starter
Complete the table using the word odd or even.
Give an example for each
×
Odd
Even
+
Odd
Odd
Even
Even
Odd
Even
starter
Complete the table using the word odd or even.
Give an example for each
×
Odd
Even
+
Odd
Even
Odd
Odd
Even
Odd
Even
Odd
Even
Even
Even
Even
Odd
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
•
•
•
•
•
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Integer
Odd
Even
Arbitrary
Variable
Addition
Multiplication
Proof
• Constant
• Factor
How to represent an even number
• All even numbers have a factor of 2
2=2×1
14= 2×7
62=2×31
• All even numbers can be represented as
2n
• Where n is any integer value
How to represent an odd number
• All odd numbers are even numbers minus 1
3=4-1
7=8-1
21 = 22 - 1
• All odd numbers can be represented as
2n – 1
or
2n + 1
• Where n is any integer value
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
Even numbers can be written
Let m, n be any integer values
odd + even = 2n - 1 + 2m
= 2n + 2m - 1
partially factorise
= 2(n + m) - 1
2n – 1
2m
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
Even numbers can be written
Let m, n be any integer values
odd × even = (2n – 1)2m
= 4mn - 2m
partially factorise
= 2(mn - m)
2n – 1
2m
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
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Success criteria
Level – all
To list
Level – most
To demonstrate
Level – some
To explain
Review whole topic
• Key questions