# Suggested activities

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```					  Suggested activities         Numbers and the number system: Prime numbers                                                     2/2a
use factors and        E     How many factors?
multiples, express           What types of numbers have 2 factors? 3 factors? 4 factors? 5 factors? Other numbers of factors?
numbers as
products of factors          - prime numbers have 2 factors, square numbers have an odd number of factors,

How many factors?
Fill in all the factor pairs on the last four cards, then use the cards to complete this task.

 Sort the cards into groups: two factors, three factors, four factors…
A cube number
 Can you find another number to fit each group?
will always have
 What do you know about the properties of the numbers in each group?                                          … factors
 Can you make any statements about the property and the number of factors
that are always true?

2 factors       3 factors      4 factors        5 factors        6 factors        7 factors        8 factors        9 factors

15                               23                                49                                16
1 × 15        3×5                     1 × 23                      1 × 49         7×7               1 × 16 2 × 8
4×4

81                               27                                31                                25
1 × 81 3 × 27                    1 × 27         3×9                      1 × 31                     1 × 25          5×5
9×9

8                               9                               10                                11
Suggested activities       Numbers and the number system: Prime numbers                                2/2a
find common factors    D   Consecutive LCM
use primes                 Lowest common multiples of sequences of whole numbers, plus 1.
LCM {1, 2} = 2                     2 + 1 = 3 ……3 is prime
LCM {1, 2, 3} = 6                  6 + 1 = 7…….7 is prime
LCM {1, 2, 3, 4} = 12              12 + 1 = 13… 13 is prime
LCM {1, 2, 3, 4, 5} =
How far does this sequence work? Why does it break down?

Consecutive LCM

Rule: write a sequence of whole numbers, starting with 1
Find their LCM then add 1

 When does this stop giving a prime number?
 Can you explain why it does not always work?

consecutive numbers = LCM                            add 1 = ?                   prime number?
1×2 =2                           2+1 =3                          
1×2×3 =6                              6+1 =7                          
1 × 2 × 3 × 4 = 12                       12 + 1 = 13                      
1×2×3×4×5 =                                     +1 =
1×2×3×4×5×6 =                                       +1 =
Suggested activities          Numbers and the number system: Prime numbers                                             2/2a
express numbers as        C   Factors of powers of 10
products of primes            How many factors does 10 have? 100? 1000?
Is there a pattern? Why?

- write each power of 10 as a product of 2, 5
- consider combinations of 2, 5 to make the various factors
eg 100 = 2²5², factors are 2050, 2051, 205², 2150, 2151, 215², 2²50, 2²51, 2²5²

Factors of powers of ten
 Use the information in the table to predict the number of factors for 1 000 000.
 Begin to fill in the table rows. Can you check your prediction without completing every row?

power of            product of                                     number
factors                                                     exploring
ten                 primes                                      of factors
10                  2×5          1 2 5 10                           4

100               2² × 5²        1 2 4 5 10                           9
20 25 50 100

1000

10 000

100 000

1 000 000
Suggested activities       Numbers and the number system: Prime numbers                                                2/2a
express numbers as     C   Generating prime numbers
products of primes          x2 + x + 41
Does this expression always generate a prime number?
For example
12 + 1 + 41 = 43 22 + 2 + 41 = 47….. continue!
What if we replace 41 with a different prime number,
e.g. x2 + x + 13; x2 + x + 23?
Do these expressions generate prime numbers?

 n² + 1
Does n² + 1 produce an infinite number of primes?

 n! + 1; n! – 1
3! = 3 x 2 x 1 = 6
So 3! + 1 = 6 + 1 = 7. 7 is prime.
Is 4! + 1 prime? What about 4! - 1?

Generating prime numbers
 Does the expression x2 + x + 41 always                            Try changing the 41 is to a different prime
generate a prime number?                                           number. Will the new expression still always
generate a prime number?

trial      x² + x + 41                   prime?                     trial       x² + x + …                   prime?
1        1² + 1 + 41         43                                     1         1² + 1 + …
2         2² + 2 + 41        47                                     2         2² + 2 + …
3         3² + 3 + 41                                                3         3² + 3 + …
4                                                                    4
5                                                                    5

 1² + 1 = 3, 2² + 1 = 5.
Does the expression n² + 1 always generate a prime number?
4! is 4 factorial, it
 1! + 1 = 2, 2! + 1 = 3, 3! + 1 = 7.                                                           means 4 × 3 × 2 × 1
Does the expression n! + 1 always generate a prime number?                                          so 4! = 12
 Does the expression n! – 1 always generate a prime number?
Suggested activities        Numbers and the number system: Equivalent fractions                          2/2c 2/2d
know fraction to        F   Matching
decimal conversion          Fractions Decimals Percentages dominoes (or other matching games)
for fractions where
the denominator is
a multiple of 10

7                                                  14
07                                         69%                                         065
.         .

100                                                 25

sixty-nine                       13                                                  10
hundredths                        20                  035                            20
7%

16                                                                   35
65%                                              75%                       080
.         .

20                                                                  100

three                              7
14%                     007                                                                      05
.       .

quarters                            50
.

4
075                    50%                       056                            5
35%

65                                                75                            7
70%                                                                 80%
.         .                                       .        .                 .       .

100                                               100                            20

one                  14
014                    069                     56%                         half                 20

065 65% 13/20 65/100                080 80% 4/5 16/20                056 56% 14/25               007 7% 7/100
014 14%. 7/50                       035 35% 7/20 35/100              075 75% 75/100 three quarters
07 70% 14/20                        05 50% 10/20 one half            069 69% sixty-nine hundredths
Suggested activities         Numbers and the number system: Equivalent fractions                                      2/2c 2/2d
know fraction to         E
decimal conversion           Find the pairs of numbers that have a product which ends in zero. Which numbers are never included?
for fractions where
the denominator is
a multiple of 2, 5, 10

Suggested activities         Numbers and the number system: Equivalent fractions                                       2/2c 2/2d
classify fractions as    C
recurring/terminating        ATM Thinkers – On the spot generalisation
using knowledge of           e.g. 28/99 = 0.282828282828… what can we deduce, predict or generalise from this?
the factors of their
denominators

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