revise&factors - the curriculum placesieve)of)eratosthenes:)) 12345678910 11121314151617181920...
TRANSCRIPT
Revise factors Explain the difference between a factor and a mul6ple. Discuss strategies for finding factors of a number e.g. divisibility rules, list of number facts, mul6plica6on grid. Iden6fy common factors of two or three numbers. -‐Learning inten+on-‐ To focus on understanding the defini+on of prime and discover the pa8ern of these numbers in the range of 1-‐100. When we use the name Prime, when referring to a number, we are talking about its special characteris+cs.
it is defined as a whole number greater than 1 that has only itself and the number 1 as factors. For example 17 is a prime number because, it can only be made using the factors of 1 and 17. Challenge students to use their +les to find another prime less than 6 (learners should model 2, 3, 5). Explain that 6 is not a prime number and underline the defini+on to emphasis greater than, itself and the number 1. Explain that mathema+cians consider that pa8erns are very important.
Pose the ques+on to the class-‐What does spagheP have in common with mathema+cs? Record ideas
Revision of the meaning of factors might need to occur (a factor is one of two or more numbers that are mul+plied together to get a product-‐for example 6 X 7 = 42, 6 and 7 are factors of 42) Check for understanding by asking learners how many factors do prime numbers have?
Mathema+cians have been fascinated by prime numbers for thousands of years. In fact, Eratosthenes (275-‐194 BC, Greece), devised a ‘sieve’ to discover prime numbers. h8p://splash.abc.net.au/media/-‐/m/1469825/primes-‐ancient-‐building-‐blocks-‐of-‐maths
Australian Curriculum Year 6 Iden+fy and describe proper+es of prime, composite numbers ACMNA122 • represen+ng composite numbers as a product of their prime factors and
using this form to simplify calcula+ons by cancelling common primes
Key Ideas • How to iden+fy numbers with only one factor pair and numbers with
more than one factor pair • What are efficient strategies to iden+fy if a number has one or more
factor pairs
Context for Learning -‐ Real life experiences: Prime numbers are good for quickly transforming a situa+on with lots of possible outcomes into an equivalent situa+on with only a handful of possible outcomes. Prime Numbers are used over the internet to encode sensi+ve informa+on.
Resources: FISH, 6le counters, 100 chart, 100 mat and number cards-‐op6onal, mul6ples
Vocabulary Factor, product, prime, composite, greater than, less than, whole number, Introductory AcBvity Process-‐Revision Define the term ‘mul6ple’. Discuss ways of finding mul6ples e.g. by mul6plying numbers, skip coun6ng or repeated addi6on. Write paKerns of mul6ples e.g. with the help of a number line, hundred board or calculator. Iden6fy common mul6ples for two or three numbers.
Ask class what happens to spagheP ader it is cooked in boiling water-‐it is drained in a container that allows the water to escape. Pose the ques+on again to the class-‐What does spagheP have in common with mathema+cs? Record ideas Prime numbers are special because they are greater than 1 and its factors are only one and itself.
The Sieve of Eratosthenes: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
A sieve is like a strainer that you drain spagheP through when it is done cooking. The water drains out, leaving your spagheP behind. Well, Eratosthenes's sieve drains out composite numbers and leaves prime numbers behind. To do what Eratosthenes did, either use a 100’s mat (blue) or hand out one hundred charts (1-‐100) or h8p://www.scootle.edu.au/ec/viewing/L3545/index.html Instruct the class to: If using a big mat
remove numbers rather than crossing them out and leave number rather than circling them. If using an individual 100 paper grid cross out 1, because it is not prime. Then, circle 2, because it is the smallest posi6ve even prime. Now cross out every mul6ple of 2; in other words, cross out every 2nd number. Then circle 3, the next prime. Then cross out all of the mul6ples of 3; in other words, every third number. Some, like 6, may have already been crossed out since they may be mul6ples of 2. Then circle the next open number, 5. Now cross out all of the mul6ples of 5, or every 5th number. Con6nue doing this un6l all the numbers through 100 are either circled or crossed out. If you have remembered your mul6plica6on tables, you have just circled all the prime numbers less than 100. Remind students to consider divisibility rules: • One whole number is divisible by another if, ader dividing, the remainder is
zero. • If one whole number is divisible by another number, then the second
number is a factor of the first number. Ask students to consider what we call the other numbers that have not been iden6fied as prime How many factors does 27 have? What strategy can we use to find out? Suggest using divisibility rules to inves6gate. 27 is an odd number so it is not divisible by 2. The sum of the digits is 2 + 7 = 9 then 27 is divisibly by 3. So 27 also has factors of 3 and 9.
If we look at the defini6on of a prime number 27 is not prime. Numbers with more than two factors are referred to as composite.
A composite number has factors other than 1 and itself. Composite numbers can be wriKen as the product of prime numbers.
AcBvity Process: How can you write a composite number as a product of prime factors Write the number 24 as a factor pair Ask students to comment on the numbers In the first 6er-‐ 4 & 6. The last set of Branches has all prime numbers. So 24 = 2 X 2 X 2 X 3 Ask students to choose a composite number that has not been demonstrated and create a factor tree. They must jus6fy how reasonable their answer is AcBvity Process: ConsolidaBon and Extension AcBviBes Why do I have to learn about Prime Numbers? Listen to Adam Spencer the Mathema6cian talking about Prime Number. Teachers should listen to this first to make a jusgement about whether to air this to the whole class of a group of students. hKp://splash.abc.net.au/media/-‐/m/86438 • As students are listening they need to be given a focus of ques6ons to share at
the end. • How does the Adam explain prime numbers? • How many prime numbers are there?
24 4 6 X 2 X 2 X 2 X 3
hKp://splash.abc.net.au/media/-‐/m/154992/prime-‐number-‐keys Is a video about the real world applica6on of encrypted messages
• Why is encryp6on important? Self Assessment: Ask learners to write ‘I can’ statements about
prime and composite numbers