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## 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

Solving 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

Infinite 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