Tuesday, August 31, 2010

Venn Diagram


Venn diagrams or set diagrams are diagrams that show all hypothetically possible logical relations between a finite collection of sets. A Venn diagram is constructed with a collection of simple closed curves drawn in the plane. John Venn, an English mathematician introduced a diagrammatic way of representing sets. The diagrams representing sets are called Venn Diagrams template.


Venn diagrams normally comprise overlapping circles. The interior of the circle symb olically represents the elements of the set, while the exterior represents elements which are not members of the set.

venn diagram maker article is the best online feature to learn about the sets and Venn diagram. Venn diagram is nothing but a pictorial representation of a set. Sets and the! ir representations is very much simple through venn diagram maker.


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Work Problem 5 Solution

This is the solution to the work problem # 5 which read: Jim can fill a pool carrying buckets of water in 30 minutes. Sue can do the same job in 45 minutes. Tony can do the same job in 1 ½ hours. How quickly can all three fill the pool together?



Ok Tony takes 1 and 1/2 hours which is 90 minutes, and then we got jim at 30 minutes and Sue at 45 minutes. Convert everything to find out how much each can do in 1 minute. Tony does 1/90 th of his job in 1 minute Jim 1/30 th and Sue 1/45 th. So lets take just one individual persons job say Sue for example. If someone says Sue takes how long to finish her job if you know that she can do 1/45 of her job in a minute what would u say? 45 minutes. Or you can look at it as x/45 = 1 which means how many over 45 would = 1? The answer of course would be 45 again. So we set this problem up just like that only were ADDING the other peoples jobs together and setting it equal to 1.

So we got x/45 + x/90 + x/30 = 1 *Multiply both sides by 90 and get

2x + x + 3x = 90

6x = 90

x= 90/6 = 15

So it would take 15 minutes if all 3 worked together to finish the job.


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LSAT Prep Book Recommendations

Here's the moment you've all been waiting for: my LSAT prep book recommendations.

Happy holidays to everyone. There WILL be more tips next week too, don't worry - not everyone is on vacation. :D Recommendations below. Stay warm!


PrepTests 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, and 56
by LSAC

Each are only sold individually, as LSAC has not yet published a book of ten exams containing them. Because the exam has changed in recent years, you'll need several of these to study effectively.

When to use: after you've completed the other PrepTests.

The Next 10 Actual, Official LSAT PrepTests
by LSAC (older edition)

Contains PrepTests 29-38 and is the newest book of ten. It's essential that you get your hands on these in order to study effectively.

When to use: during exam prep.

10 More Actual, Official LSAT PrepTests
by LSAC (older edition)

Contains PrepTests 19-28.

When to use: during exam prep.

10 Actual Official LSAT PrepTests
< br />by LSAC (older edition)

Contains most PrepTests from 7-18. Only worth doing if you're studying far enough in advance that you'll have enough time to do the most recent exams as well.

Note: These exams are really old - from 12/92 - 9/95. Don't be concerned if some of the games are a bit difficult. You'll rarely see these types on recent exams.

When to use (if at all): in early stages of preparation to familiarize yourself with the exam.

The Official LSAT SuperPrep
by LSAC (older edition)

Contains a few exams you can't find anywhere else: 2/96, 2/99, and 2/00. The explanations within are the biggest selling point, but most people find them to be confusing and technical.

When to use (if at all): in early stages of preparation.


June 2007 LSAT exam (PDF)
by LSAC

Free sample exam. Treat it as if it were PrepTest 52.5 when making your study schedule.

LSAT Logic Games Bible (2003)
by Powerscore (2008)

This book contains the best Logic Games diagramming techniques I've seen, and I've reviewed all the LSAT prep books out there. The drills are useful, the organization is clear, and the book demonstrates its techniques on real LSAT PrepTests.

Skip the section on "Pattern Games" and "The Forgotten Few" unless you've already mastered other game types. Also, don't worry too much about the book's complicated subclassifications. You'll be fine if you can distinguish between the main game-types: linear/sequencing, grouping, and combination.

Unless you have a lot of time, I recommend the 2003 edition. Why? Because the 2008 edition is nearly twice the size. It appears the author beefed it up to justify the $65 retail price tag. Also, the 2008 edition exposes you to recent games you'll want to save for later practice tests.

When to use: Before you attempt logic games from the books of PrepTests.

Logic Made Easy: How to Know When Language Deceives You
by Deborah Bennett (older edition)

Although not explicitly written for test-prep purposes, this book contains several logical reasoning-type questions and reviews several common fallacies. The author is clearly familiar with the LSAT, and this makes the book more relevant for our purposes. I highly recommend reading this because it is clear, full of simple examples, and concise. You can skip the parts on the history of logic.

American Scientist's review.
When to read: Before you begin LSAT prep or when you need! a break from practice exams.

Informal Logic: A Pragmatic Approach

by Douglas N. Walton (older edition)

Clearly explains and demonstrate multiple examples of valid and invalid arguments. Walton is obsessed with logical fallacies and covers many of the common ones appearing on the LSAT.

When to read: Before you begin studying or when you need! a break.

Elementary Logic: Revised Edition
by William V. Quine

At 144 pages, it's short and sweet. It's also the first-ever logic textbook (originally published 1941, revised 1980). It discusses many basic issues (necessary/sufficient, etc.) relevant to LSAT logic. If you have the time/inclination, feel free to look it up, but it's by no means necessary.

When to read: Before you begin studying or when you need a break.

How to Solve It: A New Aspect of Mathematical Method
by George Polya (Older editions)

Simple advice on problem solving and logical thinking. It's useful because it gives you a framework to identify and analyze the relationship between evidence and conclusion. Wikipedia, this summary, and the following will probably be enough for you.

The book gives you some questions to ask yourself about any Logic Game or Logical Reasoning Stimulus:

1. What information is unknown/provided? Does the evidence/premise satisfy the conclusion?
2. How is this game/stimulus similar to others you've done? Questions do tend to come back in future exams (with different topics, of course).
3. ! Does a restatement (the contrapositive) of the argument help?
4. What inferences can you make?
5. How can you use these inferences?

Another nice summary.

When to read it: Before you begin studying or when you need a break.

The Little Luxe Book of Sudoku: 335 Easy to Hard Puzzles and The Little Black Book of Sudoku: 400 Puzzles
by Will Shortz

Number puzzles exercise your brain and get it ready for the logic games.

When to use: Before you begin studying or when you need a break.

T! hat's it - see you next year!

---------

! Next wee k: How to study for a retake (or what to do when you've already used too many PrepTests)...
Continue Reading...»


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Trigonometric Identities

Table of Trigonometric Identities:

Trig identity solver

Reciprocal Identities: sin

cosec u inverse function will be sin u

sec u inverse function will be cos u

Cot u inverse function will be Tan u

u inverse function will be co sec u

Cos u inverse function will be sec u

Cot u is the inverse function of Tan u

more details on

Factor rules

Trinomial calculator



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Printable TAKS Tests Math Reading, Science, Social Studies All Grades

All years, all grades, both English and Spanish, with answer keys.

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Korean Automaker Made a Big Investment in the South

A few days ago, New York Times has article on Korean car maker’s investment in U.S. (see http://www.nytimes.com/2009/04/22/us/22kia.html?ref=global-home). A few things really catch my attentions. First, Korean auto manufacturers, like Japanese auto manufacturers, mostly set up their plants in the southern states of U.S (in this case, a small town of 3000 people). &n bsp;As late entrants, why did East Asian car markets choose southern states in U.S.? How did they pick those locations? Apparently union is one of the important reasons. But there are other reasons to choose southern states over other states. I will write a more detailed analysis next time. The second point is that the article mentioned that “In the case of Kia, which is based in South Korea, state and local officials doled out some $400 million in tax breaks and other incentives.”

Actually, almos! t every state has some investment tax incentives to attract business investment. Since the investment opportunity is so precious now that state and local governments also are willing to dole out additional public subsidies.

Recently, I collect some investment tax incentives from several Midwest states. I choose Midwest because this is a region that is traditionally specialized in manufacturing.
Five states that have Investment Tax Credit programs are analyzed here. While each program is unique there are some similarities (see Table 1).






Table 1








All five states have som! e Enterprise Zone or special zone programs, which are aimed to! direct investment to specific communities and locations. The other similarity is that most states award tax credits based on the additional economic activity induced by the investors. The usual measurements for the additional economic activity include the minimum investment criteria, the number of new jobs, and the average wage of new jobs. The Enterprise Zone Program in Iowa provides a basket of public incentives, such as a property tax exemption, a supplemental New Jobs Credit (260E), a sales and use tax refund to the construction period, an Investment Tax Credit, and a supplemental Research Activities Tax Credit. Illinois’s enterprise zone program offers an Investment Tax Credit equal to one half of one percent of the value of qualified property, a sales tax exemption, and an enterprise zone tax credit for new jobs created (Illinois Department of Commerce and Economic Opportunity). Indiana’s enterprise zone program only offers credits for increased wages of local emp! loyees in private companies in the enterprise zone. Nebraska has programs specially aimed to promote investments in rural areas. For Wisconsin, both investment and the new jobs are awarded with the tax credits.


Final point is that the list in the table does not include all incentive programs that promote business investment in those states. More often, the terms and conditions of public incentives are subject to negotiation. For those big name investors with deep pockets, asking the governments to tailor a new tax incentive for them is a quite normal practice.


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Celebrating teamwork and mathematical problem solving


Quest of 3 genome musketeers
Pals who transformed biomedical research share Albany Med prize


By CATHLEEN F. CROWLEY, Albany Times Union Staff writer

ALBANY -- Three good friends reunited Friday at Albany Medical Center. Three decades ago, they were the cheerleaders and the brains! behind the effort to map the human genome.

On Friday, they picked up the Albany Medical Prize in Medicine and Biomedical Research, which at $500,000 is the largest cash prize in medicine in the United States.

A few minutes into their "grand rounds" lecture -- a formal medical tradition where wisdom is shared among colleagues -- a voice announced over the Albany Med loudspeaker that fire alarm testing would begin immediately.

The scientists roared with laughter.

"We are the three genome musketeers," said Dr. Francis S. Collins, "We've been setting off fire alarms for 30 years."

In a nutshell: David Botstein, a geneticist, was one of the first people to suggest the idea of mapping the human genome; Eric Steven Lander, a mathematician, created new algorithms for analyzing genes and mapping complex multiple-gene diseases; and Collins, a physician and biologist, created faster mapping ! techniques and eventually oversaw the world-wide Human Genome ! Project from a post within the National Institutes of Health.

"They collectively unlocked and then opened the door that had previously barred us from understanding disease processes at the most basic genetic level," said James J. Barba, president of Albany Med.


Eric Lander, the mathematician on the prize winning team (pictured at left above), got a running start in developing his mathematical problem skills, as well as his teamwork and leadership skills, as a leader of his math team at Stuyvesant High School in New York City. The USA Math Olympiads were just starting up when he was in high school, and he was among the first Americans to take those contests and to represent the USA at the International Math Olympiad in 1974.

This coming Tuesday and Wednesday, 300 students across the country will be taking the USA Math Olympiads. Six of the highest scoring students will follow in Eric Landers' hig! h school footsteps and qualify to represent our country at the 2010 International Math Olympiad to be held in Kazakhstan in July.

Congratulations and best wishes for a great adventure in problem solving in the coming weeks to all the students writing the USAMO this week, especially those from New York State, listed below, and--of course, especially to the five students from Albany Area Math Circle.

We know that the problem solving skills that all our New York State mathletes have already demonstrably acquired can be put to many important uses in the future. Our state and our country and our world have many problems to solve.

Albany Area Math Circle, Niskayuna NY:
Andrew Ardito
Matthew Babbitt
David Bieber (Niskayuna High School)
Schuyler Smith
Felix Sun (Shenendahoah High School)

Columbia Grammar & Prep School New York NYReed LaFleche

Comsewogue High School Port Jefferson NY
David Lawrence

Corning-Painted Post West High School Painted Post NY
Vasily Kuksenkov

Garden City High School, Garden City NY
Jan Gong

Great Neck South High School, Great Neck NY
Keaton Stubis

Hackley School Tarrytown NY
Michael Celentano

High School for Math, Sci & Engineering New York NY
Mo Lam

Hunter College High School New York NY
Andre Arslan
Meena Boppana
Paul Handorff
Bohao Zhou

New Hyde Park Memorial High School Hyde Park NY
Michael Hodgson

Penfield High School Penfield NY
Allen Liu

Stuyvesant High School New York NY
Milo Beckman
Junghwa Cha
Lijin Chen
Daniel Mendelsohn
Joseph Park Joseph
Yevginey Rudoy
Andrew Ryba
Zachary Young

The Dalton School New York NY
Alexander Iriza

Vestal Senior High School Vestal NY
Colin Lu

solving math problems

Billionaire Aims To Solve A Math Problem

Math For America Pays Students To Get Master's Degree And Boosts Their Salaries If They Teach Math

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Slope - Intercept Form

Slope basketball game

slope intercept form

Slope Y-Intercept Form to Standard Form

My students are having some trouble with changing slope y-intercept form of a line to standard form and vice versa. You must first understand solving equations before this will make sense. Having a good grasp of variables on both sides of the equation will help since it involves moving a variable term from one side of the equation to the other.

Here is an example:

Rewrite y = 2x - 5 (slope y-intercept form) as standard form.

Move the 2x to the left side of the equation with the y. -2x + y = -5

That is all that is required for this one. It is now in standard form (Ax + By = C).

Another part of the assignment that we worked on required that the standard form be written with integers (no fractions).

Rewrite 3/5y = 2/3x + 2 in standard form using integers.

Move the x term to the left side of the equation. -2/3x + 3/5y = 2
Now to get rid of the fractions, you will multiply both sides of the equation! by 15 (the least common multiple of 3 and 5). You could multiply by any multiple of 3 and 5 but generally we use the least common multiple. As long as this multiplication is done to BOTH sides, the equation will stay balanced. Hint: Remember that you must use the distributive property on the left side of the equation.

15[-2/3x + 3/5y] = 2(15)

-30/3x + 45/5y = 30

-10x + 9y = 30 (standard form using integers)

slope intercept form

FLASH GAMES

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Adam Conclusion 2

Another area where modern study potentially impacts our discussions is in the area of brain research and psychology. Neuro-scientists still have a long way to go in mapping the brain, but it seems fairly clear that there is least some part of the brain that corresponds to each human experience. Whether it be thinking, memory, anger, personality, or even religious experience, we can say which parts of the brain "light up" when that happens. Change the structure of the brain--whether through Alzheimers, cancer, or physical injury--and you change the person.

These discoveries have led some Christians to explore whether the idea of the soul might itself be an instance of Christian language that points to something real but not exactly in the way we have thought. [1] Such Christians point out that the word soul in the Old Testament never refers to a detachable part of me that survives death. Rather, the Hebrew word soul refers to an entire li! ving thing, whether it be a human or a fish. Even in the New Testament, it is only rarely that the word seems to refer to a part of my human make-up (e.g., Heb. 4:12*). Our common conceptions of the soul have as much to do with later Greek influence on Christianity than on what the Bible actually says.

The point here is not to take a position on this issue. Our point is that if scholars are debating on this level, then we can see how far afield the old arguments about whether our carnal nature can be eradicated or suppressed were. Yet no one who pays much attention to the world around us will doubt that our world is thoroughly "enslaved to sin." Ask the social worker at the local elementary school or the prosecutor who works at the courthouse or the nurse that works the emergency room or the person who helps with foreign aid. They can perhaps identify with the pessimism of Joshua 24:19 as Joshua tells the Israelites, "You cannot serve the LORD."
In other words, you guys can't hack it. Serving God is just! too har d. Godliness and righteousness is a miracle in this world. This issue, perhaps more than any other, surfaces especially in Romans 6-7, the subject of our next section.

[1] cf. Joel Green, ***

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Factoring the time

I stumbled upon this comic that you might enjoy... factoring the time (from xkcd.com).


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Miquel s Pentagram Theorem

Interactive proof with animation and key theorems.
Auguste Miquel (France, Nantua, College des Castres) published this beautiful theorem in Journal de Mathematiques Pures et Appliquees (Liouville ‘s Journal) Tome Troisieme, Paris 1838.

Miquel's Pentagram Theorem.
Continue reading at:
gogeometry.com/miquel_pentagram1.htm


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Find This Free Statistics Help

The internet brings many advantage in our life. We can take advantage of the internet to buy and sell items online, get entertainment, find information, and many more. The internet is also the best place for students who are seeking for homework help. For instance, if you have Statistics homework, you can search for online tutoring program to get the answers for it.

TutorVista.com is a site that offers online tutoring programs for students of K-12 and college. If you are looking for Statistics help, Tutor Vista offers online Statistics tutoring program. The free Statistics help from Tutor Vista is intended to help students have better understanding for Statistics concepts. The advantage of the online tutoring program is that you can connect with a statistics tutor who will give personal attention to you. The online tutor will guide you step by step so th! at you can solve the Statistics problems and statistics questions.

The free Statistics help includes basic Statistics such as Mean, Median, Mode, range, and many more. When you have Statistics homework, you can also get the Statistics answers for the homework. So, what are you waiting for? Just go to this site to get the free Statistic help.

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March, my birthday, and I'm still here!

Hi Readers! I am still here responding to people's requests for homework help, I just don't post the answers on my blog as much anymore. But rest assured, I am still here answering your emails.

Also, to celebrate my birthday this week, I invite you to partake in this fun online gaming experience called "Auditorium"!

http://www.playauditorium.com/

I enjoyed it so much that I played it all the way through on my first sitting!


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Free algebra worksheets

Usually algebra textbooks provide lots of problems to practice the algebraic concepts and techniques, but some of you may still benefit from resources for free (or mostly so) printable algebra worksheets. Please see the list below, which I've originally compiled for my HomeschoolMath.net site.

Algebra worksheets

Worksheet Builder
Great and free worksheet maker software with nearly 7,000 built-in algebra and geometry questions.
www.jmap.org/JMAP_WORKSHEET_BUILDER_INSTALLATION_FILES.htm

Free Algebra Worksheets from KUTA Software
Free worksheets (PDF) for equations, exponents, inequalities, polynomials, radical & rational expressions and more.
www.kutasoftware.com/free.html

AlgebraHelp.com worksheets
Interactive worksheets that are checked online for most algebra 1 topics.
www.algebrahelp.com/worksheets/

Math.Com algebra worksheets generator
Generate worksheets for: linear equations, systems of equations, and quadratic equations.
www.math.com/students/worksheet/algebra_sp.htm

LessonCorner worksheets
These free worksheets include a few topics such as calculations with polynomials, factoring, and graphing lines.
www.lessoncorner.com/worksheets/

Algebra Fun Sheets
Worksheets that integrate algebra skills with fun activities including sudoku, word finds, riddles, color patterns, crosswords, games, matching cards, etc. A subscription is required.
www.algebrafunsheets.com

About.com Algebra Worksheets
An assorted collection of free algebra worksheets and answers. These pages are not very well organized, but they have lots of worksheets.
math.about.com/od/algebraworksheets/Algebra_Worksheets.htm

Algebra Worksheets from MathWorksheetCenter
Lots of worksheets for over 100 algebra topics. A few are free; most are accessible only by one-year a subscription.
www.mathworksheetscenter.com/mathskills/algebra/

A few fun algebra worksheets
These are for graphing linear equations and linear inequalities.

Online Math Work
Free multiple-choice worksheets for pre-algebra and algebra 1 topics. You can do them online, or copy to a word processor to print.
www.mathonlinework.com




Lastly... my own algebra worksheet collections, which aren't free but there are many free samples:

Algebra 1-A worksheets cover Algebra 1-B worksheets cover Math Mammoth Algebra 1 Worksheets Collection
A two-part collection (A and B) of 137 quality algebra worksheets covering all the topics in a typical algebra 1 curriculum. These worksheets are hand-crafted and contain lots of word problems and other variable problems. Free samples available. $11.50.
www.mathmammoth.com/worksheets/algebra_1.php


Summer Math Program


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A free download of a digital Algebra 1 book

Kinetic Books Algebra 1 looks really interesting! It is not really just a book, but software, or a digital interactive textbook.

It contains text, interactive problems and activities, and a scoring system all on the computer. Students can get step-by-step assistance in the form of audio hints and one-click access to relevant examples.

See a demo here. But the best is that the company Kinetic Books is even offering a free download of the product till September 30!

That really sounds fantastic, so if you have algebra 1 student(s), don't fail to take advantage of this tremendous offer.


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Level of measurement

The level of measurement has been classified into basically four categories. It is important for the researcher to understand that the level of measurement is determined partly by arithmetic operations and statistical operations.

Statistics Solutions is the country's leader in level of measurement and dissertation statistics. Contact Statistics Solutions today for a free 30-minute consultation.

Sorted in an ascending order of precision, the four different levels of measurement are the nominal, the ordinal, the interval and the ratio scale.

The first among the four levels of measurement is the nominal level. This level of measurement basically refers to those cases in which the numbers are used to organize the data. The use of words and letters is also done in this level of measurement. Suppose there is data that has two categories of students, na! mely weak students and strong students. Using this level of measurement, the researcher can easily classify the weak category of students with the letter ‘W,’ and the strong category of students can be denoted with an ‘S.’ This assigning of letters to distinguish the classification is the nominal level of measurement.

The second type of level of measurement is the ordinal level. This level of measurement generally involves those measurements that signify some kind of ordered associations between the number items. If four teams participate in a match, the team that has beaten all three teams would win the match and would be assigned the first rank. Then, the team performing right below the first team would be assigned the second rank, and so on. Thus, this level of measurement also assigns the reasons behind the rank assigned to any particular item. So, this level of measurement indicates the appropriate ordering of the measurements. T! he researcher should note that in this type of level of measur! ement, t he change or the share between any two types of rankings does not remain the same along the scale.

The next type of level of measurement is that of the interval level of measurement. In this level of measurement, the researcher categorizes and assigns orders to the measurements and also reveals that the distances between each interval on the scale is equivalent along the scale from the low interval to the high interval. One such example is the measurement of anxiety of a student that is in between the score of 10 and 11 is same as if the student is in between the score of 40 and 41. Another appropriate example for this type of level of measurement is that while measuring the temperature in centigrade, the distance between 940C and 960C is similar to the distance between 1000C and 1020C.

The last level of measurement is the ratio level of measurement. In this type of level of measurement, the researcher can observe a value of actual zero as well. This k! ind of phenomena is quite unlike the other types of level of measurement. However, the researcher should note that this level of measurement has the same property as that of the interval level of measurement. The divisions between the points on the scale have the equivalent distance between them, and the rankings assigned to the items are according to their size in this level of measurement.

The researcher should note that among these levels of measurements, the nominal level is simply used to classify the data, whereas the levels of measurement described by the interval and the ratio are much more exact.

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Digital Fundamentals
Basic English II
English Composition I & II
American Literature I
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Western Civilization II
US History II
Developmental Algebra
Elementary Algebra
Intermediate College Algebra
Pre-Calculus Algebra
Music Appreciation
College Life 101
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General Psychology
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Intro Spanish I & II
Fundamentals of Public Speaking
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How to divide irrational numbers?

Many times students just accept what they are told in math class without much questions; it's perhaps boring, they don't want to take time to investigate and delve deeper, or whatever the reasons.

The teacher feeds them with knowledge and they take it in: "Oh, okay, there's such a thing as Pi. Oh, okay, some numbers are irrational."

But a question such as this shows that the person is wondering, wanting to understand the material more. So it's a good question!

I will divide the 'division' question to two parts.

1) When exact answer is desirable, often we do cannot technically divide. For example, let's take Pi ÷ 2. Well, we just write it as Pi/2 or π/2 and leave it at that.

If we can simplify the answer, we do that. For example, √15/√5 can be simplified to √3.

We could simplify √2/2 this way: since 2 = √22, then √2/2 = √2/(√22) = 1/√2.

But often the expression √2/2 is left as is, since there is a convention that we should consider a root in the numerator to be "prettier" than a root in the denominator. Or I don't know why it is; I just remember this little rule from school math.

2) When you need a numerical answer, then you use the decimal approximation of the irrational number, and divide normally as you would decimals.
For example, to find Pi/2, you take the decimal appro! ximation to Pi as, say, 3.14159, and go 3.14159 ÷ 2 = 1.570795.


How to divide numbers

Adding and Subtracting Mixed Numerals - A Survey of Methods Taught

Now that our family health crisis has abated (my daughter is doing well), I guess it's time to jump back in with both feet. A math program leader in a district with which I am consulting, asked for my opinion on an important issue of curriculum and instruction.

How much time should middle school teachers spend on the traditional vertical algorithm for adding and subtracting mixed numerals vs. converting to improper fractions immediately?

I assumed that both methods are still commonly taught with about equal time given to each, but I wasn't all that sure about how that was across the country. This is where I need the help of my informed readers.

First, my thoughts. From a practical perspective of those who utilize fractions in their occupation, I would guess that the mixed numeral form is most commonly employed. Whether it's the carpenter tak! ing measurements to see how many board feet of wood must be ordered (or for precise measurement to the nearest sixteenth of an inch) or someone following a recipe in the kitchen, I can't imagine that converting to improper fractions would be their first choice. On the other hand when I personally need to add fractions in a math problem, I usually use improper. I took an informal survey of one of the groups I'm working with and the majority stated they were taught both methods and some preferred working with mixed numerals and others said it's more complicated that way.

Are the number of steps roughly the same?

Mixed Numerals Algorithm for Subtraction:
(1) Convert the proper fractions to common denominator form.
(2) If needed, regroup, i.e., "borrow" 1 from the whole nu! mber part of the larger mixed number, convert the 1 into commo! n denomi nator form and combine this with the other fraction (of course students are shown short-cuts for this which they blithely and mechanically follow without much thought).
(3) Subtract the whole numbers and the proper fractions.
(4) If the resulting fraction is improper convert it and add the whole number part to the previous result.

Improper Fraction Algorithm:

(1) Convert each mixed numeral to an improper fraction by the traditional algorithm (again blithely and mechanically without much thought).
(2) Determine a common denominator (or the lcd) and convert each fraction.
(3) Subtract the fractions.
(4) Convert the answer to mixed numeral form by the traditional division algorithm.

Now I may have combined steps or there are oversights but essentially they appear to be roughly the same number of steps. However, the difficulty or complexity level of th! e steps
may not be equivalent.

I also feel that the mixed numeral form requires somewhat more conceptual understanding even if the child does it routinely. It may also prepare the youngster for working with algebraic expressions like A + B/C, but that's debatable. Further there seems to me to be a strong connection between the Mixed Numerals Algorithm and adding and subtracting denominate numbers. For example:
Subtract
15 hr 37 min
9 hr 46 min
I doubt that we would encourage students to convert both to minutes first, subtract, then convert back to hrs and min. I could be wrong there!

I feel there are arguments on both sides here. My instinct is that both need to be taught but it's not clear to me how much time should be spent on each method. Certainly some youngsters could handle both with facility while some would struggle mightily with at least one of these methods.

Further, I suspect there are some youngster! s who convert mixed numerals to improper fractions procedurall! y withou t full conceptual understanding that a mixed numeral is an addition problem!

Your experiences and thoughts...


How to add and subtract mixed fractions