Algebra 1 Articles - Wyzant Tutor Blogshttps://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.Fri, 24 Nov 2017 22:24:00 -0600https://www.wyzant.com/images/logos/wyzant-logo.pngAlgebra 1 Articles - Wyzant Tutor Blogshttps://www.wyzant.com/resources/blogs/algebra_1https://www.wyzant.com/resources/blogs/algebra_1466938https://www.wyzant.com/resources/blogs/466938/oops_i_did_it_again_living_environment_us_and_global_history_algebra_coreGilant P.https://www.wyzant.com/resources/users/view/77505480Oops I did it again!! Living Environment, US and Global History, Algebra Core...<div><br />We did it! With hard work, determination, my high school students passed their regents exams. I tutored US and Global History, Living Environment, Earth Science, Algebra Core, Algebra 2/Trigonometry, Geometry and Chemistry and the students passed. One student passed with a 70%, another 75%, 76% and another 79%. All the other students scored 80% and up. </div>
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<div>I am so proud of my students. Well done students and parents, we did it!</div>Tue, 28 Jun 2016 14:54:08 -05002016-06-28T14:54:08-05:00402290https://www.wyzant.com/resources/blogs/402290/things_your_teacher_didn_t_tell_you_or_you_didn_t_catchDouglas G.https://www.wyzant.com/resources/users/view/85769345Things Your Teacher Didnt Tell You, Or You Didn't Catch<div>As an experienced teacher of over 15 years, it's easy to recognize frustration in students. Some of that frustration is admittedly self-imposed, but let's face it; some is teacher/environment imposed. Not all students learn the same way. As a teacher and tutor, I modify my approach to meet the needs of individual students. This task can be quite daunting when you have a classroom full of 25, less than fully engaged pupils; however, when tutoring one on one or in a small group dynamic the task is quite masterfully attained. </div>
<div>I love teaching, I love seeing those "light bulb" moments. Successful teaching/tutoring is measured by student success and learning is gauged by how well mastery has been achieved. That's my goal.</div>Mon, 16 Nov 2015 14:16:13 -06002015-11-16T14:16:13-06:00377412https://www.wyzant.com/resources/blogs/377412/i_left_astoriaAlexei M.https://www.wyzant.com/resources/users/view/79167790I left Astoria<div>Hello, Manhattan! As you may have noticed, I moved to Manhattan. I hope you would welcome me with more students. Please see my profile for more information. I only would like to add that I am flexible with my schedule. Thank you very much. </div>
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<div>Alexei M.</div>Mon, 07 Sep 2015 18:49:13 -05002015-09-07T18:49:13-05:00377050https://www.wyzant.com/resources/blogs/377050/all_my_grade_8_9_students_passed_the_algebra_core_regents_examGilant P.https://www.wyzant.com/resources/users/view/77505480ALL MY GRADE 8 & 9 STUDENTS PASSED THE ALGEBRA CORE REGENTS EXAM<h3><strong>All my grade 8 & 9 students (10 students) passed the Algebra Core Regents exam</strong>. Only one student had to retake it in August and she passed with an 83%. In June she scored 53%. My two Trigonometry students passed the Regents, but only 2 out 4 students passed the Geometry Regents exams.</h3>Sun, 06 Sep 2015 10:51:47 -05002015-09-06T10:51:47-05:00371883https://www.wyzant.com/resources/blogs/371883/do_you_like_a_challenge_have_a_go_at_this_calculationGaurav W.https://www.wyzant.com/resources/users/view/85919779Do you like a challenge? Have a go at this calculation...<div>In the calculation below the mathematical symbols have been removed.</div>
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<div>Using only +, -, x and / can you make it correct?</div>
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<div><strong>7 32 6 14 9 12 = 112</strong></div>
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<div>Best regards,</div>
<div>Gaurav</div>Sat, 15 Aug 2015 23:58:41 -05002015-08-15T23:58:41-05:00353905https://www.wyzant.com/resources/blogs/353905/summertime_with_math_planningDave R.https://www.wyzant.com/resources/users/view/85370801Summertime with Math - Planning<div>With the school year winding down, arranging for summer break Math time starts!<br /> <br />Why Math?<br /><br />Consider: <br /> 1) Not practicing newly acquired math skills will allow for knowledge to erode<br /> 2) Not practicing previously acquired math skills will expedite knowledge erosion<br /> 3) Not having other non-math course work will allow for</div>
<div> - focusing on math remedial work, or</div>
<div> - getting a jump on next year’s math academic growth. <br /> <br />Math needs are the same per subject, whether the learning setting is for advanced placement, over-age/under-educated, middle school, high school, or Veterans. BUT, the instructional approach should be different. Differentiating the approach to each student’s situation addresses learning styles (do we not all have different learning styles, which, if catered to, maximize results?).<br /> <br />Also, a subtle, but critical, issue during summer instruction is matching the substance and strategy of each learner's teacher’s curriculum. The need to do this is clearly seen when comparing:<br /> - public vs. private schools, <br /> - pre- vs. post- Common Core materials, <br /> - Blue Ribbon vs. state run district improvement plan curriculum.<br /><br />This means that it is critical to be very sensitive to the subtle differences in terminology, sequencing and pacing which each teacher and district tracks.<br /><br />Now is the time to arrange for the appropriate educational scenario, starting with a precise understanding of the Math subject needs and goals, and the student’s learning style and knowledge base.</div>Tue, 12 May 2015 09:49:24 -05002015-05-12T09:49:24-05:00331313https://www.wyzant.com/resources/blogs/331313/welcome_parents_students_and_educatorsPatrick D.https://www.wyzant.com/resources/users/view/85614378Welcome Parents, Students, and Educators!<p>Hello <strong>Wzyant Academic Community</strong> and welcome to my blog section! This is where I am available for <strong>online chit-chat</strong>, <strong>educational assistance</strong> free of charge, <strong>business discussions & arrangements</strong>, and <strong>more!</strong> I am always eager to help and love to talk turkey with all realms of academia, so don't be shy and feel free to ask many questions!!! </p>
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<p>P.S. ∫∑∞√−±÷⁄∇¾φΩ</p>Fri, 13 Mar 2015 03:26:23 -05002015-03-13T03:26:23-05:00318866https://www.wyzant.com/resources/blogs/318866/my_favorite_math_wordsElias H.https://www.wyzant.com/resources/users/view/79074590My Favorite Math Words<div>Here are 48 of my favorite math words in 12 groups of 4. Each group has words in it that can be thought of at the same time or are a tool for doing math.</div>
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<div>between</div>
<div>on</div>
<div>over</div>
<div>in</div>
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<div>each</div>
<div>multiply</div>
<div>of</div>
<div>many</div>
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<div>ratio</div>
<div>divisions</div>
<div>distribution</div>
<div>compartments</div>
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<div>limit</div>
<div>neighborhood</div>
<div>proximity</div>
<div>boundary</div>
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<div>infinite</div>
<div>infitesmal</div>
<div>mark</div>
<div>differentiation</div>
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<div>graph</div>
<div>width</div>
<div>height</div>
<div>depth</div>
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<div>circle</div>
<div>sphere</div>
<div>point</div>
<div>interval</div>
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<div>hyper</div>
<div>extra</div>
<div>spacetime</div>
<div>dimensional</div>
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<div>geometry</div>
<div>proportion</div>
<div>sketch</div>
<div>spatial</div>
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<div>four</div>
<div>table</div>
<div>cross</div>
<div>squared</div>
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<div>target</div>
<div>rearrange</div>
<div>outcome</div>
<div>result</div>
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<div>area</div>
<div>volume</div>
<div>space</div>
<div>place</div>
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<div>What are your favorite math words? If you aren't sure, search for "mathematical words" and pick a few.</div>Mon, 02 Feb 2015 09:58:40 -06002015-02-02T09:58:40-06:00310191https://www.wyzant.com/resources/blogs/310191/anyone_can_learn_anything_if_they_put_their_mind_to_itPhillip R.https://www.wyzant.com/resources/users/view/76595060Anyone Can Learn Anything if they put their mind to it!<div>I do believe that any subject can be learned if one decides that they want to learn that subject. Its been my way of thinking throughout my career. If you want to learn and have an open mind, then it can happen!</div>
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<div>Positive thinking is what it takes to succeed in this life. Believe in yourself and it will happen!</div>
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<div>Phil</div>
<div> </div>Fri, 19 Dec 2014 10:33:35 -06002014-12-19T10:33:35-06:00306466https://www.wyzant.com/resources/blogs/306466/why_study_math_algebraPeter A.https://www.wyzant.com/resources/users/view/78575920Why Study Math - Algebra<div>A question that I have heard many times from my own students and others is this: "When am I ever going to use this?" In this post and future posts, I'm going to address possible answers to this question, and I'm going to also take a look at what mathematics educators could learn from the question itself.</div>
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<div>Let's look at the answer first. When I was in school myself, the most common response given by teachers was a list of careers that might apply the principles being studied. This is the same response that I tend to hear today. </div>
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<div>There is some value in this response for a few of the students, but the overwhelming majority of students just won't be solving for x, taking the arcsine of a number, or integrating a function as part of their jobs. Even as a total math geek, I seldom use these skills in practical ways outside my tutoring relationships.</div>
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<div>Can we come up with something better, that will apply to every student? I say that we can. Let's start with algebra.</div>
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<div>Algebra is the point in the study of mathematics where students discover that problems can be solved even when information is missing. In fact, it is possible to create formulas where all the information is missing. These formulas can be applied when the information arrives. In life, this is something that is done all the time. We have a task to do, and we need a process for that task. In practice, we often find a similar task that we have done previously, and modify its process for the new task. The process is analogous to the formulas used in algebra.</div>
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<div>The real skill that is learned in algebra is one that is widely applicable. Solutions to specific problems can be generalized, and if necessary, adapted to work for new problems. Troubleshooting, investigation, and efficiency are all issues that are touched by the back-and-forth process of abstraction and application.</div>
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<div>Here is an illustration that may be helpful to you or your students: Let's say there is a student in gym class who asks why they need to do pushups and sit-ups. "When will I ever use this in my career"? The answer is that they probably won't, unless they go into one of a small number of careers. The answer to this student is very clear: "You probably won't use this." Why do it? It's exercise. Having strong muscles and a healthy body is of benefit to anyone, regardless of chosen career.</div>
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<div>In a similar way, the study of algebra strengthens one's ability to apply existing processes in new situations.</div>
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<div>Rather than rambling on for a long time in one post, I'm going to write about other math disciplines and their universal applications in other posts. Also, I'll address what I have learned from the existence of this common question.</div>
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<div> </div>Wed, 03 Dec 2014 11:21:24 -06002014-12-03T11:21:24-06:00301376https://www.wyzant.com/resources/blogs/301376/looking_for_worksheets_that_are_free_and_great_for_math_reviewKristin C.https://www.wyzant.com/resources/users/view/80970430Looking for Worksheets that are free and great for Math review?<div>This is my all time favorite website for Math worksheets.</div>
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<div><a href="https://www.kutasoftware.com/freeipa.html">kutasoftware.com</a></div>Mon, 10 Nov 2014 13:21:59 -06002014-11-10T13:21:59-06:00299452https://www.wyzant.com/resources/blogs/299452/how_to_read_area_of_a_circle_formulaAvery A.https://www.wyzant.com/resources/users/view/85292630How To Read Area of a Circle Formula<div>Reading Formulas can make or break how a student comprehends their formula when alone - outside the presence of the teacher, instructor, tutor, or parent.<br /><br /> Formula for Area of Circle: A = π * r^2<br /><br />Ineffective ways to read the area of a circle formula are as follows:<br /><br />Area is π times the radius squared.<br />Area is π times the radius of the circle squared.<br />Area of a circle is π times the radius squared.<br />A equals π times r squared.<br /><br />>>>> Why are these ways NOT effective ways to read this formula? <<<<<<br />1. Students will recall and repeat what they hear their educators say.<br /><br />2. If students recall letters (A) versus words (Area of a Circle) they will not realize the connection with word problems.<br /><br />3. Half way reading the formula (radius versus radius of a circle) creates empty pockets or disconnects in student thinking.<br /><br />4. Train students to read formulas in a way that improves recall and applications (i.e. for word problems)<br /><br />***This minor adjustment will produce significant results and has application across different disciplines that use formulas!!!!! <br /><br />Let me know your thoughts.</div>Sun, 02 Nov 2014 10:43:14 -06002014-11-02T10:43:14-06:00299296https://www.wyzant.com/resources/blogs/299296/great_website_for_algebra_helpSharon W.https://www.wyzant.com/resources/users/view/77887090Great website for algebra help.<div>Hi all algebra students. I found a great website, algebra-class.com that has an algebra calculator that you can use to check your homework. It has been very useful in our algebra classes as a tool for homework help. </div>
<div> </div>Sat, 01 Nov 2014 07:28:34 -05002014-11-01T07:28:34-05:00293709https://www.wyzant.com/resources/blogs/293709/excellent_middle_school_encode_decode_problem_rigorousElias H.https://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 -05002014-10-05T15:02:21-05:00293128https://www.wyzant.com/resources/blogs/293128/the_transition_points_of_grade_school_3rd_4th_6th_9th_gradesGary B.https://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 -05002014-10-02T13:29:33-05:00289553https://www.wyzant.com/resources/blogs/289553/mathematical_journeys_inverse_operations_or_the_answer_is_always_3Ellen S.https://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 -05002014-09-16T11:35:25-05:00276519https://www.wyzant.com/resources/blogs/276519/the_importance_of_s_t_e_m_science_technology_engineering_mathJacob C.https://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 -05002014-06-30T16:15:12-05:00270826https://www.wyzant.com/resources/blogs/270826/mathematical_journeys_what_does_the_function_look_likeEllen S.https://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 -05002014-04-29T08:20:23-05:00270661https://www.wyzant.com/resources/blogs/270661/fun_math_sites_for_middle_school_and_high_schoolGwen R.https://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 -05002014-04-27T23:55:17-05:00262225https://www.wyzant.com/resources/blogs/262225/mathematical_journeys_the_unperformed_operationEllen S.https://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 -06002014-02-18T18:36:42-06:00