GRE WorkshopQuantitative ReasoningFebruary 13 and 20, 2018

OverviewWelcome and introduction

Tonight: arithmetic and algebra

6-7:15 – arithmetic

7:15 – break

7:30-8:45 – algebra

Time permitting, we’ll start some geometry tonight

Next Tuesday, 2/20: (more) geometry and data analysis

Throughout: think-pair-share as we do problems. Get to know a couple of your neighbors.

I’ll have Brian Eikenhout post my PPT slides for you on BB.

General Test Resources

www.ets.org/gre Test Taker Prepare for the TestPowerPrep II Program (Two full-length practice tests)Math Review (Broken down by topic with practice problems) Instructional Videos on Khan AcademyMath Conventions

GVSU.edu/library Databases Learning Express Library Featured Resources GRE Preparation Two practice tests

number2.com – Practice problems based on category

majortests.com/gre – Practice problems based on type

It is essential that you DO PROBLEMS. You can use the Math Center for support, too.

Quantitative Reasoning

• Includes:

Arithmetic, algebra, geometry, and data analysis

• Excludes:

Trigonometry, calculus, and other college-level math

• Question formats:

Multiple choice – one answer

Multiple choice – one or more answers

Numeric entry – type correct answer in box

Quantitative Comparisons – compare two quantities.

Data Interpretation set – 2 or more sets of questions concerning a display of data.

• You should read and study the “GRE Math Conventions” … it’s 11 pages, and the notation and ideas there are important to refresh

• be sure you spend time exploring www.ets.org/gre and look at various question forms

Quantitative Comparisons - styleIn quantitative comparison questions you will be asked to compare two quantities and determine the relationship between them.

You should familiarize yourself with the answer choices.

A Quantity A is greater.

B Quantity B is greater.

C The two quantities are equal.

D The relationship cannot be determined from the information given.

Four main topic areasArithmetic

Algebra

Geometry

Data Analysis

1. Arithmetic – working with numbers• Integers (whole numbers, positive and negative)

• Fractions, ratios, and percentages

• Basic exponents and Roots

• Decimals

• Real Numbers

Column A Column B

64.3 62.3

A Quantity A is greater.

B Quantity B is greater.

C The two quantities are equal.

D The relationship cannot be determined from the information given.

Quantitative Comparisons -suggestions• Make one quantity look like the other.

• Compare quantities piece by piece.

• Do the same thing to both quantities.

• Don’t forget to consider other possibilities for variables (Think – negative)

• Think about simpler examples that are comparable

• Try to demonstrate two different relationships between the quantities (D).

Column A Column B

13)14(14)13(

Hint: If the two quantities are completely numeric the answer cannot be D. (That is, there’s

no way the answer is “can’t determine” in an all-number problem like this one.)

Column A Column B

)112)(61( )111)(62(

Hint: Try to make the two expression look alike.

Column A Column B

53 x 82 x

Hints: try a few small values of x; think about the graphs of the linear functions, too

Can also explore making the two expressions have overlapping parts

1x

Order of Operations• Parentheses

• Exponentiation (includes square roots)

• Multiplication and Division (from left to right)

• Addition and Subtraction (from left to right)

• Order of operations are recognized by the built-in calculator.

https://www.ets.org/gre/revised_general/prepare/quantitative_reasoning/calculator/

Column A Column B

55510 20

Column A Column B

0005.0

025.050

Column A Column B

6

1

5

1 4.0

Common Fractions

1

20= 0.05

1

10= 0.1

1

8= 0.125

1

5= 0.2

1

4= 0.25

1

3= 0.3333…

1

2= 0.5

1

100= 0.01

Using the Calculator

• Consider options for mental arithmetic or estimation first as most questions do not require difficult computations.

• Use the calculator for more complicated computations for example products of larger numbers, long division or evaluating square roots

• Entry may be done with the keyboard or mouse

• Has four arithmetic functions, square root, and parentheses

• The Transfer Display button will transfer a number to a Numeric Entry question with a single answer box

• Follows order of operations

• You can download a sample version to experiment with at http://calc.greatestprep.com/

http://calc.greatestprep.com/

Numeric Entry

Make sure you are answering the question that is being asked.

Enter your answer as an integer or a decimal if there is a single answer box

Enter it as a fraction if there are two separate boxes—one for the numerator and one for the denominator.

For a single answer box, a number can be transferred to the box from the on-screen calculator

Enter the exact answer unless the question requires you to round your answer and be sure to round to the required degree of accuracy.

Determine if your answer is reasonable using estimation.

Numeric Entry (single box)

• Directions: Enter your answer as an integer or a decimal if there is a single answer box

• To enter an integer or a decimal, either type the number in the answer box using the keyboard or use the Transfer Display button on the calculator.

• First, click on the answer box — a cursor will appear in the box — and then type the number.

• To erase a number, use the Backspace key.

• For a negative sign, type a hyphen. For a decimal point, type a period.

• To remove a negative sign, type the hyphen again and it will disappear; the number will remain.

• The Transfer Display button on the calculator will transfer the calculator display to the answer box.

• Equivalent forms of the correct answer, such as 2.5 and 2.50, are all correct.

• Enter the exact answer unless the question asks you to round your answer.

Numeric Entry (two boxes)

• Directions: Enter your answer as a fraction if there is are two answer boxes

• To enter a fraction, type the numerator and the denominator in the respective boxes using the keyboard.

• For a negative sign, type a hyphen; to remove it, type the hyphen again. A decimal point cannot be used in a fraction.

• The Transfer Display button on the calculator cannot be used for a fraction.

• Fractions do not need to be reduced to lowest terms, though you may need to reduce your fraction to fit in the boxes.

A retailer marks up the purchase price of an item 20% to

determine the selling price. After a while the item is discounted

10% of the selling price. Determine the profit as percentage of

purchase price.

%

A committee of 20 people originally consisted of three times as

many men as women. If 20% of the men are replaced by

women what is the ratio of women to men on the new

committee expressed as a fraction?

Percentagespart = percent x whole

What is 8 percent of 80?

32 is what percent of 40?

12 is 2 percent of what number?

PercentagesPercent increase = amount of increase x 100%

original number

A company had sales in January of $3,000.

Their sales in February were $3,200. If they

expect sales to increase by the same

percentage, what should sales be in March?

Mary earns 20% more than Bill. Rachel earns

10% less than Bill. If Bill earns $120, how

much more does Mary earn than Rachel?

Multiple Choice (Multiple Answer)• The number of choices will vary with the question.

• Pay attention to the number of answers expected.

Some questions will specify the number of choices

Questions that do not specify the number of solutions can be from 1 to all answers selected.

• Credit is given only for choosing all of the correct answer and no extras.

• Consider limiting values to eliminate answer choices.

• Look for patterns in the correct answer choices.

Integers – whole numbers, positive & negative

• Factors/Divisors (include negatives)

• Multiples

• Least common multiple

• Greatest common divisor/factor

• Even Integers

• Odd Integers

• Prime Numbers

• Prime Factorization

• Composite Numbers

Which of the following integers are multiples of

both 2 and 3? Indicate all such integers.

(a) 8

(b) 9

(c) 12

(d) 18

(e) 21

(f) 36

Given that a is an even number and b is odd number

determine which of the following values are even. Select all

that are even.

a) 𝑎 + 𝑏b) 𝑎 − 𝑏c) 𝑎𝑏d) 𝑎𝑏

e) 𝑏𝑎

f) 3𝑎 + 2𝑏

Which of the following could be the units digit of 23 ,where n is a

positive integer? Indicate all such integers.

(a) 0

(b) 1

(c) 2

(d) 3

(e) 4

(f) 5

(g) 6

(h) 7

(i) 8

(j) 9

n

P and Q are prime numbers, with P < Q. R is the smallest prime

number greater than P and S is the smallest prime number

greater than Q.

Column A

P – R

Column B

Q – S

Fractions• Adding & subtracting fractions

• Multiplying fractions

• Dividing fractions

• Mixed number fractions

• Positive and Negative Exponents

• Properties of Exponents

• Roots

Exponents

Ratio• The ratio of one quantity to another is a way to express their

relative sizes.

• Notations:

a to b

a:b

𝑎

𝑏

Arithmetic Exercises

p 13-14

You should do all of these on your own.

We’ll focus on #4, 5, 14, 15

Break

2. Algebra – working with unknown quantities (variables)

• Operations with Algebraic Expressions

• Rules of Exponents

• Solving Linear Equations

• Solving Quadratic Equations

• Solving Linear Inequalities

• Functions

• Applications

• Coordinate Geometry

• Graphs of Functions

Working with algebraic expressions• Permitted Operations:

Addition

Subtraction

Multiplication by positive numbers

Division by positive numbers

• Avoid (or use carefully) the following operations:

Squaring both quantities

Only use when quantities are known to be positive

Multiplication/division by negative numbers

Switches the statement of an inequality

Rules of ExponentsProperties of Exponents:

If m and n are counting numbers, then

Understanding negative exponents and zero as an exponent:

If and n is a counting number, then

m n m nb b b

mm n

n

bb

b

n n nbc b c

n n

n

b b

c c

n

m mnb b

0b

0 1b 1nn

bb

1 n

nb

b

Solve the linear equation for x.

3𝑥 + 2 = 4 𝑥 − 1 + 5

Solve the given system find the value of x + y

3x + 2y = 2

5x – y = 25

Given the system, find the ratio of xto y:

3x + 2y = 19

x = 3y – 12

Quadratic Equations

Two options:

1. Factor, and use the zero product property

2. Use the quadratic formula

Solve the following equations:

2 3 10 0x x

2 9 0x

22 15 0x x

Ben can walk at a speed of 6 mph and bike at a speed of 15 mph.

How many miles can he travel in four hours without biking more

than three times as long as he walks?

(A) 66

(B) 33

(C) 51

(D) 60

(E) 58

Ben can walk at a speed of 6 mph and bike at a speed of 15 mph.

How many miles can he travel in four hours without biking more

than three times as long as he walks?

(A) 66

(B) 33

(C) 51

(D) 60

(E) 58

If it takes 3 days for 10 workers to finish building one

house, how long will it take 15 workers to finish 4

houses?

(A) 15

(B) 10

(C) 8

(D) 6

(E) 4

In applied problems, we often introduce a variable to represent

the quantity we are seeking.

How many ounces of pure hydrochloric acid should be added to

20 ounces of a 25% solution of hydrochloric acid to obtain a 40%

solution of hydrochloric acid.

(A) 5

(B) 10

(C) 15

(D) 18

(E) 25

At a concession stand candy can be purchased for $0.25

each and bags of popcorn for $0.50 each. If 18 items

were purchased for $6, how many bags of popcorn were

purchased.

(A) 0

(B) 4

(C) 6

(D) 10

(E) 12

Use the definition

to evaluate the following expressions

( )( )a b a b a b

5 4 2 2

Use the definition

to evaluate the following expressions

2 2a ba b

a b

5 4 2 2

Coordinate Geometry• Graphs will be drawn to scale (but pay attention to scale).

• Terms to know:

x-axis

y-axis

Origin

Quadrants

Coordinates

Intercepts

• Reflections

Reflection about x-axis (symmetric about the x-axis)

Reflection about y-axis (symmetric about the y-axis)

Reflection about the origin (symmetric about the origin)

Coordinate Geometry• Equations:

• Lines y = mx + b

Parallel Lines

Perpendicular Lines

• Parabolas 𝑦 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐

Vertex

Line of Symmetry

• Circles (𝑥 − ℎ)2+(𝑦 − 𝑘)2= 𝑟2

Center

Radius

• Piecewise-defined functions

Determine which of the following lines intersect the line 𝑦 = −1

3𝑥 + 2 in the first quadrant.

Select all that apply.

(A) 𝑦 = −2

3𝑥 − 1

(B) 𝑦 = −1

3𝑥 + 1

(C) 𝑦 = 𝑥 − 1

(D) 𝑦 =1

3𝑥 + 1

(E) 𝑦 =2

3𝑥 + 2

Algebra Exercises

p 40-42

You should do all of these on your own.

We’ll focus on part (a) in each of #1-8 and

try to work through several examples

from 11-21.