Week 1, Day 2: Intro to Exam & Counting Methods (Sets, Permutations, Combinations)
The GRE Exam
The GRE has three primary measures.
Analytical Writing
2 essays - 30 minutes each
Focus on the skill of directly responding to the presented
tasks
2 sections of 20 questions each - 35 minutes each
Focus on basic math concepts & skills
Verbal
2 sections of 20 questions each - 30 minutes each
Focus on reading & reasoning skills
1 minute break between all sections except for a 10 minute
break after section 3
Labeled research section possible; unlabeled experimental
section possible
•
Onscreen calculator during Quantitative sections
(reduces emphasis on computation, increases emphasis on logic)
•
You can change/edit/review answers, i.e. “Mark
& Review” feature
•
Visit ets.org/gre for news
•
Visit “Bulletins & Forms” section: https://www.ets.org/gre/revised_general/about/bulletin/
•
Official information in the ETS guide
•
Review question types, answer types, scoring,
and skills.
•
An “endurance” test where you are tested for
hours.
•
Incorrect answers do NOT subtract but do not add
to scores either.
•
It is best to answer every question – at least
guess!
•
Once a section is completed, you may not go back
through it.
•
Scratch paper are provided, and you
can always request more. Use them on all sections!
•
Verbal & Quantitative sections are adaptive;
the raw score ≈ number of correct answers; scaled scores are generated by
equating questions with difficulty levels and on comparison with the scores of
other test-takers in your cohort.
•
Scores are reported within 10-15 days after test
date. See each exam’s details to see more about how score reports work.
History of Standardized Testing:
Standardized tests are loosely based on the IQ tests
developed initially to determine the difference between officers and infantry
in the military. Questions that were “easy” should have been answerable by
everyone; questions that were “difficult” should have been answerable only by
the very intelligent. These produce a bell curve of scores, and the “average”
does not “change.”
However, where IQ tests are supposed to measure innate
abilities, standardized tests for university programs are supposed to measure
acquired skills. In reality, these tests measure one skill: your ability
to take a standardized test.
Schools use test results to distinguish between applicants with similar GPAs. A high GPA tends to indicate a hard worker, while a high test score tends to indicate someone with a lot of skills necessary for the school programs. In addition to the scaled score, you also have a percentile ranking, which further differentiates between high-scoring tests.
Do not merely aim for your “best”! This is not a good goal. Good goals are SMART: specific, measurable, achievable, realistic, and time-based. In other words, aim to raise your current score by a certain number of points within a specific amount of time, or to increase the number of correct answers you have within the 8-week session we have.
The test makers: ETS = Educational Testing Service
The test customers: Admissions departments and
mailing lists!
The test writers & graders: Computers and
graduate students
To prepare for standardized tests:
•
Have a strategic plan
•
Practice regularly, methodically – cramming is
worse than useless!
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Take simulated tests on the computer – use the
free downloadable “PowerPrep” software from the ETS website
•
Pick the order with the “easy” test first –
questions you like, concepts you are good at, and then do the others later
•
Do NOT approach like a fact-based test
•
Learn to think like the test-maker so you can
avoid common errors
•
Remember that the test has to be predictable,
otherwise it would not be standardized, and this means you can improve your
test-taking strategies!
•
The test uses the same principles for every
test-taker, and the changes are normed through exhaustive repetition over
random groups of test-takers.
This is an adaptive test:
•
Questions will start with medium difficulty
level
•
If you guess repeatedly, your score will drop
dramatically
•
Getting several questions right will increase
the difficulty level
•
Getting several questions wrong will decrease
the difficulty level
•
There is NOT a one-to-one correspondence between
right or wrong answers and changes in difficulty level
•
Do not waste time trying to figure out the
difficulty level of each question
•
Do NOT worry if the questions suddenly seem
easier, you will reach an equilibrium
•
DO take heart when the questions get more
difficult, because this means you are doing well!
•
Unanswered questions = WRONG every time
•
Pace yourself – never take more than 2 minutes
for any question, and remember that all questions are equally important for
your score
•
Practice at a higher difficulty level than you
are at currently to improve your overall level
•
Make educated guesses through Process of
Elimination (PoE) – this will improve your chances of guessing correctly
•
Every answer has an equal probability of being
the right answer – do NOT just guess C or the longest answer
•
Always check your answers before completing a
section
•
Verbal does NOT affect Quantitative nor vice
versa
•
DO NOT CANCEL YOUR SCORES unless you are
extremely sick or the building catches fire
Math concepts included:
•
Arithmetic & number properties
•
Proportions, fractions, percents, & decimals
•
Algebra & coordinate geometry
•
Geometry of forms and solids
•
Probability & statistics
•
Word problems
•
Logic & critical thinking
•
Pattern recognition
Most basic strategies for all quantitative sections:
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Do the “easy” parts first
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Educated guesses/PoE
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USE the scratch paper
•
Double-check your answers
•
Make sure you are answering the question they
asked!
•
Leave NO question unanswered
•
Read ALL answer choices
•
Plug in the answers to test them
•
Try to come up with your own answers and test
them
•
Memorize (turn the memorization sheet into flash
cards) basic formulas and computations as well as math vocabulary – see also
Kaplan’s appendix D: Math Reference (p. 663-687)
•
Use the calculator as little as possible
Week 1, Day 2
Notes from Chapter 9
·
No trigonometry or calculus
·
Algebra, geometry, arithmetic, data
interpretation
·
Reasoning: apply given information, think
logically, draw conclusions
·
All necessary for graduate work regardless of
your area of study
·
Math reference appendix
·
Question Types
o
Quantitative Comparison – compare two values
using logic
o
Problem Solving – solve word problems
o
Data Interpretation – interpret information in
charts & graphs
·
Rely on the calculator as little as possible –
practice without!
·
Calculator image
·
Math Vocabulary
Chapter 14: Math Foundations – Counting Methods & Probability Review
Counting Methods: Sets
Set – a group of values having a common property
e.g.,
negative odd integers greater than -10, or all positive integers evenly
divisible by 3
Elements – items in sets, aka members; 1 Î A,
where A = {1, 2, 3, …}
Finite set – countable, comes to a stop
Infinite set – uncountable, limitless, never ends
Empty set – sets with no elements, aka null set; {}, Æ, or
{Æ}
Non-empty set – a set that has elements
Subset – A is a subset of B if and only if all members of A
are members of B; the null set is a
subset of every set! A = {2, 4, 6},
B = {1, 2, 3, 4, 5, 6}, so A Ì B
Sets have unique properties:
·
No members repeat
·
Order does not matter
·
Usually given in {} (brackets)
Contrast with lists:
·
Can repeat
·
Order does matter
·
Not usually given in {}
Set operations:
Union – A È B – members of either or both sets, mutually
inclusive, no double-counting; A = {1, 2, 3} and B = {3, 4, 5}, so A È B =
{1, 2, 3, 4, 5}
Intersection – A Ç B – members of both sets only, just the overlapping
members; given the above sets, A Ç B = {3}
If sets have no members in common, they are mutually
exclusive and their intersection is the empty set.
Venn diagrams can help visualize sets.
The universal set (U)
holds all sets.
The inclusion-exclusion principle: counting members of an
intersection is easy because it is just those that overlap; counting members of
a union is trickier because there is overlap between both sets and we must avoid
double-counting. Therefore we use this principle:
|AÈB| = |A| + |B| - |AÇB|
Lastly note, that when sets are mutually exclusive (i.e.,
have no overlapping members), then |AÈC| = Æ
The multiplication principle: when choices or possibilities
occur one after the other and independently of each other, we multiply the
number of possibilities; when the choices are a matter of either/or, we add the
numbers. If the choices are dependent on previous choices, each subsequent
choice is minus however many iterations there were in previous choices.
Counting Methods: Combinations and Permutations
In the case that we have to make a number of successive
choices, we have to do either permutations or combinations of those choices.
The simplest successions are just multiplication problems and factorials.
Simple combinations (and permutations) are factorials (x!) where x is an integer and ! is an operation such that x = x * x-1 * x-2 * …* 1
In other words, 1! = 1, 2! = 2 * 1, 3! = 3 * 2 * 1, 4! = 4 *
3 * 2 * 1, etc.
I list the first 10 factorials on our memorization chart.
It is easier and less time-consuming to memorize these than to calculate them anew each time you encounter them.
When there are actually very few options (only 2 or 3), it
may be easier to list the possible combinations. For example, three people are
on the schedule to wash dishes; how many ways can they be ordered? Call those
people A, B, and C. The possible orders are: ABC, ACB, BAC, BCA, CAB, and CBA,
so there are only 6 possible orders. This list is not very time-consuming, but
when there are more than three possible choices, the list gets more
complicated, lengthy, and error-prone.
Solving more complex combinations (these are groups where
order does NOT matter) and permutations (these are groups where order DOES
matter) can be very time consuming; e.g., possible codes with 6 letters out of
all 26 letters in the alphabet would be difficult to compute: abcdef, bcdefg,
cdefgh, etc., and errors would creep in quickly. Remember: There are no
small mistakes. So instead of trying to figure out all possible
configurations by rote effort, we use a formula to simplify the process, make
it shorter, and make fewer mistakes:
When making k choices out of n possible options, we take n!
and divide by (n-k)! (where order matters) OR we take n! and divide by k! *
(n-k)! (where order does not matter).
Permutations are where order matters (for example, when three contestants win a race and one receives a gold medal, the other a silver, and the third a bronze).
Combinations are where order does not matter (for example, when just three contestants win, but the problem does not specify placement).
These formulas also work when k = n or k = 0 because only 1 subset has zero elements (the null set) and only 1 subset has exactly the same number of elements as the set of all possible choices (the set of n itself).
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