2 2 Factors, multiples and primes
Specification
GCSE 2010
N c (part) Use the concepts and
vocabulary of factor (divisor), multiple,
common factor,… prime number and
prime factor decomposition
FS Process skills
Select the mathematical information
to use
Examine patterns and relationships
FS Performance
Level 1 Select mathematics in an
organised way to find solutions
2 2.1 Factors, multiples and prime numbers
Concepts and skills
•
•
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Recognise even and odd numbers.
Identify factors, multiples and prime numbers from a list of numbers.
Find the prime factor decomposition of positive integers.
Find common factors and common multiples of two numbers.
Functional skills
•
L1 … multiply and divide whole numbers using a range of strategies.
Prior key knowledge, skills and concepts
•
Tables up to 10 × 10.
Starter
Resources
CD Resources
Resource sheet 2.1
The factor game
There are various ways of playing this game but the class elimination version illustrates the principles.
As there is an element of luck with when the question reaches you there is no real shame in dropping out.
Choose a number. Although any number will do, one with several factors is best.
2.
The first student names a factor of that number.
3.
The next student names a different factor of the number.
4.
Once all the factors have been exhausted the only acceptable answer from the next student is ‘no further factors’ or words to
that effect.
5.
Any student unable to offer a correct answer is eliminated.
6.
Continue until a winner is found. As you get towards the end students may be difficult to eliminate, so have some numbers
with tricky factors like 13, 17 up your sleeve.
Counting aloud in 2s starting from, say, 30. Then from, say, 250. Would it be any more
difficult from, say, 2310? What is the pattern?
•
•
Continue similarly from an odd number start. What is the pattern?
Although this is the introduction to odd and even numbers, it can also be used for
multiples.
Main teaching and learning
Resource sheet 2 2.1
1.
•
An alternative version is to play as a team game, scoring points for correct answers. You may prefer to use nominated team
members to answer rather than throwing it open. This avoids the game being dominated by able individuals.
Another possibility if you wish to emphasise that factors occur in pairs is to require answers as factor pairs. In this instance the
class will need to be advised about square numbers as having a repeated factor.
•
•
•
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Even numbers always end in 2, 4, 6, 8 or 0.
•
•
•
Prime numbers only have 1 and the number itself as factors.
Odd numbers always end in 1, 3, 5, 7 or 9.
Multiples are obtained by multiplying the number by a positive whole number.
The factors of a number are whole numbers that divide exactly into the number,
including 1 and the number itself.
Numbers can be written as products of prime numbers.
Explain the tree method of decomposition.
Common misconceptions
Links
http://nrich.maths.org/public/search.
php?search=factors
http://www.bbc.co.uk/schools/
gcsebitesize/maths/number/
primefactorsact.shtml
ActiveTeach resources
Qualifying heats 1 video
Word problems quiz 2
HCF and LCM interactive
Ladder method interactive
factor
Students sometimes confuse factors and multiples. (Tell them that multiples come from
multiplying.)
•
Thinking that 1 is a prime number.
Enrichment
•
Most numbers have an even number of factors. There are some numbers that have an
odd number of factors. Find some of these and say what is special about them.
•
•
Factors – Packing and tiling problems.
Multiples – How many biros costing 19p can I buy with £5? (Multiples of 19 are not
easy, but using multiples of 20 you can buy 25 biros and get 25p change. Enough to buy
another biro. Answer: 21 biros.)
Plenary
Follow up
4.2 Equivalent fractions
64
•
•
multiple
Ask students to give examples of: prime numbers, factors of a given number (say 72 (1,
2, 3, 4, 6, 8, 9, 12, 18, 24, 36), multiples of, say, 4 (e.g. 8, 12, 16, 20), and to write a number
such as 144 as a product of its prime factors. (144 = 2 × 2 × 2 × 2 × 3 × 3 or 144 = 24 × 32)
common factor
common multiple
prime factor
prime number
2 2 Factors, multiples and primes
Specification
GCSE 2010
N c (part) Use the concepts and
vocabulary of … Highest Common
Factor (HCF), Least Common Multiple
(LCM)….
FS Process skills
Select the mathematical information
to use
Examine patterns and relationships
FS Performance
Level 1 Select mathematics in an
organised way to find solutions
Resources
2 2.2 Finding lowest common multiple (LCM)
and highest common factor (HCF)
Concepts and skills
•
•
Find common factors and common multiples of two numbers.
Find the LCM and HCF of two numbers.
Functional skills
•
L1 … multiply and divide whole numbers using a range of strategies.
Prior key knowledge, skills and concepts
•
Mental division by 2, 3, 5 and 7.
Starter
•
A stamp measures 6 cm by 3 cm. There are to be 60 stamps to a sheet, which has to be
rectangular. Work out possible sizes for the sheet. (3 cm by 360 cm, 6 cm by 180 cm,
12 cm by 90 cm, 9 cm by 120 cm, 18 cm by 60 cm, 24 cm by 45 cm, 15 cm by 72 cm,
30 cm by 36 cm.)
Main teaching and learning
•
Explain how to find the LCM by listing families of multiples. The product of the numbers
is always a common multiple but not necessarily the least. (List the first few multiples of
the numbers. Pick out the smallest that is in all lists. This is the LCM.)
•
Ask students to write out lists of up to twenty multiples of each of the following
numbers: (a) 2 and 3 (b) 5 and 7 (c) 6 and 4 (d) 12 and 8 (e) 9 and 6.
ActiveTeach resources
Multiples and factors quiz
•
Follow up
4.7 Adding and subtracting fractions
For each pair of numbers, circle the smallest number that appears in both lists of
multiples. Is the circled number always the product of the two original numbers? For (d)
and (e) ask What are the factors? What are the common factors?
•
Explain how to find the HCF. (List all the factors of the numbers. Pick out the smallest
that is in all lists. This is the HCF.)
•
Some students might benefit from knowing that the HCF of any two numbers must be a
factor of the difference between the two numbers.
Links
http://www.bbc.co.uk/education/
mathsfile/shockwave/games/gridgame.
html
Common misconceptions
•
Confusing HCF with LCM.
Enrichment
•
Question 2 of the guided practice worksheet may be used here.
Plenary
•
•
66
Discuss question 5 on the guided practice worksheet.
Discuss question 15 of the Review exercise. Explore the possibilities if the box contains
48 chocolates.
lowest common multiple (LCM)
highest common factor (HCF)
2 2 Factors, multiples and primes
Specification
GCSE 2010
N d Use the terms square, positive and
negative square root, cube and cube root
2 2.3 Finding square numbers and
cube numbers
Concepts and skills
•
•
•
FS Process skills
Use appropriate mathematical
procedures
FS Performance
Level 1 Use appropriate checking
procedures at each stage
Recall integer squares up to 15 × 15 and the corresponding square roots.
Recall the cubes of 2, 3, 4, 5 and 10.
Find squares and cubes.
Functional skills
•
L1 … multiply … whole numbers using a range of strategies.
Prior key knowledge, skills and concepts
•
Resources
CD Resources
Resource sheet 2.3
Starter
•
Alice in Mathsland
This is an investigative activity that looks at the link between square numbers and geometry.
The way you use it could be as an investigation or as a class teaching exercise.
•
•
Teach students about square numbers.
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•
•
Teach students about squaring.
•
Teach students about cubing.
Alice was making patterns with tiles.
The patterns she was making were stair patterns.
9 tiles
16 tiles
She knew that 4, 9 and 16 were all square numbers and wondered if all the patterns would use a number of tiles that was a
square number.
Launch investigation
Plenary
She made some more patterns and then had a brainwave.
If she cut off the right-hand side and turned it
upside down it would exactly fit the left-hand
side of her pattern and make a square.
This would always be true.
Her result had been proved.
She then experimented some more and added tiles in a different way to make squares.
1+3
(1 + 3) + 5
Use the Alice in Mathsland resource sheet as a class activity.
Main teaching
Resource sheet 2 2.3
4 tiles
Long multiplication
(1 + 3 + 5) + 7
She realised that she had found another interesting result.
The sum of consecutive numbers starting with 1 always made a square number.
Show the number 9 represented by nine dots in a square. Ask students to say what
other numbers, represented by dots, would form a square. Without drawing squares,
how can we work out the first 10 square numbers? (2 × 2; 3 × 3 etc).
Teach students about cube numbers.
If we calculate square numbers by multiplying a number by itself, how would we
calculate a cube number? (Write the number down three times then multiply.) Ask
students to calculate the first 5 cube numbers.
Common misconceptions
ActiveTeach resources
RP KC Factors and multiples knowledge
check
RP PS Factors and multiples problem
solving
Follow up
3.5 Squares and square roots, cubes and
cube roots
Unit 3 section 1.3 Using a calculator –
working out powers and roots
68
square number
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•
Students sometimes double numbers instead of squaring them.
Students sometimes multiply by 3 instead of cubing.
Enrichment
•
1 000 000 = 10002 = 1003 1 million is a square number and a cube number. Can you find
another number which is both a cube number and a square number? There are nine of
them less than 1 million. (1, 64, 729, 4096, 15 625, 46 656, 117 649, 262 144, 531 441)
Plenary
•
Devise a quiz based on the chapter review. For example, ask students to define a factor.
cube number