year 9 – end of year revision

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ζ Year 9 – End of Year Revision Dr Frost

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ζ. Dr Frost. Year 9 – End of Year Revision. Percentages. Be careful: Are you trying to find the new value or the old value? In the first case, you multiply , in the second case, you divide . Percentage change is based on the old value. - PowerPoint PPT Presentation

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Page 1: Year 9 – End of Year Revision

ζYear 9 – End of Year RevisionDr Frost

Page 2: Year 9 – End of Year Revision

PercentagesBe careful: Are you trying to find the new value or the old value? In the first case, you multiply, in the second case, you divide. Percentage change is based on the old value.

A jumper is bought in for £30 and marked up by 40%. What is it sold for?

Answer: 30 x 1.4 = £42

After one year the value of a care fell by 20% to £9600. What was its original value?

Answer: £12000

I put £15,000 into a savings account. It accrues 2.6% interest. What is in my account in one year’s time?

Answer: £15390

Lucy made 20% profit on the picture frame she sold at £35. What did she buy it in for?

Answer: £29.17

?

?

?

?

Page 3: Year 9 – End of Year Revision

Percentages

The interest rate for a savings account is 2.5% p.a. with compound interest. The principal is £1500. How much do I have in 10 years time?

Answer: £1500 x 1.02510 = £1920.13

My Bentley depreciates in value 10% each year. It is bought new for £150,000. How much is it worth in 5 years time?

Answer: £150,000 x 0.95 = £88573.50

? ?

Page 4: Year 9 – End of Year Revision

Compound Measures

A cat travels at 15km/s. It races around a 50km track. How much time did it take him?

Answer = 3.33s

The density of a hamster is 1.3kg/m3. Its volume is 0.03m3. What is the hamster’s mass?

Answer = 0.039kg

?

?

Page 5: Year 9 – End of Year Revision

GraphsMatch the graphs with the equations, and identify what type of equation it is.

1

2

3

4

5

6

7

8

9

10

11

y = -2x3 + x2 + 6xy = 4x

y = 2x - 3y = x2 + x – 2y = 5 – 2x2

y = 2x3

y = 5 – xy = x3 – 7x + 6y = -3x3

6 Cubic11 Exponential9 Straight Line1 Quadratic2 Quadratic

8 Reciprocal

5 Cubic10 Straight Line3 Cubic4 Cubic

7 Reciprocal

???????????

Page 6: Year 9 – End of Year Revision

Graphs

y = x3 – 2x2 - 5x + 6

x -3 -2 -1 0 1 2 3 4y -24 0 8 6 0 -4 0 18

When sketching, ensure you sketch a curvy line (i.e. don’t join up your points with lines), or you’ll lose a mark.

? ? ? ? ? ? ? ?

Page 7: Year 9 – End of Year Revision

Changing the SubjectChange the subject of the formula to the indicated letter.

??????????

(b)

Page 8: Year 9 – End of Year Revision

Changing the Subject

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Page 9: Year 9 – End of Year Revision

Changing the SubjectThe following require you to factorise at some point.

Make a the subject of the formula:

n = _3a_a+1

a = n2-PnP-1? ?

? ?

Page 10: Year 9 – End of Year Revision

Simultaneous Equations

You can either use elimination or substitution.

3x + 2y = 105x – 2y = 14

3x + 2y = 44x + 3y = 7

x = 3, y = 0.5 x = -2, y = 5? ?

Page 11: Year 9 – End of Year Revision

Probability

Question: Give there’s 5 red balls and 2 blue balls. What’s the probability that after removing two balls from the bag, we have a red ball and a blue ball?

R

B

R

B

R

B

57

27

46

26

56

16

?

?

?

?

?

?Answer =

1021?

Page 12: Year 9 – End of Year Revision

Probability

What’s the probability that when I roll 10 dice, I see the same number on every die? 𝒑 (𝒔𝒂𝒎𝒆𝒏𝒖𝒎𝒃𝒆𝒓 )= 𝟔

𝟔𝟏𝟎=𝟏𝟔𝟗

What’s the probability that when I roll 10 dice, the total of the dice is 10?

𝒑 (𝒔𝒖𝒎𝒊𝒔𝟏𝟎)= 𝟏𝟔𝟏𝟎?

Difficult: What’s the probability that when I roll 3 dice, I see exactly two sixes. 𝒑 (𝒕𝒘𝒐𝟔 𝒔 )=𝟏𝟓

𝟔𝟑 =𝟓𝟕𝟐?

Total outcomes

Matching outcomes

?

Page 13: Year 9 – End of Year Revision

6 29 3

Probability

If I have two dice, one numbered 1, 2, 3 and the other numbered 2, 3, 4, what’s the probability the sum is at least 5?

+ 2 3 4

1 3 4 5

2 4 5 6

3 5 6 7

Second DieFi

rst D

ie

p(sum ≥ 5) = =

?

?

Page 14: Year 9 – End of Year Revision

Sequences

Determine the formula for the following sequences.

5, 8, 11, 14, 17, ... 10, 8, 6, 4, 2, 0, -2, ...

3, 9, 17, 27, 39, ...

Un = n2 + 3n - 1

Un = 3n + 2 Un = 12 – 2n

1, 3, 6, 10, 15, ...

Un = 0.5n(n+1)

? ?

? ?

Page 15: Year 9 – End of Year Revision

Expanding brackets

Expand the following.

(x+1)(x-2) = x2 – x – 2(x-4)(x-8) = x2 – 12x + 32(x+1)(y+1) = xy + x + y + 1(x2+1)(y2-1) = x2y2 – x2 + y2 – 1(2x+1)(2x-1) = 4x2 – 1(x + 1)(x + y + 1) = x2 + xy + 2x + y + 1 x(y-x)-y(x-y) = y2 – x2

?

?

This is known as the: difference of two squares

?

??

??

?

Page 16: Year 9 – End of Year Revision

Factorisation

Factorise the following

x2 + 7x + 12 = (x + 4)(x + 3)x2 + 2x – 3 = (x – 1)(x + 3)x2 – 10x + 24 = (x – 4)(x – 6)2x2 – 5x – 12 = (2x + 3)(x – 4)12x2 + 5x – 3 = (4x + 3)(3x – 1)x2 – 9 = (x + 3)(x – 3)4 – y2 = (2 + y)(2 – y)x3 – x = x(x + 1)(x – 1)16x2y2 – 9z4 = (4xy + 3z2)(4xy – 3z2)x4 + 2x2 – 143 = (x2 + 13)(x2 – 11)

????????

??

Page 17: Year 9 – End of Year Revision

x -5 -4 -3 -2 -1 0 1 2 3 4 5 6

y4

3

2

1

-1

-2

-3

-4

Object

Enlarge the shape by a scale factor of 2 about the point (0,-2)

Enlargement

Image

Page 18: Year 9 – End of Year Revision

x -5 -4 -3 -2 -1 0 1 2 3 4 5 6

y4

3

2

1

-1

-2

-3

-4

Object

Enlarge the shape by a scale factor of -1 about the point (0,2)

Enlargement

Page 19: Year 9 – End of Year Revision

x -5 -4 -3 -2 -1 0 1 2 3 4 5 6

y4

3

2

1

-1

-2

-3

-4

Object

Enlarge the shape by a scale factor of -0.5 about the point (0,2)

Enlargement

Page 20: Year 9 – End of Year Revision

Trigonometry

60 ° x

12

30 °

4

xx = 13.96 x = 3.46

65 °x

15x = 6.99

? ?

?

Page 21: Year 9 – End of Year Revision

2

3

θ

1

3

1 1θ

6

θ8

a

b c

d

θ

Trigonometry

θ = 33.69° θ = 70.53°

θ = 45°

θ = 48.59°? ??

?

Page 22: Year 9 – End of Year Revision

Trigonometry

What is the cosine of the angle between the internal diagonal of a cube and the bottom face of the cube?

Answer = √2√ 3?

Page 23: Year 9 – End of Year Revision

Solving EquationsSolve the following equations for x.

x(2x + 1)(x – 2) = 0 x = 0 or -0.5 or 2x2 = 4 x = 2 or -2x2 = 3x x = 0 or 3x2 + 5x – 6 = 0 x = -6 or 1x3 = x x = -1, 0 or 1x2 + 32 = 12x x = 4 or 825x2 – 4 = 0 x = 2/5 or -2/5

???????

Page 24: Year 9 – End of Year Revision

2x + 2

x3x - 2

Determine x

Answer:By Pythagoras, x2 + (2x+2)2 = (3x-2)2

Expanding and simplifying, we get 4x2 – 20x = 0Solving, x = 5 (we reject the 0 solution).

Solving Equations

?

Page 25: Year 9 – End of Year Revision

Similarity

5

3

10 x

x = 16?

Page 26: Year 9 – End of Year Revision

Similarity

A square is inscribed in a right-angled triangle with length 4 and height 3. Find the width of the square.

3

4

Length of square = 127?

Page 27: Year 9 – End of Year Revision

Loci

AB

3km4km

A Spotted Studdert Sheep is known to be within 3km of A and 4km of B. What region could the sheep be in?

Page 28: Year 9 – End of Year Revision

Loci

AB

3km4km

Now the sheep is also known to be of equal distance from A and B. Where can it be?

Page 29: Year 9 – End of Year Revision

Loci

AB

3km4km

Now the sheep is within 3km of A, but at least 4km away from B. Where could it be?

Page 30: Year 9 – End of Year Revision

Loci

I’m equidistance from two lines AB and AC. Where could I be?

A

B

C

Page 31: Year 9 – End of Year Revision

Loci

I’m the indicated distance away from the walls of a building. Where could I be?

Circular corners.

Straight corners.

Page 32: Year 9 – End of Year Revision

Loci

My sheep is attached to a fixed point A on a square building, of 10m x 10m, by a piece of rope 20m in length. Both the sheep and rope are fire resistant. What region can he reach?

10m

20m A

Page 33: Year 9 – End of Year Revision

Dimensional Analysis

(all variables are lengths)

Expression Length Area Volume None of these

2rh

πr + 4h

(r+h)2

3b

b3 + rh

πr2(h + r)

bhr_ (b+h)

Click your choice.

? ? ?

? ? ?? ? ?

???? ? ?

? ? ?

??

??

??

Page 34: Year 9 – End of Year Revision

Ratio

My fish tank has black and yellow fish in the ratio 3:1. A fish plague, Sanjotitus, wipes out a third of my fish. I then restock my fish tank with just black fish, so that I have the same number of fish as before. What’s the new ratio of black to yellow fish?

Answer = 5 : 1?

Method 1:Suppose a full tank has 12 fish. Then 9 fish are black and 3 yellow. The plague leaves 6 black fish and 2 white. Then if we fill up the rest of the tank with black fish, we have 10 black fish and 2 yellow. This ratio is 5:1.

Method 2: of the tank is black and is yellow. The plague leaves us with of the fish, so the tank is now full of black fish and of yellow fish. Now the of the tank wiped out is replenished with black fish, so that’s black fish (and still yellow fish). This ratio is 5:1.

Page 35: Year 9 – End of Year Revision

Proportion

x 16 8 24y 10 5 15

Given that y is proportional to x, find the missing values.

?

?

Page 36: Year 9 – End of Year Revision

Inverse proportion

x 16 50 0.25L 5 2.83 40

Given that L is inversely proportional to √x, fill in the missing values in this table.

?

?

Page 37: Year 9 – End of Year Revision

Inequalities

Solve the following.

2x > x - 6

-x + 1 ≤ 6

x > - 6

x ≥ -5

?

?

1 ≤ 2x + 3 < 9 -1 ≤ x < 3?

− 𝑥2 ≤1 𝑥≥−2?

Page 38: Year 9 – End of Year Revision

Inequalities on a number line.

2 ≤ x < 4 x < -1 or x > 4

0 1 2 3 4 5

?

-1 0 1 2 3 4

?

Page 39: Year 9 – End of Year Revision

Inequalities on a number line.

2 ≤ x < 5x < 3 or x > 4

0 1 2 3 4 5

Draw the range of x on the number line given that both of these inequalities hold.

?

Page 40: Year 9 – End of Year Revision

-10 -8 -6 -4 -2 2 4 6 8 10

8

6

4

2

-2

-4

-6

-4 < y ≤ -2

Page 41: Year 9 – End of Year Revision

-10 -8 -6 -4 -2 2 4 6 8 10

8

6

4

2

-2

-4

-6

y ≤ x + 1 and x ≤ 6 and y > 2

Page 42: Year 9 – End of Year Revision

Inequalities

When would

When all of x, y and z are negative, or one of x, y and z are negative.

?

Page 43: Year 9 – End of Year Revision

Arcs and Sectors

5

Sector area = 10.91

Arc length = 4.36 Area = 20

Radius = 4.122.1cm

Sector area = 4.04cm2

Arc length = 3.85cm

?

?

??

?

50°

105°

135°

(Hint: Plug values into your formula and rearrange)

Page 44: Year 9 – End of Year Revision

The shape PQR is a minor sector.The area of a sector is 100cm2.The length of the arc QR is 20cm.

a) Determine the length PQ.

Answer: 10cm

b) Determine the angle QPR

Answer: 114.6°

P

Q

R?

?

Arcs and Sectors

Page 45: Year 9 – End of Year Revision

Volume of a prism

10cm

4cm

6cm

8cm

Volume = 400cm3 ?

1m

5m

3m

5m

6mVolume = 17m2 x 6

= 102m3 ?

Page 46: Year 9 – End of Year Revision

Surface Area

8m

4m

2m

Surface Area = 112m3?