Download 0,1,2,3… - mrmulholland

PLACE VALUE
1 000 000 000
100 000 000
10 000 000
1 000 000
100 000
10 000
1 000
100
10
1
0.1
0.01
0.00 1
0.00 01
0.00 001
0.00 000 1
0.00 000 01
0.00 000 001
0.00 000 000 1
NATURAL NUMBERS
Billions
Hundred Millions
Ten Millions
Millions
Hundred Thousands
Ten Thousands
Thousands
Hundreds
Tens
Ones
Decimal
Tenths
Hundredths
Thousandths
Ten Thousandths
Hundred Thousandths
Millionths
Ten Millionths
Hundred Millionths
Billionths
109
108
107
106
105
104
103
102
101
100
10-1
10-2
10-3
10-4
10-5
10-6
10-7
10-8
10-9
CHAINS OF OPERATIONS (BEDMAS)
Ex. [4 + 3 x (2 + 6)] ÷ (10 – 2 x 3)
[4 + 3 x 8 ] ÷
4
28
÷
4
3)2]
=7
32)
Ex. [(5 + 2) x (8 –
÷ (25 – 2 x
[(5 + 2) x (5 )2] ÷ (25 – 2 x 9 )
[(5 + 2) x 25 ] ÷ (
7
) = 25
ROUNDING A DECIMAL NUMBER
0 1 2 3 4 5 6 7 8 9 10 11
Find the place value and use the number just to
the right of it to determine whether it will round
up or down. Ex. 0.0453 to the thousandths. 5 is
the thousandths place, the 3 determines that it
will round down
Natural number P is indicated on the number
line, point P has an abscissa value of 5
PROPERTIES OF MULTIPLICATION
= {0,1,2,3…}
P
COMPARING NUMBERS on a number line
a= 2
a< 2
a> 2
a≤ 2
a> 2
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
0 1 2 3 4 5 6 7 8 9 10
Multiplication is distributive (over operations of
addition and subtraction) : 3(n + 2) = 3n = 6
GREATEST COMMON FACTOR
Find the prime factorization of the numbers
being compared. Ex. 48 + 18. 48 = 24 x 3
18 = 2 x 32. They have one 2 and one 3 in
common. GCF = 2 x 3 = 6.
To factor out the GCF. Ex. Rewrite the
following expression by factoring out the GCF,
48 + 18 = 6(8 + 3)
0 1 2 3 4 5 6 7 8 9 10
LARGEST COMMON MULTIPLE (LCM)
1. Common Multiples Method: list the multiples
of each number: ex LCM (18,24)
M18: 18, 36, 54, 72…
LCM(18, 24) = 72
M24: 24, 48, 72…
2. Prime Factorization Method
2
F18 = 2 x 32
F24 = 23 x 3 2
LCM
2
3
2
2
3
2
2
3
3
3
LCM(18, 24) =
2 x 2 x 2 x 3 x 3 = 72
EXPONENTIAL NOTATION
Ex. 3 x 3 x 3 x 3 x 3 = 35
an = a x a x … x a
n times
EXPANDED vs. STANDARD FORM
Standard form is when a number is written
normally. Ex. 95.24
Expanded form uses one of three forms
Decimal notation:
9 x 10 + 5 x 1 + 2 x 0.1 + 4 x 0.01
Fraction notation:
9 x 10 + 5 x 1 + 2 x 1/10 + 4 x 1/100
Using powers of ten:
9 x 101 + 5 x 100 + 2 x 10-1 + 4 x 10-2
RELATIVELY PRIME NUMBERS
For any two natural numbers whose GCF is 1
TRICKS FOR SOLVING WORD PROBLEMS
STATISTICS
TABLE OF VALUES
Sum = addition
Quotient = Division
Difference = subtraction
Product = multiplication
Number of
brothers or sisters
Frequency
(tally)
“At most” = ≤
“Less than” = <
“No fewer than” = ≥
“More than” = <
Population is the subject that is being studied.
Ex. In a survey of students in your school, we
choose your class, therefore the students in your
class are the population.
0
3
Relative
Freqeuncy (%)
Divide the tally
by the total to
get the %
15
1
7
35
2
3
Total
6
4
20
30
20
100
Remember: “a number” means a variable. Use
trial and error
Key words for GCF word problems: largest,
biggest, greatest, most, square
Key words for LCM word problems: smallest,
every, often, at the same time, again, fewest,
least, together
Variable is the question being asked. It is what
we are looking to compare. Variables can be a
quantity (like height) “quantitative”. They can
be a quality (like eye colour) “qualitative.”
INTEGERS
= {…-3,-2,-1,0,1,2,3…}
The positive integers. Ex.: +4, or 4
The negative integers. Ex.: -7
The null integer: 0
Two numbers are opposites if they consist of the
same natural number but different signs.
ADDITION AND SUBTRACTION OF
INTEGERS
-When we add two positive integers the sum is
positive
-When we add two negative integers the sum is
negative
-When we add integers of opposite signs, we find
the difference of the natural numbers which
comprise them, and this sum is preceded by the
sign of the integer with the largest natural
number.
-5 -4 -3 -2 -1 0 1 2 3 4 5 6
ORDER FOR INTEGERS
-When two numbers are positive, the larger one
is the larger natural number. Ex.: 8 > 2
-When two numbers are negative, the larger one
is comprised of the smaller natural number. Ex.:
-4 > -6
-When two numbers have opposite signs, the
larger one is the positive number.
-Every positive number is greater than zero,
every negative number is less than zero.
Subtracting an integer is the same as adding its
opposite
Ex.: 3 – (-5) = 3 + 5 = 8
Ex.: -4 – 5 = -4 + (-5) = -9