Number Concepts
Place Value, Numeration, Decimals,
Fractions, Percents, Ratio,
Probability, Statistics
Place Value
Trillions
Billions
Millions
Thousands Ones
Period
Period
Period
Period
Period
__ __ __,__ __ __, __ __ __,__ __ __, __ __ __.__ __ __ __
Ten thousandths
Thousandths
Hundredths
Tenths
Ones
Tens
Hundreds
Thousand
Ten thousand
Hundred thousand
Million
Ten million
Hundred million
Billion
Ten billion
Hundred billion
Trillion
Ten trillion
Hundred trillion
Read the number in each period. Then say the period’s name
except for the one’s period. The decimal point means and.
Numbers following the decimal are read and only the last place
value is said.
Comparing Numbers
> Greater Than
= Equal To
< Less Than
When comparing numbers, stack the
numbers where place values are aligned.
Add zeroes after the decimal if needed.
> Great than or equal to
< Less than or equal to
Rounding Numbers
Underline the place value that you are rounding to.
Look at the digit to the right. If that digit is 0-4, round
the number down. If that digit is 5-9, round that number
up.
1,286
To nearest ten
1,286
1,290
To nearest hundred 1,286
1,300
To nearest thousand 1,286
1,000
Rounding Numbers (2)
34,967
To nearest hundred
34,967
35,000
513.456
To nearest one
513.456
513
To nearest tenth
513.456
513.5
To nearest hundredth
513.456
513.46
Factors, Primes, & Composites
•Factors-Numbers that divide into a larger number evenly,
or numbers that multiply to get a product.
•Prime Numbers-Numbers that have only 2 factors, one
and that number.
•Composite Numbers-Numbers that have more than 2
factors.
•0 & 1 are neither prime nor composite.
•Greatest Common Factor (GCF)
12: 1, 2, 3, 4, 6, 12
CF-1, 2, 3, 6
18: 1, 2, 3, 6, 9, 18
GCF-6
Multiples and LCM
•Multiples-a number that can be divided evenly by
another number. (skip counting)
•Least Common Multiple (LCM)-the smallest
common multiple that is not zero
4:0, 4, 8, 12, 16, 20, 24, …
6:0, 6, 12, 18, 24, 30, …
CM-0, 12, 24, …
LCM-12
Rules of Divisibility
•2-Any even number (ends in 0, 2, 4, 6, or 8)
•3-The sum of the digits is divisible by 3.
(24= 2+4=6 and 6/3 = 2)
•5-The last digit is either 5 or 0
•6-The number is divisible by 2 and 3
•9- The digital root must be 9.
•10-The last digit is 0
Prime Factorization
Prime Factorization is writing a number as the
product of its prime factors. Use the rules of
divisibility.
48
6
2
2
X
X 3
3
8
X
X 2
X 2
X
X 4
2 X
2
24 X 3=2 x 2 x 2 x 2 x 3 = 48
Fractions
Fractions-show a part of something and are in
equal size parts
4 numerator-how many parts we’re talking about
5 denominator-how many parts make a whole
Equivalent Fractions-name the same amount
1
2

3
6
4
2

10
5
Multiply or divide both numerator &
denominator by the same number.
Lowest Terms
Lowest Terms-a fraction whose numerator &
denominator have no common factor greater
than 1.
8
=2
12
1. Find the gcf of the numerator &
denominator.
3
(8-1, 2, 4, 8
12-1, 2, 3, 4, 6, 12 gcf-4)
2. Divide the numerator & denominator by
the gcf. 8   4 = 2

12 4 = 3
Improper Fractions & Mixed Numbers
Improper Fractions-the numerator is
larger than the denominator
5
4
Mixed Number-a whole number and a
fractional number
1
2
4
Changing Between Improper Fractions
& Mixed Numbers
To Change an Improper Fraction to a
1. Divide the numerator by the
Mixed Number
denominator
1
5
1
2. The remainder becomes the
= 4
4
numerator and the denominator
4 5
stays the same
To Change a Mixed Number to an Improper
1. Multiply the denominator by
Fraction
1 7
3 
2 2
(2  3) + 1 = 7
the whole number
2. Add the numerator to the
product
3. Denominator stays the same
Comparing Fractions
You must have a common denominator.
1
2
3
7
1
7 3
6


2
14 7
14
7
6

14 14
1. Find the least common
denominator (lcd).
[Example: 2 & 7-lcd 14]
2. Make equivalent fractions.
3. Compare.
Probability
Probability-the likelihood of an event
favorable outcomes
possible outcomes
6
3
P(blue) =

10
5
3
P(red) =
10
1
P(yellow) =
10
Bag of Marbles
6 blue, 3 red, 1 yellow
Statistics
Mean (Average)
1. Add the numbers.
13, 19, 19, 21, 23
19
sum = 95 5 95 average = 19
2. Divide the sum by the number of addends.
Range-the difference between the largest and
smallest number.
23-13 = 10
Median-when numbers are arranged in numerical
order, the middle number 13 19 19 21 23
Mode-the number(s) that occurs most frequently
13 19 19 21 23
Operations
Addition, Subtraction, Multiplication,
Division with Whole Numbers,
Decimals, Fractions, & Integers
Addition & Subtraction-Whole
Numbers
25 addend
92 minuend
+37 addend
- 48 subtrahend
62 sum
44 difference
Before you add or subtract, line up the place
values. To check subtraction, add the difference
and the subtrahend. The sum should be the
minuend.
Addition with Decimal Numbers
1.8 + 0.65=
1. Line up the decimal points.
2. Add zeros to write an
equivalent decimal if necessary.
1.80
+0.65
2.45
3. Add from the right to the left.
4. Don’t forget to write the
decimal point in your sum by
bringing it straight down.
Subtraction with Decimal
Numbers
1.3 - 0.85=
12 1
01.30
- 0.85
0.45
1. Line up the decimal points.
2. Add zeros to write an
equivalent decimal if necessary.
3. Subtract from the right to the
left.
4. Regroup if necessary.
5. Don’t forget to write the
decimal point in your sum by
bringing it straight down.
Estimation-Addition &
Subtraction
1. Round the numbers.
2. Add or subtract the rounded numbers.
3,849
4,000
9,251
9,000
+5,207
5,000
-3,760
4,000
9,000
5,000
Multiplication & DivisionWhole Numbers
9  factor
 8  factor
72  product
quotient
remainder
4 r.2
6 26
divisor
dividend
dividend
9  8  72
9(8)  72
 means "of"
divisor quotient
26  6  4 r.2remainder
26
2
1
4 4
6
6
3
Multiplication & DivisionWhole Numbers
16
6 97r. 1
23
 35
115
 720
23 x 5
23 x 30
835
Don’t forget to write the
0 when you multiply by a
power of 10.
-6
37
- 36
1
16
x 6
96
+1
97
Divide, Multiply,
Subtract, Bring Down
Check division with
multiplication.
Estimation-Multiplication &
Division
Use estimation to see if an answer is reasonable or
when an estimate is needed.
1. Round the numbers.
2. Multiply or divide the numbers.
351
400
x 26
x30
12,000
784  4 
800  4 = 200
784  23 =
800  20  40
Properties of Addition
•Identity Property
7+0=7
•Commutative Property (Order Property)
6+7=7+6
•Associative Property
(Grouping Property)
(8 + 2) + 1 = 8 + (2 + 1)
Properties of Multiplication
•Zero Property 7 x 0 = 0
•Identity Property
7x1=7
•Commutative Property (Order Property)
8x2=2x8
•Associative Property (Grouping Property)
(8 x 2) x 3 = 8 x (2 x 3)
•Distributive Property 6 x (3 + 4) = (6 x 3) + (6 x 4)
Measurement
Metric and Customary Measure
Measurement-Customary
Length-inch (in.), foot (ft), yard (yd), mile (mi)
12 in. = 1 ft
5,280 ft = 1 mi
3 ft = 1 yd
1,760 yd = 1 mi
36 in. = 1 yd
Volume-tablespoon (tbsp), fluid ounce (fl. oz), cup (c), pint
(pt), quart (qt), gallon (gal)
2 tbsp = 1 fl. oz
2 pt = 1 qt
8 fl. oz = 1 c
4 qt = 1 gal
2 c = 1 pt
Mass-ounce (oz), pound (lb), Ton (T)
16 oz = 1 lb
2,000 lb = 1 T
Customary Conversions
When you are converting from one unit to another unit within
the customary system, think about whether the unit that you
are changing to is larger in size and therefore will be a smaller
number or if the unit is smaller in size and therefore will be a
larger number.
Example:
12 in. = 1 ft
4 qt = 1 gal
___ in. = 2 ft (inches are smaller than feet so
2 x 12 = 24 the number should be larger
24 in. = 2 ft than 2)
24 qt = __ gal (gallons are larger than quarts
24  4  6
so the number should be
24 qt = 6 gal smaller than 24)
Measurement-Metric
Basic Units for Metric Measurement are:
meter (m)-length
liter (L)-volume
gram (g)-mass
The metric system uses the same prefixes to show different
sizes.
kilo- (k) - 1,000
deci- (d) - 1/10
hecto- (h) - 100
centi - (c) - 1/100
deca- (da) - 10
milli - (m) - 1/1,000
Example: 1 km= 1,000 m
1,000mL = 1 L
Geometry
Definitions, Perimeter, Area,
Volume, Transformations
Geometry-Basic Terms
•
Point-an exact location in space
Line-a set of points extending endlessly along a straight
path.
Segment-a part of line having 2 endpoints
Ray-a part of a line having one endpoint & extending
endlessly in one direction.
Angle-2 rays that share a common endpoint called a
vertex
Plane-an endless flat surface extending in all directions
Parallel Lines-lines that are an equal distance apart in
the same plane and never intersect
Perpendicular Lines-lines that intersect to form right
angles
More Geometry Basic Terms
Perimeter-the distance around a polygon; measured in units
Area-the number of square units needed to cover a region
inside a figure; measured in square units
Circumference-the distance around a circle; measured in units
Congruent-two figures that have the same size & same shape.
Similar-two figures that have the same shape but different
size.
Symmetry-a fold line where both sides match up exactly
Angles
Right  measures 90
Acute measures less than 90
Obtusemeasures greater than 90 Straight  measures 180 (line)
Supplementary s - two angles whose measures total 180
Complimentary s - two angles whose measures total 90
2
Vertical s
1
4
3
s2&4 and s1&3 are pairs of vertical s
Vertical angles are always congruent.
Adjacent s - two angles that share a common side
Corresponding s
1
3
5
7
4
6
8
2
1  5
2  6
3  7
4  8
Polygons
Polygons-many sided figures that must be closed
figures and have straight sides.
Triangles-3 sides & 3 angles
Quadrilaterals-4 sides & 4 angles
Pentagons-5 sides & 5 angles
Hexagons-6 sides & 6 angles
Octagons-8 sides & 8 angles
Decagons-10 sides & 10 angles
Quadrilaterals
Quadrilaterals
Any four sided figure
Sum of the angles=360o
Parallelogram-2
pairs of parallel sides
Trapezoid-1 pair
of parallel sides
Perimeter -add the
lengths of the sides
Area-Rectangle-l*w
Parallelogram-b*h
Square-s2
Rhombusparallelogram with
4 congruent sides
Rectangleparallelogram with
4 right angles
Square-4 congruent
sides and angles
Solids (3 Dimensional Figures)
Solids have 3 dimensions-length, width, and height
Name
Faces Edges Vertices
Cube
6
12
8
Rectangular Prism
6
12
8
Triangular Prism
5
9
6
Triangular Pyramid
4
6
4
Rectangular Pyramid
5
8
5
The shape of the base determines the kind of prism or pyramid.
More Solids
There are other kinds of solid figures.
Cylinder
shaped like a can
Sphere
shaped like a ball or the earth
Cone
shaped like a pointed ice-cream cone
Volume
Volume is the number of cubic units within a solid figure.
Find the area of the base and multiply by the height of the
figure.
Area of base=10 x 4 = 40 sq cm
5cm
4cm Volume=40 sq cm x 5 cm=200 cu cm
10cm
Congruence, Similarity, &
Symmetry
Congruent-same shape and same size
Similar-same shape, but different size
Symmetry-a line on which you could fold a figure and both
sides would match up
Transformations
Transformation-a change in the size, shape, or position of a
figure.
Translation-slide
Reflection-flip
Rotation-turn
Graphing and Number Lines
II
Y-axis
I
The coordinate plane is divided
into 4 quadrants.
I-Positive x and positive y
II-Negative x and positive y
III-Negative x and negative y
IV-Positive x and negative y
III
(0, 0) is the point
where the x axis and
the y axis intersect.
X-axis
Ordered Pairs locate points on
the coordinate plane. The first
number always names the x
coordinate and the second
number always names the y
coordinate. (3,5)
IV
Problem Solving
Strategies
Problem Solving
When you solve problems, there are four basic steps to use:
•Restate the problem
•Select a Problem Solving Strategy
•Solve the Problem Using the Chosen Strategy
•Answer to the Problem with the Appropriate Label
Problem Solving Rap
Read and Scan,
Understand.
Underline the question,
Circle key words.
Cross out any info that’s just for the birds.
Do a doodle,
Make a plan.
Find a strategy,
I know I can.
Work, work, work!
Let persistence prevail.
Check it out,
I am on the right trail.
Preparing Students for Math in 2000
Canyon ISD Math Vertical Teams
Grades 5-8
Presentation developed by Diane Reid, CJHS
Copyright D. Reid 1998
Math Concepts
4 Number
4 Measurement
Concepts
4 Operations
4 Geometry
4 Problem
Solving
Powers of 10
100= 1
100= 1
101= 10
10-1= 0.1
102= 100
10-2= 0.01
103= 1,000
10-3= 0.001
104=10,000
10-4= 0.0001
105=100,000
10-5= 0.00001
The exponent tells how many zeroes follow the 1. Negative
exponents do not make negative numbers. Negative exponents
tell how many places are behind the decimal.
Exponents
Exponents tell how many times the
base number is used as a factor.
Exponent
23= 2 x 2 x 2 = 8
Base Number
Scientific Notation
.
X 10
Exponent represents the
number of places the decimal
has moved.
Significant digits-a number greater than or equal to 1 and
less than 10. We can use zero for a place holder but not
as the last digit to the right.
Example:
•5,402,000 = 5.402 x 106 (positive exponent for numbers > 1)
•0.00046 = 4.6 x 10-4 (negative exponent for numbers < 1)
Fractional & Decimal
Equivalents
To Change a Fraction to a Decimal Number
0.8
4
= 5 4.0
5
1
= 0.25
4
1
= 0.5
2
3
= -.75
4
1

8
3

8
5

8
7

8
0.125
0.375
0.625
0.875
1. Divide the numerator by
the denominator.
2. Divide until the
remainder is zero or it
repeats.
1
1
3
 0.3
 0.2
 0.6
3
5
5
2
2
4
 0.6
 0.4
 0.8
3
5
5
Percents
Percent-per hundred (100% represents a whole)
To Change a Fraction to a Percent
Fraction  Decimal  Percent
0.25
1
= 4 1.00 = 25%
4
To Change a Decimal to a Percent
Move the decimal point two places to the right
Decimal 
0.18
=
Percent
18%
Percents (2)
To Change a Percent to a Decimal-move the
decimal point 2 places to the left
Percent
37% =
62.5% =
Decimal
.37
.625
To Change a Percent to a Fraction-write the
percent as the numerator over the denominator
of 100. Put in lowest terms.
Percent
Fraction
65% = 65  13
100 20
Finding a Percentage of an
Amount
What is 40% of $52.00?
40%= 0.40
0.40 x 52 = $20.80
What is 25% of 16?
25
1
25% 

100
4
1 16

4
4
1
1. Change the
percentage to a decimal
number or fraction.
(Remember of means
to multiply)
2. Multiply the decimal
or fraction by the
amount.
3. Remember the
decimal point or to
write it in lowest terms.
Discount Price
Discount price is the price you would pay if
something was on sale for a discounted
amount.
Tennis Racquet sells for $65.00
1. Find the discount amount.
Discount-15% off
Multiply the price by the
Discount Amount
decimal for the percent.
.15 x $65 = $9.75
2. Subtract the discount
00.56$
57.9 52.55$
amount from the original
amount.
Ratio
Ratio is a comparison of two numbers.
Compare the number of
triangles to parallelograms.
4 triangles to 3 parallelograms
Can be written as:
4 to 3
4:3
4
3
To Read: four to three
Proportion
Proportion is two ratios that are equal.
4
3

12 9
Two ratios are equal if the crossproducts are equal.
4
3

12 9
12 x 3 = 36
4 x 9 = 36
Solving a Proportion-use cross-products
to solve for the unknown.
n
5
10  5 = 50

10
25
n  25  50
25n = 50
n = 2
(10 x 5)  25 = n n  2
Numbering System Categories
REAL NUMBERS
Rational Numbers-all positive
Irrational Numbers
•“i”-imaginary numbers
•
square root of
non-perfect squares
•non-repeating, nonterminating decimals
or negative numbers, fractions,
decimals, perfect squares
Integers {…,-3, -2, -1, 0, 1, 2, 3, …}
Whole Numbers {0, 1, 2, …}
Natural Numbers {1, 2, 3, …}
Adding & Subtracting Fractions
You must have a common denominator (c.d.).
1. Find the lcd (least common denominator).
2. Make equivalent fractions.
3. Add or subtract the numerators. The denominators stay
the same.
4. Write the answer in lowest terms.
7
7

12
12
1
3
+

4
12
10
5

12
6
7
14

10
20
1
5


4
20
9
20
Adding Mixed Numbers
2
14
4 4
3
21
5
15
3  3
7
21
29
8
7
8
21
21
1. Find the lcd.
2. Make equivalent fractions.
3. Add the fractions.
4. Add the whole numbers.
5. Regroup any improper
fraction.
6. Write the answer in lowest
terms.
Subtracting Mixed Numbers
14
1 2 2
3  3
6
12
3
9
1  1
4
12
5
1
12
1. Find the lcd.
2. Make equivalent fractions.
3. Regroup if needed.
A. Regroup the whole number.
B. Add up on the fraction.
4. Subtract the fractions.
5. Subtract the whole numbers.
6. Write the answer in lowest
terms.
Integers-Addition & Subtraction
Remember-Integers are whole numbers and their opposites.
{…, -3, -2, -1, 0, 1, 2, 3, ...}
Rules:
•If both signs are the same, add the numbers together and
keep the same sign. Example: 3 + 4 = 7 - 3 - 4 = -7
•If the signs are opposite, subtract the numbers and take the
sign of the larger number. Example: 3 + -4 = -1 -3 + 4 = 1
Decimal-Multiplication &
Division
Multiplication
When you multiply decimal numbers, multiply as you would
with whole numbers, but adjust the product (answer) to have
the same number of places behind the decimal point as in the
factors.
41 x 12 =492
4.1 x 1.2 = 4.92
Division
When you divide by a decimal divisor, you must move the
decimal in the divisor to the right to make it a whole number.
Then you must move the decimal point in the dividend to the
right the same number of spaces.
1.3
Bring the decimal point
12
. . 15
. .6
straight up in the quotient.
Fraction-Multiplication
You do not have to have a common denominator.
Multiplication
1. Multiply the numerators.
2. Multiply the denominators.
1 1 1
 
4 3 12
3. Write the answer in lowest terms.
To simplify, divide a number in the numerator and the
denominator by a common factor.
5 12
5
48

3

12
Fraction-Division
You do not have to have a common denominator.
Division
1. Write the reciprocal of the divisor. (Reciprocal-exchange
1 1
the numerator and denominator.)
 
4 2
2. Multiply the dividend and the reciprocal.
1 2 2 1
  
3. Write the answer in lowest terms.
4 1 4 2
To simplify, divide a number in the numerator and the
denominator by a common factor.
5 12
5
4 8

3

12
Mixed Numbers-Multiplication
& Division
When you multiply or divide by at least one
fractional number, all numbers must be written in
fractional form.
To write a whole number as a fraction, write the
5
number over 1. 5 
1
Change mixed numbers to improper fractions.
1
1 5 510
25
1
1 3  

4
4
3 24
3
6
6
1 8 5 8 2 16
1
82     
3
2 1 2 1 5
5
5
Integers-Multiplication
Remember-Integers are whole numbers and their opposites.
Multiplication
•If signs of both numbers are the same, the answer
will be positive.
+ 6  +7 = +42
- 6  -7 = +42
• If the signs of the numbers are opposite, the
answer will be negative
+ 6  -7 = - 42
- 6  +7 = - 42
Integers-Division
Rules for Division are the same as for
Multiplication.
•If you have an even number of negative numbers,
the answer is positive.
3
1

 18 6
- 21  -7 = 3
•If you have an odd number of negative numbers,
the answer is negative.
 20  4

36  -9 = - 4
 25
5
Rational Numbers-Multiplication
& Division
Rational number is any number which can be written as the
quotient of two integers.
Rational Numbers:
1) Use the same rules as Fractions when multiplying &
dividing.
2) Use the same rules as Decimals when multiplying &
dividing.
3) Use the same rules as Integers when multiplying &
dividing.
 4.0
 3.4 13.6
1   3   8  1
   
2  4  9  3
Order of Operations
Please Excuse My Dear Aunt Sally
When there are multiple operations to perform, do
them according to this order.
1) Parenthesis-do whatever is in parenthesis first
2) Exponents-solve any numbers written in exponent form
3)Multiplication & Division-work from left to right
4)Addition & Subtraction-work from left to right
10 2   8  2  6  1
4   2  3  7
4   5  7
20  7  13
10 2  16  6  1
100  10  1  10  1  11
Metric Conversions
To help with conversions within the metric system, you can
use King Henry.
“King Henry Danced Merrily* Down Center Main”
Main-milli
Center-centi
Down-deci
Merrily-meter
Danced-deca
Henry-hecto
King-kilo
___ ___ ___ ___ . ___
___
5
2 . ___*Use Merrily when you
are measuring meters.
*Use Lightly when you
are measuring liters.
*Use Gracefully when
you are measuring grams.
Metric conversions are easy because
you are only moving the decimal
point. Example: 52cm = .52m
Triangles
Triangles
PerimeterAreaAny three sided figure 1/2(length x width)
Add the
lengths of Sum of the angles=180o
the sides
Classify by Sides
Classify by Angles
Acute-all 3 angles acute
Equilateral-3 congruent sides
Obtuse-1 obtuse angle
Isosceles-2 congruent sides
Right-1 right angle
Scalene-No congruent sides
More About Angles
•A
1
2
3
5
7
•B
4
6
8
AB is the transversal-a line that
intersects 2 or more lines
Interior Angles-angles that are
on the inside of the lines
{angles 3, 4, 5, 6}
Exterior Angles-angles that are
on the outside of the lines
{angles 1, 2, 7, 8}
Alternate Interior Angles-angles 3 & 6 and angles 4 & 5
are pairs of alternate interior angles
Alternate Exterior Angles-angles 1 & 8 and angles 2 & 7
are pairs of alternate exterior angles
Circles
P - Center
C
AP  Radius
AB  Diameter
A
P
AC  Chord
D
APD is a central angle
AC - minor arc
ACD - major arc
Circumference - distance around a circle - d or 2 r
Area = r 2
B
Surface Area
Surface area is the number of square units that it would take
to cover all of the surfaces of a solid figure. Find the area of
each surface or face and add the areas together.
5cm
Bottom&Top=10 x 4 = 40 x 2 = 80sq cm
Front&Back=10 x 5 = 50 x 2=100sq cm
End&End= 5 x 4 = 20 x 2=40sq cm
Surface Area=80 + 100 + 40 = 220 sq cm
4cm
10cm
2cm
2cm
5cm
5cm
Circles-r2=(3.14 x 22) x 2
12.56 x 2 =25.12 cm2
Rectangle-Height x Circumference
5 x (d)=5 x (3.14 x 4)
5 x 12.56=62.8cm2
Surface Area= 25.12 + 62.8 = 87.92 cm2