Warm-Up Session
Non-Calculator Paper
Wednesday, 24 May 2017
Non-Calculator Paper80 marks in 90 minutes
IF YOU FINISH EARLY
CHECK EVERYTHING!
You have made a silly mistake somewhere.
Redo some questions
READ THE QUESTION!
• Answer the question as asked.
• Give your answer as requested e.g. 3sf or 2dp – it is silly to loose these marks.
• Round only at the very end – especially important when using a calculator.
• Show your working out.
• Mistakes require only one line through – allow the examiner to give you marks.
• Diagrams are not drawn accurately.
Quick recap
Reciprocal: multiplicative inverse
Round to decimal places: count the digits after the decimal point
Round to significant figures: Count all the digits after the first digit over 0
Solve: Find the value of the letter
Estimate: Round to 1sf (usually!)
Explain/justify: Give reasons in a sentence
Hence: Use what you have done before
Co-ordinates: Along the corridor and up the stairs e.g. (3, 4)
Number
0.0453682
0.05 2 decimal places
0.045 2 significant figures
468.493628
468.49 2 decimal places
470 2 significant figures
NEED TO KNOW
4
1= 4
4
4=
2
2
6
6
13
13
1256
12561
6
27=
2
9
÷ 3
÷ 3
NEED TO KNOW
Fractions, Percentages, Decimals!
½ 50% 0.5
¼ 25% 0.251/10 10% 0.11/20 5% 0.05
50/100
25/100
10/100
5/100
You Cannot: You Can:
2
3+
3
4
8
12+
9
12DENOMINATORS MUST BE THE SAME WHEN WE ADD AND
SUBTRACT FRACTIONS !!!
2
3x3
4
4
5÷2
3
FRACTIONS
NEED TO KNOW
Squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
Cubes: 1, 8, 27, 64, 125
½ x ½ = ¼
Writing Simple Recurring Decimals as Fractions
Z
x = 0. 41 41 41 41....
100 x = 41. 41 41 41....
99 x = 41. 41 41 41
0. 41 41 41 -
41. 00 00 00
So 99 x = 41
x = 𝟒𝟏
𝟗𝟗
Finding Simple Percentages without a Calculator
Percentage Method
1% 100% ÷ 100 = 1%
10%
25%
50%
20%
5%
Increasing/Decreasing by a % without a calc
Increase £70 by 35%35% = 3 x 10% + 5%10% = 70 ÷ 10 = £730% = 3 x £7 = £21
5% = 10% ÷ 2= 7 ÷ 2 = £3.50
35% = 21 + 3.50= £24.50Increase: 70 + 24.50
= £94.50
Decrease £340 by 85%85% = 50% + 25% + 10%50% = 340 ÷ 2 = £170
25% = 50% ÷ 2= 170 ÷ 2 = £85
10% = 340 ÷ 10 = £3485% = 170 + 85 + 34 = £289
Decrease: 340 – 289= ££51
Find the percentages you need and add/subtract
96 = 2 × 2 × 2 × 2 × 2 × 3
= 25 × 3
Write 96 as a product of prime factors
96
3 32
2 16
2 8
2 4
2 2
INDICES
r3 x r4 =
e8 ÷ e3 =
(g4)3 =
r7
e5
g12
x-8 ÷ x-3 =
(2g4)3 =
(2g2h3)3 =
x-5
8g12
8g6h9
INDICES
r0 =
8-1 =
8-2 =
1
𝟏
𝟖
𝟏
𝟔𝟒
81
3 =
82
3 =
𝟑𝟖
(𝟖𝟏𝟑)𝟐
SURDS
• √2 x √6 = √12 = 2√3
• √2 + √6 ≠ √8
• Rationalising Surds
4 x 2-√3 = 4 (2-√3) = 8 - 4√3
2+√3 2-√3 (2+√3)(2-√3)
0.0007261=
How can we write these numbers in standard form?
80 000 000 = 8 × 107
230 000 000 = 2.3 × 108
724 000 = 7.24 × 105
7.261 × 10-4
0.003152 = 3.152 × 10-4
Standard form
NEED TO KNOWPROPORTION
Y = kX
Y = kX2
Y = kX
Y = kX2
Find the highest common factor (HCF) and lowest common multiple (LCM) of 44 and 60.
4460
LCM = The product of all numbers in the diagram
HCF = The product of the numbers in the middle
Step 1: Use a factor tree to break down the numbers into theirprime factors.
Step 2: Organise the numbers into a venn diagram .
BearingsMeasured from North, Clockwise and Three Figures
048
N
AB
Averages and spreadMode
The mode is the most common
or most popular thing
Median
The middle value when the numbers
are in order
Mean
𝑠𝑢𝑚 𝑜𝑓 𝑣𝑎𝑙𝑢𝑒𝑠
𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑣𝑎𝑙𝑢𝑒𝑠
Range
𝑔𝑟𝑒𝑎𝑡𝑒𝑠𝑡 𝑣𝑎𝑙𝑢𝑒 − 𝑠𝑚𝑎𝑙𝑙𝑒𝑠𝑡 𝑣𝑎𝑙𝑢𝑒
Algebra
NEED TO KNOW
8, 11, 14, 17, ……
What is the 22nd term?
3n + 5
3(22) + 5 = 71
Expand & Factorise
𝟒(𝒅 − 𝟑) = 𝟒𝒅 – 𝟏𝟐
(𝒙 + 𝟑)(𝒙 + 𝟐) = 𝒙𝟐 + 𝟓𝒙 + 𝟔
“Solve”
𝟓(𝟑𝒙 − 𝟐) = 𝟓𝟎 𝑬𝒙𝒑𝒂𝒏𝒅
𝟏𝟓𝒙 − 𝟏𝟎 = 𝟓𝟎 (+𝟏𝟎 𝒕𝒐 𝒃𝒐𝒕𝒉 𝒔𝒊𝒅𝒆𝒔)
𝟏𝟓𝒙 = 𝟔𝟎 ÷ 𝟏𝟓 𝒕𝒐 𝒃𝒐𝒕𝒉 𝒔𝒊𝒅𝒆𝒔𝒙 = 𝟒
𝑪𝒉𝒆𝒄𝒌: 𝟓 𝐱 (𝟑 𝐱 𝟒 – 𝟐) = 𝟓𝟎
“Solve”5𝑥 + 2 = 3𝑥 + 7
5𝑥 + 2 < 3𝑥 + 7
𝑪𝒉𝒆𝒄𝒌 𝒊𝒕 𝒘𝒐𝒓𝒌𝒔!𝐼𝑛𝑡𝑒𝑔𝑒𝑟 = 𝑤ℎ𝑜𝑙𝑒 𝑛𝑢𝑚𝑏𝑒𝑟. .
2𝑥 = 5𝑥 = 2.5
2𝑥 < 5𝑥 < 2.5
Inequalities
n is an integer
−𝟐 < 𝟐𝒏 ≤ 𝟔
List all the possible values of n
Solution−𝟏 < 𝒏 ≤ 𝟑 Therefore 𝒏 = 0,1,2,3
Solving Quadratics
𝒙𝟐 + 𝟓𝒙 + 𝟔 = 𝟎
Factorise first: (𝒙 + 𝟐)(𝒙 + 𝟑) = 𝟎
Then solve: 𝒙 + 𝟐 = 𝟎 𝒙 + 𝟑 = 𝟎
𝒙 = −𝟐 𝒙 = −𝟑
Simultaneous Equations
𝟔𝒙 + 𝟐𝒚 = −𝟑𝟒𝒙 − 𝟑𝒚 = 𝟏𝟏
Same sign -Different +
Check your answer!
Using a table of valuesx
y = x2 – 3
–3 –2 –1 0 1 2 3
6 1 –2 –3 –2 1 6
The points given in the table are
plotted …
Remember (-3)2 = 9
y = (-3)2 – 3
y = (-2)2 – 3
y = (-1)2 – 3
y = (0)2 – 3
y = (1)2 – 3
y = (2)2 – 3
y = (3)2 – 3
x0–2 –1–3 1 2 3
–1
–2
1
2
3
4
5
yy
Recognising Graphs
Positive
Negative
Linear Quadratic Cubic
a
(-3,1)
Use (-4, 4) and (-2, -2)
−4 + −2
2,4 + −2
2
−4 − 2
2,4 − 2
2
Geometry
base
perpendicular height
Area of a parallelogram = bh Area of a trapezium = 1
2𝑎 + 𝑏 ℎ
perpendicular height
a
b
Area of Rectangle = b x h
h
bb
h
Area of Triangle = 𝑏ℎ
2
The circumference of a circle
C = πd3 cm
= 𝜋 × 3
= 3𝝅 cm
C = πd6 m
= 𝜋 × 12
= 12𝜋 m
The area of a circle
A = πr2
2 cm
= 𝜋 × 22
= 4𝜋 cm2
A = πr2
10 m= 𝜋 × 52
= 25𝜋 m2
𝐶 = 𝜋𝑑 or 𝐶 = 2𝜋𝑟 𝐴 = 𝜋𝑟2
Unit conversion
When converting to a smaller unit, multiply.
When converting to a larger unit, divide.
m cmx 100
m cm÷ 100
cm mmx 10
cm mm÷ 10
km mx 1000
km m÷ 1000
kg gx 1000
kg g÷ 1000
g mgx 1000
g mg÷ 1000
l mlx 1000
l ml÷ 1000
Unit conversion
Acute Obtuse Reflex
Less than 90° Between 90° and
180°
More than 180°
Straight line = 180⁰
Angles on a straight line
Angles around a point = 360⁰
Angles around a point
Angles in a quadrilateral (4-sided shape) add up to 360⁰
Angles in a Quadrilateral
Angles in a triangle add up to 180⁰
Angles in a Triangle
The two angles at the base are equal (the base is always the line without a stroke!)
Example:
All the angles in an Equilateral Triangle are equal (60°)
Example:
Special Triangles: Equilateral Triangle
Special Triangles: Isosceles Triangle
One angle is 90 degrees
Example:
Special Triangles: Right-Angled Triangle
All angles and side lengths are different
Example:
Special Triangles: Scalene Triangle
Polygons
Transformations
1. Reflection (2 marks) – State reflection and line of symmetry
2. Rotation (3 marks) – State rotation, centre of rotation, Degrees of rotation, Direction clockwise/ anti clockwise
3. Enlargement (3 marks) – State enlargement, centre of enlargement, scale factor
4. Translation (2 marks) – State translation and vector e.g. 2−4
SIMILAR SHAPES
Scale Factors are to the power of the dimensions
3cm
6cm2
6cm
24cm2
x 2
x 22
Example S.F. of 2
Length x (SF)1
Area x (SF)2
Volume x (SF)3
Prisms
Triangular-based prism Rectangular-based prism Pentagonal-based prism
Hexagonal-based prism Octagonal-based prism Circular-based prism
Cylinder
Cuboid
Prisms are 3 dimensional shapes that have a constant cross-sectional area
Volume of prisms
10cm
7cm
5cm
4cm
Volume = Area of cross-section x length
Cross section=trapezium
Area of trapezium = (7+5)x4 = 24cm2
2
Volume = 24x10=240cm3
Radius = 5cm
10cm
10cm
5cm
7cm
5cm2cm
4cm
Volume of a primsArea of the cross section x height
Pythagoras
8cm10cm
𝒙𝟐 = 𝟏𝟎𝟐 − 𝟖𝟐𝒙𝟐 = 𝟑𝟔𝒙 = 𝟔𝒄𝒎
x
3cm
4cm
x
𝒙𝟐 = 𝟑𝟐 + 𝟒𝟐
𝒙𝟐 = 𝟐𝟓𝒙 = 𝟓𝒄𝒎
Learn these two triangles
Z
1
1 2
45°
1
32
60°
Constructing a perpendicular bisector
This is the LOCUS of points that are the same distance from A as from B
Bisecting an angle
This is the LOCUS of points that are the same distance from AB as from BC
Z
Parallel Line Angles
Alternate angles are equal
a
b
a = b
Look for an
F-shape
Look for a
Z-shape
Corresponding angles are equal
a
b
a = b
Look for a
C- or U-shape
Interior angles add up to 180°
a
b
a + b = 180°
Statistics
1 4 3
1 7 9 6 6 8 5
2 1 3 3 4 4 0 2 1
2 8 6 9 5
Stem and Leaf DiagramsNow we need to put the leaves in numerical order
1 3 4
1 5 6 6 7 8 9
2 0 1 1 2 3 3 4 4
2 5 6 8 9
Key:
1 4 means 14
Key:
1 4 means 14
Median=21
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
190
200
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80
Age of member
Cu
mu
lati
ve f
req
uen
cy
0
10
20
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
190
200
0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80
Age of member
Cu
mu
lati
ve f
req
uen
cy
Cumulative Frequency graph.
We can now use this to find
the following information..
Median
Lower quartile
Upper quartile
Lowest Value
Highest Value
37
25
51
0
This information can now
be used to draw a box and
whisker diagram..
80
Interquartile range 51 - 2526
Probability
ProbabilityProbability of the target event happening =
Number of target outcomes
Total number of possible outcomes
Theoretical Probability = The expected probability if an experiment is fair
Mutually exclusive = outcomes
Two outcomes that cannot occur at the same time
Expected Frequency = Theoretical probability x number of trials
Exhaustive outcomes = All the possible outcomes of an event
P(outcome) + P(complementary outcome) = 1
1 – 30 minutes, easy marks only!2 – 30 minutes, harder but doable questionsTAKE A BREAK!!!!3 – go back over the (approx) 60 marks you’ve attempted4 – guess the rest! A blank page scores 0
You are as ready today as you can be.
Enjoy the test it is a chance to show how much you have learnt.