OBJECTIVE. REVISION MOD 3. G. Check the side of the slide to see what level you are working at!. F. E. 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
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OBJECTIVE
REVISION MOD 3
G
Check the side of the slide to see what level you are working at!
F
E
D
C
B
A
A*
G
G
e.g. the factors of 36 are:
1 & 36
2 & 18
3 & 12
4 & 9
6 & 6
1 & ITSELF.
The prime numbers less than 30 are ….
2, 3, 5, 7, 11, 13, 17, 19, 23, 29
e.g. factors of12are1, 2 ,3 ,4 , 6 & 12of these2 & 3are prime factors.
12can be written as a product of prime factors…
12 = 2 x 2 x 3 in its INDEX FORM = 22 x 3
C
e.g.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.
We use HCF’s when cancelling fractions!!!
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
C
180
2 x 2 x 3 x 3 x 5 = 180
or; in INDEX FORM
22 x 32 x 5 = 180
90
x
2
45
x
C
2
15
3
x
5
x
x
3
x
x
x
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
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
B
This is what’s left from BOTH numbers
when you take out the
HCF
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)
C
Thus5 is always factor of a series of five
consecutive numbers
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.
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.
D
Square Root
TheNUMBERthat isSQUAREDto make9is3.
3is called theSQUARE ROOTof9and 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.
D
For example:
= 53
5 x 5 x 5
= 24
2 x 2 x 2 x 2
= 75
7 x 7 x 7x 7 x 7
5 is the INDEX
7 is the BASE NUMBER
75 & 24 are numbers in INDEX FORM
26 x 24
= 210
x
2x2x2x2x2x2
2x2x2x2
35 x 37
= 312
x
3x3x3x3x3
C
3x3x3x3x3x3x3
General Rule
am x an = am+n
44÷42
= 42
÷
4x4x4x4
4x4
26÷23
= 23
÷
2x2x2x2x2x2
C
2x2x2
General Rule
am÷ an = amn
(26)2
= 26 x 26
= 212
(35)3
= 35 x 35 x 35
= 315
C
General Rule
(am)n = am x n
How could you get an answer of 30?
35÷ 35
= 355
= 30
30 =
1
C
General Rule
a0 = 1
x 2 x 2 x 2 x 2
5 x 5 x 5
= 53
x 24
We can not write this any more simply
Can ONLY do that if BASE NUMBERS are the same
26 x 24
23
= 210
23
= 27
35 x 37
34
= 312
34
= 38
C
25 x 23
24 x 22
= 28
26
= 22
2a3 x 3a4
= 2 x 3 x a3 x a4
= 6a7
8a6÷ 4a4
= (8 ÷ 4) x (a6 ÷ a4)
= 2a2
2
2
8a6
4a4
C
a6 x a4
= a10
b5 x b7
= b12
c5 x c3
c4
= c8
c4
= c4
C
a5 x a3
a4 x a6
= a8
a10
= a2
1.am x an = am+n
2.am÷ an = amn
3.(am)n = am x n
4.a1 = a
5.a0 = 1
1
2
General Rule
an =
1
an
25
32
24
16
23
8
22
4
21
2
20
1
21
22
B
General Rule
a = √a
n
9 x 9 = 91 =9
From Rule 1 & 4
So 9 = √9
A
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*
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.
But it still not any easier to handle!?!
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.
How could we turn the number 800,000,000,000 into standard index form?
So, 800,000,000,000 = 8 x 1011 in standard index form
Let’s investigate!
Converting large numbers
We can break numbers into parts to make it easier,
e.g. 80 = 8 x 10 and 800 = 8 x 100
C
800,000,000,000 = 8 x 100,000,000,000
And 100, 000,000,000 = 1011
5.3 x 10n
There will also be a power of 10
C
The first part of the number is between
1 and 10
But NOT 10 itself!!
The first number must be a value between
1 and 10
One of the most important rules for writing numbers in standard index form is:
But NOT 10 itself!!
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.
C
Notice that the number of zeros
matches the index number
2
100
10
10
10
10
3
1,000
4
10,000
5
100,000
10
So, 45,000,000,000 = 4.5 x 1010
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
4.5000000000
Count the number of digits after the decimal point.
C
This is our index number (our power of 10)
How can we convert 0.067 into standard index form?
0.067 = 6.7 x 0.01
0.01 = 102
C
0.067 = 6.7 x 102
And numbers less than 1?
How can we convert 0.000213 into standard index form?
0.000213 = 2.13 x 0.0001
0.0001 = 104
C
0.000213 = 2.13 x 104
56
567
5678
56789
0.56
0.056
0.0056
Write the following in standard form.
0.00056
23
234
4585
4.6
0.78
0.053
0.00123
How to write a number in standard form.
Place the decimal point after the first nonzero digit then multiply or divide it by a power of 10 to give the same value.
= 5.6 x 10 = 5.6 x 101
= 5.67 x 100 = 5.67 x 102
= 5.678 x 1000 = 5.678 x 103
= 5.6789 x 10 000 = 5.6789 x 104
= 5.6 10 = 5.6 x 101
C
= 5.6 100 = 5.6 x 102
= 5.6 1000 = 5.6 x 103
= 5.6 10 000 = 5.6 x 104
2.3x 101
2.34x 102
4.585x 103
4.6x 100
7.8x 101
5.3x 102
1.23x 103
Examples:
Exp/EE?
Calculate: 4.56 x 108x 3.7 x 105
+/
Sharp
Standard Form on a Calculator
You need to use the exponential key (EXP or EE) on a calculator when doing calculations in standard form.
4.56
Exp
8
x
3.7
Exp
5
=
1.6872 x 1014
1.7 x 1014(2sig fig)
Calculate: 5.3 x 104 x 2.7 x 1013
C
5.3
Exp
 4
x
2.7
Exp
 13
=
1.431 x 1016
1.4 x 1016 (2 sig fig)
Calculate: 3.79 x 1018 9.1 x 105
3.79
Exp
18
9.1
Exp
 5
=
4.2 x 1022(2 sig fig)
Calculations Using SIF
B
4 x 1018 x 3 x 104
Numbers
Powers of 10
4 x 3
x 1018 x 104
ADD powers
= 12
x 1022
NOT Std Form!
B
= 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?
398
2.39 x 1027 / 6.0 x 1024 =
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 1022
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 1024
How many times heavier is uranium than hydrogen?
A
237
3.95 x 1022/ 1.67 x 1024 =
Complex word problems involving SIF
Writing Answers in Decimal Form (Noncalculator)
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.
D
245 000
S
49
hours
S =
so T =
=
=
T
5 000
D
A
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.
Rounding.
G
6 7 8
G
20 30 40
G
400 500
Decimal Places
F
6.348
F
If this number is a 0, 1, 2, 3 or 4 we don’t have to do anything else and we have our answer.
Now look at the number immediately after where we stopped highlighting
Firstly, highlight the number to the first number after the decimal point
So we have 6.3
But is this the answer?
6.348 = 6.3 (1dp)
F
Lets look at an example
F
9.2721
F
If this number is a 5, 6, 7, 8 or 9 we increase the last digit by one.
Now look at the number immediately after where we stopped highlighting
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
9.2721 = 9.3 (1dp)
F
If this number is a 0, 1, 2, 3 or 4 we don’t have to do anything else and we have our answer, but it is not, so we round up the number in the second decimal place to give us our answer.
7.456
F
Firstly, highlight the number to the second number after the decimal point
Now look at the number immediately after where we stopped highlighting
7.46
If this number is a 0, 1, 2, 3 or 4 we don’t have to do anything else. In this case it is so we have our answer highlighted.
3.992
F
Firstly, highlight the number to the second number after the decimal point
Now look at the number immediately after where we stopped highlighting
3.99
6.348 = 6.3 (1dp)
F
Lets look at an example
F
9.2721
F
If this number is a 5, 6, 7, 8 or 9 we increase the last digit by one.
Now look at the number immediately after where we stopped highlighting
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
9.2721 = 9.3 (1dp)
F
Decimal Places (Rounding)
Numbers can be rounded to 1,2, 3 or more decimal places.
Rounding to 1 d.p
4 . 8 3 2 5
5 or bigger ?
5 or bigger ?
5 or bigger ?
4.8
4.8
4.9
F
4. 8 5 2 5
4. 8 4 2 5
No
No
Yes
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.
Decimal Places
It is often necessary/convenient/sensible to give approximations to real life situations or as answers to certain calculations.
F
Rounding to 1 d.p
4 . 8 3 2 5
4. 8 5 2 5
4. 8 4 2 5
No
No
Yes
5 or bigger ?
5 or bigger ?
5 or bigger ?
5 or bigger ?
5 or bigger ?
5 or bigger ?
4.9
4.9
4.9
4.8
4.8
4.9
F
4. 8 6 2 5
4. 8 7 2 5
4. 8 9 2 5
Yes
Yes
Yes
0.29
0.40
1.43
0.56
0.61
5.84
5 or bigger ?
5 or bigger ?
5 or bigger ?
5 or bigger ?
5 or bigger ?
5 or bigger ?
Rounding to 2 d.p
1. 4 2 6 1
5. 8 4 2 5
0. 6 0 8 3
F
No
Yes
Yes
0. 2 9 4 3
0. 5 5 5 0
0. 3 9 7 0
Yes
No
Yes
6.295
0.400
1.426
5.401
0.608
5.843
5 or bigger ?
5 or bigger ?
5 or bigger ?
5 or bigger ?
5 or bigger ?
5 or bigger ?
Rounding to 3 d.p
1. 4 2 6 1 8
5. 8 4 2 5 4
0. 6 0 8 3 4
F
Yes
No
No
6. 2 9 4 7 1
5. 4 0 0 9 7
0. 3 9 9 7 7
Yes
Yes
Yes
F
3.5
3(not 4)
Significant Figures
E
Example
E
Round
235440
To 2 significant figures
Underline the 1st 2 digits
Now look at the next digit
E
If this is 5 or more then you must round up
The 3 is changed to a 4
240000
All other digits are changed to zero
2350
2400
437900
440000
69723
70000
43490
43000
E
What about decimal numbers?
E
For example:
Round
0.004367
to 2 significant figures
Underline the 1st 2 digits which are not zero
Look at the next digit along
E
You change the 3 to a 4
If it is 5 or more you add 1 to the previous digit
0.0044
You can ignore any number after the 1st 2 digits which are not zeros
Round the following to 2 significant figures
0.05475
0.055
0.00475
0.0048
0. 45475
0.45
E
24,579 to 1 s.f could not possibly be 2
24,579 to 1 s.f is 20,000
E
49382.95 to 2 s.f. and1dp =
0.05961 to 1 s.f. and2dp =
374.582 to 3 s.f. and 1dp =
0.0009317 to 2 s.f. and 3dp =
E
49000 49383.0
0.06 0.06
375 374.6
0.00093 0.001
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
Third MULTIPLY the ratio by this answer 2 X 4 : 7 X 4
This is the answer 8 : 28
C
When sharing into a given ratio, the name to remember is: ADaM
A
D
M
and
+
÷
X
Share 32 into the the ratio 3 : 5
3 + 5 = 832÷ 8 = 4
3 X 4 = 12 : 5 X 4 = 20
Answer 12 : 20
C
C
Share these into the given ratio
C
Example – If the ratio of red beads black beads is 3 : 5, how many black beads will I need for 21 red beads?
Red : Black
3 : 5
21 : ?
C
Red : Black
3 : 5
21 : ?
C
X 7
Red : Black
3 : 5
21 : 35
C
X 7
X 7
Another Example
Red : Black
2 : 7
12 : ?
2 X 6 is 12 so you
multiply 7 by 6 to get the ?
C
X 6
Red : Black
2: 7
12 : 42
X 6
C
X 6
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
Part
NUmerator
Denominator
o
Whole
n
F
ent
♫
iv
♫
EQU = AL
=
=
=
=
S
i
E
m
=
p
L
=
i
f
=
y
+
D
=
D
1
1
+
3
4
C
Find multiples of 3 and 4
X table shows the multiples of 3
3,6,9,12,15,18,21,……..
X table shows the multiples of 4
4,8,12,16,20,24,…………
C
7
12
Change the denominators into 12
3
4
C
1/3
1/4
+
+
= ?
C
=
+
3/12 + 4/12 = 7/12
C
2
1

3
6
Find multiples of 3 and 6
X table shows the multiples of 3
3,6,……..
X table shows the multiples of 6
6,…………
C
2
1

3
6
2
2
C
1/3
1/3
1/6
1/6
4/6  1/6 = 3/6 = 1/2
2/3  1/6 = ?
C
2 X 2
1
1
3

=
=
4
+
1
=
2
2 X 3
6
6
6
6
D
D
If it’sfractionsyou’ve got tosum,
the first thing to do ischeckitsbum.
Add topstogetherif bumsare thesame,
if they’re not, then it’s a pain.
Equal bumsis what youneed,
usetimes tables, yourbumstofeed.
Take awayis thesameasadd,
times and divide are not so bad.
Fortimesdo thebottomandthenthetop,
dividedo thesamewiththe2nd bottom up.
C
.
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 = 1724
9990 4995