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Whole Numbers

Whole Numbers

Curriculum Ready

Whole NumbersMathletics Passport © 3P Learning

3HSERIES TOPIC

1

Q The ant nest below has a tunnel system that leads down to a main chamber. After one ant enters the tunnel from the top, how many different ways can it get to the main chamber if it only travels downwards the entire way?

11 1

1 2 11 3 3 1

1 4 6 4 11 5 10 10 5 1

1 6 15 20 15 6 11 7 21 35 35 21 7 1

1 8 28 56 70 56 28 8 11 9 36 84 126 126 84 36 9 1

Pascal's TriangleBlaise Pascal developed this triangle by simply adding two whole numbers together each time

Main Chamber

Work through the book for a great way to solve this

Give this a go!

It is important to be able to identify the different types of whole numbers and recognize their properties so that we can apply the correct strategies needed when completing calculations.

2 Whole NumbersMathletics Passport © 3P Learning

3HSERIES TOPIC

How does it work? Whole NumbersWhole NumbersWhole Numbers

Whole NumbersMathletics Passport © 3P Learning

How does it work?

Method 1: Multiplying the number by the multiple of 10 matching its position in the number

3 2504

4 is in the thousands position

4 1000` #

Identify the position of the 4 in the number

Remember: When multiplying by multiples of 10, just add the same number of zeros to the end 2 100 200 5 100000 500000 11 10000 110000# # #= = =

Multiply 4 by the place value

3 2504

04 000 Change all the other numbers to a zero

Ignore all zeros in front of the 4

` place value of 4 is 4000

` place value of 4 is 4000

× 1

000

000

× 10

0 00

0

× 10

000

× 10

00

× 10

0

× 10 × 1

Place values

Numbers can be separated into columns that represent different multiples of 10.The column where a number is found determines the place value of that number.

1 4 2

Millions

Hundreds of th

ousands

Tens o

f thousan

ds

Thousands

HundredsTe

nsOnes

Method 2: The place value of a number can also be found by changing all the other numbers to a 0

N U M B E R S

What is the place value of 4 in the number 34 250?

2 3HSERIES TOPIC

3HSERIES TOPIC

3Whole NumbersMathletics Passport © 3P Learning

How does it work? Whole Numbers

Multiplying the number by the multiple of 10 matching its position in the number.

Here is another example.

× 1

000

000

× 10

0 00

0

× 10

000

× 10

00

× 10

0

× 10 × 1

Millions

Hundreds of th

ousands

Tens o

f thousan

ds

Thousands

HundredsTe

nsOnes

1 0 7 2 1 3 8

(i) Using words: Six hundred and thirty one thousand, four hundred and five

Name using groups of three

(ii) Expanded form: 6 100 000 3 10 000 1 1000 4 100 0 10 5 1# # # # # #+ + + + +^ ^ ^ ^ ^ ^h h h h h h Multiply each number by the place value and add together

(i) Using words: One million, seventy two thousand, one hundred and thirty eight

Name using groups of three

(ii) Expanded form: 1 1000 000 0 100 000 7 10 000 2 1000 1 100 3 10 8 1# # # # # # #+ + + + + +^ ^ ^ ^ ^ ^ ^h h h h h h h Multiply each number by the place value and add together

Hundreds of th

ousands

Tens o

f thousan

ds

Thousands

HundredsTe

nsOnes

× 10

0 00

0

× 10

000

× 10

00

× 10

0

× 10 × 1

6 3 1 4 0 5

(i) Write 1 072 138 using words (ii) Write 1 072 138 in expanded form

(i) Write 631 405 using words (ii) Write 631 405 in expanded form


3HSERIES TOPIC

How does it work? Whole NumbersYour Turn

Place value

1 Write down the place values for each of these numbers a 1426 b 42 603

Place value of 1: Place value of 3:

Place value of 2: Place value of 4: c 560 142 d 7 380 261



2 Write each of these ordinary numbers in: (i) worded form (ii) expanded form a 2560 (i) Two thousand, five hundred and sixty

(ii) (2 # 1000) + (5 # 100) + (6 # 10)

b 1 306 211 (i)

(ii)

c 891 026 (i)

(ii)

3HSERIES TOPIC



WH

OLE NUMBERS * WHOLE NUMB

ERS * Place Value

..../...../20...

Place value

d 708 002 (i)

(ii)

e 9 011 060 (i)

(ii)

3 Write the ordinary number for each of these:

a Four hundred and thirty nine thousand, two hundred and six

b 4 1000000 2 100000 0 10000 1 1000 0 100 3 10 0 1# # # # # # #+ + + + + +^ ^ ^ ^ ^ ^ ^h h h h h h h

c Eighty one thousand and five

d 9 10000 8 1000 9 100 9 10 8 1# # # # #+ + + +^ ^ ^ ^ ^h h h h h

e Any number whose place values for 4, 5 and 2 are 4000, 5 and 200

f Three million, thirty thousand and thirty


3HSERIES TOPIC


Adding and subtracting large numbers

When adding or subtracting large numbers, make sure the place values are lined up correctly.Here are some addition examples to refresh your memory.

Calculate 13 829 + 4271

You can check your answer by simply entering the sum into your calculator.

Ensure matching place values are aligned

Calculate 317 293 + 20 011 + 102 356

13 829 4271 18100` + =

Carry over the 'tens' value

Ensure matching place values are aligned

317293 20091 102356 439740` + + =

Carry over the 'tens' value

3 1 7 2 9 3 + 2 0 0 9 1 + 1 0 2 3 5 6 = 439 740

=13 8 2 9 +

4 2 7 11 1 1

18 1 0 0

3 1 7 2 9 3 += 2 0 0 9 1

1 0 2 3 5 62 1

4 3 9 7 4 0

3 1 7 2 9 3 +317 293 + 20 091 + 102 356 = 2 0 0 9 1

1 0 2 3 5 6

13 829 + 4271 =13 8 2 9 +

4 2 7 11 1 1

Always check that your answer makes sense if using a calculator. It is easy to accidentally press a wrong button when entering numbers and operations into a calculator.

3HSERIES TOPIC



There are many different accurate ways to subtract large numbers. You should always use the method that you were taught or know best.

Here is an example using one way.

Calculate 7635 – 4829

Step 1:

Step 2:

Step 3:

Step 4:

Step 5:

7 6 3 5 – 4 8 2 9 = 2806

Line up matching place values

Since 5 91 we cannot subtract, ` place a 1 between 5 and 2

The 1 is in front of the 5 (in the tens position), making it 15

1 is traded and carried to 2 to make it 3 for using the 1 to make 15

15 – 9 equals 6 and 3 – 3 equals 0

Since 6 81 we cannot subtract, ` place a 1 between 4 and 6The 1 is in front of the 6 (in the tens position), making it 161 is traded and carried to 4 to make it 5 for using the 1 to make 16

16 – 8 equals 8 and 7 – 5 equals 2

7635 4829 2806` - =

7 6 3 5 –1 1

4 8 2 9

7 6 3 5 –1 1

4 8 2 9

7 6 3 5 –1 1

4 8 2 90 6

7 6 3 5 –1 1

4 8 2 90 6

7 6 3 5 –1 1

4 8 2 92 8 0 6


3HSERIES TOPIC


Calculate 38 234 – 21 576

Step 1:

Step 2:

Step 3:

Step 4:

Step 5:

Step 6:

Step 7:

Line up matching place values

Since 4 1 6 we cannot subtract, ` place a 1 between 7 and 4 The 1 is in front of the 4 (in the tens position), making it 14

1 is traded and carried to 7 to make it 8 for using the 1 to make 14

14 – 6 equals 8

Since 3 1 8 we cannot subtract, ` place a 1 between 3 and 5The 1 is in front of the 6 (in the tens position), making it 131 is traded and carried to 5 to make it 6 for using the 1 to make 13

13 – 8 equals 5

Since 2 1 6 we cannot subtract, ` place a 1 between 2 and 1The 1 is in front of the 2 (in the tens position), making it 121 is traded and carried to 1 to make it 2 for using the 1 to make 12

12 – 6 equals 6, 8 – 2 equals 6, 3 – 2 equals 1Do all the subtractions since there are no more columns with the top 1 bottom

38234 21576 16658` - =

Here is another example for subtraction.

3 8 2 3 4 –2 1 5 7 6

3 8 2 3 4 –1

2 1 5 7 6

3 8 2 3 4 –1

2 1 5 7 68

3 8 2 3 4 –1 1

2 1 5 7 68

3 8 2 3 4 –1 1

2 1 5 7 65 8

3 8 2 3 4 –1 1

2 1 5 7 65 8

3 8 2 3 4 –1 1

2 1 5 7 61 6 6 5 8

3 8 2 3 4 – 2 1 5 7 6 = 16 658

3HSERIES TOPIC




1 Calculate each of these addition questions showing all working .

a 5 6 2 1 0 +8 8 3 5

3 0 6 1 4

c 9 94 3 2 1

8 6 4 2

e

2 4 6 3 9 93 9 5 1 1

7 2 6 0 1 9 3

b 7 1 4 0 0 +1 0 8 0 9 4 2 0 1

d

8 4 3 1 9 5

6 0 4 6 3

f

8 7 6 3 82 1

9 3 1 0 53 4 1 2

2 Combo Time!

Calculate the sum (+ ) of three hundred and forty five thousand, two hundred and nine and eighteen thousand, seven hundred and ninety six.

Large Whole Numbers

..../...../20...

+

+

+

+


3HSERIES TOPIC



3 Calculate each of these substraction questions showing all working.

a 5 2 6 8 – 2 3 5 2

c

3 6 5 2 6 8 – 1 0 4 8 2

e

2 0 3 0 4 0 – 1 0 2 0 3

b 2 5 2 7 2 – 5 6 4 0

d

5 4 3 2 1 –1 2 3 4 5

f

7 0 0 0 0 –2 6 7 8 9

4 Combo Time!

Calculate the difference (–) between:

five hundred and seventy thousand, two hundred and seventeen

and

ninety eight thousand, four hundred and twenty one

3HSERIES TOPIC



As you did when adding and subtracting, keep your place value columns lined up neatly.You need to be aware of the place value of the number you are multiplying by.

Long multiplication

Calculate 1429 # 32

Line up matching place valuesMultiply the 1429 by 2Carry over any ‘tens’ values after multiplying

Calculate 423 # 506

1429 32 45728#` =

For 1429 # 3 tens, put a 0 in the ones column and multiply by 3 Carry over any ‘tens’ values after multiplying

1 4 2 9 3 2

+1 2 8 4 8

1 4 2 9 3 2

2 8 5 8+1 +2

3 2 6 7 0

1 4 2 9 3 2

2 8 5 84 2 8 7 04 5 7 2 8

Add the two new numbers together

Here is another example. Be careful to line up the columns correctly.

For 423 # 0 tens, put a 0 in the ones column first and multiply by 0

Add the two new numbers together

4 2 3 5 0 6

+1 +1 2 4 2 8

4 2 3 5 0 6

2 5 3 80 0 0 0 0

4 2 3 5 0 6

2 5 3 80 0 0 0 0

+1 +12 0 0 5 0 0

4 2 3 5 0 6

2 5 3 80 0 0 0 0

+12 1 1 5 0 02 1 4 0 3 8

Line up matching place valuesMultiply the 423 by 6Carry over any ‘tens’ values after multiplying

For 423 # 5 hundreds, put a 0 in the ones and tens columns and multiply by 5 Carry over the ‘tens’ value after multiplying

423 506 214038#` =

#

#

#

+

#

#

#

#

+


3HSERIES TOPIC


a 3 0 1 6 2 1

c

9 5 7 0 6 3

e

1 0 1 2 3 7

b 2 5 8 1 1 9

d

3 8 7 6 4 5

f

2 0 2 0 2 1 5

Long multiplication

1 Calculate each of these multiplication questions showing all working. Check your answers on the calculator.

WHOLE NUMBERS * WHOLE NUM

BERS *

..../...../20...

Long Multiplication

#

#

#

#

#

#

3HSERIES TOPIC



a 2 1 2 × 1 2 1

c 9 0 8 × 2 0 9

e 1 3 2 5 × 4 3 7

b 2 5 8 × 4 0 5

d

8 6 4 × 3 4 5

f

6 4 8 5 × 1 2 3

Long multiplication

2 Calculate each of these multiplication questions showing all working. Check your answers on the calculator.


3HSERIES TOPIC


Step 1:

Step 2:

Step 3:

Step 4:

Step 5:

Divide 7 by 6Put the whole number answer (1) above the 7 Make the remainder (1) the ‘tens’ digit for the next number




Divide 48 by 6Put the answer (8) above the 8

Step 1:

Step 2:

Step 3:

If there is a remainder at the end, always write it as a fraction.

518 3 172 32` ' =

the amount left overthe divisor

1 8

5 81

5

1 7

8

2

3

3

3 1

51

17

2

2

2

32

g

g

g

Short and long division

Short and long division are only different due to the size of the number you are dividing by.Here is a short division question.

1 8

5 81

5

1 7

8

2

3

3

3 1

51

17

2

2

2

32

g

g

g

1 8

5 81

5

1 7

8

2

3

3

3 1

51

17

2

2

2

32

g

g

g

1 8

5 81

5

1 7

8

2

3

3

3 1

51

17

2

2

2

32

g

g

gdivisor

amount left over

remainder fraction =

Calculate 75 408 ' 6

Calculate 518 ' 3


Divide 21 by 3Put the whole number answer (7) above the 1 There is no remainder this time

Divide 8 by 3Put the whole number answer (2) above the 8 Write the remainder as a fraction ( 3

2 ) to the right

1

1 2

1 2 5

1 2 5 6

1 2 5 6 8

6 7 5 4 0 8

6 7 5 4 0 8

6 7 5 4 0 8

6 7 5 4 0 8

6 7 5 4 0 8

1

1 3

1 3 4

1 3 4 4

1

3

4

4

g

g

g

g

g75 408 6 12 568` ' =

3HSERIES TOPIC



6 259 14 447 141` ' =

414 6 2 5 9

5 6

414 6 2 5 9

5 66 5

4 414 6 2 5 9

5 66 55 6

4 414 6 2 5 9

5 66 55 6

9 9

4 4 714 6 2 5 9

5 66 55 6

9 99 8

4 4 714 6 2 5 9

5 66 55 6

9 99 8

1

Divide 62 by 14

Put the whole number answer (4) above the 2

Multiply 14 by the answer (4) and write this underneath the 62

Subtract 56 from 62

Drop the 5 down next to the answer

Divide 65 by 14



Subtract 56 from 65

Drop the 9 down next to the answer

Divide 99 by 14



Subtract 98 from 99

Write the remainder as a fraction

Step 1:

Step 2:

Step 3:

Step 4:

Step 5:

Step 6:

g

g

g

g

g

g

Here is a long division question.

Calculate 6259 ' 14


3HSERIES TOPIC



1 Calculate each of these short division questions showing all working. Check your answers on the calculator.

a 4767 ' 3 b 6180 ' 5

c 6912 ' 4 d 12 054 ' 6

2 Calculate each of these short division questions showing all working. (psst: remember to write any remainders as a simplified fraction) Check your answers on the calculator.

a 8965 ' 7 b 3879 ' 2

c 9263 ' 8 d 5801' 6

g g

g g

g g

g g


BERS *

..../...../20...

Short & Long Division n

3HSERIES TOPIC



g g

g g


3 Calculate each of these long division questions showing all working. Check your answers on the calculator.

a 15 3 8 5 5 b 23 8 9 4 7

c 24 5 1 8 5 d 17 2 5 7 8


3HSERIES TOPIC


Arithmetic laws

• Addition

• Multiplication

4 lots of 5

4 rows of 5

4 rows

3 51

1

4

3

2

2 4 5 columns

4 # 5

4 # 5

5 # 4

5 # 4

20

20

=

=

=

=

=

= =

=

5 lots of 4

5 columns of 4

a + b = b + a

rows # columns = columns # rows

a # b = b # a

1. Commutative lawsThe order that we add or multiply numbers can be switched without changing the answer.

The commutative law for multiplication can be shown also in terms of rows and columns.

The commutative law does not work for subtraction or division as the order of the terms is important.

There are three laws of arithmetic that can be powerful tools to help with calculations.

4 and 5

4 + 5 5 + 4 9

=

=

=

=

5 and 4

+ +

4 - 5 = -1 and 5 - 4 = 1

4 ÷ 5 = 0.8 and 5 ÷ 4 = 1.25

• 4 - 5 ! 5 - 4

• 4 ÷ 5 ! 5 ÷ 4

rows

columns

3HSERIES TOPIC



= = + +

= =

Commutative laws

Shade groups of boxes to match these descriptions and write the calculation they represent.

Fill in the mathematical sentences for each of these diagrams showing the commutative law.

1

2

Two rows of four

Four columns of one

a

a

c

b

d

e

Four rows of two

Two rows of seven

c

g

Four columns of two

Three rows of six

b

f

Two columns of four

Four rows of one

d

h

+

=

+

=

#

=

#

=

#

=

#

=

=

=

= =

= #

= #

= #

= #

= #

= #

= #

= #

= =


3HSERIES TOPIC


Commutative laws

Draw two different diagrams below to demonstrate that 2 # 6 = 6 # 2 = 12.

Earn yourself an awesome stamp for this one.Draw all four different pairs of diagrams that represent the commutative law for multiplication with an answer of 24.

4

5

Write the expression for the commutative law represented by this diagram.

=

=

3

3HSERIES TOPIC



2. Associative laws

The numbers we group together when adding or multiplying can change without changing the answer.

The associative law can make adding/multiplying terms easier.

What about subtraction or division? There are only a few very special cases where the associative law works for subtraction and division.When a 0 is involved like this: When a 1 is involved like this: • (6 - 2) - 0 = 6 - (2 - 0)• (0 ' 7) ' 5 = 0 ' (7 ' 5)

• (18 ' 3) ' 1 = 18 ' (3 ' 1)

In all other cases, the associative law does not work for subtraction and division.

31 + 25 + 9 = (31 + 9) + 25 = 40 + 25 = 65

13 # 25 # 4 = 13 # (25 # 4) = 13 # 100 = 1300

• Addition

• Multiplication

(a + b) + c = a + (b + c)

(a # b) # c = a # (b # c)

3 lots of 4 # 5

3 # 4 lots of 5

(3 # 4) # 5

3 # (4 # 5)

12 lots of 5

3 lots of 20

12 # 5

3 # 20

60

60

=

=

=

=

=

=

=

=

(3 + 4) + 5 7 + 5 12= =

=

=

3 + (4 + 5) 3 + 9 12= =

+

+

=

=

=

=

+ +

#

#

+ +

• (3 - 4) - 5 ! 3 - (4 - 5)• (3 ' 4) ' 5 ! 3 ' (4 ' 5)

(3 - 4) - 5 = -6 and 3 - (4 - 5) = 4(3 ' 4) ' 5 = 0.15 and 3 ' (4 ' 5) = 3.75


3HSERIES TOPIC


Associative laws

Complete these equations for the associative law of addition shown in each diagram.

a

c

b

d

+ += = 12

+

+

=

+

+

2 24 66 4

+

=

+

=

86 46 12

+ +=

+ +=

+

+

=

+

+

+

+

=

+

+

+

+

=

+

+

+

=

+

=

+

=

+

=

+

=

+

=

+ +=

1

3HSERIES TOPIC



Associative laws

The associative law lets us choose what we want to do first to make calculations easier and quicker.

Pairing up terms that make nicer whole numbers before adding to other terms is a great trick. Use the associative law for addition to simplify these calculations:

25 + 91 + 75 = +

+ 91

=

+ 91

=

122 + 163 + 37 = +

+

=

+

=

83 + 52 + 18 = +

+ 83

=

+ 83

=

102 + 43 + 25 = +

+

=

+

=

For example:14 + 9 + 16 is made easier by adding the 14 and 16 together first to make it 9 + 30 = 39

2

a

c

b

d

37 + 14 + 56 + 23 = 111 + 80 + 19 + 45 =e f


3HSERIES TOPIC


Associative laws

Complete these equations for the associative law of multiplication shown in each diagram.

a

c

b

d# #=

# #= = 6

# #= = 40

# #= = 168

#

#

=

#

#

#

#

=

#

#

#

=

#

= 40

#

=

#

= 168

#

#

=

#

#

#

#

=

#

#

#

=

#

= 6

#

=

#

=

3

3HSERIES TOPIC



Associative laws

Shade groups of squares in these grids to match the associative law of multiplication underneath.

Use the associative law for multiplication to simplify these calculations:

4

5

5 # 28 # 20 = #

# 28

=

# 28

=

4 # 9 # 75 = #

#

=

#

=

12 # 50 # 7 = #

# 7

=

# 7

=

15 # 12 # 11 = #

#

=

#

=

a

a

c

c d

b

b

d

# #=

# #=

# #= =

# #= =

# #=2 3(3 # 3) (2 # 3)

# #=4 6(3 # 6) (4 # 3)

#= =#4 243(2 # 3) (4 # 2)

#= =#3 302(5 # 2) (3 # 5)


3HSERIES TOPIC


Associative laws

Tick the box ‘true’ or ‘false’ to show which of these subtraction or division statements are special cases where the associative law does work.

Larger strings of numbers can also have the associative law applied to them.

Write an expression that represents these dot diagrams and use the associative law to simplify.

Group terms together in these expressions that make the calculation much easier.

12 + 34 + 8 + 4 + 2

12 # 2 # 5 # 3

Use the associative law to rearrange into three pairs to make the calculation easier.

23 + 11 + 37 + 24 + 16 + 9

(12 ' 4) ' 1 = 12 ' (4 ' 1)

(20 - 11) - 0 = 20 - (11 - 0)

(0 - 4) - 3 = 0 - (4 - 3)

(0 ' 1) ' 1 = 0 ' (1 ' 1)

(1 - 1) - 1 = 1 - (1 - 1)

(5 ' 0) ' 3 = 5 ' (0 ' 3)

(5 ' 1) ' 5 = 5 ' (1 ' 5)

(0 ' 16) ' 2 = 0 ' (16 ' 2)

(8 - 0) - 1 = 8 - (0 - 1)

(0 - 1) - 0 = 0 - (1 - 0)

True False True False





6

7

8

a

a

a

b

c

e

c

g

i

b

f

d

h

j

#b #

Explain why you grouped the numbers you did.

3HSERIES TOPIC



• Distributing the multiplication to each term within the brackets:

These signs must always match

These signs must always match

3. Distributive law

This law allows you to spread a multiplication out (or expand) into smaller parts to make it simpler.

Also works if a subtraction is inside the brackets.

Using this law in reverse makes the multiplication of large numbers easy to calculate mentally.

For example, we can use the distributive law to calculate 127 # 4 using either of these:

You could have split 127 up into any sum/subtraction you found easiest to use.

So the distributive law is:a # (b + c) = a # b + a # c

a # (b - c) = a # b - a # c

• Using the order of operations and calculating the brackets first:

+# #= =

# #= =3 3 27(4 + 5) (9)

# # #= = = 27+ +3 3 3 54 12 15(4 + 5)

# # #= = = 2- -2 2 2 34 8 6(4 - 3)

+ + +# # #= =

#

- - -# # #= =

#

127 # 4 = (120 + 7) # 4 = 120 # 4 + 7 # 4 = 480 + 28

= 508

127 # 4 = (130 – 3) # 4 = 130 # 4 – 3 # 4 = 520 – 12

= 508

120 # 4 + 7 # 4 or 130 # 4 – 3 # 4


3HSERIES TOPIC


Distributive law

Fill in the missing values for each of these examples of the distributive law.

Use the distributive law to simplify and evaluate these multiplications:

Explain why both have the same value even though the terms in the brackets are different.

Use the distributive law to expand these multiplications:

Write another similar expression to those in a that will also give the same answer.

1

3

2 a

b

c

a

a

c

c

e

b

b

d

d

f

6 # 25 = 6 #

+

= 6 #

+ 6 #

=

+

=

11 # 32 = 11 # +

= 11

#

+ 11 #

=

+

=

15 # 19 = 15 # -

= 15

#

- 15 #

=

-

=

8 # 98 = 8 # -

= 8 #

- 8 #

=

-

=

3 # 5 +

= 3 # + 3 # 6

5 #

8 +

2 = 5 # + 5 #

= +

=

5 # 4 +

6 = 5 # + 5 #

= +

=

6 # 13 + 6 # 9 = # +

# 8 -

4 = 7 # 8 - 7 #

# 9 +

= 45 + 5 # 11 12 #

-

= 36 - 12

# 7 - # 10 = 15 # 7 -

a) b)

3HSERIES TOPIC



You can apply the distributive law more than once to simplify a calculation:

Apply the distributive law twice to simplify and calculate these:

For example:

25 # 59 = 25 # (60 - 1) = 25 # 60 - 25 # 1 = 25 # 60 - 25 OR

= 25 # (20 + 40) - 25

= 25 # 20 + 25 # 40 - 25

= 500 + 1000 - 25

= 1475

= (20 + 5) # 60 - 25

= 20 # 60 + 5 # 60 - 25

= 1200 + 300 - 25

= 1475

14 # 37 45 # 82

22 # 75 25 # 112

83 # 35 120 # 108

4

a

c

e

b

d

f

Distributive law

= 25 # 60 - 25


3HSERIES TOPIC


Divisibility tests

Divisibility tests are used to see if a small whole number will be a factor of a larger composite number.

A number is always divisible by 2 if the last digit is an even number (i.e. 0 , 2 , 4 , 6 or 8)

234 is divisible by 2 as the last digit (4) is even

A number is always divisible by 3 if the sum (+) of all its digits is divisible by 3

234 is divisible by 3 because 2 + 3 + 4 = 9 (which is divisible by 3)

A number is always divisible by 4 if the number formed by the last two digits is divisible by 4

1324 is divisible by 4 because the last two digits form the number 24 (which is divisible by 4)

A number is always divisible by 5 if the last digit of the number is a 0 or 5

265 is divisible by 5 because the last digit is a 5

A number is always divisible by 6 if it is divisible by both 2 and 3

234 is divisible by 6 because it is even (so divisible by 2) and 2 + 3 + 4 = 9 (which is divisible by 3)

A number is always divisible by 8 if the number formed by the last three digits is divisible by 8

1328 is divisible by 8 because the last three digits form the number 328 (which is divisible by 8)

A number is always divisible by 9 if the sum (+) of all its digits is divisible by 9

234 is divisible by 9 because 2 + 3 + 4 = 9 (which is divisible by 9)

A number is always divisible by 10 if the last digit of the number is a 0

1840 is divisible by 10 because the last digit is 0

Investigate the divisibility tests for 7 and 11. They are a little more involved but interesting!

3HSERIES TOPIC



Use the divisibility tests to determine whether each of these numbers are divisible by the numbers listed on the right hand side. Draw a line to all the numbers each one is divisible by.

The first number is completed for you.

Divisibility tests* D

IVISIBIL

ITY TESTS FOR NUMBERS

..../...../20...

620

136

96

1491

345

207

512

588

738

1 001 001

312 756

8640

12 871

6030

2

3

4

5

6

8

9

10


3HSERIES TOPIC


When a number is multiplied by itself twice, this is called cubing the number.

The same pattern continues for any number of multiplications

(iii) 3 3 3 3 3 3# # # # #

A mixture of numbers multiplied together can also be simplified using exponent notation

(iv) 54 5 5 5 4# # # # #

Doing the reverse to simplifying is called expanding.

Simplify these products by using exponent notation and then calculate:

Exponent notation for numbers

Exponent notation uses a small number called a ‘power’ or ‘exponent’ to show how many times a number is multiplied by itself.

16

4 4 42# =

=

Two 4s in the multiplication, so the exponent is 2We say ‘4 squared’

Three 2s in the multiplication, so the exponent is 3We say ‘2 cubed’

3 3 3 3 3 3 3

729

6# # # # # =

=

Six 3s in the multiplication, so the exponent is 6We say ‘3 to the power of 6’

4 5 5 5 4 5 4 4 5 5 5 5

4 5

16 625

10 000

2 4

# # # # # # # # # #

#

#

=

=

=

=

Group identical numbers

We say ‘4 squared times 5 to the power of 4’

Write these in expanded form:

7 7 7 774# # #= The exponent is 4, so four 7s multiplied together

9 9 9 9 9 9 9 97# # # # # #= The exponent is 7, so seven 9s multiplied together

8

2 2 2 23# # =

=

(i)

(ii)

When a number is multiplied by itself once, this is called squaring the number

(i) 4 4#

74

97

(ii) 2 2 2# #

Be careful: A lot of people make this mistake: 7 7 44#= , which is NOT true.

7 7 44#!

Make sure you can see the difference.

3HSERIES TOPIC



a 5 # 5 b 4 # 4 # 4

c 2 # 2 # 2 # 2 # 2 d 11 # 11 # 11 # 11

e 7 # 7 # 7 # 7 # 7 # 7 f 3 # 3 # 3 # 3 # 3 # 3 × 3 # 3

2 Write each of the mixed products using exponent notation and then calculate.

a 2 # 2 # 2 # 3 # 3 b 5 # 5 # 4 # 4

c 6 # 6 # 6 # 6 # 7 # 7 # 7 d 2 # 1 # 2 # 1 # 2

e 2 # 8 # 8 # 2 # 8 # 8 # 8 f 4 # 3 # 3 # 4 # 3 # 2 # 2 # 2

3 Change each of these to expanded form.

a 33 b 84

c 65 d 127

e 5 73 2# f 2 34 2

#

g 275 4# h 2 3 52 4 2

# #

Exponent notation for numbers

1 Write each of these products using exponent notation.

EXPON

ENT NOTATION *

EXPONENT NOT

ATION *

..../...../20...


3HSERIES TOPIC


Puzzle time

The area of a square can be written using exponent notation:

2 units

2 unitsArea = 2 # 2

= 22 units2

= 4 units2

Using each of the different grey squares below twice and the black square only once, form a rectangle on the grid above. You can do this by shading in the squares using a pencil or cut some similar-sized squares out of another sheet of paper and try to complete like a jigsaw.

The top left-hand corner of the rectangle is already completed for you, so only one more 62 grey square can be used.

12

32use twice 42

5262

62

use once

use twiceuse twice

use twice

When finished, have a go at writing two different expressions for the total area of the rectangle using exponent notation.Hint: For one expression multiply the side lengths together.

Area expression 1 Area expression 2

3HSERIES TOPIC



The little ‘root’ number indicates how many times the same number appears in the multiplication.

You could be asked to write a value using square or cube root notation.

Calculate the square root of these whole numbers

Square roots and cube roots

Finding the square root or cube root of a number is the opposite operation to squaring or cubing.The radical symbol ( ) is used for roots, with used for square root and 3 for cube root.

The square root sign is asking: What number multiplied by itself once will get the number inside me?

36 6

6 6 6 362#

=

= =Because

Calculate the cube root of these whole numbers

8 2

2 2 2 2 8

3

3# #

=

= =

343 7

7 7 7 7 343

3

3# # =

=

=

Rewrite these numbers

4 4 4 16

4 16

2#

`

= =

=

3 3 3 3 27

3 27

3

3

# #

`

= =

=

9 3

3 3 3 92#

=

= =

Because

Because

The cube root sign is asking: What number multiplied by itself twice will get the number inside me?

3333

Because

(i) 9

(ii) 36

(i) 8

(ii) 343

(i) 4 as a square root

(ii) 3 as cube root

9 3 3

3

3

2

#=

=

=or

36 6 6

6

6

2

#=

=

=

9 written as a product of its prime factors


2 28 2

2

2

3 3

33

# #

=

=

=

or

343 7 7 7

7

7

3 3

33

# #=

=

=


343 written as a product of its prime factorsor

We look closely at prime factors next


3HSERIES TOPIC


Square roots and cube roots

1 Calculate each of these square roots

a 4 b 16

c 2 5 d 4 9

e 81 f 121

2 Calculate each of these cube roots

a 2 73 b 6 43

c 2163 d 5123

3 Write each of these values using square root notation

a 3 b 8

c 6 d 12

4 Write each of these values using cube root notation

a 1 b 2

c 5 d 7


BERS *

..../...../20...

Square roots & Cube roots

3HSERIES TOPIC


Where does it work? Whole NumbersWhole Numbers


BERS *

..../...../20...

Square roots & Cube roots

Express 18 as a product of its prime factors

Split 18 into two smaller factors

Solid circle around prime numbers to stop that branch



ALWAYS at the prime number.

Don’t ever do this

Once every branch has reached a prime number, multiply all the prime numbers together

Simplify answer

18 2 3 3

2 32

# #

#

` =

=

Where does it work?

18

6

2 3

3

3

1 3

because 1 is NOT a prime number

Remember: A prime number has two factors, itself and 1

Factor trees

Composite numbers can be divided exactly (with no remainder), by other smaller or equal whole numbers called factors.

Composite numbers: 15 9 12 4 24

Factors: 1 , 3 , 5 , 15 1 , 3 , 9 1 , 2 , 3 , 4 , 6 , 12 1 , 2 , 4 1 , 2 , 3 , 4 , 6 , 8 , 12 , 24

Prime numbers only have 1 and themself as factors.

Prime numbers: 2 3 17 11 31

Factors: 1 , 2 1 , 3 1 , 17 1 , 11 1 , 31

All composite numbers can be written as the product (#) of prime factors (all the prime numbers that divide exactly into them). Let’s see how.

‘Express’ is a another way of saying ‘write’ in Mathematics.


3HSERIES TOPIC

Where does it work? Whole Numbers





38 19 2#` =

Here are some more examples.


Split 6 and 8 into two smaller factors

Solid circle around prime numbers to stop that branchSplit 4 into two smaller factors



There is often more than one way to create a factor tree for numbers with a lot of factors.


Simplify answer2 3

48 2 2 2 3 24

# # # #

#

` =

=

38

19 2

48

8

22

4 2

6

3 2

3HSERIES TOPIC


Where does it work? Whole NumbersYour Turn

Factor trees

1 Fill in the missing values on the following factor trees and write the number as a product of its primes.

a b

c d

e f ..../...../20...

* PRIME FACTOR TREES * PRIME FAC

TOR TREES

12 8` =

8 4` =

12` = 18` =

3 2` =

5 6` =

12

4

2

18

2

3

32

22

4

4

2 2

56

14

84

12

32

128

2

4


3HSERIES TOPIC


2 Complete a factor tree for each number below and express them as a product of their prime factors.

a 8 b 20

8` = 20` =

c 24 d 60

24` = 60` =

e 96 f 144

96` = 144` =

Factor trees

3HSERIES TOPIC



Greatest common factor (GCF)

The GCF is the largest number that divides exactly into two or more composite numbers.

Write all the factors of each number then circle the largest one which appears in both lists.

(i) 6 and 8

Factors of 6: 1 , 2 , 3 , 6

Factors of 8: 1 , 2 , 4 , 8 ` The GCF for 6 and 8 is: 2

(ii) 18 and 12

Factors of 18: 1 , 2 , 3 , 6 , 9 , 18

Factors of 12: 1 , 2 , 3 , 4 , 6 , 12

` The GCF for 18 and 12 is: 6

Find the greatest common factor for these pairs of numbers

List all the factors for each number

Circle the largest number common to both lists

List all the factors for each number

Circle the largest number common to both lists

We can use the list of prime factors for larger numbers to find the GCF.

(i) 72 and 96

Factors of 72: 2 , 2 , 2 , 3 , 3

Factors of 96: 2 , 2 , 2 , 2 , 2 , 3

` The GCF for 72 and 96 is: 2 2 2 3 24# # # =

(ii) 528 and 624

Factors of 528: 2 , 2 , 2 , 2 , 3 , 11

Factors of 624: 2 , 2 , 2 , 2 , 3 , 13

` The GCF for 528 and 624 is: 2 2 2 2 3 48# # # # =

Find the GCF for these pairs of larger numbers

List all the prime factors for each number

List all the prime factors for each number


3HSERIES TOPIC


a 8 and 12 b 6 and 15

c 10 and 18 d 18 and 24

e 14 and 28 f 16 and 36

2 Use the prime factors to find the GCF for these larger numbers.

a 42 and 84 b 92 and 72

c 280 and 490 d 256 and 640

Greatest common factor (GCF)

1 Find the greatest common factor for these pairs of numbers.

* GREA

TEST

COMMON

FACTORS * GREATEST

COMMON FACTORS

..../...../20...

GCFs

3HSERIES TOPIC



Lowest common multiple (LCM)

The LCM is the smallest number that is common to the multiplication tables of two or more numbers.

Write down the multiples of the numbers and stop once you find the lowest common multiple.

Find the lowest common multiple for these pairs of numbers

List some multiples of the first number

List the multiples of the second number until there is a match

List some multiples of the first number

List the multiples of the second number until there is a match

We can use the list of prime factors for larger numbers to find the LCM by looking at the differences.

(i) 30 and 100

Prime factors of 30: 2 , 3 , 5

Prime factors of 100: 2 , 2 , 5 , 5

` The LCM for 30 and 100 is: 100 3003# =

(ii) 24 and 388

Prime factors of 24: 2 , 2 , 2 , 3

Prime factors of 388: 2 , 2 , 97

` The LCM for 15 and 388 is: 388 23282 3# # =

Find the LCM for these pairs of larger numbers

2 2# 4 2#

51 #

6 2#

1 2#

2 5#

3 2# 5 2# 7 2#

(i) 2 and 5

Multiples of 2: 2 , 4 , 6 , 8 , 10 , 12 , 14 ,...

Multiples of 5: 5 , 10 ,...

` The LCM for 2 and 5 is: 10

(ii) 6 and 8

Multiples of 6: 6 , 12 , 18 , 24 , 30 ,...

Multiples of 8: 8 , 16 , 24 ,... ` The LCM for 6 and 8 is: 24

List all the prime factors for both numbers

Circle all the different factors in the smaller number

Multiply the larger number by the different factor

List all the prime factors for both numbers

Circle all the different factors in the smaller number

Multiply the larger number by the different factors


3HSERIES TOPIC


a 3 and 9 b 5 and 10

c 4 and 6 d 5 and 6

e 6 and 7 f 12 and 16

2 Use the prime factors to find the LCM for these larger numbers.

a 60 and 108 b 42 and 150

c 168 and 180 d 210 and 385

Lowest common multiple (LCM)

1 Find the lowest common multiple for these pairs of numbers.

* LOW

EST COMMON M

ULTIPLE * LOWEST

COMMON MULTIPLE

..../...../20...

LCMs

3HSERIES TOPIC


What else can you do? Whole Numbers


H 3SERIES TOPIC

10 + 1 1 + 0

1 10 + 1 1 + 1 1 + 0

1 2 10 + 1 1 + 2 2 + 1 1 + 0

1 3 3 10 + 1 1 + 3 3 + 3 3 + 1 1 + 0

1 4 6 4 10 + 1 1 + 4 4 + 6 6 + 4 4 + 1 1 + 0

1 5 10 10 5 10 + 1 1 + 5 5 + 10 10 + 10 10 + 5 5 + 1 1 + 0

1 6 15 20 15 6 1

Pascal’s triangle

This amazing triangle developed in 1653 by French mathematician Blaise Pascal uses the addition of two whole numbers to create it. The number pattern forms the shape of a triangle and contains many mathematical applications.

To create Pascal’s triangle, each number on the line is obtained by adding the two numbers above it.

The first seven lines of Pascal’s Triangle

The pattern continues in the same fashion for each added row of numbers

The second diagonal of Pascal’s triangle contains all the counting numbers

11 1

1 2 11 3 3 1

1 4 6 4 11 5 10 10 5 1

1 6 15 20 15 6 1

Counting numbersCounting numbers



3HSERIES TOPIC


Here are some more patterns found within Pascal’s triangle.

The third diagonal of Pascal’s triangle contains triangular numbers

The Fibonacci sequence is also within Pascal’s triangle and is found by adding terms along the lines shown

11 1

1 2 11 3 3 1

1 4 6 4 11 5 10 10 5 1

1 6 15 20 15 6 1

Triangular numbersTriangular numbers

Triangular numbers are formed by creating equilateral triangles using dot diagrams starting from 1 dot

, , , , ,...1 3 6 10 15

A very well known number pattern which occurs frequently in nature is the Fibonacci Sequence.

1 1 1 + 1 = 2 1 + 2 = 3 1 + 3 + 1 = 5 3 + 4 + 1 = 8 1 + 6 + 5 + 1 = 13

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

Each number in a Fibonacci Sequence is found mathematically by adding the two numbers before it

1 , 1 , 2 , 3 , 5 , 8 , 13 , 21 ,... 0 + 1 1 + 1 1 + 2 2 + 3 3 + 5 5 + 8 8 + 13

Sunflowers contain a Fibonacci sequence

3HSERIES TOPIC


What else can you do? Whole NumbersYour Turn

The Fibonacci sequence is also within Pascal’s triangle and is found by adding terms along the lines shown

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

1 8 28 56 70 56 28 8 1

1 9 36 84 126 126 84 36 9 1

1 10 45 120 210 252 210 120 45 10 1

1 11 55 165 330 462 462 330 165 55 140 1

1 12 66 220 495 792 924 792 495 220 66 12 1

1 13 78 286 715 1287 1716 1716 1287 715 286 78 13 1

1 14 91 364 1001 2002 3003 3432 3003 2002 1001 364 91 14 1

1 15 105 455 1365 3003 5005 6435 6435 5005 3003 1365 455 105 15 1

Pascal’s triangle

Another special pattern is called the Sierpinkski Triangle. This is a special fractal pattern made using triangles.Each dark equilateral triangle is split into four smaller equilateral triangles at every step.

This pattern can be reproduced using Pascal’s triangle by simply separating the odd and even numbers.In Pascal’s triangle below, colour in all the odd numbered hexagons to see this pattern emerge!

* PA

SCAL’S TRI

ANGLE * PASCAL’S

TRIANGLE 1

1 11 2 1

..../...../20...


3HSERIES TOPIC


Applications of Pascal’s triangle

Pascal’s triangle can often be useful when solving problems like the ones shown here. Each number in Pascal’s triangle represents the number of paths that can be taken to get to that point.

Show all the different downward paths that can be taken to get to the circled number in the triangle

For this four-line Pascal triangle:

The number circled is 3, so there are 3 different downward paths leading to this point

The total number of different paths to the bottom of a Pascal triangle is found by adding the numbers across.

The total number of different paths 1 3 3 1

8

= + + +

=

Path 2Path 1 Path 3

11 1

1 2 11 3 3 1

11 1

1 2 11 3 3 1

11 1

1 2 11 3 3 1

11 1

1 2 11 3 1

11 1

1 2 11 3 3 1

3

(i) How many different paths can be taken to reach the bottom of the triangle below?

(ii) How many paths to reach the bottom if one more line was added?

11 1

1 2 11 3 3 1

1 4 6 4 1

The total number of different paths 1 4 6 4 1

16

= + + + +

=

3HSERIES TOPIC



Applications of Pascal's triangle

Write down how many different downward paths there are to each of the points circled on this triangle.

Number of downward pathways to:

1

2 Show the six different downward paths that lead to the circled point on this triangle from the top.

B

C D

A

E F

D

B

E

C

F

=

Start

=

=

=

=

=

Start

Path 5

Start

Path 4

Start

Path 6

Path 2

Start

Path 1

Start

Path 3

Start

A

(i) How many different paths can be taken to reach the bottom of the triangle below?

(ii) How many paths to reach the bottom if one more line was added?

6


3HSERIES TOPIC


Applications of Pascal's triangle

The ant nest below has a tunnel system that leads down to a main chamber. After one ant enters the tunnel from the top, how many different ways can it get to the main chamber if it only travels downwards the entire way?

Hint: Fill in Pascal’s triangle values.

3

Remember me?

Main Chamber

3HSERIES TOPIC


Cheat Sheet Whole Numbers

Here is a summary of the important things to remember for whole numbers.

× 1

000

000

× 10

0 00

0

× 10

000

× 10

00

× 10

0

× 10 × 1

Millions

Hundreds of th

ousands

Tens o

f thousan

ds

Thousands

HundredsTe

nsOnes

N U M B E R S

Place value• Writing numbers using words, name using groups of three digits. • To write in expanded form, multiply each number by the place value

and add together.• The place value of a numeral in a large number is found by

multiplying the numeral by the matching position value.

Adding and subtracting large numbersWhen adding or subtracting large numbers, make sure the place values are lined up correctly first.

Long multiplication• Make sure the place values are lined up correctly first.• Add zeros on each line to match the place value of the number you are multiplying by.• Add together the new numbers formed after multiplying.

Short and long division• Keep all place values lined up neatly.• Be careful and methodical with each step.• Always write the remainder as a fraction.

Exponent notation for numbersExponent notation is used to show how many times a number is multiplied by itself. 3 3 3 3 3 35

# # # # =

Square and cube roots • The square root or cube root of a number is the opposite operation to squaring or cubing.• The symbols used are for square root and 3 for cube root. 9 3= because 3 93 # = and 27 33 = because 3 3 3 27# # =

Factor trees These are used to write any composite number as the product of prime number factors only.

Greatest common factor (GCF)The GCF is the largest number that divides exactly into two or more composite numbers.

Lowest common multiple (LCM)The LCM is the smallest number that is common to the multiplication tables of two or more numbers.

Pascal’s triangle• Each number in Pascal’s triangle is the sum of the two numbers above it.• Each number is the number of different downward paths that can be taken to get to that point.• The number of different downward paths to the bottom of a Pascal’s triangle is found by adding

together all the values across the bottom.


3HSERIES TOPIC

Whole Numbers Notes

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