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Adapted from NZ Maths unit A prime search
Adaptions for ESOL students: Margaret Kitchen
|Duration: 1 week|
Students will be able to:
Language learning focus
Focus on building vocabulary and mathematical literacy
How To achieve the vocabulary and oracy language outcomes:
In this unit we use rectangular models or arrays to explore numbers from one to fifty. We practice expressing numbers as the product of two smaller numbers and in doing so identify the factors of a number. We are also introduced to prime numbers.
This unit looks at the basic number concepts of factors and prime numbers. These relatively straightforward ideas have a surprisingly wide range of applications. Searching for certain types of prime number has become a test for the speed of new computers. Prime numbers are an integral part of modern coding theory. This allows the easy encryption of words and numbers but means that decoding is quite difficult. Codes based around the fact that large numbers are hard to factorise are used by banks and the military. Such codes are very difficult to break.
Finding factors of a given number can always be done by a systematic search. Starting at 1, we simply test each consecutive number to see if it is a factor of the number we are looking at. So to find the factors of 18, say, we first check 1, then 2, then 3, then 4, and so on until we get to 18. That way we can find all the factors of 18, or any other number for that matter. Systematic searches are useful throughout mathematics and are useful things for students to know about.
But we can do better than the search above. We don't have to test all of the numbers from 1 up to the number in question. Look at 18 for example. We needn't test any numbers above 9. This is because any number above 9 will be paired with a factor less than 9 and we have already tested all of these. So we only have to search half of the numbers less than the given number. For 18 this means that we get 1 and 18 by testing for 1; 2 and 9 by testing for 2; 3 and 6 by testing for 3; 6 and 3 by testing for 6; 9 and 2 by testing for 9.
However, can you see that we can actually do better than this? When we were trying to find the factors of 18 we could have stopped at 3! In fact, in general, we only have to check up to the square root of a number. Take 18 as an example again. The square root of 18 is just over 4, so we only have to check up to 4. Once we get past 4 we can be sure that we'll only meet the factors that we have already found.
So factors and prime numbers are pretty important and can be fun to play with. What more can you ask?