Whose has bought some exercise books We may change how we use them actually On with maths Factorise Simplify Make the subject of Chapter 12 Writing Mathematics ID: 647517
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Slide1
Did you all manage to get on to integral?Slide2
Whose has bought some exercise books?
We may change how we use them actually….Slide3
On with maths….Slide4
Factorise
Simplify
Make
the subject of:
Slide5
Slide6
Chapter 1.2
Writing Mathematics
“It’s not that I’m so smart, it’s just that I stay with problems longer.”
- Albert EinsteinSlide7
Implies
Chapter 1.2
Implies
Therefore
NecessarySufficientConverseTheoremThe symbol means ‘leads to’ or ‘Implies’.It is used when you want to present a mathematical argument, step by step.
AOB is a straight line
Slide8
Therefore
Chapter 1.2
Implies
Therefore
NecessarySufficientConverseTheoremAnother way of expressing the same idea is to use the symbol which means ‘therefore’.
AOB is a straight line
Slide9
If…then…
Chapter 1.2
Implies
Therefore
NecessarySufficientConverseTheoremA third way of writing the same thing is to use the words ‘if … then …’If AOB is a straight line, then
Slide10
Is Implied By
Chapter 1.2
Implies
Therefore
NecessarySufficientConverseTheoremYou can write the symbol the other way round, as .In that case it means ‘is implied by’ or ‘follows from’.
AOB is a straight line
In this case, it is still true.Slide11
Implies & Is Implied By
Chapter 1.2
Implies
Therefore
NecessarySufficientConverseTheoremIn some situations, both the symbols and give true statements. The two symbols can than be written together as .This means ‘implies and is implied by’.
AOB is a straight line
Slide12
Writing Mathematics
Chapter 1.2
Implies
Therefore
NecessarySufficientConverseTheoremWrite one of the symbols , and between the two statements A and B. A: Nimrod is a catB: Nimrod has whiskers
You can write A
B. Since all cats have whiskers and Nimrod is a cat, it must be true that Nimrod has whiskers.
However, you cannot write A
B. That is saying ‘Nimrod has whiskers’, implies ‘Nimrod is a cat’. However, there are other sorts of animals that have whiskers, so you cannot conclude that Nimrod is a cat.
Slide13
Writing Mathematics
Chapter 1.2
Implies
Therefore
NecessarySufficientConverseTheoremWrite one of the symbols , and between the two statements A and B. A: The hands of the clock are at right anglesB: The clock is showing 3 o’clock
You can write A
B because it is true that at 3 o’clock the hands are at right angles.
However, you cannot write A
B because there are other times, for example 9 o’clock, when the hands are at right angles.
Slide14
Necessary
Chapter 1.2
Implies
Therefore
NecessarySufficientConverseTheoremNecessary is often used when giving the conditions under which a statement is true. Thus a necessary condition for a living being to be a spider is that it has eight legs. However, it is not a sufficient condition because there are other creatures with eight legs.A living being is a spider
it has eight legs
Slide15
Sufficient
Chapter 1.2
Implies
Therefore
NecessarySufficientConverseTheoremThe word sufficient is also used when giving conditions under which you can be certain that a statement is true. Thus being a spider is a sufficient condition for a creature to have eight legs (but not a necessary one).A living being has eight legs it is a spider Slide16
Necessary & Sufficient
Chapter 1.2
Implies
Therefore
NecessarySufficientConverseTheoremIf A is both a necessary and sufficient condition for B, what symbol should connect A and B?Can you come up with statements for A and B that would make this true?
A
B
Example:
A: a shape has 5 sides
B: a shape is a pentagonSlide17
Converse
Chapter 1.2
Implies
Therefore
NecessarySufficientConverseTheoremWhen theorems are involved, it is quite common to use the word ‘converse’ to express the idea behind the symbol.
Angle
Its converse is:
Angle
It is more usual to write the converse the other way round:
Pythagoras Theorem:Slide18
Converse
Chapter 1.2
Implies
Therefore
NecessarySufficientConverseTheoremOne theorem states:If AB is the diameter of a circle through points A, B and C, then .Write down the converse of this theorem.Is the converse of the theorem true?
The converse is: If
, then AB is the diameter of a circle through points A, B and C.
This is also true.
Slide19
Converse
Chapter 1.2
Implies
Therefore
NecessarySufficientConverseTheoremExample:If a number is divisible by 6, then the number is divisible by 3Can you think of a mathematical statement where the converse is not true?Slide20
Homework reminder.
A) problem solving level 2 activity from integral – make sure you mark it in a different colour
B) follow the walkthroughs for surds and indices and make notes / examples
etcSlide21
Additional QuestionsSlide22
In each case, write one of the symbols
,
or
between the two statements A and B.
A: The object is a cubeB: The objects has six facesA: Jasmine has spotsB: Jasmine is a leopard
A: The polygon has four sides
B: The polygon is a quadrilateral
A: Today is January 1st
B: Today is New Year’s Day
Slide23
2. In
each case, write one of the symbols
,
or
between the two statements A and B.A: B:
A: This month has exactly 28 days
B: This month is February and it is not a leap year
A:
is a rectangle
B:
is a parallelogram
A: The three sides of triangle
are equal
B: The three angles of triangle
are equal
Slide24
3. For
each of the following statements, state the converse and state whether the converse is true.
If a triangle has two sides equal, then it has two angles equal
If Fred murdered Alf, then Alf is dead
is a square Each of the angles of is A triangle with three equal sides has three equal angles
If it is sunny, then Struan goes swimming
Slide25
4. In
each case, write one of the symbols
,
or
between the two statements P and Q.P:
Q:
P:
Q:
and
P:
Q:
P:
Q:
Slide26
5. In
each case, write one of the symbols
,
or
between the two statements P and Q.P: Q:
P: The number
is a positive integer greater than 1
Q:
has exactly two factors
P:
and
are odd integers
Q:
is an even integer
Note:
and
are points
P:
Q:
lies inside the sphere with centre
and radius 5cm
Slide27Slide28
7. To
show that an integer
is divisible by 5, is
it
NecessarySufficient to show that it ends in zero?