Wednesday, July 20, 2022

Week 8, Day 2 (Final Exam)

Today is the day of the final exam. You must take the exam with the camera turned on and pointed at your face at all times to ensure the validity of the testing environment. You must keep your microphone muted so all students have a quiet testing experience. You may turn your camera off during breaks and when you have completed the exam.

You will receive an email with a link to the exam. You have the entire class period to complete all questions. If everyone finishes before class time is over, we will review the exam during the remainder of class. If everyone needs the full time, you may email your instructor with questions about your exam.

Please email your instructor if you have difficulty connecting to Zoom, receiving the exam link, or accessing the exam. Please send your instructor direct messages on Zoom if you have questions during the exam.

Grades for the final exam will only be calculated once ALL students have submitted their exams. Final course grade reports will be emailed out by the end of week 8 no later than 2 PM on Friday. Please contact the Registrar and the Dean of Students for counseling about which forms are necessary to change from TP to IEP, to request specific TP courses, to go on Vacation, or to Transfer out. 

Thank you all for a great session!

Monday, July 18, 2022

Week 8, Day 1 - Review for Final

Review for Final

 

Our final exam:

50 questions total

25 – from Cliff’s Notes

24 – from ETS

1 – from GMAC

 The Final Exam will cover all areas of math on the syllabus: arithmetic and number properties, data analysis (statistics & probability, permutations & combinations), algebra, geometry, and the different special question types of the GRE and the GMAT (Quantitative Comparison and Data Sufficiency), as well as the special answer types of the GRE (Multiple Choice, Numeric Entry, Select All that Apply).

Review your strategies and formulas, USE your scratch paper for partial credit, and show me your flash cards for bonus points!

Strategy Review

·       Take the "easy" test first

·       Never leave questions blank - ALWAYS at least guess!

·       Be sure to use scratch paper/note boards/whiteboard

·       Memorization:

o   1) formulas

o   2) terminology

o   3) equivalent numerical values

·       Process of Elimination (PoE - educated guesses with statistical strength!)

·       Read all answer choices, also notice formats

·       Check your work carefully for:

o   1) calculation

o   2) what the question asked

o   3) no typos

·       Use visualizations: draw diagrams, draw pictures, tables, charts, etc.

·       Estimate wherever possible/necessary

·      

      General Method:

1) Analyze the question

2) Identify the task

3) Approach Strategically

4) Check your work

General Test Prep Tips:

(some of these may not apply to online testing during pandemic conditions)

The weeks & days before:

·       Study regularly daily – at least 10 practice problems a day, especially in struggle areas!

·       Review the testing website for any possible changes to policies & procedures

·       Review terminology, use flashcards, etc.

·       Have a “study outfit” you can wear to the test center when the time comes; this is useful for getting in the right mindset and triggering your memory during testing

·       Sign up with the name on your legal ID so it is exactly identical

o   My horror story: my married name was different from my maiden name, and my old ID did not include my new marital patronymic, but I signed up with the new name. I was forbidden from taking the test and they refused to permit any refund. If you are rich, this might not be an issue for you, but most of us are not rich!

o   Be very careful about hyphens, spelling, double last names, etc.

·       Make a practice drive to the test center at the same day and time for which your test is scheduled – online testing? Familiarize yourself with the testing procedures early and often!

·       NO major life changes (new medications [unless absolutely necessary], breakups, quitting or starting anything important, etc.)

·       If you have an early test, start going to bed and getting up earlier

 

The night before:

·       STOP STUDYING – cramming is worse than useless, it is actually harmful!

·       Lay out what you will need to bring with you: study outfit, registration confirmation, keys, water bottle, legal ID, wallet – online testing? Consider having this stuff ready in your testing space

·       Eat well (healthy, safe, non-irritable food)

·       Drink well (plenty of water, NO alcohol nor other intoxicants including “smart” drugs)

·       Sleep well (start going to bed early a week ahead if your test is early in the morning)

 

The morning of:

·       NO STUDYING – a waste of time, will only stress you out more

·       Eat well (healthy, safe, non-irritable food; brain food includes whole grains, yogurt, eggs)

·       Drink well (caffeine only if you regularly have it, plenty of water, etc.)

·       Put on your study outfit and bring your necessities with you

·       Arrive at least 30 minutes early – online testing? I suggest at least 15 minutes early

·       Bring nothing in except the necessities: legal ID, water bottle, car keys, wallet, confirmation of test registration

 

During the test:

·       Use your breaks to get extra scratch paper/noteboards, use the restroom, drink a little water

·       When stuck on a problem try one of two methods:

o   Mark and Review / ‘easy’ test first (GRE only)

o   Stare at the ceiling and think about anything else for 30 seconds or less (both exams)


Wednesday, July 13, 2022

Week 7, Day 2 (Elimination Strategies, Quantitative Comparison, & Data Sufficiency)

Chapter 6 Mathematical Reasoning p. 329-351, 
Practice:
Arithmetic & Statistics p. 352-356, 
Algebra p. 356-360, 
Geometry p. 360-365, 
Word Problems p. 366-367; 

Chapter 7 Quantitative Comparison p. 389-295, 
Practice: 
Arithmetic & Statistics p. 396-400, 
Algebra p. 400-402, 
Geometry p. 403-405; 

Chapter 8 Data Sufficiency p. 419-423, 
Practice: 
Arithmetic p. 424-425, 
Algebra p. 425-426, 
Geometry p. 426-428

Basic Strategies

·       Note key words

·       Pull out given information

·       Plug in numbers (aka Picking Numbers), use FROZEN

o   Fractions

o   Repeats

o   Ones

o   Zeros

o   Extremes

o   Negatives

·       Work from the answers (aka Backsolving)

·       Approximate/estimate

·       Make comparisons

·       Mark diagrams (draw your own)

·       Procedure Problems

·       Multiple-Multiple Choice (Process of Elimination, PoE, aka Educated Guesses)

See p. 392 for the “Plan of Attack”

 

Quantitative Comparison Strategies (GRE)

·       Cancel out equal amounts

·       Make partial comparisons

·       Keep perspective

·       FROZEN

·       Simplify

·       Mark/draw diagrams

·       Use easier numbers

See p. 448 for the “Plan of Attack”

 

Data Sufficiency Strategies (GMAT)

·       12TEN

o   1 alone is sufficient

o   2 alone is sufficient

o   Together they are sufficient

o   Either alone is sufficient

o   Neither is sufficient together or alone

·       Determine what information is necessary

·       Don’t solve unless you have to

·       Use a simple marking system for PoE

·       Use ONLY common knowledge

·       Mark/draw diagrams

See p. 472 for the “Plan of Attack”

Monday, July 11, 2022

Week 7, Day 1 (Word Problems)

Chapter 5: Word and Graph Interpretation Problems p. 285

Technique p. 287

1.     Identify what is being asked – and write it down!

2.     Pull out given information – draw pictures, write down key words

3.     Set up an equation – or a chart, a table, etc.

4.     Identify whether all given information is necessary – ignore irrelevant information

5.     Solve the equation – check units, formulae, etc.

6.     Check that you answer the question asked – go back to the question and make sure you answer what they asked for and aren’t stopping too soon or answering the wrong part of the question

7.     Check your computation – plug in your answers, check your calculations, etc.

Remember: There are no small mistakes!

 

Key words for addition, subtraction, multiplication, and division  p. 289-290

 

Formulae

Memorize these formulas for efficiency:

·       Simple Interest

o   Interest = Principle * Rate * Time, or I = prt

§  Principle: the amount invested

§  Rate: the interest rate, usually given as a percentage or as a decimal

§  Time: usually annual (once a year), but could be biannual (every two years), semiannual (twice a year), quarterly (four times a year)

·       Compound Interest

o   Time consuming to calculate because at the end of each period of time, the interest is added to the principle and then recalculated for the next period of time

o   Compute simple interest for the first year (or first time period if interest is compounded at a time other than annually)

o   Add the interest to the principle

o   Use the new principle to compute the next year (or second time period)

o   Repeat until you have compounded as many times as necessary

Formula from 3rd edition: 

Future Total Value Amount = Principal * (1 + [interest rate/frequency number per year])^(frequency*number of years)

or in other words: the compounded total value equals the principal times the sum of 1 and the quotient of the interest rate divided by the frequency number per year, raised to the power of the product of frequency times the number of years

·       Ratio & Proportion

o   Usually set up as two fractions equal to each other (e.g., x/y = a/b)

o   Solve through cross-multiplication

·       Motion

o   Distance = Rate * Time or d = rt (“dirt”)

o   Rate is velocity, speed, etc.

o   Equivalent equations:

§  Rate = Distance / Time or r = d/t

§  Time = Distance / Rate or t = d/r

o   Be careful about time! Most often, you are looking for mph or miles per hour, but sometimes time is given in days, minutes, etc. Remember: 24 hours to a day, 60 minutes to an hour, 60 seconds to a minute

o   Be careful about conversions! If they give you information in mph, but want an answer in kph, look for the suggested conversion rate or refer to your memorized conversion: 1 km » 0.6 mi

·       Percent

o   Remember: is/of = % or in other words, x% of y is z, so z/y = x%

o   All percentages are out of 100, and can be set up in proportions with 100 to find the missing value

·       Percent Change

o   The formula is change/starting point = % change

o   Calculate the change from the starting point to the new value through subtraction

o   Put the change over the starting point and divide

o   Your answer is the percent increase or decrease from the original value

·       Number

o   These are logic problems that require careful attention to the wording and use of our arithmetic key words that we have memorized

o   Note down what you are looking for so you can double-check at the end

o   Use the problem to set up an equation

o   Solve the equation

o   Check your computation

o   Check that you solved for what you are looking for



·       Age

o   Similar to number problems

o   Using a table to track the information can be useful

o   Double check your computation

o   Double check that you solved for what the question is asking

·       Geometry

o   Use memorized geometry formulas

o   Draw pictures!

o   Double check your computations

o   Double check that you solved for what the question is asking



·       Work

o   The combined work formula is given as the sum of fractions equal to a third fraction, which must be inverted to answer the question “how long does it take them working together”

o   You are solving for T, a quantity of Time

o   A, B, C, etc. are the workers involved

o   1/A + 1/B = 1/T (for two workers) or 1/A + 1/B + 1/C = 1/T (for three workers)

§  This equation requires you to find a common denominator to solve the equation

§  This equation assumes you have 1 unit of product per every unit of time worked for each individual

§  Other ratios may be necessary if, e.g., one person makes 5 pies for every 11 hours of work

§  When solving for T, you have to invert 1/T

o   For two people (or machines, or animals, etc.) the simplified version will give you T without inversion: AB/(A+B) = T

o   For three people, the simplified version is ABC/(AB + AC + BC) = T

·       Mixture

o   To my knowledge, the only formula is: the sum of the total costs of each type equals the total cost of the mixture, or cost(type 1) + cost(type 2) = cost(mixture)

o   However, you can set up a table to help you compute the missing information:

 

Cost per unit ($ per lb)

Amount of units (# of lbs)

Total cost of each ($)

Type 1

 

 

 

Type 2

 

 

 

Mixture

 

 

 

 

Alternatively:

 

Rate

Amount of solution

Amount of substance

Solution 1

 

 

 

Solution 2

 

 

 

Mixture

 

 

 

 

Another alternative:

Coin Type

Number of coins

Value of each

Total Value

Type 1

 

 

 

Type 2

 

 

 

Mixture

 

 

 

Etc.

Glossary p. 328

Wednesday, July 6, 2022

Links to Many, Many Practice Problems

 Manhattan Review: https://www.manhattanreview.com/free-gre-practice-questions/








Week 6, Day 2 (Geometry Part 2)

Polygons p. 232

Commonly known as shapes, polygons are many-sided figures (poly = many, gon = side) on a plane, or flat, 2D shapes. These include: triangles (three sides, three angles), quadrilaterals (four sides), pentagons (five sides), hexagons (six sides), heptagons (seven sides), octagons (eight sides), nonagons (nine sides), decagons (ten sides), enneagons (eleven sides), dodecagons (twelve sides) … etc.

Regular polygons have equal angles and equal sides; regular triangles are also known as equilateral triangles, while regular quadrilaterals are also known as squares.

Polygons have invisible diagonals, or lines connecting one vertex or angle to another vertex or angle through the middle of the shape. A convex polygon has all diagonals interior to the shape (and each interior angle is less than 180°), while a concave polygon has one or more diagonal exterior to the shape (and one or more interior angles is greater than 180°).

 

Triangles p. 233

Triangles are named by their three corners or vertices (e.g. rABC). The angles always add up to exactly 180°. There are several types of triangles:

Equilateral – triangles having equal sides and equal angles.

Isosceles – triangles having two equal sides and two equal angles.

Scalene – triangles having no equal sides and no equal angles.

Right – triangles having one right angle (90°).

Obtuse – triangles having one angle larger than 90° but less than 180°.

Acute – triangles having all angles smaller than 90°.

 

The base of the triangle can be any side of the triangle. The height or altitude is the perpendicular measure from the base to the opposite vertex, so it might be interior or exterior to the triangle depending on the type of triangle.


By contrast, a median goes from the vertex to the midpoint of the opposite side, and an angle bisector evenly divides an angle. There are three medians and three angle bisectors to every triangle. Also note the interesting fact: if the segment is two of these (median, altitude, angle bisector) it is automatically the third.

If two sides are equal, the opposite angles will also be equal, e.g., if AB = AC, then ÐC will equal ÐB. This is the case for all isosceles triangles.


 

For equilateral triangles, all sides and angles are equal, so the angles have to equal 60°.

 

For other triangles, the longest side will always be opposite the largest angle, and the other angles will be proportionally smaller and opposite the proportionally smaller sides. For example, as in a right triangle, where the longest side is the hypotenuse and is opposite the 90° angle.

 

Also note, for every triangle, no matter the type, the sum of any two sides must be larger than the third side.

 

An exterior angle will always be equal to the sum of the two opposite angles. In the example below, x = y + z:

(Alternatively, you can subtract x from 180°.)

 


The Pythagorean Theorem states that a2 + b2 = c2 for all right triangles. This means that the sum of the squares of the legs of a right triangle will equal to the square of the hypotenuse.

Pythagorean triples are sets of three numbers that show up as a pattern for right triangles’ legs and hypotenuses. We have several on our memorization chart, including those that apply to isosceles right triangles (or 45°-45°-90° triangles) and special right triangles (or 30°-60°-90° triangles).


The most common triple is 3, 4, 5 and its multiples (6, 8, 10 and 9, 12, 15, etc.), followed by the 5, 12, 13. Less common are 7, 24, 25 and 8, 15, 17. Standardized tests use triples like these frequently.

The isosceles right triangle triple is 1, 1, Ö2 and its multiples (x, x, xÖ2, for example 2, 2, 2Ö2 or 8, 8, 8Ö2, etc.). Remember that Ö2 » 1.3, so this side will always be the hypotenuse.

The other special right triangle triple is 1, Ö3, 2, and its multiples (x, xÖ3, 2x, for example 2, 2Ö3, 4 or 8, 8Ö3, 16, etc). Remember that Ö3 » 1.7, so it is the measure of the longest leg, while the 2x side will be the measure of the hypotenuse.

 

Quadrilaterals p. 251

All four-sided figures are quadrilaterals (quadri = 4, lateral = side). The sum of the interior angles will always equal 360°. The most common types of quadrilaterals are the square, rectangle, parallelogram, rhombus, and trapezoid, but there are irregular quadrilaterals that do not fit into these categories.

Square – all equal parallel sides and equal angles, the only truly regular quadrilateral; the diagonals are equal, bisect each other, bisect the angles, and are perpendicular at their intersection.


Rectangle – two opposite pairs of equal parallel sides, all equal angles; diagonals are equal and bisect each other.


Parallelogram – opposite sides are parallel and equal, opposite angles are equal, consecutive angles are supplementary; diagonals bisect each other but are not equal unless it is a rectangle.


Rhombus – a parallelogram with equal sides but not necessarily equal angles though the opposite angles will be equal; the diagonals need not be equal but they will bisect, be perpendicular, and be angle bisectors.  


Trapezoid – will have a pair of parallel sides, but everything else is undetermined. Note that an isosceles trapezoid has legs of equal measure and thus diagonals which are equal, and thus also base angles will be equal as well.

 

 

Sum of Interior Angles p. 255

The sum of the interior angles of any polygon can always be determined based on the number of sides: (n – 2)180° = S(Ðint)

This is why triangles have sums of 180° and quadrilaterals have sums of 360°, etc.

 

Perimeter & Area p. 256

These formulas are included on our memorization chart. The perimeter of any polygon is always the sum of all sides, but these formulas work as well:



For any square where x is the length of one side, P□ = 4x. The area is the square of that side, or A□ = x2.

For any parallelogram, including rectangles and rhombuses, where l is the length and w is the width (alternatively, use b for base and h for altitude or height), P = 2l + 2w or 2(l + w), and A = lw, or length times width (alternatively, use bh or base times height particularly for non-rectangular parallelograms).

For any triangle where a, b, and c are the sides, Pr = a + b + c, and where b is the measure of the base and h is the altitude or height, Ar = ½bh, or one-half base times height. 

Trapezoids are special cases; again, the perimeter is the sum of the sides, i.e., two bases (b1 and b2) and two legs (x and y), or P    = b1 + b2 + x + y; the area will be A     = ½h(b1 + b2) or one-half the height times the sum of the bases.

 

Circles p.260

Circles are single-sided figures (or two-sided, if you count them as having an inside and an outside) that include the set of all points equidistant from a center point by which it is named.

The parts of the circle are the radius, diameter, chord, and arc. The perimeter is called the circumference.

 

We use the measure p or pi (pronounced like “pie” for United States English) to calculate the circumference and the area. Pi is approximately equal to 3.14 (p » 3.141592658… - pi is a transcendental number and has an infinite, non-repeating decimal value; as such, it is an irrational real number), and though our textbook often calculates pi into the answers, most of the time standardized tests just have p as part of the answer.

 

Radius – the measure from the center to the edge of the circle, represented with a lowercase r.

Diameter – the measure from one edge to the other passing through the center, represented with a lowercase d. The diameter equals twice the radius, or d = 2r.

Chord – any line segment whose endpoints are on the edges of the circle. It need not pass through the center; the longest chord is always the diameter.

Arc – the measure around the edge of the circle from one point to another, measured in either length or degrees, and named by its endpoints.

 

The formulas for circles are on our memorization sheet:

 

The circumference equals twice the radius times pi, or the diameter times pi, i.e., C = 2pr or pd.

The area equals the radius squared times pi, or half of the diameter squared times pi, i.e., A○ = pr2 or p(½d)2.

 

Angles in a circle are used to calculate the measure of arcs. Central angles are formed by two radii, while inscribed angles have their vertices at the edge of the circle. The degree measure of a central angle is equal to the degree measure of the arc, but the degree measure of an inscribed angle is half the measure of the arc. Contrast the two following examples:

          


 

 

Some other terminology for circles:

Concentric circles are circles with the same central point.

Tangents are lines touching a circle at one point and are always perpendicular to the radius.


Note that similarity means having proportional measures: all circles are similar and all squares are similar. However, congruence means having identical measures.

 

Volume and Surface Area p. 267-269

Volume is the three-dimensional measure of the interior of any solid figure or prism, while surface area measures the two-dimensional area of each surface of a solid figure or prism. The formulas are also included on the memorization sheet. Volume is always given in cubic units, and surface area, like regular area, is given in square units.

The most important solid figures are the cube, the rectangular solid, and the cylinder (also known as a right circular cylinder on most exams). I have included other shapes such as the cone, pyramid, sphere, and other prisms just in case.


 



For cubes, volume is equal to one side measure cubed, or V = x3 where x is the measure of one side. The surface area is six times that side measure squared, or SA = 6x2, because there are six sides for any cube. For rectangular solids, the principles are the same but the formulas are slightly different. Volume will be the product of the three dimensions or V = lwh, for length times width times height, and surface area will be the sum of twice the area of each of the three pairs of similar sides, or A = 2lw + 2lh + 2hw. A right circular cylinder has bases which are circular and a lateral side that is perpendicular to the bases. The volume will be the product of the area of the base and the height, or V = pr2h. The lateral surface area is the measure of the area of the rectangle wrapped around the cylinder, which equals the product of the base’s circumference and the height of the cylinder, or LSA = 2prh, while the total surface area of a cylinder is the sum of the area of the two bases and the lateral surface area, or total SA = 2prh + 2pr2.





The book gives most of these formulas in a chart on p. 272-274.