Number 2
Place value
Number 2 level descriptors
• LEVEL 3: WRITE NUMBERS
IN WORDS
• LEVEL 4: MULTIPLY BY
POWERS OF 10
• ORDER WHOLE NUMBERS
• MULTIPLY BY MULTIPLES OF
10
• LEVEL 5: ROUND NUMBERS
TO NEAREST 1, 10, etc
• ROUND NUMBERS TO ONE
AND TWO DECIMAL PLACES
• LEVEL 6: ORDER DECIMAL
NUMBERS
• APPROXIMATE DECIMALS
TO A SENSIBLE DEGREE OF
ACCURACY
• LEVEL 7: ESTIMATE TO 1, 2
AND 3 SIGNIFICANT
FIGURES SF
• USE SF TO ESTIMATE
CALCULATIONS
• LEVEL 8:
• SOLVE PROBLEMS USING
STANDARD FORM
KEY WORDS
•
•
•
•
•
•
•
ESTIMATE
INTEGER
MULTIPLY
DIVIDE
APPROXIMATE
ROUNDING
MULTIPLES
•
•
•
•
•
PLACE VALUE
POWERS
ORDER
DECIMAL PLACE
SIGNIFICANT
FIGURES
• STANDARD FORM
NUMBERS IN WORDS
• OBJECTIVE
• LEVEL 3:
UNDERSTAND HOW
TO WRITE
NUMBERS IN
WORDS
• SUCCESS CRITERIA
• WRITE NUMBERS
OF INCREASING
MAGNITUDE IN
WORDS
• CONVERT
NUMBERS WRITTEN
IN WORDS INTO
DECIMAL NUMBERS
TYPES OF NUMBER
•
•
•
•
•
•
•
Counting numbers
Natural numbers
Odd numbers
Even numbers
Integers
Positive integers
Negative integers
1, 2, 3, 4, 5, …
1, 2, 3, 4, 5, …
1, 3, 5, 7, 9, …
2, 4, 6, 8, 10, …
…,-2, -1, 0, 1, 2, …
+1, +2, +3, +4, …
-1, -2, -3, -4, -5, …
Match the words to the numbers
Match the words to the numbers
Sixty two thousand
and five
60025
Sixty two thousand
and five
60025
Sixty thousand two
hundred and five
625
Sixty thousand two
hundred and five
625
Six thousand two
hundred and fifty
60205
Six thousand two
hundred and fifty
60205
Sixty thousand and
twenty five
6025
Sixty thousand and
twenty five
6025
Sixty two thousand
five hundred
600025
Sixty two thousand
five hundred
600025
Six hundred and
twenty five
62005
Six hundred and
twenty five
62005
Six thousand and
twenty five
6205
Six thousand and
twenty five
6205
Six thousand two
hundred and five
62500
Six thousand two
hundred and five
62500
Six hundred
thousand and
twenty five
6250
Six hundred
thousand and
twenty five
6250
Complete the table
Ten
thousands
thousands
hundreds
tens
units
words
Hundred
thousands
millions
3
4
0
9
0
thirty four thousand and ninety
three million and fifty six
4
0
0
0
0
9
3
three hundred and fifty six thousand
3
3
6
2
9
9
0
4
5
2
9
2
4
7
1
8
4
2
1
8
5
CREATE YOUR OWN
NUMBER
whole numbers in words
IN WORDS
What does the 9 represent
in each case
9 653
Nine thousand or 9000
85 096
49 632
839
9 828 400
96 000 000
What does the 9 represent
in each case
9 653
Nine thousand or 9000
85 096
Nine tens or 90
Nine thousands or 9000
49 632
Nine units or 9
839
9 828 400 Nine million or 9000 000
96 000 000 Ninety million or 90 000 000
Write out the decimal numbers in
words
5.96
Five point nine six
98.7
2.945
52.87
96.709
0.009
0.06
Note: the most common mistake is to say 5.96 is
five point ninety six. This is five point nine six
Write out the decimal numbers in
words
5.96
Five point nine six
98.7
2.945
Ninety eight point seven
52.87
Two point nine four five
Fifty two point eight seven
96.709
Ninety six point seven zero nine
0.009
Zero point zero zero nine
Zero point zero six
0.06
Note: the most common mistake is to say 5.96 is
five point ninety six. This is five point nine six
CREATE YOUR OWN
NUMBER
decimal numbers in words
IN WORDS
What does the 4 represent
in each case
53.47
8.504
9.642
83.49
6.84
93.004
Four tenths or 4/10
Four thousandths or 4/1000
Four hundredths or 4/100
What does the 4 represent
in each case
53.47
8.504
9.642
83.49
6.84
93.004
Four tenths or 4/10
Four thousandths or 4/1000
Four hundredths or 4/100
Four tenths or 4/10
Four hundredths or 4/100
Four thousandths or 4/1000
NUMBERS IN WORDS REVIEW
• WRITE NUMBERS OF INCREASING
MAGNITUDE IN WORDS
• CONVERT NUMBERS WRITTEN IN WORDS
INTO DECIMAL NUMBERS
• IDENTIFY MAGNITUDE BY POSITION
MULTIPLY AND DIVIDE BY
POWERS OF 10
• OBJECTIVE
• LEVEL 4: UNDERSTAND
HOW TO MULTIPLY AND
DIVIDE BY POWERS
OF 10
• SUCCESS CRITERIA
• MULTIPLY BY 10 AND
ADD A ZERO
• MULTIPLY BY 100 AND
ADD TWO ZERO’S
• MULTIPLY A DECIMAL
BY 10 AND THE
NUMBERS MOVE TO
THE LEFT
• DIVIDE A DECIMAL BY
10 AND THE NUMBERS
MOVE TO THE RIGHT
Starter
a) 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 =
b) 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 + 5 =
c) 8 + 8 + 8+ 8 + 8 + 8 + 8 + 8 + 8 + 8 =
d) Is there a quick way to add the same
number ten times?
Modulus
• MODULUS means size irrespective of sign
|-4| = 4
|6| = 6
|-0.06| = 0.06
The decimal point needs to be put
back into the number
7850
16300
47600
233000
650000
650000
650000
185730
31200
07500
Seven point eight five
Sixteen point three
Four point seven six three
Two hundred and thirty three
Sixty five thousand
Six thousand five hundred
Six hundred and fifty
Eighteen point five seven three
Three hundred and twelve
Zero point seven five
The decimal point needs to be put
back into the number
7 850
16 300
4 7630
233 000
65000 0
6500 00
650 000
18 5730
312 00
0 7500
Seven point eight five
Sixteen point three
Four point seven six three
Two hundred and thirty three
Sixty five thousand
Six thousand five hundred
Six hundred and fifty
Eighteen point five seven three
Three hundred and twelve
Zero point seven five
Multiply numbers by ten
• When a number is multiplied by ten the modulus of
the number increases.
• If a whole number is multiplied by 10 a zero is placed
on the right hand end of the number
16 × 10 = 160
7 × 10 = 70
• If a decimal number is multiplied by 10 then the
numbers move one place to the left of the decimal
point and empty spaces are replaced by zero’s
2.76 × 10 = 27.6
0.62 × 10 = 6.2
Multiply numbers by 10
Example
1.
2.
3.
4.
5.
6.
7.
8.
45 × 10 = 450
45 × 10 =
5 × 10 =
57 × 10 =
20 × 10 =
89 × 10 =
128 × 10 =
167 × 10 =
360 × 10 =
Example
1.
2.
3.
4.
5.
6.
7.
8.
5.7 × 10 = 57
5.1 × 10 =
9.2 × 10 =
12.8 × 10 =
10.9 × 10 =
6.056 × 10 =
0.67 × 10 =
0.061 × 10 =
0.0074 × 10 =
Multiply numbers by 10
Example
1.
2.
3.
4.
5.
6.
7.
8.
45 × 10 = 450
45 × 10 = 450
5 × 10 = 50
57 × 10 = 570
20 × 10 = 200
89 × 10 = 890
128 × 10 = 1280
167 × 10 = 1670
360 × 10 = 300
Example
1.
2.
3.
4.
5.
6.
7.
8.
5.7 × 10 = 57
5.1 × 10 = 51
9.2 × 10 = 92
12.8 × 10 = 128
10.9 × 10 = 109
6.056 × 10 = 60.56
0.67 × 10 = 6.7
0.061 × 10 = 0.61
0.0074 × 10 = 0.074
Multiply numbers by one hundred
• When a number is multiplied by one hundred the
modulus of the number increases.
• If a whole number is multiplied by 100 two zero’s are
placed on the right hand end of the number
46 × 100 = 4600
8 × 100 = 800
• If a decimal number is multiplied by 100 then the
numbers move two places to the left of the decimal
point and empty spaces are replaced by zero’s
2.6 × 100 = 260
0.38 × 100 = 38
Multiply numbers by 100
Example 37 × 100 = 3700
Example
1.
2.
3.
4.
5.
6.
7.
8.
1.
2.
3.
4.
5.
6.
7.
8.
89 × 100 =
9 × 100 =
27 × 100 =
60 × 100 =
49 × 100 =
524 × 100 =
973 × 100 =
940 × 100 =
5.7 × 100 = 570
8.1 × 100 =
3.2 × 100 =
16.8 × 100 =
14.9 × 100 =
8.056 × 100 =
0.95 × 100 =
0.039 × 100 =
0.0059 × 100 =
Multiply numbers by 100
Example 37 × 100 = 3700
Example
1.
2.
3.
4.
5.
6.
7.
8.
1.
2.
3.
4.
5.
6.
7.
8.
89 × 100 = 8900
9 × 100 = 900
27 × 100 = 2700
60 × 100 = 6000
49 × 100 = 4900
524 × 100 = 52400
973 × 100 = 97300
940 × 100 = 94000
5.7 × 100 = 570
8.1 × 100 = 810
3.2 × 100 = 320
16.8 × 100 = 1680
14.9 × 100 = 1490
8.056 × 100 = 805.6
0.95 × 100 = 95
0.039 × 100 = 3.9
0.0059 × 100 = 0.59
Divide numbers by 10
• When a number is divided by ten the modulus of the
number decreases.
• If a whole number is divided by 10 then the numbers
move one place to the right of the decimal point and
empty spaces are filled with zero’s
16 ÷ 10 = 1.6
8 ÷ 10 = 0.8
• If a decimal number is divided by 10 then the
numbers move one place to the right of the decimal
point and empty spaces are filled with zero’s
14.6 ÷ 10 = 1.46
0.46 ÷ 10 = 0.046
Divide numbers by 10
Example 49 ÷ 10 = 4.9
Example
1.
2.
3.
4.
5.
6.
7.
8.
1.
2.
3.
4.
5.
6.
7.
8.
89 ÷ 10 =
5 ÷ 10 =
87 ÷ 10 =
40 ÷ 10 =
39 ÷ 10 =
174 ÷ 10 =
925 ÷ 10 =
620 ÷ 10 =
9.5 ÷ 10 = 0.95
65.1 ÷ 10 =
43.2 ÷ 10 =
6.8 ÷ 10 =
4.9 ÷ 10 =
7.046 ÷ 10 =
0.51 ÷ 10 =
0.064 ÷ 10 =
0.0038 ÷ 10 =
Divide numbers by 10
Example 49 ÷ 10 = 4.9
Example
1.
2.
3.
4.
5.
6.
7.
8.
1.
2.
3.
4.
5.
6.
7.
8.
89 ÷ 10 = 8.9
5 ÷ 10 = 0.5
87 ÷ 10 = 8.7
40 ÷ 10 = 4
39 ÷ 10 = 3.9
174 ÷ 10 = 17.4
925 ÷ 10 = 92.5
620 ÷ 10 = 62
9.5 ÷ 10 = 0.95
65.1 ÷ 10 = 6.51
43.2 ÷ 10 = 4.32
6.8 ÷ 10 = 0.68
4.9 ÷ 10 = 0.49
7.046 ÷ 10 = 0.7046
0.51 ÷ 10 = 0.051
0.064 ÷ 10 = 0.0064
0.0038 ÷ 10 = 0.00038
Divide numbers by 100
• When a number is divided by one hundred the modulus
of the number decreases.
• If a whole number is divided by 100 then the numbers
move two places to the right of the decimal point and
empty spaces are filled with zero’s
67 ÷ 100 = 0.67
4 ÷ 100 = 0.04
• If a decimal number is divided by 100 then the
numbers move two places to the right of the decimal
point and empty spaces are filled with zero’s
67.8 ÷ 100 = 0.678
0.86 ÷ 100 = 0.0086
Divide numbers by 100
Example 49 ÷ 100 = 0.49
Example 9.5 ÷ 100 = 0.095
1.
2.
3.
4.
5.
6.
7.
8.
1.
2.
3.
4.
5.
6.
7.
8.
89 ÷ 100 =
63 ÷ 100 =
7 ÷ 100 =
90 ÷ 100 =
73 ÷ 100 =
714 ÷ 100 =
953 ÷ 100 =
630 ÷ 100 =
9.1 ÷ 100 =
6.2 ÷ 100 =
46.8 ÷ 100 =
34.9 ÷ 100 =
7.056 ÷ 100 =
0.74 ÷ 100 =
0.062 ÷ 100 =
0.0094 ÷ 100 =
Divide numbers by 100
Example 49 ÷ 100 = 0.49
Example 9.5 ÷ 100 = 0.095
1.
2.
3.
4.
5.
6.
7.
8.
1.
2.
3.
4.
5.
6.
7.
8.
89 ÷ 100 = 0.89
63 ÷ 100 = 0.63
7 ÷ 100 = 0.07
90 ÷ 100 = 0.9
73 ÷ 100 = 0.73
714 ÷ 100 = 7.14
953 ÷ 100 = 9.53
630 ÷ 100 = 6.3
9.1 ÷ 100 = 0.091
6.2 ÷ 100 = 0.062
46.8 ÷ 100 = 0.468
34.9 ÷ 100 = 0.349
7.056 ÷ 100 = 0.07056
0.74 ÷ 100 = 0.0074
0.062 ÷ 100 = 0.00062
0.0094 ÷ 100 = 0.000094
× 100
× 100
× 10
÷ 100
÷ 10
÷ 10
÷ 10
30
÷ 10
30000
÷ 10
300
× 100
3
× 10
30
× 100
÷ 100
3000
3000
÷ 10
÷ 10
3000
÷ 10
30
300
÷ 10
÷ 10
30000
300
× 100
30
Create and complete your own snake
× 10
×
÷ 10
÷
name……………………….
Multiply by Powers of 10 review
• MULTIPLY BY 10 AND ADD A ZERO
• MULTIPLY BY 100 AND ADD TWO ZERO’S
• MULTIPLY A DECIMAL BY 10 AND THE
NUMBERS MOVE TO THE LEFT OF THE
DECIMAL POINT
• DIVIDE A DECIMAL BY 10 AND THE
NUMBERS MOVE TO THE RIGHT OF THE
DECIMAL POINT
ORDER WHOLE NUMBERS
• OBJECTIVE
• LEVEL 4: UNDERSTAND
HOW TO ORDER
WHOLE NUMBERS
• SUCCESS CRITERIA
• ORDER NEGATIVE
NUMBERS
• ORDER POSITIVE
NUMBERS
• ORDER INTEGERS
Complete the Number lines
-10
-6
-60
-250
0
-20
0
-50
0
2
4
8
40
80
100
250
Order whole numbers
• We are often asked to place numbers in
order of size. We must remember that:
• Negative numbers are smaller than
positive numbers
• Zero lies between the positive and
negative numbers
• If in doubt – think of a number line
Number lines
-10
-8
-6
-4
-2
0
2
4
6
8
10
-100
-80
-60
-40
-20
0
20
40
60
80
100
-250
-200
-150
-100
-50
0
50
100
150
200
250
ORDER POSITIVE INTEGERS
• Example
• Place the following numbers in order,
smallest first
6, 8, 24, 17, 81, 12, 15
6, 8, 12, 15, 17, 24, 81
ORDER POSITIVE INTEGERS
Place the following sets
of numbers in order,
smallest first.
Place the temperatures
in order, highest to
lowest
1.
2.
3.
4.
5.
1.
2.
3.
4.
5.
12, 83, 27, 1, 5
30, 41, 5, 17, 46
41, 1, 24, 56, 28
12, 91, 72, 8, 93
81, 93, 56, 34, 9
760C, 210C, 260C
150C, 180C, 160C
910C, 120C, 390C
510C, 170C, 560C
190C, 160C, 730C
EXTENSION – calculate the totals for each question 1
ORDER POSITIVE INTEGERS
Place the following sets
of numbers in order,
smallest first.
Place the temperatures
in order, highest to
lowest
1.
2.
3.
4.
5.
1.
2.
3.
4.
5.
1, 5 , 12, 27, 83
5, 17, 30, 41, 46
1, 24, 28, 41, 56
8, 12, 72, 91, 93
9, 34, 56, 81, 93
EXTENSION – 128, 123
210C, 260C, 760C
150C, 160C, 180C,
120C, 390C, 910C
170C, 510C, 560C
160C, 190C, 730C
ORDER NEGATIVE INTEGERS
• Example
• Place the following numbers in order,
smallest first
-6, -8, -4, -7, -18, -2, -5
-18, -8, -7, -6, -5, -4, -2
ORDER NEGATIVE INTEGERS
Place the following sets
of numbers in order,
smallest first.
Place the temperatures
in order, highest to
lowest
1.
2.
3.
4.
5.
1.
2.
3.
4.
5.
-2, -8, -17, -1, -52
-3, -4, -8, -27, -16
-3, -18, -2, -76, -26
-7, -9, -2, -18, -28
-8, -56, -42, -37 -92
-150C, -20C, -160C
-120C, -80C, -60C
-90C, -150C, -190C
-50C, -70C, -90C
-120C, -60C, -80C
EXTENSION – calculate the totals for each question 1
ORDER NEGATIVE INTEGERS
Place the following sets
of numbers in order,
smallest first.
Place the temperatures
in order, highest to
lowest
1.
2.
3.
4.
5.
1.
2.
3.
4.
5.
-52, -17, -8, -2, -1
-27, -16, -8, -4, -3
-76, -26, -18, -3, -2
-28, -18, -9, -7, -2
-92, -56, -42, -37 -8
EXTENSION – -80, -33
-160C, -150C, -20C
-120C, -80C, -60C
-190C, -150C, -90C
-90C, -70C, -50C
-120C, -80C, -60C
ORDER INTEGERS
• Example
• Place the following numbers in order,
smallest first
6, 0, -8, 4, 7, 8, -2, -5
-8, -5, -2, 0, 4, 6, 7, 8
ORDER INTEGERS
Place the temperatures in order, lowest to highest
1.
2.
3.
4.
5.
6.
60C, 20C, 80C, -50C, -20C, -60C
50C, 80C, 00C, -20C, -80C, -60C
10C, 120C, 90C, -90C, -150C, -190C
50C, 70C, 60C, 00C, -70C, -90C
190C, 160C, 70C, -120C, -60C, -80C
What is the difference between the highest and lowest
values
EXTENSION – calculate the totals for each set of values
ORDER INTEGERS
Place the temperatures in order, lowest to highest
1.
2.
3.
4.
5.
6.
-60C, -50C, -20C, 20C, 60C, 80C
-80C, -60C, -20C, 00C, 50C, 80C
-190C, -150C, -90C, 10C, 90C, 120C
-90C, -70C, 00C, 50C, 60C, 70C,
-120C, -80C, -60C, 70C, 160C, 190C
14, 16, 31, 16, 31
EXTENSION – 1, -3, -21, 2,16
ORDER INTEGERS
-60C, -40C, -20C, 00C, 20C, 40C, 60C , 80C
1.
2.
3.
4.
Calculate the total for the temperatures
Which is the coldest temperature
List the three coldest temperatures
List the three highest temperatures
Extension
What is the difference between the highest and
coldest temperatures
ORDER INTEGERS
-60C, -40C, -20C, 00C, 20C, 40C, 60C , 80C
1.
2.
3.
4.
Calculate the total for the temperatures 8
Which is the coldest temperature -6
List the three coldest temperatures -6, -4, -2
List the three highest temperatures 4, 6, 8
Extension
What is the difference between the highest and
coldest temperatures 14
ORDER INTEGERS
5
7
0
3
1
1. Create the largest number using all five cards
2. Create the smallest number using all five cards
3. Create the largest number using only the four smallest
cards
4. Create a number from four cards that can be divided
by 2
Extension
Create a four digit number that is divisible by 3
ORDER INTEGERS
5
7
0
3
1
1. Create the largest number using all five cards 75310
2. Create the smallest number using all five cards10357
3. Create the largest number using only the four smallest
cards, 5310
4. Create a number from four cards that can be divided
by 2, ends in 0
Extension
Create a four digit number that is divisible by 3, digits add
up to a multiple of 3
Change in temperature
1. Calculate the temperature if it drops by 100C
from 70C
2. Calculate the temperature if it drops by 150C
from -20C
3. Calculate the temperature if it rises by 120C
from 40C
4. Calculate the temperature if it rises by 150C
from -80C
ORDER WHOLE NUMBERS
REVIEW
• ORDER NEGATIVE NUMBERS
• ORDER POSITIVE NUMBERS
• ORDER INTEGERS
Rounding numbers
• OBJECTIVE
• LEVEL 5:
UNDERSTAND HOW
TO ROUND
INTEGERS AND
DECIMAL NUMBERS
TO ONE AND TWO
DECIMAL PLACES
• SUCCESS CRITERIA
• ROUND INTEGERS
TO THE NEAREST
10 AND 100
• ROUND DECIMAL
NUMBERS TO ONE
DECIMAL PLACE
• ROUND DECIMAL
NUMBERS TO TWO
DECIMAL PLACES
Which is closer
1. Is 390 closer to 400 or 300
2. Is 7647 closer to 7640 or 7650
3. Is 4849 closer to 4900 or 4800
4. Is 38 closer to 100 or 0
Rounding Numbers
Rules for rounding numbers
• To round to the nearest 10 we look at the number in the units
column
• To round to the nearest 100 we look at the number in the tens
column
• If the number is less than 5 we round down
• If the number is 5 or greater we round up
• Example
Note:
Round 5946 to the nearest
a) 10
b) 100
c) 1000
a) 5950
c) 6000
b) 5900
27 to the nearest 100 is 0
974 to the nearest 100 is 1000
Complete the table
Extension
Nearest 10 Nearest 100
Nearest 1000
6794
9485
6328
593
786
78
82
17
78325
Complete the table
Extension
Nearest 10 Nearest 100
Nearest 1000
6794
9485
6328
6790
9490
6330
6800
9500
6300
7000
9000
6000
593
786
78
82
590
790
80
80
600
800
100
100
1000
1000
0
0
17
78325
20
78330
0
78300
0
78000
Rounding Decimal numbers
Rules for rounding decimal numbers
• To round to one decimal place we look at the number in the second
decimal place
• To round to two decimal places we look at the number in the third
decimal place
• If the number is less than 5 we round down
• If the number is 5 or greater we round up
• Example
Note:
Round 48.7852 to
a) 3 dp
b) 2 dp
c) 1 dp
a) 48.765
c) 48.8
b) 48.79
3.96 rounded to 1 decimal place is 4.0
4.796 rounded to 2 decimal places is 4.80
Complete the table
1 dp
5.7626
94.7549
6.8463
8.7638
16.482
47.839
38.978
42.73
18.48
2 dp
Extension
3 dp
Complete the table
1 dp
2 dp
Extension
3 dp
5.7626
94.7549
6.8463
5.8
94.8
6.8
5.76
94.75
6.85
5.763
94.755
6.846
8.7638
16.482
47.839
38.978
8.8
16.5
47.8
39.0
8.76
16.48
47.84
38.98
8.764
16.482
47.839
38.978
42.73
18.48
42.7
18.5
42.73
18.48
42.730
18.480
Round the number in the centre in different ways
5649.4
1 decimal
place
5650
Nearest
10
5649.3976
Write a number in the centre and round in different ways
Extension
• Is it possible to round a number to a given
number of decimal places by rounding in
stages.
• Example round 7.3241 to 1 dp
• 7.3241
7.324
7.32
7.3
change to 3 dp then2dp then 1dp
Will this method work for any number 5.247
Rounding numbers review
• ROUND INTEGERS TO THE NEAREST 10
AND 100
• ROUND DECIMAL NUMBERS TO ONE
DECIMAL PLACE
• ROUND DECIMAL NUMBERS TO TWO
DECIMAL PLACES
Order decimal numbers
• OBJECTIVE
• LEVEL 6:
UNDERSTAND HOW
TO ORDER
DECIMAL NUMBERS
• SUCCESS CRITERIA
• LOOK AT PLACE VALUE
TO ORDER DECIMAL
NUMBERS
• MULTIPLY BY 10, 100,
1000 WHEN ORDERING
DECIMALS
STARTER ORDER INTEGERS
Place the temperatures in order, lowest to highest
1.
2.
3.
4.
5.
6.
60C, 20C, 80C, -50C, -20C, -60C
50C, 80C, 00C, -20C, -80C, -60C
10C, 120C, 90C, -90C, -150C, -190C
50C, 70C, 60C, 00C, -70C, -90C
190C, 160C, 70C, -120C, -60C, -80C
What is the difference between the highest and lowest
values
EXTENSION – calculate the totals for each set of values
ORDER INTEGERS
Place the temperatures in order, lowest to highest
1.
2.
3.
4.
5.
6.
-60C, -50C, -20C, 20C, 60C, 80C
-80C, -60C, -20C, 00C, 50C, 80C
-190C, -150C, -90C, 10C, 90C, 120C
-90C, -70C, 00C, 50C, 60C, 70C,
-120C, -80C, -60C, 70C, 160C, 190C
14, 16, 31, 16, 31
EXTENSION – 1, -3, -21, 2,16
Multiply by 1000
Multiply each set of numbers by 1000
1.
2.
3.
4.
5.
15.798, 22.782, 35.562, 18.386, 39.784
23.42, 49.02, 89.34, 12.73, 27.56
38.12, 39.94, 29.48, 16.89, 27.39
3.82, 9.03, 6.37, 1.93, 7.48, 2.83
2.3, 4.7, 8.3, 2.5, 0.5, 0.6, 0.2, 0.4
Multiply by 1000 answers
Multiply each set of numbers by 1000
1.
2.
3.
4.
5.
15798, 22782, 35562, 18386, 39784
23420, 49020, 89340, 12730, 27560
38120, 39940, 29480, 16890, 27390
3820, 9030, 6370, 1930, 7480, 2830
2300, 4700, 8300, 2500, 500, 600, 200, 400
ORDER DECIMAL NUMBERS
Rules to order decimal numbers method 1
• First look at the whole number part of the
decimal number and place in order of size
• Then look at the first decimal place and
place in order based on this number
• Then move to the next decimal place and
place in order based on this number
ORDER DECIMAL NUMBERS
Example method 1
Place the following numbers in order, largest to smallest
1.607, 1.67, 0.6, 0.7, 1.7, 0.76, 0.607
Look at the whole number part and arrange by whole
number
1.607, 1.67, 1.7, 0.6, 0.7, 0.76, 0.607
Look at the first decimal place and arrange in order
1.7, 1.607, 1.67, 0.7, 0.76, 0.6, 0.607
Look at the next decimal place and arrange in order
1.7, 1.67, 1.607, 0.76, 0.7, 0.6, 0.607
Look at the next decimal place and arrange in order
1.7, 1.67, 1.607, 0.76, 0.7, 0.607, 0.6
ORDER DECIMALS
0.85 2.508 2.5
0.58 0.805 0.08
1. Which is the largest number
2. Which is the smallest number
3. Place the numbers in order of size, smallest first
Extension
What is the difference between the smallest and largest
ORDER DECIMALS
0.85 2.508 2.5
0.58 0.805 0.08
1. Which is the largest number 2.508
2. Which is the smallest number 0.08
3. Place the numbers in order of size, smallest first
0.08, 0.58, 0.805, 0.85, 2.508
Extension
What is the difference between the smallest and largest
2.5
ORDER DECIMAL NUMBERS
Place the following sets of
numbers in order,
smallest first.
Place the following sets of
numbers in order, smallest
first.
1.
2.
3.
4.
5.
6.
7.
1.
2.
3.
4.
5.
6.
7.
0.8, 0.4, 0.6, 0.5, 0.3
0.7, 0.77, 0.76, 0.07
0.45, 0.05, 0.4, 0.04
0.9, 0.2, 0.18, 0.28
0.56, 0.42, 0.37, 0.92
1.7, 1.8, 1.07, 1.08
3.5, 0.05, 2.05, 2.5
0.507, 0.57, 0.705, 0.75
0.604, 0.46, 0.406, 0.405
0.704, 0.074, 0.477, 0.774
1.507, 1.705, 1.075, 2.1
3.701, 2.509, 1.909, 4,39
2.009, 0.009, 0.034, 1.001
1.607, 1.76, 1.067, 1.007
EXTENSION – calculate the totals for each question 1
ORDER DECIMAL NUMBERS
Place the following sets of
numbers in order,
smallest first.
Place the following sets of
numbers in order, smallest
first.
1.
2.
3.
4.
5.
6.
7.
1.
2.
3.
4.
5.
6.
7.
0.3, 0.4, 0.5, 0.6, 0.8
0.07, 0.7, 0.76, 0.77
0.04, 0.05, 0.4, 0.45
0.18, 0.2, 0.28, 0.9
0.37, 0.42, 0.56, 0.92
1.07, 1.08, 1.7, 1.8
0.05, 2.05, 2.5, 3.5
0.507, 0.57, 0.705, 0.75
0.405, 0.406, 0.46, 0.604
0.074, 0.477, 0.704, 0.774
1.075, 1.507, 1.705, 2.1
1.909, 2.509, 3.701, 4,39
0.009, 0.034, 1.001, 2.009
1.007, 1.067, 1.607, 1.76
EXTENSION – calculate the totals for each question 1
ORDER DECIMAL NUMBERS
Rules to order decimal numbers method 2
Look at the numbers to be ordered, identify the highest
number of decimal places from the set of numbers
If the highest number of decimal places is one then multiply
all numbers by 10
If the highest number of decimal places is two then multiply
all numbers by 100
If the highest number of decimal places is three then
multiply all numbers by 1000
This method converts all the decimal numbers into whole
numbers first, we then order the numbers. We must
remember to convert back at the end.
ORDER DECIMAL NUMBERS
Example method 2
Place the following numbers in order, largest to smallest
1.607, 1.67, 0.6, 0.7, 1.7, 0.76, 0.607
Look at the number of decimal places, the biggest number
of decimal places is three
We must multiply every number by 1000 to give
1607, 1670, 600, 700, 1700, 760, 607
We now place this set of numbers in order
1700, 1670, 1607, 760, 700, 607, 600
We now divide all numbers by 1000 to give
1.7, 1.67, 1.607, 0.76, 0.7, 0.607, 0.6
ORDER DECIMALS
0.907 1.6
1.69 0.609 0.9
0.96
1. Multiply all numbers by 1000
2. Place the multiplied numbers in order, smallest first
3. Place the original numbers in order, smallest first
Extension
Add the numbers together
6.666
ORDER DECIMALS
0.907 1.6
1.69 0.609 0.9
Multiply all numbers by 1000
907, 1600, 1690, 609, 900, 960
Place the multiplied numbers in order, smallest first
609, 900, 907, 960, 1600, 1690
Place the original numbers in order, smallest first
0.609, 0.9, 0.907, 0.96, 1.6, 1.69
Extension
Add the numbers together 6.666
0.96
ORDER DECIMAL NUMBERS
Multiply each number by 100.
Place the following sets of
numbers in order, smallest
first.
1.
2.
3.
4.
5.
6.
7.
0.8, 0.84, 0.86, 0.08
0.7, 0.77, 0.76, 0.07
0.45, 0.05, 0.4, 0.04
0.9, 0.2, 0.18, 0.28
0.56, 0.42, 0.37, 0.92
1.7, 1.8, 1.07, 1.08
3.5, 0.05, 2.05, 2.5
Multiply each number by 1000. Place
the following sets of numbers in
order, smallest first.
1.
2.
3.
4.
5.
6.
7.
0.507, 0.57, 0.705, 0.75
0.604, 0.46, 0.406, 0.405
0.704, 0.074, 0.477, 0.774
1.507, 1.705, 1.075, 2.1
3.701, 2.509, 1.909, 4,39
2.009, 0.009, 0.034, 1.001
1.607, 1.76, 1.067, 1.007
EXTENSION – calculate the totals for each question 1
ORDER DECIMAL NUMBERS
Multiply each number by 100.
Place the following sets of
numbers in order, smallest
first.
1.
2.
3.
4.
5.
6.
7.
0.8, 0.84, 0.86, 0.08
0.7, 0.77, 0.76, 0.07
0.45, 0.05, 0.4, 0.04
0.9, 0.2, 0.18, 0.28
0.56, 0.42, 0.37, 0.92
1.7, 1.8, 1.07, 1.08
3.5, 0.05, 2.05, 2.5
Multiply each number by 1000. Place
the following sets of numbers in
order, smallest first.
1.
2.
3.
4.
5.
6.
7.
0.507, 0.57, 0.705, 0.75
0.604, 0.46, 0.406, 0.405
0.704, 0.074, 0.477, 0.774
1.507, 1.705, 1.075, 2.1
3.701, 2.509, 1.909, 4,39
2.009, 0.009, 0.034, 1.001
1.607, 1.76, 1.067, 1.007
EXTENSION – calculate the totals for each question 1
Place numbers in the empty boxes so they increase in size
NAME
1
2
1.1
1.9
0.5
0.9
0.1
0.4
2.1
2.2
0.1
0.2
Order decimal numbers review
•
•
•
•
•
MULTIPLY BY 10, 100, 1000 TO ORDER DECIMALS
USE PLACE VALUE TO ORDER DECIMALS
ORDER DECIMALS WITH ONE DECIMAL PLACE
ORDER DECIMALS WITH TWO DECIMAL PLACES
ORDER DECIMALS WITH THREE DECIMAL PLACES
SENSIBLE DEGREE OF
ACCURACY
• OBJECTIVE
• LEVEL 6:
UNDERSTAND HOW
TO APPROXIMATE
DECIMALS TO A
SENSIBLE DEGREE
OF ACCURACY
• SUCCESS CRITERIA
• DETERMINE THE
SENSIBLE DEGREE
OF ACCURACY
• GIVE ANSWERS TO
A SENSIBLE
DEGREE OF
ACCURACY
Starter – how many decimal places
are in each number
a) 4.78
b) 5.059
c) 0.05
d) 0.0008
DETERMINE THE DEGREE OF
ACCURACY
• We are sometimes asked to give answers to a
sensible degree of accuracy.
• To determine the degree of accuracy we must
look at the accuracy of the values given in the
question
• If values are given to 3 decimal places then the
answer should be given to two decimal places
• If the values given in the question are currency
then the answer should be given to 2 decimal
places to indicate pounds and pence
Determine the degree of accuracy
• Example 1
• A room is measured for
carpet fitting. The
length of the room is
4.3 m and the width is
3.62 m
• Calculate the area of
the carpet required to a
sensible degree of
accuracy
• Answer
Area = length × width
Area = 4.3 × 3.62
Area = 15.566 m2
As the length and width
were given to an
accuracy of up to 2
decimal places the area
should be given to the
same degree of
accuracy
Area = 15.57 m2
Determine the degree of accuracy
• Example 2
• A 4.6 m length of
timber is required to
build a fence. The
timber costs £4.68 per
metre
• Calculate the cost of
the length of timber
• Answer
Cost = length × price per m
Cost = 4.6 × £4.68
Cost = £21.528
As the final answer is
pounds and pence then a
sensible degree of
accuracy would be 2
decimal places.
Cost = £21.53
Complete the table
Calculation
4.6 × 7.25
2.8 × £3.50
3.52 × £2.43
7.642 × 7.2
14.6 × 37.2
4.63 × £1.24
5.216 × £0.63
Level of
accuracy
2 dp
Answer
Complete the table
Calculation
4.6 × 7.25
2.8 × £3.50
3.52 × £2.43
7.642 × 7.2
14.6 × 37.2
4.63 × £1.24
5.216 × £0.63
Level of
accuracy
2 dp
2 dp
2 dp
3 dp
1 dp
2 dp
3 dp Ans. 2dp
Answer
33.35
£9.80
£8.55
55.022
543.12
£5.74
£3.29
SENSIBLE DEGREE OF
ACCURACY REVIEW
• DETERMINE THE SENSIBLE DEGREE OF
ACCURACY
• GIVE ANSWERS TO A SENSIBLE DEGREE
OF ACCURACY
SIGNIFICANT FIGURES
• OBJECTIVE
• LEVEL 7:
UNDERSTAND HOW
TO APPROXIMATE
NUMBERS USING
SIGNIFICANT
FIGURES
• SUCCESS CRITERIA
• EXPRESS
NUMBERS USING
SIGNIFICANT
FIGURES
• APPROXIMATE
CALCULATIONS
USING SIGNIFICANT
FIGURES
Starter
Why would the number one hundred and
seventy three be written like this?
17
3
Significant figures
The numbers 127, 6.24, 0.0278, 809, 0.504 and
62500 all have three significant figures
• 0.0278 has three significant figures as the
zero’s at the front don’t count
• 62500 has three significant figures as the zero’s
at the end don’t count
• 809 has three significant figures as the zero’s
between other digits count
Significant figures
How many significant
figures do these
numbers have?
1. 2
2. 91
3. 183
4. 408
5. 87000
6. 2408
How many significant
figures do these
numbers have?
1. 3.1
2. 5.09
3. 0.7
4. 0.083
5. 0.508
6. 0.0006
Significant figures
How many significant
figures do these
numbers have?
1. 2
1 sf
2. 91
2 sf
3. 183
3 sf
4. 408
3 sf
5. 87000
2 sf
6. 2408
4 sf
How many significant
figures do these
numbers have?
1. 3.1
2 sf
2. 5.09
3 sf
3. 0.7
1 sf
4. 0.083
2 sf
5. 0.508
3 sf
6. 0.0006
1 sf
Rounding numbers using
significant figures
Numbers can be rounded using significant
figures in much the same way as we round
decimals.
• If the digit in the next column of
significance is 5 or greater then we round
up, else it stays the same
• 347 rounded to 2 significant figures is 350
Estimate the following to one
significant figure
1.
2.
3.
4.
5.
6.
7.
8.
9.
29
68
72
14
172
149
184
227
678
1.
2.
3.
4.
5.
6.
7.
8.
9.
17.5
128.9
0.67
0.37
0.92
0.029
0.079
0.052
0.0068
Estimate the following to one
significant figure
1.
2.
3.
4.
5.
6.
7.
8.
9.
30
70
70
10
200
100
200
200
700
1.
2.
3.
4.
5.
6.
7.
8.
9.
20
100
0.7
0.4
0.9
0.03
0.08
0.05
0.007
Estimate the following to two
significant figure
1.
2.
3.
4.
5.
6.
7.
8.
9.
562
672
272
146
138
128
1560
37600
45200
1.
2.
3.
4.
5.
6.
7.
8.
9.
45.9
386.9
0.268
0.382
0.578
0.0946
0.0942
0.00638
0.000836
Estimate the following to two
significant figure
1.
2.
3.
4.
5.
6.
7.
8.
9.
560
670
270
150
140
130
1600
38000
45000
1.
2.
3.
4.
5.
6.
7.
8.
9.
46
39
0.27
0.38
0.58
0.095
0.094
0.0064
0.00084
Rounding numbers using
significant figures
number
346
2894
2.476
19.382
14.975
0.2784
0.003864
0.07399
0.799
1 sf
2 sf
3 sf
Rounding numbers using
significant figures
number
346
2894
2.476
19.382
14.975
0.2784
0.003864
0.07399
0.799
1 sf
300
3000
2
20
10
0.3
0.004
0.07
0.8
2 sf
350
2900
2.5
19
15
0.28
0.0039
0.074
0.80
3 sf
346
2890
2.48
19.4
15.0
0.278
0.00386
0.0740
0.799
Estimating using significant figures
• Estimating is when we change numbers
into numbers we can use to calculate
answers in our heads. Round all numbers
to 1 significant figure
37 × 9 ≈ 40 × 10 ≈ 400
• We can now see that 37 × 9 should give
a value near 400
Estimate the answer example
Estimate the answers
to the following
problem
317 . 9  56 . 3
47 . 2
Answer
317 . 9  56 . 3
47 . 2

300  60
50

360
50

360
60
 6
Estimate answers to the following problems
1) Estimate the answers
a)
34  1 56 . 3
to the following
b)
19 . 7
c)
8 .2  7 .8
29 . 7
37 . 9  56 . 4
18 . 3
e)
d)
9 .2  1 5 .7
29 . 5
18 . 2  68 . 3
f)
46 . 2
2 ) Calculate
problems
278  28
17 . 4
the actual answers
and compare
answers
1
a) 10
d) 5
b) 0.6
e) 28
c) 5
f) 450
2
a) 9.66
d) 4.9
b) 0.54
e) 26.9
c) 5.2
f) 447
Create your own problem showing how you would calculate an estimate of the answer.
calculate the actual answer. Then compare answers and comment
SIGNIFICANT FIGURES REVIEW
• EXPRESS NUMBERS USING SIGNIFICANT
FIGURES
• APPROXIMATE CALCULATIONS USING
SIGNIFICANT FIGURES
STANDARD FORM
• OBJECTIVE
• LEVEL 8: UNDERSTAND
HOW TO USE STANDARD
FORM
• SUCCESS CRITERIA
• KNOW WHY WE USE
STANDARD FORM
• CONVERT NUMBERS INTO
STANDARD FORM
• CONVERT NUMBERS FROM
STANDARD FORM
• WHEN TO ADD AND
SUBTRACT INDEX
NUMBERS
• USE STANDARD FORM IN
CALCULATIONS (NON CALC)
• USE STANDARD FORM IN
CALCULATIONS
(CALCULATOR)
STARTER COMPLETE THE TABLE
NUMBER
TEN TIMES
TEN DIVIDE
POWER OF 10
WRITTEN 10n
10
100
1 × 10 × 10
1000
1 × 10 × 10 × 10
102
10000
100000
0.1
0.01
1 ÷ 10 ÷ 10
0.001
1 ÷ 10 ÷ 10 ÷ 10
0.0001
0.00001
10-2
Laws of indices
ya × yb = ya + b
ya ÷ yb = ya - b
(ya )b = yab
y0 = 1
y1 = y
Laws of indices
Example
a
y
×
72 × 74 = 72 + 4 = 7 6
Exercise Simplify
1. 32 × 34 =
2. 54 × 55 =
3. 88 × 84 =
4. 61 × 65 =
5. 92 × 96 =
b
y
=
a
+
b
y
Laws of indices
Example
a
y
÷
76 ÷ 74 = 76 - 4 = 7 2
Exercise Simplify
1. 36 ÷ 34 =
2. 59 ÷ 55 =
3. 88 ÷ 84 =
4. 68 ÷ 65 =
5. 94 ÷ 96 =
b
y
=
a
b
y
Laws of indices
Example
(ya)b
ab
=y
(76)4 = 76 × 4 = 724
Exercise Simplify
1. (36)4 =
2. (59)5 =
3. (83)4 =
4. (68)5 =
5. (94)6 =
= ya × b
Laws of indices cont
y
a
1

y
a
1
yn 
n
y
a
y
b

b
y
a
 (
b
y)
a
Laws of indices cont
y
a
1

y
a
1
yn 
n
y
a
y
b

b
y
a
 (
b
y)
a
KNOW WHY WE USE STANDARD
FORM
Standard form is used to express very large and very
small numbers in a different way
• 620000000000000 is a very large number
• In standard form 620000000000000 = 6.2 × 1014
• 0.000000000017 is a very small number
• In standard form 0.000000000017 = 1.7 × 10-12
CONVERT NUMBERS INTO
STANDARD FORM
To write numbers in standard form they must conform
to these rules
A × 10n
1 ≤ A <10
and n is an integer
• This means A must be greater than or equal to 1 but
less than 10. It must have a single digit (1, 2, 3, 4,
5, 6, 7, 8, 9) before the decimal point.
• n is a positive or negative whole number
CONVERT NUMBERS INTO
STANDARD FORM
• Example 1
• Convert 267000 into
standard form
• Example 2
• Convert 34000 into
standard form
• The decimal point
must move 5 places
to give 2.67
• The decimal point
must move 4 places
left to give 3.4
• 267000 = 2.67 × 105
• 34000 = 3.4 × 104
CONVERT NUMBERS INTO
STANDARD FORM
A × 10n
•
•
•
•
•
•
290
7300
68000
72000
45100
742000
=
=
=
=
=
=
2.9 × 102
7.3 × 103
6.8 × 104
7.2 × 104
4.51 × 103
7.42 × 105
CONVERT NUMBERS INTO
STANDARD FORM
1
1.
2.
3.
4.
5.
6.
7.
8.
Convert into standard
form
120 = 1.2 × 102
450
270
780
6300
3400
2300
7800
2 Convert into standard
form
1. 620 = 6.2 × 102
2. 950
3. 1700
4. 9830
5. 2390
6. 54000
7. 83900
8. 18000
CONVERT NUMBERS INTO
STANDARD FORM
1
1.
2.
3.
4.
5.
6.
7.
8.
Convert into standard
form
1900 = 1.9 × 103
2600
7900
8300
16000
39000
43000
28000
2 Convert into standard
form
1. 6500 = 6.5 × 103
2. 9300
3. 13000
4. 94000
5. 390000
6. 240000
7. 98000000
8. 59000000
CONVERT NUMBERS INTO
STANDARD FORM
• Example 1
• Convert 0.00056
into standard form
• Example 2
• Convert 0.0058 into
standard form
• The decimal point
must move 4 places
to give 5.6
• The decimal point
must move 3 places
to give 5.8
• 0.00056 = 5.6 × 10-4
• 0.0058 = 5.8 × 10-3
CONVERT NUMBERS INTO
STANDARD FORM
A × 10n
•
•
•
•
•
•
0.037
0.0028
0.00038
0.00029
0.00582
0.0000283
=
=
=
=
=
=
3.7 × 10-2
2.8 × 10-3
3.8 × 10-4
2.9 × 10-4
5.82 × 10-3
2.83 × 10-5
CONVERT NUMBERS INTO
STANDARD FORM
Convert into standard
form
1. 0.038
= 3.8 × 10-2
2. 0.049
3. 0.068
4. 0.039
5. 0.0032
6. 0.0084
7. 0.0067
8. 0.00074
Convert into standard
form
1. 0.098
= 9.8 × 10-2
2. 0.063
3. 0.0048
4. 0.0019
5. 0.000062
6. 0.000084
7. 0.000000062
8. 0.00000000293
CONVERT NUMBERS INTO
STANDARD FORM
Convert into standard
form
1. 0.0028 = 2.8 × 10-3
2. 0.0049
3. 0.00068
4. 0.00039
5. 0.00032
6. 0.000084
7. 0.000067
8. 0.0000074
Convert into standard
form
1. 0.0063 = 63 × 10-3
2. 0.0042
3. 0.00093
4. 0.00049
5. 0.000056
6. 0.00000748
7. 0.000000000412
8. 0.0000000000843
CONVERT NUMBERS FROM
STANDARD FORM
• Example 1
• Convert 5.39 × 104
into an ordinary
number
• Example
• Convert 6.73 × 105
into an ordinary
number
• The decimal point
must move 4 places
to the right
• The decimal point
must move 5 places
to the right
• 5.39 × 104 = 53900
• 6.73 × 105 = 673000
CONVERT NUMBERS FROM
STANDARD FORM
• Convert into ordinary
numbers
1. 5.3 × 102 = 530
2. 3.8 × 102
3. 3.5 × 102
4. 2.4 × 102
5. 3.5 × 103
6. 9.3 × 103
7. 5.27 × 103
8. 5.034 × 103
•
1.
2.
3.
4.
5.
6.
7.
8.
Convert into ordinary
numbers
6.5 × 103 = 6500
8.1 × 103
3.9 × 103
6.2 × 104
7.4 × 104
8.5 × 104
7.23 × 105
2.307 × 105
CONVERT NUMBERS FROM
STANDARD FORM
• Example 1
• Convert 6.7 × 10-3
into an ordinary
number
• Example 2
• Convert 8.31 × 10-3
into an ordinary
number
• The decimal point
must move 3 places
to the left
• The decimal point
must move 3 places
to the left
• 6.7 × 10-3 = 0.0067
• 8.31 × 10-3 = 0.00831
CONVERT NUMBERS FROM
STANDARD FORM
• Convert into ordinary
numbers
1. 5.9 × 10-2 = 0.059
2. 7.9 × 10-2 =
3. 3.6 × 10-2 =
4. 4.7 × 10-2 =
5. 2.6 × 10-3 =
6. 4.9 × 10-3 =
7. 3.1 × 10-3 =
8. 8.63 × 10-3 =
•
1.
2.
3.
4.
5.
6.
7.
8.
Convert into ordinary
numbers
8.3 × 10-3 = 0.0083
2.9 × 10-3 =
1.8 × 10-3 =
9.5 × 10-3 =
8.4 × 10-4 =
9.8 × 10-4 =
2.8 × 10-4 =
4.35 × 10-4 =
WHEN TO ADD AND SUBTRACT
INDEX NUMBERS
• The number 107 is written in index form.
The number 10 is the base number and the
number 7 is the index number
Index number
7
10
Base number
WHEN TO ADD INDEX NUMBERS
When two index form numbers are multiplied
together and they have the same base we
can add the index numbers
103 × 105 = 103 + 5 = 108
WHEN TO ADD INDEX NUMBERS
• Example 1
• Simplify
•
•
Example 2
Simplify
10-2 × 107
•
The base numbers are
the same so we can add
the index numbers
104 × 103
• The base numbers are
the same so we can add
the index numbers
104 × 103 = 104 + 3 = 107
10-2 × 107 = 10-2 + 7 = 105
WHEN TO ADD INDEX NUMBERS
• Simplify
• Simplify
1. 104 × 105 = 109
2. 102 × 105
3. 106 × 103
4. 107 × 104
5. 104 × 107
6. 106 × 104
7. 104 × 102
8. 103 × 105
1. 104 × 107 = 1011
2. 103 × 105
3. 106 × 105
4. 108 × 104
5. 10-4 × 107
6. 10-6 × 1014
7. 10-4 × 107
8. 10-3 × 105
WHEN TO ADD INDEX NUMBERS
• Simplify
• Simplify
1. 10-4 × 107 = 103
2. 10-2 × 105
3. 106 × 10-2
4. 107 × 10-3
5. 10-4 × 107
6. 106 × 10-2
7. 104 × 10-2
8. 10-3 × 105
1. 10-2 × 10-7 = 10-9
2. 10-3 × 10-5
3. 10-4 × 10-5
4. 10-8 × 10-3
5. 10-4 × 10-8
6. 10-6 × 10-4
7. 10-4 × 10-6
8. 10-3 × 10-8
WHEN TO SUBTRACT INDEX
NUMBERS
When two index form numbers are divided and
they have the same base we can subtract
the index numbers
107 ÷ 105 = 107 – 5 = 102
WHEN TO SUBTRACT INDEX
NUMBERS
• Example 1
• Simplify
•
•
Example 2
Simplify
10-5 ÷ 103
•
The base numbers are
the same so we can add
the index numbers
108 ÷ 103
• The base numbers are
the same so we can add
the index numbers
108 ÷ 103 = 108 – 3 = 105
10-5 ÷ 103 = 10-5 – 3 = 10-8
WHEN TO SUBTRACT INDEX
NUMBERS
• Simplify 1
• Simplify 2
1. 109 ÷ 107 = 102
2. 108 ÷ 105
3. 107 ÷ 103
4. 105 ÷ 104
5. 108 ÷ 103
6. 105 ÷ 103
7. 104 ÷ 102
8. 102 ÷ 105
1. 109 ÷ 104 = 105
2. 108 ÷ 103
3. 1011 ÷ 103
4. 107 ÷ 104
5. 10-5 ÷ 10-2
6. 10-2 ÷ 103
7. 104 ÷ 10-2
8. 10-8 ÷ 10-3
WHEN TO SUBTRACT INDEX
NUMBERS
• Simplify 3
• Simplify 4
1. 10-5 ÷ 106 = 10-11
2. 10-8 ÷ 105
3. 10-7 ÷ 103
4. 10-5 ÷ 104
5. 108 ÷ 10-3
6. 105 ÷ 10-4
7. 104 ÷ 10-2
8. 10-2 ÷ 10-5
1. 10-7 ÷ 10-5 = 10-2
2. 10-6 ÷ 10-2
3. 10-1 ÷ 10-3
4. 10-7 ÷ 10-3
5. 10-8 ÷ 10-4
6. 10-2 ÷ 10-8
7. 10-9 ÷ 10-2
8. 10-8 ÷ 10-5
USE STANDARD FORM IN
CALCULATIONS (NON CALC)
•
We can simplify expressions given in
standard form using the skills we have
gained.
•
•
•
•
•
Example
Simplify
3.0 × 105 × 4.0 × 103
Rearrange 3.0 × 4.0 × 105 × 103
Multiply together
Answer
12 × 108 = 1.2 × 109
USE STANDARD FORM IN
CALCULATIONS (NON CALC)
•
•
•
•
•
•
•
•
•
Simplify 1
2.0 × 105 × 4.0 × 103
3.0 × 106 × 2.0 × 102
5.0 × 102 × 3.0 × 103
4.0 × 103 × 3.0 × 104
2.0 × 107 × 2.5 × 103
1.5 × 104 × 4.0 × 105
2.0 × 105 × 1.5 × 106
2.5 × 104 × 3.0 × 103
•
•
•
•
•
•
•
•
•
Simplify 2
2.0 × 106 × 2.5 × 104
1.5 × 105 × 4.0 × 106
2.0 × 106 × 1.5 × 107
2.5 × 105 × 3.0 × 108
2.0 × 104 × 4.3 × 102
3.4 × 103 × 2.0 × 104
5.0 × 104 × 2.1 × 105
4.0 × 103 × 3.1 × 104
USE STANDARD FORM IN
CALCULATIONS (NON CALC)
•
We can simplify expressions given in
standard form using the skills we have
gained.
•
•
•
•
•
Example
Simplify
Rearrange
Divide
Answer
8.0 × 105 ÷ (4.0 × 103)
8.0 ÷ 4.0 × 105 ÷ 103
2.0 × 102
USE STANDARD FORM IN
CALCULATIONS (NON CALC)
•
•
•
•
•
•
•
•
•
Simplify 1
8.0 × 105 ÷
8.0 × 107 ÷
6.0 × 103 ÷
4.0 × 105 ÷
9.0 × 109 ÷
2.0 × 105 ÷
5.0 × 104 ÷
3.0 × 103 ÷
(4.0 × 103)
(2.0 × 102)
(3.0 × 107)
(2.0 × 106)
(3.0 × 102)
(4.0 × 103)
(2.5 × 104)
(1.5 × 106)
•
•
•
•
•
•
•
•
•
Simplify 2
6.0 × 105 ÷
9.0 × 107 ÷
6.0 × 103 ÷
4.0 × 105 ÷
5.0 × 109 ÷
2.0 × 105 ÷
9.0 × 104 ÷
7.0 × 103 ÷
(3.0
(3.0
(2.0
(2.0
(2.5
(4.0
(4.5
(3.5
×
×
×
×
×
×
×
×
106)
107)
105)
109)
105)
107)
108)
104)
USE STANDARD FORM IN
CALCULATIONS (CALCULATOR)
• To enter 105 into a calculator we must use the
raise to power button xy
• 8.6 × 105 when entered into a calculator we
press the buttons in this sequence
• 8.6
x 10 xy 5 = 860000
USE STANDARD FORM IN
CALCULATIONS (CALCULATOR)
Example
Simplify
3.2 × 10
3.2 × 108 × 4.0 × 1012
xy
8 × 4.0 × 10
xy
12 = 25.6 × 1020
3.2 × 108 × 4.0 × 1012 = 2.56 × 1021
USE STANDARD FORM IN
CALCULATIONS (CALCULATOR)
•
•
•
•
•
•
•
•
•
Simplify 1
2.0 × 105 × 4.0 × 103
3.0 × 106 × 2.0 × 102
5.0 × 102 × 3.0 × 103
4.0 × 103 × 3.0 × 104
2.0 × 107 × 2.5 × 103
1.5 × 104 × 4.0 × 105
2.0 × 105 × 1.5 × 106
2.5 × 104 × 3.0 × 103
•
•
•
•
•
•
•
•
•
Simplify 2
2.0 × 106 × 2.5 × 104
1.5 × 105 × 4.0 × 106
2.0 × 106 × 1.5 × 107
2.5 × 105 × 3.0 × 108
2.0 × 104 × 4.3 × 102
3.4 × 103 × 2.0 × 104
5.0 × 104 × 2.1 × 105
4.0 × 103 × 3.1 × 104
USE STANDARD FORM IN
CALCULATIONS (CALCULATOR)
• Example
• simplify
4.8 × 108
2.4 × 1012
4.8 × 10 xy 8 ÷ (2.4 × 10 xy 12) = 2.0 × 10-4
4.8 × 108 = 2.0 × 10-4
2.4 × 1012
R
USE STANDARD FORM IN
CALCULATIONS (NON CALC)
•
•
•
•
•
•
•
•
•
Simplify 1
8.0 × 105 ÷
8.0 × 107 ÷
6.0 × 103 ÷
4.0 × 105 ÷
9.0 × 109 ÷
2.0 × 105 ÷
5.0 × 104 ÷
3.0 × 103 ÷
(4.0 × 103)
(2.0 × 102)
(3.0 × 107)
(2.0 × 106)
(3.0 × 102)
(4.0 × 103)
(2.5 × 104)
(1.5 × 106)
•
•
•
•
•
•
•
•
•
Simplify 2
6.0 × 105 ÷
9.0 × 107 ÷
6.0 × 103 ÷
4.0 × 105 ÷
5.0 × 109 ÷
2.0 × 105 ÷
9.0 × 104 ÷
7.0 × 103 ÷
(3.0
(3.0
(2.0
(2.0
(2.5
(4.0
(4.5
(3.5
×
×
×
×
×
×
×
×
106)
107)
105)
109)
105)
107)
108)
104)
STANDARD FORM REVIEW
•
•
•
•
•
KNOW WHY WE USE STANDARD FORM
CONVERT NUMBERS INTO STANDARD FORM
CONVERT NUMBERS FROM STANDARD FORM
WHEN TO ADD AND DIVIDE INDEX NUMBERS
USE STANDARD FORM IN CALCULATIONS (NON
CALC)
• USE STANDARD FORM IN CALCULATIONS
(CALCULATOR)
Number 2 level descriptors
• OBJECTIVE
• LEVEL 7: UNDERSTAND
HOW TO ESTIMATE TO
1, 2 AND 3 SIGNIFICANT
FIGURE
• SUCCESS CRITERIA
• KNOW WHY WE
ESTIMATE NUMBERS
• WRITE NUMBERS TO 1,
2 AND 3 SIGNIFICANT
FIGURES
• KEEP ZERO’S TO
MAINTAIN MAGNITUDE
Number 2 level descriptors
• OBJECTIVE
• LEVEL 7:
UNDERSTAND HOW
TO ESTIMATE TO
SIGNIFICANT
FIGURES TO FIND
APPROXIMATE
ANSWERS TO
PROBLEMS
• SUCCESS CRITERIA
• FIND APPROXIMATE
SOLUTIONS TO
PROBLEMS BY
ESTIMATING
NUMBERS
Number 2 level descriptors
• OBJECTIVE
• LEVEL 8:
UNDERSTAND HOW
TO WRITE
NUMBERS IN
STANDARD FORM
• SUCCESS CRITERIA
• CONVERT A NUMBER
WRITTEN IN DECIMAL FORM
INTO STANDARD FORM
Number 2 level descriptors
• OBJECTIVE
• LEVEL 8:
UNDERSTAND HOW
TO WRITE
STANDARD FORM
NUMBERS IN
DECIMAL FORM
• SUCCESS CRITERIA
• CONVERT A NUMBER
WRITTEN IN STANDARD
INTO DECIMAL FORM
Number 2 level descriptors
• OBJECTIVE
• LEVEL 8:
UNDERSTAND HOW
TO SOLVE
PROBLEMS USING
STANDARD FORM
WITH A
CALCULATOR
• SUCCESS CRITERIA
• USE THE XY BUTTON
ON A CALCULATOR
• USE BRACKETS TO
KEY IN PROBLEMS
Number 2 level descriptors
• OBJECTIVE
• LEVEL 8:
UNDERSTAND HOW
TO SOLVE
PROBLEMS USING
STANDARD FORM
• SUCCESS CRITERIA
• USE THE LAWS OF INDICES
TO MULTIPLY AND DIVIDE
NUMBERS WRITTEN IN
STANDARD FORM
• ADD AND SUBTRACT
NUMBERS WRITTEN IN
STANDARD FORM
NUMBER STARTERS
Write numbers in words
• Objectives
• Success Criteria
• Understand how to
round off integers
• Be able to round off
integers to the nearest
10, 100 or 1000
• Be able to tell if an
answer is of the
correct order of
magnitude.
• Check results
Key Words
•
•
•
•
•
•
•
•
Integers
Natural numbers
Decimal numbers
Decimal place
Significant figures
Approximate
Estimate
Rounding
•
•
•
•
•
•
•
•
Standard form
Place value
Multiply
Divide
Rounding
Powers
Nearest
Zero
Rounding integers
• Objectives
• Success Criteria
• Understand how to
round off integers
• Be able to round off
integers to the nearest
10, 100 or 1000
• Be able to tell if an
answer is of the
correct order of
magnitude.
• Check results
School Bus Journey
1
3
2
5
4
7
6
9
8
School Bus Journey
10
30
20
50
40
70
60
90
80
School Bus Journey
• Which bus stop would you get
on if you lived:
• 18km from school
• 33km from school
• 11km from school
• 75km from school
• 94km from school
• 35km from school
• 87km from school
• 25km from school
School Bus Journey
100
300
200
500
400
700
600
900
800
School Bus Journey
• Which bus stop would
you get on if you lived:
• 418km from school
• 303km from school
• 630km from school
• 750km from school
• 946km from school
• 135km from school
• 597km from school
• 825km from school
Rounding Off – nearest 10
Round off the following numbers to the nearest 10:
1)
2)
3)
4)
5)
6)
7)
8)
9)
6
13
17
321
925
1369
8394
6929
12937
ans =
ans =
ans =
ans =
ans =
ans =
ans =
ans =
ans =
Rounding Off – nearest 100
Round off the following numbers to the nearest 100
1)
2)
3)
4)
5)
6)
7)
8)
9)
27
53
88
145
150
1983
2943
12948
40385
ans =
ans =
ans =
ans =
ans =
ans =
ans =
ans =
ans =
Rounding Off – nearest 1000
Round off the following numbers to the nearest 1000
1)
2)
3)
4)
5)
6)
7)
8)
9)
500
1200
2499
3501
6700
37814
67194
93167
48560
ans =
ans =
ans =
ans =
ans =
ans =
ans =
ans =
ans =
Rounding decimals
• Objectives
• Success Criteria
• Understand how to
round off decimals
• Be able to round off
decimals to the
nearest whole number
or to 1 decimal place.
• Check results
• Be able to tell if an
answer is of the
correct order of
magnitude.
School Bus Journey
1
3
2
5
4
7
6
9
8
School Bus Journey
1km
2km
2
3km
4
4km
6
5km
8
School Bus Journey
• Which bus stop would
you get on if you lived:
•
•
•
•
•
•
•
•
1.8km
3.3km
1.1km
4.5km
5.4km
3.5km
4.7km
2.5km
from
from
from
from
from
from
from
from
school
school
school
school
school
school
school
school
School Bus Journey
• Which bus stop would
you get on if you lived:
•
•
•
•
•
•
•
•
4.18km
3.03km
2.35km
4.54km
3.46km
1.35km
4.97km
3.25km
from
from
from
from
from
from
from
from
school
school
school
school
school
school
school
school
Rounding Off – nearest whole number
Round off the following decimals to the nearest whole number:
1)
2)
3)
4)
5)
6.3
1.3
1.7
2.8
5.4
ans =
ans =
ans =
ans =
ans =
Rounding Off – 0ne decimal place
Round off the following decimals to one decimal place
1)
2)
3)
4)
5)
2.74
5.36
8.81
1.45
1.58
ans
ans
ans
ans
ans
=
=
=
=
=
Rounding Off – one decimal place
Round off the following decimals to one decimal place
•
•
•
•
•
1)
2)
3)
4)
5)
5.09 ans =
12.24ans =
24.89ans =
35.11 ans =
67.06ans =
Rounding Off – two decimal places
Round off the following decimals to two decimal places
•
•
•
•
•
1)
2)
3)
4)
5)
5.093ans =
12.247ans =
24.891ans =
35.117ans =
67.065ans =
Standard form intro
• Objectives
• Understand how to
write numbers in
standard form
• Success Criteria
• Be able to write
numbers in standard
form.
• Be able to express
numbers given in
standard form as
normal decimal form.
Standard form harder
• Objectives
• Understand how to
calculate with
numbers in standard
form
• Success Criteria
• Be able to use numbers
in standard form to:
• Multiply
• Divide.
Significant figures
• Objectives
• Success Criteria
• Understand how to
write numbers using
significant figures
• Be able to round off
integers to the nearest
10, 100 or 1000
• Be able to tell if an
answer is of the
correct order of
magnitude.
• Check results
Calculations with significant figures
• Objectives
• Success Criteria
• Understand how to
write numbers using
significant figures
• Be able to round off
integers to the nearest
10, 100 or 1000
• Be able to tell if an
answer is of the
correct order of
magnitude.
• Check results
Multiply by multiples of 10
• Objectives
• Success Criteria
• Understand how to
write numbers using
significant figures
• Be able to round off
integers to the nearest
10, 100 or 1000
• Be able to tell if an
answer is of the
correct order of
magnitude.
• Check results
Estimation
• Objectives
• Success Criteria
• Understand how to
write numbers using
significant figures
• Be able to round off
integers to the nearest
10, 100 or 1000
• Be able to tell if an
answer is of the
correct order of
magnitude.
• Check results