linear higher: warm-up session - ark helenswood academy - non-calc warm up [2017].pdf · how can we...

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

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0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80

Age of member

Cu

mu

lati

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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.