# Number Sense Notes

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```Number Sense Vocabulary
Periods: The groups of three place values on the number chart. Commas are
used to separate the periods when writing large numbers.
__ __ __ ,__ __ __,
__ __ __,
__ __ __
Billions
millions
thousands
ones
Standard Form - a way to write numbers using one digit in each place value.
Example: 324, 501
Word Form: A way to write the value of a number using words only.
Example: three hundred twenty-four thousand five hundred one
Short Word Form: a way to write numbers using both digits and words.
Example: 324 thousand 501
Expanded Form: A way to write numbers showing the value of each digit.
Example: 300, 000 + 20, 000 + 4, 000 + 500 + 1
Expanded Notation: a way to write numbers showing each digit times the
corresponding place value
Example: (3 100,000) + (2  10,000) + (4 1,000) + ( 5  100 ) + (1 1 )
Symbols
< less than
> greater than
= equal to
≠ not equal to
≈ is approximately equal t0
Expanded Notation, Expanded Form and
Standard Form
Expanded Notation: (D  PV) + (Digit  Place Value)
Expanded Form: Value + Value
Standard Form: All Digits
Write 5,030,718 in:
Expanded Notation. Write the highest valued digit (5), count the
number of place values to the right of that digit, and that is the
number of zeroes that will be in your place value. (D  PV) + (D 
PV)
(5 1, 000,000) + ( 3  10,000) + ( 7  100) +( 1  10) +( 8  1) +
Expended Form: Find the highest valued digit, write it and add as
many zeroes after it as there are place values to the right.
5,000,000 + 30,000 + 700 + 10 + 8
Write in Standard Form:
(6  100,000,000) + (5  100,000) + (4  1,000) + (8  10).
Go to the highest valued digit, write the number, and then count
the number of zeroes to the right of that digit. Once you have
that amount, place that many __ to the right of the digit. From
there, plug in the rest of the digits in their corresponding PV.
Empty PVs will be filled in with 0s.
6_0_ _0_, _5_ _0_ _4_, _0_ _8_ _0_
Renaming Numerals:
to say a number in a different way. You use 4 different operations (addition,
subtraction, multiplication and division).
Rename: 3,600,000
3,000,000 + 600,000
3,800,000 – 200,000
600,000 x 6
7,200,000 ÷ 2
Step 1: Box out the digits in the number. Box out 36.
Step 2: count the number of zeroes and put it above.
Addition: What plus what equals 36?
 name TWO addends whose sum is 36. Add ALL the zeroes in the given
Subtraction: What minus what gives me 36?
 Add any number to 36. The sum becomes the minuend (1st number);
the zeroes in the given number to both the minuend and the
subtrahend. The number 0 cannot be in the equation.
Multiplication: What times what equals 36?
 Name TWO factors of 36; put a factor on each side of the
multiplication symbol. Split the number of zeroes between the
factors (you can put them all on one side). 1 cannot be a factor.
Division: What divided what what equals 36?
 Multiply any number by 36. The product will be your dividend (first
number); the number you chose will be your divisor (second number).
ALL of the zeroes will go to your dividend (first number). 1 cannot be
in the equation.
Exponents:
Definition: Exponent tells the base how many times to be a factor
Exponent
Base
4²
Base: factor(s) that are used.
Exponent: “little voice” that tells the base how many times it has
to multiply itself - Repeated multiplication.
10 5 = 10  10 10  10 10 = 100,000
54 =5555
We say: 5 to the 4th power.
SPECIAL EXPONENTS:
4² = 4 squared or 4 to the 2nd power.
4³= 4 cubed or 4 to the 3rd power.
**Any number to the zero power ALWAYS equals 1** EXCEPT
zero to the zero power is undefined. ***
Proof: Base 10
* As the EXPONENT
10 4 = 10  10 10  10 ( ÷ 10 )
10 3 = 10  10  10 (÷ 10)
10² = 10  10 (÷ 10)
10¹ = 10 (÷ 10)
10º = 1
63 = 6  6  6 ( ÷ 6 )
6² = 6  6 ( ÷ 6 )
6¹ = 6 ( ÷ 6 )
6º = 1
decreases by one, you
DIVIDE by the BASE. *
2³ = 8 ( ÷ 2 )
2² = 4 ( ÷ 2 )
2¹ = 2 ( ÷ 2 )
2º = 1
Exponential Notation:
A way to write a number using exponents.
*Remember* notation means that we use symbols like
parenthesis (), multiplication dots ●, and addition signs +
Example: 657,009,104
Write in Expanded Notation (D  PV) + (D  PV):
(6  100,000,000) + (5  10,000,000) + (7  1,000,000) + (
9  1,000) + ( 1  100) + ( 4  1)
**Think about the relationship between the exponent and PV **
Write the numeral for 10 4 =
Write the exponent for 1,000,000 =
Step 1: Think about the exponent that represents each PV. Starting
all the way to the right, place a zero above 4 (10= 1), a 1 above the
zero, a 2 above the 1, and so on until you get to the 6.
Step 2: Take the highest valued digit (6), put it in parenthesis and
multiply it by 10 to the power that you wrote above it. Close the
parenthesis and continue with the rest of the number.
(6 ● 10 8) + (5 ● 10 7) + (7 ● 10 6) + (9 ● 103) + (1 ● 10 2) + (4 ● 10 0)
Practice:
7,200,803,649
_____________________________________________________
_____________________________________________________
Convert Exponential Notation to Standard Form and Standard
From to Exponential Notation (in order AND mixed up)
Convert to Exponential Notation: 8,752,403
(8  1,000,000) + (7  100,000) + (5  10,000) + (2  1,000)
+ (4  100) + (3  1)
Convert to Standard Form (in order):
(9 10 9) + (6  10 7 ) + (4  10 6) + ( 5  10 5) + ( 7  104 ) + ( 6 
10²) + ( 8  10)
9, 0 6 4, 5 7 0, 6 0 8
Step 1: Find the greatest power and add 1- that will tell you
how many digits you need in the number. Then make the
corresponding dashes for the number of digits you need.
Step 2: Put each digit in it’s appropriate PV. The exponent
represents the number of PVs to the right of the digit.
Convert to Standard Form (mixed up order):
(7  10¹) + (8  10 4) + (5  1010) + (2  108) + (9  10 6 ) + (6  10º)
5 0, 2 0 9, 0 8 0, 0 1 6
(6  10 9) + (5  10 5) + (6  10 0) + (4  10 11) + (9  10 3) +
(7  10 2) + (1  108) 4 0 6, 1 0 0, 5 0 9,7 0 6
Rounding- Rolling 9s
Rounding is a type of estimation, but in it, you have an EXACT
answer to a specific place value.
When you round a number, you find the digit in the place value
you are rounding to and underline it. Then you look at it’s “righthand man” – and circle it.
Round to the hundredths PV: 5, 7 3 0
*the 3 tells the 7 to stay the same*
If the “right hand man” is:
 0,1,2,3, 4 = the digit stays the same
 5,6,7,8,9 = the digit goes up by 1
TO THE RIGHT OF THE PV YOU ARE ROUNDING TO, ALL
DIGITS BECOME “0”. ALL DIGITS TO THE LEFT STAY THE
SAME.
Round to the nearest hundred thousand:
57,984,320 = ________________
 The 8 tells the nine to go up by 1, so you have to ROLL OVER
each 9 until you reach a digit that is 8 or less. Add 1 to that
digit and rewrite the number.
Practice: Round to the nearest:
million: 954,602,728 ________________
hundred thousand: 579,957,321 _____________________
ten thousand: 689,996,017 _____________________
ten million: 10,589,990,451 ___________________________
Creating the 4 Largest and 4 Smallest Numbers from a List
of Digits
9, 0, 5, 1, 4, 8
FOUR LARGEST
1) Put the digits in order from
greatest to least. LOCK IN the
thousands period for all 4 numbers you
create.
2) Reverse the digits in the ONES
and TENS PVs.
3) In the ONES PERIOD, write the 2nd
LARGEST number in the HUNDREDS PV.
Place the other two digits in the ones
and tens PV.
4) Reverse the digits in the ONES and
TENS PVs.
FOUR SMALLEST
1) Put the digits in order from
least to greatest. LOCK IN the
thousands period for all 4
numbers you create.
2) Follow the same steps as above.
3) Write the 2nd SMALLEST digit
in the HUNDREDS PV.
1) 9 8 5, 4 1 0
2) 9 8 5, 4 0 1
3) 9 8 5, 1 4 0
4) 9 8 5, 1 0 4
1) 1 0 4, 5 8 9
2) 1 0 4, 5 9 8
3) 1 0 4, 8 5 9
4) 1 0 4, 8 9 5
Integers
Integers: The set of whole numbers {0, 1, 2, 3, 4, 5,…} and their
opposites. The roster of the set of integers –
{… -3, -2, -1, 0, 1, 2, 3,…}.
Opposites: Opposites are two numbers the same distance away
from zero on opposite sides of the number line. 5 and -5 are
opposites.
 What is the opposite of -8? 8
Positive Numbers: The further away from zero, the LARGER the
number.
Negative numbers: The further away from zero, the smaller the
number.
Comparing Integers: < , >, =
38 < 42
2 > -1
-22 < -21
-25 > -28
Order the integers from least to greatest:
-7, 32, 41, -36, -27, -2
-36, -27, -7, -2, 32, 42
Real Life Applications of Integers
II. Jeopardy
- Loss (negative)
- Profit (positive
-Loss of money (negative)
-gain of money (positive)
III. Temperature -below zero degrees (negative) -10
-above zero degrees (positive) 10
IV. Bank Accounts
V. Elevation
-withdrawal (negative)
-deposit (positive)
- below sea level (negative) -25 feet
-above sea level (positive) +15 feet
Ordering and Comparing Numerals
Whole Numbers (0 and all positive numbers):
Step 1: Count the number of digits in each numeral and write it
above the numeral. If one number has more digits than the
other, it is larger.
5
4
4 3, 0 0 8 ___>___ 4, 3 0 8
Step 2: If the have the same number of digits, go to the
largest PV and compare each digit; work from left to right.
When you come to a PV where the digits are different, BOX
OUT the two digits and compare. That will determine the larger
number.
6
6
5 3 2, 7 5 3 __>__ 5 3 2, 5 7 3
RECTANGLE
a)
b)
c)
d)
4 sides
opposite sides are congruent
2 pairs of parallel sides
4 right angles
RHOMBUS
a)
b)
c)
d)
4 sides
4 congruent sides
2 pairs of parallel sides
opposite angles are congruent
SQUARE
a)
b)
c)
d)
4 sides
4 congruent sides
2 pairs of parallel sides
4 right angles
PARALLELOGRAM
a)
b)
c)
d)
4 sides
opposite sides are congruent
2 pairs of parallel sides
opposite angles are congruen
TRAPEZOID
a) 4 sides
b) 1 pair of parallel sides
Whole Numbers
Unique Number (#1): The number 1 has only one factor, therefore it is
unique.
1 {1}
Prime Number: A number with two factors: the number 1 and itself.
Groups of 10s
Numbers in the single digits (4)
Numbers in the 10s (4)
Numbers in the 20s (2)
Numbers in the 30s (2)
Numbers in the 40s (3)
PRIME NUMBERS
2, 3, 5, 7
11,13,17,19
23, 29
31, 37
41, 43, 47
Factors of 11 {1, 11}
Composite Number: A number with more than two factors.
16 { 1, 2, 4, 8, 16}
Square Number: The product of a number multiplied by itself. A square
number has an odd number of factors.
1x1=1
7 x 7 = 49
2x2=4
8 x 8 = 64
3x3=9
9 x 9 = 81
4 x 4 = 16
10 x 10 = 100
5 x 5 = 25
11 x 11 = 121
6 x 6 = 36
12 x 12 = 144
5  5 = 25
25 {1, 5, 25}
Array
Factor: The numbers used in a multiplication problem OR a factor of a given
number is any number that divides evenly into that given number with no
remainder.
18 { 1, 2, 3, 6, 9, 18}
Array: (Rows/Columns): A rectangular arrangement of objects with an equal
number of objects in each row.
1
10










1  10
PRIME NUMBER 10 { 1,2, 5, 10}










10
10  1
1.
2.
3.
4.
DRAW the array
LABEL the sides of the array
NAME the array (row x column) THINK RC COLA
WRITE the FACTORS within braces { } and separated by commas.
Numbers are to be written in order (rainbow shows factor pair)
Area: The measure covering inside the figure. It is measured in square units.
Area = L x W
= 10 x 1 = 10 units squared
PeRIMeter: The distance around the RIM of a figure.
Perimeter = 2L + 2 W
= 2 (10 ) + 2 (1)
= 20 + 2 = 22 units
```