Algebra 1 Articles - WyzAnt Tutor Blogshttp://www.wyzant.com/resources/blogs/algebra_1This is an aggregate of all of the Algebra 1 articles in WyzAnt.com's Tutors' Blogs. WyzAnt.com is your source for tutors and students.Sat, 01 Nov 2014 01:53:16 -0500http://www.wyzant.com/images/WyzAnt_white200.gifAlgebra 1 Articles - WyzAnt Tutor Blogshttp://www.wyzant.com/resources/blogs/algebra_1http://www.wyzant.com/resources/blogs/algebra_1293709http://www.wyzant.com/resources/blogs/293709/excellent_middle_school_encode_decode_problem_rigorousElias H.http://www.wyzant.com/resources/users/view/79074590Excellent Middle School Encode-Decode Problem (Rigorous)<div>I am taking from The Official Hunter College High School Test: problem 76 on page 20. We read the following.<br /><br />In the expression below, each letter represents a one digit number. Where the same letter appears, it represents the same number in each case. Each distinct letter represents a<br />different number. In order to make the equation true, what number must replace C?<br /><br /> AAA<br /> AAB<br />+ ABC<br /> 2012<br /><br />A great start is to decode each AAA, AAB, and ABC. It helps to look at this problem wholly; particularly we look at the leading sum on the left wall (of the same types). We glean that either: (1) A + A + A = 20, (2) A + A + A + 1 = 20 or (3) A + A + A + 2 = 20: its very important to remember that given three numbers each less than ten, the sum of them which is great, is at most 2 in the tens place. This means that each row can only donate a 1 or 2 to the next. We can conclude that our line is line (3). That, A + A + A + 2 = 20 is the correct choice of the three since A may only equal an integer and…that 20/3 and 19/3 are not integers. <br /><br />We have shown that A = 6.<br /><br />If A = 6, then A+A+B, as a sum is almost solved for. The variable B is either a 3, 2, 1, or 0 depending on the sum on the right wall. See that A+B+C is bounded (between) 10 and 27. Recall our reminder. Thus we have two equations, where either A + A + B + 1 = 21 or A + A + B + 2 = 21. Substituting A = 6 we get B = 7 or B = 8. We now simply fill in our data into two scenarios and see which is correct. We simultaneously solve for C using the ones place in the bottom right of the equation.<br /><br />Scenario 1:<br /> 666<br /> 668<br />+688<br />2012<br /><br />Scenario 2:<br /> 666<br /> 667<br />+679<br />2012<br /><br />Only scenario 2 is correct: we conclude that A=6, B=7, and C=9. The answer is 9.<br /><br />In conclusion, we address the following question: did we encode or did we decode? You can actually say that we did either. We encoded into a numerical alphabet {0,….,10}. We decoded out of a three letter alphabet {A,B,C}.<br /><br />Thanks for your attention, you guys.</div>Sun, 05 Oct 2014 15:02:21 -05002014-10-05T15:02:21-05:00293128http://www.wyzant.com/resources/blogs/293128/the_transition_points_of_grade_school_3rd_4th_6th_9th_gradesGary B.http://www.wyzant.com/resources/users/view/79646890The Transition Points of Grade School (3rd, 4th, 6th, & 9th grades)<div>There are several points in grade school that involve a critical shift in the thinking that is required in the school work. Parent's should be aware of these points as they navigate through the abyss of raising a school-aged child and supporting the child as he/she moves forward through the grades.</div>
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<div><strong>3rd Grade -</strong> The third grader is transitioning from whole number thinking into understanding the concepts of parts. They are exposed to fractions, decimals and percentages. This is a major paradigm shift. Students are also exposed to long division at this point. Supporting children in this phase requires an emphasis on helping the child conceptualize whole things being split into parts. In addition to homework support, tutoring, and supplementary work, parents should introduce cooking chores to children at this time, and make them follow a recipe that has precise measurements. Reading comprehension and writing is also an issue here. 3rd grade is the last year where children are "learning to read," (I'll explain this as I discuss 4th grade), so there is a big push for the 3rd grader to read a passage and recall facts from the passage, and understanding the main idea. Students will spend a lot of time writing, and they will learn to write in cursive (although many schools are moving away from this).</div>
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<div><strong>4th Grade -</strong> In 4th grade, the student transitions from "learning to read," to "reading to learn." This will be apparent not only in their reading and language arts classes, but also in other classes. Science and social studies classes will have more to read, and mathematics word problems will require more reading comprehension abilities. In mathematics, students will begin performing operations on "the parts" (fractions, decimals, percentages).</div>
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<div><strong>6th Grade -</strong> The sixth grade is the transitioning from the real time, in-the-moment mathematical thinking of elementary school arithmetic into the "bigger-picture" that is algebra. Integers are introduced at this point. Just as children had to re-adjust for understanding parts of a whole in 3rd grade, the 6th grader will have to adjust to understanding "less than nothing." Essay writing at this point will become more complex and the students will be expected to show deeper thought than exhibited in elementary school. And for the first time, science and social studies will have with their own teachers in different classrooms (although some elementary schools do this beginning in 4th and 5th grade).</div>
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<div><strong>9th Grade -</strong> The high school freshman year can be an awkward one. Everything is more challenging. Classic novels will be required reading on a monthly or semi-monthly basis. Science labs start to get serious at this point. And Algebra presents a whole different ballgame, where the teenager is exposed to more abstract, bigger-picture mathematical concepts.</div>
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<div>All K-12 students could use assistance during the school year, as well as the summer. But these stages are very critical because of the changes that accompany them.</div>Thu, 02 Oct 2014 13:29:33 -05002014-10-02T13:29:33-05:00289553http://www.wyzant.com/resources/blogs/289553/mathematical_journeys_inverse_operations_or_the_answer_is_always_3Ellen S.http://www.wyzant.com/resources/users/view/75479140Mathematical Journeys: Inverse Operations, or "The Answer is Always 3"<div> </div>
<div><em>Four years ago, I came up with this math trick. Take a look at it, and at the end I'll show you why it works!</em><br /><br />~<br /><br />Let's play a game. I’m going to let you make up a math problem, and I will be able to tell you the answer from here. I can’t see what you’re doing, I’m not even in the same room as you, but I will still be able to tell you the correct answer.<br /><br />Trust me. I’m a professional. Ready?<br /><br />Okay. First, pick a number. It can be any number you wish, large or small. Now add 5 to that number. Got it? Okay, now double your new number (multiply by 2). Alright, now subtract 4 from the double.<br /><br />Next, divide your new number by 2. Now, finally, subtract your original number from this new quotient. Got it? Okay. Here comes the cool part. Ready?<br /><br />The answer is 3. Nifty, huh? What’s that? How’d I do it? Oh, magic.<br /><br />Okay, okay, it’s not magic. The answer will always be 3, no matter what number you pick. Let’s illustrate this by writing it out as an algebraic expression.<br /><br />Pick a number, any number. Since your number could be anything and is therefore a variable, we’ll call it b.<br /><br />Add 5.<br /><br />b + 5<br /><br />Double that.<br /><br />2(b + 5)<br /><br />Subtract 4.<br /><br />2(b + 5) – 4<br /><br />Divide by 2.<br /><br />[2(b + 5) – 4] / 2<br /><br />Now subtract your original number.<br /><br />([2(b + 5) – 4] / 2) — b<br /><br />Okay, so let’s simplify this expression and see what we get.<br /><br />([2(b + 5) – 4] / 2) — b<br /><br />Let’s get that fraction out of there. Divide each term in the numerator by 2.<br /><br />(b + 5) – 2 – b<br /><br />That’s better. Now simplify that.<br /><br />b – b + 5 – 2<br /><br />5 – 2<br /><br />3<br /><br />See? It doesn’t matter what number you pick, because the variable cancels itself out at the end. The answer is always 3. Now, go forth and amaze your friends!<br /><br />~<br /><br /><em>This game is a perfect example of the concept of inverse operations. Inverse operations are operations that cancel each other out; what I sometimes refer to as “undoing” each other. Addition undoes subtraction and vice versa, multiplication undoes division. Early in the problem you double your mystery number, and then later on you divide it by two. Those two actions cancel each other out – one makes the number larger and the other shrinks it back down. </em><br /><br /><em>In an algebraic equation, you can effectively move a term from one side of the equals sign to the other by performing the inverse operation to both sides. Y = x + 5 becomes y – 5 = x, which can tell you the value of x instead of y. Algebra, at its heart, is the process of using these inverse operations to rewrite an equation so that it tells you the piece of information you want to know.</em></div>Tue, 16 Sep 2014 11:35:25 -05002014-09-16T11:35:25-05:00276519http://www.wyzant.com/resources/blogs/276519/the_importance_of_s_t_e_m_science_technology_engineering_mathJacob C.http://www.wyzant.com/resources/users/view/85224961The Importance of S.T.E.M. (Science, Technology, Engineering, Math)<div>Today, the future depends on you as much as it does on me. The future also depends on educating the masses in Science, Technology, Engineering, and Math, otherwise known as STEM. As a new tutor to WyzAnt, I hope to instill the importance of these subjects in student's lives, as well as, the lives around them. </div>
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<div>Besides the fact that, "the average U.S. salary is $43,460, compared with the average STEM salary of $77,880," (Careerbuilder) these subjects are interesting and applicable to topics well beyond the classroom. Success first starts with you; I am only there to help you succeed along the way. STEM are difficult subjects. Yet when you seek out help from a tutor, like myself, you have what it takes to master them. </div>
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<div>Please enlighten me on students looking to achieve and succeed rather than live in the past and think I can't as opposed to I can. We can take the trip to the future together, one question at a time</div>Mon, 30 Jun 2014 16:15:12 -05002014-06-30T16:15:12-05:00270826http://www.wyzant.com/resources/blogs/270826/mathematical_journeys_what_does_the_function_look_likeEllen S.http://www.wyzant.com/resources/users/view/75479140Mathematical Journeys: What Does the Function Look Like?<div><em>This week's Math Journey builds on the material in <a href="http://www.wyzant.com/resources/blogs/234308/mathematical_journeys_the_function_machine">The Function Machine</a>. If you have not yet read that journey, I suggest you do so now.</em><br /><br />In <strong><em>The Function Machine</em></strong> we discussed why graphing a function is possible at all on a conceptual level – essentially, since every x value of a function has a corresponding y value, we can plot those corresponding values as an ordered pair on a coordinate plane. Plot enough pairs and a pattern begins to emerge; we join the points into a continuous line as an indication that there are actually an infinite number of pairs when you account for all real numbers as possible x values.<br /><br />But plotting point after point is a tedious and time-consuming process. Wouldn't it be great if there was a quick way to tell what the graph was going to look like, and to be able to sketch it after plotting just a few carefully-chosen points?<br /><br />Well, there is! Mathematicians look for an assortment of clues that help to determine the shape of a function's graph from the equation itself – and it's those clues that we'll be talking about today. They come in four basic flavors: the power, the sign, the co-efficient, and the constant.<br /><br />The Power<br /><br />Let's start with our old standby from the previous journey: y = x + 4. When we talk about “power” in this context, we're referring specifically to the highest exponent on an x value. The highest power in this problem is one; there are no exponents so the x is simply raised to the first power. This means that for every value of y, there is exactly one corresponding value of x. If x is 1, y is 5. If x is 2, y is 6, and so on. For every given increase in x, there is a proportional increase in y, in this case it's 1 to 1. And that means that this graph is a straight line. Easy enough, right?<br /><br />Well, let's throw a bit of a wrench into the works here, shall we? Your new function is y = x<sup>2</sup>. Now, if I turned the machine around backwards and told you that y was 4, what would you give me for x? You might give me 2, right? 2 squared is 4. But hang on, there's more than one thing you can square to get 4.<br /><br />Not seeing it?<br /><br />How about negative 2? When you square a negative number it goes positive, right? So your x value could just as easily have been – 2 as positive 2. And the same thing would have been true for any value of y, right – the corresponding x value could be either the square root of y or the negative square root of y. So in this case, there is more than one corresponding x value for any given value of y – in fact there's exactly 2 corresponding x values for each y (with the exception of 0, of course). That means that this graph is NOT a straight line.<br /><br />Turns out, it's actually a parabola. All functions with x<sup>2</sup> as their highest power (known as quadratic functions) graph out as parabolas. The specific parameters of each parabola are determined by the other categories of clues, but the power tells us that this graph will be some kind of parabola. In the same way, the powers of higher-power functions also tell us the type of shape they will graph; third power functions (ones with a cube as their highest power) will form hyperbolas, and so on. This holds true with functions that include radicals as well; the type of power indicates the rough shape of the graph.</div>
<div><br />The Sign<br /><br />Let's take our quadratic function of y = x<sup>2</sup>. When you plot some points it becomes clear that this is a parabola opening upwards; the larger the x values become, the exponentially larger the y value becomes. But what if I made one slight change to this equation?<br /><br />Y = - (x<sup>2</sup>)<br /><br />Now I'm asking you, essentially, to take each of those y values and invert it. If x is 2 (or negative 2), y would now be negative 4. This holds true for every value of y, so if you plot a few of those sets of points it quickly appears that you've just flipped the parabola upside down. And indeed, the sign on the highest-power x value dictates which direction the graph will be facing (at least in terms of up-and-down; the side-to-side graphs are usually dictated by higher powers in the first place or by radicals or other more complex types of functions). If we were dealing with a straight line, the negative sign would indicate that the line travels downward as it moves to the right, rather than upward. Y = -x, for example, is a line with a negative slope, which means it moves down and to the right rather than up and to the right as y = x does. If you graphed both of those line functions, they'd come out to be mirror images of each other. So the sign on the highest-power x value dictates direction.<br /><br />The Co-Efficient<br /><br />When we talk about a co-efficient in math, we're generally referring to the number that is multiplied by a variable. Take, for example, the function y = 3x<sup>2</sup>. How would this differ from our original y = x<sup>2</sup>?<br /><br />Well, let's follow the problem through. With a co-efficient, each time we get the square we'll need to multiply it by 3 before it becomes the y value. This will mean that each y value is quite a bit larger than the y value in our original problem - three times larger in fact. The curve will be quite a bit steeper, since using 2 for x will give us 12 for y instead of 4. So with a co-efficient above 1, the graph will show up steeper/skinnier/more closed. With a co-efficient that is a fraction, however, the graph will show up shallower or more open. Think about y = (1/3)x<sup>2</sup>. With 2 for x, you'd now end up with 4/3 for y; even less than with the original problem. So the co-efficient tells us how steep or sharp the progression of the curve is. Higher numbers mean sharper curves, while smaller fractions mean more gentle progressions.<br /><br />The Constant<br /><br />The constant is my favorite clue. A constant is a number that does not involve a variable. In our original y = x + 4, that +4 is the constant. That constant is the y-intercept – the value at which x is 0. If x were 0, all terms with x's in them would become zeros and all you'd have left would be the constant. So with a quick look at the constant you can figure out one of your points with no work at all. But here's the really fun part. Since it doesn't involve a variable, the constant doesn't actually change the shape of the curve itself. What it does do is move it around the plane. Take a look at y = x<sup>2</sup> versus y = x<sup>2</sup> + 4. That +4 on the end simply means that every y value you normally would have gotten is now 4 places higher on the graph. The whole curve has been lifted up four places on the graph. If it were a negative 4 – you guessed it – it would have moved down four places. <br /><br />So the natural next question is: what if you want to move it right or left on the plane? Well, that involves getting a second co-efficient into play. Let's change our equation to x<sup>2</sup> + 2x + 4. That 2x will shift the graph horizontally – but it's a little bit more complicated than you might think. The signs here are actually reversed – adding 2x moves the graph to the left, and subtracting it moves the graph to the right. Also, it's not a one-to-one ratio; in fact the ratio varies depending on the equation itself. Remember, too, that <em>the constant is still the y-intercept</em>, so if you get sideways transposition involved the center won't necessarily be cleanly at an easily-discernible value anymore; but the curve will still cross the y-axis at 4. Combining those two pieces of information, along with the power, sign, and leading co-efficient to tell you the shape of the curve, will get you well on your way to knowing what the graph looks like.<br /><br />Remember back at the beginning when I told you that using these clues would allow you to plot just a few points and sketch the graph more quickly? Well, here's how we put it all together. Let's take a new equation:<br /><br />y = 3x<sup>2</sup> + 5x – 2 <br /><br />What can we tell about the graph from the clues presented here?<br /><br />First, the power. This is a quadratic function, which means we're dealing with a parabola. The leading sign is positive, so it'll open upward. The leading co-efficient is 3, which is greater than 1, so it'll be a sharper, steeper curve, 3 times steeper than the basic parabola. We're adding 5x, so the graph will be transposed to the left, and the y-intercept is at – 2. We'd still need to work out and plot a couple of points (personally, I'd factor the quadratic to find the x-intercepts and work from there - more on that next time), but now we have a better idea of what the graph would look like – and we can see all of that just from the equation alone!</div>Tue, 29 Apr 2014 08:20:23 -05002014-04-29T08:20:23-05:00270661http://www.wyzant.com/resources/blogs/270661/fun_math_sites_for_middle_school_and_high_schoolGwen R.http://www.wyzant.com/resources/users/view/82407210FUN Math Sites for Middle School and High School!<div>My recommendationa:</div>
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<div><strong>Vi Hart, website: vihart.com<br /></strong></div>
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<div><strong>Sal Khan, https://www.khanacademy.org/math/algebra</strong></div>
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<div><strong>Mamikon Mnatsakanian, www.its.caltech.edu/.../calculus.html‎</strong></div>
<div> </div>Sun, 27 Apr 2014 23:55:17 -05002014-04-27T23:55:17-05:00262225http://www.wyzant.com/resources/blogs/262225/mathematical_journeys_the_unperformed_operationEllen S.http://www.wyzant.com/resources/users/view/75479140Mathematical Journeys: The Unperformed Operation<div>Come with me on a journey of division.<br /><br />I have here a bag of M&Ms, which you and I and two of your friends want to share equally. I'm going to pour the bag out on the table and split it into four equal piles. For this example, “one bag” is our whole, and the best number to represent that whole would be the number of M&Ms in the bag. Let's say there were 32. If I split those 32 M&Ms into four equal piles and asked you how many were in one pile, you could certainly just count them. But a quicker way would be to take that 32 and divide it by the number of piles I'd made, which in this case is 4. You'd probably write that as:<br /><br /> 32 ÷ 4 = 8<br /><br />So there are 8 candies in each pile.<br /><br />Seems easy enough with a large number of M&Ms, right? But what if there were less candies – what if our “whole” was less than the entire bag? Well, for a while we'd be okay – if there were 16, for example, we'd do the same thing and come up with piles of 4 instead of piles of 8. But what if there were less candies than we needed piles – if there were less than 4 candies in the whole? What if, in fact, we had only one candy? There's still four of us, and we still need to share equally – I guess we're each getting less than one candy, right? <br /><br />If we set up our equation the same way we did above, we'll get:<br /><br /> 1 ÷ 4 = ?<br /><br />Now hold on a minute. See that dividing sign there? That looks an awful lot like a fraction, doesn't it? If you just replaced the dots with numbers? That's because it is. The dividing sign they teach you first actually came after the invention of fractions. It's a way of indicating that the first number goes on top of the fraction and the second number goes on the bottom. But once you get into higher level math classes, that dividing sign disappears. Instead, we write division as a fraction – because that's what it is!<br /><br />The reason we stop using the dividing sign is because writing division as a fraction allows you to deal with our one-M&M scenario from earlier. You simply write the division itself as a fraction, and that fraction becomes the result of the division.<br /><br />So instead of writing that equation as:<br /><br /> 1 ÷ 4 = ?<br /><br />And being confused by your lack of an answer, you'd write:<br /><br /> <span style="text-decoration: underline;">1</span><br /> 4<br /><br />And that fraction would <em>become</em> the answer. <br /><br />What's important to remember here is that a fraction is not just a number. While it IS a number – there IS exactly one point on the number line that that fraction represents – it's also an indicator of an unperformed operation. By writing that number as a fraction, you are saying “I'm supposed to divide this number by this other number, but I don't want to do that calculation just yet.” <br /><br />This might seem a bit strange, especially given that ¼ is an easy calculation to make, and that its decimal form, 0.25, is equally easy to work with. Okay, fine – I'm going to make one of your friends disappear!<br /><br />Now there are only three of us fighting over that one candy. Your new fraction would be:<br /><br /> <span style="text-decoration: underline;">1</span><br /> 3<br /><br />If you try to convert that into decimal form by dividing one by three, you'll get 0.33333333333... on into infinity. Now, I don't know about you, but I don't particularly like the idea of trying to work with a number that stretches on into infinity – my arms aren't that long! So I just won't let it out of the box – I'll keep it as a fraction as long as I possibly can, thus acknowledging the existence of another operation while refraining from performing it until I'm really ready.<br /><br />You see this concept of the unperformed operation a lot once you get into higher level math concepts, particularly in the use of named constants. Take pi, for instance. Pi is a constant, described as the result of a specific calculation involving circles. No matter what dimensions you give a circle, when you perform this calculation you end up with the same number. So clearly it's important, and it makes sense that we should be able to work with it. Only one problem – it's an incredibly unwieldy number, a non-repeating, non-terminating decimal that stretches out into infinity. Working with such a number would be downright impossible unless we are willing to approximate and chop off most of the digits. So what do we do? We give it a name, assigning it to a letter of the greek alphabet and using this letter to represent the constant in full.<br /><br />To make sure we are always working with the entire non-terminating number and not an approximation, we leave operations involving this number unperformed. We simply carry the symbol through the problem, attached to whatever other number it was supposed to be multiplied or divided by. Only at the very end do we ever actually perform the operation, and even then only if we need a numerical estimation. Much more frequently we simply express our answer “in terms of” this constant, leaving the symbol intact for the next mathematician to pick up and work with later.</div>Tue, 18 Feb 2014 18:36:42 -06002014-02-18T18:36:42-06:00260028http://www.wyzant.com/resources/blogs/260028/factoring_without_the_guess_and_check_methodShawil D.http://www.wyzant.com/resources/users/view/84848050Factoring without the "Guess and Check" method<div>Factoring can be quite difficult for those who are new to the concept. There are many ways to go about it. The guess and check way seems to be the most common, and in my mind, it is the best, especially if one wants to go further into mathematics, than Calculus 1. But for those just getting through a required algebra course, here is another way to consider, that I picked up while tutoring some time ago:</div>
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<div>If you have heard of factor by grouping, then this concept will make some sense to you. Let's use an example to demenstrate how to do this operation:</div>
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<div>Ex| x<sup>2</sup> + x - 2</div>
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<div>With this guess and check method, we would use (x + 1)(x - 2) or (x + 2)(x - 1). When we "foil" this out, we see that the second choice is the correct factorization. But, instead of just using these guesses, why not have a concrete way to do this.</div>
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<div>Let's redo the example, with another method.</div>
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<div>Ex| x<sup>2</sup> + x - 2</div>
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<div>First notice the -2, the negative shows us that the only way to this is (x - ?)(x + ?)</div>
<div>Now we look for the factors of 2. The only factors are 1,2 and 2, 1.</div>
<div>(In another case in which there are more factors, the difference or addition of these two factor must equal the middle term. That determines the correct pair of factors.</div>
<div>Now notice the middle term is positive. That means the larger of the two factors we choose must be positive also.</div>
<div>So, we can now write this as:</div>
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<div>x<sup>2</sup> + 2x - 1x - 2</div>
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<div>What we just did was replace the original "+x" with "2x - 1x". These two statements are equivalent, so this is fair game.</div>
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<div>Now, we group the left and right sides together. Note that when they are grouped, the negative stays with the 1, as shown below:</div>
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<div>(x<sup>2</sup> + 2x) + (-1x - 2)</div>
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<div>Now we factor out common terms from each:</div>
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<div>x is common on the left, and -1 is common on the right, so we factor out each of these:</div>
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<div>x(x + 2) - 1(x + 2)</div>
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<div>Now we check to see if our terms in parenthesis are equal. Since they are, we can group them into one term, and and the outer terms together. These two expressions are then multiplied:</div>
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<div>(x + -1)(x + 2)</div>
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<div>Which equals:</div>
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<div>(x - 1)(x + 2)</div>
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<div>If we foil this out, we get:</div>
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<div>x<sup>2</sup> + x - 2</div>
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<div>Which is our original problem. Therefore we have correctly factored this. And in so doing, we have learned a new way to factor that requires no guessing.</div>Mon, 10 Feb 2014 12:53:30 -06002014-02-10T12:53:30-06:00257170http://www.wyzant.com/resources/blogs/257170/all_my_students_passed_the_ny_state_regents_examsGilant P.http://www.wyzant.com/resources/users/view/77505480All my students passed the NY State Regents Exams...<div>I am happy to announce that all my students have passed the NY State Regents examinations, except one student. The subjects varied from Algebra 1, Algebra 11/Trigonometry, English, US and Global History and Living Environment. I am so proud of them. Most of these students are students who struggled quite a bit. It was a long journey but one I would do again. </div>
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<div>I am very proud of them as most of them will be graduating this year. The NY State Common Core examinations are next.</div>Wed, 05 Feb 2014 09:52:04 -06002014-02-05T09:52:04-06:00255740http://www.wyzant.com/resources/blogs/255740/high_school_geometry_for_some_why_is_this_more_challenging_than_algebra_1Bruce H.http://www.wyzant.com/resources/users/view/76725150High School Geometry: For Some, Why Is This More Challenging Than Algebra 1? <div>Several of my current Geometry students have commented on this very distinction. This has prompted me to offer a few possible reasons.</div>
<div> </div>
<div>First, Geometry requires a heavy reliance on explanations and justifications (particularly of the formal two-column proof variety) that involve stepwise, deductive reasoning. For many, this is their first exposure to this type of thought process, basically absent in Algebra 1.</div>
<div> </div>
<div>Second, a large part of Geometry involves 2-d and 3-d visualization abilities and the differences in appearance between shapes even when they are not positioned upright. Still further, for a number of students, distinguishing the characteristic properties amongst the different shapes becomes a new challenge.</div>
<div> </div>
<div>Third, in many cases Geometry entails the ability to form conjectures about observed properties of shapes, lines, line segments and angles even before the facts have been clearly established and stated. This level of abstract thinking is rarely encountered in Algebra 1.</div>
<div> </div>
<div>Implications based upon the aforementioned are significant. For those following a Science/Technology/Engineering/Math (STEM) course track and headed into Precalculus, Calculus, General Chemistry and Organic Chemistry, the ability to visualize shapes and chemical structures in 2-d and 3-d space is extremely important.</div>
<div> </div>
<div>In order to better prepare students for dealing with Geometry's rigors, many educators have believed that certain math instructional practices even going back to the elementary school level will have to be improved within the classroom. In the meantime, all students can still take steps to increase their current mental capacities and reasoning abilities by following a brain-enhancing diet (such as my "Maximum Mental Health Diet" recommendations offered when requested at any tutoring session). Among other things, this includes daily consumption of long-chain Omega-3 essential fatty acids through fish or supplements and an abundance of colorful, antioxidant-rich plant foods.</div>Tue, 04 Feb 2014 14:04:24 -06002014-02-04T14:04:24-06:00250445http://www.wyzant.com/resources/blogs/250445/easy_way_to_remember_the_quadratic_formulaKatie H.http://www.wyzant.com/resources/users/view/80680710Easy way to remember the quadratic formula!<div>As a student, I found that I remembered information a lot easier when the information was in a song. I learned the 'quadratic formula song' in one of my math classes and have not forgotten the formula since. Several of my students have also found this song helpful (and catchy!), so I though I'd share:</div>
<div> </div>
<div>The 'Quadratic Formula Song' (sung to the lyrics of 'Pop Goes the Weasel')</div>
<div>The quadratic formula is negative b</div>
<div>plus or minus the square root</div>
<div>of b squared minus four a c</div>
<div>all over 2a!</div>
<div> </div>
<div>(Warning, this will get stuck in your head!) </div>Sun, 12 Jan 2014 00:02:49 -06002014-01-12T00:02:49-06:00249113http://www.wyzant.com/resources/blogs/249113/favorite_math_resources_k_12_gedJessica S.http://www.wyzant.com/resources/users/view/84101090Favorite Math Resources, K-12 + GED<div>Here are some of my favorite Math resources. Check back again soon, this list is always growing! I also recommend school textbooks, your local library, and used bookstores. <br /><br />As a note, college-level math textbooks are often helpful for high school math students. Why is that? Isn't that a little counter-intuitive? Yes, it would appear that way! However, many college-level math textbooks are written with the idea that many college students may not have taken a math class in a year or more, so they are written with more detailed explanations. This can be particularly helpful for high school students taking Algebra, Geometry, and Trig. I have a collection of college-level math books that I purchased at a local used bookstore. The most expensive used math book I own cost $26 used. Books that focus on standardized test prep (such as the SAT, AP, or GED prep) can be helpful for all core subjects, as they summarize key ideas more succinctly than 'normal' textbooks. These are GREAT for test review and studying as a supplemental text, whethre you're studying for that standardized test or not.</div>
<div> </div>
<div><br />(K-12) <a href="http://www.kahnacademy.org">KahnAcademy.org</a> – All math subjects, with video tutorials and practice problems (plus answers!)</div>
<div><br />(K-Algebra\Geom.) <a href="http://www.mathplayground.com">MathPlayground.com</a> – Fun math tutorials and games to help reinforce what you’ve learned!</div>
<div><br />(K-Geometry) <a href="http://www.mathisfun.com">Mathisfun.com</a> – Number Value (K/Gr. 1) to Geometry</div>
<div><br />(K-Geometry) <a href="http://www.aaamath.com">Aaamath.com</a> – Explanations, practice problems (with answers), and more!</div>
<div><br />(Gr. 2+) iPhone app – Math Help – Division, multiplication, division with remainders</div>
<div><br />(Algebra) <a href="http://www.purplemath.com">Purple Math.com</a> – All things Algebra, with detailed explanations and (a few) practice problems.</div>
<div><br />(Gr. 9-12) iPhone/Droid app- GED Math Lite – All high school math topics, with quizzes and answer explanations.<br /><br /></div>
<div>(Gr 3-12) <a href="http://www.convert-me.com/en/">Convert-Me.com</a> - Not sure how to convert kilometers to meters? Gallons to quarts? Give this site a try!</div>Tue, 31 Dec 2013 14:09:48 -06002013-12-31T14:09:48-06:00246136http://www.wyzant.com/resources/blogs/246136/i_am_in_need_of_some_studentsAshley M.http://www.wyzant.com/resources/users/view/84377950I am in need of some students. <div>Hi,</div>
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<div>I would be honored in having the opportunity of working with students and parents. The education and success of students are very important to me and I would love to do what I can to help. I am a math and education major with an Associate's of Arts and Teaching Degree from Lee College and I am seeking a teaching career. I live in the Baytown area and I am not able to provide my own transportation due to the fact that I have a disability which prevents me from driving, so I can only rely on public transportation and I am limited to how far I can travel. Therefor, communication is much needed. I am available until 4:30 p.m. Monday through Friday. Anyone needing a private tutor, please contact me. I would be happy to help you at any time.</div>Sat, 23 Nov 2013 19:47:33 -06002013-11-23T19:47:33-06:00245878http://www.wyzant.com/resources/blogs/245878/christmas_presentAndrew L.http://www.wyzant.com/resources/users/view/75937410Christmas Present!!<div>Hi All!</div>
<div> </div>
<div>In the spirit of giving, starting on 11/29/2013, I will be offering a few brainteasers/ trivia questions where the first 3 people to email me the correct answer will receive a free, one hour, tutoring session in any subject that I offer tutoring for (via the online platform)! That's right free! Get your thinking hats on everyone!</div>
<div> </div>
<h1><span class="greenText"><strong>Merry <span class="orange">Christmas</span>!! </strong></span></h1>
<div><span class="greenText"><strong><a href="http://www.wyzant.com/Tutors/AndrewLane">Andrew L. Profile</a></strong></span></div>
<div> </div>Thu, 21 Nov 2013 01:35:13 -06002013-11-21T01:35:13-06:00243335http://www.wyzant.com/resources/blogs/243335/math_equations_formulas_and_vocabularyKayleigh T.http://www.wyzant.com/resources/users/view/84291860Math Equations, Formulas and Vocabulary<div>
<ul>
<li>Area, Volume and Circumference equations:</li>
<li>Area of a Square</li>
</ul>
</div>
<div>A=S<sup>2</sup></div>
<div>
<ul>
<li>Area of a Triangle</li>
</ul>
</div>
<div>A=1/2bh</div>
<div>
<ul>
<li>Area of a Rectangle</li>
</ul>
</div>
<div>A=LW</div>
<div>
<ul>
<li>Right Triangle/Pythagorean Theorem</li>
</ul>
</div>
<div>a<sup>2</sup>+b<sup>2</sup>=c<sup>2</sup></div>
<div>
<ul>
<li>Area of Parallelogram</li>
</ul>
</div>
<div>A=bh</div>
<div>
<ul>
<li>Area of a Trapezoid</li>
</ul>
</div>
<div>A=1/2h(a+b)</div>
<div>
<ul>
<li>Area of a Circle</li>
</ul>
</div>
<div>A=πr<sup>2</sup></div>
<div>
<ul>
<li>Circumference of a Circle</li>
</ul>
</div>
<div>c=πd or c=2πr</div>
<div>
<ul>
<li>Volume of a Sphere</li>
</ul>
</div>
<div>V=4/3πr<sup>3</sup></div>
<div>
<ul>
<li>Surface Area of a Sphere</li>
</ul>
</div>
<div>SA=4πr<sup>2</sup></div>
<div>
<ul>
<li>Volume of a Cube</li>
</ul>
</div>
<div>V=s<sup>3</sup></div>
<div>
<ul>
<li>Volume of a Rectangular Solid</li>
</ul>
</div>
<div>V=lwh</div>
<div>
<ul>
<li>Slope of a line Equations</li>
<li>Slope-intercept form</li>
</ul>
</div>
<div>y=mx+b</div>
<div>m is the slope</div>
<div>b is the y-intercept</div>
<div>y is a y coordinate on the graph (that coincides with the line)</div>
<div>x is an x coordinate on the graph (that coincides with the line)</div>
<div>
<ul>
<li>Horizontal line</li>
</ul>
y=b</div>
<div>
<ul>
<li>Vertical line</li>
</ul>
x=a</div>
<div>
<ul>
<li>Finding slope of line containing points (x<sub>1</sub>,y<sub>1</sub>) to (x<sub>2</sub>,y<sub>2</sub>)</li>
</ul>
m=<span style="text-decoration: underline;">y<sub>2</sub>-y<sub>1</sub></span></div>
<div> x<sub>2</sub>-x<sub>1</sub></div>
<div>
<ul>
<li>Distance from one point to another</li>
</ul>
(x<sub>1</sub>,y<sub>1</sub>) to (x<sub>2</sub>,y<sub>2</sub>)<span style="font-size: 9px;"><br /></span></div>
<div>d=√(x<sub>2</sub>-x<sub>1</sub>)<sup>2</sup>+(y<sub>2</sub>-y<sub>1</sub>)<sup>2</sup></div>
<div>
<ul>
<li>Midpoint of a Line Segment</li>
</ul>
( <sup><span style="text-decoration: underline;">x1+x2 </span> , <span style="text-decoration: underline;">y1+y2</span> </sup>)</div>
<div> <sup>2 2</sup></div>
<div>
<ul>
<li>Quadratic Formula</li>
</ul>
x=<span style="text-decoration: underline;">-b±√b<sup>2</sup>-4ac</span></div>
<div> 2a</div>
<div>
<ul>
<li>Math Vocabulary</li>
</ul>
<ol>
<li>Equilateral Triangle- Three sides of equal length and 3 angles 60° each</li>
<li>Scalene Triangle- Three unequal sides and 3 unequal angles</li>
<li>Isosceles Triangle- Two sides of equal length and two base angles are equal</li>
<li>Commutative Property- a+b=b+a; ab=ba</li>
<li>Associative Property- a=(b+c)=(a+b)+c; a(b+c)=ab+ac</li>
<li>Distributive Property- a9b+c)=ab+ac</li>
<li>Additive Identity- a+0=0+a=a</li>
<li>Multiplicative Identity- a•1=1•a=a</li>
<li>Additive Inverse- -a+a=a+(-a)=0</li>
<li>Multiplicative Inverse- a•1/a=1, a≠0</li>
</ol></div>Wed, 06 Nov 2013 00:09:02 -06002013-11-06T00:09:02-06:00243067http://www.wyzant.com/resources/blogs/243067/why_i_hate_foilDevin F.http://www.wyzant.com/resources/users/view/78876240Why I Hate FOIL<div>Let's use our imagination a bit. Picture yourself in math class (Algebra I to be exact), minding your own business, having fun playing with the axioms (aka rules) of algebra, and then one day your teacher drops this bomb on you:<br /><br />"Expand (x+3)(x-1)"<br /><br />And you might be thinking, "woah now, where did come from?"<br /><br />It makes sense that this would shock you. You were just getting used to the idea of expanding 3(x-1), and you probably would have been fine with x+3(x-1), but (x+3)(x-1) is a foreign idea all together.<br /><br />Well, before you have much time to think about it on your own and discover anything interesting, your teacher will probably tell you that even though you don't know how to solve it now, there is a "super helpful", magical technique that will help you…<br />FOIL<br /><br />For those of you lucky enough never to have heard of FOIL, I will explain. FOIL stands for First Outside Inside Last and is a common mnemonic device used to confuse children about a fairly easy concept.<br /><br />If you remember the distributive property a(b+c) = ab+ac, then it might seem odd that all of a sudden we put two grouping next to each other and now we are doing something "new".<br />But is FOIL really new?<br /><br />The answer is of course no, what we are actually doing is just a short cut for the distributive property. And if you were aloud to try and solve it before being told what to do, you might actually have figured that out. For example, if we have (a+b)(c+d), we could distribute (a+b) as if it was a whole quantity. So (a+b)(c+d) = c(a+b) + d(a+b) and then we distribute again and get ac + bc + ad + bd.<br /><br />To me this seems much simpler than having to learn a mnemonic device, and remember how to draw our "rainbow lines" and remember where to put a plus and a minus, and so son and so forth. We merely follow a simple rules we already know.<br /><br />Another useful reason not to teach FOIL is because in only works for expressions similar to (a+b)(c+d). But what about expressions that look like (a+b)(c+d+e), or even (a-b+c)(d+e)(f-g-h+i+j)(k-l+p)? You can't use FOIL for these, but of course, you use the distributive property.<br /><br />So please, if you are a math teacher, the next time you have a chance to teach FOIL… don't. Spare your students the confusion and teach them what is really going on. FOIL might be quicker, but math isn't about the destination, it's about the journey.</div>Sun, 03 Nov 2013 10:08:24 -06002013-11-03T10:08:24-06:00242939http://www.wyzant.com/resources/blogs/242939/algebraic_equations_and_expressionsMichael L.http://www.wyzant.com/resources/users/view/82542570Algebraic Equations and Expressions<div>One of the major differences between algebraic equations and algebraic expressions consist of the equal sign because the equal sign consitutes for a solution that can be checked to verify that it is the solution. Expressions are meant to be simplified so common factors are important in simplifying expressions. Equations give a way to actually check the answer by subsitution for the variable while expressions are normally checked by multiplication or another type of operation.</div>Fri, 01 Nov 2013 19:40:00 -05002013-11-01T19:40:00-05:00241993http://www.wyzant.com/resources/blogs/241993/a_few_thoughts_on_why_we_learn_algebraEllen S.http://www.wyzant.com/resources/users/view/75479140A Few Thoughts on Why We Learn Algebra<div>I recently responded to a question on WyzAnt's “Answers” page from a very frustrated student asking why he should bother learning algebra. He wanted to know when he would ever need to use it in the “real world” because it was frustrating him to tears and “I'm tired of trying to find your x algebra, and I don't care y either!!!”<br /><br />Now, despite that being a pretty awesome joke, I really felt for this kid. I hear this sort of complaint a lot from students who desperately want to just throw in the towel and skip math completely. But what bothered me even more were the responses already given by three or four other tutors. They were all valid points talking about life skills that require math, such as paying bills, applying for loans, etc., or else career fields that involve math such as computer science and physics. I hear these responses a lot too, and what bothers me is that those answers are clearly not what this poor student needed to hear. When you're that frustrated about math, the last thing you want to hear is that you'll “need” to be able to do math in order to live. That might cause you to take those responses as a sign that you should not venture anywhere near computer science or physics, or ever rent an apartment or own a car. What this student needed to hear was the larger picture of why math is useful even if you never touch another x or y in your life. So here are my thoughts, in the form of an example from my own life.<br /><br />When you get right down to it, at its most basic level, algebra centers around the idea that you can add, subtract, multiply, or divide both sides of an equation by the same number and the equation will continue to be true. That very basic concept of “doing the same thing to both sides” has the implication of allowing you to rewrite the same equation a multitude of different ways without changing its value. In essence, algebra is problem solving at its most basic. You start with what you know, and by working step by step through rewriting the equation while maintaining its value, you arrive at a version of the equation that makes the unknown element clear. The whole time, the laws of algebra remind you that you're not changing what the equation means; you're just rewriting it in a way that's easier to work with and understand. This type of step-by-step problem solving has a multitude of uses in everyday life that don't involve a single number.<br /><br />Here's my example. One evening in college, I arrived back at my dorm building after a long day of classes, only to find that my wallet was not in my bag. I had no idea how long I'd been without my wallet, and even less idea where it was. On top of that, I had a small window of time in which I was supposed to go home and change out my books before heading out again, so I needed to get into the dorm NOW, which I couldn't do without my student ID card, which was – you guessed it – in my wallet. So what do I do?<br /><br />Well, I'll be honest, I began to panic slightly. But I worked through the panic and figured out my first plan: retrace my steps until I found my wallet. Fortunately, all of my classes that day had been in the same building, so I didn't have far to go. Unfortunately, my wallet was not anywhere on the path I'd taken from the dorm to the classroom, the path home, or anywhere in between. The wallet was lost.<br /><br />Having hit a dead-end on that front, I decided to set that problem aside and deal with the second issue: I still needed to get into the dorm to change out my textbooks. I figured I'd work on getting into the dorm, and perhaps once I was there more options for the lost wallet would present themselves. So instead of heading for the back door, which required an ID swipe to get in, I walked around to the front entrance and went into the lobby (a public area). I then headed over to the door that led to my wing, and killed time by pretending to read the bulletin board on the wall nearby. Soon, another student came by and swiped her card to open the door. I hurriedly slipped in behind her before the door closed, knowing that most people ignored the signs saying to not let anyone else in after you. I ran up to my room, opened the door (thankfully I still had my keys!) and there was my wallet, lying on the floor in the middle of the room.<br /><br />So what does any of this have to do with algebra? Well, compare my problem-solving strategies to the process of solving a system of equations. In my case, I had two variables: I needed to get into the dorm, and my wallet was gone. I started by trying to find my wallet – when solving a system, you start by solving one equation for one variable. I got as far as I could go on that path and eventually ended up with wallet = gone. I had to set that equation aside for a moment and deal with the other variable, just as you then switch equations in the system. I plugged “I don't have my ID” into the equation of “getting into the dorm” and solved that problem using what I knew about the building and the residents' laziness, and managed to get into my room (I solved for “I need to get into the dorm”). Once in my room, the first equation became solvable again, since my wallet turned out to be there – right where it had fallen out of my bag before I left the room that morning.<br /><br />This may sound way too coincidental, but the truth is that algebraic reasoning is incredibly important for a lot of tasks that have nothing to do with numbers. The ability to rewrite an equation while maintaining its value until the answer presents itself is at the heart of all problem-solving abilities. I often remind my students of the larger usefulness of the skills learned in math class by encouraging them to “take the numbers out of it.” What exactly are you doing in a broader sense, and how might you be able to use those skills in other situations? Give it a try – you might find that you like math more than you thought.</div>Wed, 23 Oct 2013 08:47:38 -05002013-10-23T08:47:38-05:00239146http://www.wyzant.com/resources/blogs/239146/how_to_be_successful_in_mathematicsAdrienne J.http://www.wyzant.com/resources/users/view/81955960How to be Successful in Mathematics<p><strong>How to be Successful In Mathematics</strong><br /><br />Math is a complicated subject. Students struggle with it, parents don’t feel comfortable helping with homework, and teachers find it impossible to “re-teach” every year. It is for these reasons that I feel having a good foundation in math is imperative. Students that have a great foundation feel confident and are not afraid of tackling a problem until they figure it out.</p>
<p> <br /><strong>What do students need to know to have a good foundation? </strong></p>
<p>Well, I think the most basic concepts they need to master are the concepts learned in pre-algebra. Most parents would be shocked to hear that students begin to learn these concepts as early as second grade. <br />Some are those concepts include properly using the order of operations; being able to add, subtract, multiply and divide negative numbers, fractions and decimals; and working problems with more than one variable. I encounter students “freezing” all the time when they encounter fractions, negative numbers, and variables. If they had more confidence in these concepts, they would feel comfortable trying a more complicated problem without hesitation.</p>
<p><br />My recommendation for mastering these concepts is practice, practice, practice. Identify where you are weak and do as many problems as you can to work those concepts. It is amazing how working the same kind of problem over and over again can drive a concept home.</p>
<p><br /><strong>Where can you find extra problems??</strong></p>
<p><br />Many students don’t realize that most textbooks supply extra problems in the back of their textbook. Also, there is a mixed review at the end of each chapter that combines all of the concepts of that chapter in one place. Feel free to go back to previous chapter and work problems for a review. The index is also a great tool to help you find extra problems.</p>Wed, 25 Sep 2013 20:39:42 -05002013-09-25T20:39:42-05:00237366http://www.wyzant.com/resources/blogs/237366/the_ordering_of_algebraic_operationsYingda Z.http://www.wyzant.com/resources/users/view/81959800The ordering of algebraic operations<div>When both writing down and reading the algebraic expressions, the binary operation (including addition+, subtraction-, multiply*, divide/, exponential^) follow a conventional order:<br /> <br />0) Parenthesis, including {}, [], ()<br />1) Exponent, multiply and divide<br />2) Addition and subtraction<br /> <br />The ordering is 0)>1)>2). Then there is no ordering within each group, eg multiply and divide are at the same level of priority except that 0) comes in such as a parenthesis.<br /> <br />Let's take a look at one quick example: 3+(8-2)*6.<br />First compute (8-2)=6;<br />Then compute (8-2)*6=6*6=36;<br />Finally compute 3+(8-2)*6=3+36=39.<br /> <br />Another example: 3^2+3/(5-2)<br />First compute (5-2)=3;<br />Then do 3/(5-3)=3/3=1;<br />Next compute 3^2=3*3=9;<br />Finally add 3^2+3/(5-2)=9+1=10.<br /> <br />Hope it helps!</div>Mon, 16 Sep 2013 09:22:10 -05002013-09-16T09:22:10-05:00