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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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Presentation Transcript
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

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
Integer

Odd

Even

Arbitrary

Variable

Multiplication

Proof

Constant

Factor

Key words
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

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

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

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

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

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

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

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

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
• 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)

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

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

Objective

To

Success criteria

Level – all

To list

Level – most

To demonstrate

Level – some

To explain

Title - review
Review whole topic
• Key questions