calderglen mathematics department

Calderglen Mathematics Department

Blue Course

Revision Sheets

Block F

BF1 Brackets, equations and inequalities

BF2 Pythagoras’ Theorem and Significant Figures

BF3 Scientific Notation, Indices and Surds

BF4 Statistics, graphs, charts and probability

BF5 Rotations and transformations

BF1 Brackets, Equations and Inequalities

BF1.1 I have revised the use of algebraic shorthand.

BF1.2 Substitution into expressions involving negative numbers.

1. Evaluate when and .

2. Evaluate when and .

3. Evaluate

when , and .

BF1.3 I can multiply out brackets of the form: ( )ax bx cy

Multiply out the brackets:

1. 2.

3. 4.

BF1.4 I can multiply out brackets of the form : ( )( )ax by cx dy


1. 2.

3. 4.

5. 6.

.

BF1.5 I can multiply out brackets of the form : 2( )( )ax b cx dx c


1. 2.

BF1.6 I can multiply out brackets in more complex expressions and gather

like terms.

Simplify:

1.

2.

BF1.7 I can solve equations that contain brackets.

Solve:

1.

2.

BF1.8 I can solve equations which contain fractions.

Solve:

1.

2.

.

.

BF1.9 I can solve inequalities which may contain a change of direction of

inequality sign.

Solve:

1. 2.

BF1.10 I can use equations and inequalities to make mathematical models

1. The shape shown below is made from a small rectangle cut from a larger

rectangle. If the shaded area is 87 square centimetres, find the value of x.

2. The heights of 10 plants were measured as 5cm, 3cm, 4cm, xcm, 5cm, 3cm,

xcm, xcm, 2cm, 5cm.

a) Write down an expression in x for the mean height of a plant.

b) If the mean height of the plants is greater than 3∙9cm.

Write down an inequality for the above information and solve it for x.

c) Explain your answer to b) in the context of this problem

BF2 Pythagoras and Significant Figures

BF2.1 I can use Pythagoras to find the length of a hypotenuse

Calculate the length of the missing side in each triangle:

a)

b)

c)

x

15

8

x

4∙5

20

x

12 12

BF2.2 I can use Pythagoras to find the length of a shorter side

Calculate the length of the missing side in each triangle:

a)

b)

c)

10

x

8

60

x

50

4.8

x

1∙4

BF2.3 I can use Pythagoras to solve problems

1. In the isosceles

triangle shown, find

the length of AB

2. For the kite shown, find

the length of the side

marked x.

3. Fairy lights are strung

across a river in the shape

of an isosceles triangle with

a base length of 60 metres.

If the length of the string of fairy lights is 90 metres, calculate the width of

the river to the nearest centimetre.

60m

16m

A

B C

27∙5cm

18cm

36∙5cm x

12

m

12∙5m

A B

BF2.4 I can use Pythagoras to find the distance between two coordinate

points

On squared paper plot these pairs of points and calculate the distance between them.

a) O ( 0 , 0 ) and M ( 8 , 6 ) b) P ( 1 , 2 ) and Q ( 9 , 8 ) c) R ( 3 , 6 ) and S ( 8 , -6 )

BF2.5 I can use the Converse of Pythagoras to prove or disprove that a

triangle is right angled.

1. Use the Converse of Pythagoras to decide which of these triangles are

right angled:

a)

b)

c)

4

3∙1

2.4

25

24

7

15

9

12

BF2.6 I can apply the theorem of Pythagoras to construct mathematical

models of real life situations.

1. A rope has to be fed through a pipe in the ground for the telephone wire to

be connected from the house to the telephone pole.

John has a 40 metre

long rope to complete

the job.

Is the rope long enough?

You must justify your answer with appropriate working.

2. A loop of rope is used to mark

out a triangular plot, PQR.

The loop of rope measures 24 metres.

Pegs are positioned at P and Q such that PQ is 10 metres.

The third peg is positioned at R such that QR is 8 metres.

Prove that angle PRQ = 90 .

Do not use a scale drawing.

P Q

R

Pipe

Telephone

Pole

30 m

25 m

House

House

3. The top of a crane is in the shape of a triangle, shown as PQR.

PQ = 39m, PR = 15m, RQ = 36m.

(a) Prove that angle PRQ is a right angle.

(b) Hence calculate the area of PQR .

(c) Calculate the length of altitude RM.

P M

39m

Q

R

36m 15m

BF2.7 I can round to a specified number of significant figures

1) Round each of these numbers correct to 2 sig fig.

a) 49483 b) 365∙4 c) 1∙789 d) 7∙77

2) Round each of these numbers correct to 1 sig fig.

a) 44 b) 6∙08 c) 0∙909 d) 17∙5

3) Complete the following calculations and give your answers correct to 3 sig

fig.

a) 17 ÷ 9

b) 7% of £125000

c) Find the circumference of a circle with diameter 4.15cm

4) A plane departs Newtown and flies 65 miles north followed by 40 miles

west, as shown, until it reaches Rivercity.

Calculate the direct distance from Newtown to Rivercity, giving your

answer to 3 significant figures.

Newtown

Rivercity

65 miles

40 miles

BF1 Scientific Notation, Indices and Surds

BF3.1 I can convert large and small numbers to and from scientific notation.

1. Write the following numbers in Scientific Notation:

a) 8,000 b) 70,000 c) 5,600 d) 72,000

e) 6,700,000 f) 8,250,000 g) 38,600,000 h) 6,700

i) 42,000,000 j) 3,810 k) 6,340 l) 700

m) 943 n) 32,000,000 o) 7,321 p) 627

q) 8,125 r) 720 s) 173,100,000 t) 15,562,000

u) 176,000,000 v) 324,000,000 w) 464,000 x) 17


a) 4 million b) 12 million c) 6∙5 million

d) 9½ million e) 8∙46 million f) 5¼ million

g) 68∙75 million h) 12¾ million i) 23∙648 million

3. Write the following numbers out in full:

a) b) c)

d) e) f)

g) h) i)

j) k) l)

m) n) o)

p) q) r)


a) 0.0071 b) 0.00024 c) 0.000031 d) 0.000057

e) 0.00076 f) 0.0241 g) 0.00382 h) 0.000711

i) 0.0000324 j) 0.00675 k) 0.000038 l) 0.00028

m) 0.0000629 n) 0.000054 o) 0.00000068 p) 0.0005002


a) b) c)

d) e) f)

g) h) i)

j) k) l)

m) n) o)

p) q) r)


a) 370,000 b) 5,620,000 c) 0.0024 d) 0.000721

e) 3,221,000 f) 0.000023 g) 172,130,000 h) 0.0000923


a) b) c)

d) e) f)

g) h) i)

BF3.2 I can solve problems involving multiplication and division of numbers

expressed in scientific notation with and without a calculator.

1. Using your scientific calculator calculate the following, leaving your

answer in scientific notation:

a) b)

c) d)

e) f)

g) h)

i) j)

k) l)

m) n)

o) p)

q) r)

s) t)

u)

v)

w)

x)

y)

z)

2. During the year 2009, twenty thousand „CHOCPOPS‟ were eaten every

minute. How many „CHOCPOPS‟ were eaten in total during 2009.

Give your answer in scientific notation.

3. The annual profit of a company was pounds for the year 2011.

What profit did the company make per second.

Give your answer correct to three significant figures.

4. The moon has an approximate mass of kilograms.

The planet Earth has a mass of 81 times that of the moon.

Calculate the mass of the Earth.


5. Large distances in space are measured in light years.

The Canis Major Dwarf galaxy is the closest galaxy to the planet Earth, a

distance of approximately 25,000 light years away.

A light year is approximately kilometres.

Calculate the distance of the galaxy to Earth in kilometres.


6. A newspaper report stated “Concorde has now flown miles.

This is equivalent to 300 journeys from the earth to the moon.”

Calculate the distance from the earth to the moon.

Give your answer in scientific notation correct to two significant figures.

7. The total number of visitors to an exhibition was .

The exhibition was open each day from 5 June to 29 September inclusive.

Calculate the average number of visitors per day to the exhibition.

BF3.3 I can use the rules of indices ,

and

abbba xkkx )( ‟ 0 1a and

1n

na

a

applying them to my

previous learning.

1. Using the rule simplify the following:

a) b) c)

d) e) f)

g) h) i)

j) k) l)

m) n) o)

p) q) r)


a) b) c)

d) e) f)

g) h) i)

j)

k)

l)

m)

n)

o)

p)

q)

r)


a) b) c)

d) e) f)

g) h) i)

j) k) l)

m) n) o)

p) q) r)

4. Using all of the above rules simplify the following:

a) b) c)

d) e) f)

g) h) i)

BF3.4 I know that ( )m

mnna a and can apply this knowledge in problems.

1. Write the following as surds.

3

4

3

4

2

3

3

2

3

1

3

4

3

2

2

3

4

1

2

1

3

1

2

1

)()()()()()()()()()()()(

100)(64)()8()(8)(27)(16)(

7)(13)(16)(4)(8)(25)(

27

82

4

35

2

13

3

2

8

11

2

1

10

rqponm

lkjihg

fedcba

2. Simplify each of the following

2

3

2

1

2

3

2

1

2

1

2

2

2

32

2

2

222

3

2

24

)12()()(

3)(

1)(

1)(

1)(

1)(

1)(

12)(

1)()()()1()(

x

xl

x

xxk

x

xj

xx

xix

xhxxx

g

xxfxx

xe

xx

xd

xxx

cxxxbxxa

BF3.6 I can simplify, add, subtract, multiply and divide surds.

1. (a) 5 5 (b) 2 2 (c) 3 5 (d) 6 2

(e) 3 6 (f) x y (g) 8 2 (h) 32 2

(i) 25 35 (j) 32 27 (k) 43 23 (l) 5 32

(m) 26 33 (n) 82 12 (o) 53 35 (p) 48 22

2. (a) 2(1 - 2) (b) 3(3 + 1) (c) 5(5 - 1)

(e) 2(3 + 6) (f) 23(8 + 1) (g) 3(6 - 28)

(i) 46(26 - 8) (j) 8(2 + 4) (k) 212(3 + 6)

3. (a) (2 + 3)(2 - 1) (b) (5 + 1)(25 - 4)

(d) (3 + 1)(3 - 1) (e) (2 + 5)(2 - 5)

(g) (2 - 4)(32 - 1) (h) (8 + 2)(8 + 1)

(j) (2 + 3)2 (k) (2 + 3)2

BF3.7 I can rationalise a surd denominator.

Rationalise the surd denominator

54

32)(

24

23)(

8

4)(

12

10

50

1

23

32

23

8

2

5

3

4

65

12

23

2

25

4

52

3

2

20

5

3

3

2

5

10

3

6

5

1

3

1

2

1

)()()()(

)()()()()()()(

)()()()()()(

utsrqpo

nmlkjih

gfedcb(a)

BF4 Stats, Graphs, Charts and Probability

BF4.1 I have revised my knowledge of: average (mean, median and mode) and spread

(range) including using Extended Frequency Tables and Cumulative Frequency

Tables.

1. Calculate the mean for each of the following data sets:

a) 11, 12, 14, 17, 17, 19 b) 21, 23, 23, 26, 36, 81

c) 0∙1, 0∙2, 0∙4, 0∙5, 0∙7, 0∙7, 0∙9 d) 12, 17, 9, 16, 22, 8, 17, 11, 12, 3

2. Calculate the median for each of the following data sets:

a) 5, 8, 4, 2, 1, 6, 3, 9, 7 b) 11, 21, 14, 16, 27, 9, 15

c) 11, 7, 8, 6, 4, 7, 3, 10 d) 1∙3, 1∙4, 0∙8, 1∙7, 2∙3, 1∙6, 0∙9, 1

3. Calculate the mode for each of the following data sets:

a) 11, 22, 13, 54, 11, 13, 31, 10, 13 b) 1∙7, 2∙1, 2∙3, 1∙4, 2∙1, 6∙0, 2∙8

c) 131, 210, 113, 124, 21, 120, 124

d) , , , , , , ,

4. Calculate the mean, median, mode and range for each of the following

data sets:

a) 107, 106, 93, 114, 106, 98 b) 5∙6, 2∙2, 4∙3, 4∙3, 5∙0, 4∙3, 37

c) 30, 32, 23, 41, 55, 36, 27, 30 d) 15, 15, 13, 14, 17, 16, 17, 17

5. Copy and complete each of the following tables, add a third column

and calculate the mean, median and mode.

a) b)

6. Copy and complete each of the following tables, add a cumulative

frequency column and calculate the median.

a) b)

BF4.2 I can construct and interpret: a pie chart and a scatter graph.

1. In a local government election, four candidates stood for election in the

Murraywood ward. There were 720 votes cast and the candidates received the

number of votes shown below:

T. Green 342 J. Black 186

R. White 102 K. Brown 90

Construct a pie chart which displays these results (calculate the angle of each

sector, clearly showing all working).

2. Some pupils in 2S2 sat a Literacy test and a Numeracy test. The results are

shown below in the table.

Pupil A B C D E F G H

Numeracy 22 18 8 30 22 14 18 26

Literacy 16 20 10 28 24 12 20 24

a) Display these results on a scattergraph.

b) Describe the correlation between the Literacy and Numeracy marks.

3. The pie chart shows the share of

the votes received by candidates

in the Gleniston constituency at

the general election in 2005.

A total of 30 960 people voted in

the Gleniston constituency.

How many people voted for the

Liberal candidate?

BF4.3 I have investigated the existence of discrete and continuous data.

Which of the following are examples of discrete data, and which are examples of

continuous data.

a) The number of red cars on a road.

b) The weight of a blue whale.

c) The height of a two year old child.

d) The number of people in 3S1 who like salt and vinegar crisps.

BF4.4 I can find: the five figure summary and interquartile range for a

sample and illustrate this information with a box plot.

Give the five figure summary and the interquartile range for each of the

following sets of data.

a) 13, 17, 25, 36, 39, 42, 51, 60

b) 6, 7, 12, 22, 35, 36, 38, 43, 51, 53, 62, 69, 71

c) 5, 9, 12, 15, 17, 23, 27

BF4.5 I can find the Standard deviation of a sample and use it as an

alternative measure of spread using both methods.

1. Use the formula in the following examples.

(a) Calculate the mean and standard deviation of

(i) 14, 15, 18, 20, 23, 18 (ii) 41, 45, 34, 45, 46, 47, 50

(b) The costs of a can of diet coke in 6 different shops are

67p, 69p, 60p, 54p, 58p, 54p

Calculate the mean and standard deviation of these costs.

(c) The prices of a bag of sugar in 6 different shops are

86p, 88p, 84p, 79p, 81p, 86p

Calculate the mean and standard deviation of these prices.

2. Use the formula

in the following examples.

a) Scientists are studying the differences between crocodiles and alligators.

The lengths of 6 crocodiles are recorded in feet. The results are shown below.

18∙2, 23∙0, 17∙3, 22∙0, 20∙8, 18∙1

Calculate the mean and standard deviation of these lengths.

b) Calculate the mean and standard deviation of 10 numbers where

Σ x = 180 and Σx2 = 3356

c) The cost of a printer in 6 different British shops is

£66, £55, £70, £53, £61, £55

Calculate the mean and standard deviation of these costs.

BF4.6 I can compare two sets of data using average and spread and

investigate the most appropriate measure of average in a given

context.

1.

2.

BF4.7 I can predict the number of desired outcomes given the probability of

an outcome occurring.

1.

2.

3.

4.

5.

BF5 Rotations and Transformations

BF5.1 I can describe the order of rotational symmetry of a shape.

1. Write down the order of rotational symmetry for each shape

a) b)

c) d)

2. For each diagram write down the smallest angle of rotation about the

centre of the shape so that it fits its outline.

BF5.2 I can create a shape by rotating a template around a point.

1. Copy the diagram and

complete the shape so that it

has quarter turn symmetry

about the dot.

BF5.3 I can translate points and shapes using displacement (translation)

vectors.

1. Plot the points A(-3, 4), B(3, 5) and C(4, -3) on a Cartesian (coordinate)

diagram and join them to form triangle ABC.

(i) On the same diagram show A’B’C’, the image of ABC under the

translation

.

(ii) Also on the same diagram show A’’B’’C’’ the image of ABC

under the translation .

2. P, Q and R have the coordinates (3, 4), (-2, 1) and (-3, -4) respectively.

State the translation which maps PQR onto P’Q’R’ where P’(1, 5), Q’(-4, 2)

and R’(-5, -3).

calderglen mathematics department

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