## TL;DR?

## WHAT IS BACKWARD FADING?

*'In backward faded worked examples, students are required to try to find a solution in the last step on problem 1, the last two steps on problem 2, and so on. In other words, students are required to continue the steps given to solve the problem.'*

Omitting an additional step in each worked example allows pupils to build up to independent problem solving, having seen (and worked through) each solution step multiple times.

This is in contrast to 'forward fading' where the fading occurs in the first step on problem 1, the first two steps on problem 2, and so on. In this setup, pupils are required to fill in the missing steps in the solution.

## why is backward fading beneficial?

According to Cognitive Load Theory, faded examples can assist pupils in developing more advanced problem solving skills. The gradual introduction of parts for pupils to complete lessens the cognitive demand of the task, enabling pupils to attend to a specific element of the problem solving procedure and develop strategies across the worked examples.

Backward fading is preferable to forward fading as the cognitive demand of the latter steps in problem solving solutions are typically lower than earlier steps. As a result the steps with a higher cognitive demand are shown multiple times (when using backward fading) before pupils are expected to complete this step themselves.

Backward fading is preferable to forward fading as the cognitive demand of the latter steps in problem solving solutions are typically lower than earlier steps. As a result the steps with a higher cognitive demand are shown multiple times (when using backward fading) before pupils are expected to complete this step themselves.

## WHAT ARE PROMPTS?

There are 'significant learning gains' of backward fading, which can be further built upon by replacing the faded steps with prompts, which you'll find examples of in the resources below.

Prompts direct the attention of the learner to the relevant information in the problem, which helps to foster the development of problem solving skills. Having faded the worked examples, pupils are attending to a limited number of aspects of solving the problem, and so the load imposed on the cognitive ability of the pupil is minimised, enabling pupils to fully process each step of the problem in turn. The best way for pupils to do this is to self-explain each step, but this is inadequately done by the majority of learners.

The use of prompts addresses these failings, making use of the freed up cognitive capacity by asking questions of pupils to aid them in identifying the important parts of the posed problem, as well as their applications within the given context.

Prompts direct the attention of the learner to the relevant information in the problem, which helps to foster the development of problem solving skills. Having faded the worked examples, pupils are attending to a limited number of aspects of solving the problem, and so the load imposed on the cognitive ability of the pupil is minimised, enabling pupils to fully process each step of the problem in turn. The best way for pupils to do this is to self-explain each step, but this is inadequately done by the majority of learners.

The use of prompts addresses these failings, making use of the freed up cognitive capacity by asking questions of pupils to aid them in identifying the important parts of the posed problem, as well as their applications within the given context.

## what's out there already?

'Backward fading' is something that I came across in my CPD with Complete Mathematics, and they come up quite regularly when I read a maths book. The problem is that there's not a great lot of detail in these books about what backward fading is, and/or where to find activities which incorporate backward fading.

If you Google 'backward faded examples' you get a return of (as of January 7, 2022) a worksheet from Chris McGrane, and links to research into backward fading. Craig Barton has pages on his web site which give key takeaways from research papers, in case you're interested in reading those, but the key message about backward faded examples is that 'gradually removing the later steps in a multi-step problem brings about significant learning gains' (Atkinson et al, 2003) when compared to example-problem pairs.

If you Google 'backward faded examples' you get a return of (as of January 7, 2022) a worksheet from Chris McGrane, and links to research into backward fading. Craig Barton has pages on his web site which give key takeaways from research papers, in case you're interested in reading those, but the key message about backward faded examples is that 'gradually removing the later steps in a multi-step problem brings about significant learning gains' (Atkinson et al, 2003) when compared to example-problem pairs.

## HOW MIGHT YOU USE THE RESOURCES?

The resources are free to use as you wish (obviously), but in many cases my suggestion would be to give pupils 5 minutes in silence to study the worked example and continue to work through the faded examples, following this up with a discussion to identify any misconceptions that have developed. I would progress the discussion through the sheet as far as pupils have worked, before setting them off on the sheet again and then on to a follow up exercise to develop greater levels of fluency.

## HOW DO I SUBMIT SOME RESOURCES THAT I'VE PUT TOGETHER?

When I wrote

I was wrong.

I'd like to invite everyone to submit whatever resources they've created, and I'll share them on this page. If you want to use a template, use one of these (three questions on 1-side, four questions on 2-sides, six questions on 1-side, follow-up activities) and email them to d.taylor3142 @ gmail.com (no space, obviously) or DM me on Twitter at @taylorda01.

If you want a 'how-to' video, this might be useful:

*my*Increasingly Difficult Questions, I felt that I needed to create them in order to share them. I wanted to be the one who made, uploaded, amended, ... did everything with them.I was wrong.

I'd like to invite everyone to submit whatever resources they've created, and I'll share them on this page. If you want to use a template, use one of these (three questions on 1-side, four questions on 2-sides, six questions on 1-side, follow-up activities) and email them to d.taylor3142 @ gmail.com (no space, obviously) or DM me on Twitter at @taylorda01.

If you want a 'how-to' video, this might be useful:

## SHOW ME THE RESOURCES!

OK... Here you go...

## Number AND RATIO & PROPORTION

**Adding and Subtracting Using Fractions**

by Dave Taylor

Solutions: a) 39/88, b) 9/20, c) 11/18, d) 8/15

Follow-up Activity: .pdf

Solutions: 29/45, b) 9/10, c) 3/4

**Calculations With Bounds**

by Dave Taylor

Solutions: a) 16, b) 15, c) 23

Follow-up Activity: .pdf

Solutions: a) 22, b) lower bound = 17.42 (2 d.p.),

upper bound = 18, c) 300 (to the nearest hundred)

**Contextual Fractions**

by Dave Taylor

Solutions: a) 288m, b) 150g, c) 750ml

Follow-up Activity: .pdf

Solutions: a) 2.4km, b) 40 chocolates, c) 24 biscuits

**Contextual Lowest Common Multiple**

by Dave Taylor

Solutions: a) 10:10, b) 14:30, c) 14:58, d) 13:37

Follow-up Activity: .pdf

Solutions: a) 120 seconds, b) 3 tins of hot dogs and 4 packets of hot dog buns, c) 24 pupils

**Coordinates and Ratio**

by Dave Taylor

Solutions: a) (32,8), b) (29,5), c) (19,12), d) (24,20)

Follow-up Activity: .pdf

Solutions: a) (26, 27.75), b) (14.5, 21), c) (3,0)

**Dividing Fractions**

by Bob Jackson

Solutions: a) 5/14, b) 5/14, c) 7/22, d) 3/28, e) 7/8, f) 27/28

**Electricity Readings**

by Dave Taylor

Solutions: a) £73.60, b) £266.38, c) £225.54, d) £144.54

Follow-up Activity: .pdf

Solutions: a) £206.46, b) £312.29, c) £894.52

**Exchange Rates**

by Connor Rollo

Solutions: a) $72, b) £74.56, c) 356.80 Lev, d) £200,

e) 476 Swiss Franc, f) £239.35

**Inverse Proportion**

by Dave Taylor

Solutions: .png

Follow-up Activity: .pdf

Solutions: a) y = 160/x³, b) y = 3, c) x = 100

**Inverse Proportion**

by Bob Jackson

Solutions: a) i) P = 28/Q, ii) P = 4, b) i) P = 36/Q, ii) P = 4,

c) i) P = 48/Q, ii) P = 16, d) i) P = 60/Q, ii) P = 4,

e) i) P = 20/Q, ii) P = 2.5, f) i) P = 40/Q, ii) P = 5

**Maximum Servings From a Recipe**

by Dave Taylor

Solutions: a) 25 biscuits, b) 24 pancakes, c) 10 servings,

d) 4 servings

**Money and Coins**

by Dave Taylor

Solutions: a) 4 coins, b) 5 coins, c) 4 coins, d) 5 coins

Follow-up Activity: .pdf

Solutions: a) (20p, 5p, 1p, 1p, 1p) or (10p, 10p, 5p, 2p, 1p) or (20p 2p, 2p, 2p, 2p) b) (20p, 5p, 2p, 1p) and (10p, 10p, 5p, 2p, 1p) = 2 ways, c) (50p, 10p, 2p, 1p) = 4 coins

**Money, Percentages and Ratio**

by Dave Taylor

Solutions: a) 8 : 3, b) 4 : 3, c) 5 : 3, d) 1 : 2

Follow-up Activity: .png

Solutions: a) 1 : 2, b) 4 : 5, c) 25 : 13, d) 30 : 17

**More Complex Calculation with Bounds**

by Dave Taylor

Solutions: a) 20.6, b) 24.2, c) 11, d) 7

Follow-up Activity: .pdf

Solutions: a) 99/61, b) 130/7 ≤ (a-b)/c < 70,

c) 13.95 ≤ ac/(b-c) < 39.29

**Multiplying Fractions in Context**

by Dave Taylor

Solutions: a) 2.5% decrease, b) 32% increase, c) 0.25% decrease, d) 25.44% increase

**Percentage Change**

by Dee Vijayan

Solutions: a) 25%, b) 20%, c) 12%, d) 10.8% (1d.p.),

e) 20%, f) 25%

**Percentage Profit**

by Bob Jackson

Solutions: a) 4.76%, b) 57.14%, c) 15.79%, d) 19.05%, e) 17.69%,

f) 26.52%

**Proportional Reasoning**

by Dave Taylor

Solutions: a) 2 hours 15 minutes, b) 2 hours 15 minutes,

c) 3 hours 45 minutes, d) 2 hours 40 minutes

Follow-up Activity: .pdf

Solutions: a) 1 hour 20 minutes, b) 1 hour 41 minutes,

c) 13 hours 20 minutes

**Rates in Context**

by Dave Taylor

Solutions: a) 260 minutes, b) 601 minutes,

c) 68 minutes, d) 93 minutes

Follow-up Activity: .pdf

Solutions: a) 208 minutes, b) 1694 minutes,

c) Machine B

**Ratio, Fractions and Percentages**

by Dave Taylor

Solutions: a) 30, b) 26, c) 72, d) 28

Follow-up Activity: .pdf

Solutions: a) 70, b) 155, c) 71

**Reverse Percentage Decrease**

by Dave Taylor

Solutions: a) £16 000, b) £17 000, c) £16 500, d) £16 200

Follow-up Activity: .pdf

Solutions: a) £30 000, b) £399 000, c) £18 500

**Reverse Percentages**

by Dave Taylor

Solutions: a) £495, b) £720, c) 750ml

Follow-up Activity: .pdf

Solutions: a) £480, b) £160,000, c) 480g

**Reverse Percentages Twice**

by Dave Taylor

Solutions: a) 90 minutes, b) 60 minutes, c) 80 minutes,

d) 40 minutes

Follow-up Activity: .pdf

Solutions: a) 50 minutes, b) £620, c) $3099.92

**Sharing Amounts in a Ratio**

by Dave Taylor

Solutions: a) 12:8, b) 15:5, c) 9:15, d) 6:30, e) 40:16, f) 16:20

**Using Inverse Proportion**

by Dave Taylor

Solutions: a) 4 hours, b) 5 hours, c) 6 hours,

d) 8 hours

Follow-up Activity: .pdf

Solutions: a) 3 hours 12 minutes, b) 6 hours, c) 1280 units

## ALGEBRA

**Completing The Square**

by Chloe Bennett

Solutions: a) (x+2)²+3, b) (x+3)²+8, c) (x+2)²+6,

d) (x+1)²+8, e) (x+4)²+9, f) (x+5)²+47

**Converting Recurring Decimals to Fractions**

by Dave Taylor

Solutions: a) 5/9, b) 6/11, c) 49/90, d) 14/33

**Differentiation From First Principles**

by Sam Blatherwick

Solutions: a) f'(x) = 8x, b) f'(x) = 6x², c) f'(x) = x, d) f'(x) = 2x - 1

**Direct Proportion**

by Bob Jackson

Solutions: a) i) P = 5Q, ii) P = 55 , b) i) P = 3Q, ii) P = 33,

c) i) P = 4Q, ii) P = 44, d) i) P = 2.5Q, ii) P = 27.5,

e) i) P = 1.5Q, ii) P = 16.5, f) P = 16Q, ii) P = 176

**Direct Proportion With Percentage Increase/Decrease**

by Dave Taylor

Solutions: a) 44%, b) 125%, c) 33.1%, d) 48.8%

**Equation of a Circle**

by Dave Taylor

Solutions: a) x²+y²=9, b) x²+y²=25, c) x²+y²=29, d) x²+y²=81

Follow-up Activity: .pdf

Solutions: a) x²+y²=29, not 21, b) Inside, c) (3,8) and (3,-8)

**Equation of a Line - Given Gradient and a Point on the Line**

by Chloe Bennett

Solutions: a) y = 2x - 7, b) y = 3x + 4, c) y = 3x - 26,

d) y = 8x - 19, e) y = 8x + 29, f) y = -7x - 26

**Equation of a Tangent to a Curve**

by Sam Blatherwick

Solutions: a) y = 6x - 4, b) y = 15x - 12,

c) y = 8 - 4x, d) y = (5x + 5)/2

**Equations of Perpendicular Lines**

by Dave Taylor

Solutions: a) y=(-1/2)x+8, b) y=(-1/4)x+4, c) y=17-2x, d) y=-(1/3)x+8

**The Factor Theorem**

by Dave Taylor

Solutions: a) (2x-1)(x-2)(x+4), b) (x-1)(x+1)(x+3),

c) (3x-1)(2x-1)(x+3), d) (3x+1)(x-5)(x+2)

Follow-up Activity: .pdf

Solutions: a) (3x-5)(2x+1)(x+2), b) a = 13, c) x = -3, x = 2 and x = 5

**Factorising Quadratics (ac method)**

by Chloe Bennett

Solutions: a) (2x+3)(x+2), b) (3x+2)(x+2), c) (5x-3)(x+3),

d) (2x-1)(x+3), e) (3x+5)(x+4), f) (3x-4)(x+4)

**Factorising Quadratics (grid method)**

by Chloe Bennett

Solutions: a) (2x+3)(x+2), b) (3x+2)(x+2), c) (5x-3)(x+3),

d) (2x-1)(x+3), e) (3x+5)(x+4), f) (3x-4)(x+4)

**Forming Equations and Ratio**

by Dave Taylor

Solutions: a) 1.5, b) 2.5, c) 3.2, d) 2

Follow-up Activity: .pdf

Solutions: a) 4, b) 2.8, c) 72

**Inverse Proportion**

by Bob Jackson

Solutions: a) i) P = 28/Q, ii) P = 4, b) i) P = 36/Q, ii) P = 4,

c) i) P = 48/Q, ii) P = 16, d) i) P = 60/Q, ii) P = 4,

e) i) P = 20/Q, ii) P = 2.5, f) i) P = 40/Q, ii) P = 5

**nth Term of Arithmetic/Linear Sequences**

by Kieran McConville

Solutions: E) 2n + 3, 1) 4n - 1, 2) 3n + 4, 3) 5n + 2, 4) 3n - 2,

5) 2n + 5

**nth Term of a Quadratic Sequence**

by Dave Taylor

Solutions: a) 2n²+3n+5, b) n²+3n+2, c) 4n²+2n-5, d) 2n²-2n+9

**Quadratic Sequences**

by Ranjit Kaur and Tara Atefi, St Paul's School for Girls

Solutions: a) 3n²+2n+5, b) 6n²+7n+1, c) 2n²+3n-4, d) n²+5n+3,

e) 5n²+3n-2, f) 7n²+3n+8, g) 9n²+2n-4, h) an²+bn+c

**Solving Linear Equations with One Bracket (A)**

by Zoe Nye

Solutions: a) x = 4/3, b) x = 5/2, c) x = 23/3, d) x = 6/5,

e) x = 47/5, f) x = 7/3

**Solving Linear Equations with One Bracket (B)**

by Zoe Nye

Solutions: a) x = 5, b) x = 1, c) x = 18, d) x = -1,

e) x = 12, f) x = 4

**Solving Linear Equations with Brackets**

by Dee Vijayan

Solutions: a) x = 5, b) x = 13, c) x = 18, d) x = -1,

e) x = 12, f) x = 4

**Solving Linear Equations with Brackets**

by Chloe Bennett

Solutions: Example) x = 4, a) x = 2, b) x = 3,

c) x = 7, d) x = 5, e) x = 3

**Solving Linear Equations with Unknowns on Both Sides**

by Dee Vijayan

Solutions: a) x = -1, b) x = -4, c) x = 7, d) x = -8,

e) x = -5, f) x = -7

**Solving Linear Equations with Unknowns on Both Sides**

by Richard Dare

Solutions: 1) x = 2, 2) x = 2, 3) x = 7, 4) x = 8, 5) x = 8,

6) x = 2, 7) x = 7, 8) x = 4

**Solving Linear Equations with Unknowns on Both Sides (with Brackets)**

by Richard Dare

Solutions: 1) x = 11, 2) x = 2, 3) x = 4, 4) x = 5,

5) x = 3, 6) x = -7, 7) x = 9, 8) x = 3, 9) x = 5, 10) x = 4,

11) x = -4, 12) x = -2

**Solving Quadratic Inequalities**

by Paul Rochester

Solutions: 1) x < -3, x > 1, 2) 5 < x < 7, 3) -3 ≤ x ≤ 8,

4) x ≤ 3, x ≥ 4, 5) -4 < x < 5, 6) x ≤ -9, x ≥ -7, 7) x < -1, x > 9,

8) -3 < x < 9

**Solving Simultaneous Equations**

by Alex Hughes

Solutions: a) x = 2, y = 14, b) x = 6, y = 24, c) x = 5, y = 10, d) x = 4, y = 12, e) x = 10, y = 20

a) x = 5, y = 1, b) x = 1, y = 2, c) x = 8, y = 3, d) x = 4, y = 3, e) x = 1, y = -2

**Solving Simultaneous Equations by Elimination**

by by Ranjit Kaur and Tara Atefi, St Paul's School for Girls

Equal Coefficients

Solutions: a) x = 7, y = 2, b) x = 8, y = 4, c) x = 5, y = 2,

d) x = 6, y = 3, e) x = 9, y = 5, f) x = 7, y = 4, g) x = 10, y = 8,

h) x = 9, y = 6

Non-equal Coefficients

Solutions: a) x = 1, y = 4, b) x = 5, y = 2, c) x = 1, y = 3, d) x = 5, y = 4, e) x = 5, y = 1,

f) x = 7, y = 3, g) x = 4, y = 3, h) x = 5, y = 2, i) x = 3, y = 4, j) x = 2, y = 4, k) x = -2, y = -1,

l) x = 7, y = 6

Using Algebra Discs

Solutions: a) x = 7, y = 2, b) x = 8, y = 4, c) x = 5, y = 2, d) x = 6, y = 3, e) x = 9, y = 5,

f) x = 7, y = 4, g) x = 10, y = 8, h) x = 9, y = 6

**Solving Simultaneous Equations with Equal Coefficients**

by Chloe Bennett

Solutions: 1. x = 7, y = 2,

2. x = 8, y = 4,

3. y = 5, x = 2

4. y = 6, x = 3

5. x = 7, y = 4

6. x = 10, y = 8

**Solving Simultaneous Equations with a Circle (Substitution)**

by Dave Taylor

Solutions: a) x = -5, y = -3 and x = 3, y = 5,

b) x = 4, y = 1 and x = -1, y = -4,

c) x = 7, y = 8 and x = -6 , y = -5,

d) x = 6, y = 1 and x = -1, y = -6

**Solving Simultaneous Equations with a Quadratic (Equality)**

by Dave Taylor

Solutions: a) x = -3, y = -5 and x = 1, y = -1,

b) x = -2, y = -3 and x = 1, y = 0,

c) x = -1.5, y = -3.5 and x = 2 , y = 7,

## GEOMETRY

**Angles in Regular Polygons**

by Dave Taylor

Solutions: a) 150°, b) 72°, c) 135°, d) 162°

Follow-up Activity: .pdf

Solutions: a) 228°, b) 72°, c) 20°

**Angles in Regular Polygons at a Point**

by Dave Taylor

Solutions: a) 132°, b) 105°, c) 76°, d) 81°

Follow-up Activity: .pdf

Solutions: a) 27°, b) 36°, c) 150°

**Area of a Circle**

by Alex Hughes

Solutions: a) 153.94cm², b) 314.16cm², c) 380.13cm², d) 50.27cm², e) 201.06cm², f) 50.27cm², g) 113.10cm²

**Area of a Triangle**

by Dee Vijayan

Solutions: a) 14cm², b) 22.5cm², c) 16cm², d) 15.75cm²,

e) 10cm², f) 31cm²

**Circumference of a Circle**

by Alex Hughes

Solutions: a) 21.99cm, b) 34.56cm, c) 50.27cm, d) 12.57cm,

e) 21.99cm, f) 50.27cm, g) 56.55cm

**Combining Average Speeds**

by Dave Taylor

Solutions: a) 10mph, b) 12mph, c) 11.25mph, d) 8mph

Follow-up Activity: .pdf

Solutions: a) 21mph, b) 20mph, c) 20.3km/h (to 1d.p.)

**Contextual Area of a Semi-Circle**

by Dave Taylor

Solutions: a) £142.20, b) £74.00, c) £45.00, d) £40.80

Follow-up Activity: .pdf

Solutions: a) £17.10, b) £67.80

**Contextual Area of a Triangle Using Sine**

by Dave Taylor

Solutions: a) 5000kg, b) 4156.8kg, c) 3129.6kg, d) 4925.5kg

Follow-up Activity: .pdf

Solutions: a) 13522.96kg, b) 13950kg

**Contextual Volume of a Sphere**

by Dave Taylor

Solutions: a) 7.82cm, b) 10.46cm, c) 8.77cm, d) 6.87cm

Follow-up Activity: .pdf

Solutions: a) 7.10cm, b) 8.56cm, 10.16cm

**Density, Mass and Volume in Context**

by Dave Taylor

Follow-up Activity: .pdf

Solutions: a) 49.8cm², b) 13.1g/cm³, c) 30.3

**Interior Angle Sum of Polygons**

by Dave Taylor

Solutions: a) 1080°, b) 1800°, c) 1440°, d) 1260°, e) 720°, f) 540°

**Matrix Multiplication - Combinations of Transformations**

by Dave Taylor

Solutions: a) A reflection in the line y = -x, b) A reflection in the y-axis, c) A reflection in the line y = x, d) A reflection in the x-axis

Follow-up Activity: .pdf

**Pythagoras' Theorem - Calculating the Hypotenuse**

by Alex Hughes

Solutions: a) 5, b) 10.82, c) 11.18, d) 13.89, e) 8.06, f) 6.32

**Right-Angled Trigonometry - Finding a Side Length, Unknown in the Numerator**

by Simon Job

Solutions: .pdf

**Right-Angled Trigonometry - Finding a Side Length, Unknown in the Denominator**

by Simon Job

Solutions: .pdf

**Sine or Cosine Rule**

by Dave Taylor

Solutions: a) 7.00cm, b) 10.09cm, c) 3.61cm, d) 3.11cm

Follow-up Activity: .pdf

Solutions: a) 7.70cm, b) 5.21cm, c) 3.61cm

(All solutions given to two decimal places)

**Speed, Distance and Time**

by Dave Taylor

Solutions: a) 3.64m/s, b) 3.10m/s, c) 3.96m/s

(All given to two decimal places)

Follow-up Activity: .pdf

Solutions: a) 3.07m/s (to 2 d.p.), b) 40mph , c) 4:09

**Using Exact Trigonometric Values**

by Dave Taylor

Solutions: a) x = 8, b) x = 2, c) x = 4root(3), d) x = 16

**Vector Arithmetic**

by Dave Taylor

Solutions: a) k = -9, b) k = -7, c) k = 3, d) k = 5

Follow-up Activity: .pdf

Solutions: a) k = -6, b) k = -3.5 , c) p = 1, r = -1

**Vector Geometry (Ratios)**

by Dave Taylor

Solutions: a) 2b - 1.8a, b) 4.8b - 0.4a, c) 3.75b - a, d) 3b - 3.2a

Follow-up Activity: .pdf

Solutions: a) 4.8b - 2a, b) 2.25c - a - 1.25b, c) 1.2a + 3b

**Volume of a Cuboid**

by Dave Taylor

Solutions: a) 9.6cm, b) 6cm, c) 6.66666...cm, d) 3.75cm

Follow-up Activity: .pdf

Solutions: a) 12cm, b) 10cm from the top of the cuboid, in any position on any face or edge

**Volume of a Cylinder**

by Bob Jackson

Solutions: a) 502.7cm³, b) 565.5cm³, c) 1131.0cm³, d) 1570.8cm³, e) 377.0cm³, f) 15393.8cm³

**Volume of Pyramids and Spheres**

by Dave Taylor

Solutions: a) 20.11cm, b) 32.17cm, c) 6.98cm, d) 19.88cm

Follow-up Activity: .pdf

Solutions: a) 24.43cm, b) 5.23cm, c) 5.86cm

## STATISTICS AND PROBABILITY

**Conditional Probability**

by Dave Taylor

Solutions: a) 0.5625, b) 0.51, c) 0.788, d) 0.0975

Follow-up Activity: .pdf

Solutions: a) 0.75, b) 0.824, c) 53/150

**Estimating Mean From Grouped Data**

by Dave Taylor

Solutions: a) 25.8, b) 27.75, c) 31.46, d) 35.4, e) 22.75, f) 46

**Expected Value Using Probability**

by Dave Taylor

Solutions: a) £80, b) £75, c) £31

Follow-up Activity: .pdf

Solutions: a) £110, b) £43

**Mean Average - Missing Value Problems**

by Dave Taylor

Solutions: a) 9, b) 88, c) 66, d) 69

Follow-up Activity: .pdf

Solutions: a) 7, b) 29, c) 42

**Mean of a Frequency Table**

by Diana Page at Wootton Academy Trust

Solutions: .pdf

Follow-up Activity: .pdf

Solutions: .pdf

**Mean of a Frequency Table (In Reverse)**

by Dave Taylor

Solutions: a) x = 6, b) x = 15, c) x = 2, d) x = 5

**Probability**

by Dave Taylor

Solutions: a) 60, b) 56, c) 60, d) 48

Follow-up Activity: .pdf

Solutions: a) 60, b) 80, c) 100

**Probability of Successive Events**

by Dave Taylor

Solutions: a) 0.3, b) 0.36, c) 0.425, d) 0.2946

Follow-up Activity: .pdf

Solutions: a) 0.315, b) 0.29, c) 1/6

**Tree Diagrams**

by Dave Taylor

Solutions: a) 10/21, b) 16/33, c) 91/380, d) 60/119

Follow-up Activity: .pdf

Solutions: a) 51/100, b) 3 toffees and 7 mints, c) 2/15

**Using Probabilities**

by Dave Taylor

Solutions: a) 120, b) 90, c) 80, d) 30

Follow-up Activity: .pdf

Solutions: a) 60, b) 24, c) 40

**Using Relative Frequencies**

by Dave Taylor

Solutions: a) No (154), b) Yes (164), c) Yes (272), d) No (489)

**Venn Diagrams and Equations**

by Dave Taylor

Solutions: .png

Follow-up Activity: .pdf

Solutions: a) 13/20, b) 19