Number
http://www.youtube.com/watch?v
=52CzD31SqaM&feature=related
(maths is confusing II funny)
http://www.youtube.com/wat
ch?v=MQuHQKkNldk
(maths is confusing funny)
SLO
To find multiples of a number
Copy into
note book
Multiples
A multiple is formed by multiplying a given
number by the counting numbers; 1, 2, 3, 4, 5,
6, etc.
Copy into
note book
Example: List the multiples of 4:
4x1=4
4x2=8
4 x 3 = 12
4 x 4 = 16
4 x 5 = 20
4 x 6 = 24
Counting Numbers
So, the multiples of 4 are:
4, 8, 12, 16, 20, 24, 28, etc.
What are the first five multiples of 13?
13 x 1 =13
13 x 2 = 26
13 x 3 = 39
13 x 4 = 52
13 x 5 = 65
13, 26, 39, 52, 65
Your Turn: Find the Missing Multiples
30
24 ____
6, 12, 18, ____,
18 21
3 6, 9, 12, ____,
15 ____,
___,
12
72
___, 24, 36, 48, 60, ____
Web Resource
http://www.youtube.com/watch?v=Aphgn84e6ao&feature=gvrec (multiples video)
Copy into
note book
SLO
Find Common Multiples
How do we know if a number is a multiple of:
2
5
number ends in 0, 2, 4, 6, 8
number ends in 0 or 5
Your Turn:
Are the following numbers multiples of 2?
246
Yes
1067 No
20000 Yes
1235
No
Your Turn:
Are the following numbers multiples of 5?
1000 Yes
25456 No
5555 Yes
1025 Yes
Three million two hundred and fifty six No
Copy into
note book
Which of the numbers below are
multiples of both 2 AND 5?
20
40
45
90
76
105
200
305
40
42
These are called COMMON MULTIPLES
Copy into note book
Method to Find Common Multiples
E.g. Find the common multiples of
8
and
10
Step 1: Write the times table (multiples) for each number
8
16
24
32
40
48
56
64
72
80
10
20
30
40
50
60
70
80
90
100
Step 2: find numbers common to both lists
Numbers 40 and 80 are in both lists so they are common multiples
What do you think the next common multiple of 8 and 10 is?
120
Your Turn:
Find the first two common multiples of
2 and 3
6 and 12
4 and 5
20 and 40
5 and 10 10 and 20
8 and 16 16 and 32
SLO
Find Lowest Common Multiple
Copy into note book
Finding the Lowest Common Multiple (LCM)?
The lowest common multiple is the smallest number
which is the multiple of both.
Multiples of 2 are: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20…
Multiples of 5 are: 5, 10, 15, 20, 25, …
10 is the LOWEST COMMON MULITPLE.
Find the Lowest Common Multiple (LCM) of 4 and 6
The multiples of 4 are 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44, ...
The multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66, ...
12, 24, and 36 are common multiples of both 4 and 6.
So the Lowest Common Multiple (LCM) of 4 and 6 is 12
17
Your Turn:
Find the Lowest Common Multiple (LCM) of
2 and 3
6
4 and 5
20
5 and 10 10
8 and 16 16
Extension question:
• Decide with your partner the answer to the
question below. You must EXPLAIN your
answer and give examples.
• Is the lowest common multiple of two numbers
always found by multiplying those two
numbers together?
• E.g. LCM of 2 and 7 is 2 x 7 = 14
Web Resources
http://www.youtube.com/watch?v=JXeZ2Ezo1pM (you tube:
intro to common multiples and LCM)
SLO
To find the factors of a number
Copy into
note book
Definition
Factor – a number that is
multiplied by another to give
another number.
7 x 8 = 56
Factors
Copy into
note book
For example, 24 has several factors.
24 = 1 x 24
24 = 2 x 12
24 = 3 x 8
24 = 4 x 6
The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24
Copy into
note book
Finding Factors
You will need to know your times tables!!!
Starting with number 1, Check each times
table until you either repeat your factors or
you get doubles (such as 4 x 4).
E.g. 1: What are the factors of 16?
1 x 16
2x8
3 x ??
4x4
1 and 16 are factors
2 and 8 are factors
3 is not a factor, so cross it out
4 is a factor
doubles = done
The factors of 16 are: 1,2,4,8,16
Copy into note book
E.g. 2: What are the factors of 18?
1
2
3
4
5
6
x
x
x
x
x
x
18
9
6
??
??
3
1 and 18 are factors
2 and 9 are factors
3 and 6 are factors
4 is not a factor, so cross it out
5 is not a factor, so cross it out
Repeat! Cross it out! We’re done!
The factors of 18 are: 1,2,3,6,9,18
E.g. 3: What are the factors of 7?
1x7
2 x ??
3 x ??
4
5
6
7
x
x
x
x
??
??
??
1
1 and 7 are factors
2 is not a factor, so cross it out
3 is not a factor, so cross it out
4 is not a factor, so cross it out
5 is not a factor, so cross it out
6 is not a factor, so cross it out
Repeat so ignore.
The factors of 7 are:1,7
Your Turn: List the factors of
8
1, 2, 4, 8
12
1, 2, 3, 4, 6, 8, 12
10
1, 2, 5, 10
27
1, 3, 9, 27
Web Resources
http://www.youtube.com/watch?v=UVRkhr0FiqA&feature
=g-vrec (multiple and factors video)
http://www.mathsmaster.org/types-of-number/factors/
(video: factors)
http://www.youtube.com/watch?v=Q8ehxzOBzQU&featur
e=relmfu (you tube: factors)
SLO
Find Highest Common Factor (HCF)
Copy into note book
Highest Common Factor HCF
E.g. 1) Find the Highest Common Factor of
8
and
The factors of 8 are: 1, 2, 4, 8
The factors of 10 are 1, 2, 5, 10
The common factors of 8 and 10 are: 1 and 2
The largest number that goes into 8 and 10 is 2
So 2 is the biggest FACTOR
or the Highest Common Factor
10
Highest Common Factor (HCF)
E.g. 2) Find the Highest Common Factor of 12 and 18
The factors of 12 are: 1, 2, 3, 4, 6 and 12
The factors of 18 are: 1, 2, 3, 6, 9 and 18 :
The common factors of 12 and 18 are: 1, 2, 3 and 6
The largest number that is a factor of 12 and 18 is: 6
So the Highest Common Factor of 12 and 18 is
6
Your Turn: Find the HCF of
12 and 18 6
16 and 24 8
10 and 25 5
12 and 24 12
Web Resources
http://www.youtube.com/watch?v=HM9lyY9QSFg (youtube:
HCF)
SLO
Know what a prime number is
Copy into
note book
Definition of a prime number
A prime number has exactly 2 factors
Number
7
6
2
1
Factors
1,7
1, 2, 3, 6
1, 2
1
Prime
YES
NO
YES
NO
Prime Numbers
Eratosthenes’ Sieve
Eratosthenes
(ehr-uh-TAHS-thuh-neez)
Eratosthenes was the librarian at
Alexandria, Egypt in 200 B.C.
Note every book was a scroll.
Eratosthenes
(ehr-uh-TAHS-thuh-neez)
Eratosthenes was a Greek
mathematician, astronomer, and
geographer.
He invented a method for finding
prime numbers that is still used
today.
This method is called Eratosthenes’
Sieve.
Eratosthenes’ Sieve
A sieve has holes in it and is used to
filter out the juice.
Eratosthenes’s sieve filters out
numbers to find the prime numbers.
Hundreds Chart
Make a chart of the numbers from 1
to 100, with 10 numbers in each row.
An example is on the next slide
Copy into
WORK book
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
Hundreds Chart
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
100
The idea is to cross out all numbers on this
chart that are not prime, leaving only the
prime numbers.
1 is not prime so cross it out.
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
100
Apart from number 2, even numbers are not prime
2 is a prime number so leave the number 2.
All other even numbers can be divided by 2 so
cross out all even numbers.
How can you tell if a number is even?
It ends in 0, 2, 4, 6, 8
Leave 2; cross out multiples of 2
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
100
Hint For Next Step
To find multiples of 3, add the digits
of a number; see if you can divide
this number evenly by 3; then the
number is a multiple of 3.
267
Total of digits (2 + 6 + 7) = 15
3 divides evenly into 15
267 is a multiple of 3
Leave 3; cross out multiples of 3
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
100
Hint For the Next Step
To find the multiples of 5 look
for numbers that end with the
digit 0 or 5.
385 is a multiple of 5
& 890 is a multiple of 5
because the last digit
ends with 0 or 5.
Leave 5; cross out multiples of 5
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
100
Hint for next step
Sorry but no easy way of crossing out the
multiples of 7.
Just start at 7 and count on 7 again, again and
again etc
Leave 7; cross out multiples of 7
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
100
All the numbers left are prime
1
11
21
31
41
51
61
71
81
91
2
12
22
32
42
52
62
72
82
92
3
13
23
33
43
53
63
73
83
93
4
14
24
34
44
54
64
74
84
94
5
15
25
35
45
55
65
75
85
95
6
16
26
36
46
56
66
76
86
96
7
17
27
37
47
57
67
77
87
97
8
18
28
38
48
58
68
78
88
98
9
19
29
39
49
59
69
79
89
99
10
20
30
40
50
60
70
80
90
100
The Prime Numbers from 1 to
100 are as follows:
2, 3, 5, 7, 11, 13, 17,
19, 23, 31, 37, 41, 43,
47, 53, 59, 61, 67, 71,
73, 79, 83, 89, 97
Your Turn: Explain why the following
are or are not prime numbers
8
Not prime as it has more than 2 factors: 1, 2, 4, 8
17
Prime as it has only 2 factors: 1 and 17
15
Not prime as it has more than 2 factors: 1, 3, 5, 15
23
Prime as it has only 2 factors: 1 and 23
Web resources
http://hoodamath.com/games/primelanding.php (Prime number game:
state yes or no if the number on the left is prime)
http://www.mathsmaster.org/types-of-number/prime-numbers/
(video: prime numbers)
http://www.youtube.com/watch?v=cRz4hW9SPPc (prime rap song)
http://www.youtube.com/watch?v=9m2cdWorIq8
(youtube: sieve explained)
SLO
Know what composite numbers are
Copy into
note book
Composite Numbers
If a number is not prime it is called composite
E.g. 12 is not prime as it has more then 2
factors therefore it is a composite number
NB: This similar to vowels (a, e, i, o, u )
and consonants in the English language.
Recap the difference between
Prime and Composite Numbers
Prime numbers are
numbers that only have
two factors.
Composite numbers
have more than two
factors.
EXAMPLES:
3, 5, 7, 11, 31
EXAMPLES:
6, 18, 30, 100
http://www.youtube.com/watch?v=KgU4FYp_f7E
(youtube: prime and composite numbers)
Your Turn: State if each of the
following is a prime or composite
7
9
11
19
25
31
Prime
Composite
Prime
Prime
Composite
Prime
SLO
To draw a factor tree
FACTOR TREES
To grow a tree for 180
First find a FACTOR PAIR for 180
This should be two numbers that multiply together to
give 180.
You might see that 180 is an EVEN NUMBER and
that means that 2 is a factor…
2 x•
90 = 180
Or you might notice that 180 has a ZERO in its ONES
PLACE which means it is a multiple of 10.
10 x 18 = 180
OR, OR, OR (there are lots to choose from)
I have chosen
the factor pair
10 and 18
180
18
10
NOW
You have to find
FACTOR PAIRS
for 10 and 18
We “grow” this
“tree” downwards
since that is how we
write in English (and
we can’t be sure how
big it will be - we
could run out of
paper if we grew
upwards).
Copy into note book
Factor Tree for 180
180
10 x 18 = 180
18
10
6 x 3 = 18
2 x 5 = 10
2
5
6
3
180
Since
2 and 3 and
5 are
PRIME
NUMBERS
they do not
grow “new
branches”.
They just
grow down
alone.
18
10
2
2
5
6
3
5 2 3 3
6 is NOT a
prime number
- it is a
COMPOSITE
NUMBER - it
still has
factors. Since
it is an EVEN
NUMBER we
see that:
6=2x3•
… and if
we flip it
over we
can see
why it is
called a
tree
2
2
5 2 3 3
5
3
6
10
18
180
Your Turn: Draw a factor tree for
24
36
24
8
4
2
36
3
2
2
18
3
2
9
3
3
2
2
3
2
2
2
Your trees may be different but the numbers at the end
should be the same
Your Turn: Draw a factor tree for
90
150
150
90
45
9
3
5
3
30
2
10
2
5
2
2
5
3
5
5
3
5
Your trees may be different but the numbers at the end
should be the same
SLO
Know what a prime factor is
Copy into
note book
Prime factors
A factor that is a prime number is called
prime factor e.g.
4 x 5 = 20
Both 4 and 5 are factors, but as only 5 is a prime number
5 is one of a prime factors of 20
Your turn: Answer the following
1) Is 7 a prime factor of 14? Yes
2) Is 5 a prime factor of 15? Yes
3) Is 4 a prime factor of 12? No
4) Is 11 a prime factor of 20? No
5) Are 3 and 5 both prime factors of 30? Yes
SLO
To write numbers as a product of primes
Copy into
note book
Product of Primes
Multiplication
What does product mean?
Numbers with
two factors
What does prime mean?
Every number can be written as a
product of primes.
For example,
12 = 2 x 2 x 3
100 = 2 x 2 x 5 x 5
140 = 2 x 2 x 5 x 7
Copy into
note book
Using factor trees to find the
product of primes
To find the product of primes
Step 1) Draw the factor tree
Step 2) Multiply the end branches together
E.g.
Copy into note book
Find the product of primes for 150
150
Step 1:
30
10
2
5
3
5
5
3
5
150 written as a product of primes is
x
x
x
Step 2:
A better way to write this is: 2 𝑥 3 𝑥 52
E.g. 2) Product of Primes for 60
Step 1
Step 2
60 as a product of
primes is
2 x 2 x 3 x 5
OR
2
2 𝑥3𝑥5
60
30
2
15
2
3
5
E.g. 3) Find the product of primes for72
72
Step 1
36
2
Step 2
72 is the prime numbers
2 x 2 x 2 x 3 x3
multiplied together
3
2
OR 2 𝑥 3
18
2
2
9
3
3
Insert Lesson Title Here
Your Turn:
Use a factor tree to find the product of primes for.
1. 27 33
2 x 32
2
2. 36
3. 28
22 x 7
4. 132 22 x 3 x 11
5. 52 22 x 13
6. 108 22 x 33
Web Resources
http://www.youtube.com/watch?v=WlcLgCOGOtU
(prime + factor tree song)
Extension work of HCF and LCM
You will need to know how a Venn Diagram works
Using Venn diagrams to find HCF and LCM
E.g. To find the LCM and HCF of 20 and 30
Write both numbers as their product of primes
20 = 2 x 2 x 5 and
30 = 2 x 3 x 5
Write the
factors of 20
in one circle
2
2
3
5
Write the factor of 20
and 30 in the centre
Write the
factors of 30
in one circle
20
2
30
2
3
5
The HCF is found by multiplying all the
numbers in the middle i.e. 2 x 5 = 10
The LCM is found by multiplying all numbers
in Venn Diagram i.e. 2 x 2 x 5 x 3 = 60
http://www.teacherled.com/resources/vennfactors/vennfactorload.html
(Interactive venn diagrams to find HCF, LCM)
Your Turn: Use the Venn diagram to help you find the
HCF and LCM of 42 and 49
Your Turn: Use the Venn diagram to help you find the
HCF and LCM of 42 and 49
Your Turn: Use the Venn diagram to help you find
the HCF and LCM of 48 and 84
Your Turn: Use the Venn diagram to help you find
the HCF and LCM of 48 and 84
Web Resources
http://www.mathsmaster.org/types-of-number/highest-common-factor-andlowest-common-multiple-part-2-using-prime-factorisation/ (product of primes
to find HCF + LCM, take care video uses the words prime factorising)
http://www.youtube.com/watch?v=KhW9P9Zn_HU (you tube HCF
from factor trees)
SLO
To understand how exponents work
Copy into
note book
base
Definitions
8
5
exponent
index form
Squared: If the exponent is two we say the number is squared.
Cubed: if the exponent is three we say the number is cubed.
http://www.youtube.com/watch?v=HNMDklg4uis
(Youtube: what to squared and cubed mean)
Square/Squared
The term ‘to square’ a number is used as this would be
the area of the square e.g.
3
A square with a side
of 3 would make a
square with 9 small
squares.
i.e. 3 x 3 = 9
9 is a square number
3
The first 10 square numbers:
12 = 1 × 1 = 1
22 = 2 × 2 = 4
32
=3×3= 9
42
= 4 × 4 = 16
52
= 5 × 5 = 25
62
= 6 × 6 = 36
72 = 7 × 7 = 49
82
= 8 × 8 = 64
92 = 9 × 9 = 81
102
+3
+5
+7
+9
+ 11
+ 13
+ 15
+ 17
= 10 × 10 = 100
+ 19
Cube/Cubed
The term ‘to cube’ a number is used as this would be
the volume of the cube e.g.
A cube with a side of 3
would make a cube
with 27 small cubes.
i.e. 3 x 3 x 3 = 27
27 is a cube number
The first 5 cube numbers
13 = 1 × 1 × 1 = 1
‘1 cubed’ or ‘1 to the power of 3’
23 = 2 × 2 × 2 = 8
‘2 cubed’ or ‘2 to the power of 3’
33 = 3 × 3 × 3 = 27
‘3 cubed’ or ‘3 to the power of 3’
43 = 4 × 4 × 4 = 64
‘4 cubed’ or ‘4 to the power of 3’
53 = 5 × 5 × 5 = 125
‘5 cubed’ or ‘5 to the power of 3’
Extension work: Triangular numbers
The tenth
first triangular
second
third
fourth
fifth
sixth
seventh
eighth
ninth
triangular
triangular
triangular
triangular
triangular
triangular
triangular
number
number
number
number
number
number
number
number
isis
is
isis
1.
15.
is
6.
21.
45.
55.
isis
10.
36.
3.
28.
1+2+
=3+
=4
6+
=5
10+
=6
15+
=7
21+
=8
28+
=9
36+
= 10
45 = 55
Copy into
note book
Exponents
5
1
2
3
4
5
2 =2x2x2x2x2
3
6 =6x6x6
4
7 =7x7x7x7
2
9 =9x9
Video exponent to expanded form:
http://www.virtualnerd.com/pre-algebra/factors-fractionsexponents/exponential-to-expanded-form-conversion.php?&sid=
Your Turn:
Write the following in index form
3
1) 5 x 5 x 5 5
4
2) 2 x 2 x 2 x 2 2
5
3) 8 x 8 x 8 x 8 x 8 8
4
4) 3 x 3 x 3 x 3 3
Write the following out in full
5) 64 6 x 6 x 6 x 6
6) 75 7 x 7 x 7 x 7
SLO
Know what a square root is
Square roots
The area of this square is 64 cm2.
? cm
8
? cm
8
What is the length of the sides?
The area of the square is 9, what is the length?
3
The square root of 9 ( 9) is
3
The area of the square is 16, what is the length?
4
The square root of 16 is
4
The area of the square is 25, what is the length?
5
The square root of 25 is 5
Copy into
note book
Square roots
Finding the square root is the inverse of finding the square:
squared
8
64
square rooted
We write
64 = 8
The square root of 64 is 8.
Numbers like 25, which have whole numbers for their
square roots, are called perfect squares
Square
root
Perfect
square
1
1 = 1
81
4
4 = 2
100
100 = 10
9
9 = 3
121
121 = 11
16
16 = 4
144
144 = 12
25
25 = 5
169
169 = 13
36
36 = 6
196
196 = 14
49
49 = 7
225
225 = 15
64
64 = 8
Perfect
square
Square
root
81
= 9
Extension work: Cube roots
Finding the cube root is the inverse of finding the cube:
cubed
5
125
cube rooted
We write
125 = 5
3
The cube root of 125 is 5.
Web Resources
http://www.youtube.com/watch?v=ROIfbUQrSY4
(youtube: square root)