Maths

LEAVING CERTIFICATE MATHS
HIGHER LEVEL
Power of
Maths
ACTIVITY BOOK
PAPER
1
SAMPLE
SECTION 1
SAMPLE
Tony Kelly and
Kieran Mills

Introduction
Introduction
Power of Maths is the first new Leaving Certificate Higher Level series since the full implementation
and examination of the new Maths syllabus. The new syllabus encourages teachers and students to
engage deeply with mathematical content. Power of Maths promotes this engagement by developing
students’ mathematical knowledge and problem-solving skills.
Optional activities are signposted in the margin of the Power of Maths Paper 1 Higher Level textbook.
The activities for Section 1 Higher Level are included in this Sample Activity Book. There are also
tables listing all of the activities that will be included in the complete Power of Maths Paper 1 Higher
Level Activity Book at the end of this Sample Activity Book.
The activities are designed to allow students to discover mathematical concepts on their own and to
acquire a deep understanding of the material. This enhances the enjoyment of the subject, allowing
students to learn complex material as effortlessly as possible.
Enjoy the experience of learning a beautiful subject.
Tony Kelly and Kieran Mills

1 Number
SECTION
Activity 1: Identifying composite and
prime numbers
OBJECTIVE: To investigate whether numbers are composite or prime
1. Insert these numbers into the Venn diagram below: 1, 7, 11, 15, 25, 32.
N
Primes
2. Complete the table below.
Natural number N
(a) 8
(b) 13
(c) 19
(d) 22
(e) 37
(f) 81
(g) 96
Divisors
Prime number
(tick)
Composite
number (tick)
(h) 111
3. Complete the statements with regard to prime numbers.
(a) All even natural numbers except _________________ are _________________.
(b) All natural numbers except _________________ with a last digit of 5 are _________________
because __________________________________________________________________.
(c) All natural numbers with a last digit of 0 are _____________________________________
because __________________________________________________________________.
1

Power of Maths: Paper 1 – Section 1
(d) A composite number is a natural number greater than _________________ that is not
_________________.
(e) With the exception of _________________, if the sum of the digits of a natural number is
divisible by 3, then the number itself is divisible by 3. Verify this for:
(i) 531: Sum of digits = , 531 = 3 ×
(ii) 5121: Sum of digits = , 5121 = 3 ×
(iii) 14 733: Sum of digits = , 14 733 = 3 ×
(iv) 5 162 154 627: Sum of digits = , 5 162 154 627 = 3 ×
These numbers are not prime because ______________________________________________
_____________________________________________________________________________.
Activity 2: Deciding which numbers
are prime
OBJECTIVE: To pick out the prime numbers from the first 100 natural numbers
The table shows all primes less than 100. Use all your tricks to cross out the obvious non-primes. Then
use Eratosthenes’ technique and your calculator to find the prime numbers.
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
2

Number
1
Activity 3: Working with prime factors
OBJECTIVE: To draw factor trees in order to work out prime factors
Prime factors
• A prime number is a natural number, excluding 1, which can only be divided by itself and 1 exactly.
• A composite number is a natural number greater than 1 with factors other than 1 and itself.
• 1 is neither prime nor composite.
1. You can make a factor tree to find the prime factors of a number.
Two factor trees for 36 = 12 × 3 are shown.
36
4
12 × 3
× 3 × 3
36
9 × 4
2 × 2
×
3
×
3
3 × 3 2 × 2
36 = 2 × 2 × 3 × 3
36 = 3 × 3 × 2 × 2
Complete the following factor trees and then write out the prime factors.
48
8 × 6
64
8 × 8
32
16 × 2
48 = × × × × 64 = × × × × × 32 = × × × ×
2. Three girls have the same birthday. The product of their ages is 36 and the sum of their ages is 13.
By completing the following factor table, find their ages. The youngest girl has brown eyes.
A B C Product Sum
(a) 36 1 1 36 38
(b) 18 2 1 36 21
(c) 12
(d) 9
(e) 9
(f) 6
(g) 6
(h) 4
3

Power of Maths: Paper 1 – Section 1
Activity 4: Working with rationals
OBJECTIVE: To write numbers in decimal form as rationals
Complete the following:
1. 0∙17 = 1 × _____ + 7 × ________ = ________ + ________ = ________
100
2. 3∙29 = 3 × _____ + 2 × _____ + 9 × ________ = ________ + ________ + ________ = ________
100
3. 0∙623 = 6 × _____ + 2 × _____ + 3 × ________ = ________ + ________ + ________ = ________
1000
4. −61∙2 = −6 × _____ − 1 × _____ − 2 × ________ = − ________ − ________ − ________ = − ________
10
5. 0∙101 = 1 × _____ + 0 × _____ + 1 × ________ = ________ + ________ = ________
1000
Activity 5: Working with number systems
OBJECTIVE: To understand to which set a particular number belongs
1. Write down two examples of the following:
(a) Natural numbers
(b) Positive integers
(c) Negative integers
(d) Integers
(e) Rational numbers which are positive and
not natural numbers
(f) Rational numbers which are also integers
(g) Irrational numbers
(h) Real numbers
(i) Real numbers which are rational numbers but not integers
(j) Numbers in the set Z \ N
4

Number
1
2. State which real numbers below are rational and which are irrational by ticking the appropriate box.
__
(a) √ 7
__
Rational Irrational
(b) √ 8
__
Rational Irrational
(c) √ 9
___
Rational Irrational
(d) − √ 16 Rational Irrational
(e) 3π
__
Rational Irrational
(f) 2 √ 2 Rational Irrational
(g) √ ___
___ 10
9
Rational
Irrational
(h) 1∙ 3 ˙
Rational Irrational
(i) 2⋅497612 Rational Irrational
(j) −2∙ ˙ 4 ˙ 7 Rational Irrational
(k) −4⋅721431 Rational Irrational
(l) 1⋅2 Rational Irrational
(m) 2⋅71314259... Rational Irrational
(n) 0 Rational Irrational
__
___
(o) √ 3
Rational
Irrational
7
3. Place the following numbers in their correct positions in the Venn diagram below.
__
___ 15
5, √ 5 ,
13 , 5·3, − __ 2 __
7 , 0· 3 ˙ , 0· 2 ˙ 9 ˙ , √ 4 , π, −3, 0, 1
R
Q
Z
N
5

Power of Maths: Paper 1 – Section 1
8__
4. Show, by long division, that
5 is a terminating decimal and that ___ 11
is a recurring decimal.
9
Check your answers on the calculator.
____
________
5 | 8·00 9 | 11·00000
Using your calculator [Model: Casio fx-85GT PLUS ], press the fraction button:
_____
Input the fraction and press equals.
To turn the fraction into a decimal press: S⇔D
Activity 6: Constructing lines of length
root 2 and root 3
OBJECTIVE: To learn how to use your compass and set square to draw lines with irrational lengths
__ __
Follow the instructions on page 7 to construct √ 2 and √ 3 on the same diagram.
O
–2 –1 0 1
2
6

Number
1
__
Constructing a line of length √ 2 :
1. Starting at O, draw a line of 1 unit along the number line going in the positive direction. Call this line OA.
2. Draw a line segment [AB] perpendicular to [OA] of length 1 unit at A.
B
3. Join O to B as shown to the right.
| OB | 2 2 2
= + = ⇒ | OB | =
1
4. Draw a circle of centre O and radius | OB | .
O A
5. Mark where the circle cuts the number line on the positive side as C.
1
__
(d) 1 2 to 0 (h) 1·2 to 0
__
| OB | = | OC | = √ 2 =
__
[Use the graph to calculate correct to one decimal place.]
Calculator: √ 2 = [Use the calculator to calculate correct to three decimal places.]
__
C
Constructing a line of length √ 3 :
1. Draw a line segment [BD] perpendicular to [OB] of length 1 unit.
1
2. Join O to D as shown to the right.
B
| OD | 2 =
2
+
2
= ⇒ | OD | =
3. Draw a circle of centre O and radius | OD | .
1
4. Mark where the circle cuts the number line on the positive side as E.
__
O 1 A
| OD | = | OE | = √ 3 = [Use the graph to calculate correct to one decimal place.]
__
Calculator: √ 3 = [Use the calculator to calculate correct to three decimal places.]
OBJECTIVE: To position various numbers on the number line
1.
−4 −3 −2 −1 0 1 2 3 4 5
Using the number line, find the distance of
(a) 5 to 0
__
(e) − 2 3 to 0
__
(b) −4 to 0
(f) − √ 2 to 0
(c) 3 to 0
(g) π to 0
Activity 7: Working with absolute values
7

Power of Maths: Paper 1 – Section 1
2. If | number | stands for the distance of the number to 0, complete the following:
(a) | 4 | =
(b) | −3 | =
(c) | − √
__
2 | =
(d) | −0·5 | =
(e) | −2·8 | =
(f) | 1__ 3 | =
(g) | − 2__ 5 | =
__
(h) | −3 √ 2 | =
Activity 8: Working with fractions,
decimals and percentages
OBJECTIVE: To work with fractions, decimals and percentages and, when presented with various shapes,
to write down the shaded area as a fraction, decimal and percentage
1. Write the following as percentages:
(a) 0·5 = %
(b) 0·25 = %
(c) 0·6 = %
(d) 0·48 = %
(e) 1·6 = %
(f) 7·65 = %
(g) 0·23 = %
(h) 2 = %
(i) 0·7 = %
(j) 0·18 = %
2. Write the following as decimals and then as fractions in their lowest form:
(a) 20% =
= _____
__
(e) 66 2
3 % = = _____
(b) 32% =
= _____
(f) 4% =
= _____
(c) 67% =
= _____
(g) 2·3% =
= _____
(d) 18% =
= _____
(h) 11% =
= _____
3. Calculate what percentage is:
(a) 5 of 20 = %
(b) 15 of 45 = %
(c) 10·2 of 27·2 = %
(d) 520 of 400 = %
8

Number
1
(e) 1 __
3 of 1 __
6 = %
(f)
1__
of 0·8 = %
5
(g) 0·8 of
1__
5 = %
(h) 2__ 3 of 5__
6 = %
(i) 0·2 of 0·3 = %
(j) 12·5 of 120 = %
4. Write the fraction that the shaded area is of the whole area as a rational number, as a decimal and
as a percentage.
(a) Circle
________
Rational
120°
Decimal
Percentage %
(b) Square
________
Rational
Decimal
Percentage %
(c) Regular hexagon
________
Rational
Decimal
Percentage %
(d) Rhombus
________
Rational
(e) Circles
6 cm
Decimal
Percentage %
________
Rational
4 cm
Decimal
Percentage %
9

Power of Maths: Paper 1 – Section 1
Activity 9: Using scientific notation
OBJECTIVE: To carry out a number of calculations in your head on numbers in scientific notation
1. Write the following in scientific notation:
(a) The number of stars in the Andromeda galaxy (the closest galaxy to our Milky Way galaxy):
1000000000000 =
(b) The mass of the Higgs boson (God particle):
0·0000000000000000000000002222 kg = kg
(c) The speed of light: 300 000 000 m/s =
m/s
(d) The population of China: 1 400 000 000 =
(e) The mass of the Earth: 6 000 000 000 000 000 000 000 000 kg =
kg
(f) A quintillion: 1 000 000 000 000 000 000 =
(g) A ZB (zettabyte) of memory: 1 000 000 000 000 000 000 000 =
Bytes
(h) Planck’s constant: 0·00000000000000000000000000000000062 J s =
J s
(i) The temperature at the centre of the Sun: 15 700 000 K =
(j) The diameter of the electron: 0·0000000000000056 m =
K
m
2. Write the following in the form a × 10 n , n ∈ Z, 1 ≤ a < 10.
(a) 0·023 =
(b) 1000 =
(c) 0⋅00014 =
(d) 2 =
(e) 0·0000243 =
(f) 12600000 =
(g) 0·000012 =
(h) 14 =
(i) 267 =
(j) 0·0000000057 =
10

Number
1
3. Write the following numbers, which are in scientific notation, in standard decimal form:
(a) 3·0 × 10 −2 =
(b) 4·12 × 10 −4 =
(c) 3·57 × 10 5 =
(d) 2 × 10 6 =
(e) 5·6 × 10 −7 =
(f) 6 × 10 −5 =
(g) 1 × 10 −4 =
(h) 1 × 10 5 =
(i) 3·3 × 10 0 =
(j) 7·8 × 10 8 =
4. Simplify the following without using your calculator. Give all answers in the form
a × 10 n , 1 ≤ a < 10, n ∈ Z.
(a) 2 × 10 3 × 3 × 10 2 =
(b) 4·2 × 10 5 × 3 × 10 −3 = =
(c) 7 × 10 3 × 8 × 10 5 = =
(d) 1·6 × 10 −19 × 6 × 10 23 =
(e) (3 × 10 8 ) 2 =
(f) 6 × 10 −26 × (3 × 10 8 ) 2 =
______
(g) 4 × 10 8
2 × 10 = 4
_______
(h) 1 × 10 5
2 × 10 = =
−3
(i)
6 × 1 0 −4 _______
3 × 1 0 −5 =
× = =
(j) 3 × 10 −2 + 2 × 10 −3 = + =
11

Power of Maths: Paper 1 – Section 1
Activity 10: Finding orders of magnitude
OBJECTIVE: To write down the orders of magnitude of given numbers
1. Write the following numbers in scientific notation and hence write down their order of magnitude.
(a) 2·3 = _________________ Order of magnitude:
(b) 35642 = _________________ Order of magnitude:
(c) 0·00013 = _________________ Order of magnitude:
(d) 4765314 = _________________ Order of magnitude:
(e) 0·0000563 = _________________ Order of magnitude:
(f) 730 × 10 4 = _________________ Order of magnitude:
(g) 0·056 × 10 −2 = _________________ Order of magnitude:
(h) 0·512 × 10 −7 = _________________ Order of magnitude:
(i) 1 × 10 5 = _________________ Order of magnitude:
(j) 6·4 × 10 −9 = _________________ Order of magnitude:
2. Fill in the boxes.
(a) Which is 5⋅6 nearer to, 1 or 10?
Its order of magnitude is .
(b) Which is 725 nearer to, 100 or 1000?
Its order of magnitude is .
(c) Which is 0⋅017 nearer to, 0·01 or 0⋅1?
Its order of magnitude is .
(d) Which is 9563 nearer to, 1000 or 10000?
Its order of magnitude is .
(e) Which is 37 nearer to, 10 or 100?
Its order of magnitude is .
12

Number
1
Activity 11: Understanding errors
OBJECTIVE: To understand how errors accumulate when results are added and subtracted
1. For l = (47·3 ± 0·1) cm and d = (34·7 ± 0·1) cm, complete the table.
Accumulated absolute error
in cm
Percentage error correct to
one decimal place
l + d =
cm
l − d =
cm
2. For m 1
= (32 ± 5) g and m 2
= (27 ± 5) g, complete the table.
Accumulated absolute error
in g
Percentage error correct to
nearest percentage
m 1
+ m 2
=
g
m 1
− m 2
=
g
3. For T 1
= (27·6 ± 0·2)°C and T 2
= (15·3 ± 0·2)°C, complete the table.
Accumulated absolute error
in °C
Percentage error correct to
nearest percentage
T 1
+ T 2
= °C
T 1
− T 2
= °C
4. The lengths l and d are measured in cm.
l = (4·2 ± 0·1) cm
l
d = (5·3 ± 0·1) cm
Evaluate the following, giving your answers with
their absolute errors.
d
(i) d + l =
(ii) d − l =
13

Power of Maths: Paper 1 – Section 1
5. The radius r 2
of the circular disc s 2
below is measured to be
(8·7 ± 0·1) mm. A circular disc of radius r 1
is removed. r 1
is measured
to be (3·8 ± 0·1) mm. O is the centre of both discs.
Find the width of d, giving your answer with the percentage error in d.
s 2
r 2
O
d =
s1
r 1
d
Absolute error in d =
Percentage error in d correct to the nearest percentage =
∴ d = ( ± % ) mm
Activity 12: Using approximation and
estimation
OBJECTIVE: To round off a variety of numbers to a certain number of decimals or significant figures
1. Evaluate using your calculator.
(a) π 2 correct to two decimal places
(b) (tan 18°) 2 correct to one decimal place
__
(c) 1 ___
3 + √ 2∙5 correct to one decimal place
____
(d) 3 √ 38∙3 + 2 0·5 correct to three decimal places
___
___________
(e) √ 7∙3 × 62∙56
correct to four decimal places
8∙62 3
2. (a) If A = π r 2 find A correct to one decimal place, if r = 3·7 cm. cm 2
(b) If u =
____ vf
, find u correct to two decimal places if f = 23·4 and v = 43·8.
v − f
(c) If f = 1 __
2l √ __ T __μ
, find f correct to the nearest whole number if l = 0·503,
T = 9·8 and μ = 0·00032
(d) If x = y 3 , find y correct to three decimal places if x = 27·32.
(e) If A = π r 2 + 2π rh, find A correct to one decimal place if r = 5·2 and h = 11·47.
14

Number
1
3. Round off the following:
(a) 479357 to four significant figures
(b) 1·3274 to three significant figures
(c) 57·321 to four significant figures
(d) 6·023 to two significant figures
(e) 0·0005734 to three significant figures
(f) 5·1795 × 10 −12 to two significant figures
(g) 6·00057 × 10 4 to three significant figures
(h) 0·056325 to three significant figures
(i) 7800 to three significant figures
(j) 82500 to two significant figures
4. Estimate the value of each of the following, giving a whole number answer in each case.
(a)
609 000 _______
14 500 = ≈
(b) 31·28 × 14·7 =
≈
(c) 21% of 5⋅32 =
≈
(d) 504·2 × (6·89 + 42·9) =
≈
____
√ 79∙3 × 3 ____
_____________ √ 1051
(e)
(3∙1) 2 = ≈
(f )
6∙1 × 1 0 3 × 4∙7 × 1 0 5
_________________
2∙9 × 1 0 4 × 2∙1 × 1 0 2 =
≈
(g) (2·9 × 10 8 ) 2 × (5·9 × 10 −12 ) =
________
_________
(h) √ 3∙9 × 10 4
2∙1 × 10 2 =
≈
≈
(i) 81% of 500 =
≈
(j)
____
__________ √ 0∙62 × 5∙1
(1∙9) 2 = ≈
15

Power of Maths: Paper 1 – Section 1
Activity 13: Understanding direct
proportion
OBJECTIVE: To draw a graph from a table of results and to answer questions that involve direct
proportionality
1. p, q, r, s are four straight lines through the origin. Complete the table below.
y
m = 3
p
q
r
s
m = 2
m = 1
1
m = –
2
x
The equation of
the line is
The constant of
proportionality is
Fill in the words
y is
p q r s
to x
y is
Fill in the symbol y x y x y x y x
to x
y is
to x
y is
to x
2. Write down a formula connecting the independent variable and the dependent variable in each of the
following graphs:
(a) P
(b) Q
Slope = 2·5
Slope = 3 × 10 −6
h
V
Formula:
Formula:
16

Number
1
(c)
R
(d) H
Slope = 4·8
Slope = 1·6
l
Q
Formula:
Formula:
3. The following table shows the relationship between the distance s in metres travelled by a body after
time t (seconds) at a steady speed.
t 1 2 3 4 5 6
s 10 20 30 40 50 60
(a) Complete the following table:
t 1 2 3 4 5 6
s 10 20 30 40 50 60
s_
t
(b) On the grid below plot s against t.
s(m)
Distance
60
50
40
30
20
10
1 2 3 4 5 6 7
Time
t(s)
(c) What can you say about the points you have plotted on the graph?
(d) Write down an equation connecting s and t.
Equation:
(e) What is s when t = 4·2 s?
s =
17

Power of Maths: Paper 1 – Section 1
(f) What is t when s = 35 m?
t =
(g) Complete the sentence: s is _________________________________________ proportional to t.
⇒ s =
⇒ s _
t =
× t
Activity 14: Understanding inverse
proportion
OBJECTIVE: To draw a graph from a table of results and to answer questions that involve inverse
proportionality
The following table shows the relationship between the frequency f (Hz) of a sound wave and its
wavelength λ (m).
k 2 4 6 8
f 180 90 60 45
1. Complete the table below:
k 2 4 6 8
f 180 90 60 45
f k
1__
k
2. On the grid below plot f against 1 __
λ .
f(Hz)
180
160
140
120
100
80
60
40
20
18
0·1 0·2 0·3 0·4 0·5 0·6 1
– (m –1 )
λ

Number
1
(a) What can you say about the points you have plotted on the graph?
(b) Write down an equation connecting f and 1 __
λ .
Equation:
3. What is f when λ = 10 m?
f =
4. What is λ when f = 50 Hz?
λ =
5. Complete the statement:
f is ____________________________________ proportional to λ.
__
∴ f =
× 1 λ ⇒ f λ =
19

Power of Maths: Paper 1 – Answers
Answers
20
Section 1
Number
Activity 1
1.
2.
N
1
Primes
7
Natural
number
N
Divisors
11
15
25
Prime
number
32
Composite
number
(a) 8 1, 2, 4, 8
(b) 13 1, 13
(c) 19 1, 19
(d) 22 1, 2, 11,
22
(e) 37 1, 37
(f) 81 1, 3, 9,
27, 81
(g) 96 1, 2, 3, 4,
6, 8, 12,
16, 24,
32, 48, 96
(h) 111 1, 3, 37,
111
3. (a) 2, composite ( b) 5, composite, they all are
divisible by 5 (c) composite, they are all even
(d) 1, prime (e) 3 (i) 9, 531 = 3 × 177
(ii) 9, 5121 = 3 × 1707 (iii) 18, 14 733 = 3 × 4911
(iv) 39, 5 162 154 627 = 3 × 1 720 718 209;
they are all divisible by 3
Activity 2
2, 3, 5, 7
11, 13, 17, 19
23, 29
31, 37
41, 43, 47
53, 59
61, 67
71, 73, 79
83, 89
97
Activity 3
1. 2 × 2 × 2 × 2 × 3 2 × 2 × 2 × 2 × 2 × 2
48 = 2 4 × 3, 64 = 2 6 ,
2.
2 × 2 × 2 × 2 × 2
32 = 2 5
A B C Product Sum
(a) 36 1 1 36 38
(b) 18 2 1 36 21
(c) 12 3 1 36 16
(d) 9 4 1 36 14
(e) 9 2 2 36 13
(f ) 6 6 1 36 13
(g) 6 3 2 36 11
(h) 4 3 3 36 10
Activity 4
1. ____ 17
100
Activity 5
2. ____ 329
100
3. _____ 623
1000
4. − ____ 612
10
2. (a) Irrational ( b) Irrational (c) Rational
(d) Rational (e) Irrational (f ) Irrational
( g) Irrational ( h) Rational ( i) Rational
( j ) Rational (k) Rational (l) Rational
( m) Irrational (n ) Rational (o) Irrational
3. R
Q
Z
5
··
0·29
4. 8 __
5
= 1·6
N
–3
5
0
5·3
11 ___
9 = 1· ˙ 2
15 __
13
4
2
– –
7
1
π
0·3·
5. _____ 101
1000
Activity 7
1. (a) 5 ( b) 4 (c) 3 (d)
2 1__ 2__
__
(e)
3 ( f) √ 2 ( g) π
__
(h) 1·2 2. (a) 4 (b) 3 (c) √ 2 (d) 0⋅5 (e) 2⋅8
(f )
3 1__ 2__
__
(g)
5 ( h) 3 √ 2

Number
1
Activity 8
1. (a) 50% ( b) 25% (c) 60% (d) 48% (e) 160%
(f ) 765% (g) 23% ( h) 200% (i) 70% ( j) 18%
2. (a) 0⋅2 = 1__ 8
5
( b) 0⋅32 = ___ 67
25
(c) 0⋅67 = ____
100
(d) 0⋅18 = ___ 9
50
(e) 0· 6 ˙ = 2__ 1
3
(f) 0⋅04 = ___
25
_____ 23
( g) 0⋅023 =
11
1000
(h) 0⋅11 = ____
100
3. (a) 25%
(b) 33 1_ 3 % (c) 37 1_ 2
% (d) 130% (e) 200% (f ) 25%
(g) 400% ( h) 80% (i) 66 2_ 5
3
% ( j) 10 __
12 %
4. (a) 1__
3 = 0· 3 ˙ = 33 1_ 1__
3
% ( b)
4
= 0·25 = 25%
(c) 1__
6 = 0·1 6 ˙ = 16 2_ 1__
3
% (d)
4
= 0·25 = 25%
(e) 5__
9 = 0· ˙ 5 = 55 5_ 9 %
Activity 9
1. (a) 1 × 10 12 ( b) 2⋅222 × 10 –25 kg (c) 3 × 10 8 m/s
(d) 1⋅4 × 10 9 (e) 6 × 10 24 kg (f ) 1 × 10 18
(g) 1 × 10 21 Bytes ( h) 6⋅2 × 10 –34 Js (i) 1⋅57 × 10 7 K
( j) 5⋅6 × 10 –15 m 2. (a) 2⋅3 × 10 –2 ( b) 1 × 10 3
(c) 1⋅4 × 10 –4 (d) 2 × 10 0 (e) 2⋅43 × 10 –5
(f ) 1·26 × 10 7 (g) 1⋅2 × 10 –5 ( h) 1⋅4 × 10 1
(i) 2⋅67 × 10 2 ( j) 5⋅7 × 10 –9 3. (a) 0⋅03
( b) 0⋅000412 (c) 357 000 (d) 2 000 000
(e) 0⋅00000056 (f ) 0⋅00006 (g) 0⋅0001 ( h) 100 000
(i) 3⋅3 ( j) 780 000 000 4. (a) 6 × 10 5 ( b) 1⋅26 × 10 3
(c) 5⋅6 × 10 9 (d ) 9⋅6 × 10 4 (e) 9 × 10 16 (f ) 5⋅4 × 10 –9
(g) 2 × 10 4 ( h) 5 × 10 7 (i) 2 × 10 1 ( j) 3⋅2 × 10 –2
Activity 10
1. (a) 2⋅3 × 10 0 , 0 ( b) 3⋅5642 × 10 4 , 4
(c) 1⋅3 × 10 –4 , –4 (d) 4⋅765314 × 10 6 , 6
(e) 5⋅63 × 10 –5 , –4 (f ) 7⋅3 × 10 6 , 7 (g) 5⋅6 × 10 –4 , –3
( h) 5⋅12 × 10 –8 , –7 (i) 1⋅0 × 10 5 , 5 ( j) 6⋅4 × 10 –9 , –8
2. (a) 10, 1 ( b) 1000, 3 (c) 0⋅01, –2 (d) 10 000, 4
(e) 10, 1
Activity 11
Accumulated
Percentage error
absolute error
1. l + d = 82 cm 0·2 cm 0·2%
l – d = 12·6 cm 0·2 cm 1·6%
2. m 1
+ m 2
= 59 g 10 g 17%
m 1
– m 2
= 5 g 10 g 200%
3. T 1
+ T 2
= 42·9 °C 0·4 °C 1%
T 1
– T 2
= 12·3 °C 0⋅4 °C 3%
4. (i) d + l = (9⋅5 ± 0⋅2) cm (ii) d – l = (1⋅1 ± 0⋅2) cm
5. d = 4⋅9 mm
Absolute error = ±0⋅2 mm Percentage error = ±4%
d = (4⋅9 ± 4%) mm
Activity 12
1. (a) 9⋅87 (b) 0⋅1 (c) 1⋅9 (d) 4⋅785 (e) 0⋅2639
2. (a) 43⋅0 cm 2 ( b) u = 50⋅24 (c) f = 174
(d) y = 3⋅012 (e) A = 459⋅7
3. (a) 4⋅794 × 10 5 = 479 400 (b) 1⋅33 × 10 0 = 1⋅33
(c) 5⋅732 × 10 1 = 57⋅32 (d) 6⋅0 × 10 0 = 6⋅0
(e) 5⋅73 × 10 –4 = 0·000573 (f) 5⋅2 × 10 –12
(g) 6⋅0 × 10 4 = 60 000 (h) 5⋅63 × 10 –2 = 0⋅0563
(i) 7⋅8 × 10 3 = 7800 ( j) 8⋅3 × 10 4 = 83 000
4. (a) 40 (b) 450 (c) 1 (d) 25 000 (e) 10 (f ) 500
(g) 540 000 (h) 1 (i) 400 ( j) 1
21

Power of Maths: Paper 1 – Answers
Activity 13
1.
p q r s
The equation of
the line is
The constant of
proportionality is
y = 3x y = 2x y = 1x y = 1__ 2 x
3 2 1 1__ 2
Fill in the words
directly
proportional
directly
proportional
directly
proportional
directly
proportional
Fill in the symbol y ∝ x y ∝ x y ∝ x y ∝ x
2. (a) P = 2⋅5h ( b) Q = 3 × 10 −6 V (c) R = 1⋅6l (d) H = 4⋅8Q
3. (c) All the points lie on a straight line through the origin of slope 10. (d ) s = 10t (e) 42 m (f ) 3⋅5 s
(g) directly, constant, constant
Activity 14
2. (a) All the points lie on a straight line through the origin of slope 360. (b) l = ____ 360
l
3. 36 Hz 4. 7⋅2 m 5. inversely, constant, consta nt
22

Activities in
Power of Maths Paper 1
Higher Level Activity Book
Activities
Section 1: Number
Chapter 1 : Number Systems
Activity 1
ACTION Identifying composite and prime numbers
OBJECTIVE To investigate whether numbers are composite or prime
Activity 2
ACTION Deciding which numbers are prime
OBJECTIVE To pick out the prime numbers from the first 100 natural numbers
Activity 3
ACTION Working with prime factors
OBJECTIVE To draw factor trees in order to work out prime factors
Activity 4
ACTION Working with rationals
OBJECTIVE To write numbers in decimal form as rationals
Activity 5
ACTION Working with number systems
OBJECTIVE To understand to which set a particular number belongs
Activity 6
ACTION Constructing lines of length root 2 and root 3
OBJECTIVE To learn how to use your compass and set square to draw lines with irrational lengths
Activity 7
ACTION Working with absolute values
OBJECTIVE To position various numbers on the number line
Chapter 2: Arithmetic
Activity 8
ACTION
OBJECTIVE
Working with fractions, decimals and percentages
To work with fractions, decimals and percentages and, when presented with various shapes,
to write down the shaded area as a fraction, decimal and percentage
23

Power of Maths: Paper 1
Activity 9
ACTION
OBJECTIVE
Activity 10
ACTION
OBJECTIVE
Activity 11
ACTION
OBJECTIVE
Activity 12
ACTION
OBJECTIVE
Activity 13
ACTION
OBJECTIVE
Activity 14
ACTION
OBJECTIVE
Using scientific notation
To carry out a number of calculations in your head on numbers in scientific notation
Finding orders of magnitude
To write down the orders of magnitude of given numbers
Understanding errors
To understand how errors accumulate when results are added and subtracted
Using approximation and estimation
To round off a variety of numbers to a certain number of decimals or significant figures
Understanding direct proportion
To draw a graph from a table of results and to answer questions that involve direct
proportionality
Understanding inverse proportion
To draw a graph from a table of results and to answer questions that involve inverse
proportionality
Section 2: Algebraic Expressions
Chapter 3: Working with Algebraic Expressions
Activity 1
ACTION
OBJECTIVE
Activity 2
ACTION
OBJECTIVE
Activity 3
ACTION
OBJECTIVE
Activity 4
ACTION
OBJECTIVE
Activity 5
ACTION
OBJECTIVE
Writing out the terms and coefficients in algebraic expressions
To recognise the terms and coefficients in algebraic expressions
Understanding the meanings of algebraic expressions
To write an algebraic expression in its simplest form
Multiplying algebraic terms
To understand the processes involved in multiplying algebraic terms, including
commutativity and associativity
Combining terms using magic squares
To use magic squares to combine terms in algebraic expressions
Expanding brackets and combining like terms
To multiply out brackets term by term and then combine terms
24

Activities
Activity 6
ACTION
OBJECTIVE
Activity 7
ACTION
OBJECTIVE
Activity 8
ACTION
OBJECTIVE
Activity 9
ACTION
OBJECTIVE
Activity 10
ACTION
OBJECTIVE
Activity 11
ACTION
OBJECTIVE
Activity 12
ACTION
OBJECTIVE
Finding the value of an algebraic expression
To find the value of an expression when given a value of a variable
Working with the binomial theorem
To explore every aspect of the binomial theorem
Expanding binomials
To apply the binomial theorem to expand a number of binomials raised to a certain power
Picking out terms in a binomial expansion
To show you understand the binomial by picking out individual terms in the expansion
Factorising algebraic expressions
To use your techniques to revise factorising
Factorising the difference of two cubes using a geometrical approach
To arrive at the formula for the difference of two cubes by a geometric approach
Algebraic modelling
To warm up with some solid basics before getting down to the business of algebraic
modelling
Chapter 4: Polynomial and Rational Expressions
Activity 13
ACTION
OBJECTIVE
Activity 14
ACTION
OBJECTIVE
Activity 15
ACTION
OBJECTIVE
Activity 16
ACTION
OBJECTIVE
Activity 17
ACTION
OBJECTIVE
Recognising the different types of polynomial expressions from their equations and graphs
To explore linear, quadratic and cubic equations and graphs
A modelling approach involving the multiplication of terms
To use a practical problem involving the multiplication and combination of algebraic terms
Recognising patterns when multiplying brackets
To understand that when each bracket has its terms lined up in descending powers,
multiplying the first terms by the first terms yields the first term in the answer. The same
idea applies to the last terms.
Dividing polynomials
To follow the procedure step by step for dividing polynomials
Adding algebraic fractions
To follow the procedure of finding a lowest common denominator (LCD) and simplifying
algebraic expressions
25

Power of Maths: Paper 1
Activity 18
ACTION
OBJECTIVE
Activity 19
ACTION
OBJECTIVE
Activity 20
ACTION
OBJECTIVE
Multiplying and dividing algebraic fractions
To factorise and cancel brackets in order to simplify fractions that are multiplied and divided
Reducing algebraic fractions to their lowest form
To reduce algebraic fractions by factorising and cancelling
Simplifying double-decker fractions
To simplify more complicated fractions called double-deckers
Chapter 5: Exponentials and Logs
Activity 21
ACTION
OBJECTIVE
Activity 22
ACTION
OBJECTIVE
Activity 23
ACTION
OBJECTIVE
Activity 24
ACTION
OBJECTIVE
Activity 25
ACTION
OBJECTIVE
Activity 26
ACTION
OBJECTIVE
Activity 27
ACTION
OBJECTIVE
Activity 28
ACTION
OBJECTIVE
Activity 29
ACTION
OBJECTIVE
Activity 30
ACTION
OBJECTIVE
Learning the multiplication rule for exponentials
To understand that when numbers to the same base are multiplied, you add their powers
Learning the division rule for exponentials
To understand that when numbers to the same base are divided, you subtract their powers
Learning the power of a power rule
To understand that when numbers to a power are raised to a power, you multiply their powers
Learning about powers of products and quotients
To manipulate more complicated expressions raised to a power
Working with negative powers
To learn how to work with negative powers and the procedure for turning them into positive
powers
Working with fractional powers
To follow the procedure for working step by step with fractional powers
Recognising surds
To identify whether numbers are surds or rational numbers
Simplifying surds
To break surds down into their simplest form
Adding surds
To combine surds into their simplest form
Multiplying surds
To multiply and break down surds to their simplest form
26

Activities
Activity 31
ACTION
OBJECTIVE
Activity 32
ACTION
OBJECTIVE
Activity 33
ACTION
OBJECTIVE
Activity 34
ACTION
OBJECTIVE
Activity 35
ACTION
OBJECTIVE
Activity 36
ACTION
OBJECTIVE
Activity 37
ACTION
OBJECTIVE
Rationalising the denominators of surds
To rationalise by multiplying above and below by the conjugate of the denominator
Understanding logs
To develop a deeper understanding of logs
Getting out of logs (‘hooshing’)
To learn how to effortlessly turn logs into their exponential expressions
Adding logs
To apply the rules of logs to simplify the addition of logs to the same base
Subtracting logs
To apply the rules of logs to simplify the subtraction of logs to the same base
Multiplying logs by a number
To apply the rules of logs to simplify logs that have been multiplied by a number
Changing the base of logs
To learn how to go quickly from the log of one base to another
Section 3: Algebraic Equations
Chapter 6: Polynomial Equations
Activity 1
ACTION
OBJECTIVE
Activity 2
ACTION
OBJECTIVE
Activity 3
ACTION
OBJECTIVE
Activity 4
ACTION
OBJECTIVE
Interpreting linear equations
To write linear equations in the form y = ax + b and hence find the x and y intercepts
Learning techniques for modelling linear problems
To learn some basic techniques to aid the modelling of linear problems
Completing the square
To bring you through the steps for completing the square on a variety of quadratics
Using the quadratic formula
To practise using the quadratic formula by solving a number of equations
27

Power of Maths: Paper 1
Activity 5
ACTION
OBJECTIVE
Activity 6
ACTION
OBJECTIVE
Using techniques for modelling quadratic problems
To learn some basic techniques to aid the modelling of quadratic problems
Working with cubic equations
To learn the techniques needed to solve cubic equations
Chapter 7: Other Types of Equations
Activity 7
ACTION
OBJECTIVE
Activity 8
ACTION
OBJECTIVE
Activity 9
ACTION
OBJECTIVE
Activity 10
ACTION
OBJECTIVE
Activity 11
ACTION
OBJECTIVE
Activity 12
ACTION
OBJECTIVE
Activity 13
ACTION
OBJECTIVE
Activity 14
ACTION
OBJECTIVE
Working with surd equations
To learn the techniques needed to solve surd equations
Working with literal equations
To learn the techniques needed to solve literal equations
Working with exponential equations
To learn the techniques needed to solve exponential equations
Understanding e
To understand the nature of base e
Working with log equations
To learn the technique needed to solve log equations
Working with absolute values
To calculate the absolute values of various numbers using the number line
Working with absolute value equations
To learn the techniques needed to solve absolute value equations
Plotting graphs with absolute value functions
To draw graphs of absolute value functions and write down the linear equations
Chapter 8: Systems of Simultaneous Equations
Activity 15
ACTION
OBJECTIVE
Activity 16
ACTION
OBJECTIVE
Working with equivalent equations
To learn the techniques needed to solve linear simultaneous equations
Using simultaneous equations involving a linear and an equation of order two
To learn the techniques needed to solve linear and order two simultaneous equations
28

Activities
Chapter 9: Inequalities
Activity 17
ACTION Understanding basic inequality operations
OBJECTIVE To learn the techniques needed to carry out basic inequality operations
Activity 18
ACTION Working with linear inequalities
OBJECTIVE To learn the techniques needed to solve linear inequalities
Section 4: Sequences and Series
Chapter 10: Patterns
Activity 1
ACTION Completing the pattern
OBJECTIVE To recognise patterns and to fill in the missing terms in the sequence
Activity 2
ACTION Writing down terms from sequences
OBJECTIVE To write down specified terms from given sequences
Activity 3
ACTION Generating a rule from a sequence
OBJECTIVE To recognise patterns in sequences and to write down a rule for the sequence
Activity 4
ACTION Using the Fibonacci sequence
OBJECTIVE To learn to construct the Fibonacci sequence and to generate its general term
Chapter 11: Analysing Sequences and Series
Activity 5
ACTION
OBJECTIVE
Activity 6
ACTION
OBJECTIVE
Activity 7
ACTION
OBJECTIVE
Activity 8
ACTION
OBJECTIVE
Activity 9
ACTION
OBJECTIVE
Examining convergent and divergent sequences
To examine a number of sequences and to state whether they converge to a particular value
or diverge
Working with limits
To evaluate a limit (using a calculator)
Working with sequences and series
To write the corresponding series when presented with a sequence
Exploring partial sums of series
To calculate the partial sums of a number of series
Working with Σ
To practise using the Σ notation
29

Power of Maths: Paper 1
Chapter 12: Arithmetic Sequences and Series
Activity 10
ACTION
Understanding an arithmetic sequence
OBJECTIVE To understand an arithmetic sequence
Activity 11
ACTION Working with arithmetic sequences (1)
OBJECTIVE To write down the first term and common difference of a number of arithmetic sequences
Activity 12
ACTION Working with arithmetic sequences (2)
OBJECTIVE
Activity 13
ACTION
OBJECTIVE
Activity 14
ACTION
OBJECTIVE
To explore various arithmetic sequences and to gain an understanding of how these
sequences operate
Exploring consecutive terms in an arithmetic sequence
To practise a number of problems involving three consecutive terms in an arithmetic
sequence
Finding the sum of the first n natural numbers
To verify Gauss’ formula by adding up the first 10 natural numbers
Chapter 13: Geometric Sequences and Series
Activity 15
ACTION Understanding a geometric sequence
OBJECTIVE To gain an understanding of a geometric sequence
Activity 16
ACTION Working with geometric sequences (1)
OBJECTIVE To write down the first term and common difference of a number of geometric sequences
Activity 17
ACTION Working with geometric sequences (2)
OBJECTIVE To explore various geometric sequences and to gain an understanding of how these
sequences operate
Activity 18
ACTION Exploring consecutive terms in a geometric sequence
OBJECTIVE To practise a number of problems involving three consecutive terms in a geometric
sequence
Activity 19
ACTION Working with a geometric series
OBJECTIVE To add the terms in a geometric series and to come to an understanding of how a summing
formula simplifies the task
Activity 20
ACTION Working with the Σ notation
OBJECTIVE To become familiar with how the Σ notation works
30

Activities
Section 5: Financial Maths
Chapter 14: Financial Maths 1
Activity 1
ACTION
OBJECTIVE
Activity 2
ACTION
OBJECTIVE
Activity 3
ACTION
OBJECTIVE
Working with unit rates
To understand the idea of solving problems by finding the unit value first
Converting to different units
To understand the idea of solving problems by converting to different units
Calculating tax on earnings
To calculate the tax to be paid on a single person’s and a married person’s salary
Chapter 15: Financial Maths 2
Activity 4
ACTION
OBJECTIVE
Activity 5
ACTION
OBJECTIVE
Activity 6
ACTION
OBJECTIVE
Activity 7
ACTION
OBJECTIVE
Working with simple interest
To go through the steps to calculate the simple interest on a sum of money
Mastering the compound interest formula
To go through the steps to calculate the compound interest on a sum of money
Proving the amortisation formula
To use the amortisation formula to calculate the amount of annual repayments
Compiling a repayment schedule
To use the amortisation formula to compile a repayment schedule and draw a graph to
illustrate the debt and interest portions of the repayments over time
Section 6: Complex Numbers
Chapter 16: Complex Numbers 1
Activity 1
ACTION
OBJECTIVE
Activity 2
ACTION
OBJECTIVE
Asking: Is it real or imaginary?
To recognise whether numbers are real or imaginary
Exploring powers of i
To write various powers of i in their simplest form
31

Power of Maths: Paper 1
Activity 3
ACTION Understanding complex numbers
OBJECTIVE To examine statements about complex numbers and to decide if they are true or false
Activity 4
ACTION Working with real and imaginary numbers
OBJECTIVE To pick out the real and the imaginary parts of complex numbers and to put them into sets
Activity 5
ACTION Drawing Argand diagrams
OBJECTIVE To plot complex numbers on Argand diagrams and to explore the properties of these numbers
Activity 6
ACTION Adding and subtracting complex numbers
OBJECTIVE To carry out the operations and to explore their geometric meaning
Activity 7
ACTION Multiplying a complex number by a scalar
OBJECTIVE To investigate the effect of multiplying complex numbers by scalars (real numbers)
Activity 8
ACTION Exploring the modulus of a complex number
OBJECTIVE To find the moduli of complex numbers and see the effects by graphing on an Argand
diagram
Activity 9
ACTION Exploring the argument of a complex number
OBJECTIVE To find the argument of complex numbers in degrees and radians
Activity 10
ACTION Finding the argument and modulus of z (1)
OBJECTIVE To find the argument and the modulus of a complex number in the first quadrant
Activity 11
ACTION Finding the argument and modulus of z (2)
OBJECTIVE To find the argument and modulus of a complex number in the second quadrant
Activity 12
ACTION Exploring the conjugate of a complex number
OBJECTIVE To find the conjugate of complex numbers and to deduce some general results
Activity 13
ACTION Proving properties of conjugates
OBJECTIVE To prove certain conjugate properties by showing the left-hand side (LHS) is equal to the
right-hand side (RHS)
Activity 14
ACTION Proving modulus and conjugate properties
OBJECTIVE To prove certain modulus and conjugate properties by showing the left-hand side (LHS) is
equal to the right-hand side (RHS)
Activity 15
ACTION Exploring the geometric effect of multiplying numbers by powers of i (1)
OBJECTIVE To explore the effects of multiplying numbers by powers of i
32

Activities
Activity 16
ACTION Exploring the geometric effect of multiplying numbers by powers of i (2)
OBJECTIVE To explore more difficult examples of the effects of multiplying numbers by powers of i
Activity 17
ACTION Exploring the geometric effect of multiplying complex numbers
OBJECTIVE To explore the effects of multiplying complex numbers by each other
Activity 18
ACTION Exploring the geometric effect of dividing complex numbers (1)
OBJECTIVE To explore the geometric effects of dividing complex numbers by each other
Activity 19
ACTION Exploring the geometric effect of dividing complex numbers (2)
OBJECTIVE To explore more difficult examples of the geometric effects of dividing complex numbers by
each other
Chapter 17: Complex Numbers 2
Activity 20
ACTION
OBJECTIVE
Activity 21
ACTION
OBJECTIVE
Activity 22
ACTION
OBJECTIVE
Activity 23
ACTION
OBJECTIVE
Activity 24
ACTION
OBJECTIVE
Activity 25
ACTION
OBJECTIVE
Activity 26
ACTION
OBJECTIVE
Activity 27
ACTION
OBJECTIVE
Exploring the equality of complex numbers
To work with some simple examples involving the equality of complex numbers
Solving quadratic equations with complex numbers
To solve a variety of quadratic equations using the magic formula
Forming quadratic equations given the roots
To write down a quadratic equation given a variety of roots
Working in polar form
To write down the modulus and argument of complex numbers written in polar form
Writing complex numbers in polar and Cartesian form
To write complex numbers in polar and Cartesian form given the modulus and argument
Exploring the reference angles of complex numbers
To explore the relationship between four different complex numbers
Writing the polar form of complex numbers given in Cartesian form
To consider situations where you are given complex numbers in Cartesian form and are
asked to write them in their polar form
Changing complex numbers to general polar form
To change complex numbers to general polar form by adding 2nπ, n ∈ 0
, to the angle
33

Power of Maths: Paper 1
Activity 28
ACTION
OBJECTIVE
Activity 29
ACTION
OBJECTIVE
Activity 30
ACTION
OBJECTIVE
Working with de Moivre Objects (DMOs)
To apply your knowledge of DMOs to simplify all types of complex numbers
Using de Moivre’s theorem to work out powers of complex numbers
To use de Moivre’s theorem to work out powers of complex numbers and to solve equations
Proving trigonometric identities using de Moivre’s theorem
To use de Moivre’s theorem to prove trigonometric identities
Section 7: Functions
Chapter 18: Relations and Functions
Activity 1
ACTION
OBJECTIVE
Activity 2
ACTION
OBJECTIVE
Activity 3
ACTION
OBJECTIVE
Activity 4
ACTION
OBJECTIVE
Determining if curves are functions
To use the vertical line test to determine if curves are functions
Working with domains, ranges and functions
To write down the domains and ranges and state if the relation is a function
Knowing the type of function
To decide what type of function is present for given mappings of functions
Drawing inverse functions
To find inverse functions graphically
Chapter 19: Linear Functions
Activity 5
ACTION
OBJECTIVE
Activity 6
ACTION
OBJECTIVE
Plotting linear functions
To plot a number of linear functions in a given domain
Intersecting linear functions
To plot two linear functions and find their point of intersection both graphically and
algebraically
34

Activities
Chapter 20: Quadratic Functions
Activity 7
ACTION
OBJECTIVE
Activity 8
ACTION
OBJECTIVE
Activity 9
ACTION
OBJECTIVE
Activity 10
ACTION
OBJECTIVE
Plotting quadratic graphs
To draw a number of quadratic functions. (Calculator instructions are provided for
generating the function values.)
Sketching graphs given quadratic functions
To draw a quadratic graph by looking at the equation
Writing the equation of a quadratic function from its graph
To write the equation of a function by looking at its graph
Intersecting quadratic functions
To plot two functions (either a linear and quadratic or two quadratics) and find their points
of intersection both graphically and algebraically
Chapter 21: Cubic Functions
Activity 11
ACTION
OBJECTIVE
Activity 12
ACTION
OBJECTIVE
Activity 13
ACTION
OBJECTIVE
Plotting cubic functions
To draw a number of cubic functions
Intersecting cubic functions
To plot two functions (where at least one is cubic) and find their points of intersection
Transforming functions
To transform (shift) various functions
Chapter 22: Exponential and Logarithmic Functions
Activity 14
ACTION
OBJECTIVE
Activity 15
ACTION
OBJECTIVE
Activity 16
ACTION
OBJECTIVE
Plotting exponential functions
To draw a number of exponential functions
Intersecting exponential functions
To plot two functions (where at least one is exponential) and to find their points of
intersection
Plotting log functions
To draw a number of log functions and intersecting log functions
35

Power of Maths: Paper 1
Section 8: Differentiation
Chapter 23: Techniques of Differentiation
Activity 1
ACTION Investigating instantaneous rates of change
OBJECTIVE To investigate instantaneous rates of change by sliding a point closer and closer to another
point
Activity 2
ACTION Exploring first principles (1)
OBJECTIVE To find the slopes of various functions as a first step towards differentiating from first
principles
Activity 3
ACTION Exploring first principles (2)
OBJECTIVE To find the derivative of some simple functions from first principles
Activity 4
ACTION Proving the sum rule from first principles
OBJECTIVE To prove the sum rule from first principles
Activity 5
ACTION Preparing to differentiate
OBJECTIVE To manipulate algebraic expressions to get them ready to differentiate and to tidy up
afterwards
Activity 6
ACTION Differentiating algebraic expressions
OBJECTIVE To differentiate a number of algebraic expressions and to tidy up afterwards, writing them
with positive powers
Activity 7
ACTION Tidying up algebraic expressions
OBJECTIVE To learn to tidy up complex algebraic expressions that are the results of differentiating
Activity 8
ACTION Preparing to differentiate trigonometric functions
OBJECTIVE To manipulate trigonometric expressions to get them ready to differentiate and to tidy up
afterwards
Activity 9
ACTION Graphing inverse trigonometric functions
OBJECTIVE To reflect trigonometric functions in the line y = x to obtain their inverse functions
Activity 10
ACTION Preparing to differentiate exponential functions
OBJECTIVE To manipulate exponential expressions to get them ready to differentiate and to tidy up
afterwards
Activity 11
ACTION Preparing to differentiate log functions
OBJECTIVE To manipulate log expressions to get them ready to differentiate and to tidy up afterwards
36

Activities
Chapter 24: Applications of Differentiation
Activity 12
ACTION
OBJECTIVE
Activity 13
ACTION
OBJECTIVE
Activity 14
ACTION
OBJECTIVE
Activity 15
ACTION
OBJECTIVE
Activity 16
ACTION
OBJECTIVE
Activity 17
ACTION
OBJECTIVE
Activity 18
ACTION
OBJECTIVE
Activity 19
ACTION
OBJECTIVE
Exploring slopes of tangents
To calculate the slopes of a number of tangents to curves
Exploring increasing and decreasing curves
To examine a number of functions to determine where their curves are increasing and
decreasing
Understanding the nature of slopes
To explore the nature of slopes by finding their first and second derivatives
Sketching graphs of first and second derivatives
To sketch the graphs of the first and second derivative of functions
Sketching exponential curves
To sketch the graphs of exponential functions
Sketching log curves
To sketch the graphs of log functions
Exploring rates of change
To understand some simple ideas about rate of change
Exploring modelling and optimisation
To consider some basic ideas about modelling and optimisation
Section 9: Integration
Chapter 25: Techniques of Integration
Activity 1
ACTION Introducing the idea of integration (anti-differentiation)
OBJECTIVE To think backwards from the derivative to the original function
Activity 2
ACTION Using rational power integration 1
OBJECTIVE To think backwards from the derivative to the original function with functions involving
rational powers
Activity 3
ACTION Using rational power integration 2
OBJECTIVE To progress to more complex rational powers (a follow-on activity to Activity 2)
37

Power of Maths: Paper 1
Activity 4
ACTION
OBJECTIVE
Activity 5
ACTION
OBJECTIVE
Activity 6
ACTION
OBJECTIVE
Activity 7
ACTION
OBJECTIVE
Using more difficult rational power integration
To integrate rational powers of brackets with linear expressions
Using exponential integration
To think of functions which, when differentiated, give exponential functions
Using log integration
To think of functions which, when differentiated, give log functions
Using trigonometric integration
To think of functions which, when differentiated, give trigonometric functions
Chapter 26: Applications of Integration
Activity 8
ACTION
OBJECTIVE
Finding the area under curves
To use geometric and integration techniques to find the areas under curves
Section 10: Mathematical Induction
Chapter 27: Proof by Induction
Activity 1
ACTION
OBJECTIVE
Activity 2
ACTION
OBJECTIVE
Activity 3
ACTION
OBJECTIVE
Preparing series for mathematical induction
To explore some of the techniques that prove series by induction
Preparing divisibilities for mathematical induction
To explore some of the techniques that prove divisibilities by induction
Preparing inequalities for mathematical induction
To explore some of the techniques that prove inequalities by induction
38