as g(f(x)), and that (f o g)(x) is not equal to (g o f)(x). Neverwinter Nights: She nearly forgot to check the common room before leaving the main academy complex, but here she found a nice pair of gloves from one of the cases, the Gloves of Swordplay, which are COMPOSED of magically enhanced materials to provide a bonus to the Parry skill; such is augmentation of fencing skills produced when wearing these gloves that they enable Zanzi, who is already naturally very dexterous, to use the tip of her sword to score an almost perfect CIRCLE (ring) on any surface.
NWN: 0017 Algebra Mind Map 1: 9.0 (West passageway (south)): Function notation is a shorthand way of specifying functions. E.g. if we have two graphs y = 3x-6 and y = -x^2 +2, we will have no way of specifying which graph we are referring to unless we write out the full equation for each. If instead we define f(x) = 3x-6 and g(x) = -x^2 +2 we can specify which graph we mean just by referring to f(x) or g(x). Neverwinter Nights: In the west passageway (south) she looks at the SHORTHAND notes on her map sees that all rooms have been explored except for one. Her map has fulfilled a NOTABLE FUNCTION in this regard.
NWN: 0016 Algebra Mind Map 1: 8.1 (Study): The domain of a function is the set of all possible values of x and the range of the same function is the corresponding set of all possible values of y. A function is like a mathematical machine: values of x get put into the function and after the function does something to those values it spits them out as values of y. So another way domain and range are referred to are as function inputs and outputs. Inequality notation works well when expressing range and domain, as does interval notation. Remember that x>=0 is a fact whenever you're asked to find the domain of x when x is inside a square root, fourth root, etc.- any even radical. Neverwinter Nights: She now enters the alchemist's study, where obtains her first Potion of Cat's Grace. The function of this potion is to increase her dexterity so that her skills are more like those of a RANGER.
NWN: 0015 Algebra Mind Map 1: 8.0 (South corridor (minor)): Neverwinter Nights: After turning right out of the Antechamber she enters the short south corridor which, without SLOPE or INTERCEPTion, leads in a STRAIGHT LINE to the alchemist's study. (Here we learn the slope intercept equation of a line: y = mx + b, where m is the slope and b is the y intercept. There are also a fair number of other topics we learn here, some more obviously related to straight line graphing then others).
NWN: 0014 Algebra Mind Map 1: 7.2 (Antechamber): The slope of a graphed line indicates how steeply it is inclined and is also referred to as the gradient - the larger the value of the slope the steeper it is - a positive slope goes up from left to right and a negative slope goes down from left to right. It's very tempting to describe the slope or gradient of a line in terms of its angle relative to horizontal, although it can of course be thought of in those terms, it's just that slope is not measured in degrees or radians but rather as the ratio of the difference of the second coordinates of two points to the first. One of the equations used for finding the slope this is called the "point slope form" and is written: m = (y2-y1) / (x2-x1) (the digits 1 and 2 are subscripted), where m is the slope. It doesn't matter which two points you choose, just don't separate the coordinates for each point, i.e. if point (1, 2) is chosen then designate this either (x1,y1) or (x2 ,y2). The slope, m, can be an integer or fraction but as the equations for calculating it rely on point coordinates, integer slopes should be thought of as fractions with a denominator of one. From the point slope form equation we can see that the numerator is the change of y coordinates, so is a measure of the vertical up and down component, and the denominator is the change of x coordinates, so is a measure of the horizontal left and right component of the slope. Neverwinter Nights: Through the large archway leading from the auditorium she enters the antechamber, where she finds no adversaries. She did, however, find her first bottle of Cure Serious Wounds. This is a seriously powerful potion and she makes a POINT to remember to use this if on the slippery SLOPE to death or in poor FORM during a losing battle.
NWN: 0013 Algebra Mind Map 1: 7.1 (Auditorium): The point where a graphed line crosses the x axis is called the x intercept and the y, the y intercept. These two points can provide a speedy and accurate way to draw a straight line graph without having to enter other ordered pairs on an x y chart. To find the x and y intercepts set one variable to zero, then the other. E.g. 5x + 7y = 35, when x = 0 y = 5 (y intercept); when y = 0, x =7 (x intercept). Use a ruler to join those two points and the graph is done. Neverwinter Nights: On entering the auditorium from the west passageway she does battle with numerous adversaries, including the mysterious mage whom she successfully dispatches and obtains from his corpse the Gloves of the Minstrel. With these gloves on her lute playing skills increase, as she can now INTERCEPT four strings with just TWO POINTS of contact.
NWN: 0012 Algebra Mind Map 1: 6.1 (Central Store Room): Neverwinter Nights: Here in the central store room she found some arrows from the chest, which reminded her that a bow without arrows is nothing and she must always have some arrows in her quiver. This is like the two special cases for equations, where you can eliminate the variable completely, leaving either a false or true statement. 7.0 (West passageway (north)): If a linear equation has two variables, x and y for example, the equation can be graphed on a square or rectangular grid plane, or "coordinate plane", and the x variable and y variable are described as an "ordered pair" of point coordinates, written (x,y). The first point coordinate represents where a point on a graph will be in relation to the x axis and the second point coordinate represents where the same point on the graph will be in relation to the y axis. Neverwinter Nights: As she enters the west passageway (north) leading from the central store room, she hears the sound of goblins and PLOTS her course of action accordingly.
NWN: 0011 Algebra Mind Map 1: 6.0 (North corridor): Things to note when solving absolute value equations. Firstly any equation |x| = -y is always false. Secondly because |x| indicates how far a number is from zero on either side of zero on the number line, x itself (not |x|) could be a positive or negative number of equal magnitude, so you will always have two possible outcomes to your absolute value equation, except when |x|=0. Thirdly you must isolate the absolute value expression on one side of the equation before attempting to solve it. When that's done you change |x| to x and solve each of the resulting two equations separately, one for the positive value and one for the negative. Neverwinter Nights: In the north corridor she once again defeated a goblin, and received her second Potion of Bless, which EQUATES to ABSOLUTE bliss when fighting strong enemies.
NWN: 0010 Algebra Mind Map 1: 3.2 (Geldar's study): The addition property of equality means that if we add a number to one side of an equation we must also add the same number to the other side of the equation to keep the equation true. Down the hall Zanzi enters an ADDITION to Geldar's room, namely his study. Here she finds books that teaches her how to learn how to solve linear equations with one variable where the primary operations used to isolate the variable are addition and subtraction. As these books are Geldar's PROPERTY, they must all EQUALLY be handled with care. 4.1 (Dining room): The multiplication property of equality means that if we multiply one side of an equation by a number we must multiply the other side of the equation by the same number to keep the equation true. In the room off the south corridor, the dining room, she found a bottle containing Potion of Bless which will EQUALLY augment her attack and damage and Will saving throw against fear - a MULTIPLICATION PROPERTY of the potion. (We learn here about the multiplication property of equality and how to use it to isolate the variable in linear equations with one variable). 4.2 (Store): In one of the crates of the store room off the dining room she obtained a couple of bottles of Potion of Cure Light Wounds and our first bottle of Bull's Strength, which is a very powerful combination of potions, capable of ADDING health AND MULTIPLYING strength in EQUAL measure when about to begin combat. (We rest and try a few more linear equations where we used both the addition property and multiplication property of equality to isolate the variable). 4.3 (Lecture room): When solving equations that have parentheses, first multiply out the terms within the bracketed expression by using the distributive property for multiplication and then combine like terms and use the addition and multiplication properties of equality to isolate the variable. For example, solve 5(2x-4)+3=4x-5. Step one, after multiplying out the bracketed terms we have 10x-20 +3=4x-5. Step two, combine like terms: 10x-17=4x-5. Step three, use the addition and multiplication properties of equality to isolate the variable: 10x-4x=-5 +17 leads to 6x=12 leads to x=2. Check the answer by plugging it into the original expression: 5(2.2 -4)+3=(4.2) -5 leads to 5(0)+3=8 -5 leads to 3=3 which is a true statement, so x=2 is correct. In the lecture room at the bottom of the south corridor she sees a lectern, which is used to DISTRIBUTE information to an EMBRACED audience. 4.4 (Reading room): When solving equations that have fractional terms we generally want to eliminate fractions by multiplying by the LCD for those fractions, but it's not always necessary to do this and for simple equations it's often just as easy to solve these by working with the fractions. For example: Solve 2x+1/2=3/4. Method one, multiply the equation by the LCD: 4(2x+1/2=3/4) leads to 8x+2=3 leads to x=(3 -2)/8 = 1/8. Method two, keep the fractions: 2x+1/2 = 3/4 leads to 2x=3/4 -2/4 leads to 2x=1/4 leads to x=1/4.2/4= 2/16=1/8. In the reading room beyond the lecture room she found her first Greenstone, and looking at all these books in this room on various aspects of mathematics she realises that although she's learning fast, she's researched only a tiny FRACTION of this subject - she's still very GREEN to mathematics. 5.0 (West corridor): Not much to say about working with equations with one variable that have decimal terms, other than to treat them as you would fractional terms - you can either work with the decimals or multiply them out. Having defeated a goblin in the west corridor, she now obtains her first copper ring. So far she's DECIMATED the goblins roaming the academy, and for little more than copper rings, bows and arrows and the odd gold coin or two - such a waste of life!
NWN: 0009 Algebra Mind Map 1: 3.1 (Geldar): Neverwinter Nights: Via the south hallway Zanzi enters a room where Geldar teaches her about the Neverwinter Nights system of leveling up. He also ASSIGNS A VALUE to her the next level of studies, which is how to find the value of (evaluate) an expression when assigning values to variables as a prelude to learning about equations. At this time Zanzi learns three new spells when she levels up to a level two bard, namely Flare (a cantrip of Evocation), Sleep (a level one spell of Enchantment) and Charm Person (a level one spell of Enchantment). Elves are generally biased towards anything that involves charm or illusion, and Zanzi is no exception.
NWN: 008 Algebra Mind Map 1: 2.1 (East hallway): When simplifying radicals we need to find the largest perfect square factor of the number under the radical and then take that perfect square out from under the radical. E.g. 20^(1/2) = (4x5)^(1/2) = 2.5^(1/2). Neverwinter NIghts: At the ROOT of the SQUARE hallway leading east from the library Zanzi defeats a goblin that was carrying a SQUARE shield known as the Shield of the Watch, which boosts her AC when combating humans. And indeed the root of the square hallway reminds us of our next study, that of simplifying radicals.
NWN: 0007 Algebra Mind Map 1: 1.0 (Main Library Entrance): The commutative property of addition means that a+b = b+a and for multiplication that ab = ba. This property does not hold for subtraction (a-b does not equal b-a) or division (a/b does not equal b/a). Neverwinter Nights: When Zanzi exits the storage room she sees Pavel at the entrance to the main academy library. Pavel tells her that his brother, Bim, has been killed at which point she realises that she cannot go back on her quest to confront the evil facing Neverwinter. Her situation is non-commutative (like subtraction and division), in that she cannot go back because the situation in her mind has changed (we get a different result), unlike addition and multiplication, which are commutative (we get the same result). 1.1 (First Bookshelf): The associative properties of addition and multiplication mean that if we have expression involving three or more numbers or variables all grouped by the same operation, then changing how those numbers or variables are grouped will not change the result of the calculation. Neverwinter Nights: After speaking with Pavel she looks at the first bookshelf which is brimming with books on basic algebra, but because the information contains information she already knows, it doesn't really matter which order she reads the books, she will gain no new knowledge. This is like the associative properties for addition and multiplication: a+(b+c)=(a+b)+c=(a+c)+b and a(b.c)=(a.b)c=(a.c)b. 1.2 (Second Bookshelf): Neverwinter Nights: Looking at this next bookshelf she notices how the books are distributed differently on the shelves, and this reminds her of the distributive property for multiplication. The fact that a(b+c)=a.b+a.c. So what the distributive property does is change a calculation that is predominately multiplication-based to a calculation that is predominately addition-based. 1.3 (Third Bookshelf): Neverwinter Nights: On the third bookshelf she finds books on three different subjects: absolute values, opposite values and reciprocal values of numbers. Absolute values are scalar numbers that indicate how far those numbers are away from zero on the number line. Opposite values are numbers that have opposite sign but equal magnitude. A reciprocal value of x is that specific rational expression of x which is equal to 1/x. NOTE: Be very careful with the term "inverse", because sometimes it is correct to think of an inverse in relation to reciprocal values (as in the multiplicative inverse of 2 is 1/2 and 1/2 is the reciprocal of 2), but sometimes it is incorrect to think of an inverse in relation to a reciprocal, specifically the reciprocal of a function is not the same as the inverse of a function (i.e. 1/f(x) or f(x)^-1 is not the same as f^-1(x), where 1/f(x) inverses both the doman and range values of f(x) and f^-1(x) swaps round the ordered coordinate pairs of f(x) so that (x,y) of f(x) becomes (y,x) of f^-1(x), the power -1 of the inverse of a function is cosmetic and is for notation purposes only). 1.4 (Fourth Bookshelf): Neverwinter Nights: The corpse of a dead man distracts her from looking at the fourth bookshelf, which is possibly just as well because the information in the books on these shelves is very basic. Here we note that order of operations generally means calculate from left to right, but if the calculation involves addition or multiplication it may be possible to rearrange the calculation, because of the associative property of these operations.
NWN: 0006 Basic Maths Mind Map 3: 1.1 (Mysterious Mage): Percentage problems that ask such questions as "What number is 35% of 65?" or "19 is what percent of 44?", can be solved by remembering that is/of = %/100. These questions will undoubtedly also be in the form of word problems, where you will have to identify the "is", "of" and percent numbers. Cross multiplying to isolate the unknown value often results in fairly large numbers, so pause to see if you can factor the numbers to lower (if not lowest) terms before reaching for a calculator. As always, you should check your answer makes sense (e.g. you should never get an "is" number that's larger than an "of" or a % answer that's larger than 100). Zanzi exits the graduation chamber and is immediately confronted by a mysterious mage who summons two goblins and then teleports himself out of danger like the cowardly scoundrel he is. She duly dispatches the goblins with some ease. Taking out two of these enemies isn't bad - a 66.7% kill percentage. 1.2 (Amulet of Will): Being able to solving percent change basically relies on remembering that % change = (new value - original value) / original value. Now that she has killed the two summoned goblins, she roams the academy training halls collecting loot and various bits and bobs she can find, killing numerous goblins along the way. Eventually she uses the key given to her by Lady Aribeth to exit the training halls and enter the main academy complex. Here, in the storage room, she once again sees the mysterious mage, who true to form summons two more goblins and then transports himself out of danger. She kills the two goblins and then obtains an Amulet of Will from a chest, which will add + 1 to her will saving throws - a small percentage change, but every little helps.
NWN: 0005 Basic Maths Mind Map 2: 1.1 (Lady Aribeth): A proportion is two equal ratios (or fractions) such that a/b = c/d and each number or variable in the equation has a term designation as follows: a is the first term, b is the second term, c is the third term and d is the fourth term. We also can designate a proportion in terms of means and extremes, such that b and c are the means and a and d are the extremes. Neverwinter Nights: Having crept out of the divine magic school room, Zanzi is now allowed entry into the graduation chamber where she meets Lady Aribeth for the first time, and it is instantly apparent to Zanzi that Aribeth is also an elf who is even taller than herself (Zanzi is unusually tall for an elf at 5.5 feet), but Aribeth is a good two or three inches taller. When speaking to Aribeth it soon becomes very clear to Zanzi how extreme the crisis is facing Neverwinter and the means by which Lady Aribeth intends to use her to help find defeat the enemy and find a cure for the Wailing Death: The academy has been invaded by interlopers and Zanzi must rush to save the Waterdhavian creatures that will be used to cure the Wailing Death. To this end Aribeth gives her a key to the locked door in the academy training halls. 2.1.2 (Fonlim): The fundamental property of proportions states that in any proportion the product of the means is equal to the product of the extremes and vice versa. E.g. if 3/4 = 6/8 then 3.8 = 4.6 and 4.6 = 3.8, which is true as the products of the means and extremes equate to 24 = 24. So the fundamental property of proportions allows us to change an equation of two equal fractions to an equation without fractions. Neverwinter Nights: In some panic after the attack on Aribeth was overcome, she speaks to one of the survivors, Fonlim, and he directs her to urgently defend the academy as its survival is FUNDAMENTAL to defeating the enemy.
NWN: 0004 Basic Maths Mind Map 1: 5.3.1 (Ketta): When multiplying mixed numbers you must first convert the mixed numbers into improper fractions and then multiply those improper fractions together (dividing out any common factors first to make the numbers of the resulting product as small as possible). E.g. 2 3/4 x 3 1/5 = 11/4 x 16/5 = 11/1 x 4/5 = 44/5 = 8 4/5. It's not always necessary to convert the resulting product into a mixed number, and in fact in algebra it's often more convenient to keep the improper fraction showing. After visiting the arcane magic study room, Zanzi makes her way to the rogue study room, where she meets the chief instructor Ketta. As Zanzi is a bard who plays her lute with considerable dexterity she can actually make use of some of the thieving skills taught here - her MIX of skills is MULTIPLYING. 5.3.2 (Bruno): The procedure for dividing mixed numbers requires that we first convert the mixed numbers into improper fractions and then calculate the product of those improper fractions by converting the divisor fraction into its reciprocal. E.g. (1 3/5) / (2 4/5) = (8/5) / (14/5) = (8/5)(5/14) = 4/7. The rogue school is DIVIDED into a highly diverse MIX of individuals, from lowly simpletons to high class intellectuals. Unfortunately, Bruno is quite clearly of the former category. (See Olgerd for a more information on how to divide fractions). 6.1.1 (Elynwyd): A ratio is equivalent to a fraction, such that fraction a/b can be written as a ratio a:b and we describe this as "the ratio of a to b". E.g. The ratio of 16 to 48 is equivalent to the fraction 16/48 (which we can describe as "16 over 48") and in lowest terms is equivalent to 1/3 ("one over three" or "one third"). So the ratio of 16 to 48 is the same as the ratio of 1 to 3. Perhaps Zanzi's visit to the rogue school room has brought out the devil in her, because from there she enters the academy school of divine magic, which is a big no no for bards. When she meets the head tutor of this school, Elynwyd, he is polite to Zanzi but quite firm in his assertion that she can learn NOTHING NEW from him. And in fact this is true, as learning all about fractions is all Zanzi needs to progress in her study of ratios.
NWN: 0003 Basic Maths Mind Map 1: 5.1 (Hewwet): A mixed number is an integer and fraction combined (e.g. 2¾) and is another way or writing an improper fraction (e.g. 11/4). In this example, 2¾ is equal to 2/1+3/4 and this calculates as 8/4 + 3/4 = 11/4. The quick way to convert a mixed number to an improper fraction is to multiply the integer by the denominator of the fraction and then add the numerator. Hewwet is the range weapon instructor at the academy and he gives us practice in using crossbows, longbows, etc. He also introduces Zanzi to mixed numbers and shows us how to convert mixed numbers into improper fractions. Note that knowing how to convert mixed numbers into fractions is key for when we want to multiply and divide mixed numbers. 5.2.1 (Jaroo): Adding mixed numbers. There are two methods you can use to add mixed numbers. The first method is usually the longer of the two. Here you change the mixed numbers to be added into improper fractions, convert them to equivalent fractions that each have the lowest common denominator, carry out the addition operation on those equivalent fractions and then, if you want, change the summed improper fraction back into a mixed number. E.g. 3 2/3 + 4 1/5 = 11/3 + 21/5. Then (11/3)(5/5) + (21/5)(3/3) = 55/15 + 63/15. Then (55 +63)/15 = 118/15 = 7 13/15. The second method generally takes less time because the numbers during the calculation remain small. Here you add the whole numbers and fractions of the mixed numbers separately and then add them together. E.g. 3 2/3 + 4 1/5 = 7 + [(2/3)(5/5)+(1/5)(3/3)] = 7 + (10/15 + 3/15) = 7 13/15. Another example of the second method where the summed fractions result in an improper fraction: 5 3/4 + 9 5/6 = 14 + [(3/4)(3/3) + (5/6)(2/2)] = 14 + (9/12 + 10/12) = 14 + 19/12 = 14 + 1 7/12 = 15 7/12. So if you end up with an improper fraction using this second method, you convert it to a mixed number and then add the whole number to the whole number. Jaroo is the arcane magic instructor, and because he knows Zanzi is a very talented bard he is more than happy to introduce her to this discipline. He is also happy to teach her the two methods for addition of mixed numbers. 5.2.2 (Chandra): The methods for subtracting mixed numbers are the same as those for addition of mixed numbers. E.g. 12 7/10 - 8 3/5 = 12 7/10 - 8 6/10 = 4 1/10, using the second method shown in addition of mixed numbers. Now consider 8 3/5 - 12 7/10 = 8 6/10 - 12 7/10 = -4 - 1/10 = -4 1/10. So mixed numbers can be defined in two ways depending on whether they are positive or negative numbers, e.g. 4 1/10 means add 1/10 to 4, or 4 + 1/10 and -4 1/10 means subtract 1/10 from -4, or -4 - 1/10. Chandra is the first of Jaroo's students of arcane magic. She does not like Zanzi at all, as she sees her as a competitor for her place in Jaroo's very prestigious class and therefore has a NEGATIVE reaction. 5.2.3 (Zedir): Subtracting a mixed number from an integer. Suppose we want to subtract 5 2/7 from 10. In the first method we convert 10 into 70/7 and 5 2/7 into 37/7. So 70/7 - 37/7 = 33/7 = 4 5/7. In the second method we convert 10 into 10 0/7 and 5 2/7. So 10 0/7 - 5 2/7 = 5 - 2/7 = 4 + 7/7 - 2/7 = 4 5/7. The second method is a little awkward and can be made simpler by converting 10 into 9 7/7 at the start of the calculation. So 9 7/7 - 5 2/7 = 4 5/7. Zedir is the second of Jaroo's arcane magic students. He is less hostile towards Zanzi than Chandra was, but still finds Zanzi's presence unwelcome and the conversation is AWKWARD and the atmosphere is still NEGATIVE.
NWN: 0002 3.0 (Berna): The procedure for adding and subtracting fractions is quite restricted when compared to multiplication and division of fractions, and from the get go we need to know that the only time we can add or subtract fractions is when those fractions have the same denominator. Once we have fractions with the same denominator we can then go ahead and calculate the sum of those fractions, and we do so by adding the numerators only, not the denominators (e.g. 2/7 + 3/7 = 5/7). The reason why we don't add the denominators can be seen when we consider that 2 divided by 7 is the same as 2 multiplied by 1/7; likewise 3 divided by 7 is the same as 3 multiplied by 1/7. So we now have 2(1/7)+3(1/7)=5(1/7) which is 5/7. So when adding fractions we add x of one thing to y of the same thing to get x+y of the same thing. Likewise with subtraction to get the difference of the same thing. Neverwinter Nights: After Zanzi has finished purchasing items from Olgerd she exits the store and enters the academy library, where she meets Berna. Berna is a knowledgeable lady who can teach us how to make use of our journal in Neverwinter Nights. So she ADDS to our knowledge and SUBTRACTS from our ignorance. 4.1 (Herban): The next step in adding and subtracting fractions is knowing how to manipulate fractions with different denominators and turn them into fractions that have the same denominator so you can then calculate their sum or difference, and the key to doing this is multiplication by the number one in the form of a fraction. Anytime we want to obtain a certain denominator for a fraction we can multiply the numerator and denominator by 2/2 or 3/3 or 4/4, etc. to get an equivalent fraction with whatever quotient value is convenient for the calculation. E.g. 1/3 + 5/6 can't be added as is, but changing 1/3 to 2/6 by multiplying 1/3 by 2/2 allows us to perform the calculation 2/6 + 5/6 = 7/6. Neverwinter Nights: From the library she enters a large room where combat techniques are practiced. The first person she meets here is the main combat instructor, Herban. He explains that fractions with different denominators are like apples and pears and are too dissimilar to allow us to add or subtract them. 4.2 (Dendy): Now that we know that we can use the number one in the form of a fraction to produce an equivalent fraction that will allow us to calculate the sum or difference of two or more fractions, the next question is how do we decide which version of our number one fraction to choose in order to produce any desired equivalent fraction? Our choice is guided by the fact that we want to obtain the smallest number for each equivalent fraction's denominator that is divisible by all of the denominators. This is called the lowest (or least) common denominator (LCD) and you obtain this by factoring denominators into the product of their prime factors and then building the LCD by using just those prime factors that give the smallest number divisible by those denominators. E.g. 5/12 + 7/18, the LCD of 12 and 18 is built from (2.2.3) for 12 and (2.3.3) for 18. The 2 and one of the 3's for 18 can be disregarded because they're included in the prime factors of 12, so the LCD is 2.2.3.3 = 36. The calculation now becomes (5/12)(3/3)+(7/18)(2/2)=15/36 + 14/36 = 29/36. Neverwinter Nights: Herban instructed Zanzi to go and speak to Dendy who will teach her how to parry in combat. Dendy also taught her about the LCD, which she remembers as "Lowest Common Dendy".
NWN: 0001 Basic Maths Mind Map 1: 1.1 (Zanzi's bedroom): In the beginning we learn what it is that defines a fraction. We learn that a fraction is any real number that has a numerator (top number) and denominator (bottom number) separated by a horizontal bar or diagonal slash, e.g. ½, ¼, ¾. An improper fraction has a greater value for the numerator than the denominator, e.g. 5/3. The first thing Zanzi looks at after waking up in her bedroom in the academy is the chest by the wall. Inside this chest are items that define why she is at the academy, but it also can be used to define fractions: the openable lid is like the flexible upper part of a fraction (the numerator, which can be any number), and the fixed base is like the bottom part of the fraction (the denominator, which cannot be zero and sometimes has other restrictions). The join line between the two is like the fraction bar separating numerator from denominator. 1.2 (Bedroom two): Equivalent fractions are fractions of the same quotient value but with numerators and denominators that vary in the same proportion. The most important number in obtaining equivalent fractions is the number one in the form of a fraction (e.g. 5/5) because one in this form can be used to change one fraction into another, equivalent fraction. From her bedroom she walks into the adjacent second bedroom. Here she notices the desk by the wall. The two drawers of the desk are identical in size and function - they are equivalent. 1.3 (Bedroom three): Reduce fractions to lowest terms by obtaining the factors of the numerators and denominators and dividing out any common factors (use of the commutative property of multiplied numbers to rearrange factors often required). There are two properties of the number one which make it very important in this regard: the fact that any numerator that has a denominator of equal value and sign will always produce a quotient of one (e.g. 5/5 = 1, 6/6 = 1, etc.) and the fact that any number multiplied an infinite number of times by the number one does not in any way change the value of that number (e.g. 3x1=3, 3x1x1=3, etc.). She walks from the second bedroom into the third adjacent bedroom. Here she sees the armoire by the wall, this is furniture designed to REDUCE clutter. 1.0 (Senior barracks): Multiplying fractions means multiplying all numerators and all denominators (very simple and intuitive). Algebraically, (a/b)(c/d)=(ac)/(bd). Again the number one is important here because any numerator with a denominator of positive one results in a quotient equal in sign and value to the numerator (e.g. 8/1 = 8, 13/1 = 13, etc.). This means that any whole number can be represented as a fraction with a denominator of one, which often helps us when multiplying fractions. Note though that because we usually want fractions expressed in lowest terms, we often reduce the numerator numbers and denominator numbers by dividing out their common factors first before multiplying. After visiting the third bedroom she walks into the main open area of the senior barracks, where she sees two academy guards, Parvel and Bim. They are brothers (2 x brothers). 2.0 (Olgerd): Division of fractions is achieved by multiplication of the dividend fraction by the reciprocal of the divisor fraction, (e.g. 1/3 (dividend fraction) divided by 1/4 (divisor fraction) is the same as 1/3 multiplied by 4/1). So the quotient of 1/3 divided by 1/4 is the same value as the product of 1/3 multiplied by 4/1, which is 4/3. Algebraically, (a/b)/(c/d)=(a/b)(d/c). After speaking with Pavel and Bim, she exits the senior barracks and enters the academy store, where she meets the storekeeper, Olgerd. He can sell us weapons to DIVIDE and conquer. Note that knowledge of division of fractions is key to understanding ratios and knowing how to divide mixed numbers.
INTRODUCTION: In the flickering candlelight a young elf lays her head on her pillow and awaits fatigue to carry her slowly into sleep. Through her half open almond-shaped green eyes she sees outlined in soft shadow the lute, once so beautifully played by her mother, resting by itself in the corner of her bedroom at the Neverwinter Academy. It is her mother’s naturally gifted playing of this magically imbued instrument that has driven Zanzi’s obsession to want nothing more than to become a bard, and throughout her childhood her education in many disciplines had consequently suffered. However she has in recent years discovered that she has a strong propensity for mastering the skills involved in casting arcane magic. And it is this coupled with a wanderlust characteristic to many of her race that has motivated her to travel north to Neverwinter to help the citizens of this city as they succumb in large numbers to a virulent plague, the Wailing Death. She knew that persuading her parents to let her travel to Neverwinter in such circumstances would be an impossible task, so she didn’t even try. Instead she explained her departure in a note of few words and with a cruelty of purpose left home for the last time, taking her mother’s lute with her. As she is about to drift into sleep she hears her mind asking why she has sacrificed so much to bring herself to this city of such tragedy and suffering, to which she whispers, “Because ...” (Beatles – “Because”).
I need help on creating a conversation dialog scenario for HCA/220, here is the detail. Scenario 3: You are a medical records technician and have just received the STAT reading report from a radiologist. After reading the medical report, you take it to the physician for a signature. Discuss with the physician the results of the report. Your classmates can continue the dialogue by suggesting additional testing to be ordered or any additional referrals you might need to make to a specialist.
Hi all. My Brightstorm maps are derived from several mind maps that are linked sequentially. This is a very big project that’s nowhere near complete, so a lot of the Brightstorm maps may appear weak on content or puzzling until I’ve fleshed them out a bit and made them more relevant to my Brightstorm maps. The idea behind these maps is to use computer games to make learning as fun as possible. Besides fun, the primary reason for doing this is mnemonic – you remember situations, items, places visited, etc. in the game and tie them to the thing you are learning about (flashcards are an excellent way to tie the two together). I’m working on maths at the moment and my game of choice for this subject is Neverwinter Nights. This is a huge game and can be used to follow the main protagonist, “Zanzi”, from basic maths at the beginning of the game to advanced maths topics. In interval notation: [0, ?). Hope you have fun with this. Peter.
