Success step english 1 potx

Integer An integer is simply a number with no fraction or dec-imal attached {.. Integers include both negative −5 and positive 9,687 numbers.. Zero is also an integer, but is considered

8 8

1 Read the graph carefully Read all around the graph, including the title and the key.

2 Some questions may try to trick you by leaving out numbers If all the numbers are not given, it is a very good

idea to fill in all the missing numbers on the graph To do this, you will need to know the value of each increment

3 Sometimes, instead of reading bars or lines, you can compare differences by using a piece of your test

book-let to measure from one point to another or from the end of one bar to the end of another

Line Graphs

3 In what year is the increase in student

popula-tion projected to be less than the increase in

number of new homes built?

a 1998

b 1999

c 2000

d 2001

e 2002

Answer

The answer is c A look at the graph shows that during

the year 2000 there was a sharper increase in the

num-ber of new homes built than in student population

The line slopes up steeper there for houses than it does

for student population Percent of increase is a

differ-ent question and might yield a differdiffer-ent answer Check

Ratios, Proportions, and Percents (p 109) on percents

for details

Picture Graphs

4 How many MTAC members were there in 1990?

a 312

b 350

c 700

d 1,750

e 2,250 Answer

It is important to read the key at the bottom of the graph Each piano represents 500 members.12a piano represents 250 members 1990 has 312pianos This

rep-resents 1,750 members The answer is d.

Circle Graphs

5 For which category was the income for the Save

the Caves Foundation approximately 14?

a Mailings

b Sales

c Thrift Shop

d Phone

e Other

Answer

Look for the section that takes up approximately 14of

the circle The answer is a.

Here’s a different kind of circle graph question

6 Which of the following could include a

represen-tation of expenditures of 25% and 33%?

Answer

25% is the same as 14 33% is close to 13 The only answer with areas of14and 13of the circle is answer d Choice b is close, but the larger area seems a little large

for 13

e

d

c

b

a

Some graphs are just plain odd

7 The circles above represent the dials on an

elec-tric meter What reading do they represent?

a 476

b 465

c 466

d 486

e 487

Answer

There are no clues whatsoever on the graphs A look at

the answer choices reveals that all are three digits and

that all begin with the digit 4 Maybe each circle

rep-resents a digit If so, the first dial has to represent 4 If

the first dial represents 4, and the arrow is nearly

half-way around (assuming the dials go clockwise), then

half is probably 5 In that case, the last circle represents

5, which tells you your answer, choice b It makes sense

that the middle digit of 465 is 6, since the middle

cir-cle’s arrow goes just a bit past the 5 mark In order to

correctly answer this question, it was important that

you didn’t assume the graphs represent clocks

Ques-tions similar to this have been on the test before

All right, here’s a challenge

8 Which number best represents the speed

indi-cated on the speedometer above?

a 1212

b 3012

c 35

d 4012

e 45 Answer

It is very important to label the graph in order to answer this question If there are 11 segments between

10 and 120, each segment must represent 10 mph The arrow is pointing halfway between 30 and 40 Halfway

between 30 and 40 is 35 The answer is c.

 M a t h 1 : Wo r d s , Wo r d s , Wo r d s

Many times, an otherwise simple problem may seem difficult merely because the test writers have used terms that you are not familiar with An understand-ing of mathematical terms will enable you to under-stand the language of the problem and give you a better chance of solving that problem This lesson presents a list of words used when speaking about numbers

Here is a sample question that will show you why knowing these words is important Once you’ve learned the words in this sample question, you should

be able to answer it You can check your answer against the explanation given later in this lesson

9 0

Sample Definition Question

1 If x is a whole number and y is a positive integer,

for what value of x MUST x < y be true?

a. −3

b 0

c. 12

d 3

e Any value of x must make the statement true.

Definitions

Below are definitions of words you need to know to

answer CBEST math questions If some of these words

are unfamiliar, put them on flash cards: the word on

one side, and the definition and some examples on the

other side Flash cards are handy because you can carry

them around with you to review at odd moments

dur-ing the day If you use this method, it won’t take you

long to learn these words

Integer

An integer is simply a number with no fraction or

dec-imal attached { .−2, −1, 0, 1, 2 } Integers include

both negative (−5) and positive (9,687) numbers Zero

is also an integer, but is considered neither negative nor

positive in most mathematical texts

Positive Integer

A positive integer is an integer, according to the

defi-nition above, that is greater than zero Zero is not

included Positive integers begin with 1 and continue

infinitely {1, 2, 3 } Examples of positive integers are

5, 6,000 and 1,000,000

Negative Integer

A negative integer is an integer that is less than zero.

Zero is not included The highest negative integer is

negative one (−1) The negative integers go down

infi-nitely { .−3, −2, −1} Some examples of negative

inte-not fit the definition of a negative integer:−4.5 (not an integer because of the decimal), 0, and 308

Zero

Zero is an integer that is neither positive nor negative

Whole Numbers

Whole numbers include all positive integers, as well as zero {0, 1, 2, 3 } Like integers, whole numbers do not include numbers with fractions or decimals

Digit

A digit is a single number symbol In the number 1,246, each of the four numerals is a digit Six is the

ones digit, 4 is the tens digit, 2 is the hundreds digit, and

1 is the thousands digit Knowing place names for

dig-its is important when you’re asked on a test to round

to a certain digit Rounding will be covered in the third math lesson

Real Numbers

Real numbers include all numbers: negative, positive, zero, fractions, decimals, most square roots, and so on Usually, the numbers used on the CBEST will be real numbers, unless otherwise stated

H O T T I P

Negative numbers appear smaller when they are larger To help you make sense of this concept, think of the degrees below zero on a thermometer Three degrees below zero

is hotter than 40 degrees below, so −3 is greater than −40, even though 40 appears to be a larger number Test mak-ers like to test your grasp of this principle.

Variables are letters, such as x and y, that are used to

replace numbers The letter is usually a letter of the

alphabet, although occasionally, other symbols are

used When a math problem asks you to “solve for y,”

that means figure out what number the letter is

replac-ing At other times, the problem requires you to work

with the letters as if they were numbers Examples of

both of the above will be covered in the lesson on

algebra on page 114

Reciprocal

The reciprocal of a fraction is the fraction turned

upside down For example, the reciprocal of38is 83and

vice versa The reciprocal of an integer is one over the

integer For example, the reciprocal of 2 (or 21) is 12and

vice versa To get the reciprocal of a mixed number

such as 312, first change the number to an improper

fraction (72) and then turn it over (27)

Numerator and Denominator

The numerator of a fraction is the number on top, and

the denominator is the number on the bottom The

numerator of67is 6 and the denominator is 7

= and 

The symbol = is called an equal sign It indicates that

the values on both sides of the = are equal to each

other For example, 7 = 2 + 5 A line drawn through an

equal sign (≠) indicates that the values on either side

are not equal: 8 ≠ 4 + 5

 and ;  and 

The symbol < means less than and the symbol > means

greater than The number on the closed side of the

symbol is smaller, and the number on the open side is

larger Thus, 3 < 5 and 10 > 2 Remember: The

alliga-tor eats the bigger number

The symbol ≤ means less than or equal to, and the symbol ≥ means greater than or equal to These two

symbols operate the same way as the < and >, but the added line means that it’s possible that the two sides

are equal Thus, in the equation x ≥ 3, x can represent

3 or any number greater than 3

Answer to Sample Definition Question

Using the definitions above, can you solve sample

question 1 from the previous page? The variable x can

be any whole number including zero The variable y

can be any positive integer, which doesn’t include zero.

The question reads “ for what value of x MUST x <

y be true?” Must means that x has to be less than y

under all circumstances You are being asked to replace

x with a number that will be less than any positive

inte-ger that replaces y The only whole number that would make x < y true, no matter what positive integer is put

in place of y, is zero Therefore, b is the correct answer.

Try another sample question Again, the defini-tions above will be useful in solving this problem

Sample Digits Question

2 In a certain two-digit number, the tens digit is

four more than the ones digit The sum of the two digits is ten What is the number?

a 26

b 82

c 40

d 37

e 73

9 2

There are two requirements for the unknown number:

The tens digit has to be four more than the ones digit,

and the two digits have to add up to 10 The best way

to solve the problem is to eliminate answers that don’t

meet these two requirements Consider the second

cri-terion first A glance at the answers shows that the

dig-its in the answers a and c do not add up to 10 They can

be eliminated Next, consider the first requirement

Answer b contains a tens digit that is six, not four,

more than the ones digit Answer d has the ones digit

four more than the tens, reversing the requirement

Therefore, e is the only number that correctly meets

the requirements

Practice with Definitions

Match the word on the left with the description or

example on the right You may want to write these

def-initions on flash cards

Answers

 M a t h 2 : N u m b e r s Wo r k i n g

To g e t h e r

In the last lesson, you learned the definitions of several mathematical terms This lesson will discuss ways in which numbers work together You will need this information to solve simple algebra and perform cer-tain arithmetic functions that will be part of some CBEST problems

Adding and Subtracting Integers

The following sample questions are examples of the types of problems about adding and subtracting you may see on the CBEST The answers are given later in this lesson

Sample Integer Questions

1 Every month, Alice’s paycheck of $1,500 goes

directly into her bank account Each month, Alice pays $800 on her mortgage payment and

$500 for food and all other monthly expenses She spends $1,650 per year on her car (insur-ance, gas, repairs, and maintenance), $500 per year for gifts, and $450 per year for property tax What will be her bank balance at the end of a year?

a $500

b $300

c 0

d.−$200

3 e.

4 c.

5 f.

6 a.

7 b.

8 g.

9 i.

10 h.

11 k.

12 j.

13 d.

3 integer

5 zero

6 negative integer

7 positive integer

8 digit

9 <

10 >

11. ≤

12. ≥

13 real number

a { .−3, −2, −1}

b {1, 2, 3 }

c {0, 1, 2, 3 }

d number set includ-ing fractions

e { .−3, −2, −1, 0, 1,

2, 3 }

f neither negative nor positive

g one numeral in a number

h greater than

i less than

j greater than or equal to

k less than or equal to