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The Ultimate SAT Math

Strategies Guide

Created by Sherman Snyder

Fox Chapel Tutoring

Pittsburgh, PA

412-352-6596

Go to Success Model

Math SAT Success Model

Math Definitions & Concepts

Test Taking Tips

Math Strategies

Student Success

Return to Introduction

SAT Test Taking Tips

Two Rules

Back to Success Model

Math Definitions and ConceptsThe Top 25


Math Strategies

Math Topics


Absolute ValueBack to

Top 25

Definition: How far a number is from zero. An alternative

definition is the numeric value of a quantity without regards to its

sign. The absolute value of a number is always positive or zero.

The symbol “|….|” is used to denote absolute value of a quantity.

Applications:

• Values: |6.5| = 6.5; |- 3.2| = 3.2; |0| = 0

• Solving equations: |x - 5| = 3

• Solving inequalities: See math strategy

• Graphs of functions: See math strategy

Arc

Definition: An unbroken part of the circumference of a circle. An

arc can be measured by its length or by its central angle. When

measured by its central angle, the arc has the same degree

measure as the central angle.

arccentral

angle

Applications:

• Finding the length of an arc

• Finding area of a sector

• Finding internal angles of an isosceles triangles with one vertex

at the central angle

isosceles

triangle

Back to

Top 25

Average (arithmetic mean)

Applications:

• Usually involves values expressed in terms of variables, not

numerical values. See math strategy

• Note: You will never be asked to calculate the mean of a list of

numbers. Such questions always ask for the median, not the

mean of the list.

Definition: The most commonly used type of average on the SAT

sum of values

number of valuesaverage (arithmetic mean) =

Back to

Top 25

Average Speed

Applications:

• Word problems that involve the motion of an object

• Caution: If a question involves the motion of an object at two

different rates and asks for the overall average speed of the

object, the correct answer will be the average of the two given

rates if and only if each segment of motion occurs over the same

time period. If the motion of each segment occurs over the same

distance, the above definition of average speed must be applied.

Definition: The total distance traveled by an object divided by the

total time traveled

total distance traveled

total timeAverage speed =

Back to

Top 25

BisectorBack to Math

Definitions

Definition: A line segment, line, or plane that divides a geometric

figure into two congruent halves.

Applications:

• Most common application involves angle bisectors.

angle

bisector

Central Angle

Definition: An angle whose vertex is at the center of a circle. The

measure of a central angle is also the measure of the arc that the

angle encloses.

Applications:

• See

Applications:

• Finding the length of an arc

• Finding area of a sector

• Finding internal angles of an isosceles triangles with one vertex

at the central angle

• Note: You will never be asked questions about inscribed angles

isosceles

triangle

central

angle

inscribed

angle

Back to

Top 25

Diagonal

Definition: A line segment joining two non-consecutive vertices of

a polygon. In the figure, the three dashed lines are diagonals

Applications:

• Finding the number of diagonals in a polygon of “n” sides (see

example)

• Finding the number of possible triangles formed by all diagonals

from one vertex of the polygon

Back to

Top 25

Digit

Definition: The set of integers from “0” to “9” in the decimal

system that are used to form numbers.

The digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9

Note: The number zero is contained in the set of digits

Applications:

• Formation of integers

Back to

Top 25

Directly Proportional

Definition: A relationship between two variables in which the ratio

of the value of the dependent variable to the value of the

independent variable is a constant. If y is proportional to x, then

y/x is a constant. This can be written in equation form as y =kx

where k is a proportionality constant.

Applications:

• See math strategy

• Proportions, ratios, and probability are closely related in many

applications. See math strategy

Back to

Top 25

Distance Between Points

Applications:

• Any question that contains the words distance, points, and

number line requires the application of the above definition.

Definition: The distance between two points on a number line is

the absolute value of the difference between the two points. The

order of subtraction does not affect the result.

3-4 0

Distance = |3 - (-4)| = |7| = 7

or

Distance = |-4 - 3| = |-7| = 7

Distance = 7

Back to

Top 25

Divisor

Definition:

• A number or quantity to be divided into another number or

quantity (the dividend)

• A number that is a factor of another number

Applications:

• Questions involving long division and remainders. See math

strategy

• For some questions the word “divisor” can be replaced with the

word “factor”.

Back to

Top 25

Factor

Applications:


Definition: A factor of a number or expression, N, is a number or

expression that can be multiplied by another number or expression

to get N. When a number or expression is written as a product of its

factors, it is said to be in factored form.

Example: (2)(4)(15) = 120 Example: (x + 1)(x + 2) = x2 + 3x +2

Factors Factors

Back to

Top 25

Function

Definition: A special relationship between two quantities in which

one quantity, the argument of the function, also known as the

input, is associated with a unique value of the other quantity, the

value of the function, also known as the output. A function

assigns exactly one output to each input. The notation f(x) is said

“F of X”. An example of a function is f(x) = 2x, a function which

associates with every number twice as large.

Applications:


Back to

Top 25

Inversely Proportional

Applications:

• Questions that begin with the words “If “y” is inversely

proportional to “x” and…”

• Questions that contain a table of “x” and “y” values that have a

constant product.

Definition: The product of the value of the independent variable

and the value of the dependent variable is constant. Can be

written as k = xy, or y = k/x. The relationship between “x” and “y”

can be expressed graphically as

Back to

Top 25

Median

Applications:


Definition: The middle number in a sorted list of numbers. Half

the numbers are less and half the numbers are greater. If the

sorted list contains an even number of values, the median is the

average of the two numbers in the middle of the list.

Example: 2, 3, 3, 6, 8, 9, 9

Example: 2, 3, 3, 3, 5, 6, 7, 9

Median = 4

Back to

Top 25

Multiple

Definition: The product of an integer by an integer. For example,

the multiples of 4 are 4, 8, 12, 16, 20, …..For any positive integer

there are an infinite number of multiples.

Applications:

• Finding the value of a term in a repeating sequence.

• Variety of questions that require understanding of the multiple

definition

Back to

Top 25

Percent

Definition: A ratio that compares a number to 100. Percent

means “out of one hundred”. For example:

10% means 10/100, 750% means 750/100, “k%” means k/100

Applications:


Back to

Top 25

Percent Change

Applications:


Definition: The amount of change in a quantity divided by the

original amount of the quantity times 100%.

% change =amount of change

original amountx 100%

Back to

Top 25

Probability

Definition: The likelihood of the occurrence of an event. The

probability of an event is a number between 0 and 1, inclusive. If

an event is certain, it has a probability of 1. If an event is

impossible, it has a probability of 0.

Applications:

• Elementary probability

• Probability of independent/dependent events

• Geometric probability

Back to

Top 25

Proportional

Applications:

• Proportions, ratios, and probability are closely related in many

applications. See math strategy

Definition: An equation showing that two ratios are equal

• Two variables are proportional if their ratio is constant.

If a is proportional to b, then a/b is a constant.

Can be written in equation form as a = kb

where k is a proportionality constant.

Back to

Top 25

Rate

Definition: A rate is a ratio that compares two quantities

measured with different units. For example, the speed of a car is

a rate that compares distance and time.

Applications:

• When the word “rate” is contained in a question, create a ratio of

the two given quantities identified in the question. Such questions

usually vary the value of one of the given quantities and ask for

the value of the second quantity that will maintain the given rate.

To solve efficiently, create a proportion of the two ratios and solve

for the unknown quantity. See math strategy

Back to

Top 25

Sector

Applications:

• Area of sector

• Length of arc AB

Definition: A sector of a circle is the portion of a circle bounded

by two radii and their intersected arc.

Sector

Back to

Top 25

Sequences

Definition: A sequence is an ordered set of numbers. Four types

of sequences on the SAT.

• Arithmetic sequence: A sequence of numbers that has a

common difference between each number. 3,7,11,15,19,23

• Geometric sequence: A sequence of numbers that has a

common ratio between each number. 3, 6,12, 24, 48, 96

• Repeating sequence: A sequence of numbers that form a

repeating pattern. See math strategy

• “Other” sequence: A sequence that does not fit any of the above

three categories. A formula is usually provided that can be used to

determine each value of the sequence.

Applications:

• Any or all of the above types of sequences will be found on

every SAT. However, the sequence names used above will never

be found in any SAT questions. Instead, a description of the

sequence is used. Bottom line….know the sequence definitions.

Back to

Top 25

Similar Triangles

Definition: Two triangles are similar if and only if all pairs of

corresponding angles are congruent and all pairs of

corresponding sides are proportional.

76

4

3.5 3

2

Applications: (See figure at right)

• When a smaller triangle is completely

inside a larger triangle such that

corresponding angles are congruent or one

pair of corresponding sides are parallel, the

two triangles are similar. Congruent

angles

Back to

Top 25

Slope of a Line

Applications:

• Slope of a line when two points are known

• Identification of the “x” or “y” value of a point when the coordinates

of a second point are known and the slope of the line is given.

• Slope of a line parallel or perpendicular to another line

• Linear relationships or functions that ask for the change in the

value of a quantity as the independent variable is changed.

Slope = ∆y

∆x = = y2 - y1

x2 - x1

rise

run

Definition: Slope is a measure of the tilt or steepness of a line.

Slope is calculated as the vertical distance divided by the

horizontal distance between two points.

Slope is also a measure of the amount that the dependent variable

(often “y”) changes as the independent variable (often “x”)

changes by one unit.

y

x

Back to

Top 25

Venn Diagram

Definition: A diagram (usually made of circles) that shows all

possible relations between sets.

Applications:

• Venn diagrams (2 sets): See math strategy

• Venn diagram (3 sets): See math strategy

Back to

Top 25

Zero

Definition: Zero is an even integer (thus it is divisible by 2) that is

neither positive nor negative. As a result, zero is the smallest non-

negative number. Zero is also the smallest of 10 digits. Zero is a

whole number, a rational number, and a real number. Division by

zero results in an undefined value.

Applications:

• Questions that ask for the number of integers, the number of

even integers, or the number of positive integers that are

contained in a solution set.

• Questions that ask for a specific value of “x” for which a function

is not defined.

Back to

Top 25

Questions You Can Count On

• A figure that is rotated, flipped, reflected, taken apart, unfolded

is usually either question 3, 4, or 5 in the 20 multiple choice

section of math.

• Parallel lines cut by one or more transversals: See strategy

• Tangent lines to a circle: See strategy

• The “If…..then what is the value?” question: See strategy

• Equation of a line or slope of a line perpendicular to another

line: See strategy

• Formation of even/odd numbers: See strategy

Learn More

Back to Tips


• Average (arithmetic mean) questions: See strategies

• Sequence questions

• Two types of definition questions

• Substitution into an expression: See strategy

• Words in quotations: See strategy

• Rate and or ratio questions: See strategy

• Rules of exponents: See strategy

• System of equations: See strategy

• Questions that contain an inequality: See strategy

Previous Learn More

Back to Tips


• Area of irregular shapes and area of sectors

• Counting problems including the number of ways to pair

objects: See strategy

• Geometric probability: See strategy

• Probability of events occurring: See strategy

• Use of function notation and function translations or

reflections: See strategy

• Percentage questions: See strategies

• Long division and remainder questions: See strategies

Previous Learn More

Back to Tips


• Overlap of data sets (Venn diagram applications): See strategy

• Patterns of number or shapes/objects: See strategy

• Similar shapes (usually triangles): See strategy

• Directly or indirectly proportion questions

• Absolute value equation or inequality: See strategy

• Median of a list of numbers: See strategy

• Creation of a cost equation for the purchase of an item or

service

Previous

Back to Tips

The Two Test Taking Rules

• Keep it simple. View each question through the lens of

simplicity, not the lens of complexity. The math portion of the

SAT is not a two headed monster. With good reasoning skills

and an understanding of basic math definitions and content,

every question can be solved with little difficulty. Having this

mindset will often lead to increased confidence.

• Answer the question. Make sure you answer the question

being asked, not the question being assumed. Before choosing

an answer, read the last half of the last sentence. If the

questions asks for the cost of three pounds of bananas, do not

choose the per pound cost. If a question asks for the value of

the “y” variable, do not choose the value of the “x” variable. If

the question asks for the value of the largest of three

consecutive integers, do not choose the smallest integer. If the

questions asks for the value of “4x”, do not choose the value of

“x”. Answer the question being asked!

Back to Tips

The Three Questions

• What piece of information do I need? This is a crucial question

to ask. SAT questions are asked in ways that are more abstract

than a typical math question. The answer to this question will

ensure you are heading down the correct path toward the answer.

• What do I do with the information? This is the math step that

usually requires using a formula.

• What is the strategy for finding this information? This is where

most students have difficulty. A good strategy is usually needed at

this point. If none can be identified, students will go to Plan B

(substitution of answers, elimination and guess), or skip the

question.

Back to Tips

Test Day Tips

• Replace calculator batteries. Replace the batteries in your

calculator (usually four AAA batteries) with fresh, out of the package

batteries. Do not replace with the batteries that are rolling around in

your desk drawer…..the ones that should have been tossed out the

last time you replaced batteries.

• Take a watch to the testing center. You do not have control over

the amount of time for each test section. However, with a watch, you

are in a position to control the use of your time. If the testing room

has a clock on the wall, your watch may not be needed.

• Have your admission ticket and photo ID. This a common sense

issue.

• Prepare a survival kit. In a lunch bag, pack bottled water and

many snacks. Include one chocolate bar to be consumed between

sections seven and eight of the ten section test. Fatigue will be high

at this point during the test. Eat the chocolate bar for a burst of

energy and tough it out until the end.

Back to Tips

Learn More

Test Day Tips

• Take plenty of No. 2 wood pencils. Mechanical pencils are not

permitted apparently due to cheating issues.

• Proctors are not your friend. The test proctor is there to make a

few bucks on a Saturday morning. They are not there to help you in

anyway. They are prone to making mistakes with the timing of

sections, have been observed talking on the phone causing noise

issues, and often have a nasty disposition. They are not your

friend!

• Four math sections…do not panic. The SAT is comprised of ten

sections: three writing, three reading, three math, and one

“experimental section”. The experimental section will be an

additional writing, reading, or math section that will not be part of

your final score. The experimental section is not identified. Do your

best on all sections!

• Bubble in the student-generated response answers: Some

students forget to bubble the answers.

Back to Tips

Previous

What Study Guides Will Never Reveal

• Be prepared to reason: Math content is plentiful in study guides, however

math strategies are virtually nonexistent. To be successful on the SAT,

reasoning skills are as important as having basic math content knowledge and

basic computational skills.

• Answer the easy questions first. All questions are equally weighted. Do not

try the hard questions first. Attempt the questions in the order they are presented.

• Basic calculations should be done without a calculator: Calculators are

absolutely, positively not needed for the SAT, however, you should absolutely,

positively use one…..sparingly. Avoid using the calculator for basic addition

and multiplication operations, especially those involving negative numbers.

Student calculator input errors often lead to costly mistakes that are absolutely

avoidable.

• Complex computational skills not required: The SAT is a test of quantitative

reasoning skills, not computational skills. With strong reasoning ability, only

basic calculations are needed to answer most questions.

Back to Tips

Learn More


• No need to memorize formulas: There is no need to memorize

formulas….they are all provided. If a formula is needed and is not contained on

the list of formulas at the beginning of each math section, then the formula will

be provided in the text of the question. The bottom line is this….if you believe a

formula is needed to solve a specific problem and the formula is not provided,

look for an alternative way (and often more efficient way) to solve the problem.

• Never enter a value for “pi” into your calculator: Entering “pi’ into your

calculator will often result in a close approximation to the correct answer, not the

exact answer. Solve questions in terms of “pi”, especially the student-produced

response questions that require exact answers.

• Cross multiplication is your best friend: The solution to many questions is

made easier by using cross multiplication. Look for opportunities to use it.

• Need to know math definitions: Definitions are not provided. You are

expected to know all math definitions. Examples include slope of a line, average

(arithmetic mean), percent, percent change, average speed, etc.

Back to Tips

Learn MorePrevious


•The words “arithmetic” and “geometric” sequence are not used: Students

are not expected to know the definition of these sequences, as suggested by

study guides. Instead of using the words “arithmetic” and “geometric”

sequences, SAT questions describe the characteristics of these sequences.

• Do not need to use permutations or combinations: Although both topics are

discussed in most study guides, you can always use Fundamental Counting

Principles to solve counting problems.

• Inscribed shape questions: When a shape is inscribed inside a second shape,

their centers always coincide. This is often useful when developing a strategy to

solve this class of questions.

• Never asked to calculate the average of a list of numbers: When a list of

values is provided, analysis of the median (sometimes mode) is always asked. Do

not be fooled into making a lengthy calculation of the mean of a list of

numbers….it is never asked for.

• Never asked to find the domain of a function: This topic is discussed in study

guides, however, it is not found on the SAT reasoning test. More likely to find this

topic on the SAT math subject test.

Back to Tips

Previous

The Usual Study Guide Tips

• Use the figure when figuring. All figures are drawn to scale unless

stated otherwise. Use this to your advantage. If there is a note stating

the figure is not drawn to scale, you must stick to the facts when

drawing conclusions about the answer.

• Student produced response answers must be non-negative

rational numbers. All non-negative integers (including zero) and all

fractions are acceptable answers.

• Guess on student generated response questions. No penalty is

given for missing a student produced response question. If the

answer is not known, take a guess.

• To guess or not to guess. There is a ¼ point penalty for each

missed multiple choice question. The conventional wisdom is to

guess if one answer choice can be eliminated. My recommendation

is to guess if two of five choices can be eliminated.

Back to Tips

Math Strategies

Table of Contents

Number and OperationsOrdering of Negative Numbers


Venn Diagrams (2 sets)


Ratios and their Multiples

Ratios, Proportion, Probability

Rate

Counting Problems

The Pairing Strategy

Long Division and Remainders

Percent Change

Dealing With Percentages

Repeating Sequences

Consecutive Integers

Even/Odd Integer Creation

AlgebraUsing New Definitions: Type 1

Using New Definitions: Type 2

Solving Simple Inequalities

Equivalent Strategy

System of Equations

Matching Game

Factoring Strategy

Word problems

Basic Rules of Exponents

Additional Rules of Exponents

Absolute Value Inequalities

Creation of Math Statements

Parabolas

Single Term Denominators

Making Connections

Geometry and MeasurementDividing Irregular Shapes

Line Segment Length in Solids

Putting Shapes Together

3-4-5 Triangle

30-60-90 Triangle

45-45-90 Triangle

Distance Between Two Points

Midpoint Determination in x-y Coordinate

Midpoint Determination on Number Line

Exterior Angle of a Triangle

Perpendicular Lines

Interval Spacing - Number Line

Triangle Side Lengths

Similar Triangle Properties

The Slippery Slope

Parallel Lines and Transversals

Tangent Line to a Circle

Data Analysis, Statistics,

and ProbabilityAverage (Arithmetic Mean)

Median of Large Lists

Elementary Probability

Probability of Independent Events

Geometric Probability

The Unit Cell

FunctionsUsing Function Notation

Reflections - x axis

Reflections - y axis

Reflections - Absolute Value

Translations - Horizontal Shift

Translations - Vertical Shift

Translations - Vertical Stretch

Translations - Vertical Shrink

Math Strategies

Table of Contents

Lesson 1

Algebra Strategies




Equivalent Strategy

System of Equations

Matching Game

Factoring Strategy

Word problems





Parabolas

Single Term Denominators

Making Connections

Back to Success Model Back to Math Topics

Math Strategies

Table of Contents

Lesson 2

Geometry and Measurement

Strategies

Dividing Irregular Shapes

Line Segment Length in Solids


3-4-5 Triangle

30-60-90 Triangle

45-45-90 Triangle


Midpoint Determination in x-y Coordinate

Midpoint Determination on Number Line


Perpendicular Lines

Interval Spacing - Number Line



The Slippery Slope


Tangent Line to a Circle


Lesson 3

Number and Operations

Strategies

Ordering of Negative Numbers




Ratios and their Multiples

Ratios, Proportion, Probability

Rate

Counting Problems



Percent Change


Repeating Sequences


Even/Odd Integer Creation

Math Strategies

Table of Contents


Math Strategies

Table of Contents

Lesson 4

Functions Strategy

Using Function Notation

Reflections - x axis

Reflections - y axis

Reflections - Absolute Value

Translations - Horizontal Shift

Translations - Vertical Shift

Translations - Vertical Stretch

Translations - Vertical Shrink


Math Strategies

Table of Contents

Lesson 5

Data Analysis, Statistics,

and Probability Strategies

Average (Arithmetic Mean)





The Unit Cell

It’s Absolutely Easy!


All The Equations You Need!

Strategy: Great news! The equations on

this page are the only ones you need to

be successful on the SAT.

Return to Table of Contents See example of strategy

Reasoning: If the equation is not on this

page, you do not need to use it. Hooray!

Examples include quadratic formula,

combinations, permutations, equation of

a line or circle, surface area and volume

of a cone, pyramid, or sphere. If one of

these equations is needed to solve a

problem, it will be provided.

Application: There are plenty of

questions on the SAT for which these

formulas are used. To save time when

taking the SAT, it is recommended that

you memorize these basic formulas.

Area of rectangle = lw

Area of Circle = π r2

Circumference of Circle = 2π r

Area of triangle = ½ bh

Volume of rectangular solid = lwh

Volume of cylinder = π r2h

Pythagorean theorem c2 = a2 + b2

30 - 60 - 90 Triangle Click for details

45 - 45 - 90 Triangle Click for details


Example 1

Question: Under construction

Return to Table of Contents Return to strategy page See another example of strategy

What essential information is needed?

What is the strategy for identifying

essential information?:

Solution Steps


Example 2


Return to Table of Contents Return to strategy page Return to previous example



essential information?

Solution Steps

The Important Definitions You

Need!

Strategy: These definitions are

extremely important for you to memorize.

Unlike formulas, definitions are not

provided on the SAT.


Reasoning: Students often consider

these definitions to be formulas. They

are not formulas! Formulas are derived

in geometry using proofs.

Application: These definitions are

extremely valuable resources when

solving a variety of problems on the SAT.

The definition of empty set, integer,

positive and negative numbers, even

and odd numbers, digits, and

percentages are also important to know.

Average speed = total distance traveled

total time

Average (arithmetic mean) = sum of values

number of values

Click for more details

Percent change = amount of change

original amountX 100%


Slope = ∆y

∆x = =

y2 - y1

x2 - x1

rise

run


Distance between two points = 2 1x x

The Important Definitions

Example 1






Solution Steps

The Important Definitions

Example 2






Solution Steps


Strategy: Visualize the position of a

single negative value or a list of negative

values as they would appear on a

number line


Reasoning: As you move left on a real

number line, the values get smaller. This

property is especially useful when

ranking negative numbers.

Application: Any question that requires

you to rank the values of negative values

from smallest to largest or vice-versa.

Also useful when assigning values to

positions on a number line.

-7 -4 -1-10

A B C D E

A

-⅜ -¼ -⅛-½

B C D E

On the number line shown below,

which letter best represents the

location of the value -2/5?

Click to see answer

On the number line shown below,

which letter best represents the

location of the value -5/2?

Click to see answer


Example 1

Question: If a < 0, which of the four

numbers is the greatest?

A) a + 2 B) 2a + 2

C) 4a + 2 D) 8a + 2

E) It cannot be determined from the

information given




Solution Steps

Return to strategy page See another example of strategyReturn to Table of Contents


Example 2






Solution Steps


Strategy: Often given values for “x1” and

“y1” and asked to find value for “x2” when

given “y2”. Use the following proportion:

Reasoning: The ratio of y:x is constant

for any two points. Click to see

properties of directly proportional

Application: Any relationship that can

be expressed as ratios. In addition to

points on a line, examples include

amount of ingredients in recipes, number

of marble colors in a container, and

segment lengths of a number line.

y

x

“y” is directly proportional to “x”

(x2, y2)

Properties of a directly proportional include

the following:

1) Graph of “y” versus “x” is linear and passes

through the origin. Has the form of y = kx.

2) Slope of line is the ratio of y:x for any point

on the line

3) Slope of line is equal to proportionality

constant “k”.

(x1, y1)constant

2

2

1

1 x

y

x

y

Back to

Definition

y = kx

kxy

constantx

yk



Example 1

Question: A machine can produce 80

computer hard drives in 2 hours. At this

rate, how many computer hard drives

can the machine produce in 6.5 hours?

Return to strategy page See another example of strategy


Rate of computer hard drives per hour.


essential information?: Ratio the

number of computer hard drives to the

number of hours required to produce

them. With this ratio, create a linear

proportion to answer the question.

Solution Steps

1) Create a ratio representing rate

of computer hard drive production:

80 hard drives

2 hours

2) Create a linear proportion to

solve for number of hard drives

produced in 6.5 hours:

80 hard drives

2 hours

“n” hard drives

6.5 hours=

3) Solve for ‘n”:2n = (80)(6.5)

n = 260

Return to Table of Contents


Example 2

Question: If y varies directly as x, and if

y = 10 when x = n and y = 15 when x = n

+ 5, what is the value of n?

Return to previous example


A link between y and x that can be used

to solve for n.


essential information? The ratio y/x is

a constant. Create a proportion and

solve for n.

Solution Steps

1) Create a linear proportion to

solve for n. 10

n

15

n + 5=

2) Solve for n using cross

multiplication:

15n = 10(n + 5)

15n = 10n + 50

5n = 50

n = 10

Return to strategy pageReturn to Table of Contents

Venn Diagram (2 sets)

Strategy: To determine the overlap

(intersection) of members in two groups

(sets), use the following approach:

Step 1: add the total number of

members from both groups

Step 2: subtract the sum consisting of

the total number of members in one

group only and both groups from the

number of members in step 1

See example of strategy

Reasoning: By eliminating the overlap of

members, the sum of three numbers in

the Venn diagram will equal the total

number of members being counted.

Application: Used when members of two

or more groups (sets) have common

members.

18 22 10

Total number of students = 50

Number of students

that study math only:

40 – 22 = 18

Number of students

that study history only:

32 – 22 = 10

Number of students

that study history = 32

Number of students

that study math = 40

Number of students that study

math and history = 22

Step 1

40 + 32 = 72

Step 2

72 – 50 = 22

Math History

18 + 22 + 10 = 50

Back to

Definition

Back to

Frequent

Questions

Return to Table of Contents


Example 1

Question: The Venn diagram to the right

shows the distribution of students who

play football, baseball, or both. If the

ratio of the number of football players to

the number of baseball players is 5:3,

what is the value of n?


Connection between the number of

players in each sport to “n”, the number

of players that participate in both sports.


essential information?:Use the

properties of Venn diagrams and

proportions to find the value of “n”

Solution Steps

Football Baseball

28 14n

1) Create a proportion of the number

of football players to baseball players

n + 28

n + 14

5

3=

2) Solve for “n” using cross

multiplication: 5n + 70 = 3n + 84

2n = 14

n = 7



Example 2

Question: The 350 students at a local

high school take either math, music, or

both. If 225 students take math and 50

take both math and music, how many

students take music?


Connection between the multitude of

given information and the unknown

quantity.


essential information? Use the

properties of Venn diagrams to help

“visualize” the given information.

Solution Steps

Math Music

175 m50

1) Create an appropriate Venn diagram

to help visualize the given information.

2) Find the value of m, the number of

students that take music only

175 + 50 + m = 350 m = 125

3) Find the value of m + 50, the

number of students that take music

m + 50 = 125 + 50 = 175

Return to previous exampleReturn to strategy pageReturn to Table of Contents


Strategy: When analyzing the overlap of

three data sets, it is important to

understand the meaning of each section

of the resulting Venn diagram (see

example)

Reasoning: The interpretation of data in

each section is determined by the rules of

logic

Application: Data sets in which there is

overlap of members of two or more sets.

Applications include student choices of

school classes or sport activities, and

overlapping properties of various real

numbers

3

7

4 5

6 8

9

Football

Soccer

Baseball

The number of athletes that

play all three sports = 3


play two sports only = 16


play one sport only = 23


play two sports. Example:

football and baseball = 10

The number of athletes that play football only

(6), baseball only (8), or soccer only ( 9)

Back to

Definition

See example of strategyReturn to Table of Contents


Example 1






Solution Steps


Example 2






Solution Steps

Ratios and Their Multiples

Strategy: When given the total number

of several different objects and a ratio

that describes their distribution, create

an equation to find the exact number of

each object. (click to see example)

Reasoning: For discrete objects like

marbles, bowling balls, and people, the

total number of each object in the group

must be a multiple of their respective

ratio value.

Application: Questions that ask for the

distribution of angles in a triangle or the

distribution of objects among containers.

A jar contains a total of 30 red,

yellow, and blue marbles. The

number of each marble color in the

jar follows the ratio 3 red: 2 yellow:

1 blue. How many of each color are

there in the jar.?

3x + 2x + x = 30 marbles

6x = 30 marbles

x = 5 blue marbles

2x = 10 yellow marbles

3x = 15 red marbles

Total = 30 marbles



Example 1

Question: The measures of the interior

angles in a triangle are in the ratio 9:4:2.

What is the measure of the largest angle

in the triangle?


The measure of each individual angle.


essential information? Create and

solve an equation using the angle ratios

and the fact that the sum of the interior

angles is 180 degrees in a triangle.

Solution Steps

1) Create equation using ratio values

9x + 4x + 2x = 180 degrees

2) Solve equation for “x”. Multiply by

nine to find measure of largest angle.

9x + 4x + 2x = 180 degrees

15x = 180 degrees

x = 12 degrees

9x = 108 degrees



Example 2

Question: Cookies are distributed within

four separate jars in the ratio of 7:5:3:1.

The total number of cookies contained in

the four jars is 48. How many cookies

are contained in the jar with the greatest

number of cookies?


The number of cookies in each jar.


essential information? Create and

solve an equation using the given ratios

and the fact that the total number of

cookies contained in the four jars is 48.

Solution Steps

1) Create equation using ratio values

7x + 5x + 3x + x = 48 cookies

2) Solve equation for “x”. Multiply by

seven to find measure of largest angle.

7x + 5x + 3x + x = 48 cookies

16x = 48 cookies

x = 3 cookies

7x = 21 cookies


Ratios, Proportions,

Probability Connections

Strategy: When the whole consists of

two parts and the parts are expressed

as a ratio of each other, there are

several connections between ratios,

proportions, and probability that are

useful to solve a variety of problems.

Reasoning: For the example shown to

the right, three out of every four marbles

in the can are blue. To maintain this

ratio, the total number of marbles in the

can must remain a multiple of four. As a

result, the probability of selecting a blue

marble is ¾.

Application: Problems involving lengths

of line segments, rate/time, areas and

perimeters, sizes of angles

The ratio of red

to blue marbles

is 1 to 3.

Connection 1: The total number of

marbles in the can must be a multiple

of four marbles (1 + 3 = 4).

Connection 2: The probability of

randomly selecting a blue marble

from the can is ¾.

Connection 3: To maintain this ratio

when adding to or removing marbles

from the can, a proportion should be

used.

Back to

Definition

Back to

Frequent

Questions


Ratios, Proportions, Probability

Example 1

Question: During the month of

February (28 days) the city of Pittsburgh

had two days on which it snowed for

every five days on which it did not snow.

For the month of February, the number

of days on which it did not snow was

how much greater than the number of

days on which it snowed?


Need to determine the number of days in

which it snowed and the number of days

in which it did not snow.


essential information?: Use proportions

to determine essential information.

Solution Steps

1) Set up a proportion using the

following strategy: For every seven

days (2 + 5 = 7) during the month of

February, it snowed 2 days. Find the

number of days it snowed.

2

7

n

28= n = 8 days of snow

2) Find the number of days in which

it did not snow.

28 days - 8 days = 20 days

3) Subtract the result of Step 1 from

the result of Step 2

20 days – 8 days = 12 days greater


Ratios, Proportions, Probability

Example 2

Question: The ratio of almonds to

cashews in a mixture is 2:3. How many

pounds of almonds are there in a seven

pound mixture of almonds and cashews.


The number of pounds of almonds

required to maintain proper mixture ratio.


essential information? Use proportions

to determine essential information.

Solution Steps

1) Set up ratio of almonds to mixture.

2 pounds almonds + 3 pounds

cashews = 5 pounds mixture

2 pounds of almonds

5 pounds of mixtureRatio:

2) Create proportion to solve problem.

2

5

n pounds of almonds

7 pounds of mixture=

5n = 14Cross

multiply

14

5n =

pounds of

almonds


Rate Strategy

Strategy: For all questions that require

the rate of two quantities to be held

constant, create a proportion to solve for

the new value of one quantity when the

value of a second quantity is changed a

given amount.

Reasoning: A proportion is an equation

stating that two ratios are equivalent.

Application: Any question that requires

the rate to be held constant. Examples of

constant rate include speed of an object,

rate of work, rate of flow of a liquid, rate

of growth of money, etc.

Definition: A rate is a ratio that

compares two quantities measured

with different units. For example, the

speed of a car is a rate that compares

distance and time.

Note: When you read the word

rate in a question, think ratio!


Back to

Definition

Rate Strategy

Example 1




Solution StepsQuestion: The rate of motion of a

baseball is k feet per 2 seconds. In

terms of k, how many seconds will it

take a baseball to move k + 50 feet if

the rate of motion is constant?

A) B) C)

D) E)

1002

k

k

1002

100

2 k

k2

50

k2

100


Rate Strategy

Example 2

Question: The rate of flow of water from

a hose is 4 gallons per 20 seconds. At

this rate, how many gallons of water can

the hose provide in 5 minutes?




Solution Steps


Counting Problems

Strategy: Use “Fundamental Counting

Principles” (FCP) and reasoning to solve

many counting problems that do not

involve pairing of objects. For pairing

problems, see Handshake/ Pairing

strategy.

Reasoning: FCP represent a broad

class of counting principles that include

permutations and combinations. Some

counting problems will have constraints.

Such problems, along with reasoning,

can be solved using these principles.

Application: Any problem asking you to

figure the number of ways to select or

arrange members of a group. Examples

include numbers, letters of the alphabet,

or officers of a club.

Fundamental Counting Principles: If

one event can happen in n ways, and a

second, independent event can

happen in m ways, the total number of

ways in which two events can happen

is n times m.

A restaurant uniform consists of a hat,

shirt, and pants. If a worker has two

hats, four shirts, and three pair of pants

to choose from, how many uniforms can

the worker create?

Step 1: Choice of a

hat, shirt, or pants is

independent of each

other .

Step 2: Multiply the

number of each together

to find the total number of

uniforms.

2 x 4 x 3 = 24 uniforms

Back to

Frequent

Questions


Counting Problems

Example 1

Question: Five individual pictures of the

Jones family consists of the Jones

parents and each of the four Jones

children. The individual pictures are to

be arranged vertically on a living room

wall. How many arrangements of

pictures can be made if the parent

picture must be placed at the top of the

arrangement?


The number of ways the five pictures can

be arranged vertically on the wall.


essential information?: Use

fundamental counting principles.

Solution Steps

1) Determine the number of arrangements

of pictures. Take into account there is a

constraint: the top picture must be the

Jones parents.

2) Multiply each number together to find

the total number of arrangements

Top position → 1 picture to choose

Second position → 4 pictures to choose

Third position → 3 pictures to choose

Fourth position → 2 pictures to choose

Fifth position → 1 picture to choose

1 x 4 x 3 x 2 x 1 = 24 arrangements


Counting Problems

Example 2

Question: A certain restaurant offers ice

cream specials that consist of two

scoops of ice cream and one topping. If

there are four toppings to choose from

and four flavors of ice cream, how many

different ice cream specials can be

created if the two scoops of ice cream

must be different flavors?


A special consists of two groups → the

number of toppings and the number of

ways to pair up four flavors of ice-cream.


essential information? Use

fundamental counting principles to

identify the number specials.

Solution Steps

1) Determine the number of ways to pair

scoops of ice cream if there are four

flavors to choose from.

2) Multiply the number of toppings (4)

and number of pairs of flavors (6) to find

the total number of ice cream specials

4 x 6 = 24 specials

Vanilla StrawberryChocolate Peach

1 2 34 56



Strategy: The total number of ways to

pair “n” objects is equal to ½n(n -1).

Reasoning: For a total of “n” objects,

each object can be paired with “n -1”

other objects. However, each pair is

shared by two objects. Click to see an

example of the total number of

handshakes exchanged by 6 people.

Application: Examples include

determining the total number of games

played in a sport league, or the number

of ways a two scoop ice cream cone can

be created from a known number of

available flavors.

Alternative Solution: Total number of

handshakes can be found by addition of

the number of handshakes exchanged

by each individual person.

5 + 4 + 3 + 2 + 1 + 0 = 15 handshakes

½n(n -1) = ½(6)(5) = 15 total handshakes

shared by a group of 6 people

n = 6

people

n - 1 = 5

handshakes

per person



Example 1

Question: In a baseball league with 6

teams, each team plays exactly 4 games

with each of the other 5 teams in the

league. What is the total number of

games played in the league?


How many games are played between

the eight teams.


essential information?: Find the

number of games played between the 6

teams using the handshake problem

strategy. Multiply the result by 4 to

account for the fact that each team

plays exactly 4 games with each of the

other 5 teams.

Solution Steps

1) Find the number of games played

between the 6 teams

½(6)(5) = 15 individual games

played without repeats

2) Multiply by 4 to account for the fact

that each team plays exactly four

games with each of the other 5 teams

Total number of games played:

15 x 4 = 60 games



Example 2

Question: How many diagonals can be

drawn inside a regular polygon with 6

congruent sides.


The total number of diagonals drawn

from the 6 vertices of the polygon.


essential information? Use the pairing

strategy with modifications. Polygons

have sides that do not require lines

connecting adjacent vertices. To

account for this, multiply the total

number of vertices “n” by “n - 3” rather

than “n - 1”. Total number of diagonals

is ½n(n - 3).

Solution Steps

n = 6

sides n -3 = 3

diagonals

½n(n - 3) = ½(6)(6 - 3) = 9 diagonals

can be drawn in a regular polygon

with 6 sides

Back to

Diagonal

Definition



Strategy: Find a value for the unknown

variable k by adding the given divisor to

the given remainder. Process the value

found for k as specified in the question.

Divide this result by the new divisor to

find the desired remainder. Click to see

a review of long division.

Reasoning: Long division questions

always involve analysis of the

remainder, not the quotient. All long

division questions provide a value for

the divisor and remainder. By choosing

a value of 1 for the quotient, a value for

the dividend (unknown variable k) can

be easily and quickly found.

Application: Any long division question

that expresses the dividend as a variable

rather than a numerical value.

137

1

-07

6

dividend divisor x remainderquotient +=

13 = 7 x 1 + 6

Back to

Frequent

Questions

Example: When the positive integer

k is divided by 7, the remainder is 6.

What is the remainder when k + 8 is

divided by 7 ?


Back to

Divisor

Definition


Example 1

Question: When d is divided by 9, the

remainder is 7. What is the remainder

when d + 4 is divided by 9?


Find a number for d that satisfies the

requirements. Add 4 to d, divide by 9,

and find the remainder.


essential information? Add remainder

to the divisor. This will quickly provide a

possible value for d.

Solution Steps

Find a possible value for d by adding

the remainder to divisor:

d = 7 + 9 = 16

The new remainder is 2

Add d = 16 to 4:

d + 4 = 20

Divide 20 by 9:

20 / 9 = 2 with

remainder 2



Example 2

Question: When n is divided by 7, the

remainder is 5. What is the remainder

when 3n is divided by 7?


Find a number for n that satisfies the

requirements. Multiply n by 3, divide by

7, and find new remainder.


essential information? Add remainder

to the divisor. This will quickly provide a

possible value for n.

Solution Steps

Find a possible value for n by adding

the remainder to divisor:

n = 5 + 7 = 12

Multiply n = 12 by 3:

3n = 36

Divide 36 by 7:

36 / 7 = 5 with

remainder 1

The new remainder is 1



Strategy: When a percentage is quoted

as a number or variable, express the

percentage as a ratio with the

percentage in the numerator and the

number 100 in the denominator.

Reasoning: Percentages are expressed

as a ratio of a number over 100 in

mathematics. This strategy will avoid

issues related to expressing a

percentage as a decimal when the given

percentage is a variable rather than a

numerical value.

Application: Any question that contains

a percentage expressed as a variable.

10 % should be written as

k % should be written as

100

10

100

k

Note: If a question expresses

percentages as a numerical value

only, it is okay to use the decimal

form of a percentage.

Back to

Definition

Back to

Frequent

Questions



Example 1

Question: If k% of 60% of 180 is 54,

what is the value of k?


A mathematical statement is needed that

properly describes the given information

and provides a way to solve for the value

of “k”.


essential information?: Two strategies

are required:

•Creation of Mathematical Statements

•Percentages Strategy

Solution Steps

1) Create a mathematical statement

that properly expresses k%

2) Solve for “k” using algebra

k% should be expressed as k

100

Math statement is:

k

100

60

100x x 180 = 54

k

100

60

100x x 180 = 54

Eliminate

zero’s

k(6)(18) = 5400Multiply

by 100k = 50



Example 2

Question: If the length of a rectangle is

increased 40% and the width is

decreased 40%, how does the new area

compare to the original area?


Rectangle lengths and widths that meet

the percent change requirements.


essential information? Start with

convenient length and width values.

Apply the required percentage changes

to each value. Calculate new rectangle

area and compare to original value.

Solution Steps

1) Choose convenient values for

length and width

2) Apply percentage changes

•Note: A square is a rectangle. Great

shape to use for area calculations

•Convenient original area is 100. Use

length of 10 and width of 10

New length = 10 + 4 = 14

New width = 10 - 4 = 6

3) Calculate new area and compare

New area = (14)(6) = 84

Area is reduced by 16%



Example 3

Question: What is ½ percent of 8?


Need to convert ½ percent into an

appropriate form to answer question.


essential information? Use percentage

strategy. Express percentage as a

fraction over 100 rather than decimal

form.

Solution Steps

1) Express percentage in proper form

2) Determine answer to question

•Recommended form is: ½

100= 1

200

•Multiply recommended form by 8

1

200x 8 =

25

1 1

25= .04

Return to example 1Return to strategy pageReturn to Table of Contents

Percent Change

Strategy: Percent change is defined as

the amount of change in the quantity

divided by the original amount of the

quantity times 100%.

Reasoning: This a well known definition

in mathematics. Mostly used in chemistry

and physics.

Application: Can be used for any

question involving percent increase or

decrease.

Caution: Do not divide amount of

change by the final amount

% change =amount of change

original amountx 100%

Back to

Definition


Percent Change

Example 1

Question: Elliot’s height at the end of

third grade was 48 inches. His height at

the end of sixth grade was 60 inches.

What was the percent change in Elliot’s

height?

a) 12 b) 15 c) 20

d) 25 e) 30


The change in height is essential to

determining percent change.


essential information?: Determine the

change in height from the end of third

grade to the end of sixth grade using

subtraction.

Solution Steps

1) Determine the change in Elliot’s

height

Change in height = height at end of 6th

grade - height at end of 3rd grade

Change in height = 60 inches - 48 inches

Change in height = 12 inches

2) Determine the percent change in

Elliot’s height

Percent change = 12 inches

48 inchesx 100%

Percent change = 25%


Percent Change

Example 2


The change in projected population is

essential to determining percent change.


essential information? Using the

function equation, determine the

population in 1990 and 2005. Subtract

the two values to determine the change

in population.

Solution Steps

Question: For the years 1990 to 2005,

the function above expresses the

projected population of Mathville. What

is the projected percent increase in

population of Mathville from 1990 to

2005?

P(t) = 500t + 25,000

1) Determine the population in 1990

and 2005 using function equation.

P(t) = 500t + 25,000

P(0) = 500(0) + 25,000 = 25,000

P(15) = 500(15) + 25,000 = 32,500

2) Determine the percent change in

population from 1990 and 2005.

Percent change = 7,500

25,000x 100%

Percent change = 30%

Change in population = 7,500 people

Note: t = 0 for 1990 and t = 15 for 2005


Repeating Sequence

Strategy: For any sequence that

repeats, the value of the last term before

the sequence repeats will always be

repeated for any multiple of its term

number.

Reasoning: The letter “T” is the last

letter before the sequence repeats. “T”

appears as the 4th, 8th, 12th,.. 20th,….40th

term value. Term numbers that are a

multiple of 4 will always have the letter

“T” as its value for this sequence.

Application: Used when any sequence

of numbers or objects repeat. Examples

include numbers or letters, days of the

week, hours on the clock, remainders

from long division.

A C F T A C F T A C…….

4th term 8th term

2nd term 6th term 10th term

The term number of letter “C” will

always be the following:

4n + 2

where “n” is an integer value and 2

is the remainder when the term

number is divided by the multiple 4

Back to

Definition


Repeating Sequence

Example 1

Question: If the day of the week is

Friday and it is assigned the value of

one, what day of the week would be

assigned the value one hundred?


Identify the appropriate multiple number

for the repeating sequence.


essential information?: Identify the day

of week at end of cycle, apply the

multiple of seven to this day, identify the

day assigned the value of one hundred.

Solution Steps

1) Identify the day at end of cycle

2) Find the remainder when one hundred is

divided by the value seven

•If Friday is day one of the cycle, Thursday

is the end of the weekly cycle and is

assigned the value of seven

•Apply multiple of seven to Thursday

100

7= 14 with a remainder of 2

3) Identify day assigned the value of one

hundred•For remainder of two, day one hundred is

two days beyond Thursday → Saturday


Repeating Sequence

Example 2Question: A pattern consisting of three

red circles, two blue circles, three

yellow circles, and three green circles

was painted side by side along the

perimeter of a rectangular box. If the

color of the last painted circle was blue,

which of the following could be the total

number of circles painted on the box?

a) 80 b) 83 c) 86

d) 89 e) 92


Multiple number for sequence and

possible remainders for a blue circle


essential information? Use repeating

sequence principles to identify essential

information

Solution Steps

1) Identify multiple number for sequence

2) Identify possible remainders for blue

circle

3 red + 2 blue + 3 yellow + 3 green = 11

•Add total number of circles in pattern:

•Multiple number is 11 for sequence

•Blue circles are located at positions

four and five in sequence.

•Correct choice is a value that is 4 or 5

greater than a multiple of 11

•Correct choice is (11)(8) + 4 = 92

•Conclusion: Third green circle is

always a multiple of 11 in sequence.


Using New DefinitionsType 1

Example: For all positive integers x, let x @ be

defined to be (x+1)(x+2). What is the value of 4@ ?

Strategy: Read and apply the new

definition carefully before choosing

answers.

Reasoning: The new definition will

typically break down to a simple

application involving basic math

operations.

What does x@ mean? How

do I determine a value?

4@ = (4+1)(4+2)

4@ = (5)(6)

4@ = 30

Apply the definition

in given form

Operation is easy to

apply for any value of “x”

Final answer

Caution: Do not foil (x+1)(x+2).

More efficient to apply definition in

factored form.

Back to

Frequent

Questions


Using New DefinitionsType 2

Example: A positive integer is said to be “bi-factorable” if it

is the product of two consecutive integers. How many

positive integers less than 100 are bi-factorable?

Strategy: Read and apply the new

definition carefully before choosing

answers. Note the defined word is in

“quotations” and there is no math

expression as in Type 1.

Reasoning: Requires reasoning to apply

the intended meaning due to lack of a

math expression as in Type 1. Type 2

“New Definition” questions are usually

more difficult to solve than Type 1.

What does the definition

“bi-factorable” mean? How

do I determine a value?

1 x 2 = 2

2 x 3 = 6

8 x 9 = 72

9 x 10 = 90

Smallest integer less than

100 that is “bi-factorable”

Largest integer less than

100 that is “bi-factorable”

Result: There are nine positive integers

less than 100 that are “bi-factorable”

Back to

Frequent

Questions


Using New Definitions

Example 1Question: Let <x> be defined as the sum

of the positive integers from 1 to x,

inclusive. What is the value of <53> -

<50>?


Find the value of each quantity and

perform the subtraction operation.


essential information?: Carefully apply

the definition of <x> to each quantity.

Look for opportunities to simplify the

solution process through cancellation of

like terms.

Solution Steps

Apply the definition to each quantity:

<53> = 53+52+51+50+49+…+1

<50> = 50+49+…+1

Look for cancellation opportunities:

<53> - <50> =

(53+52+51+50+49+…) – (50+49+…)

<53> - <50> = 53+52+51

<53> - <50> = 156

Note: No calculator needed due to

cancellation of like terms. Without

cancellation strategy, problem would

be consume too much time.



Example 2

Question: Let ©(x) be defined as ©(x) =

(10-x) for all values of x. If ©(b) = ©(2b-2)

what is the value of b?


Find the value of b that satisfies the

given equation.


essential information? Carefully apply

given definition to the expressions on

each side of the equation. Set both

expressions equal to each other and

solve for b using simple math operations.

Solution Steps

Apply definitions to each expression:

©(b) = 10-b

©(2b-2) = 10-(2b-2)

Set both expressions equal to each

other and solve:

10-b = 10-(2b-2) Distribute (-)

10-b = 10-2b+2 Subtract 10

-b = -2b+2 Add 2b

b = 2



Example 3


Find the value of b that satisfies the

given equation.


essential information? Carefully apply

the given definition using the values in

each answer choice.

Solution Steps

Apply definitions to each expression:

Set both expressions…

Question: For positive integers a and b,

let a b be defined by a b = ba .

Which of the following is equal to 243.

A) 3 5 C) 9 27 E) 81 3

B) 5 3 D) 3 81

Return to Table of Contents Return to strategy page Return to example 1

Solving Simple InequalitiesBack to

Frequent

Questions

Example: For all values of x, what must be true

about the value of “n” in the inequality k – n < k + 2?

Strategy: Always solve the inequality

directly by eliminating like terms and/or

factors before analyzing answer choices.

Reasoning: By eliminating like terms or

factors, the inequality often simplifies to

one of the answer choices. Without

simplification, each answer choice

typically requires time consuming

analysis to determine correct choice.

Recommended

Solution

Step 1: Eliminate like

terms by subtractionk – n < k + 2

Step 2: Solve for “n” n > - 2

Caution: Do not choose values for

“k” and use guess and check

methods. Can be time consuming.



Example 1

Question: For all values of x, what is a

possible value of x that satisfies the

inequality x + 5 > x + 7?


All possible values of x that will make the

left expression greater than the right

expression.


essential information?: Look for like

term cancellation opportunities that

eliminate the need to do time consuming

guess and check steps.

Solution Steps

2) Evaluate remaining terms of

inequality:

5 > 7

This result is impossible

The correct answer is the empty set.

1) Cancel x term from both sides of

inequality:

x + 5 > x + 7

Note: Cancellation of like terms by

subtraction provides a clear result to

analyze.



Example 2

Question: If a + b > a - b, which of the

following statements must be true?

a) b < a b) a < b c) a = b

d) b > 0 e) a > 0


From answer choices it is clear a method

is needed to condense the number of

variables to one on each side of the

inequality.


essential information?: Look for like

term cancellation opportunities that

reduce the number of variables and

eliminate the need to do time consuming

guess and check steps.

Solution Steps

1) Simplify inequality by elimination and

consolidation of like terms

Correct answer choice is “d”

a + b > a - b

b > -b Add “b” to

both sides

2b > 0

b > 0

Eliminate “a”

from both sides

Divide “b” from

both sides


Equivalent Strategy

525 xx

Example: What is equivalent to the following

equation?

Strategy: When a question asks for an

equivalent form of an equation or

expression, review all answer choices for

guidance on ways to process the given

equation/expression.

Reasoning: Equations or expressions

can be expressed in an infinite number

of equivalent forms. The answer

choices often provide valuable guidance

on how to transform the equation or

expression into the correct answer

choice. Click to see equivalent forms

Equivalent Forms

xx 525 15

25

x

x

xxx 102 101 x

All of the above are equivalent forms of the

original equation. Answer choices on the

SAT will typically include one of the above

equivalent forms and four incorrect choices.


Equivalent Strategy

Example 1


Guidance on how the expression should

be transformed into “correct” equivalent

form


essential information?: Review answer

choices for guidance on “correct”

equivalent form.

Solution StepsQuestion: For x ≠ 0, which

of the following is equivalent to

a) 6x b) 12x c) 24x

d) 6x2 e) 12x2

?

8

14

3

x

x

1) Review answer choices for clues

Conclusion: Answer choices suggest

equivalent form requires elimination

of fractions in numerator and

denominator

2) Eliminate fractions by multiplying

numerator by reciprocal of denominator

1

8

4

3

8

14

3xx

x

x 2

26x


Equivalent Strategy

Example 2

Question: If k is a positive integer, which

of the following is equivalent to

2k + 2k + 2k + 2k ?

a) 24k b) 4k c) 42k

d) 2k+2 e) 2k+4


Need clues that better define equivalent

form of expression.


essential information? Review answer

choices for guidance on correct solution

path.

Solution Steps


2) Simplify radical using proper rules


equivalent form requires simplification

of radical expression

2k + 2k + 2k + 2k = 4(2k)

22(2k)

2k+2


Equivalent Strategy

Example 3


Need clues that better define equivalent

form of expression.


essential information? Review answer

choices for guidance on correct solution

path.

Solution Steps



equivalent form requires squaring of

radical

2) Square radical using proper rules

3) Transform equation and factor

Question: For all x > -2, which of the

following expressions is equivalent to

?

a) x + 2 = 10x b) x + 2 = 20x

c) x + 2 = 10x2 d) x + 2 = 20x2

e) x(100x - 1) = 2

xx

25

2

xx 102

21002 xx

2100 2 xx

2)1100( xx


System of EquationsBack to

Frequent

Questions

Example: What is the value of “w” in the following

system of equations?

Strategy: Solve a system of equations

using elimination method or by

reasoning. Do not use substitution .

Reasoning: A system of three or more

equations takes considerable time to

solve using substitution methods. The

questions are typically designed to be

quickly solved by reasoning or by

elimination of unwanted variables by the

elimination method.

3w = x – y + 4

w = z – x – 9

2w = y – z + 11

w = 1

Strategy: Use elimination method.

Reasoning method not practical

without more information about

the values of or relationships

between the variables.

3w = x – y + 4

w = z – x – 9

2w = y – z + 11

6w = 6


System of Equations

Example 1

Question: At a used book sale, Hillary

paid $5.25 for 2 paperback books and 3

hardback books, while Ally paid $6.75 for

4 paperback books and 3 hardback

books. At these prices, what is the cost,

in dollars, for 3 paperback books?


The unit price for a paperback book.


essential information?: Can use

system of equations to develop two cost

equations. An alternative method is to

apply reasoning skills.

Solution Steps

1) Solution using reasoning skills

•The only difference between Hillary’s

book order and Ally’s book order is the

number of paperback books purchased.

•Ally spent $1.50 more than Hillary to

purchase 2 additional paperback books.

2) Find the unit cost for paperback books

Unit cost = $1.50/2 paperback books

Unit cost = $0.75

3) Find the cost for 3 paperback books

Total cost = $2.25


System of Equations

Example 2

Question: In the system of equations

below, what is the value of x + y?

x + y - 4z = 400

x + y + 6z = 1200


Need a value for the expression x + y or

separate values of x and y.


essential information? Use elimination

to determine value of expression x + y.

Solution Steps

1) Subtract second equation from first

equation and solve for the value of z:

x + y - 4z = 400

x + y + 6z = 1200

-10z = -800

z = 80

2) Substitute the value of z into first

equation and solve for x + y:

x + y -4(80) = 400

x + y -320 = 400

x + y = 720


The Matching Game for

Equalities

Example: If k is a constant and 2(kx + 4) = 6x + 8

for all values of x, what is the value of k?

Strategy: When two equivalent

expressions are set equal to each other,

match corresponding terms and solve for

the unknown constant.

Reasoning: Terms on each side of the

equal sign can be easily matched and

common factors and/or terms can often

be eliminated. This will allow the

possibility of quickly identifying the value

of the unknown constant.

2(kx + 4) = 6x + 8

2kx + 8 = 6x + 8

Equivalent expressions

Distribute

“k” is unknown

constant

2kx + 8 = 6x + 8Match

corresponding

terms

Set equal and solve for “k”

2kx = 6x 2kx

2x

6x

2x=

k = 3


The Matching Game

Example 1

Question: If xy2 + 5 = xy + 5, which of

the following values of y are solutions to

the equation?

I -1 II) 0 III) 1

a) I only b) II only c) III only

d) II and III only e) I, II, and III


All possible values of “y” that make the

left side of equation equal to the right

side.


essential information? Look for like

term and common factor cancellation

opportunities that eliminate the need to

do time consuming guess and check

steps.

Solution Steps

1) Cancel like terms from both sides of

equation.

xy2 + 5 = xy + 5

2) Cancel common factors from both

sides of equation.

xy2 = xy

3) Evaluate y2 = y for possible

solutions

Solutions are 0 and 1.

Subtract 5

Divide out “x”

Correct answer choice is d


The Matching Game

Example 2

Question: In the equation below, k and

m are constants. If the equation is true

for all values of x, what is the value of m?

(x + 6)(x – k) = x2 - 4x + m


Need value of “m” that will make

expression on right side of equal sign

equivalent to the expression on left side.


essential information? Match terms in

expression on left side of equal sign to

corresponding terms in expression on

right side.

Solution Steps

1) Convert expression on left side

to trinomial form by distributing:

x2 - kx + 6x - 6k = x2 - 4x + m

x2 - (k – 6)x - 6k = x2 - 4x + m

2) Match like terms on each side:

x2 - (k – 6)x - 6k = x2 - 4x + m

m = - 6k -(k – 6) = -4

3) To solve for “m” need value of “k”

-(k – 6) = -4

-k + 6 = -4

k = 10

m = - 6k

m = - 6(10)

m = - 60


Factoring Strategy

Strategy: If an expression is in factored

form, generally leave it that way. If an

expression can be factored, it is always

to your advantage to factor it.

Reasoning: Working in factored form

provides opportunities to quickly reason

through problems with little computation.

Example: Can

be factoredStrategy

Given equation

is in factored

form. Reason

through problem

in this form.

Example: In

factored formStrategy

What conditions

must be true for the

following expression

to be odd?

I. a is odd

II. b is odd

III. a + b is odd

a2 +ab

Reason through

problem with the

expression in

factored form

a(a + b)

For the following

expression, what is the

largest integer value

for which the

expression is positive?

(4a - 2)(4 - a)

Back to

Definition


Factoring Strategy

Example 1

Question: If x2 – y2 = 92 and x + y = 23,

what is the value of x – y?

What is the essential information

needed?: Need values for x and y.

Better approach is to directly find a value

for the expression x – y.


essential information?: x + y and x – y

are factors of x2 – y2 . Write x2 – y2 in

factored form. Divide the value of x2 – y2

by the value of x + y.

Solution Steps

x2 – y2 = (x + y)(x – y)

1) Write in factored form

92 23 ?

2) Solve for x - y

x – y =92

23

x - y = 4


Factoring Strategy

Example 2

Question: If (x + 2)(x – 5) < 0, how many

integer values of x are possible?

What is the essential information

needed?: Need to identify specific

integer values of x that result in a value

less than zero for the left side of the

inequality.


essential information?: It can be

reasoned that the two linear binomial

factors on the left side of the inequality

describe a parabola. Use the properties

of parabolas to determine answer.

Solution Steps

When considered a parabola, two

properties are useful to answer question:

1) The parabola opens upward

2) The parabola has roots at x = -2 and

x = 5

There are six integer values

between -2 and 5 that result

in a value less than zero

-2 5

{-1, 0, 1, 2, 3, 4}


Word Problems

How many days are there in h hours and m minutes?

Strategy: Use a two-step strategy to

solve most word problems:

1) Eliminate choices that do not properly

model the situation (often obvious).

2) Eliminate choices that do not provide

proper units (dimensions)for the solution.

Reasoning: By reasoning, some choices

will obviously not appear to be proper

solutions. Of those remaining, some will

likely have wrong or inconsistent units.

144024

mh

mh

144024

mh 144024

mh

144024

mh

144024

Step 1: Both minutes and hours

should be smaller than days,

not greater. Likely need to

divide both terms in answer by

a number or variable.

No

No

No

No

Yes

Step 2: To end with units of

days, divide hours by 24 hours

per day. Also, divide minutes by

1440 minutes per day.

This choice properly converts

hours and minutes into days.


Word Problems

Example 1

Question: Water from a leaking roof is

collected in a bucket. If n ounces of

water are collected every m minutes,

how many ounces of water are collected

in z minutes?


Need to establish relationships between

the given variables that provide

dimensionally correct answer.


essential information? Determine the

proper units of final expression that are

consistent with question being asked.

Create an expression that is consistent

with the required units.

Solution Steps

1) Determine units of correct answer

• Final answer represents quantity

of water collected

•Units of final answer should be

ounces

2) Arrange the three variables in

proper way that provides correct units

ounces

minuteminutesx

Units of minutes

cancel - ounces

remain

Replace units with

corresponding variables

n

m(z) = nz

m


Word Problems

Example 2

Question: In a certain grocery store,

there are b stockcases with c shelves in

each stockcase. If a total of d cans is to

be stored on each of the shelves, what is

the number of cans per shelf?


Need to establish relationships between

the given variables that provide

dimensionally correct answer.


essential information? Determine the

proper units of final expression that are

consistent with question being asked.

Create an expression that is consistent

with the required units.

Solution Steps

1) Determine units of correct answer

•Final units should be cans per shelf

2) Divide the total number of cans (d)

by the total number of shelves

b stockcases x c shelves

stockcase= bc shelves

Number of cans per shelf = d

bc


Rules of ExponentsBasic Rules

Back to

Frequent

Questions

Strategy: When the bases of two

powers are the same in an equation,

use these three basic rules to combine

the two powers into a power with a

single base. The value on the right hand

side of the equation should be

converted into a power with the same

base as the power on the left hand side

of the equation.

Reasoning: The three basic rules of

exponents evolve from the fundamental

definition of “exponentiation” that states:

xa means “x” multiplied “a” times.

Example: If x and y are positive integers and

(23x )(23y) = 64, what is the value of x + y?

Product of Two

Powers Rule:

Quotient of Two

Powers Rule:

Power of a

Power Rule:

ba

b

a

xx

x

baba xxx

baba xx

Caution: The product and power rules

are often confused for one another.



Example 1

Question: If p and n are positive

integers, and 32p = 2n , what is the value

of p/n?

What essential information is

needed? Need to establish a

relationship between expressions on left

side and right side of equal sign that

clarify the relationship between p and n.


essential information?:Use rules of

exponents to covert 32 to a power with a

base of 2.

Solution Steps

1) Convert 32p to a power with base 2

32p = 2n

(25)p = 2n

25p = 2n

2) Set exponents equal to each

other and solve for p/n.

5p = n

p

n

1

5=



Example 2

Question: If 28x+2 = 643 , what is the

value of 4x?


needed? Need to establish a

relationship between expressions on left

side and right side of equal sign that

clarify the relationship between the two

exponents.


essential information? Use rules of

exponents to convert 64 to a power with

base 2.

Solution Steps

1) Convert 643 to a power with base 2.

28x+2 = 643

28x+2 = (26)3

28x+2 = 218

2) Set exponents equal to each other

to solve for the value of “4x”

8x + 2 = 18

8x = 16

4x = 8

Note: No need to solve for “x”. Can

solve directly for the value of 4x.


Rules of ExponentsAdditional Rules

Strategy: Use these additional rules of

exponents when needed. Of the four

additional rules, the negative exponent

and rational (fractional) exponent rules

are utilized most.

Reasoning: When an equation contains

a variable with a negative exponent and

rational exponent, follow a two step

process to isolate variable:

Application: Questions with expressions

that contain negative exponents and/or

rational exponents.

Negative Exponent

Rule:

Zero Exponent

Rule:

Power of a

Product Rule:

n

n

xx

1

10 x

aaayxxy

Rational (fractional)

Exponent Rule:a

b

a b xx

1) Convert the negative exponent to a

positive exponent using rule2) Raise both sides of equation to the

reciprocal of the rational exponent.



Example 1

Question: Positive integers a, b, and c

satisfy the equations a-½ = ¼ and b-¾ =

⅛. What is the value of a + b?


The values of a and b are needed.


essential information?: Use negative

exponent rule and raise both sides of

each equation to the reciprocal of the

rational exponent.

Solution Steps

1) Apply negative exponent rule to

each equation

2) Raise both sides of each equation

to the reciprocal of the rational

exponent

a-½ = ¼ 1

a½ = ¼

a½ = 4

b-¾ = ⅛1

b¾= ⅛

b¾ = 8

(a½ )2 = 42

a = 16

(b¾ )4/3 = 84/3

b = 16

a + b = 32



Example 2

Question: If 4-y/2 = 16-1 , then y = ?


Need to directly solve for the value of “y”


essential information? Use negative

exponent rule first. Solve for value of “y”

by converting powers on both sides of

equation to the same base.

Solution Steps

1) Apply negative exponent rule to

both sides of equation

2) Convert to same powers

4-y/2 = 16-1

1

4y/2= 1

16

4y/2 = 16

4y/2 = 16

4y/2 = 42

y

2= 2

y = 4



Strategy: To solve absolute value

inequalities quickly, use a three step

approach: 1) Using given information eliminate choices

representing the wrong solution type

2) Remove absolute value, evaluate positive

solution, and eliminate choices

3) With remaining choices evaluate negative

solution and choose correct answer

Reasoning: Absolute value inequalities

have properties that can be used to

eliminate wrong choices.

Example: A manufacturer produces picture frames between

28 and 42 inches in width. If x represents the size, in inches,

of the picture frames produced by the manufacturer, which

of the following represents all possible values of x ?

28 < x < 42

| x – 35 | < 7

x – 35 < 7 x < 42

x > 28

Possible

Solution Types:

Possible

Inequality:

Solution details for | x – 35 | < 7

28 < x < 42

+/-(x – 35) < 7

x – 35 > -7

x < 28 or x > 42

| x – 35 | > 7

Example of

Solution:

x < a or x > ba < x < b

Remove absolute value:

Positive solution:

Negative solution:

Overall solution:

Back to

Definition

Back to

Frequent

Questions



Example 1

Question: For a certain airline

company, the weight of pilots must be

between 140 and 200 pounds. If w

pounds is the acceptable weight of a

pilot for this airline company, which of

the following represents all possible

values of w?

a) │w - 170│= 30 b) │w + 140│< 60

c) │w - 170│> 30 d) │w -170│< 30

e) │w - 140│< 60


The correct answer must be the solution

to 140 < w < 200.


essential information?: Use the

absolute value strategy to identify answer

Solution Steps

1 & 2) Remove absolute value sign,

solve positive solution, eliminate choices

that do not meet the solution w < 200.

a) w = 200 Not a solution b) w < -80 Not a solution c) w > 200 Not a solution d) w < 200 Possible solution e) w < 200 Possible solution

3) Evaluate negative solution

d) w - 170 > -30 w > 140 Solution

e) w - 170 > -60 w > 110 Not a

solution



Example 2Question: A certain manufacturer of

pencils requires all pencils to meet a

length specification between 6.9 and

7.0 inches inclusive. If x is the length of

a pencil that meets the specification,

which of the following represents the

length of pencils that do not meet the

specification?

a) │x - 6.0│< 1.0 b) │x - 6.0│> .05

c) │x - 6.0│> 1.0 d) │x - 6.95│> .05

e) │x + 6.0│> 13.0


The correct answer will be the solution to

x < 6.9 or x > 7.0


essential information? Use the absolute

value strategy to identify answer

Solution Steps

1 & 2) Remove absolute value sign,

solve positive solution, eliminate choices

that do not meet the solution x > 7.0

a) x < 7.0 Not a solution b) x > 6.5 Not a solution c) x > 7.0 Possible solution d) x > 7.0 Possible solution e) x > 7.0 Possible solution

3) Evaluate negative solution

c) x - 6.0 < -1.0 x < 5.0Not a

solution

d) x - 6.95 < -.05 x < 6.9 Solution

e) x + 6.0 < -13.0 x < -7.0Not a

solution



from Words

Strategy: Use the information in the table

to the right to translate words into

mathematical expressions and equations.

Reasoning: These are common words

that are utilized in questions. When

properly translated, the solution to a

question is usually straightforward.

Words Symbol Translation

Is, the same as, is

equal to= Equals

Sum of, more

than, greater than+ Addition

Less than,

difference, fewer- Subtraction

Of, product, times × Multiplication

For, per ÷ Division

Example: If three times a number x is twelve less than x,

what is x ?

Translation: 3x = x – 12

Solution: x = -6



from Words Example 1

Question: If ¾ of 3x is 15, what is ½ of

6x?


Create a math statement that properly

describes the given information.


essential information?: Use the table

of words to convert the given information

into the proper math statement.

Recognize that ½ of 6x equals 3x.

Solving for the value of 3x will provide

correct answer to question.

Solution Steps

1) Create the proper math statement

2) Solve for the value of “3x”

¾ · 3x = 15

¾ of 3x is 15times equals

¾ · 3x = 15 multiply by 4

3

[¾ · 3x] = [15] 4

3

4

3

5

3x = 20 Correct

answer



from Words Example 2

Question: Which of the following

expresses the number that is 15 less

than the product of 4 and x + 1?

a) -4x + 14

b) -4x + 16

c) 4x - 11

d) 4x - 13

e) 4x - 15


Create a math statement that properly

describes the given information.


essential information? Use the table of

words to convert the given information

into the proper math statement.

Solution Steps

1) Create the proper math statement

from given information

Product of 4 and x + 1

2) Simplify the math statement to

match answer choices

4(x + 1)

15 less than product

of 4 and x + 14(x + 1) - 15

4(x + 1) - 15 Distribute 4

4x + 4 - 15 Subtract 15

4x - 11 Correct answer


The Parabola

Strategy: Many questions about the

parabola (sometimes called “the

quadratic function”) require an

understanding of the impact of constants

“a”, “b”, and ‘c” on the graph of f(x).

Reasoning: 1) The coefficient or

constant “a” directly influences the x2

term of the function. When f(x) = x2, the

parabola opens up. When f(x) = -x2, the

parabola opens in the opposite direction

or down. 2) The constant “c” is the

function value for f(0) = “c”. This is the

definition the y-intercept. 3) The impact

of “b” is more complicated and usually

not important.

Example: The quadratic function f is given by f(x) = ax2 + bx + c,

where “a” is a positive constant and “c” is a negative constant.

Which of the figures could be the graph of f?

Standard form of a parabola

f(x) = ax2 +bx + c

“a” positive

opens up

“a” negative

opens down

“c” positive

positive “y”

intercept

“c” negative

negative “y”

intercept

Click to show

answer


The Parabola

Example 1

Question: The quadratic function f is

given by f(x) = ax2 + bx + c, where “a” is

a positive constant and “c” is equal to

zero. Which of the figures could be the

graph of f?


Need to know the effects of constants “a”

and “c” on the graph of a parabola.


essential information?: Use parabola

strategy to determine effects of “a” and

“c”.

Solution Steps

A

D

B C

E

“a” positive

“c” positive

“a” positive

“c” negative

“a” positive

“c” zero

“a” negative

“c” zero“a” negative

“c” zero

What is the correct choice?

(click to verify choice)


The Parabola

Example 2

Question: The quadratic function f is

given by f(x) = ax2 + bx + c, where the

product “ac“ is a positive constant.

Which of the figures could be the graph

of f?


Need to know the effects of constants “a”

and “c” on the graph of a parabola.


essential information? Use parabola

strategy to determine effects of “a” and

“c”.

A B C

D E

“a” positive

“c” zero

“a” positive

“c” negative

“a” positive

“c” zero

“a” negative

“c” zero“a” negative

“c” negative



Solution Steps

“ac” =

positive


Single Term Denominator

Equations

Strategy: When an expression contains

two or more variable terms in the

numerator and a single variable term in

the denominator, expand the expression

by placing each term in the numerator

over the variable in the denominator

Reasoning: The expression will often

easily simplify into the form required to

directly answer the question.

5x

x

y

x

26

5=+

5x

x

y

x

26

5= -

y

x

1

5=

Example: If , what is the value of ? 5x + y

x

26

5=

y

x

Alternative Solution: This problem can

also be solved using cross multiplication.

Although the algebra is straightforward,

students often struggle to isolate the

answer when a ratio is required. Try it!

5x + y

x

26

5=



Example 1


Need values of each variable or find way

to simplify the expression using the given

ratio values.


essential information? Use single term

denominator strategy to simplify the

expression without the need to identify

values of each variable.

Solution StepsQuestion: What is the value of

if and ?

7x + y + z

yy

x = 14 z

y = 5

1) Expand the expression

7x

y

y

y

z

y + +

2) Substitute given ratio information

and simplify y

y 1=

x

y

1

14=

z

y5=

Note: The value of each ratio is given

7[ ]+ 1 + 51

14

+ 1 + 5 1

2

6.5



Example 2


Need values of each variable or find way

to simplify the expression using the given

ratio values.


essential information? Use single term

denominator strategy to simplify the

expression without the need to identify

values of each variable.

Solution StepsQuestion: If , what is the value

of ?6

76

y

yx

y

x


Dividing Irregular Shapes in

Polygon Shapes

x

y

5

450

Incorrect strategy

Note: You are setting a

“trap” when a shape is

divided into a trapezoid.

Triangle

Trapezoid

Rectangle

Triangle

Example: Which of the following represents the

area of the five-sided figure shown to the right? Correct strategy

Strategy: Always divide irregular polygon

shapes into rectangles (or squares) and

right triangles. Do not divide the shape

into trapezoids or parallelograms. Click

to see the animation.

Reasoning: The area and perimeter of

rectangles and right triangles are usually

easy to determine from the given

information. In particular, right triangles

can be solved using Pythagorean

theorem or properties of 30-60-90 and

45-45-90 triangles.



Example 1


Sides AB and BC easy to determine.

Need to divide figure into shapes that will

provide an efficient way to find the length

of segment AC


essential information?: Divide the

shape into a rectangle and right triangle.

Solution Steps

Question: In the figure above, what is

the perimeter of triangle ABC?

A

B

C

4

3

8

6

Figure not

drawn to scale 1) Divide the shape into a

rectangle and right triangle

(see original figure) 4

9

4

9

A

C

2) Determine the length of each side of

triangle ABC•Determine length of sides AB and BC

from properties of 3-4-5 triangle

AB = 5 and BC = 10

•Determine length of side AC from

Pythagorean Theorem

9749AC 22



Example 2

Question: In the rectangle above, the

sum of the areas of the shaded region is

14. What is the area of the unshaded

region?



essential information? Divide the shape

into a rectangle and right triangle.

Solution Steps

xyx

x

y


Line Segment or Diagonal

Length in a Geometric Solid

a

b

c

Right Triangle

Line Segment

c2 = a2 + b2

Pythagorean Theorem

Example: In the figure shown to the right, the

endpoints of the line segment are midpoints of

two edges of a cube of volume 64cm3. What is

the length of the line segment?

Strategy: To find the length of a diagonal

or a line segment that connects two

edges of a geometric solid, create a right

triangle within the solid that uses the

unknown segment as the hypotenuse.

Click to see the animation.

Reasoning: By finding a right triangle

within the solid, Pythagorean Theorem

can be used to find the segment or

diagonal length. Helpful Hint: The diagonal of any cube is

equal to the cube side length times √3

Caution: Does not apply for rectangular

solids (shoe box shape)


Line Segment Length in Solid

Example 1

Question: What is the volume of a cube

that has a diagonal length of 4√3?


Side length of the cube is needed to find

the volume.



properties of a cube, the diagonal length,

and Pythagorean theorem to find the

side length.

Solution Steps

1) Establish relationships between

cube diagonal length and side length

using properties of a cube

a

a√2

a

a

•Let “a” be the side

length of cube

•The longer side length

of right triangle found

using properties of

45-45-90 triangle

4√3

2) Apply Pythagorean theorem to find

side length a2 + (a√2)2 = (4√3)2

a = 4

Volume = a3 = 43 = 64


Line Segment Length in Solid

Example 2


A connection between given side lengths,

the center of solid, and the midpoint of AB


essential information? Half the length

of diagonal BD is equivalent to the

desired distance. Use Pythagorean

theorem.

Solution Steps

Question: In the figure above, if AB =

24, BC = 12, and CD = 16, what is the

distance from the center of the

rectangular solid to the midpoint of AB?

A

C

B

D

E

1) Diagonal BD is the hypotenuse of

right triangle BCD. Find the length of

BD.A

C

B

D

E

24

12

16

Can easily find the length of BD by

recognizing that triangle BCD is a

multiple of the 3-4-5 triangle. The

length of BD is 20. (12-16-20)

2) Half the length of diagonal BC is

20/2= 10 (shown in white on diagram)



Strategy: When asked to piece together

several regular shapes into one shape,

sum together the areas of individual

pieces. The final shape will have the

same area as the sum of the individual

pieces.

Reasoning: The area must be

conserved provided there is no overlap

when the individual pieces are combined

into one shape. Click to see the

animation of the correct choice.

Which of the shapes below could be

made from the three individual shapes

shown above?

Area = 9 Area = 10 Area = 8

Area = 2Area = 3

Area = 5

Total area of three

shapes = 10

Unit Area = 1 block



Example 1

Question: Page under construction





Solution Steps


Example 2






Solution Steps

3-4-5 Triangle

Strategy: Recognizing the 3-4-5 right

triangle in a figure can save time and

reduce the possibility of error when

determining side lengths of a triangle.

Reasoning: Recognizing triangles as 3-

4-5 do not require calculation of the third

side using Pythagorean Theorem.

Triangles with common multiple lengths

of a 3-4-5 are similar to the 3-4-5.

Application: Look for right triangles

with side lengths that are multiples of 3-

4-5. Common examples include 6-8-10,

9-12-15, 12-16-20, and 15-20-25

triangles. Use similar triangle properties

to determine unknown side lengths, not

Pythagorean Theorem.

3

45

6

810

9

1215


3-4-5 Triangle

Example 1


Side length BC is needed to find the

triangle area.


essential information?: Can use

Pythagorean theorem, however, more

efficient to use properties of 3-4-5

triangle.

Solution Steps


the area of ∆ABC?

100

80A

B

C1) Use properties of 3-4-5 triangle to find

length of BC

•Side CA has a length of 80. This is a

multiple of four (4 x 20 = 80)

•Side AB (hypotenuse) has a length of 100.

This is a multiple of five (5 x 20 = 100)

•Conclusion: Side BC is a multiple of 3 and

will have a length of 60. (3 x 20 = 60)

2) Calculate the area of ∆ABC

Area = ½(base)(height) = ½(80)(60)

Area = 2400


3-4-5 Triangle

Example 2


The length of side XZ is needed to find

perimeter.


essential information? ∆XYZ is a right

triangle. Can use Pythagorean theorem,

however, it is easier and more efficient to

use 3-4-5 triangle relationships.

Solution Steps


the perimeter of ∆XYZ?

x

y

z

55

33

1) Use properties of 3-4-5 triangle to find

length of XZ

2) Calculate the perimeter of ∆XYZ

•Side YZ has a length that is a multiple of

three (3 x 11 = 33)

•Side XY has a length that is a multiple of

five (5 x 11 = 55)

•Conclusion: Side XZ is a multiple of four

and will have a length of 44. (4 x 11 = 44)

Perimeter = XY + YZ + XZ

Perimeter = 55 + 33 + 44

Perimeter = 132


30-60-90 Triangle

Reasoning: This relationship is derived

by splitting an equilateral triangle into

two congruent 30-60-90 triangles. The

relationships between sides are derived

using Pythagorean Theorem. The

formula for this relationship is found on

the SAT formula sheet.

Application: Consider using for any

triangle that has a 300 or 600 angle. Also,

use for any right triangle that has a 300

or 600 angle.

Strategy: If the leg of a right triangle is

expressed in terms of , the triangle is

likely a 30-60-90. The coefficient

associated with the is the length of

the shorter leg. The hypotenuse is twice

the length of the shorter leg.

3

3

Coefficient

600

300

5√3

510

Note: The 30-60-90 triangle is not

a 3-4-5 triangle


30-60-90 Triangle

Example 1


A connection between side lengths that

justifies calling triangle ABC a right

triangle.


essential information?: Use properties

of 45-45-90 triangle or 30-60-90 triangle

to establish connection to right triangle.

Solution Steps

Question: In triangle ABC shown

above, the length of side BC is half the

length of side AB. The length of side AC

is 4√3. What is the length of side AB?

C A

B

1) Identify connection to right triangle

2) Use properties of 30-60-90 triangle

to find length of AB

•Triangle side BC = ½ side AB

•Triangle side AC has length 4√3

Conclusion: ∆ABC is a 30-60-90 triangle

•Side BC is short leg of triangle

•Side AC is long leg of triangle

•Side AB is hypotenuse of triangle

3) Determine length of side AB

AC = 4√3 BC = 4 AB = 2 x 4 = 8


30-60-90 Triangle

Example 2


Need a connection between side length

AB (value of 4), AD (base of ∆ABD), and

BD (height of ∆ABD)


essential information? The altitude of

an equilibrium triangle divides the

triangle into two 30-60-90 triangles. Use

properties of 30-60-90 triangle to make

connection.

Solution Steps

Question: Equilateral triangle ABC has

a side length of 4. If BD is an altitude of

∆ABC, what is the area of ∆ABD?

AD

B

C

4

1) Find the length of AD (base of ∆ABD)

and length of BD (height of ∆ABD)

2) Find the area of ∆ABD

Note: ABD is a 30-60-90 triangle with

angle BAD = 600 and angle ABD = 300

Conclusion: Side AD = 2; half the

length of hypotenuse ABConclusion: Side BD = 2√3; √3 times

the length of the short side AD

Area = ½(base)(height)

Area = ½(2)(2√3)

Area = 2√3


45-45-90 Triangle

Reasoning: This relationship is a

property of the 45-45-90 triangle. It can

be derived using Pythagorean Theorem.

The formula for this relationship is found

on the SAT formula sheet.

Application: Consider using for any

triangle that has a 450 angle . Also, any

right triangle that is isosceles will be a

45-45-90 triangle.

Strategy: If the hypotenuse of a right

triangle is expressed in terms of √2 , the

triangle is likely a 45-45-90. The

coefficient associated with the √2 is the

length of each triangle leg.

Coefficient

450

450

5

5

5√2


45-45-90 Triangle

Example 1


Side length of square is needed to

calculate area.


essential information?: Most efficient

strategy is to recognize that the diagonal

of a square divides the square into two

congruent, isoceles triangles. Each

triangle is a 45-45-90.

Solution StepsQuestion: In the figure below, what is

the area of the square?

101) Use properties of 45-45-90 triangle

to find side length

2) Calculate area of square

(Side length ) √2 = 10

Area = (side length)2

Side length = 10

√2

(10)

(√2)Area = (10)

(√2)

Area = 50

= 100

2


45-45-90 Triangle

Example 2


Need to make a connection between the

value of DC and the value of BC.


essential information? The two right

triangles share a common side AC. Use

properties of 30-60-90 and 45-45-90

triangles to make connection.

Solution Steps

Question: In the figure above, if DC =

2√6, what is the value of BC?

B

C

A

D

450

300

1) Find the length of AC using properties

of 30-60-90 triangle

2) Find the length of BC using properties

of 45-45-90 triangle

Note: AC is twice the length of AD

and DC is √3 times the length of AD

AD(√3) = DC = 2√6

AD = 2√6

√3= 2√2

Conclusion: AC = 2(2√2) = 4√2

Note: BC is √2 times the length of AC

BC = (4√2)(√2)

BC = 8


Distance Between Two Pointsx-y Coordinate Plane

Strategy: Draw the x-y coordinate, plot

the points, and find a right triangle.

Calculate the distance as shown.

Reasoning: As shown to the right, the

distance formula is an outcome of

applying Pythagorean Theorem in the x-y

coordinate plane. The distance “formula”

is not given on the SAT formula sheet.

Application: Multitude of problems

involving lines and points in the x-y

coordinate plane. See examples for

specific applications.

212

2

12 yyxxd

x2 - x1

y2 - y1

(3, 3.5)

(-5, -2.5)

10100

5.25.35322

d

d

d = 10



Example 1

Question: If points A (6, 2), B(12, 2), and

C(9, 9) are endpoints of triangle ABC,

what is the perimeter of the triangle?


Need to find the length of each side of

triangle ABC.


essential information?: A quick sketch

of the triangle reveals an isosceles

triangle with the non-congruent side AB

parallel to the x-axis. The remaining two

sides are congruent and require use of

the distance formula to find side length.

Solution Steps

1) Find the length of side AB using

distance formula for a number line

2) Find the length of congruent sides

AC and BC using distance formula for

x-y coordinate plane

d = │12 - 6│ = 6

212

2

12 yyxxd

58296922 BCAC dd

3) Find the perimeter of triangle ABC

Perimeter = 6 + √58 + √58

Perimeter = 6 + 2√58

A(6, 2) B(12, 2)

C(9, 9)



Example 2






Solution Steps

Midpoint DeterminationNumber Line

Strategy: The midpoint (xm) between two

endpoints on a number line is found by

averaging the two endpoints.

Reasoning: The midpoint is equidistant

from either endpoints. This is consistent

with the properties of the average (mean)

of two numbers.

Application: Number line applications

that requires the determination of

midpoint or endpoint values. The

midpoint “formula” is not given on the

SAT formula sheet.

Midpoint

5.12

74

2

21

xx

xm

xm

5.5 5.5

- 4

x1

7

x2

0

xm = 1.5


Midpoint Determination

Example 1

Question: If 3n and 3n+4 are end points

on a number line, what is the midpoint?

a) 3n+1

b) 3n+2

c) 3n+2.5

d) 3n+3

e) 41(3n)


Find the point that is located midway

between the two endpoints.



midpoint determination strategy for

finding midpoint on a number line

Solution Steps

1) Find the sum of the two endpoints

3n + 3n ·34 Factor 3n

3n (1 + 34 ) = 3n (1 + 81)

2) Divide the sum by two to find

midpoint 82(3n )

2= 41(3n )

82(3n )

3n + 3n+4 Expand 3n+4



Example 2

Question: If x - 2 and y are endpoints

on a number line and x + 6 is the

midpoint, which of the following

expressions represents y?

a) x

b) x + 2

c) x + 12

d) x + 14

e) x + 16


Find the endpoint that has x + 6 as the

midpoint when x - 2 is the other endpoint.


essential information? Apply the

midpoint determination strategy to find

the endpoint “y”.

Solution Steps

1) Apply the midpoint strategy to set

up the solution.

x + 6 =(x - 2) + y

2

2) Solve for the endpoint “y”

x + 6 =(x - 2) + y

2

Cross

multiply

2(x + 6) = (x - 2) + y Simplify and

solve for “y”2x + 12 = x - 2 + y

x +14 = y


Midpoint Determinationx-y Coordinate Plane

Strategy: The midpoint (xm , ym )

between two endpoints on the x-y

coordinate plane is found by averaging

the x-coordinates and y-coordinates of

the two endpoints.

Reasoning: The midpoint of each x-y

coordinate point is equidistant from either

endpoint. This is consistent with the

properties of the average of two numbers

Application: In addition to the x-y

coordinate, questions could ask for the

midpoint on a number line. Some

questions will give the midpoint and one

end point and ask for the unknown end

point. The midpoint “formula” is not

given on the SAT formula sheet.

1

2

68

2

21

m

m

x

xxx

Midpoint

(xm , ym )

x1 + x2

y1 + y2

(6, 6)

(-8, -4)

1

2

64

2

21

m

m

y

yyy

(-1, 1)



Example 1

Question: In the x-y coordinate plane,

the points (2, 8) and (12, 2) are on line

m. The point (7, y) is also on line m.

What is the value of y?


A method for determining the value of “y”


essential information?: Can use two

known points to find the equation of line

m and use equation to find y. Equation

of line not on SAT formula sheet. As a

result, likely not the most efficient

approach. As an alternative, midpoint

analysis can be used.

Solution Steps

1) Midpoint analysis of “x” values

2) Find the midpoint of 2 and 8

Conclusion: The “y” value must be

the midpoint of 2 and 8

72

122

mx

The “x” value of 7 is the

midpoint of 2 and 12

52

82

my

“y” value = 5

Note: Same result using equation of

line……less efficient method.



Example 2

Question: In the x-y coordinate plane,

the midpoint of AB is (2, 3). If the

coordinates of point A are (-1, 1), what

are the coordinates of point B?


Need to connect coordinates of endpoint

to the coordinates of midpoint.



midpoint formula to connect the

coordinates of endpoints to the midpoint.

Solution Steps

1) Find the endpoint by using the

midpoint formula

Coordinates of endpoint are (5, 5)

2

21 xxxm

2

21 yyym

2

12 2x

2

13 2y

214 x 216 y

52 x 52 y



Strategy: Any exterior angle of a triangle

is equal to the sum of the two remote

interior angles

Reasoning: The sum of the two remote

interior angles is supplementary to the

third interior angle. Likewise, the exterior

angle is supplementary to the third

interior angle.

Application: This strategy is a useful

way to save time and potential

calculation errors when an exterior angle

of any triangle is needed.

450

750

x0

Exterior angle

Remote interior

angles



Example 1


A strategy is needed to connect the

known angle values to the unknown

variables.


essential information?: Can easily find

the value of x + y using exterior angle of

triangle strategy. Can also find the value

of y. From alternate interior angles, x = z.

Solution Steps

Question: In the figure above, line m is

parallel to line k. What is the value of z?

x0 y0

1100

m

k

z0

1000

1) Find the value of x + y

2) Find the value of y

3) Find the value of z

1100 is an exterior angle; x and y are

the remote interior angles

Conclusion: x + y =1100

y is a linear pair with angle 1000

Conclusion: y = 800 and x = 300

From alternate interior angles, z = x

Conclusion: z = 300



Example 2


Need a connection between the given

angle value of 950 and the unknown

angle variables.


essential information? The given angle

of 950 is an exterior angle to both

triangles.

Solution Steps


the sum of a + b + c + d?

a0 b0

d0c0

950

1) Find the value of a + b

2) Find the value of c + d

950 is an exterior angle; a and b are


Conclusion: a + b = 950

950 is an exterior angle; c and d are


Conclusion: c + d = 950

3) Find the value of a + b + c + d

a + b + c + d = 2(950 )

a + b + c + d = 1900


Perpendicular Lines

Strategy: The slopes of perpendicular

lines are opposite reciprocals of each

other.

Reasoning: This is a fundamental

relationship developed in coordinate

geometry

Application: All questions involving

perpendicular lines require comparison of

slopes

l

q

q

lm

m1

Back to

Frequent

Questions


Perpendicular Lines

Example 1

Question: In the xy-plane above, the

equation of line m is 4x + 3y = 12. Which

of the following is an equation of a line

that is perpendicular to line m?

a) y = x + 3 b) y = -4x + 3

c) y = 4x - 3 d) y = ¾x + 6

e) y = -¾x - 6


The slope of line m is needed to

determine the slope of line perpendicular

to line m


essential information?: Slope of line m

can be determined from equation of line

m or directly from figure.

Solution Steps

42

2 4

1) Slope of line m

2) Equation of line perpendicular to

line m

= -4

3

•Slope using figure

•Slope using equation of line m

Slope = ∆y

∆x

4 - 0

0 - 3=

4x + 3y = 12

3y = -4x + 12

y x + 12= -4

3

•Correct choice is y = ¾x + 6

•Line must have slope = ¾


Perpendicular Lines

Example 2


Need to identify a line perpendicular to

line q and determine the slope of the new

line.


essential information? Draw a line from

origin to point of tangency. This line is a

radius and is perpendicular to line q.

Solution Steps

Question: Line q is tangent to the circle

at the point (4, -3). What is the slope of

line q?

(4, -3)

q

1) Find slope of new line

2) Find the slope of line q

•Slope of a line that passes through

origin can be determined from the ratio

of y/x for any point on the line.

•Slope of new line is -¾

•Slope of line q is the opposite

reciprocal of slope of new line

Slope of line q is 4

3


Interval Spacing

Strategy: The interval spacing on a

number line is found by a two-step

process:

1. Determine the distance between

two known points on the number line

2. Divide the distance by the number of

intervals separating the two known

points (Caution: Do not divide by the

number of tick marks)

Reasoning: By design, the number line

has equal distance between each tick

mark on the line

Application: Used to identify an

unknown coordinate on number line. Also

used to identify the value of specific term

in an arithmetic sequence.

3 18 23

What is

this value?

2.5

|18 - 3|

6= 2.5

18 + 2(2.5) = 23


Interval Spacing

Example 1

Question: The value of each term of a

sequence is determined by adding the

same number to the term immediately

preceding it. The value of the third term

of a sequence is 4 and the value of the

eighth term is 16.5. What is the value of

the tenth term?


The common value added to each term of

the sequence.


essential information? Use interval

spacing strategy to identify the common

value. Add twice this value to the eighth

term to find value of tenth term.

Solution Steps

1) Find the common value.

16.5 - 4

5 intervals=

12.5

5 intervals= 2.5

2) Add twice the common value of 2.5

to the eighth term value of 16.5.

Tenth term = 16.5 + 2.5 + 2.5

Tenth term = 21.5


Interval Spacing

Example 2

Question: On the number line above,

what is the value of point P?

a) 2n+½ b) 2n+¾ c) 3·2n

d) 3·2n+1 e) 3·2n+2


The interval spacing can be used to find

the value of “P”.


essential information? Find the interval

spacing by dividing the difference of the

two endpoints by the number of intervals

(six). Multiply the interval spacing by

three and add to the value of the left

endpoint.

Solution Steps2n+1 2n+2P

1) Find the interval spacing

2) Find the value of “P”

2n+2 - 2n+1 Expand the powers

2n ·22 - 2n ·21 Common factor is 2n

2n (22 - 21) Simplify 22 - 21

2n (2) Divide by six intervals

2n (2)

6=

2n

3

Interval

spacing

2n+1 + (3) 2n

3= 2n+1 + 2n Expand the

powers and

factor2n ·21 + 2n = 2n (21 + 1)

3∙ 2n Value of point “P”

3



Strategy: The 3rd side of any triangle is

greater than the difference and smaller

than the sum of the other two sides

Reasoning: A side length of 15 would

require the formation of a line, not a

triangle. A side length of 3 would also


triangle

Application: Given two sides, choose

the smallest or greatest integer value of

third side. Given three sides as answer

choices, which will not form a triangle.

9

6

3 < x < 15

9 6

15

9

63



Example 1

Question: If the side lengths of a triangle

are 8 and 23, what is the smallest integer

length of the third side?

a) 14 b) 15 c) 16

d) 30 e) 31


The smallest possible length of the third

side of the triangle


essential information?: The third side

of a triangle must be greater than the

difference of the given two sides of the

triangle.

Solution Steps

1) Find the smallest possible length of

the third side

2) Determine the smallest integer

length of third side of triangle

Length of third side > 23 - 8

Length of third side > 15

Smallest integer length is 16



Example 2

Question: Each choice below

represents three suggested side lengths

for a triangle. Which of the following

suggested choices will not result in a

triangle?

a) (2, 5, 6) b) (3, 7, 7) c) (3, 8, 12)

d) (5, 6, 7) e) (6, 6, 11)


The range of possible triangle side

lengths for each answer choice.


essential information? Evaluate the

first two numbers of each answer choice

using triangle side length strategy. Test

the third number of each answer choice

by comparing to range of possibilities

based on first two numbers.

Solution Steps

1) Determine range of possible side

lengths using first two numbers

2)Test third number of each answer choice

a) 5 - 2 < x < 5 + 2 3 < x < 7

b) 7 - 3 < x < 7 + 3

c) 8 - 3 < x < 8 + 3

d) 6 - 5 < x < 6 + 5

e) 6 - 6 < x < 6 + 6

4 < x < 10

5 < x < 11

1 < x < 11

0 < x < 12

a) (2, 5, 6) b) (3, 7, 7) c) (3, 8, 12)

d) (5, 6, 7) e) (6, 6, 11)

Correct answer choice is “c”

yes

yes

no

yes

yes



Strategy: Under construction

Reasoning:

Application:

Back to

Frequent

Questions



Example 1

Question: In the figure to the right, what

is the value of “a” ?




Solution Steps

x

x

a

4

3

8



Example 2




Solution Steps

Question: In the figure to the right,

, , , and

What is the length of ?

DEAC || 2BD 4DA 3DE

AC

ED

C

B

A


The Slippery Slope

Strategy: When given linear equations

as answer choices and a question about

the amount of change in the “y” variable

as the “x” variable is changed a given

amount, use the properties of slope to

quickly select the correct choice.

Reasoning: Slope is a measure of the

amount of change in the “y” value when

the “x” value is changed by one unit. The

constant in the equation has no impact

on the amount of change in the

dependent variable value.

Application: Any question

a) d = 50t - 100

e) d = -500t + 10000

b) d = 40t + 1000

c) d = 40t + 100 d) d = -50t + 1000

If d represents the distance measured in

meters from a particular coffee shop and

t is time measured in minutes, which of

the following equations describes the

greatest increase in distance from the

coffee shop during the period from t = 5

minutes to t = 8 minutes?

Caution: Do not calculate distance values

by direct substitution into each equation.

Use properties of slope to quickly

determine answer. Click for correct

choice.


The Slippery Slope

Example 1

Question: The table to the right gives

the value in dollars of five different

investments at t years after the

investment was started. The value of

which investment falls the greatest

amount during the period t = 4 to t = 9 ?




Solution Steps

Investment Value at t

Years

A -30t + 50

B -10t - 50

C -10t + 50

D 10t - 50

E 30t - 50


The Slippery Slope

Example 2






Solution Steps


Strategy: Replace the variable in the

function expression (right side of equal

sign) with the value, letter, or expression

that has replaced the variable (usually x)

in the function notation (left hand side of

equal sign)

Reasoning: Function notation is a road

map or guide that directly connects the

“x” value for a given function with one

unique “y” value.

Application: Function notation can be

applied in many different ways on the

SAT. See examples for details. Function

notation is commonly used to describe

translations and reflections of functions.

See Table of Contents for additional

strategies that use function notation.

Function notation such as f(x), g(x), and

h(x) are useful ways of representing the

dependent variable “y” when working

with functions. For example, the function

y = 2x + 5 can be written as f(x) = 2x + 5,

g(x) = 2x + 5, or h(x) = 2x + 5.

Introduction

Important Note: Function notation is

not a mathematical operation.

See example of commonly made

mistake.

Back to

Definition

Back to

Frequent

Questions



Example of Common Mistake

Question: At a certain factory, the cost

of producing control units is given by the

equation C(n) = 5n + b. If the cost of

producing 20 control units is $300, what

is the value of “b”?

Common mistake: Function notation

should not be used as a math operation.

C(n) should be replaced with 300 when

n = 20. Do not multiply 300 and 20 as in

a math operation.

Correct use of function notation: C(n)

is replaced with 300 when n is replaced

with 20 in the function equation.

Solution Steps for

Commonly Made Mistake

1) Replace “C” with 300 and replace

“n” with 20

C(n) = 5n + b

300(20) = 5(20) + b

6000 = 100 + b

b = 5900 (incorrect answer)

C(n) = 5n + b

Correct Solution Steps

300 = 5(20) + b

300 = 100 + b

b = 200 (correct answer)

Return to Table of Contents Return to strategy page See example of strategy


Example 1

Question: If f(x) = x + 7 and 5f(a) =15,

what is the value of f(-2a)?


The value of “a” is needed to determine

the value of f(-2a).


essential information?: Use the given

information and properties of function

notation to identify the value of “a”. Use

this value to evaluate f(-2a).

Solution Steps

1) Find the value of “a”

Given 5f(a) = 15 Divide both sides by 5

Result f(a) = 3

Given f(x) = x + 7 Evaluate f(a)

f(a) = a + 7 = 3

Result: a = -4

2) Use a = -4 to find f(-2a)

f(-2a) = f[(-2)(-4)] = f(8) Evaluate f(8)

f(8) = 8 + 7

f(-2a) = 15



Example 2

Question: The graph of y = f(x) is shown

to the right. If the function y = g(x) is

related to f(x) by the formula g(x) =

f(2x) + 2, what is the value of g(1)?


The math expression g(1) from which the

value of g(1) can be determined


essential information? Find the

expression for g(1) by substitution and

the value of g(1) using the graph of

y = f(x).

Solution Steps

y = f(x)2

2

-2

-2

1) Find the expression for g(1)

g(x) = f(2x) + 2

g(1) = f(2) + 2

2) Find the value of f(2) from the

graph of y = f(x)f(2) = 2

g(1) = 2 + 2 g(1) = 4



Example 3

Question: Using the table to the right, if

f(3) = k, what is the value of g(k)?


The value of “k” is needed to find g(k).


essential information? Use the table of

function values to find “k”. Once known,

find g(k) using the table of function

values.

Solution Steps

x f(x) g(x)

1 3 8

2 4 10

3 5 8

4 6 6

5 7 4

1) Find the value of “k” using table.

f(3) = k

2) Find the value of g(5) using table.

f(3) = 5

g(5) = 4



Example 4

Question: If f(x) = x + 8, for what value of

x does f(4x) = 4?


Need to determine the value of “x” that

satisfies f(4x) = 4.


essential information? Use function

notation principles to determine an

expression for f(4x). Set the expression

equal to the value of 4.

Solution Steps

1) Determine an expression for f(4x)

2) Set the expression for f(4x) equal to

4 and solve for the value of “x”

f(x) = x + 8

f(4x) = 4x + 8

f(4x) = 4x + 8 = 4

4x + 8 = 4

4x = -4

x = -1


Function Reflectionsx - Axis

Strategy: The reflection of a function y =

f(x) around the x-axis is easily performed

by graphing the opposite (negative) of

each y-value. Using function notation,

this can be communicated as y = - f(x).

Reasoning: The reflection of a function

around the x-axis can be viewed as a

mirror image of the original reflection.

Imagine the x-axis as a flat mirror that

reflects and produces an image of the

original function on the opposite side of

the x-axis.

Application: x-axis reflections can be

performed for any function using the

strategy described above.

y = f(x)

y = - f(x)

Reflection of f(x)


Function Reflections: x - Axis

Example 1

Question: If point (a, b) is reflected over

the x-axis, what are the coordinates of

the point after the reflection?


Must determine which, if any, coordinate

signs will be affected.


essential information?: For an x-axis

reflection, use the function notation

y = -f(x) as a guide.

Solution Steps

A reflection over the x-axis is

described by y = -f(x). To accomplish

the reflection, change the sign of the

y-coordinate only.

Correct answer is (a,-b)

Note: Do not get confused by the

original sign of the y-coordinate. If

the original sign is “-y”, the reflected

point will have the sign “+y”.


Function Reflections: x - Axis

Example 2






Solution Steps

Function Reflectionsy - Axis

Strategy: The reflection of a function y =

f(x) around the y-axis is easily performed

by graphing the opposite (negative) of

each x-value. Using function notation,

this can be communicated as y = f(-x).

Reasoning: The reflection of a function

around the y-axis can be viewed as a

mirror image of the original reflection.

Imagine the y-axis as a flat mirror that

reflects and produces an image of the

original function on the opposite side of

the y-axis.

Application: y-axis reflections can be



y = f(x)y = f(-x)

Reflection of f(x)


Function Reflections: y - Axis

Example 1


Must determine which, if any, coordinate

signs will be affected.


essential information?: Helps to

recognize that f(x) = f(-x) describes a

reflection about the y - axis.

Solution Steps

1) Reflect f(x) about the

y - axis ( click to show

reflection)

2) Identify the point for which f(x) = f(-x)

Question: For the graph of the function

f shown above, for what point does f(x)

= f(-x)?

(-1, 0)

(0, 1)

(2, 2)

(5, 0)

•The only point that remains the same

after reflection is the y intercept

f(0) = 1 and f(-0) = 1

Correct choice is (0, 1)


Function Reflections: y - Axis

Example 2






Solution Steps

Function ReflectionsAbsolute Value

Strategy: The absolute value of function

y = f(x) is easily created by graphing the

opposite (negative) of each y-value that

is negative on the original function.

Using function notation, this can be

communicated as y = |f(x)|.

Reasoning: The absolute value of a

function is a reflection of y = f(x) around

the x-axis for those intervals of x that

have negative y values.

Application: Absolute value can be

created for any function using the


y = f(x)y = |f(x)|

Back to

Definition


Function Reflections: Absolute

Value Example 1


Need to determine the effect of absolute

value on the graph of f(x)


essential information?: The absolute

value strategy should be used.

Solution Steps

A B C

D E

Question: The graph of y = f(x) is

shown above. Which of the choices

could be the graph of y = │f(x)│?

The absolute value reflects the graph of

y = f(x) about the x- axis for intervals of

“x” where f(x) < 0.

Correct answer is choice C


Function Reflections: Absolute

Value Example 2






Solution Steps

Function TranslationsHorizontal Shift

Strategy: A horizontal shift of a function

y = f(x) is easily performed by sliding the

function right or left parallel to the x-axis

a specified distance. Using function

notation, a shift to the right of 2 units

can be communicated as y = f(x-2). A

shift to the left of 4 units can be

communicated as y = f(x+4)

Reasoning: A horizontal shift described

by y = f(x-2) has the same y-value at x =

2 as the original function f(x) at x = 0.

Application: Horizontal shifts can be



y = f(x)

y = f(x-2)

y = f(x+4)

2

2


Function Horizontal Shift

Example 1

Question: The graph of y = f(x) is shown

to the right. Which of the following could

be the graph of y = -f(x+1) ? Click to see

answer choices


Need to interpret the impact of -f(x+1) on

the original function y = f(x).



function notation strategy and the

properties of function reflections and

translations to choose the correct

answer.

Solution Steps



A

Horizontal shift left

y = f(x+1)

E

Horizontal shift left

x-axis reflection

y = -f(x+1)

C

Horizontal shift right

y = f(x-1)

D

Horizontal shift right

x-axis reflection

y = -f(x-1)

B

x-axis reflection

y = -f(x)

-1 2

y = f(x)


Function Horizontal Shift

Example 2






Solution Steps

Function TranslationsVertical Shift

Strategy: A vertical shift of the function

y = f(x) is easily performed by sliding the

function up or down parallel to the y-axis

a specified distance. Using function

notation, a shift down of 2 units can be

communicated as y = f(x)-2. A shift up of

4 units can be communicated as y =

f(x)+4

Reasoning: A vertical shift described by

y = f(x)-2 decreases the y-value by 2

units for each value of x on the original

function y = f(x).

Application: Vertical shifts can be



y = f(x)

y = f(x)- 2

2

y = f(x)+4


Function Vertical Shift

Example 1

Question: The figure to the right shows

the graph of function f(x) in the x-y

coordinate plane. If the area between

f(x) and x-axis is 10, what is the area

between the function f(x)+2 and x-axis ?




Solution Steps

y = f(x)

5


Function Vertical Shift

Example 2






Solution Steps

Function TranslationsVertical Stretch

Strategy: A vertical stretch of the

function y = f(x) is easily performed by

multiplying each y-value by a specified

amount greater than one. Using function

notation, a vertical stretch of 2 units can

be communicated as y = 2f(x).

Reasoning: A vertical stretch described

by y = 2f(x) multiplies each y-value by 2

units for each value of x on the original

function y = f(x).

Application: Vertical stretches can be



y = f(x)

y = 2f(x)

Multiply each

y-value by 2


Function Vertical Stretch

Example 1


Need to understand the impact on y = f(x)

when f(x) is multiplied by 2.


essential information?: y = 2f(x)

describes a vertical stretch. Apply the

properties of a vertical stretch to y = f(x).

Solution Steps

Question: The graph of y = f(x) is

shown above. Which of the choices

could be y = 2f(x)?

A B C

D E

A vertical stretch multiplies each “y”

value on f(x) by two. As a result, the x-

intercepts remain the same on y = 2f(x).

The correct answer choice is E


Function Vertical Stretch

Example 2






Solution Steps

Function TranslationsVertical Shrink

Strategy: A vertical shrink of the function

y = f(x) is easily performed by multiplying

each y-value by a specified amount

between zero and one. Using function

notation, a vertical shrink of ½ units can

be communicated as y = ½f(x).

Reasoning: A vertical shrink described

by y = ½f( x) multiplies each y-value by

½ units for each value of x on the original

function y = f(x).

Application: Vertical shrinks can be



y = f(x)

y = ½f(x)

Multiply each

y-value by ½


Function Vertical Shrink

Example 1






Solution Steps

Function Vertical Shrink

Example 2






Solution Steps


Problems

Strategy: Apply the basic definition of

average (arithmetic mean) to solve this

class of problems.

Reasoning: Information will typically be

given for the average and the number

of values. The sum of values will be

always be needed to reason through

question and will typically consist of an

expression with unknown variable(s).

Application: 1) Problems that ask for

an unknown value when given

remaining values in the list and the

average value of the list. 2) Problems

that provide the average of a list of

numbers, removes a number from the

list, gives the new average, and asks for

the value of the removed number.

sum of values

number of valuesaverage =

Caution: You will rarely be asked

to find the average of a list of

values. Instead, you will typically

be asked to find the median of a

list of values.

Often used form:

sum of values =

(average)( number of values)

Back to

Definition

Back to

Frequent

Questions



Example 1

Question: If the average of 6 and x is 12,

and the average of 5 and y is 13, what is

the average of x and y?


Need values of x and y to determine

average value.


essential information?: Apply basic

definition of average to find values of x

and y separately.

Solution Steps

1) Determine the values of x and y:

6 + x

2= 12 5 + y

2= 13

6 + x = 24 5 + y = 26

x = 18 y = 21

2) Find average of x and y using

basic definition of average:

18 + 21

2Average =

Average = 19.5



Example 2

Question: The average of five positive

odd integers is 15. If n is the greatest of

these integers, what is the greatest

possible value of n?


The sum of the five positive odd integers

is needed and a strategy to determine

the greatest possible value of “n”


essential information? Apply the

definition of average to find sum. Use

reasoning skills to determine greatest

possible value of “n”

Solution Steps

1) Find the sum of the five positive

odd integers.

Sum of values = (15)(5) = 75

2) Determine the greatest possible

value of “n” using reasoning skills

•The four smallest positive integers are

1, 1, 1, 1 with a sum of four.

•The greatest possible value of “n” is

n = 75 - 4 = 71



Strategy: The middle value in a list of

ascending or descending ordered values

is the median. Large lists of values

(more than 7 values) are usually

structured in table form or bar chart

form. Either form will not require

rewriting of the order by the student.

Reasoning: Values provided in table

form are similar to values provided in

histogram form. In both forms it is easy

to determine the cumulative total number

of values starting with the lowest value.

Application: When values are organized

in tables, questions will generally ask for

the median directly or will give the

median and ask for the value of an

unknown variable.

Caution: Do not confuse median with

mean. When presented a table of

values or a list of values, the question

typically requires determination of the

median, not the mean.

Additional Helpful Hints

1) For an ordered list with an odd

number of values, the median is the

middle value. 2) For an ordered list with an even

number of values, the median is the

average of the two middle values.

Back to

Definition

Back to

Frequent

Questions



Example 1

Question: The scores on a recent

physics test for 20 students are shown in

the table to the right. What is the median

score for the test?


When the test scores are ordered from

largest to smallest, find the middle score

for the list.


essential information?: With the test

scores in table form, no additional

ordering is needed. With 20 students,

the median is the average of the scores

of the 10th and 11th students.

Solution Steps

Score Number of Students

100 0

95 1

90 1

85 2

80 3

75 4

70 3

65 2

60 4

The 8th ,9th ,10th ,and 11th

students each received a

score of 75 on the test

Median score is 75

0

1

1

2

3

Sum = 7

0

1

1

2

3

4

Sum = 11

The 5th , 6th , and 7th

students each received a

score of 80 on the test



Example 2

Question: If the median of 10

consecutive odd integers is 40, what is

the smallest integer among these

integers?




Solution Steps



Strategy: Divide the number of values

that meet the given criteria by the total

number of values in the set.

Reasoning: This is the basic definition

of probability. The probability of an event

is a number between 0 and 1, inclusive.

If an event is certain, the probability is 1.

If an event is impossible, the probability

is 0.

Application: Additional applications

include finding the probability of choosing

a particular object (marbles, cookies,

coins) from a container with more than

one type of object.

Given information:

{10, 12, 13, 18, 21, 23, 25, 29}

Question:

What is the probability of choosing

a prime number at random from

the above set?

Essential information:

1)The number of values meeting

the question criteria is 3

2)The total number of values in

the set is 8

Solution:

Probability = ⅜

Back to

Definition



Example 1

Question: A jar contains red, blue, and

yellow marbles in the ratio 9:4:2. If a

marble is selected at random, what is the

probability of selecting a blue marble?


The ratio of number of blue marbles to

the total number of marbles.



properties of ratios to determine the

essential information. Use the ratio to

determine the probability.

Solution Steps

1) Determine the ratio of blue marbles

to total number of marbles

2) Determine the probability

•For every 15 total marbles in the jar (9

+ 4 + 2 = 15) there are 4 blue marbles

•The probability can be found by using

the ratio of blue marbles to total

marbles.

Note: It is not necessary to know the

exact number of each marble in the

jar. Ratios are sufficient for probability.

Probability =4

15



Example 2

Question: A certain bowling center has

two sizes of bowling balls, twelve

pounds and sixteen pounds. For every 3

twelve pound bowling balls there are 4

sixteen pound bowling balls. If a bowling

ball is chosen at random, what is the

probability that a sixteen pound bowling

ball will be selected?


The ratio of the number of sixteen pound

bowling balls to the total number of

bowling balls.



properties of ratios to determine the

essential information. Use the ratio to

determine the probability.

Solution Steps

1) Determine the ratio of blue marbles

to total number of marbles

2) Determine the probability

•For every 7 bowling balls (3 + 4 = 7),

there are 4 sixteen pound bowling balls

•The probability can be found by using

the ratio of sixteen pound bowling balls

to the total number of bowling balls

Probability =4

7

Note: The strategy for this problem is

identical to the previous example. The

questions are slightly different, however

both involve ratios


Probability of Independent

Events

Strategy: Multiply the probabilities of the

individual events together to find the

overall probability.

Reasoning: Each individual first event

must be paired with each individual

second event. To account for the total

number of outcomes meeting the given

criteria (value in numerator) and the

total number of possible outcomes

(value in denominator), the individual

probabilities must be multiplied together.

Application: Popular applications

include the probability of an outcome

when a coin is flipped multiple times and

the probability of passing multiple

academic courses

Definition: Two events are independent if

the outcome of the first event has no effect

on the outcome of the second event

Example: David has a red, yellow, blue,

and green hat. He also has a red and blue

shirt. If an outfit consists of a hat and shirt,

what is the probability that David will wear

an all red outfit?

Solution: The probability of choosing a

red hat is ¼ and the probability of

choosing a red shirt is ½.

The overall probability is (¼)(½) = ⅛

Back to

Definition

Back to

Frequent

Questions



Example 1

Question: Adam has a 90% chance of

passing history and a 60% chance of

passing calculus. What is the probability

that Adam will pass calculus and not

pass history?


Are these events independent of each

other?


essential information?: It can be

assumed that passing history is

independent of passing calculus. The

two events are independent and the

individual probabilities can be multiplied

together.

Solution Steps

1) Determine the probability that Adam

will pass calculus


will not pass history


will pass calculus and not pass history

Overall probability = 1

10

6

10 x =

6

100 =

3

50

Probability = 60

100

6

10 =

Probability = 10

100 =

1

10



Example 2

Question: The three cards shown to the

right were taken from a box of ten cards,

each with a different integer from 0 to 9.

What is the probability that the next two

cards selected from the box will both

have an even integer on it?




Solution Steps

1 5 7



Strategy: Divide the area of the smaller

geometric shape by the area of the larger

geometric shape.

Reasoning: For planar geometrical

shapes, area is the proper quantity to

compare when selecting a point inside

the figure.

Application: Usually involve simple

shapes such as circles, rectangles, and

squares. In all cases there is a smaller

shape inside the larger shape and the

analysis requires calculation of shape

area.

Definition: Geometric probabilities involve

the use of two or more geometric figures.

Example: A small circle with radius 3 is

completely inside a larger circle with

radius 6. If a point is chosen at random

from the large circle, what is the probability

that the point will be in the small circle?

Essential information:

1) Area of small circle is π(3)2 = 9π

2) Area of large circle is π(6)2 = 36π

Solution:

Probability = ¼

Back to

Definition

Back to

Frequent

Questions



Example 1

Question: In the figure above, each of

the small circles has a radius of 3 and

the large circle has a radius of 9. If a

point is chosen at random inside the

larger circle, what is the probability that

the point does not lie in the shaded

area?


Need the area of the large circle and area

of each of the smaller circles.



formula for area of a circle to find areas

of each circle. To find probability, ratio

the area of the shaded region to the area

of the large circle.

Solution Steps

1) Find the area of each circle

2) Find the geometric probability

Area of each small circle = π(3)2 = 9π

Area of large circle = π(9)2 = 81π

Probability = 81π - 2(9π)

81π=

63π

81π

Probability =7

9



Example 2


Need to determine the area of triangle

ABC and the area of the rectangle. The

length of AB is needed to find both areas.


essential information? Side AB is twice

the radius of circle C. Knowing AB, use

Pythagorean theorem to find AC and CB.

Solution Steps

Question: The rectangle above with

side length 4 contains circle C that has

a radius of 1. If a point is chosen at

random inside the rectangle, what is the

probability that the point will lie in

triangle ABC?

C

4 B

A

1) Find the area of triangle ABC

2) Find the probability

•Triangle ABC is a 45-45-90 triangle

•AB is twice the radius of circle C and

has a length of 2

•AC and CB are congruent and are

each equal to √2

Area = ½(√2)(√2) = 1

Probability = area of triangle

area of rectangle = 1

(2)(4)

Probability = ⅛


The Unit Cell

Strategy: Divide the given end of the

metal strip into a smaller shape, called a

“unit cell”, that can used to easily and

quickly answer the question. Click to

show the unit cell!

Reasoning: The unit cell is a repeating

shape that comprises the entire object

shape. Ten unit cells comprise the entire

metal strip. Click to see calculation.

The top horizontal section and the

bottom notched section of each unit cell

contributes 3 + 1 + 3 + 1 = 8 inches to

the perimeter.

Application: Any question that provides,

in the form of a figure, a representative

section of a longer object.

One end of a 30-inch long metal strip is shown

in the figure above. The lower edge was formed

by removing a 1-in square from the end of each

3-inch length on one edge of the metal strip.

What is the total perimeter, in inches, of the 30-

inch metal strip?

The “Unit Cell”

1 in

3 in

1 in

1 in

The total perimeter is equal to:

10 unit cells x 8-in/unit cell + 2 vertical sides x 3-in

Perimeter = 86 inches

Three “Unit Cells” shown

Leftover section

Not a unit cell

30-in strip

3-in unit cell= 10 unit cells

Back to

Frequent

Questions


The Unit Cell

Example 1






Solution Steps

The Unit Cell

Example 2






Solution Steps


Strategy: Under construction


Reasoning:

Application:


Example 1






Solution Steps


Example 2






Solution Steps

Making ConnectionsThe “if…” Statement

Back to

Frequent

Questions

Strategy: For questions that begin with

“If…” and end with “what is the value

of…”, or “which of the following must

equal…”, find a straightforward

connection that links the given

information (usually an equation) to the

desired answer (usually the value of an

expression).

Reasoning: The questions are designed

to be solved in a straightforward way,

provided the connection between the

given information and the desired

answer is made. To find the connection

typically requires out of the box thinking.

Example 1: If 4x2 = 18y = 36, what is the

value of 2x2y?

Example 2: If 2x + 7y = y, which of the

following must equal 4x + 12y ?

Example 1

4x2 = 18y = 36 2x2y?

Connection #1: Set 4x2 = 36. Solve for 2x2

Connection #2: Set 18y = 36. Solve for y

Connection?

Example 2

2x + 7y = y 4x + 12y

Connection: Subtract y from both sides of

equation. Result is 2x + 6y = 0. Multiply

both sides of equation by 2.

Connection?


Making Connections

Example 1

Question: If x is positive and x(x-1) = 30,

what is the value of x(x+1) ?


Need to find a connection between the

factored form of the expression on the

left side of the equal sign and the value

of 30 on the right side.


essential information?: The factors on

the left side are consecutive integers.

Determine if the value 30 has factors

that are consecutive positive integers.

Note: Not necessary to foil the

expression and solve as a quadratic

equation x2 - x - 30 = 0

Solution Steps

1) Identify the factors of 30 that are

consecutive integers:

6(6-1) = 6(5) = 30

x = 6

2) Find the value of x(x+1) for x = 6

6(6+1) = 6(7) = 42


Making Connections

Example 2

Question: If x and y are positive

numbers and , then what is the

value of ?


Need to find a connection between the

equation and the expression.


essential information? Solve directly

for and substitute the result into the

expression

Solution Steps

1) Solve directly for

2) Substitute result into expression

y

x 99 yx

9x

9x

9 yx

09 yx

yx 9

19

y

y

y

x



Strategy: If uncertain of parallel line

properties, use the diagram appearance

to determine the relationship between

pairs of angles. Note: This strategy is

valid if and only if the figure is drawn to

scale.

Reasoning: Any pair of angles will either

be congruent (equal measure) or

supplementary (sum to 180 degrees).

Using the figure given in a question, it is

usually obvious when angles are

congruent. If they do not appear

congruent, they are supplementary.

Application: Many questions contain

parallel lines with two transversals (see

example 2).

In the figure shown above, pairs of

red or pairs of blue angles are

congruent. A pair consisting of a red

and blue angle are supplementary.

Parallel

Lines

Back to

Frequent

Questions



Example 1

Question: In the figure to the right, if m is

parallel to n, what is the value of x ?


Determine the measures of the two

remaining angles inside the triangle that

contains angle x.



properties of parallel lines and

transversals to determine the measures

of the two angles.

Solution Steps

1) The two remaining angles inside

the triangle are 50o (congruent to the

50o angle) and 65o (supplementary to

the 115o angle). Click again to see

animation of the angles.

2) Calculate the measure of angle x:

x = 180 - (50 + 65)

x = 65o

50o 65o

115o

115o

50o

xo

n

m

q

p



Example 2

Question: In the figure to the right, if m is

parallel to n, what is the value of x + y ?


Need to define the two remaining angles

inside the triangle in terms of x and y.



properties of parallel lines and

transversals to define the measures of

the two angles in terms of x and y.

Solution Steps

1) The two remaining angles inside the

triangle are 180 - x (supplementary to

angle x) and 180 - y (supplementary to

angle y). Click again to see

animation of the angles.

2) Calculate the measure of angle x:

(180 - x) + (180 - y) + 55 = 180

x + y = 235o

yoxo

55o

180 - yo180 - xo

180 - xo 180 - yo

m

n


Even/Odd Integers

Strategy: Use the table of properties to

the right to determine if an operation

between two integers will result in an

even or odd integer.

Reasoning: These integer formation

properties eliminate the need to use the

“plug in a number” strategy that is often

more time consuming than applying the

integer properties.

Application: There is always at least one

question that can be easily solved using

these integer formation properties.

Addition or

Subtraction

Multiplication

odd + odd = even

odd - odd = even

odd x odd = odd

even + even = even

even - even = even

even x even = even

odd + even = odd

odd - even = odd

odd x even = even

Back to

Frequent

Questions


Even/Odd Integers

Example 1

Question: If a + b is an even integer,

which of the following must be even?

a) 2a + b b) 2a - b

c) ab d) (a + 1)(b + 1)

e) a2 - b2




Solution Steps


Even/Odd Integers

Example 2

Question: If 2a + b is an odd integer,

which of the following must be true?

I. a is odd

II. b is odd

III. 2a2 - b2 is odd




Solution Steps



Strategy: Express the sum of three

consecutive integers, consecutive odd

integers, or consecutive even integers as

the sum of the expressions shown to the

right.

Reasoning: When you count by one’s

from any number in the set of integers,

consecutive integers are obtained. If you

count by two’s beginning with any

even/odd integer, consecutive even/odd

integers are obtained.

Application: Questions that ask for the

smallest of three consecutive integers or

consecutive odd/even integers when

their sum is a specified value. Any

question that begins with the phrase

“Given three consecutive integers”.


n, n + 1, n + 2

Where n is any integer

Consecutive Odd Integers

n, n + 2, n + 4

Where n is an odd integer

Consecutive Even Integers

n, n + 2, n + 4

Where n is an even integer



Example 1

Question: The average of a set of 5

consecutive even integers is 20. What is

the smallest of these 5 integers?


Find the sum of the 5 consecutive even

integers. Use the sum to find the smallest

integer.



definition of average to find the sum. Use

the sum and the consecutive integer

strategy to find the smallest integer.

Solution Steps

1) Find the sum of the 5 integers

using the definition of average

2) Find the smallest integer using

consecutive even integer strategy

205

valuesofsum

valuesofnumber

valuesofsumaverage

100 valuesofsum

)8()6()4()2( nnnnnvaluesofsum

100205 nvaluesofsum

16n



Example 2

Question: What is the median of 7

consecutive integers if their sum is 42?


The fourth value in a list of seven

consecutive integers.



consecutive integer strategy to find the

smallest integer. Add three to the

smallest integer to find the value of the

fourth integer. This will be the median

value.

Solution Steps

1) Find the smallest integer in a list

of seven integers.

2) Find the median value by adding

three to the smallest integer.

42)6()5()4()3()2()1( nnnnnnn

42217 n

3n

3 nvaluemedian

6 valuemedian


Tangent To A Circle

Strategy: If a line is drawn tangent to a

circle, draw the radius of the circle to the

point of tangency with the line.

(Click again to draw radius)

Reasoning: A tangent line and the radius

always form a right angle at the point of

tangency. The right angle relationship will

be used in all applications involving

tangent lines to circles.

Application: Find the slope of the

tangent line when given the coordinates

of the point of tangency with the circle

and the center of the circle. Find the

perimeter of a shape when a circle is

inscribed inside the given shape.

Tangent line

Back to

Frequent

Questions


Tangent To A Circle

Example 1

Question: In the figure to the right, a

circle is centered at the origin and is

tangent to the line at point P. If the radius

of the circle is 15, what is the slope of

line?


needed? The radius and line are

perpendicular to each other. Find the

radius slope and use the relationship that

the slope of perpendicular lines are

opposite reciprocals of each other.


essential information?:Use the radius

length and the x-coordinate of point P to

find b, the y-coordinate of point P. This is

accomplished using Pythagorean

Theorem.

Solution Steps

P(9, b)

1) Using Pythagorean Theorem, the

y-coordinate, b, has a value of -12.

The slope of the radius is:

9

15

12

2) Find the slope of line using the

relationship between the slopes of

perpendicular lines. Slope of line is

3

4

9

12

09

012

4

3

34

1

P(9, -12)


Tangent To A Circle

Example 2

Question: In the figure to the right, a

circle is tangent to the side of equilateral

triangle xyz and the radius equals 5.

What is the perimeter of triangle xyz ?


The length of a side of the triangle.


essential information? The circle

radius and the equilateral triangle side

are perpendicular at the tangent point.

Draw a right triangle and use the

properties of the 30-60-90 triangle to

find the side length.

Click again to show the right triangle

Solution Steps

3) The perimeter is three times the

triangle side length:

x

y

z

30

60

35

5

Radius

1) Using properties of the 30-60-90

triangle, the length of half the triangle

side is .

2) The triangle side length is .

35

310352

3303103


Strategy Section Concluded

Return to Table of Contents Return to first strategy Return to Introduction

This is the end of the Strategy

section. Please select one of the

options at the bottom of this page

Sample Strategy

Strategy: The 3rd side of any triangle is

greater than the difference and smaller

than the sum of the other two sides


Reasoning: A side length of 15 would


triangle

Application: A side length of 3 would

also require the formation of a line, not a

triangle

Sample Strategy

1) The total cost of 4 equally priced notebooks is $5.00. If the price is

increased by $0.75, how much will 6 of these notebooks cost at the new rate?

(A) $7.50

(B) $8.00

(C) $10.00

(D) $12.00

(E) $14.00





Sample Strategy

2) If Jim traveled 20 miles in 2 hours and Sue traveled twice as far in twice

the time, what was Sue’s average speed, in miles per hour?

(A) 5

(B) 10

(C) 20

(D) 30

(E) 40





Sample Strategy

3) In the figure below, if CD is a line, what is the value x ?

(A) 45

(B) 60

(C) 90

(D) 100

(E) 120





C Dx0

x0 x0x0

x0x0

y0

Note: Figure not drawn to scale.

Sample Strategy

4) For which of the following functions is f(-2) > f(2) ?

(A) 3x2

(B) 3

(C) 3/x2

(D) x2 + 2

(E) 3 - x3





Sample Strategy

5) The energy required to stretch a spring beyond its natural length is

proportional to the square of how far the spring is being stretched. If an

energy of 20 joules stretches a spring 4 centimeters beyond its natural

length, what energy, in joules, is needed to stretch this spring 8 centimeters

beyond its natural length?

(A) 10

(B) 40

(C) 80

(D) 100

(E) 120


What essential information is needed?What is the strategy for identifying


Sample Strategy

6) The average (arithmetic mean) of x and y is 10 and the average of x, y,

and z is 12. What is the value of z ?

(A) 2

(B) 4

(C) 12

(D) 16

(E) 26





Sample Strategy

7) If Z is the midpoint of XY and M is the midpoint of XZ, what is the length of

ZY if the length of MZ is 2 ?

(A) 2

(B) 4

(C) 6

(D) 8

(E) More information is needed to answer question





Sample Strategy

8) In the figure below, line L is parallel to line m. What is the value of x ?

(A) 110

(B) 120

(C) 130

(D) 140

(E) 150





x0

600

1100

M

L

Sample Strategy

9) If a and b are odd integers, which of the following must also be an odd

integer?

(A) I only

(B) II only

(C) III only

(D) I and II

(E) II and III





I. (a + b)b

II. (a + b) +b

III. ab +b

Sample Factoring Strategy

Example 1

Question:





Solution Steps

Sample Factoring Strategy

Example 2

Question:





Solution Steps

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