SUM

1
SUM
The answer to an addition problem is called the
SUM.
Example 15 5 20
2
DIFFERENCE
The answer to a subtraction problem is called the
DIFFERENCE.
Example 15 - 6 9
3
FACTOR
A numbers being multiplied in a multiplication
problem are called FACTORS.
Example 8 x 6 48
4
PRODUCT
The answer to a multiplication problem is called
the PRODUCT.
Example 8 x 6 48
5
QUOTIENT
The answer to a division problem is called the
QUOTIENT.
Example 6 ) 4 8
8
6
DIVISOR
The number being divided into the dividend in a
division problem is called the DIVISOR.
Example 6 ) 4 8
8
7
DIVIDEND
The number being divided in a division problem is
called the DIVIDEND.
Example 6 ) 4 8
8
8
PLACE VALUE
.
W H O L E N U M B E
R S
DECIMALS
MILLION PERIOD ___ ___ ___ ,
BILLION PERIOD ___ ___ ___ ,
THOUSAND PERIOD ___ ___ ___ ,
(UNIT) PERIOD ___ ___ ___ .
___ ___ ___
AND
ONES
ONES
ONES
ONES
TENS
TENS
TENS
TENS
HUNDREDS
HUNDREDS
TENTHS
HUNDREDS
HUNDREDS
HUNDREDTHS
THOUSANDTHS
9
ROUNDING
The digit in the place being rounded will either
stay the same or jump upit never goes
down. Look at the number to the right of the
place being rounded. If the number to the
right of the place being rounded is 0 - 4, ignore
it. The digit in the place being rounded will
stay the same. If the number to the right of the
place being rounded is 5 - 9, add 1 to the
digit in the place being rounded. 0 - 4 ?
Just ignore 5 - 9 ? Add 1 more
10
WHEN COMPARING ( gt lt ), ADDING, OR
SUBTRACTING DECIMALS, YOU MUST FIRST LINE UP THE
DECIMALS IN THE TWO NUMBERS PLACE THE DECIMAL
POINT OF EACH NUMBER ON THE MIDDLE LINE. PLACE
THE WHOLE NUMBER PART OF EACH NUMBER ON THE
LEFT. (REMEMBER THE WHOLE NUMBER PART IS ALWAYS
THE PART OF THE NUMBER THAT IS ON THE LEFT SIDE
OF THE DECIMAL.) PLACE THE DECIMAL PART OF EACH
NUMBER ON THE RIGHT. (REMEMBER THE DECIMAL PART
IS ALWAYS THE PART OF THE NUMBER THAT IS ON THE
RIGHT SIDE OF THE DECIMAL.) FILL IN ZEROES WHERE
NEEDED. THEN COMPARE, ADD, OR SUBTRACT.
?
?
?
11
WHAT PAIR OF NUMBERS IS __________
BETWEEN? DRAW A NUMBER LINE. PLACE ALL
NUMBERS IN THE PROBLEM AND ALL THE NUMBERS IN
EVERY ANSWER CHOICE ON THE NUMBER LINE. USE THE
NUMBER LINE TO ANSWER THE QUESTION.
12
BOX MULTIPLICATION
Example 165 X 42 6930
100 60 5
40 2

4200 2520 210 6930
13
TRADITIONAL MULTIPLICATION
Example 165 X 42
2
2
1
1
Think 2 x 5 10. Write the 0 of the 10 below
the 2. Carry over the 1 of the 10 and write it
above the 6. Think 2 x 6 12. Add the 1
that you wrote above the 6 to 12 and get 13.
Write the 3 of the 13 below the 4. Carry over
the 1 of the 13 and write it above the
7. Think 2 x 7 14. Add the 1 that you
wrote above the 7 and get 15. Write the 15
beside the 3. Think I need a place holder to
show that I have already multiplied the 2 times
every digit in 765. So I will place a 0 in the
same column as the 2 before multiplying 4 times
every digit in 765. Think 4 x 5 20.
Write the 0 of the 20 below the 3. Carry over
the 2 of the 20 and write it above the 6. (Be
sure to cross out or ignore the old carry over
of 1 first.) Think 4 x 6 24. Add the 2
that you wrote above the 6 to 24 and get 26.
Write the 6 of the 26 below the 5. Carry over
the 2 of the 26 and write it above the 7. (Be
sure to cross out or ignore the old carry over
of 1 first.) Think 4 x 7 28. Add the 2
that you wrote above the 7 and get 30. Write
the 30 beside the 6. Think Add 1530 30600
and get the final answer of 32130.
7 6 5 x 4 2

0
3
1 5
0
0
6
3 0
3 2 1 3 0
14
NON-TRADITIONAL DIVISION
Think How many groups of 51 can I subtract
from 932? Think 1 x 51 51
10 x 51 510 100 x 51 5100 (too
much). Ill subtract 10 groups of 51. Think
How many groups of 51 can I subtract from
422? Think 10 groups (510) will be too
manyIll try 5 groups 5 x 51 255. That
works. Think How many groups of 51 can I
subtract from 167? Think 5 groups of 51
255. Thats too muchIll try 3 groups. 3 x 51
153. That works. Think How many groups of
51 can I subtract from 14? Think NoneI have
subtracted 10 5 3 18 groups of 51 and my
remainder is 14.
51
932
- 510
10
422
- 255
5
167
- 153
3
14
18

Answer 18 r. 14
15
TRADITIONAL DIVISION
Think Can 51 go into 9? Since it cannot,
write a 0 above the 9. Think How many times
can 51 go into 93? Since 51 can go into 93 one
time, write a 1 above the 3 of the 93. Then
multiply 1 x 51 51. Write the 51 below the
93. Think 93 51 42. Write the
42 below the 51. Since my remainder (42) is
smaller than my divisor (51), I am ready to bring
down the next digit in 932. Think Bring
down the 2 of the 932 and write it beside the 42.
Now, how many times can 51 go into 422? Since
51 can go into 422 eight times, write an 8 above
the 2 of the 932. Then multiply 8 x 51 408.
Write the 408 below the 422. Think 422
408 14. Write the 14 below the 408. Since
there are no more digits in the 932 to bring
down, and since my remainder (14) is smaller than
my divisor (51), I am ready to bring down the
next digit in 932 but since there are no more
digits to bring down, I am finished. My answer
is 18, remainder 14.
0
1
8
5 1
9 3 2
5 1
-
4 2
2
4 0 8
-
1 4
Answer 18 r. 14
16
PRIME NUMBER
A prime number is any number with ONLY two
factors 1 and the number itself. Examples 3
1 x 3 (no other factors) 7
1 x 7 (no other factors)
17
COMPOSITE NUMBER
A composite number is any number that has more
than 2 factors. Any number that is NOT a prime
number is a composite number. Examples 4 1 x
4 and 2 x 2, so the factors of 4
are 1, 2, and 4
16 1 x 16, 2 x 8, and 4 x 4, so the
factors of 16 are
1, 2, 4, 8, and 16 Both 4 and 16 are composite
numbers because they have more than 2 factors.
18
DIVISIBILITY RULES
2 Rule Every even number has 2 as a factor.
3 Rule If the sum of the digits of a
number has 3 as a factor,
then the original number has 3 as a factor. 5
Rule Every number that ends in 0 or 5 has 5
as a factor. 9 Rule If the sum of the
digits of a number has 9 as a
factor, then the original number has 9 as a
factor. 10 Rule Every number that ends in 0
has 10 as a factor. 11 Rule Every 2 digit
number in which both digits are the
same (for example 22, 33, 44, etc.) has 11
as a factor.
19
COMMON FACTORS
Common factors are the factors that two numbers
have in common.
Example Factors of 4 1, 2, and 4
Factors of 8 1, 2, 4, and 8 Since 1, 2, and 4
are factors of both 4 and 8, they are said to be
the common factors of 4 and 8.
20
GREATEST COMMON FACTOR (GCF)
Recall that common factors are the factors that
two numbers have in common.
Example Factors of 4 1, 2, and 4
Factors of 8 1, 2, 4, and 8
Since 1, 2, and 4 are factors of both 4 and 8,
they are said to be the common factors of 4 and
8. The greatest common factor (GCF) in this case
is 4 since it is the largest factor that the two
numbers have in common.
21
SIMPLIFYING (REDUCING) FRACTIONS
Recall that the greatest common factor (GCF) is
the largest factor that two numbers have in
common.
TO SIMPLIFY (OR REDUCE) A FRACTION Find the GCF
of the numerator and denominator of the
fraction. Divide both the numerator and the
denominator by the GCF to get the simplified or
reduced fraction.

22
NUMERATOR The NUMERATOR of a fraction is the top
number in a fraction.
¾
The NUMERATOR represents the number of pieces of
each whole entity that are shaded. In the
picture above, the numerator (3) represents the
number of the whole rectangle that is shaded.
23
DENOMINATOR The DENOMINATOR of a fraction is
the bottom number in a fraction.
¾
The DENOMINATOR represents the number of pieces
into which each whole entity is divided. In the
picture above, the denominator (4) represents the
number of pieces into which the whole rectangle
is divided.
24
PROPER FRACTION A PROPER FRACTION is a number
that represents part of a whole. A PROPER
FRACTION has both a NUMERATOR and a DENOMINATOR.
¾
A PROPER FRACTION is LESS THAN 1 WHOLE. In a
PROPER FRACTION, the NUMERATOR is always SMALLER
THAN THE DENOMINATOR.
25
IMPROPER FRACTION An IMPROPER FRACTION is a
number that represents more than a whole. An
IMPROPER FRACTION has both a NUMERATOR and a
DENOMINATOR.
4/3
An IMPROPER FRACTION is MORE THAN 1 WHOLE. In
an IMPROPER FRACTION, the NUMERATOR is always
LARGER THAN THE DENOMINATOR.
26
MIXED NUMBER A MIXED NUMBER consists of both a
WHOLE NUMBER AND A FRACTION.
1 ¾
In the picture above, the 1 represents the whole
rectangle that is shaded. The ¾ represents the
shaded part of the second rectangle.
27
HOW TO CHANGE A MIXED NUMBER INTO AN IMPROPER
FRACTION
1 ¾
1) Divide each whole entity into the same number
of pieces. In the picture above, the first
rectangle will be divided into 4 equal pieces
since the second rectangle is already divided
into 4 pieces. 2) Count the total number of
pieces that are shaded. Place that number in
the numerator of the improper fraction. In the
picture above, a 7 will be written in the
numerator because there are now 7
shadedpieces. 3) Write the number of pieces
into which each whole entity is divided (from 1
above) in the denominator. In the above picture,
a 4 will be written in the denominator since each
rectangle is divided into 4 pieces.
1 ¾
7
4
28
HOW TO CHANGE AN IMPROPER FRACTION INTO A MIXED
NUMBER
7
4

• 1) Count the number of whole entities that are
  completely shaded. Write that number as the
  whole number part of the mixed number. In the
  picture above, 1 will be written as the whole
  number since 1 whole rectangle is completely
  shaded.
• Look at the entity that is not completely shaded.
  Count the number of pieces of that entity that
  are shaded. Place that number in the numerator
  of the proper fraction part of the mixed number.
  In the picture above, a 3 will be
• written in the numerator because there are 3
  shaded pieces of the rectangle
• that is not completely shaded.
• 3) Write the number of pieces into which each
  whole entity is divided in the denominator. In
  the above picture, a 4 will be written into the
  denominator since each rectangle is divided into
  4 pieces.

1
7
3
4
4
29
EQUIVALENT FRACTIONS
EQUIVALENT FRACTIONS ARE EQUAL IN AMOUNT EVEN
THOUGH THEIR NUMERATORS AND DENOMINATORS LOOK
DIFFERENT.
EXAMPLE 1 / 2 2 / 4 1 / 2 AND 2 / 4 ARE
EQUIVALENT FRACTIONS BECAUSE THEY REPRESENT THE
SAME QUANTITY, BUT THEIR NUMERATORS AND
DENOMINATORS LOOK DIFFERENT.
30
EQUIVALENT FRACTIONS

• THERE ARE TWO WAYS TO TELL IF TWO FRACTIONS ARE
• EQUIVALENT
• SEE IF YOU CAN SIMPLIFY ONE OR BOTH OF THE
• FRACTIONS SO THAT THEIR DENOMINATORS MATCH.
• THEN COMPARE THEM TO SEE IF THEY ARE EQUAL.
• OR
• 2) IF SIMPLIFYING IS NOT POSSIBLE, OR IF
  SIMPLIFYING
• DID NOT GIVE MATCHING DENOMINATORS, FIND THE
• LEAST COMMON DENOMINATOR OF THE TWO
  FRACTIONS. THEN COMPARE THE TWO FRACTIONS WITH
  MATCHING DENOMINATORS TO SEE IF THEY ARE EQUAL.

31
LEAST COMMON DENOMINATOR (LEAST COMMON MULTIPLE)
To find the LEAST COMMON DENOMINATOR between two
fractions, multiply the numerator and denominator
of each fraction by the same number to make
equivalent fractions.
Stop when the denominators of both fractions are
the same.
The LEAST COMMON DENOMINATOR of 2/3 and 5/6 is
6. 2/3 is equivalent to 4/6, so 4/6 can replace
2/3.
x 1/1
x 2/2
2/3
5/6
32
COMPARING, ADDING, AND SUBTRACTING FRACTIONS
FRACTIONS MUST HAVE THE SAME DENOMINATORS BEFORE
THEY CAN BE COMPARED, ADDED, OR SUBTRACTED.

• THERE ARE ONLY TWO WAYS TO GET THE SAME
  DENOMINATORS FOR TWO FRACTIONS WITH UNLIKE
  DENOMINATORS
• SEE IF YOU CAN SIMPLIFY ONE OR BOTH OF THE
  FRACTIONS SO THAT THEIR DENOMINATORS MATCH
• OR
• 2) FIND THEIR LEAST COMMON DENOMINATOR
• THEN COMPARE, ADD, OR SUBTRACT THE NUMERATORS OF
  THE FRACTIONS WITH MATCHING DENOMINATORS.
• IF ADDING OR SUBTRACTING THE FRACTIONS, PUT THE
  LEAST COMMON DENOMINATOR AS THE DENOMINATOR IN
  YOUR ANSWER.

33
How to Read Ordered Pairs
First, put your finger on the point, and move it
down to the x-axis. Read the x-value. In this
problem, x 2. Next, find the y-value.
6
(2, ?)
5
4
y
3
2
1
0
x
1
2
3
4
5
0
6
34
How to Read Ordered Pairs
After you record the x-value, put your finger
back on the point, and move it across to the
y-axis. Read the y-value. In this problem, x
2, y 3.
6
5
(2, 3)
4
y
3
2
1
0
x
1
2
3
4
5
0
6
35
How to Plot Ordered Pairs
In this problem, you are asked to plot (2, 3).

First, find the x-value (2) by
starting at 0 and moving across horizontally.
( x , y )
6
5
4
y
3
2
1
0
x
1
2
3
4
5
0
6
36
How to Plot Ordered Pairs
In this problem, you are asked to plot (2, 3).

( x, y
) Starting from your new position at the x-value
(2), find the y-value by moving up vertically.
Mark the point.
6
5
4
y
3
2
1
0
x
1
2
3
4
5
0
6
37
QUADRILATERAL FAMILY
parallelogram
rectangle
trapezoid
square
rhombus
38
PERIMETER

PERIMETER is the distance AROUND a figure.
5
9 in.
9
5
5 in.
5 in.
9

28 in.
9 in.
39
AREA

AREA is the measure, in square units, of the
SURFACE of a figure.
The AREA of this rectangle is 8 square units.
40
HOW TO FIND THE AREA OF A SQUARE OR A RECTANGLE
Multiply the length (l) times the width (w).
(Area l x w)
5 cm. (l)
3 cm.
3 cm. (w)
x
15 sq. cm.
5 cm.
Write the square units beside the calculated area.
41
VOLUME

VOLUME is the measure, in cubic units, of the
SPACE that a 3-dimensional object occupies.
The VOLUME of this rectangular prism is 60 cubic
units.
42
VOLUME

Multiply the length (l) times the width (w)
times the height (h). (Volume l x w x h)
Write the cubic units beside the calculated
volume.

3 cm.
4 cm.
5 cm.
Volume 5 cm. x 4 cm. x 3 cm. 60 cubic cm.
(l) (w) (h)
43
PRIME FACTORS
USE A FACTOR TREE TO FIND THE PRIME FACTORS (OR
PRIME FACTORIZATION) OF ANY COMPOSITE NUMBER.
18
What are the prime factors of 18?
What is the prime factorization of 18?
6
3
2
3
2 x 3 x 3 18
44
TO FIND THE RANGE Subtract the smallest number
in the list of numbers from the largest number in
the list of numbers. The difference between
the largest number and the smallest number is the
RANGE. EXAMPLE 7 4 8 9 11 3 7 is
the list of numbers. 3 is the smallest number in
the list. 11 is the largest number in the
list. 11 3 8. 8 is the RANGE for this
list of numbers.
45

• TO FIND THE MEDIAN
• Put the list of numbers in order from least to
  greatest.
• Scratch off one number from the left side and one
  number from the right side. Repeat this process
  over and over again until there is only one
  number left in the middle of the list. The
  number left over in the middle is the MEDIAN.
• 3) Be careful! Before you decide what the number
  in the middle is, be sure you have crossed off
  the same number of numbers on the left side as
  you have on the right side.
• EXAMPLE 7 4 8 9 11 3 7 is the list
  of numbers.
• 3 4 7 7 8 9 11 is the list of
  numbers put into least to greatest order.
  (Notice that 7 is listed twice in the reordered
  list because it is listed twice in the original
  list.).
• 3 4 7 7 8 9 11. 7 is the
  MEDIAN because it is the number left in the
  middle after I crossed out three numbers on the
  left side of the list and three numbers on the
  right side of the list.

46

• TO FIND THE MODE
• Look at your list of numbers to find the number
  in the list that is listed the greatest number of
  times.
• The number that occurs in the list the greatest
  number of times is the MODE. Sometimes there is
  no mode, sometimes there is more than 1 mode.
• EXAMPLE 7 4 8 9 11 3 7 is the list
  of numbers. In
• this list, 7 is the MODE because it appears in
  the list more
• times than any of the other numbers in the list.

47

• TO FIND THE MEAN (AVERAGE)
• Add the list of numbers together.
• Take the total from 1 and divide it by the
  number of numbers in the list. The answer you
  get will be the MEAN (or AVERAGE).
• EXAMPLE 7 4 8 9 11 3 7 is the list
  of numbers.
• 7 4 8 9 11 3 7 49
• There are 7 numbers in the list. (If a number is
  listed
• more than once, you count it more than
  once.) 49 7
• 7. 7 is the MEAN.