factor multiple common factor common multiple prime factor prime number ● 2 2 Factors, multiples and primes 64 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 Specification CD Resources Resource sheet 2.1 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 Follow up 4.2 Equivalent fractions Resources 2.1 Factors, multiples and prime numbers Concepts and skills • 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 • 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 • Even numbers always end in 2, 4, 6, 8 or 0. • 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. • Prime numbers only have 1 and the number itself as factors. • Numbers can be written as products of prime numbers. • Explain the tree method of decomposition. Common misconceptions • 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 • 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 = 2 4 × 3 2 ) Resource sheet 2 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. 1. 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. An alternative versionis 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. 2
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factor multiple common factor common multiple prime factor prime number
�2 2 Factors, multiples and primes
64
GCSE 2010N c (part) Use the concepts and vocabulary of factor (divisor), multiple, common factor,… prime number and prime factor decomposition
FS Process skillsSelect the mathematical informationto useExamine patterns and relationships
FS PerformanceLevel 1 Select mathematics in an organised way to fi nd solutions
Specifi cation
CD ResourcesResource sheet 2.1
Linkshttp://nrich.maths.org/public/search.php?search=factorshttp://www.bbc.co.uk/schools/gcsebitesize/maths/number/primefactorsact.shtml
ActiveTeach resourcesQualifying heats 1 videoWord problems quiz 2HCF and LCM interactiveLadder method interactive
Follow up4.2 Equivalent fractions
Resources
2.1 Factors, multiples and prime numbers
Concepts and skills
• 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
• Counting aloud in 2s starting from, say, 30. Then from, say, 250. Would it be any more diffi cult 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
• Even numbers always end in 2, 4, 6, 8 or 0.
• 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.
• Prime numbers only have 1 and the number itself as factors.
• Numbers can be written as products of prime numbers.
• Explain the tree method of decomposition.
Common misconceptions
• 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
• 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)
Resource sheet 2 2.1The factor gameThere 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.
1. 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.
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.
2
lowest common multiple (LCM) highest common factor (HCF)
�2 2 Factors, multiples and primes
66
GCSE 2010N c (part) Use the concepts and vocabulary of … Highest Common Factor (HCF), Least Common Multiple (LCM)….
FS Process skillsSelect the mathematical information to useExamine patterns and relationships
FS PerformanceLevel 1 Select mathematics in an organised way to fi nd solutions
Specifi cation
Linkshttp://www.bbc.co.uk/education/mathsfi le/shockwave/games/gridgame.html
ActiveTeach resourcesMultiples and factors quiz
Follow up4.7 Adding and subtracting fractions
Resources
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 fi nd the LCM by listing families of multiples. The product of the numbers is always a common multiple but not necessarily the least. (List the fi rst 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.
• 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 fi nd 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 benefi t from knowing that the HCF of any two numbers must be a factor of the difference between the two numbers.
Common misconceptions
• Confusing HCF with LCM.
Enrichment
• Question 2 of the guided practice worksheet may be used here.
Plenary
• Discuss question 5 on the guided practice worksheet.
• Discuss question 15 of the Review exercise. Explore the possibilities if the box contains 48 chocolates.
2
square number cube number
�2 2 Factors, multiples and primes
68
GCSE 2010N d Use the terms square, positive and negative square root, cube and cube root
FS Process skillsUse appropriate mathematical procedures
FS PerformanceLevel 1 Use appropriate checking procedures at each stage
Specifi cation 2.3 Finding square numbers and cube numbers
Concepts and skills
• 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
• Long multiplication
Starter
• Use the Alice in Mathsland resource sheet as a class activity.
Main teaching
• Teach students about square numbers.
• 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 fi rst 10 square numbers? (2 × 2; 3 × 3 etc).
• Teach students about squaring.
• 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 fi rst 5 cube numbers.
• Teach students about cubing.
Common misconceptions
• 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 fi nd 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 defi ne a factor.
CD ResourcesResource sheet 2.3
ActiveTeach resourcesRP KC Factors and multiples knowledge checkRP PS Factors and multiples problem solving
Follow up3.5 Squares and square roots, cubes and cube rootsUnit 3 section 1.3 Using a calculator – working out powers and roots
Resources
Resource sheet 2 2.3Alice in MathslandThis 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.
Alice was making patterns with tiles.The patterns she was making were stair patterns.
4 tiles 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 investigationPlenaryShe 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 (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.
2
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