These slides can be used as a learning resource for students. . Some answers are broken down into steps for understanding and some are “final answers” that need you to provide your own method for.. ID: 576502
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Presentations text content in GCSE Further Maths (AQA)
Slide1
GCSE Further Maths (AQA)
These slides can be used as a learning resource for students.
Some answers are broken down into steps for understanding and some are “final answers” that need you to provide your own method for.
Slide2
Coordinate Geometry GCSE FM
Find the midpoint of the line segment from A(4, 1) to B(5, -2)
Solution: M =
Slide3
Coordinate Geometry GCSE FM
Find the gradient of the line segment from C(-3, 2) to D (0, 11)
Solution:
Write down the value of the gradient of line that is
perpendicular
to this line.
Slide4
Coordinate Geometry GCSE FM
Simplify the following surds
(a)
(c)
(b)
Slide5
Coordinate Geometry GCSE FM
Express 0.45454545… as a fraction
Slide6
Ratios, decimals & fractions GCSE FM
A is 40% of B. Express A in terms of B.(ii) C is 80% of A. Express C in terms of B
Solution:
A = 0.4 x B
A = 0.4B
(
i
)
(ii)
C = 0.8A
C = 0.8(0.4B)C= 0.32B
So this means C is 32% of B
Slide7
Ratios, decimals & fractions GCSE FM
C and D are in the ratio 2:5 Express C in terms of DE is 60% of C. Write E in terms of D.
Solution:
(
i
)
(ii)
So this means E is 24% of D
Slide8
Ratios, decimals & fractions GCSE FM
Simplify the following.
(
i
)
(ii)
(
i
)
(ii)
Slide9
Ratios GCSE FM
Simplify the following ratio.
Slide10
Algebraic fractions GCSE FM
Slide11
Algebraic fractions GCSE FM
Expand & Simplify…..
Solution:
Slide12
Sequences GCSE FM
Find the nth term for this quadratic sequence….
6, 15, 28, 45, 66….
n
th term = 2n² +
3n + 1
9 13 17 21..
4 4 4 4..
nth term = 2n²….?
Subtracting 2n² leaves
4, 7, 10, 13,
1
6….
Slide13
Equations GCSE FM
Solve the following equations :
a)
b)
Slide14
Simultaneous Equations GCSE FM
Solve the following equations :
Slide15
Equations GCSE FM
Solve the following equations :
a)
b)
Slide16
Coordinate Geometry GCSE FM
P is the point (a, b) and Q is the point (3a, 5b)
Find in terms of a and b,
(
i
) The gradient of PQ
(iii) The midpoint of PQ
(ii) The length of PQ
Slide17
Rearanging formulae GCSE FM
Make ‘t’ the subject of the following formulae
(a)
(b)
Slide18
Solving inequalities GCSE FM
Solve the following inequalities
(a)
(b)
Slide19
Solving inequalities GCSE FM
Solve the following inequality
x² - 3x < 0
y
= x² - 3x
Slide20
Solving inequalities GCSE FM
Given that
Write out an inequality for
(
i
) a + b
(ii) a - b
Slide21
Rearranging formulae GCSE FM
Make r the subject
(a)
(b)
Make
l
the subject
Slide22
Equations GCSE FM
Solve the following equation :
Slide23
SURDS GCSE FM
RATIONALISE the following
(remove the surd from the denominator) :
(a)
(b)
Slide24
SURDS GCSE FM
RATIONALISE the following
(remove the surd from the denominator) :
(b)
Slide25
Infinite sequences GCSE FM
Find the first 5 terms using the following nth term :
What is happening to the terms as n increases to
∞
(infinity) ?
n12345T
As n
∞
The
limiting value
is
4
Slide26
Infinite sequences GCSE FM
Find the limit for the nth term as n ∞:
As n
∞
The
limiting value is ..
Slide27
Infinite sequences GCSE FM
Find the first 5 terms using the following nth term :
What is happening to the terms as n increases to
∞
(infinity) ?
n
1
2345T
As n
∞
The
limiting value
is 2/3
Slide28
Simultaneous equations GCSE FM
Substitute y for x and hence solve the equations :
Slide29
Simultaneous equations GCSE FM
Substitute y for x and hence solve the equations :
Slide30
Simultaneous equations GCSE FM
Find the point of intersection for these two lines :
y = 2x + 1
4
4
x
+ y = 4
y = 2x + 1
x + y = 4
x + (2x + 1) = 4
3x + 1 = 4
x = 1
So y = 3
Slide31
Rearranging formulae GCSE FM
Make a the subject of the following formulae….
Slide32
Linear (straight line) graphs GCSE FM
Find the equation of the line with gradient 3 and passing through (1, -2)
(1, -2)
Find the equation of the line that is
perpendicular
and passing through (1, -2)
Slide33
Linear (straight line) graphs GCSE FM
Distance between 2 points?
We need PQ , QR and PR.
Which two are the same?
Use Pythagoras’ Theorem.
Two equal lengths so isosceles.
Slide34
Simultaneous equations GCSE FM
Substitute y for x and hence solve the equations :
Slide35
Simultaneous equations GCSE FM
Substitute y for x and hence solve the equations :
Slide36
Simultaneous equations GCSE FM
Slide37
Simultaneous equations GCSE FM
Substitute y for x and hence solve the equations :
Slide38
Simultaneous equations GCSE FM
Substitute y for x and hence solve the equations :
Slide39
Simultaneous equations GCSE FM
Substitute y for x and hence solve the equations :
x
= 3 and y = 2
x
= 3 and y = -2
x
= 3 and y = 3
x
= -3 and y = -3
x
= 1 and y = 2
x
= -1 and y = 0
Slide40
Pythagoras 3D GCSE FM
Find : (i) AD (ii) CE (iii) AC
A
B
C
D
E
F
Slide41
Linear (straight line) graphs GCSE FM
1. Find the equation of the line with gradient ½ and passing through (-4, 6)
2. Find the equation of the line
perpendicular
to y = 2x - 1 and passing through (2, 3)
Slide42
Sequences GCSE FM
Find the nth term for this quadratic sequence….
1
, 0, -3, -8, -15….
n
th term = -n² + 2n
-1 -3 -5 -7..
-2 -2 -2 -2..
nth term = -n²….?
Subtracting -n² (adding n²) leaves
2,
4
, 6, 8, 1
0
….
Slide43
Proof GCSE FM
Show that f(x) = x² - 4x + 5
Hence
explain
why f(x) > 0 for all values of x.
Let f(x) = (x – 2)² + 1
f(2) = (2 – 2)² + 1
= 0² + 1 = 1
f(-2) =
= -4² + 1 = 17
(
-2
– 2)² + 1
Slide44
Proof GCSE FM
Show that f(x) = x² - 4x + 5
E
xplain
why f(x) > 0 for all values of x.
Let f(x) = (x – 2)² + 1
(x – 2)² + 1 = x² - 4x + 4 + 1
= x² - 4x + 5
Squaring
always
makes a positive value so adding 1 is still positive.
Slide45
Proof GCSE FM
Show that f(x) = 9x²
Hence
explain
why f(x) is a square number.
Let f(x) = 2x³ - x²(2x – 9)
2x³ - 2x³ + 9x²
9x² = (3x)² so you are squaring 3x. This makes a square number
= 9x²
Slide46
Proof GCSE FM
Show that f(n + 1) = n² + 2n + 1
Let f(n) = n²
for all positive integer values of n
f(n+1) = (n+1)²
= (n+1)(n+1)
= n² + 2n + 1
Slide47
Proof GCSE FM
Show that f(n + 1) + f(n – 1) is always even
Let f(n) = n²
for all positive integer values of n
(n+1)²
+ (n-1)²
= (n+1)(n+1)+(n-1)(n-1)
= n² + 2n + 1 + n² - 2n + 1
= 2n² + 2
Slide48
Proof GCSE FM
Proving something is even means you have to show it is in the
2 times table
(or a multiple of 2)
2n² + 2
2(n² + 1)
Since we are multiplying by 2, this must be even
Slide49
Functions
2f(x) =
Let f(x) = x²
f(2) = (
2
)² =
4
f(-2) =
= 4
(-2)²
2x²
f(2x) =
= (2x)²
Slide50
Using Functions
2f(x) =
Let f(x) = 3 – x. Sketch this graph
f(0) =
=
3
3- 0
2(3-x)
f(2x) =
= 6–2x
3-(2x)
= 3–2x
1 - f(x) =
1 - (3–x)
= -2 + x
= x - 2
What does this represent?
Slide51
EXAM REVISION (FM)
Review
TEST 1
Review TEST 2
Recent
Ratio and percentages.
Ex 1A
and 1D
Equations of lines
Ex 3D, 5B, 5C
Simultaneous equations by substitution Ex 4B
Midpoints, length of line segments
Expanding brackets further
Circles and lines
(equation of a circle)
Gradients of lines
(including
perpendicular lines) Ex 3C
Proof (to be taught)
Ex 4G
3D Pythagoras
(including the diagonal of
a cuboid) Ex 6E (part only!)
Algebraic
fractions
Ex 1B, Ex 2C, 2D
Linear Sequences
Ex 4H
Solving quadratic equations by factorising Ex 4A q. 1 only
Factorising Ex 2A
(including
quadratics)
Quadratic Sequences
Ex 4I
Quadratic
graphs
Ex 3E
SURDS (including rationalising) Ex 1F and 1G
Limiting value of a sequence
Ex 4J
Rearranging formulae
Ex 2B
Algebraic
fractional equations
Ex 1C, Ex 2E
CHECK and REVIEW ALL HOMEWOR
K tasks.
Area of a triangle Ex 7A
Linear & Quadratic inequalities
Ex 4D, 4E.
Slide52
FOCUS YOUR EXAM REVISION (FM)
Topic
CHAPTER 1Decimals, Fractions and percentagesEx 1 ASimplifying algebraEx 1BSolving equationsEx 1CRatiosEx 1DFurther algebra – expanding bracketsEx 1ESURDS Ex 1F and Ex 1G
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DownloadNote - The PPT/PDF document "GCSE Further Maths (AQA)" is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.
Presentations text content in GCSE Further Maths (AQA)
GCSE Further Maths (AQA)
These slides can be used as a learning resource for students.
Some answers are broken down into steps for understanding and some are “final answers” that need you to provide your own method for.
Slide2Coordinate Geometry GCSE FM
Find the midpoint of the line segment from A(4, 1) to B(5, -2)
Solution: M =
Slide3Coordinate Geometry GCSE FM
Find the gradient of the line segment from C(-3, 2) to D (0, 11)
Solution:
Write down the value of the gradient of line that is
perpendicular
to this line.
Slide4Coordinate Geometry GCSE FM
Simplify the following surds
(a)
(c)
(b)
Slide5Coordinate Geometry GCSE FM
Express 0.45454545… as a fraction
Slide6Ratios, decimals & fractions GCSE FM
A is 40% of B. Express A in terms of B.(ii) C is 80% of A. Express C in terms of B
Solution:
A = 0.4 x B
A = 0.4B
(
i
)
(ii)
C = 0.8A
C = 0.8(0.4B)C= 0.32B
So this means C is 32% of B
Slide7Ratios, decimals & fractions GCSE FM
C and D are in the ratio 2:5 Express C in terms of DE is 60% of C. Write E in terms of D.
Solution:
(
i
)
(ii)
So this means E is 24% of D
Slide8Ratios, decimals & fractions GCSE FM
Simplify the following.
(
i
)
(ii)
(
i
)
(ii)
Slide9Ratios GCSE FM
Simplify the following ratio.
Slide10Algebraic fractions GCSE FM
Slide11Algebraic fractions GCSE FM
Expand & Simplify…..
Solution:
Slide12Sequences GCSE FM
Find the nth term for this quadratic sequence….
6, 15, 28, 45, 66….
n
th term = 2n² +
3n + 1
9 13 17 21..
4 4 4 4..
nth term = 2n²….?
Subtracting 2n² leaves
4, 7, 10, 13,
1
6….
Slide13Equations GCSE FM
Solve the following equations :
a)
b)
Slide14Simultaneous Equations GCSE FM
Solve the following equations :
Slide15Equations GCSE FM
Solve the following equations :
a)
b)
Slide16Coordinate Geometry GCSE FM
P is the point (a, b) and Q is the point (3a, 5b)
Find in terms of a and b,
(
i
) The gradient of PQ
(iii) The midpoint of PQ
(ii) The length of PQ
Slide17Rearanging formulae GCSE FM
Make ‘t’ the subject of the following formulae
(a)
(b)
Slide18Solving inequalities GCSE FM
Solve the following inequalities
(a)
(b)
Slide19Solving inequalities GCSE FM
Solve the following inequality
x² - 3x < 0
y
= x² - 3x
Slide20Solving inequalities GCSE FM
Given that
Write out an inequality for
(
i
) a + b
(ii) a - b
Slide21Rearranging formulae GCSE FM
Make r the subject
(a)
(b)
Make
l
the subject
Slide22Equations GCSE FM
Solve the following equation :
Slide23SURDS GCSE FM
RATIONALISE the following
(remove the surd from the denominator) :
(a)
(b)
Slide24SURDS GCSE FM
RATIONALISE the following
(remove the surd from the denominator) :
(b)
Slide25Infinite sequences GCSE FM
Find the first 5 terms using the following nth term :
What is happening to the terms as n increases to
∞
(infinity) ?
n12345T
As n
∞
The
limiting value
is
4
Slide26Infinite sequences GCSE FM
Find the limit for the nth term as n ∞:
As n
∞
The
limiting value is ..
Slide27Infinite sequences GCSE FM
Find the first 5 terms using the following nth term :
What is happening to the terms as n increases to
∞
(infinity) ?
n
1
2345T
As n
∞
The
limiting value
is 2/3
Slide28Simultaneous equations GCSE FM
Substitute y for x and hence solve the equations :
Slide29Simultaneous equations GCSE FM
Substitute y for x and hence solve the equations :
Slide30Simultaneous equations GCSE FM
Find the point of intersection for these two lines :
y = 2x + 1
4
4
x
+ y = 4
y = 2x + 1
x + y = 4
x + (2x + 1) = 4
3x + 1 = 4
x = 1
So y = 3
Slide31Rearranging formulae GCSE FM
Make a the subject of the following formulae….
Slide32Linear (straight line) graphs GCSE FM
Find the equation of the line with gradient 3 and passing through (1, -2)
(1, -2)
Find the equation of the line that is
perpendicular
and passing through (1, -2)
Slide33Linear (straight line) graphs GCSE FM
Distance between 2 points?
We need PQ , QR and PR.
Which two are the same?
Use Pythagoras’ Theorem.
Two equal lengths so isosceles.
Slide34Simultaneous equations GCSE FM
Substitute y for x and hence solve the equations :
Slide35Simultaneous equations GCSE FM
Substitute y for x and hence solve the equations :
Slide36Simultaneous equations GCSE FM
Slide37Simultaneous equations GCSE FM
Substitute y for x and hence solve the equations :
Slide38Simultaneous equations GCSE FM
Substitute y for x and hence solve the equations :
Slide39Simultaneous equations GCSE FM
Substitute y for x and hence solve the equations :
x
= 3 and y = 2
x
= 3 and y = -2
x
= 3 and y = 3
x
= -3 and y = -3
x
= 1 and y = 2
x
= -1 and y = 0
Slide40Pythagoras 3D GCSE FM
Find : (i) AD (ii) CE (iii) AC
A
B
C
D
E
F
Slide41Linear (straight line) graphs GCSE FM
1. Find the equation of the line with gradient ½ and passing through (-4, 6)
2. Find the equation of the line
perpendicular
to y = 2x - 1 and passing through (2, 3)
Slide42Sequences GCSE FM
Find the nth term for this quadratic sequence….
1
, 0, -3, -8, -15….
n
th term = -n² + 2n
-1 -3 -5 -7..
-2 -2 -2 -2..
nth term = -n²….?
Subtracting -n² (adding n²) leaves
2,
4
, 6, 8, 1
0
….
Slide43Proof GCSE FM
Show that f(x) = x² - 4x + 5
Hence
explain
why f(x) > 0 for all values of x.
Let f(x) = (x – 2)² + 1
f(2) = (2 – 2)² + 1
= 0² + 1 = 1
f(-2) =
= -4² + 1 = 17
(
-2
– 2)² + 1
Slide44Proof GCSE FM
Show that f(x) = x² - 4x + 5
E
xplain
why f(x) > 0 for all values of x.
Let f(x) = (x – 2)² + 1
(x – 2)² + 1 = x² - 4x + 4 + 1
= x² - 4x + 5
Squaring
always
makes a positive value so adding 1 is still positive.
Slide45Proof GCSE FM
Show that f(x) = 9x²
Hence
explain
why f(x) is a square number.
Let f(x) = 2x³ - x²(2x – 9)
2x³ - 2x³ + 9x²
9x² = (3x)² so you are squaring 3x. This makes a square number
= 9x²
Slide46Proof GCSE FM
Show that f(n + 1) = n² + 2n + 1
Let f(n) = n²
for all positive integer values of n
f(n+1) = (n+1)²
= (n+1)(n+1)
= n² + 2n + 1
Slide47Proof GCSE FM
Show that f(n + 1) + f(n – 1) is always even
Let f(n) = n²
for all positive integer values of n
(n+1)²
+ (n-1)²
= (n+1)(n+1)+(n-1)(n-1)
= n² + 2n + 1 + n² - 2n + 1
= 2n² + 2
Slide48Proof GCSE FM
Proving something is even means you have to show it is in the
2 times table
(or a multiple of 2)
2n² + 2
2(n² + 1)
Since we are multiplying by 2, this must be even
Slide49Functions
2f(x) =
Let f(x) = x²
f(2) = (
2
)² =
4
f(-2) =
= 4
(-2)²
2x²
f(2x) =
= (2x)²
Slide50Using Functions
2f(x) =
Let f(x) = 3 – x. Sketch this graph
f(0) =
=
3
3- 0
2(3-x)
f(2x) =
= 6–2x
3-(2x)
= 3–2x
1 - f(x) =
1 - (3–x)
= -2 + x
= x - 2
What does this represent?
Slide51EXAM REVISION (FM)
Review
TEST 1
Review TEST 2
Recent
Ratio and percentages.
Ex 1A
and 1D
Equations of lines
Ex 3D, 5B, 5C
Simultaneous equations by substitution Ex 4B
Midpoints, length of line segments
Expanding brackets further
Circles and lines
(equation of a circle)
Gradients of lines
(including
perpendicular lines) Ex 3C
Proof (to be taught)
Ex 4G
3D Pythagoras
(including the diagonal of
a cuboid) Ex 6E (part only!)
Algebraic
fractions
Ex 1B, Ex 2C, 2D
Linear Sequences
Ex 4H
Solving quadratic equations by factorising Ex 4A q. 1 only
Factorising Ex 2A
(including
quadratics)
Quadratic Sequences
Ex 4I
Quadratic
graphs
Ex 3E
SURDS (including rationalising) Ex 1F and 1G
Limiting value of a sequence
Ex 4J
Rearranging formulae
Ex 2B
Algebraic
fractional equations
Ex 1C, Ex 2E
CHECK and REVIEW ALL HOMEWOR
K tasks.
Area of a triangle Ex 7A
Linear & Quadratic inequalities
Ex 4D, 4E.
Slide52FOCUS YOUR EXAM REVISION (FM)
Topic
CHAPTER 1Decimals, Fractions and percentagesEx 1 ASimplifying algebraEx 1BSolving equationsEx 1CRatiosEx 1DFurther algebra – expanding bracketsEx 1ESURDS Ex 1F and Ex 1G
Topic
Questions
Factorising
quadratics
Ex 2 A
Rearranging
formulae
Ex 2B and 2C
Simplifying
algebraic fractions
Ex 2D
Equations with fractions
Ex 2E