starter

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

+ Odd Even

Odd

Even

× Odd Even

Odd

Even

starter

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

+ Odd Even

Odd Even Odd

Even Odd Even

× Odd Even

Odd Odd Even

Even Even Even

Proof of Odd and Even

For addition and multiplication

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

Key words

• Integer• Odd• Even• Arbitrary• Variable• Addition• Multiplication• Proof

• Constant• Factor

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

2n

2=2×1 14= 2×7 62=2×31

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

2n – 1 or 2n + 1

3 = 4 - 1 7 = 8 - 1 21 = 22 - 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 - 2factorise = 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 - 1partially 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 + 1partially 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 - 2mpartially 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

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)

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

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

Title - review

• Objective • To

• Success criteria• Level – all• To list• Level – most• To demonstrate• Level – some• To explain

Review whole topic

• Key questions