Transcript OBJECTIVE - plannerLIVE

OBJECTIVE
REVISION MOD 3
G
F
Check the side of the slide to see
E
what level you are working at!
D
C
B
A
A*
INTEGERS
• INTEGER is a whole number.
• HCF / LCM simple numbers – C
• HCF / LCM complex or more than two
numbers – B
• Recognise prime numbers – C
• Write a number as product of its prime
numbers – C
• Find the reciprocal of a number - C
G
Multiples
• These are all of the integers that appear in
your number’s times table!
• 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36.
• 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84
G•
Factors
These are all of the integers that will divide into
your number and leave no remainder!
• They are usually listed in pairs!
e.g. the factors of 36 are:
1 & 36
2 & 18
3 & 12
4&9
6&6
Prime Numbers
&
Prime Factors
• A PRIME NUMBER has TWO DIFFERENT FACTORS
1 & ITSELF.
The prime numbers less than 30 are ….
2, 3, 5, 7, 11, 13, 17, 19, 23, 29
• A PRIME FACTOR, is a factor that is also a prime number.
C
e.g. factors of 12 are 1, 2 ,3 ,4 , 6 & 12 of these 2 & 3 are
prime factors.
12 can be written as a product of prime factors…
12 = 2 x 2 x 3 in its INDEX FORM = 22 x 3
Highest common Factor
• The highest common factor (HCF) of two
numbers, is the largest factor common to both.
factors of 18 are 1,2,3,6,9,18
factors of 30 are 1,2,3,5,6,10,15,30
The highest factor common to both numbers is 6.
e.g.
C
We use HCF’s when cancelling fractions!!!
Lowest Common Multiple
• The Lowest Common Multiple (LCM) of
two numbers, is the smallest number that
appears in both time tables.
• The example below is for the 9 & 15 times table…..
C
e.g. the multiples of 9 are 9,18,27,36,45,54,63,….
the multiples of 15 are 15,30,45,60,…
45 is the lowest common multiple of each sequence of
numbers
Prime factor product trees
• Products of prime numbers can be written
as “trees”.
2 x 2 x 3 x 3 x 5 = 180
180
or; in INDEX FORM
90
C
2
x
2
45
x
3
x
15
x
3
x
x
22 x 32 x 5 = 180
5
x
x
HCF and LCM
• We can use prime factors to find the HCF and
LCM…
e.g. 504 = 2 x 2 x 2 x 3 x 3 x 7
700 = 2 x 2 x 5 x 5 x 7
HCF is 2 x 2 x 7 = 28
LCM is 2 x 2 x 2 x 3 x 3 x 5 x 5 x 7 = 12600
B
504 = 23 x 32 x 7
700 = 22 x 52 x 7
HCF is 22 x 7
LCM is 23 x 32 x 52 x 7
This is what’s left from BOTH numbers
when you take out the
HCF
Consecutive Numbers
C
• A set of 5 consecutive numbers will increase
by 5 each time, or are divisible by 5.
e.g. 1+2+3+4+5 = 15
2+3+4+5+6 = 20
If n = starting number, then the next is (n+1),
etc.
n + (n+1) + (n+2) + (n+3) + (n+4) = 5n +10
= 5(n+2)
Thus 5 is always factor of a series of five
consecutive numbers
INDICES
• INDEX is another word for POWER.
• Recall integer squares / square roots to 15 –
D
D
Use index laws for positive powers – C
Use index laws for negative powers – B
• Recall integer cube / cube roots to 5 –
•
•
A
Use index laws with complex fractional powers – A*
• Use index laws with simple fractional powers –
•
Square Numbers
&
Cube Numbers
• A SQUARE NUMBER is a NUMBER x ITSELF.
D
1 x 1 = 1, 2 x 2 = 4, 3 x 3 = 9, 4 x 4 = 16 and so on
Remember the first 15 Square Numbers ….
1,4,9,16,25,36,49,64,81,100,121,144,169,196,225.
• A CUBE NUMBER is a NUMBER x ITSELF x ITSELF.
1 x 1 x 1 = 1, 2 x 2 x 2 = 8, 3 x 3 x 3 = 27, and so on
Remember the first 5 Cube Numbers ….
1, 8, 27,64,125.
Square Root
The
D
NUMBER that is SQUARED to make 9 is 3.
3 is called the SQUARE ROOT of 9 and is written √9.
Remember the square roots as the reverse of the
square numbers.
SO √1,√4,√9,√16,√25,√36,√49,√64,√81,√100,√121,√144,√169,√196,√225
are the numbers from 1 to 15.
What are Indices?
• An Index is often referred to as a power
For example:
5 x 5 x 5 = 53
2 x 2 x 2 x 2 = 24
7 x 7 x 7x 7 x 7 = 75
5 is the INDEX
7 is the BASE NUMBER
75 & 24 are numbers in INDEX FORM
Rule 1 : Multiplication
26
x
x
2x2x2x2x2x2
35
C
3x3x3x3x3
x
x
24
= 210
2x2x2x2
37
= 312
3x3x3x3x3x3x3
General Rule
am x an = am+n
Rule 2 : Division
44
4x4x4x4
26
C
2x2x2x2x2x2
÷
÷
÷
÷
42
= 42
4x4
23
= 23
2x2x2
General Rule
am ÷ an = am-n
Rule 3 : Brackets
(26)2
= 26 x 26 = 212
(35)3
= 35 x 35 x 35 = 315
C
General Rule
(am)n = am x n
Rule 4 : Index of 0
How could you get an answer of 30?
35 ÷ 35 = 35-5 = 30
C
30 = 1
General Rule
a0 = 1
Combining numbers
5x5x5 x2x2x2x2
= 53 x 24
We can not write this any more simply
Can ONLY do that if BASE NUMBERS are the same
Putting them together?
26 x 24 = 210 = 27
23
23
C
35 x 37 = 312 = 38
34
34
25 x 23 = 28
24 x 22
26
= 22
..and a mixture…
2a3 x 3a4 = 2 x 3 x a3 x a4 = 6a7
8a6 ÷ 4a4 = (8 ÷ 4) x (a6 ÷ a4) = 2a2
C
2
6
28a
4
4a
Works with algebra too!
C
a6 x a4
b5 x b7
c5 x c3
c4
a5 x a3
a4 x a6
= a10
= b12
= c8 = c4
c4
= a8 = a-2
a10
Summary of rules.
1. am x an = am+n
2. am ÷ an = am-n
3. (am)n = am x n
4. a1 = a
5. a0 = 1
More rules….. Rule 6 negative
indices
25
24
23 22 21 20 2-1
32 16 8
4
2
1




1
B
2-2
2
General Rule
a-n
=
1
an
Rule 7 – Fractional Indices
From Rule 1 & 4


1
9 x 9 = 9 =9

So 9 = √9
General Rule
A
a
n
= √a
Rule 8 – Complex Fractional Indices

81 = (4√81)³ = (3)³ = 27
General Rule
Treat the bottom as a fractional index so find
root, then use top part as a normal index.
A*
Standard Index Form
• SIF is a way of writing big or small
numbers using indices of 10.
• Convert numbers to and from SIF – C
• Use SIF in simple number problems – B
• Use SIF in complex word problems – A
Why is this number very difficult to use?
999,999,999,999,999,999,999,999,999,999
Too big to read
Too large to comprehend
Too large for calculator
To get around using numbers this large,
we use standard index form.
Look at this
100,000,000,000,000,000,000,000,000,000
At the very least we can describe it as 1 with 29 noughts.
But it still not any easier to handle!?!
Let’s investigate!
Converting large numbers
How could we turn the number 800,000,000,000 into
standard index form?
We can break numbers into parts to make it easier,
C
e.g. 80 = 8 x 10 and 800 = 8 x 100
800,000,000,000 = 8 x 100,000,000,000
And 100, 000,000,000 = 1011
So, 800,000,000,000 = 8 x 1011 in standard index form
Standard Form (Standard Index Form)
5.3 x 10
n
There will
also be a
power of 10
C
The first part of the
number is between
1 and 10
But NOT 10 itself!!
One of the most important rules for writing
numbers in standard index form is:
The first number must be a value between
1 and 10
But NOT 10 itself!!
C
For example, 39 x 106 does have a value but it’s
not written in standard index form.
The first number, 39, is greater than 10.
3.9 x 107 is standard index form.
Indices of Ten
Notice that the number of zeros
matches the index number
2
10
3
10
4
10
5
10
100
1,000
10,000
100,000
Quick method of converting
numbers to standard form
For example,
Converting 45,000,000,000 to standard form
Place a decimal point after the first digit
C
4.5000000000
Count the number of digits
after the decimal point.
10
This is our index number (our power of 10)
So, 45,000,000,000 = 4.5 x 1010
And numbers less than 1?
How can we convert 0.067 into
standard index form?
0.067 = 6.7 x 0.01
C 0.067 = 6.7 x 10-2
0.01 = 10-2
And numbers less than 1?
How can we convert 0.000213
into standard index form?
0.000213 = 2.13 x 0.0001
0.0001 = 10-4
C 0.000213 = 2.13 x 10-4
How to write a number in standard form.
Place the decimal point after the first non-zero digit then
multiply or divide it by a power of 10 to give the same value.
56 = 5.6 x 10 = 5.6 x 101
567 = 5.67 x 100 = 5.67 x 102
5678 = 5.678 x 1000 = 5.678 x 103
56789 = 5.6789 x 10 000 = 5.6789 x 104
0.56 = 5.6  10 = 5.6 x 10-1
C
0.056 = 5.6  100 = 5.6 x 10-2
0.0056 = 5.6  1000 = 5.6 x 10-3
0.00056 = 5.6  10 000 = 5.6 x 10-4
Write the following in standard form.
23
234
4585
4.6
0.78
0.053
0.00123
2.3x 101
2.34x 102
4.585x 103
4.6x 100
7.8x 10-1
5.3x 10-2
1.23x 10-3
Standard Form on a Calculator
You need to use the exponential key (EXP or EE) on a
calculator when doing calculations in standard form.
Examples:
Exp/EE?
Calculate: 4.56 x 108 x 3.7 x 105
4.56
C
Exp
8 x 3.7
Exp
5 = 1.6872 x 1014
1.7 x 1014 (2sig fig)
Calculate: 5.3 x 10-4 x 2.7 x 10-13
5.3
Exp
Sharp
- 4 x 2.7
Exp
- 13
= 1.431 x 10-16
1.4 x 10-16 (2 sig fig)
+/-
Calculate: 3.79 x 1018  9.1 x 10-5
3.79
Exp
18  9.1
Exp
-5
= 4.2 x 1022 (2 sig fig)
Calculations Using SIF
B
Multiply two numbers
4 x 1018 x 3 x 104
Numbers
4x3
B
NOT Std
Form!
Powers of 10
x 1018 x 104
= 12 x
1022
ADD powers
= 1.2 x 101 x 1022
= 1.2 x 1023
Complex word problems involving SIF
The mass of the Earth is approximately
6 000 000 000 000 000 000 000 000 kg.
Write this number in standard form.
6.0 x 1024
The mass of Jupiter is approximately
2 390 000 000 000 000 000 000 000 000 kg. Write
this number in standard form.
2.39 x 1027
A
How many times more massive is Jupiter than Earth?
2.39 x 1027 / 6.0 x 1024 =
398
Complex word problems involving SIF
The mass of a uranium atom is approximately
0. 000 000 000 000 000 000 000 395 g.
Write this number in standard form.
3.95 x 10-22
The mass of a hydrogen atom is approximately
0. 000 000 000 000 000 000 000 001 67 g.
Write this number in standard form.
1.67 x 10-24
A
How many times heavier is uranium than
hydrogen?
3.95 x 10-22 / 1.67 x 10-24 =
237
Complex word problems involving SIF
Writing Answers in Decimal Form (Non-calculator)
Taking the distance to the moon is 2.45 x 105 miles and the average
speed of a space ship as 5.0 x 103 mph, find the time taken for it to
travel to the moon. Write your answer in decimal form.
A
D
S
S =
so T =
T
D
245 000
=
= 49 hours
5 000
Rounding.
Rounding to nearest integer (whole number). G.
Rounding to nearest 10 or 100. G.
Rounding to given number of decimal places. F.
Rounding to given number of significant figures. E.
G
Rounding
to the nearest whole number
• Is the arrow nearer to 6, 7 or 8?
• If it is halfway between, then round UP
6
7
8
G
Rounding
to the nearest 10
• Is the arrow nearer to 20, 30 or 40?
• If it is halfway between, then round UP
20
30
40
G
Rounding
to the nearest 100
• Is the arrow nearer to 400 or 500?
• If it is halfway between, then round UP
400
500
F
Decimal Places
Round the following number to 1dp
F
6.348
Firstly, highlight
the number to
the first number
after the
decimal point
So we have
6.3
But is this the
answer?
If thisNow
number
look is
at
a 0, 1,
the2,number
3 or 4
we don’t
immediately
have
to doafter
anything
where
else we
andstopped
we
havehighlighting
our
answer.
Round the following number to 1dp
F
6.348 = 6.3
(1dp)
What if the red number was a 5,
6, 7, 8 or 9?
F
Lets look at an example
Round the following number to 1dp
F
If thisNow
number
look is
at
a 5, 6,
the7,number
8 or 9
we increase
immediately
the
last digit
afterby
where
one. we stopped
highlighting
9.2721
Firstly, highlight
the number to
the first number
after the
decimal point
So we have
9.2
But is this the
answer?
So 9.2 becomes
9.3
Round the following number to 1dp
F
9.2721 = 9.3
(1dp)
Round the following number to 2dp
F
7.456
If this number is
a 0, 1, 2, 3 or 4
we don’t have
to do anything
Firstly, highlight Now look at
else and we
the number to the number
have our
the second
immediately answer, but it is
number after
after where not, so we
the decimal
we stopped round up the
point
highlighting number in the
second decimal
place to give us
our answer.
7.46
Round the following number to 2dp
F
3.992
If this number is
a 0, 1, 2, 3 or 4
we don’t have
Firstly, highlight Now look at to do anything
the number to the number else. In this
the second
immediately case it is so we
number after
after where have our
the decimal
we stopped answer
point
highlighting highlighted.
3.99
Round the following number to 1dp
F
6.348 = 6.3
(1dp)
What if the red number was a 5,
6, 7, 8 or 9?
F
Lets look at an example
Round the following number to 1dp
F
If thisNow
number
look is
at
a 5, 6,
the7,number
8 or 9
we increase
immediately
the
last digit
afterby
where
one. we stopped
highlighting
9.2721
Firstly, highlight
the number to
the first number
after the
decimal point
So we have
9.2
But is this the
answer?
So 9.2 becomes
9.3
Round the following number to 1dp
F
9.2721 = 9.3
(1dp)
Decimal Places (Rounding)
Numbers can be rounded to 1,2, 3 or more decimal places.
F
Rounding to 1 d.p
4.8325
4. 8 4 2 5
4. 8 5 2 5
5 or bigger ?
5 or bigger ?
5 or bigger ?
No
No
Yes
4.8
4.8
4.9
Decimal Places
F
It is often necessary/convenient/sensible to give approximations
to real life situations or as answers to certain calculations.
For example if a case of wine containing 6 bottles costs £25
then you could price a single bottle by calculating £25  6 =
£4.166666667. It would be pointless to write out all the
numbers on your calculator display. Since we are dealing with
money (pounds and pence) we only need 2 decimal places (2 d.p.)
So it would be much better to write down £4.17.
Rounding to 1 d.p
4.8325
F
5 or bigger ?
4. 8 4 2 5
5 or bigger ?
No
4. 8 5 2 5
5 or bigger ?
No
Yes
4.8
4.8
4.9
4. 8 6 2 5
4. 8 7 2 5
4. 8 9 2 5
5 or bigger ?
5 or bigger ?
Yes
4.9
5 or bigger ?
Yes
4.9
Yes
4.9
Rounding to 2 d.p
1. 4 2 6 1
5. 8 4 2 5
F
5 or bigger ?
0. 6 0 8 3
5 or bigger ?
No
Yes
1.43
5.84
0. 2 9 4 3
5 or bigger ?
Yes
0.61
0. 5 5 5 0
0. 3 9 7 0
5 or bigger ?
5 or bigger ?
5 or bigger ?
No
Yes
Yes
0.29
0.56
0.40
Rounding to 3 d.p
1. 4 2 6 1 8
5. 8 4 2 5 4
F
5 or bigger ?
0. 6 0 8 3 4
5 or bigger ?
Yes
5 or bigger ?
No
5.843
1.426
6. 2 9 4 7 1
5. 4 0 0 9 7
No
0.608
0. 3 9 9 7 7
5 or bigger ?
5 or bigger ?
5 or bigger ?
Yes
Yes
Yes
6.295
5.401
0.400
Take Care!
F
• Round 3.48 to 1 d.p
3.5
• Round 3.48 to the nearest whole number
3
(not 4)
E
Significant Figures
Example
Round
E
235440
To 2 significant figures
235440
E
Underline the 1st
2 digits
The 3 is
changed to a 4
If this is 5
or more
then you
must round
up
Now look at the
next digit
240000
All other digits are changed
to zero
• What are these numbers to 2 significant
figures?
437900
69723
43490
2350
E
440000
70000
43000
2400
What about decimal numbers?
E
For example:
Round
0.004367
to 2 significant figures
0.004367
E
Underline the 1st 2
digits which are
not zero
Look at the next
digit along
You change
the 3 to a 4
0.0044
If it is 5 or
more you add
1 to the
previous digit
You can ignore any number after the 1st 2 digits which are
not zeros
Round the following to 2 significant figures
E
0.05475
0.00475
0. 45475
0.055
0.0048
0.45
Significant figures
E
• When first identifying significant numbers,
zeros at the beginning or end don’t usually
count, but zeros ‘inside’ the number do.
• Digits of a number kept in place by zeros
where necessary.
• The rounded answer should be a suitable
reflection of the original number e.g.
24,579 to 1 s.f could not possibly be 2
24,579 to 1 s.f is 20,000
Write questions and answers in your books?
E 49382.95 to 2 s.f. and 1dp =
49000 49383.0
0.05961 to 1 s.f. and 2dp = 0.06 0.06
374.582 to 3 s.f. and 1dp = 375 374.6
0.0009317 to 2 s.f. and 3dp = 0.00093 0.001
Objective:
• Share a quantity into a given ratio. C.
• Find an unknown number that fits a given ratio. C.
Sharing a quantity into a given ratio
For example, share 36 into the ratio
2:7
First ADD the ratio 2 + 7 = 9
Second DIVIDE this answer into the
quantity to be shared 36 ÷ 9 = 4
C
Third MULTIPLY the ratio by this
answer 2 X 4 : 7 X 4
This is the answer 8 : 28
When sharing into a given ratio, the
name to remember is: ADaM
C
A
D
+
÷
and
M
X
Share 32 into the the ratio 3 : 5
3+5=8
32 ÷ 8 = 4
3 X 4 = 12 : 5 X 4 = 20
Answer 12 : 20
Share these into the given ratio
C
Ratio
2:3
2:5
3:7
4:5
1:2:5
Quantity
50
28
20
360
32
Share
Share these into the given ratio
C
Ratio
2:3
2:5
3:7
4:5
1:2:5
Quantity
Share
20 : 30
50
8 : 20
28
6 : 14
20
160 : 200
360
4 : 8 : 20
32
Finding an unknown number
that fits a given ratio
Example – If the ratio of red beads black
beads is 3 : 5, how many black beads
will I need for 21 red beads?
C
Red : Black
3:5
21 : ?
First find the number that
you multiply 3 by to get 21
Red : Black
C
3:5
X7
21 : ?
Red : Black
C
3:5
X7
X7
21 : 35
Another Example
Red : Black
2:7
C
X6
12 : ?
2 X 6 is
12 so you
multiply
7 by 6 to
get the ?
Red :
Black
2: 7
C
X6
12 : 42
X6
FRACTIONS
Top number is the NUMERATOR, bottom number is the
DENOMINATOR
Find equivalent fractions. F
Simplify a fraction to its lowest form. E
Add and subtract fractions with common denominator. D
Multiply and divide fractions. D
Add and subtract fractions with different denominator. C
Convert to and from fractions, decimals and percentages. D
Be able to convert a recurring decimal to a fraction. C
What makes a fraction?
Part
Umerator
D
enominator
o
N
Whole
n
Something we do with Fractions.
find EQUIVALENT
F
ent
iv
♫ EQU = AL

=

♫
Something we do with Fractions.
SIMPLIFY
E
S
im
=
8
16
p
L
=
i
4
=
8
=
= f
y
2
4
=
Adding Fractions
D
1
4
+
2
4
=
3
4
Subtracting Fractions
D
5 2

7 7

3
7
What Happens if
• The two bottom numbers are different
C
1 1
+
3 4
Find LCM (Lowest Common Multiple)
1 1
+
3 4
Find multiples of 3 and 4
X table shows the multiples of 3
3,6,9,12,15,18,21,……..
C
X table shows the multiples of 4
4,8,12,16,20,24,…………
1 1
+
3 4
Change the denominators into 12
4
C
4
1
1
 +
3 4
3
3
4 3
 +
12 12
4+3 7


12
12
+
1/4
1/3
1/4
+
= ?
1/3
1/3
1/4
C


+
=
3/12 + 4/12 = 7/12
Another example but TAKING AWAY
• The two bottom numbers are different
C
2 1

3 6
Find LCM (Lowest Common Multiple)
2 1

3 6
Find multiples of 3 and 6
X table shows the multiples of 3
3,6,……..
C
X table shows the multiples of 6
6,…………
2
C
2
 21  41
3 6 6 6

3
6
1

2
1/3
1/3
1/6
1/6
4/6
C
2/
3
2X2
2X3
- 1/6 = 3/6 = 1/2
1/
-
=
?
6
1
6
=
4
6
+
1
6
3
1
= =
6
2
Multiplying Fractions - Method
5 1  4 5 4  54
 

D
12
12
12
12 3  4
Dividing Fractions - Method
• Invert the second fraction and then multiply
D
3 5 3 8 24 6
   

4 8 4 5 20 5
1
1
5
If it’s fractions you’ve got to sum,
the first thing to do is check its bum.
Add tops together if bums are the same,
if they’re not, then it’s a pain.
Equal bums is what you need,
use times tables, your bums to feed.
Take away is the same as add,
times and divide are not so bad.
For times do the bottom and then the top,
divide do the same with the 2nd bottom up
C
NEILSEAL METHOD FOR COVERTING FPD
OUT OF THE RED AND INTO BLACK
FRACTION
DECIMAL
  Add up to
last place fraction
and cancel.
FRACTION
PERCENTAGE
%
100
(cancel if possible)
DECIMAL
PERCENTAGE
Numerator ÷
Denominator
(Numerator x 100) Decimal x 100
÷ Denominator
% ÷ 100
.
How can we write 0.3 as a fraction.
.
Let n = 0.3
.
So 10n = 3.3
. .
So 10n - n = 3.3 – 0.3
B
So 9n = 3
So n = 3 = 1
9 = 3
.
.
How can we write 0.3451 as a fraction.
.
.
Let n = 0.3451
.
.
So 10000n = 3451.451
.. ..
So 10000n - 10n = 3451.451 – 3.451
B
So 9990n = 3448
So n = 3448
9990
= 1724
4995