AFRICAN INSTITUTE FOR MATHEMATICAL SCIENCES SCHOOLS ENRICHMENT CENTRE TEACHER NETWORK Grades 6 to 9 Patterns of Multiples Seive $ Choose a colour. Put a line through 4,6,8,10 like this. Do the rest of the two times table. You do not cross through the first number in the table but it may already be crossed out. $Change colour. Put a line through 6,9,12,15 like this. Do the rest of the three times table. ! What do you notice? " Talk to a partner. 4
6
What do you notice?
What patterns can you
see?
Which numbers are
coloured twice?
Or 3 times? Why?
Would you get the
same result if you
stopped after
colouring the 7 times
table and did not do
the eight and nine
times table.? Why?
NOTES FOR TEACHERS Why do this activity? Learners should do the Patterns of Multiples Sheets 1, 2 and 3 first SEE BELOW. Spread this over several weeks with other activities like Orchestra and Clap Hands so there are both active and sitting activities and not too much of the same thing in a session. This activity offers students opportunities to explore multiples in more depth than usual, in particular looking at the links between multiples of different numbers. It also encourages students to see the connection between primes and multiples. RESOURCES: Colouring pens. At least one 1 to 100 grid for each learner. Patterns and Multiples worksheets below. Six column grid and 1 to 400 grid for high flyers. Eratosthenes Sieve This activity is named after a Greek called Eratosthenes who invented the technique. What patterns do you notice? What can you say about the numbers that aren’t coloured in? You might wonder why, unlike the clapping activity, you don’t colour in the first number of a table. That is because eventually the uncoloured numbers are the PRIME numbers, those that have no FACTORS other than themselves and 1. The first numbers of some tables (2,3,5,7) are prime, so if you coloured them in, then Eratosthenes’ sieve wouldn’t work properly. There are lots of patterns to look for – numbers coloured twice have two factors, numbers coloured three times have three factors etc. You might want to look for the way the numbers make a pattern on the grid – for example the fours are a chequerboard pattern. Key questions Which numbers get crossed out more than once, and why? Which numbers don't get crossed out at all, and why? Which possible factors do we need to consider in order to decide if a number is prime? Possible extension For the high flyers -­‐ Ask: "Imagine you want to find all the prime numbers up to 200. You could do this by crossing out multiples in a 200 number grid. Which multiples will you choose to cross out? How can you be sure that you are left with the primes?" What about making a 1 to 400 seive? Which multiples will you cross out to be sure that you are left with the primes?" “We're used to working with grids with ten columns, but you might find an interesting result if you use this six-­‐
column grid instead. Can you predict what you will see?" Possible support By working in pairs we are encouraging students to share ideas and support each other. For more information about Eratosthenes see: http://www.mathsisgoodforyou.com/artefacts/sieve.htm There is more about this activity on http://nrich.maths.org/7520 PATTERNS AND MULTIPLES 1 Name:________________________________________________________
Colour multiples of 2
Colour multiples of 3
Colour multiples of 4
Colour multiples of 5
What do you notice?
PATTERNS AND MULTIPLES 2
Name:________________________________________________________
Colour multiples of 6
Colour multiples of 7
Colour multiples of 8
Colour multiples of 9
What do you notice?
PATTERNS AND MULTIPLES 3
Name:________________________________________________________
Colour multiples of 3
Colour multiples of 6
Colour multiples of 9
Colour multiples of 12
What do you notice?