mathematical reasoning form 4 mathematics
DESCRIPTION
A powerpoint presentation about Mathematical Reasoning that helps teacher to perform their Lesson. (The Document is a sharing from a course attended by me)TRANSCRIPT
Mathematical Reasoning
IntroductionIntroduction
AllAllSomeSome
OrOrandand
If and If and only ifonly if
IFIFAllAll
ThenThen
--
InductionInductionDeductionDeduction
(a) Is ‘3 + 2 = 5’ a statement ? Explain your answer. Yes, it is a statement
because it is true
(b) Is ‘3 + 2 = 6’ a statement ? Explain your answer.
Yes, it is a statement because it is false
(c) Is ‘3 + x = 5’ a statement ?
Explain your answer.No, it is not a statement
because we can’t determine whether it is true
Antecedence(cause)
Consequence(effect)If then
If x is a factor of 12 then x is a factor of 24
If m+5 = 12 then m = 7
If you work hard then you will score in the examination.
If n is an odd number then n+1 is an even numberIf x is a prime number then 2x is an even number
If you smoke then you will suffer fr lung cancer
All triangles have 3 sides
All pentagons have 5 sides
Some triangles have 4 sides
All quadratic equations have two roots.
Some quadratic equations have two roots.
Some hexagons have 5 sides
Some prime numbers are odd numbers
All prime numbers are odd numbers
Some multiples of 3 are multiple of 6
All multiples of 3 are multiples of 6
True False
QUANTIFIER ‘‘ALL’’ AND ‘‘SOME’’
Based on the information given, construct a true statement using the quantifier ‘‘all’’ or ‘‘some’’
Object and propertyAnswer (true statement)
1 Acute angle ; less than 90o All acute angles are less than 90o
2 Negative number; smaller than zero
All negative numbers are smaller than zero
3 Triangle ; right-angled triangles Some triangles are right-angled triangles
Object and property Answer (true statement)
1 Rhombuses ; four equal sides All rhombuses have four equal sidesAll rhombuses have four equal sides
2 Odd number ; prime number Some odd number are prime numbersSome odd number are prime numbers
3 Factor of 6 ; factor of 3 Some factors of 6 are factor of 3Some factors of 6 are factor of 3
4 Isosceles triangle ; two equal sides
All isosceles triangles have two equal All isosceles triangles have two equal sidessides
5 Even number ; divisible by 10 Some even numbers are divisible by 10Some even numbers are divisible by 10
QUANTIFIER ‘‘ALL’’ AND ‘‘SOME’’
Based on the information given, construct a false statement using the quantifier ‘‘all’’ or ‘‘some’’
Object and propertyAnswer (true statement)
1 Null sets ; no elements Some null sets have no elements
2 Parallel line ; the same length
All parallel lines have the same length
3 Odd number ; perfect square
All odd number are perfect squares
Object and property Answer (true statement)
1 Orchid flower ; red in colour All orchid flower are red in colourAll orchid flower are red in colour
2 Human being ; heart Some human beings have heartSome human beings have heart
3 Quadratic equation ; 2 roots All quadratic equation have 2 rootsAll quadratic equation have 2 roots
4 2 digits number ; less than 100
Some 2 digits numbers are less than Some 2 digits numbers are less than 100100
5 Multiples of 8 ; can be exactly divided by 4
Some multiples of 8 can be exactly Some multiples of 8 can be exactly divided by 4divided by 4
Consequence(effect)
If then
Converse of a statement
Antecedence(cause)
you will score in the examination.
If Consequence(effect)
then Antecedence(cause)
If you want to score in the examination. you have to work hard
If you work hard then
then
x is a factor of 24
If x is a factor of 24 x is a factor of 12
If x is a factor of 12 then
then
If x > 5
The lines are parallel
their gradients are the same
If the gradient of two lines are the same
then
then
then x + 2 > 7
if x + 2 > 7If x > 5then
the hour and minute hands are in a stright line The time is 6 oclock then
then If the hour and minute hands are in a straight line The time is 6 o’clock
If two lines are parallel
2 + 3 = 5 3 2= 6and
or
False
True
Compound Statement
Statement I : – 4 > – 2
False
True
Combine the two statements below to form a new statement which is (a) true (b) false
Statement II : 3,4,5 is a Pythagorean triple
-4 > -2 and 3,4,5 is a Pythagorean triple
-4 > -2 or 3,4,5 is a Pythagorean triple
50 or 45 is a multiple of 10 11 00 1+0=11+0=1
6 is a factor of 16 or 24 00 11 0+1=10+1=1
A rectangle has 4 sides and a pentagon has 6 sides 11 00 1x0=01x0=0
7 is a factor of 49 and a prime number 11 11 1x1=11x1=1
12 + 22 = 32 and 32 + 42 = 52 00 11 0x1=00x1=0
2 is equal to 20 or (2 –1) –1 00 11 0+1=10+1=1
All even numbers are divisible by 2 or all odd numbers are divisible by 3
11 00 1+0=11+0=1
36 is a perfect square and a multiple of 4 11 11 1x1=11x1=1
80 is a perfect square or an even number 00 11 0+1=10+1=1
17 is a prime number and a factor of 34 11 11 1x1=11x1=1
1 m2 = 10 000 cm2 or 1 cm3 = 10 000 mm3 11 00 1+0=11+0=1
Ant is an insect and has 4 legs 11 00 1x0=01x0=0
if and only if
3x > 12
3x > 12
1.If
then
2.If then 3x > 12
1.If
then
then2.If
X > 4
X > 4
X > 4
Answ
er
Click
to sh
ow
(b) xy = 0 if and only if x = 0 or y = 0
Implication I : If xy = 0 then x = 0 or y = 0
Implication II : If x = 0 of y = 0 then xy = 0
(a) m < n if and only if 4m < 4n
Implication I : If m < n then 4m < 4n
Implication II : If 4m < 4n then m < n
All regular pentagons have five equal sides
PQRST is a regular pentagon
PQRST has five equal sides.
SolveClick to show
Premise II :
Premise I :
Conclusion :
Premise II :
Conclusion :
All even number are divisible by 2
18 is an even number
18 is divisible by 2.
SolveClick to show
Premise II :
Premise I :
Conclusion :
Premise II :
Conclusion :
If x and y are odd number then the product of x and y is odd number5 and 7 are odd numbers
The product of 5 and 7 is odd number.
SolveClick to show
Premise II :
Premise I :
Conclusion :
Premise II :
Conclusion :
Premise I : All polygons have a sum of 180o for its exterior angles.
Conclusion : Octagon has a sum of 180o for its exterior angles
Octagon is a polygon.
SolveClick to show
Premise II :
Premise I : IF x is greater than zero, then x is a positive number.
Conclusion : 8 is a positive number
8 is greater than zero.
SolveClick to show
Premise II :
Premise I :
– 8 is a negative number
– 8 is less than 0
If x is a negative number then it is less than 0
SolveClick to show
Premise II :
Conclusion :
All negative numbers are less than 0
Premise I : If JKL is an equilateral triangle, then the value of its interior angles is 60o
Premise II : the value of JKL interior angles 60o
JKL is not an equilateral triangle
5 is a number that is a factor of 5
Number 5 is not a factor of 9
5 is a number that is not a factor of 9.
SolveClick to show
Conclusion :
Premise II : 11 is not divisible by 2
Conclusion : 11 is not divisible by 6
If x is a number that is divisible by 6 then x is divisible by 2
Premise 1 :
SolveClick to show
6 = (1)3 + 5
13 = (2)3 + 5
32 = (3)3 + 5
69 = (4)3 + 5 ……………………
+ 5 n = 1,2,3 …….. n = 1,2,3, 4 n = integer
SolveClick to show
n3
6 = (1)2 + 5
9 = (2)2 + 5
14 = (3)2 + 5
21 = (4)2 + 5 ……………………
+ 5
n = 1,2,3 …….. n = 1,2,3, 4 n = integer
12 = 2(2)3 – 4
50 = 2(3)3 – 4
124 = 2(4)3 – 4
246 = 2(5)3 – 4 ……………………
– 4
n = 2,3,4 …….. n =2,3, 4,5 n = integer
n2 2 n3
SolveClick to show
– 4 = 3 (1)2 – 7
5 = 3 (2)2 – 7
20 = 3(3)2 – 7
41 = 3(4)2 – 7
……………
- 7
n = 1,2,3 ……..
5 = 5 – 2(0)3
3 = 5 – 2(1)3
– 11 = 5 – 2(2)3
– 49 = 5 – 2(3)3
……………………
5
n = 0,1,2 ……..
141 = 2(3) + 5(3)3
328 = 2(4) + 5(4)3
635 = 2(5)+ 5(5)3
1092 = 2(6) + 5(6)3
……………………
2n
n = 3,4,5 ……..
n = 1,2,3,4 n = 0,1,2,3 n = 3,4,5 6.
n = integer n = integer n = integer
SolveClick to show
3n2 – 2 n3 +5 n3