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NUMBER SENSE AND NUMERATION

REPRESENTING NUMBER

1. WRITTEN FORM Examples:

i. One ii. Twelve

iii. Six hundred eight iv. Seven thousand eight three v. Seven hundredths

vi. Four hundred thousandths vii. Thirty two thousand, four hundred ninety six

viii. Nine hundred five billion, six hundred fifteen million, two hundred eight thousand, five hundred eight.

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2. STANDARD FORM Information: Beginning with the smallest whole number

(the ones), there are THREE digits in each grouping of numbers and then a space before the next set of digits. For the largest place value, there may be one, two or three digits.

Zero is a digit which represents a place value unless it is at the beginning of a whole number, and then it just a place holder

Examples: i. 32 469

ii. 905 615 208 508

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3. EXPANDED FORM Information: Numbers written in standard form can be

expanded out to show the value of each digit

When you do this, you what each digit in the

number actually represents numerically (its

place value)

Examples:

i. 8 765 = 8 000 + 700 + 60 + 5 ii. 943 567 832.23 = 900 000 000 + 40 000 000 + 3 000 000 + 5 00 000 + 60 000 + 7 000 + 800 + 30 + 2 + 0.2 + 0.03

2 6 7 5 9 8 3

ON

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TEN

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HU

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RED

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THO

USA

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TEN

S O

F TH

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HU

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USA

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PLACE VALUE Information: our number system has ten digits. They are: 0, 1, 2 , 3, 4, 5, 6, 7, 8, and 9 How those digits are arranged makes a number. It is the POSITION of the digit in the number that gives the digit a value. The further to the right a digit is in a number) the larger the value of the digit is. The further to the left a digit is (including in the decimal portion) the smaller the value of the digit is.

Example:

88 888.88 Even though all of the digits are 8, each place value means the 8s are worth different values. The underlined 8 is in the ten thousand place value position. It represents 8 groups of 10 000 or 80 000. The boxed digit 8 is in the hundredths place value position. It represents 8 groups of a hundredth, or .08

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EXPONENTS

The exponent shows how many times you must

multiple the base number times itself.

Examples:

52 means 5 x5, which equals 25

53 means 5 x 5 x 5, which equals 125

54 means 5 x 5 x 5 x 5 = 625

5 2 EXPONENT

POWER

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POWERS OF TEN

There is NO power of one!

102 = 10 x 10 = 100

103 = 10 x 10 x 10 = 1 000

104 = 10 x 10 x 10 x 10 = 10 000

105 = 10 x 10 x 10 x 10 x 10 = 100 000

106 = 10 x 10 x 10 x 10 x 10 x 10 = 1 000 000

107 = 10 x 10 x 10 x 10 x 10 x 10 x 10 = 10 000 000

108 = 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 100 000 000

109 = 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 1 000 000 000

1010 = 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 x 10 = 10 000 000 000

Notice how the exponent

and the number of zeros

in the number are the

same! MATH IS ALL

ABOUT PATTERNS

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EXPANDED FORM USING POWERS OF TEN

Since we have looked at Powers of ten, the next step

is to show numbers in expanded form using powers

of ten!

Example:

45 678

The digit 4 represents 4 groups of 10 000,

or 40 000.

We know that 104 is 10 000

We can show 40 000 as 4 x 104

The digit 5 represents 5 groups of 1 000 or 5 000 We know that 103 is 1 000 We can show 5 000 as 5 x 103 So, in expanded for, using Power of Ten:

45 678 = 4 x 104 + 5 x 103 + 6 x 102 + 70 + 8.

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DIVISIBILITY RULES

Divisible by…? The Strategy Example

2 Must end in an even number (o, 2 , 4, 6, 8)

252, 756, 928, 4 354

3 Add all digits together until they form either a single digit or a digit you recognize as being divisible by 3. If the sum of the digits is divisible by three, then so is the number

76 825 7+6+8+2+5 = 28 2+8 = 10 10 is NOT divisible by 3, so neither is 76 825.

4 Look at the last 2 digits. If they are divisible by 2, then so is the number.

9 6 5 7 57 is not divisible by 4, so neither is 9 657.

5 Any number that ends in a zero or a five is divisible by 5

6 890 is divisible by 5 because is ends in a zero.

6 Both rules for 2 and 3 apply 3 246 – last digit is even, so divisible by 2 and sum of digits is 15 which is divisible by 3, so it is divisible by 6

9 Sum of digits is divisible by 9 48 456 4+8+4+5+6 = 27 27 is divisible by 9, so so is 48 456

10 Any number that ends in a zero is divisible by 10

56 090 – is divisible by 10

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MULTIPLES

Think of your multiplication tables…

Multiples of a number are the product of that

number times any other number.

So, for example, the first 4 multiples of 2 are:

2 x 1 = 2

2 x 2 = 4

2 x 3 = 6

2 x 4 = 8

The first 6 multiples of 3 are: 3, 6, 9, 12, 15, 18, but

300 and 252 are also multiples of 3 because they can

be divided by 3 with no remainder.

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FACTORS Factors are one of two or more numbers that, when multiplied together, produce a give product. So, for example, the factors of 6 are 2 and 3 because 2 x 3 = 6. Sometimes we are asked to find ALL of the factors of a number. That means all of the possible combinations of numbers that we could multiply together to give a specific product. EXAMPLE:

36 All of the factors of 36 are:

(1 x 36) 2 x 18 3 x 12 4 x 9 6 x 6

Because ALL numbers can be

made by multiplying the

number by 1, we do NOT

include 1 as a factor of a

number

The factors of the number 36 are: 2, 3, 4, 6, 9, 12,

18 and 36.

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FACT FAMILIES – INVERSE OPERATIONS

You will be very surprised how often you find that

fact families are helpful as you progress through

intermediate and high school math! You learned

about fact families in primary grades, but guess

what? You apply (use) this knowledge in all different

forms of more challenging math!

Fact families show opposite or INVERSE operations.

Here are two examples of a fact families.

5 + 8 = 13 8 + 5 = 13 13 – 8 = 5 13 – 5 = 8

4 + 3 = 12 3 + 4 = 12 12 ÷ 3 = 4 12 ÷ 4 = 3

Fact families show how numbers are related. They

teach us that we can use the inverse operation to

help us. The inverse operation of addition is

subtraction and the inverse operation of

multiplication is division.

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OPERATIONS IN MATH

There are 4 basic operations in math. The answers

you get by using different operations have different

names!

The answer to addition is called the

SUM

The answer to subtraction is called

the DIFFERENCE The answer to multiplication is

called the PRODUCT The answer to division is called the

QUOTIENT

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“FRIENDLY” NUMBERS

Okay, well, numbers are not really either friendly or

unfriendly, but some numbers are a bit easier to

work with than others, so we call them, “friendly

numbers.”

Examples of friendly numbers include any multiple of

10, because with a bit of practise, these numbers

(because they end in zeroes) are easy to add,

subtract, multiply and divide.

What is one person’s idea of a friendly number may

differ from another’s, but usually, the following are

considered, “friendly:”

2, 5, 10, 20, 25, 50, 100, 1 000

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INTEGERS – POSITIVE AND NEGATIVE NUMBERS

An integer is a WHOLE number that is not a fraction.

Our number system is INFINITE which means that it

never ends. You can ALWAYS add to or take away

from a number.

Numbers that are BELOW (LESS THAN) zero are

called NEGATIVE integers (or numbers!). Numbers

that are ABOVE (GREATER THAN) zero are called

POSITIVE integers (or numbers!).

We encounter negative numbers all the time in our

real lives. In the winter, in Ontario, the temperature

is often below zero! We would show 25° below zero

as -25°celcius.

If you are below sea level, then this is measured as

below zero.

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

This is below sea

level!

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PRIME AND COMPOSITE NUMBERS

We describe whole numbers in different ways. One of the ways is by looking at how many factors a whole number has.

PRIME NUMBERS Some numbers only have 2 factors, one and the number itself. These numbers are called PRIME NUMBERS. Think about a glass plate made of a number. If you throw it up and it can only break into 2 sections, (1 and the number itself), it is PRIME. Another way to think about this is to describe PRIME NUMBERS as only having TWO DIVISORS. The first 20 PRIME NUMBERS are:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59,

61, 67 and 71.

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PRIME FACTORIZATION (FACTOR TREES)

We often have to find the smallest possible

prime multiples of a number. To do this we

make a FACTOR TREE.

We can use any multiples of the target number

to begin.

So, look at the composite number 36. We can

find many sets of multiples to, “smash it up.”

36

1 x 36 2 x 18 3 x 12 4 x 9 6 x 6

Let’s use 4 x 9 as our example

36 4 x 9

2 x 2 x 3 x 3

Because 2 and 3 are the smallest

PRIME numbers when 36 is

broken down, they are the PRIME

FACTORS of 36

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Let’s try another one. Here are the PRIME

FACTORS of 210

210

10 x 21

2 x 5 x 7 x 3

So the prime factors of 210 are: 2, 3, 5 & 7

SHOWING PRIME FACTORS

IN EXPONENTIAL FORM (Gr. 7/8)

432

2 x 216

2 x 2 x 108 2 x 2 x 2 x 54

2 x 2 x 2 x 3 x 18 2 x 2 x 2 x 3 x 3 x 6

2 x 2 x 2 x 3 x 3 x 2 x 3

To simplify the

bottom row we

could show it as

24 x 33

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USEFUL PREFIXES TO HELP UP RECOGNIZE

NUMBER VALUES

(USUALLY EITHER LATIN OR GREEK BASED

Prefix Meaning uni 1 bi 2 tri 3 quad 4 penta 5 hexa 6 hepta 7 octa 8 nona 9 deca 10

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GREATEST COMMON FACTOR (GCF)

& LOWEST COMMON MULTIPLES (LCM)

Often we look at two numbers and see what might be

common (shared) between them.

GREATEST COMMON FACTOR (GCF) Example:

12

3 x 4

3 x 2 x 2

20

2 x 10

2 x 2 x 5

Both 12 and 20 share two sets of 2 (I have shown one set

circles in red and one set circled in yellow). Bring ONE of

each set down and show it as a multiplication problem

2 x 2 = 4 The GCF of 12 and 20 is 4

Notice that you have “left-over,” factors that are NOT

shared by 12 and 20. I have shown them boxed in green.

Bring down the product of the shared factors (4) and

multiply that number by any of the left over (not shared)

factors.

4 x 3 x 5 4 x 3 x 5 = 60 The LCM of 12 and 20 is 60

To find the next common multiple, double the lowest one (60 + 60 =

120). To find the next one, add another 60 (120 + 60 = 180), keep

adding the LCM to find each subsequent multiple.

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COMPOSITE NUMBERS Some numbers have more than only 2 factors. These numbers are called COMPOSITE NUMBERS. Think again about the glass plate made of a number. If you throw it up and it can break in two lots of ways, it is COMPOSITE NUMBER. Another way to think about this is to describe COMPOSITE NUMBERS as having more than TWO DIVISORS. The first 20 COMPOSITE NUMBERS are:

4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28,

30, 32, 33…

The number 1 is NEITHER

prime nor composite!

In music, a COMPOSITION is a made up of

many different notes. In math, a

COMPOSITE number is made up of more

than just 2 numbers.