Mean Cuisine: The Unknown Average Quantity

I have seen this problem so many times that I need to write about it, it is so easy that all of my students should get this one right. You should too.  It goes like this example:


The average (arithmetic mean) age of a certain group of 20 people is 35. If five additional people join the group, the new average (arithmetic mean) is 36. What is the average age of the five new people?

Solution:

Average = (the sum of the things)/(the number of the things)

What do we know?

Our average is 35;   AVG = 35  [notice that in any word problem the word "is" can be substituted with an equal sign, I always tell my students "IS is EQUAL" "IS" is "="]

Our number of things is 20

What don't we know?
The ages of our group of people. We don't need to know each one, just the sum total of them all.

Let's let S = [sum of the age of the 20 people]

fill in the known information into our equation

35 avg = (S sum of things)/ (20 number of things), solve for S by multiplying both sides by 20

S= 700

Next step the problem changes the average and the number of people. The unknown quantity it the sum of the new people's ages. No sweat, just a little algebra will get us an answer in a hurry. Let's call the new sum the variable N.

New Average = (Old Sum plus New Sum of things)/(Old number of things plus new number of things)

36 = (700 + N)/(20 + 5)

36 = (700 + N)/(25), multiply both sides by 25

36*25 = 700 + N

900 = 700 + N, subtract 700 from both sides

200 = N

To find the average age of the new people, we divide the sum of their ages, N, by their total number, 5.

200/5 = 40

Generically speaking, the question format goes like this:

The average (arithmetic mean) MEASUREMENT of a certain group of a NUMBER of THINGS is the AVERAGE.

IF SO MANY MORE  new THINGS are added to the group, then the NEW AVERAGE of the THINGS  is SECOND AVERAGE.

WHAT is the THIRD AVERAGE of the MEASUREMENT of the new THINGS?

I think that I shall write a computer program to fill in random measurements, things and numbers for practice.

Isosceles

An isosceles triangle has angles of 70 and x degrees, what is the least possible value of x?

An isosceles triangle has angles of 82 and z degrees, what is the great possible value of z?

An isosceles triangle has a perimeter of 50. Two sides of the triangle measure 20 and n in length, where n is an integer. What is the difference between the maximum and minimum value of n?

See solutions tomorrow, or answer them today.

The SAT Essay Question Types

I am copying this from Collegeconfidential.com, July 2009 posted by ObsessedOne. It is simply the single best resource on the SAT Essay that I have ever seen. Thanks for showing this to me Skanda

I have compiled every SAT essay prompt administered by the College since the essay was introduced in 2005. Because the prompts are so generic, several archetypes seem to have emerged. Could it be possible to write an essay before seeing the prompt?


Individuality

--Following the Crowd

Do people need to compare themselves with others in order to appreciate what they have?
Are widely held views often wrong, or are such views more likely to be correct?
Is there any value for people to belong only to a group or groups with which they have something in common?
Is it always best to determine one's own views of right and wrong, or can we benefit from following the crowd?
Is it more valuable for people to fit in than to be unique and different?
Are people more likely to be productive and successful when they ignore the opinions of others?

--Following Authority

Should we pay more attention to people who are older and more experienced than we are?
Should society limit people's exposure to some kinds of information or forms of expression?
Can a group of people function effectively without someone being in charge?
Is it important to question the ideas and decisions of people in positions of authority?
Should society limit people's exposure to some kinds of information or forms of expression?
Is education primarily the result of influences other than school?
Should schools help students understand moral choices and social issues?

--Following Creativity

Is it always better to be original than to imitate or use the ideas of others?
Is it better for a society when people act as individuals rather than copying the ideas and opinions of others?
Is creativity needed more than ever in the world today?
Can people ever be truly original?
Do we put too much value on the ideas or actions of individual people?
Does planning interfere with creativity?


Motivation and Success

--Hardship and Success

Do people truly benefit from hardship and misfortune?
Do we really benefit from every event or experience in some way?
Do people place too much emphasis on winning?
Do people learn more from losing than from winning?
Does true learning only occur when we experience difficulties?
Does being ethical make it hard to be successful?
Can knowledge be a burden rather than a benefit?
Is persistence more important than ability in determining a person's success?
Is the effort involved in pursuing any goal valuable, even if the goal is not reached?

--Self-Determination and Success

Is identity something people are born with or given, or is it something people create for themselves?
Is it best for people to accept who they are and what they have, or should people always strive to better themselves?
Do success and happiness depend on the choices people make rather than on factors beyond their control?
Are people more likely to be happy if they focus on goals other than their own happiness?
Is it more important to do work that one finds fulfilling or work that pays well?

--Self-Expectation and Success

Do highly accomplished people achieve more than others mainly because they expect more of themselves?
Can people achieve success only if they aim to be perfect?
Is it best to have low expectations and to set goals we are sure of achieving?

--Collaboration and Success

Is it necessary for people to combine their efforts with those of others in order to be most effective?
Are organizations or groups most successful when their members pursue individual wishes and goals?
Do people achieve more success by cooperation than by competition?

--Ethics and Success

Does fame bring happiness, or are people who are not famous more likely to be happy?
Are people's actions motivated primarily by a desire for power over others?

--Quality or Quantity and Success

Do people achieve greatness only by finding out what they are especially good at and developing that attribute above all else?
Are all important discoveries the result of focusing on one subject?


Technological “Progress”

Does a strong commitment to technological progress cause a society to neglect other values, such as education and the protection of the environment?
Are there benefits to be gained from avoiding the use of modern technology, even when using it would make life easier?
Has today's abundance of information only made it more difficult for us to understand the world around us?
Is the most important purpose of technology today different from what it was in the past?
Have modern advancements truly improved the quality of people's lives?
Do newspapers, magazines, television, radio, movies, the Internet, and other media determine what is important to most people?
Should modern society be criticized for being materialistic?


Heroes

Do we benefit from learning about the flaws of people we admire and respect?
Should we limit our use of the term "courage" to acts in which people risk their own well-being for the sake of others or to uphold a value?
Should we admire heroes but not celebrities?
Is there a value in celebrating certain individuals as heroes?


Tradition

Do all established traditions deserve to remain in existence?
Do people need to "unlearn," or reject, many of their assumptions and ideas?
Should people always prefer new things, ideas, or values to those of the past?
Do incidents from the past continue to influence the present?
Do memories hinder or help people in their effort to learn from the past and succeed in the present?
Is it always necessary to find new solutions to problems?


Loyalty

Should people always be loyal?
Do circumstances determine whether or not we should tell the truth?
Can deception—pretending that something is true when it is not—sometimes have good results?
Is it sometimes necessary to be impolite?
Is acting an essential part of everyday life?


Others (less clearly defined; separated by spaces)

Is compromise always the best way to resolve a conflict?
Should people choose one of two opposing sides of an issue, or is the truth usually found "in the middle"?

Is the main value of the arts to teach us about the world around us?
Can books and stories about characters and events that are not real teach us anything useful?

Can common sense be trusted and accepted, or should it be questioned?
Do people put too much emphasis on learning practical skills?
Should people take more responsibility for solving problems that affect their communities or the nation in general?

Should people let their feelings guide them when they make important decisions?
Can people have too much enthusiasm?
Do images and impressions have too much of an effect on people?

Are decisions made quickly just as good as decisions made slowly and carefully?
Should people change their decisions when circumstances change, or is it best for them to stick with their original decisions?
Is it better to change one's attitude than to change one's circumstances?

Is criticism—judging or finding fault with the ideas and actions of others—essential for personal well-being and social progress?

Does having a large number of options to choose from make people happy?

The Asian Honor Code

I've tutored literally hundreds of people for 40 years.  No bull, my first tutoring gig was in the 3rd grade when the teacher was so frustrated that I won every multiplication competition that she stopped allowing me to compete and instead helped others. And then there is my sister who I simply loved to help. She was a year older and I desperately wanted to read her materials to see what was next!

In the course of that time it has been my pleasure to be employed by dozens of Asian families. So the question is, "Are Asian kids smarter because of biology or because they work hard?" The answer is BOTH!

Biology is a key factor, but not genetics. There is a difference. Biology is true because a person who actively uses their cognitive powers through studying actually increases the neuron connectors in the brain, INCREASING cognitive capabilities. Like a muscle that is exercised and becomes stronger and leaner, the brain that is engaged in learning and studying gains mental "strength."

But hard work is obviously the main factor. And don't give me the typical American crap that Asian kids who are tortured by their tiger moms lack creativity, spontaneity, and blah, blah, blah. My kids joyfully work and enjoy the rewards of getting high marks. They are proud and last week when I questioned a top student on why he was going so far, he responded, "I want to do my best, it is the Asian Honor Code."

Meaning the students themselves take great pride in their own personal, individual, creative processes.

The overwhelming majority of great athletes and great musicians gain their status not because they were born that way, but because they practiced hard and often with DESIRE.

And Honor.

Even the Odds

Even the Odds

If a and b are positive single digit integers, where a is even and b is odd, which of the following CAN be true?

I.               a is a prime number

II.             b is a prime number

III.           a + b is a prime number

A). II only

B). I & II only

C). II & III only

D). I, II, & III

E). None of the above 

Take a real test as practice?

There are several questions that I have been asked over and over:

1. WHEN should a student take the SAT?

2. Should the student study before taking the first test or wait to take a test as a real SAT to "get your feet wet" and benchmark/practice; see where the student is before making any decisions?

3. When should a student start studying?

4. Should the student study on their own, in a class or with a tutor one-on-one?

5. How many times should a student take the test?

The answers are as varied as the people who ask the questions.

Generically I suggest that a student begin to practice SAT questions as a freshman and set out a long term game plan of maximizing scores to have more options. Long term studying is less stressful and is absolutely proven to be more effective. If the student is strong, take the test in the spring of sophomore year

1. When should a student take the SAT?  Sophomore year may be too early for some but if the student is serious and has lofty ambitions, 10th grade is an excellent time to start. For an AP student, much of the SAT math is very basic and was taught long ago. As she advances into AP Calculus, the old math is not being used. It gets stored away in some dusty part of the brain and gets rusty.

Junior year is most typical. Spring starting in January is what I see the most.

Senior year is a very viable option to take the test. Colleges are now accepting scores as late as January for regular admission. The October test can still be taken and results sent in time for Early Decision or Early Action.


2. Take it first or study first?  Study first. If a parent and student think it is best to take a test and "see how she does," great; baseline results are valuable. I never disagree with that approach. But why waste time and money? For anything that matters in life, practice is rewarded with better performance.

3. When to start studying?  My answer is always the same: "Yesterday!" Studying vocabulary starts from the first time you speak to your baby on the day she was born. It takes an incredible amount of concentration and effort to learn a language. Research has shown that studying increases the number of connections between neurons in the brain. Get this: studying actually makes you smarter, duh. Working with any material over a long period of time increases your ability to recall it when needed.

I have met many juniors with excellent grades who do not remember simple math formulas. For an honor roll student who wants a competitive college, that is ridiculous. Practice. Study.

4. First let me caveat this entire post: I get PAID to teach kids the SAT/ACT, so should your kid start tutoring? YES and Do you have any money? 

Forgive my bad attempt at humor. Bottom line: balance goals against abilities. If a competitive college is the goal, a student should start preparing for the SAT as soon as possible.  If your daughter wanted to be in the Olympics, when would she start practicing? Would she have a coach?

Independent study is an excellent way to go. Students should learn how to teach themselves, for that is a valuable life skill. The key consideration is if the child is self motivated. There is plenty of material available online, in the library, and in the guidance office. Go to a bookstore and walk the prep book section, look on Amazon, and read my blog.

Classes and groups are okay. But they waste a lot of time. Time is very precious to the busy honor roll student. Wasting time listening to people ask questions about things that the honor roll student already knows is not optimal. But it is training and it is more than doing nothing.

One on one is best. There is no place to hide and hopefully your tutor makes the sessions about you the student not on any agenda or schedule. You should work on what you don't know.

5. How many times? My stock answer: "Did you get a perfect? No? Take it again." If the student has reached her goals. STOP and move on in life. If she has not, practice and believe in improvement. Three is the magic number and the advice that I have given the most.

Absolute Value is an absolutely valuable skill to know

This is a graph on a recent SAT question. It asks students to identify the equation.

The answer is | x^2 - 1 |

The "absolute value of the quantity x-squared minus one."

I see student miss absolute value questions all the time. The SAT presents absolute value as equations and as graphs. Remember that absolute value is always positive or zero.  When graphed, a function "f(x)" an equation in "y = | x |" will NEVER have an negative value for the y coordinate.

They are easy to spot. Go refresh your memory of absolute value and kick this question's butt.

SAT verses ACT, lesser of two evils

Now that the ACT is as widely accepted as the SAT, students are looking to gain an edge by taking the exam that best suits them. This is an excellent idea.

The ACT is a bit more "straight forward" in that it is not as "tricksy."  But the questions are harder and it is still scored according to a normalized distribution [the famous bell curve].

The math and reading are very similar, so colleges that take both look at those scores. Both tests have a grammar section and I like the ACT's style for grammar. It follows a narrative and has a flow to it. Whereas the SAT those what seems to be random sentences at you to correct.

The ACT has a unique section, science. I love it, but I'm a test nerd. It is a narrative with charts, tables and graphs and asks you to analyze data. All of the answers are right there, you just have to find them. It is a very long section with many details. Students who score well, do so by answering quickly.

In the end there are advantages to both tests. The key is to practice both and see which one you hate the least.

QAS Test Taking Tip

I just read the May 2012 SAT Booklet that is provided to students who took the test and paid the addition $20 for the QAS, Question and Answer Service.

Go get your hands on some old QAS booklets. Your friends and your school should be collecting these. They are simply the best practice materials available.

Since I have read and practiced taking all of the old SAT QAS booklets. I was able to fly through the new test.  I could see, right triangles, regular polygons (that one actually required me to use a pencil to calculate, even though my guess was pretty good), isosceles triangles, fractions sequences... etc, etc. I did all three math sections, 54 questions total, in about 20 minutes!

How? They use the same question structure about the same content. I know because I have read all of the QAS booklets.

On the Critical Reading, I saw the passage about the Hispanic girl and was able to go straight to the questions and answer most of them without reading the passage. I read the questions  and answer choices aloud to the mother of a rising senior, then I told her the correct answer choice for a half dozen questions - WITHOUT reading the passage. She was cracking up.

How was I able to do that? I have read every passage of every SAT for seven years, along with every question they have published in their books and on their website. I bought the books used for $10 and made copies of the old tests. I studied the WRONG answers and saw a pattern. I was able to see the WRONG answers on the May test and guess wisely on the questions. I completed one entire 25 minute Critical Reading section in about 5 minutes.

Anyone who says that test preparation does not raise your test scores is full of it. I identify common question types as well as common correct and incorrect answers. I can see incorrect answers and eliminate them, because I PRACTICE looking at the right and wrong answers. You can too

Slope of a line through the origin

Here is another simple SAT lesson. 

Slope ~ Line through the Origin

In the X-Y Plane, line l passes through (0,0)  and (4,-3). Line m passes through (3,-4) and is perpendicular to line l. What is the slope of line m?

Concepts to know:

1.     Positive and Negative Slope

2.     Lines that are parallel with the X and Y axis

a.     The X-axis is the line y=0. Lines parallel the X-axis have no slope, m = 0. All lines that are parallel have the same slope, so any line that is parallel to the X-axis will be a line y = a, where a is a real number. Like y = 4. These are also called Horizontal Lines.

b.     The Y-axis is the line X=0. It has an undefined slope. These lines have the equation x = b, where b is a real number. They are called Vertical Lines. [Note: Vertical Lines are NOT functions].

I tutor one-on-one in Sewickley. Summer rates start at $375 for five lessons and are negotiable for more, but worth it considering how competitive schools are and what a student can make back through academic financial aid.
Email for more information.

SLOPE

I'm back.  After taking some time to teach high school in the city of Pittsburgh, I'm returning to the Bohemian life of an itinerant SAT Tutor.

First rambling:

On every SAT is a question from Algebra I ~ SLOPE!

You remember; Slope = [Rise]/[Run]  and Slope Intercept Form is y = mx + b; where x and y are points on the line, m is the slope, and b is the where the line goes through the y-axis or "y-intercept."

Slope is from a line on the X-Y Plane.  It is also called a "Linear Function" but that is a lesson for another day.

Things to know:
1. A line through the origin; aka "Direct Variation."  Since the line goes through the origin (0,0), the y-intercept = 0. Sometimes they like to write it as y = kx where k is the constant of variation ~ but it is still the slope.
2. Parallel lines have same slope.  y = 3x - 5 is parallel to y = 3x + 2.5
3. Perpendicular lines are lines at right angles and have slopes that are "Opposite Reciprocal"  y = 2x + 1 is perpendicular to y = -(1/2)x + 1

A special thanks to Jason Ferguson, Director of Admissions at Hampden-Sydney College, for participating in the Admissions Spotlight Series. Joseph Fernandez from Parliament Tutors conducted this interview.

As the Dean of Admissions, what are your day-to-day responsibilities?

Really? My day is never the same…which is a good thing. With that said, every day I work to recruit the best and brightest young men to our campus.

What do you consider the most significant parts of an application, the parts which applicants should prepare the most carefully?

While each part is important, applicants need to take care when filling out the application. Make sure that their application is complete and the information is correct. Make sure the essay is well-written and the teacher’s reference is from someone teaching them an academic subject.

Is there anything you frequently see on an application that you hope to never see again?

The name of another college in the essay…It has always been my dream to attend [insert the name of another school].

What common pitfalls should applicants be careful to avoid?

Trying to overload the extra-curricular activities page. Only list those activities that are important to the applicant and those activities that they have a passion for.

Are there any myths about the application process which you would like to dispel?

That the application process is a difficult one. Granted, I work for a small institution. However, the more time to spend getting to know what each college needs and wants, the easier the process will be for you. Also, most admissions officers are willing to offer assistance…all you have to do is ask.

What advice would you give to an applicant with below-average test scores but significant extra-curricular experience?

For Hampden-Sydney, get to campus for an interview. While significant extra-curricular experience will never replace academic performance, it is always good for us to have a face to go with the name. Furthermore, the interview will give us better insight into the type of candidate you really are.

Do you frequently have to turn away applicants whom you wish you could admit? If so, what could those applicants do to be admitted?

Of course we do. Those applicants just weren’t ready for the academic rigors of curriculum. At least, they had not shown that they are ready at the time for their application. While they can’t go back in time and change their high school performance, we do suggest that they go elsewhere and improve their academic record and show that they are indeed ready for the challenge.

How much faith do you have in the ability of the SAT to predict success at in college?

The SAT is a factor that we consider in conjunction with a number of other credentials. We take a holistic look at the application and make decisions that are in the best interests of the College and the applicant.

What do you look for in a recommendation letter?

Hopefully, the teacher will give us an honest assessment of the student….his work ethic, contributions to the class, and a better insight into the type of learner he is.

Joseph Fernandez is an SAT Tutor with Parliament Tutors. He is a New York Tutor.

Math Vocabulary #1

#2 KNOW the Instructions

Secret #2:

KNOW the SAT test instructions. Be MORE than fully prepared, thoroughly familar with each section and it's instructions, types of questions and their style.

Do not waste a single precision second looking back to the beginning of a section or listening to anything that the proctor has to say. This is a serious competition, so be competitive.

This is especially true on the math section where certain formulas are given, circles, triangles, etc. I have seen too many students look back to that section trying to remember the area vs. circumference of a circle.

For an exercise, practice writing down all of those from memory over and over again.

Can you outline the entire SAT exam, number of sections, time, number and types of questions? Do it

101 Secrets of the SAT #1

Ok, I've been wanting to do this for a while, write and publish a book (booklet) unleashing my accumulated knowledge on SAT to help folks out

#1. The Writing Section is still useless for admissions.

In 2005, the SAT was revamped to some things were dropped, others added. Of the biggest changes was the addition of the Writing section. It is two multiple choice sections and an essay. Colleges and universities have not been using it for admissions. Though some are looking at it for placement tests.

Much has been written on the changes, so I need not comment much. Every student should strive to do well on all tests. Since this test does not count for much, if anything at all, your efforts are best spent on improving your Math and Critical Reading scores.

I'll write more on the Essay section later.