# GCSE Further Maths (AQA) PowerPoint Presentation 2017-08-06 55K 55 0 0

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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

(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….?

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)

(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

Ex 4I

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

Area of a triangle Ex 7A

Ex 4D, 4E.

Slide52

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

Ex 2 A

Rearranging

formulae

Ex 2B and 2C

Simplifying

algebraic fractions

Ex 2D

Equations with fractions

Ex 2E