Don’t confuse the key words for the basic op-erations: Sum means the result of addition, dif-ference means the result of subtraction, product means the result of multiplication, and quo
Trang 1Integers and Real Numbers
On the SAT, you only need to deal with two kinds of
numbers: integers (the positive and negative whole
num-bers (all the numnum-bers on the number line, including
integers, but also including all fractions and
deci-mals) You don’t have to know about wacky numbers
such as irrationals or imaginaries
The SAT only uses real numbers It will never
(1) divide a number by 0 or (2) take the square
root of a negative number because both these
operations fail to produce a real number
Make sure that you understand why both these
operations are said to be “undefined.”
Don’t assume that a number in an SAT
prob-lem is an integer unless you are specifically told
that it is For instance, if a question mentions
the fact that x > 3, don’t automatically assume
that x is 4 or greater If the problem doesn’t
say that x must be an integer, then x might be
3.01 or 3.6 or the like
The Operations
The only operations you will have to use on the SAT
are the basics: adding, subtracting, multiplying,
divid-ing, raising to powers, and taking roots Don’t worry
about “bad boys” such as sines, tangents, or
logarithms—they won’t show up (Yay!)
Don’t confuse the key words for the basic
op-erations: Sum means the result of addition,
dif-ference means the result of subtraction,
product means the result of multiplication,
and quotient means the result of division.
The Inverse Operations
Every operation has an inverse, that is, another
oper-ation that “undoes” it For instance, subtracting 5 is
oper-ation and then perform its inverse, you are back to
No need to calculate!
Using inverse operations helps you to solve equations For example,
Alternative Ways to Do Operations
Every operation can be done in two ways, and one way is almost always easier than the other For instance, subtracting a number is the same
thing as adding the opposite number So
di-viding by a number is exactly the same thing
as multiplying by its reciprocal So dividing by
2/3 is the same as multiplying by 3/2 When doing arithmetic, always think about your op-tions, and do the operation that is easier! For
which is easier to do in your head
The Order of Operations
Don’t forget the order of operations:
P-E-MD-AS When evaluating, first do what’s grouped
in parentheses (or above or below fraction bars
or within radicals), then do exponents (or roots) from left to right, then multiplication
or division from left to right, and then do
ad-dition or subtraction from left to right What is
the multiplication before the division (Instead,
you mistakenly subtracted before taking care
of the multiplication and division If you said
5, pat yourself on the back!
When using your calculator, be careful to use parentheses when raising negatives to powers For instance, if you want to raise –2 to the 4th power, type “(–2)^4,” and not just “–2^4,” be-cause the calculator will interpret the latter as –1(2)^4, and give an answer of –16, rather than the proper answer of 16
Lesson 1: Numbers and Operations
Trang 2Concept Review 1: Numbers and Operations
3 When is taking the square root of a number not the inverse of squaring a number? Be specific.
6 The result of an addition is called a
7 The result of a subtraction is called a
8 The result of a multiplication is called a
What is the alternative way to express each of the following operations?
What is the inverse of each of the following operations?
Simplify without a calculator:
22 Circle the real numbers and underline the integers:
23 The real order of operations is _.
24 Which two symbols (besides parentheses) are “grouping” symbols? _
26 List the three operations that must be performed on each side of this equation (in order!) to solve for x:
5 6 8, , , and12
3 75 1 333 25 7 2
5
56
7 0
1− − −(2 (1 2)2)−2 2
1
2 6 2
1+ 22
2
−( − ÷ )
÷6 7
× −5 3
÷6 7
× −5 3
There’s a lot of detail to learn and understand to do well on the SAT For more tools and resources that will help, visit our Online Practice Plus at www.MHPracticePlus.com/SATmath.
Trang 31. Which of the following is NOT equal to 1⁄3of an
integer?
product of two consecutive even integers?
3.
and 3 is a crowd, then how many are 4 crowds
k− =3 8
1− − −(1 (1 3) ) 1 1 1 2
( )− − − −( ( ( ) ) )=
left What is half the number?
SAT Practice 1: Numbers and Operations
0 0 0
1 1 1 1
2 2 2 2
3 3 3 3
4 4 4 4
5 5 5 5
6 6 6 6
7 7 7 7
8 8 8 8
9 9 9 9
.
0 0 0
1 1 1 1
2 2 2 2
3 3 3 3
4 4 4 4
5 5 5 5
6 6 6 6
7 7 7 7
8 8 8 8
9 9 9 9
.
repeated digits is subtracted from the greatest four-digit integer without repeated digits, the result is
expressions increases as x increases?
I
II
III
10−1
x x
1
2
x
least integer greater than x.
Trang 4Concept Review 1
right on the number line.)
3 If the original number is negative, then taking a
square root doesn’t “undo” squaring the number
is the absolute value of the original number, but
not the original number itself
5 Infinitely many (If you said 3, don’t assume that
unknowns are integers!)
6 sum
7 difference
8 product
multiplica-tion/division from left to right.)
19 0
20 50
23 PG-ER-MD-AS (Parentheses/grouping (left to right), exponents/roots (left to right), multiplica-tion/division (left to right), addition/subtraction (left to right))
24 Fraction bars (group the numerator and denom-inator), and radicals (group what’s inside)
25 120 (It is the least common multiple of 5, 6, 8, and 12.)
26 Step 1: subtract 7; step 2: divide by 3; step 3: take
25( )=5 −7
Answer Key 1: Numbers and Operations
SAT Practice 1
3
4 D
Add 3:
x > 2 and x < 4
k= 11
k− =3 8
− − −( ( ))
( ) − − − −( ( ( )))
= − − −( ( ( ))) − −− − −( ( ( )))
= −( ) − −( )
= − − −( )
= − +
= −
1 3 1 2
2 1
1
num-ber line) of all the integers that are greater than
by 4⁄3:
x÷3⁄4× −2
= x × −8⁄3
(Don’t forget to find half the number!)
Don’t forget that 0 is a digit, but it can’t be the first digit of a four-digit integer
whether the expressions increase or decrease 1 and
4 are convenient values to try Also, if you can graph
move to the right of 1 on the x-axis.
y= x
y= x
Trang 5The Laws of Arithmetic
When evaluating expressions, you don’t always have to
follow the order of operations strictly Sometimes you
can play around with the expression first You can
commute (with addition or multiplication), associate
(with addition or multiplication), or distribute
(multi-plication or division over addition or subtraction)
Know your options!
When simplifying an expression, consider
whether the laws of arithmetic help to make it
easier
Example:
than using the order of operations, you use the
The Commutative and
Associative Laws
Whenever you add or multiply terms, the order
of the terms doesn’t matter, so pick a convenient
arrangement To commute means to move
around (Just think about what commuters do!)
Example:
(Think about why the second arrangement is
more convenient than the first!)
Whenever you add or multiply, the grouping of
the terms doesn’t matter, so pick a convenient
grouping To associate means to group together
(Just think about what an association is!)
Example:
(Why is the second grouping more convenient
than the first?)
Whenever you subtract or divide, the grouping
of the terms does matter Subtraction and
divi-sion are neither commutative nor associative
Example:
the numbers in a difference until you convert it to
the numbers in a quotient until you convert it to multiplication:
The Distributive Law
When a grouped sum or difference is multiplied
or divided by something, you can do the multipli-cation or division first (instead of doing what’s in-side parentheses, as the order of operations says)
as long as you “distribute.” Test these equations
by plugging in numbers to see how they work:
Example:
Distribution is never something that you have
to do Think of it as a tool, rather than a re-quirement Use it when it simplifies your task For instance, 13(832 + 168) is actually much easier to do if you don’t distribute: 13(832 + 168) = 13(1,000) = 13,000 Notice how annoy-ing it would be if you distributed
Use the distributive law “backwards” when-ever you factor polynomials, add fractions, or combine “like” terms
Example:
Follow the rules when you distribute! Avoid these common mistakes:
Example:
(Tempting, isn’t it? Check it and see!)
5 7−2 7 =3 7
b
a b
a b
+ = +
b c a
b a
c a
+
( )= +
24 1 3
1 2
1 3
1
2 24
× × = × × )
Lesson 2: Laws of Arithmetic
Trang 6Concept Review 2: Laws of Arithmetic
Simplify the following expressions, and indicate what law(s) of arithmetic you use to do it Write D for
distrib-ution, CA for commutative law of addition, CM for commutative law of multiplication, AA for associative law of addition, and AM for associative law of multiplication.
Look carefully at the following equations If the equation is always true, write the law of arithmetic that justi-fies it (D, CA, CM, AA, or AM) If it is false, rewrite the right side of the equation to make it true.
Do the following calculations mentally (no calculator!) by using the appropriate laws of arithmetic.
If a and b are not 0:
21 The distributive law says that only or can be distributed over grouped or
6
3
2 1
3
+ = +y
2
5
3
x
y
x
y
x y
Trang 71. The difference of two integers is 4 and their
sum is 14 What is their product?
to which of the following expressions?
of y?
ex-presses the correct ordering of a, b, and c?
(A) c < a < b
(B) b < c < a
(C) a < b < c
(D) c < b < a
(E) b < a < c
true for all values of x, y, and z?
SAT Practice 2: Laws of Arithmetic
Trang 8Concept Review 2
associative law of multiplication)
com-mutative law of arithmetic)
add numerators.)
6 true: distributive law
over addition Expand and “FOIL” (First +
Out-side + InOut-side + Last).)
over multiplication Use the associative and
com-mutative laws of multiplication.)
mul-tiply before doing the powers!)
7
2
x
y
10 true: distributive law (Remember you can dis-tribute division over addition!)
16 19,000 (Use the distributive law.)
18 They’re reciprocals (Their product is 1.)
19 They’re the same
20 They’re opposites (Their sum is 0.)
21 multiplication or division over addition or subtraction
22 subtraction and division
23 yes
Answer Key 2: Laws of Arithmetic
SAT Practice 2
unnecessary
= (−2 + x)(x + 2) Distributive law
addition
simpler to compare:
multi-plication are commutative and associative If you’re
not convinced, you might plug in 1, 2, and 3 for x,
y, and z, and notice that equation II is not true.
x y
y x xy xy
÷
= ×
=
= 1
Trang 9Adding and Subtracting Fractions
denominators are the same But if the denominators are
different, just “convert” them so that they are the same.
When “converting” a fraction, always multiply
(or divide) the numerator and denominator by
the same number
Example:
If the denominator of one fraction is a
multi-ple of the other denominator, “convert” only
the fraction with the smaller denominator
Example:
One easy way to add fractions is with
“zip-zap-zup”: cross-multiply for the numerators, and
multiply denominators for the new
denomina-tor You may have to simplify as the last step
Example:
Multiplying and Dividing Fractions
To multiply two fractions, just multiply
straight across Don’t cross-multiply (we’ll
dis-cuss that in the next lesson), and don’t worry
about getting a common denominator (that’s
just for adding and subtracting)
Example:
To multiply a fraction and an integer, just
mul-tiply the integer to the numerator (because an
integer such as 5 can be thought of as 5/1)
y
x
y
x
y x
5
5
3 5
× = ×× =
5
6
7
8
5
6
7 8
40 48
42 48
82 48
41 24
3
2
5 3
2 5
5 15
6 15
5 6 15
5
18
4
9
5
18
4 2
9 2
5 18
8 18
13 18 + = + ×× = + =
12 18
12 6
18 6
2 3
= ÷
÷ =
2
5
2 5
5 5
10 25
= ×
× =
Example:
To divide a number by a fraction, remember
that dividing by a number is the same as
multi-plying by its reciprocal So just “flip” the second
fraction and multiply
Example:
Simplifying Fractions
Always try to simplify complicated-looking
frac-tions To simplify, just multiply or divide top and
bottom by a convenient number or expression If
the numerator and the denominator have a
common factor, divide top and bottom by that
common factor If there are fractions within the
fraction, multiply top and bottom by the
com-mon denominator of the “little” fractions
Example:
(Notice that, in the second example, 60 is the common multiple of all of the “little denominators”: 5, 3, and 4.)
Be careful when “canceling” in fractions Don’t
“cancel” anything that is not a common factor.
To avoid the common canceling mistakes, be
sure to factor before canceling
Example:
Right: x
x
1
−
− =
+
( ) ( − )
−
( ) = +
Wrong: x
x
x
1
−
2 5
2 3 1 4
60 2 5
2 3
60 1 4
24 40 15 6
+
⎛
⎝⎜ ⎞⎠⎟
×⎛⎝⎜ ⎞⎠⎟
15
4 2 2
2 2 1
x
+ = ( + )= +
3 7
5 2
3 7
2 5
6 35
2
m m
4
7 5
4 7
5 1
4 5 7
20 7
× = × = × =
Lesson 3: Fractions
15
48
40 42
Trang 10Simplify the following expressions:
1
Convert each expression to a fraction:
17 How do you divide a number by a fraction?
18 How do you add two fractions by “zip-zap-zup”?
19 What can be canceled to simplify a fraction?
20 How do you convert a fraction to a decimal?
22 If 2/3 of a class is girls, and there are 9 boys in the class, what is the total
n
6 9 12
n n
+ =
x x
5
−
− =
3
4− =2x
1
2
1
3
1
4
+ + =
− 2 ÷ = 9
4 3
2
12 4
m m
+ + =
3 2 9 8
x
z =
4
7
2
3
x
2 9
5 4
÷ = 56
21=
5 2 3 8
=
1
7
2
5
+ =
Concept Review 3: Fractions