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starter. Complete the table using the word odd or even. Give an example for each. starter. Complete the table using the word odd or even. Give an example for each. Proof of Odd and Even. For addition and multiplication. Objective

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  1. starter Complete the table using the word odd or even. Give an example for each

  2. starter Complete the table using the word odd or even. Give an example for each

  3. Proof of Odd and Even For addition and multiplication

  4. Objective To understand how to prove if a number is odd or even through addition or multiplication Success criteria Represent an even number Represent an odd number Prove odd + odd = even Prove odd + even = odd Prove even + even = even Prove odd × odd = odd Prove odd × even = even Prove even × even = even Proof of odd and even

  5. Integer Odd Even Arbitrary Variable Addition Multiplication Proof Constant Factor Key words

  6. How to represent an even number • All even numbers have a factor of 2 • All even numbers can be represented as • Where n is any integer value 2=2×1 14= 2×7 62=2×31 2n

  7. How to represent an odd number • All odd numbers are even numbers minus 1 • All odd numbers can be represented as • Where n is any integer value 3 = 4 - 1 7 = 8 - 1 21 = 22 - 1 2n – 1 or 2n + 1

  8. Proof that odd + odd = even • Odd numbers can be written 2n – 1 • Let m, n be any integer values • odd + odd = 2n - 1 + 2m - 1 = 2n + 2m - 2 factorise = 2(n + m - 1) This must be an even number as it has a factor of 2

  9. Proof that odd + even = odd • Odd numbers can be written 2n – 1 • Even numbers can be written 2m • Let m, n be any integer values • odd + even = 2n - 1 + 2m = 2n + 2m - 1 partially factorise = 2(n + m) - 1 This must be an odd number as this shows a number that has a factor of 2 minus 1

  10. Proof that even + even = even • Even numbers can be written 2n • Let m, n be any integer values • even + even = 2n + 2m factorise = 2(n + m) This must be an even number as it has a factor of 2

  11. Proof that odd × odd = odd • Odd numbers can be written 2n – 1 • Let m, n be any integer values • odd × odd = (2n – 1)(2m – 1) = 4mn - 2m – 2n + 1 partially factorise = 2(2nm - m - n) + 1 This must be an odd number as this shows a number that has a factor of 2 plus 1

  12. Proof that odd × even = even • Odd numbers can be written 2n – 1 • Even numbers can be written 2m • Let m, n be any integer values • odd × even = (2n – 1)2m = 4mn - 2m partially factorise = 2(mn - m) This must be an even number as it has a factor of 2

  13. Proof that even × even = even • Even numbers can be written 2n • Let m, n be any integer values • even + even = (2n)2m = 4mn partially factorise = 2(2mn) This must be an even number as it has a factor of 2

  14. Exercise 1 • Prove that three odd numbers add together to give an odd number. • Prove that three even numbers add together to give an even number • Prove that two even and one odd number add together to give an odd number • Prove that two even and one odd number multiplied together give an even number

  15. Exercise 1 answers • 2n – 1 + 2m – 1 + 2r – 1 = 2n + 2m + 2r – 2 – 1 = 2(n + m + r – 1) – 1 • 2n + 2m + 2r = 2(n + m + r – 1) • 2n + 2m + 2r – 1 = 2(n + m + r) – 1 • (2n)(2m)(2r – 1) = 4mn(2r – 1) = 8mnr – 4mn = 2(4mnr – 2mn)

  16. Word match Formula that represent area have terms which have order two. --------- formula have terms that have order three. --------- that have terms of --------- order are neither length, area or volume. Letters are used to represent lengths and when a length is multiplied by another --------- we obtain an ---------. Constants are --------- that do not represent length as they have no units associated with them. The --------- letter π is often used in exam questions to represent a ---------. Formula, area, length, numbers, constant, Volume, represent, mixed, Greek

  17. Word match answers Formula that represent length have terms which have order two. Volume formula have terms that have order three. Formula that have terms of mixed order are neither length, area or volume. Letters are used to represent lengths and when a length is multiplied by another length we obtain an area. Constants are numbers that do not represent length as they have no units associated with them. The Greek letter π is often used in exam questions to represent a constant

  18. Objective To Success criteria Level – all To list Level – most To demonstrate Level – some To explain Title - review

  19. Review whole topic • Key questions

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