A detailed breakdown is of the course content can be found here.

Video Solutions

Unit 1H March 2011:

Question 1 - Questionnaires

Question 2 - Scatter Graphs

Question 3 - Ratio and Percentages

Question 4 - Two Way Tables

Question 5 - Probability

Question 6 - Mean from a Table

Question 7 - Box Plots and Comparing Data

Question 8 - Problem Solving

Question 9 - Probability

Question 10a - Mean from a Table

Question 10b - Cumulative Frequency

Question 11 - Probability Trees

Question 12 - Stratified Sampling

Question 13 - Compound Interest

Question 14 - Histograms

Unit 1 Sample Assessment Higher:

Question 1: Median interval and Frequency Polygon

Question 2: Comparing Costs for flights

Question 3: Compare and Contrast Data in Stem and Leaf Diagram

Question 4: Questionnaires

Question 5: Scattergraph - Line of Best Fit

Question 6: Ratio

Question 7: Probability and Algebra

Question 8: Averages - Mean

Question 9: Compound and Simple Interest Comparison

Question 10: Histogram - Unequal class widths

Question 11: Random and Stratified Sample

Question 12: Applying Probability

Question 13: Probability Tree Diagram

Question 14: Capture Re-capture Method

Detailed Course Content

MODULE

CONTENTS

PRIOR KNOWLEDGE

OBJECTIVES

DIFFERENTIATION & EXTENSION

NOTES

1-1

Collecting data

Identify which primary data they need to collect and in what format, including grouped data, considering appropriate equal class intervals
Select and justifying a sampling scheme and a method to investigate a population
Collect data using various method, including observation, controlled experiments, data logging, questionnaires and surveys
Design and using data-collection sheets
Gather data from secondary sources
Design and using two-way tables
Deal with practical problems such as non-response or missing data
Use stratified sampling

An understanding of why data needs to be collected.
Experience of simple tally charts.
Experience of inequality notation and signs.
Basic Fractions for help when calculating stratified samples (link with Unit 2)

By the end of the module the student should be able to:
• Design a suitable question for a questionnaire (1.5)
• Understand the difference between: primary and secondary data; discrete and continuous data (1.1 – 1.8)
• Understand when appropriate to use a stratified sample
• Understand how to calculate a stratified sample
• Design suitable data capture sheets for surveys and experiments (1.4 – 1.6)
• Understand about bias in sampling (1.7)
• Choose and justify an appropriate sampling scheme, including random and systematic sampling (1.2, 1.3)
• Deal with practical problems in data collection, such as non-response, missing and anomalous data (Throughout Ch 1)

Carry out a statistical investigation of their own to include designing an appropriate means of gathering the data
An investigation into other sampling schemes, such as cluster and quota sampling

Students may need reminding about the correct use of tallies
Emphasise the differences between primary and secondary data
Discuss sample size and mention that a census is the whole population. In the UK the census is every year that ends in a ‘1’, so 2011 is the next census
If students are collecting data as a group, they should all use the same procedure
Emphasise that continuous data is data that is measured, eg temperature
Mayfield High data from coursework task can be used to collect samples and to make comparisons in following section
Use the fact that the numbers in each year group for the Mayfield High data to introduce stratified sampling techniques

1-2

Displaying data, charts and graphs

Draw and producing a wide range of graphs and diagrams
Interpret a wide range of graphs and diagrams and drawing conclusions

An understanding of the different types of data: continuous; discrete; categorical
Experience of inequality notation and symbol
Some basic fraction work for pie Charts (Link with Unit 2)
Use a protractor to measure and draw angles (Link with Unit 2 contents)

By the end of the module the student should be able to:
• Represent data as:
- Pie charts (for categorical data) (3.1, 3.2)
- Bar charts and histograms (equal class intervals) (3.4, 3.5)
- Frequency polygons (3.6)
- Stem and leaf diagrams (3.3)
• Choose an appropriate way to display discrete, continuous and
categorical data (3.2 – 3.10)
• Understand the difference between a bar chart and a histogram (3.5)
• Compare distributions shown in charts and graphs (3.2 – 3.10)

Carry out a statistical investigation of their own and use an appropriate means of displaying the results
Use a spreadsheet to draw different types of graphs
Collect examples of charts and graphs in the media which have been misused, and discuss the implications

Clearly label all axes on graphs and use a ruler to draw straight lines
Stem and leaf diagrams must have a key
Show how to find the median and mode from a stem and leaf diagram
Angles for pie charts should be correct to within 2°. Ask students to check each others’ charts
Make comparisons between previously collected data, eg Mayfield boys vs girls or Yr 7 vs Yr 8
Encourage students to work in groups and present their charts – useful for display work in classroom/corridors
Use Excel graph wizard

1-3

Averages

Find the mean for large data sets
Estimate the mean for large data sets with grouped data
Find the mode, median, mean and range of small data sets with discrete data
Relate summarised data to the initial questions
Compare distributions and making inferences, using the shapes of distributions and measures of average and spread

Knowledge of finding the mean, median, mode & range for small data sets
Stem and leaf diagram to find mode and median
An ability to order and find the mid-point of two numbers

By the end of the module the student should be able to:
• Find the mean of data given in an ungrouped frequency distribution (2.4)
• Find the mode, median and range for a small set of data (2.2, 2.10)
• Find the modal class for grouped data (2.8)
• Find the median by using , where n is the number of data (2.2, 2.8)
• Find the mean from a frequency table by using (Sigma notation for top sets only) (2.7)
• Use the mid point of equal class intervals to find an estimate for the mean of grouped data (2.8)
• Know the advantages/disadvantages of using the different measure of average (2.5)

Use statistical functions on calculators and spreadsheets
Use statistical software to calculate the mean for grouped data sets
Estimate the mean for data sets with ill defined class boundaries
Investigate the affect of combining class intervals on estimating the mean for grouped data sets
Discuss occasions when one average is more appropriate, and the limitations of each average
Mention standard deviation (it is not in the specification, but it is useful for further comparison of data sets with similar means)

Collect data from class, eg children per family etc. Extend this activity to different classes, Year groups or use secondary data from the internet. Previous coursework tasks are a rich source of data to work with, eg Second-Hand Car Sales)
Compare distributions and make inferences, using the shapes of distributions and measures of average and spread, eg ‘boys are taller on average, but there is a much greater spread in heights’ (use date collected from previous work done or Mayfield data)
Students tend to select modal class but identify it by the frequency rather than the class itself
Explain that the median of grouped data is not necessarily from the middle class interval

1-4

Histograms

Draw and producing histograms for grouped continuous data
Interpret a histogram
Understand frequency density

Displaying Data
Grouped Data
Bar Charts

By the end of the module the student should be able to:
• Complete a histogram from a frequency table (3.5, 3.7)
• Complete a frequency table from a histogram (3.5, 3.7)
• Use a histogram to work out the frequency in part of a class interval (3.5, 3.7)

Students could carry out a statistical investigation of their own choice and decide on an appropriate means of displaying the results
Investigate how the choice of class width affects the shape of a distribution
Use a previous set of exam results to show that the distribution of marks is often not uniform.

Show an example which clearly demonstrates that a histogram represents the data in a ‘fairer’ way (A-Level texts are a rich source of suitable examples to use)
Show students the fact that the area of each bar represents the frequency and how this can be useful to solve questions with partially filled in tables or graphs

1-5

Scatter graphs and correlation

Draw and produce a scatter graph
Interpret a scatter graph
Appreciate that correlation is a measure of the strength of association between two variables
Appreciate that zero correlation does not necessarily imply ‘no correlation’ but merely ‘no linear relationship’
Distinguish between positive, negative and zero correlation and using a line of best fit
Draw a line of best fit by eye, and understanding what these represent
Find the equation of the line of best fit

Plotting coordinates
An understanding of the concept of a variable
Recognition that a change in one variable can affect another
Straight line graphs: y = mx + c (link with Unit 2)

By the end of the module the student should be able to:
• Draw and produce a scatter graph (4.4)
• Appreciate that correlation is a measure of the strength of association between two variables (4.5)
• Distinguish between positive, negative and zero correlation using a line of best fit (4.5)
• Appreciate that zero correlation does not necessarily imply ‘no correlation’ but merely ‘no linear relationship’ (4.5)
• Draw a line of best fit by eye and understand what it represents (4.6)
• Use a line of best fit to interpolate/ extrapolate (4.7)
• Find the equation of the line of best fit and using it to predict values (4.7, with 4.2)

Vary the axes required on a scatter graph to suit the ability of the class
Carry out a statistical investigation of their own including; designing an appropriate means of gathering the data, and an appropriate means of displaying the results
Use a spreadsheet, or other software, to produce scatter diagrams/lines of best fit. Investigate how the line of best fit is affected (visually) by the choice of scales on the axes
Look up the correlation coefficient using Excel (or equivalent package) functions (This not in the specification, but it is a useful introduction to further statistics)

Link with direct and inverse proportion (Unit 3)
Students should realise that lines of best fit should have the same gradient as the correlation of the data
Also mention that the line of best fit should pass through the coordinate representing the mean of the data
Clearly label all axes on graphs and use a ruler to draw straight lines
Warn students that the line of best fit does not necessarily go through the origin or ‘corner’ point of the graph

1-6

Median and interquartile range

Find the median, quartiles and interquartile range for large data sets
Draw and producing cumulative frequency tables and diagrams
Draw and producing box plots for grouped continuous data
Compare distributions and making inferences, using shapes of distributions and measures of average and spread, including median and quartiles
Compare cumulative frequency diagrams and box plots and drawing conclusions

Experience of inequality notation
Ability to plot points
Understand how to find the median and range for small data sets
Understand the difference between discrete and continuous data

By the end of the module the student should be able to:
• Find the median and quartiles for large sets of ungrouped data (3.9, 3.10)
• Draw a cumulative frequency table for grouped data (using the upper class boundary) (3.8)
• Draw a cumulative frequency curve for grouped data (3.8)
• Use a cumulative frequency diagram to find estimates for the median and quartiles of a distribution (3.9)
• Use a cumulative frequency diagram to solve problems, eg how many greater than a particular value (3.9)
• Draw a box plot to summarise information given in cumulative frequency diagrams (3.10)
• Compare cumulative frequency diagrams and box lots to make inferences about distributions (3.9, 3.10)

Understand the distinction between a cumulative frequency curve and a cumulative frequency polygon
Compare more than three distributions, eg. Yrs 7, 8, 9
Explain clearly why the IQR being small shows greater consistency or less variation in the data
Encourage students to give clear comparison statements using median and IQR Use statistical software to produce cumulative frequency diagrams and box plots
Identify and represent outliers for box plots and that a ‘long whisker’ could be due to only one extreme value
Revise and extend to for discrete data. Show counter examples when is not too useful

Start by listing discrete data and putting in arrows to mark off quartiles and median – so the data is cut up into four equal parts
No distinction is made for cumulative frequency curves and cumulative frequency polygons
Students should check that their answers for mean, median and mode lie within the given range of data
Explain that the IQR gives a better indication of the true range and the full range can be affected by extreme values (ie the top and bottom 25% of values)
Explain how an examinations grade boundary can be set by using quartiles and can vary from year to year according to the shape of the distribution and curve, eg grades A – D, A for top 25% etc

1-7

Probability

List all outcomes for single events, and for successive events, in a systematic way
Identify different mutually exclusive outcomes and know that the sum of the probabilities of all these outcomes is 1
Know when to add or multiply two probabilities
Use tree diagrams to represent outcomes of compound events, recognising when events are independent
Compare experimental data and theoretical probabilities
Understand that if they repeat an experiment they may, and usually will, get different outcomes, and that increasing sample size generally leads to better estimates of probability and population parameters

Understand that a probability is a number between 0 and 1
Know how to add, and multiplying fractions and decimals (link with Unit 2)
Experience of expressing one number as a fraction of another number (link with Unit 2)
Recognise the language of statistics, eg words such as likely, certain, impossible, etc

By the end of the module the student should be able to:
• List all the outcomes from mutually exclusive events, eg from two coins, and sample space diagrams (5.3)
• Write down the probability associated with equally likely events, eg the probability of drawing an ace from a pack of cards (5.1, 5.3)
• Know that if the probability of an event occurring is p than the probability of it not occurring is 1 – p (5.3)
• Find the missing probability from a list or table (5.3)
• Know that the probability of A or B is P(A) + P(B) (5.3)
• Know that the probability of A and B is P(A) ´ P(B) (5.3)
• Draw and use tree diagrams to solve probability problems (including examples of non-replacement) (5.7, 5.8)
• Find estimates of probabilities by considering relative frequency in experimental results (including two-way tables) (5.4, 5.5)
• Know that the more an experiment is repeated the better the estimate of probability (5.4)

Experiments with dice and spinners
Show sample space for outcomes of throwing two dice
Stress that there are 36 outcomes, as students will initially guess it’s 12 outcomes for two dice
Binomial probabilities (H or T)
Do a question ‘with’, and then repeat it ‘without’ replacement. It is a good idea to show the students the contents of the bag and then physically remove the object to illustrate the change of probability fraction for the second selection

Students should express probabilities as fractions, percentages or decimals
Fractions needed not be cancelled to their lowest terms. This makes it easier to calculate tree diagram probabilities ie easier to add like denominators
Show that each cluster of branches adds up to 1, and that all the outcomes add up to 1 too
Explain that if two objects are picked, then this is the same as one followed by another without replacement
Show that it is often easier to do a question involving multiple outcomes by considering the opposite event and subtracting from 1, eg at least 2 reds, at least 2 beads of a different colour etc

1-8

Integers

Understand place value and round to a given power of 10
Understand and use positive numbers and negative integers both as positions and translations on a number line
Order integers
Add, subtract, multiply and divide integers and then by any number
Use standard column procedures for addition and subtraction of integers
Use standard column procedures for multiplication of integers
Use a variety of checking procedures, including the problem backwards, and considering whether a result is of the right magnitude
Use brackets and the hierarchy of operations ie BIDMAS

The ability to order numbers
Appreciation of place value
Experience of the four operations using whole numbers
Knowledge of integer complements to 10
Knowledge of multiplication facts to 10 ´ 10
Knowledge of strategies for multiplying and dividing whole numbers by 10

By the end of the module the student should be able to:
• Understand and order integers (2.1)
• Add, subtract, multiply and divide integers (2.1)
• Understand simple instances of BIDMAS, eg work out 12 ´ 5 – 24 ¸ 8 (2.1)
• Round whole numbers to the nearest, 10, 100, 1000, … (1.6)
• Multiply and divide whole numbers by a given multiple of 10 (2.1)
• Check their calculations by rounding, eg 29 ´ 31 » 30 ´ 30 (2.6)
• Check answers by reverse calculation, eg if 9 ´ 23 = 207 then
207 ¸ 9 = 23

Estimating answers to calculations involving the four rules
Directed number work with two or more operations, or with decimals
Encourage effective use of a calculator

Present all working clearly
Show what is entered into your calculator, not just the answer

1-9

Decimals

Write decimal numbers in order of size
Round to a given number of significant figures (and decimal places)
Estimate answers to problems

Integers
The concepts of a fraction and a decimal

By the end of the module the student should be able to:
• Put digits in the correct place in a decimal number (2.1)
• Write decimals in ascending order of size (2.2)
• Approximate decimals to a given number of decimal places or significant figures (1.6)

Use decimals in real-life problems, eg Best Buys and other Functional Elements real-life problems
Money calculations that require rounding answers to the nearest penny
Multiply and divide decimals by decimals (more than 2 decimal places)
Round answers to appropriate degrees of accuracy to suit a particular everyday unit

Advise students not to round any decimals used in calculations until the final answer is to be declared
Link decimals to statistics and probability, eg the mean should not be rounded, the probabilities of all events add up to 1
Also link decimals to converting units and compound measures (Unit 1 and 2)

1-10

Fractions

Understand equivalent fractions; simplifying a fraction by cancelling all common factors
Order fractions by rewriting them with a common denominator

Multiplication facts
Ability to find common factors
A basic understanding of fractions as being ‘parts of a whole unit’
Use of a calculator with fractions

By the end of the module the student should be able to:
• Visualise a fraction diagrammatically (5.1)
• Understand a fraction as part of a whole (5.1)
• Recognise and write fractions in everyday situations (Throughout Ch 5)
• Write a fraction in its simplest form and recognise equivalent fractions (5.1)
• Compare the sizes of fractions using a common denominator (5.1)
• Add and subtract fractions by using a common denominator (5.1)
• Write an improper fraction as a mixed fraction
• Recognising common recurring decimals can be written as exact (revision of
fractions, eg prior knowledge)

Careful differentiation is essential for this topic dependent upon the student’s ability
Relating simple fractions to remembered percentages and vice-versa
Using a calculator to change fractions into decimals and looking for patterns
Working with improper fractions and mixed numbers
The four rules of number applied to fractions with a calculator
Solve word problems involving fractions and in real-life problems, eg find perimeter using fractional values

Understanding of equivalent fractions is the key issue in order to be able to tackle the other content
Constant revision of this topic is needed
Students should learn how to identify and use the fraction button on their calculators
Link with probability calculations using AND and OR Laws
Use of fractions for calculations involving compound units (Unit 1 and 2)
Use Functional Elements questions and examples using fractions, eg off the list price when comparing different sale prices

1-11

Percentages

Understand that ‘percentage’ means ‘number of parts per 100’
Interpret percentage as the operator 'so many hundredths of'
Use percentages (and fractions) in real-life situations
Solve percentage problems, including increase and decrease.
Represent repeated proportional change using a multiplier raised to a power
Use calculators to explore exponential growth and decay, using a multiplier and power key

Four operations of number
The concepts of a fraction and a decimal
An awareness that percentages are used in everyday life

By the end of the module the student should be able to:
• Understand that a percentage is a fraction in hundredths (5.1)
• Write a percentage as a decimal; or as a fraction in its simplest terms (5.1)
• Write one number as a percentage of another number (5.2)
• Calculate the percentage (or fraction) of a given amount (5.2)
• Find a percentage increase/decrease of an amount (5.2)
• Use a multiplier to increase by a given percent, eg 1.10 ´ 64 increases 64 by 10% (5.2)
• Calculate simple and compound interest for two, or more, periods of time (5.2)

Fractional percentages of amounts (non-calculator)
Combine multipliers to simplify a series of percentage changes
Percentages which convert to recurring decimals, eg 33 %, and situations which lead to percentages of more than 100%
Problems which lead to the necessity of rounding to the nearest penny, eg Functional Elements contexts
Comparisons between simple and compound interest calculations
Formulae in simple interest/compound interest methods

Amounts of money should always be rounded to the nearest penny where necessary, except where such rounding is premature, eg in successive calculations such as compound interest

1-12

Ratio and scale

Use ratio notation, including reduction to its simplest form and it various links to fractions notation
Divide a quantity in a given ratio
Solve word problems about ratio, including using informal strategies and the unitary method of solution

Fractions and Decimals

By the end of the module the student should be able to:
• Appreciate that, eg the ratio 1:2 represents and of a quantity (5.9)
• Divide quantities in a given ratio, eg divide £20 in the ratio 2:3 (5.9)
• Solve word problems involving ratios, eg find the cost of 8 pencils given that 6 pencils cost 78p

Currency calculations using currency exchange rates
Harder problems involving multi-stage calculations
Relate ratios to Functional Elements contexts, eg investigate the proportions of the different metals in alloys or the new amounts of ingredients needed for a recipe for different numbers of guests

Students often find three-part ratios difficult
Also link ratios given in different units to metric and imperial units (Unit 2)

1-13

Introduction to algebra

Distinguish the different roles played by letter symbols in algebra
Distinguish the meaning between the words ‘equation’, ‘formula’, ‘identity’ and expression

Experience of using a letter to represent a number
Word formulae or rules to describe everyday situations, eg time to cook a nut roast linked to weight of the nut roast

By the end of the module the student should be able to:
• Distinguish the different roles played by letter symbols in algebra (2.3)
• Understand the meaning between the words ‘equation’, ‘formula’, ‘identity’ and expression (2.3)

Extend the above ideas to the ‘equation’ of the straight line, y = mx + c
Look at word equations written in symbolic form, eg F = 2C + 30 to roughly convert temperature and compare with F = + 32

There are plenty of past exam papers with matching tables testing knowledge of the ‘Vocabulary of Algebra’
(See Emporium website)

1-14

Straight line graphs

Construct linear functions from real-life problems and plotting their corresponding graphs
Discuss and interpreting graphs modelling real-life situations
Interpret information presented in a range of linear graphs
Use the conventions for coordinates in the plane
Plot points in all four quadrants
Recognise (when values are given for m and c) that equations of the
form y = mx + c correspond to straight-line graphs in the coordinate plane
Plot graphs of functions in which y is given explicitly in terms of x, or implicitly
Find the gradient of lines given by equations of the form y = mx + c, when (a) values are given for m and c and (b) the line has been plotted

Experience at plotting points in all quadrants
Linear Sequences and basic number patterns

By the end of the module the student should be able to:
• Draw linear graphs from tabulated data, including real-world examples (4.1 – 4.3)
• Interpret linear graphs, including conversion graphs and distance-time graphs (4.2 – 4.3)
• Draw and interpret graphs in the form y = mx + c (when values for m and c are given) (4.2)
• Understand that lines are parallel when they have the same value of m (4.2)
• Find the gradient and intercept of a straight line graph (4.2)

Plot graphs of the form y = mx + c where students have to generate their own tables and set out their own axes
Use a spreadsheet to generate straight-line graphs, posing questions about the gradient of lines
Use a graphical calculator or graphical ICT package to draw straight-line graphs
Link to scatter graphs and correlation

Clear presentation with axes labelled correctly is vital
Recognise linear graphs and hence when data may be incorrect
Link to graphs and relationships in other subject areas, eg science, geography etc
Interpret straight line graphs for Functional Elements contexts
– Ready reckoner graphs
– Conversion graphs
– Fuel bills
– Fixed charge (standing charge) and cost per unit
Also link conversion graphs to converting metric and imperial units and equivalents (Unit 2)

1-15

Curved graphs

Generate points and plotting graphs of simple quadratic functions
Plot graphs of simple cubic functions, the reciprocal function y = with x 0, the exponential function y = kx for integer values of x and simple positive values of k, the circular functions y = sin x and y = cos x, using a spreadsheet or graph plotter as well as pencil and paper
Recognise the characteristic shapes of all these functions

Straight line graphs
BIDMAS

By the end of the module the student should be able to:
• Plot and recognise quadratic, cubic, reciprocal, exponential and circular functions
• Use the graphs of these functions to find approximate solutions to equations, eg given x find y (and vice versa)
• Match equations with their graphs
• Sketch graphs of given functions

Explore the function y = ex (perhaps relate this to y = ln x)
Explore the function y = tan x (Unit 3)
Find solutions to equations of the circular functions y = sin x and y = cos x over more than one cycle (and generalise)

This work could be enhanced by drawing graphs on graphical calculators and appropriate software
Group work with each group assigned a different type of graph is an effective way to explore the graphical properties
There are plenty of past exam papers with matching tables testing knowledge of the ‘Shapes of Graphs’
(See Emporium website)

1-16

Lines and angles

Measure lines and using a protractor to measure angles of all sizes
Use a protractor to draw angles accurately
Use the fact that angles at a point add to 360°

An understanding of angle as a measure of turning
The ability to use a ruler and protractor

By the end of this module the student should be able to:
• Distinguish between acute, obtuse, reflex and right angles (3.1)
• Estimate the size of an angle in degrees (revision of
prior knowledge)
• Measure and draw angle to the nearest degree (3.1)
• Measure and draw line to the nearest mm (revision of
prior knowledge)
• Use angle properties ‘at a point’ to calculate unknown angles (3.1)

Extend to other angle facts in triangles, parallel lines and/or quadrilaterals (preparation for Unit 2)

Make sure that all pencils are sharp and drawings are neat and accurate
Angles should be correct to within 2°, lengths correct to the nearest mm
Apply skills to constructing pie charts

1-17

Units and reading scales

Make estimates in everyday life
Make estimates of length using metric and imperial units
Make estimates of weight, volume and capacity using metric and imperial units
Use sensible units for measuring
Read analogue and digital clocks
Read measurements on different types of scales
Read timetables and calculating time intervals
Convert from one metric unit to another

An awareness of the Metric & Imperial system of measures
Strategies for multiplying and dividing by 10 (for converting Metric Units)
Knowledge of the conversion facts for metric lengths, mass and capacity
Knowledge of the conversion facts between seconds, minutes and hours

By the end of this module the student should be able to:
• Make estimates of: length, volume and capacity, weights (Throughout Ch 1)
• Make approximate conversions between metric and imperial units (revision of
prior knowledge)
• Decide on the appropriate units to use in real life problems (1.1)
• Read measurements from instruments: scales, analogue and
digital clocks, thermometers etc (Throughout
Chapter 1)
• Recognise that measurements given to the nearest whole unit may be inaccurate by up to one half in either direction (revision of
prior knowledge)
• Convert from one metric unit to another metric unit (1.1)

This could be made a practical activity by collecting assorted everyday items for weighing and measuring to check the estimates of their lengths, weights and volumes
Link with compound units such as speed for travel graphs (m/s or km/h or mph) and Best Buys (g/penny)

Measurement is essentially a practical activity
Use a range of everyday objects to make the lesson more realistic
Use Functional Elements problems as a rich source of practice questions for this topic area

## Past Papers

## Table of Contents

## Topic Checklist

A list of the topics for Unit 1:Mean from a table

Histograms

Box plots

Tree Diagrams

## Video Solutions

## Unit 1H March 2011:

Question 1 - QuestionnairesQuestion 2 - Scatter Graphs

Question 3 - Ratio and Percentages

Question 4 - Two Way Tables

Question 5 - Probability

Question 6 - Mean from a Table

Question 7 - Box Plots and Comparing Data

Question 8 - Problem Solving

Question 9 - Probability

Question 10a - Mean from a Table

Question 10b - Cumulative Frequency

Question 11 - Probability Trees

Question 12 - Stratified Sampling

Question 13 - Compound Interest

Question 14 - Histograms

## Unit 1 Sample Assessment Higher:

Question 1: Median interval and Frequency PolygonQuestion 2: Comparing Costs for flights

Question 3: Compare and Contrast Data in Stem and Leaf Diagram

Question 4: Questionnaires

Question 5: Scattergraph - Line of Best Fit

Question 6: Ratio

Question 7: Probability and Algebra

Question 8: Averages - Mean

Question 9: Compound and Simple Interest Comparison

Question 10: Histogram - Unequal class widths

Question 11: Random and Stratified Sample

Question 12: Applying Probability

Question 13: Probability Tree Diagram

Question 14: Capture Re-capture Method

## Detailed Course Content

Select and justifying a sampling scheme and a method to investigate a population

Collect data using various method, including observation, controlled experiments, data logging, questionnaires and surveys

Design and using data-collection sheets

Gather data from secondary sources

Design and using two-way tables

Deal with practical problems such as non-response or missing data

Use stratified sampling

Experience of simple tally charts.

Experience of inequality notation and signs.

Basic Fractions for help when calculating stratified samples (link with Unit 2)

• Design a suitable question for a questionnaire (1.5)

• Understand the difference between: primary and secondary data; discrete and continuous data (1.1 – 1.8)

• Understand when appropriate to use a stratified sample

• Understand how to calculate a stratified sample

• Design suitable data capture sheets for surveys and experiments (1.4 – 1.6)

• Understand about bias in sampling (1.7)

• Choose and justify an appropriate sampling scheme, including random and systematic sampling (1.2, 1.3)

• Deal with practical problems in data collection, such as non-response, missing and anomalous data (Throughout Ch 1)

An investigation into other sampling schemes, such as cluster and quota sampling

Emphasise the differences between primary and secondary data

Discuss sample size and mention that a census is the whole population. In the UK the census is every year that ends in a ‘1’, so 2011 is the next census

If students are collecting data as a group, they should all use the same procedure

Emphasise that continuous data is data that is measured, eg temperature

Mayfield High data from coursework task can be used to collect samples and to make comparisons in following section

Use the fact that the numbers in each year group for the Mayfield High data to introduce stratified sampling techniques

Interpret a wide range of graphs and diagrams and drawing conclusions

Experience of inequality notation and symbol

Some basic fraction work for pie Charts (Link with Unit 2)

Use a protractor to measure and draw angles (Link with Unit 2 contents)

• Represent data as:

- Pie charts (for categorical data) (3.1, 3.2)

- Bar charts and histograms (equal class intervals) (3.4, 3.5)

- Frequency polygons (3.6)

- Stem and leaf diagrams (3.3)

• Choose an appropriate way to display discrete, continuous and

categorical data (3.2 – 3.10)

• Understand the difference between a bar chart and a histogram (3.5)

• Compare distributions shown in charts and graphs (3.2 – 3.10)

Use a spreadsheet to draw different types of graphs

Collect examples of charts and graphs in the media which have been misused, and discuss the implications

Stem and leaf diagrams must have a key

Show how to find the median and mode from a stem and leaf diagram

Angles for pie charts should be correct to within 2°. Ask students to check each others’ charts

Make comparisons between previously collected data, eg Mayfield boys vs girls or Yr 7 vs Yr 8

Encourage students to work in groups and present their charts – useful for display work in classroom/corridors

Use Excel graph wizard

Estimate the mean for large data sets with grouped data

Find the mode, median, mean and range of small data sets with discrete data

Relate summarised data to the initial questions

Compare distributions and making inferences, using the shapes of distributions and measures of average and spread

Stem and leaf diagram to find mode and median

An ability to order and find the mid-point of two numbers

• Find the mean of data given in an ungrouped frequency distribution (2.4)

• Find the mode, median and range for a small set of data (2.2, 2.10)

• Find the modal class for grouped data (2.8)

• Find the median by using , where n is the number of data (2.2, 2.8)

• Find the mean from a frequency table by using (Sigma notation for top sets only) (2.7)

• Use the mid point of equal class intervals to find an estimate for the mean of grouped data (2.8)

• Know the advantages/disadvantages of using the different measure of average (2.5)

Use statistical software to calculate the mean for grouped data sets

Estimate the mean for data sets with ill defined class boundaries

Investigate the affect of combining class intervals on estimating the mean for grouped data sets

Discuss occasions when one average is more appropriate, and the limitations of each average

Mention standard deviation (it is not in the specification, but it is useful for further comparison of data sets with similar means)

Compare distributions and make inferences, using the shapes of distributions and measures of average and spread, eg ‘boys are taller on average, but there is a much greater spread in heights’ (use date collected from previous work done or Mayfield data)

Students tend to select modal class but identify it by the frequency rather than the class itself

Explain that the median of grouped data is not necessarily from the middle class interval

Interpret a histogram

Understand frequency density

Grouped Data

Bar Charts

• Complete a histogram from a frequency table (3.5, 3.7)

• Complete a frequency table from a histogram (3.5, 3.7)

• Use a histogram to work out the frequency in part of a class interval (3.5, 3.7)

Investigate how the choice of class width affects the shape of a distribution

Use a previous set of exam results to show that the distribution of marks is often not uniform.

Show students the fact that the area of each bar represents the frequency and how this can be useful to solve questions with partially filled in tables or graphs

Interpret a scatter graph

Appreciate that correlation is a measure of the strength of association between two variables

Appreciate that zero correlation does not necessarily imply ‘no correlation’ but merely ‘no linear relationship’

Distinguish between positive, negative and zero correlation and using a line of best fit

Draw a line of best fit by eye, and understanding what these represent

Find the equation of the line of best fit

An understanding of the concept of a variable

Recognition that a change in one variable can affect another

Straight line graphs: y = mx + c (link with Unit 2)

• Draw and produce a scatter graph (4.4)

• Appreciate that correlation is a measure of the strength of association between two variables (4.5)

• Distinguish between positive, negative and zero correlation using a line of best fit (4.5)

• Appreciate that zero correlation does not necessarily imply ‘no correlation’ but merely ‘no linear relationship’ (4.5)

• Draw a line of best fit by eye and understand what it represents (4.6)

• Use a line of best fit to interpolate/ extrapolate (4.7)

• Find the equation of the line of best fit and using it to predict values (4.7, with 4.2)

Carry out a statistical investigation of their own including; designing an appropriate means of gathering the data, and an appropriate means of displaying the results

Use a spreadsheet, or other software, to produce scatter diagrams/lines of best fit. Investigate how the line of best fit is affected (visually) by the choice of scales on the axes

Look up the correlation coefficient using Excel (or equivalent package) functions (This not in the specification, but it is a useful introduction to further statistics)

Students should realise that lines of best fit should have the same gradient as the correlation of the data

Also mention that the line of best fit should pass through the coordinate representing the mean of the data

Clearly label all axes on graphs and use a ruler to draw straight lines

Warn students that the line of best fit does not necessarily go through the origin or ‘corner’ point of the graph

Draw and producing cumulative frequency tables and diagrams

Draw and producing box plots for grouped continuous data

Compare distributions and making inferences, using shapes of distributions and measures of average and spread, including median and quartiles

Compare cumulative frequency diagrams and box plots and drawing conclusions

Ability to plot points

Understand how to find the median and range for small data sets

Understand the difference between discrete and continuous data

• Find the median and quartiles for large sets of ungrouped data (3.9, 3.10)

• Draw a cumulative frequency table for grouped data (using the upper class boundary) (3.8)

• Draw a cumulative frequency curve for grouped data (3.8)

• Use a cumulative frequency diagram to find estimates for the median and quartiles of a distribution (3.9)

• Use a cumulative frequency diagram to solve problems, eg how many greater than a particular value (3.9)

• Draw a box plot to summarise information given in cumulative frequency diagrams (3.10)

• Compare cumulative frequency diagrams and box lots to make inferences about distributions (3.9, 3.10)

Compare more than three distributions, eg. Yrs 7, 8, 9

Explain clearly why the IQR being small shows greater consistency or less variation in the data

Encourage students to give clear comparison statements using median and IQR Use statistical software to produce cumulative frequency diagrams and box plots

Identify and represent outliers for box plots and that a ‘long whisker’ could be due to only one extreme value

Revise and extend to for discrete data. Show counter examples when is not too useful

No distinction is made for cumulative frequency curves and cumulative frequency polygons

Students should check that their answers for mean, median and mode lie within the given range of data

Explain that the IQR gives a better indication of the true range and the full range can be affected by extreme values (ie the top and bottom 25% of values)

Explain how an examinations grade boundary can be set by using quartiles and can vary from year to year according to the shape of the distribution and curve, eg grades A – D, A for top 25% etc

Identify different mutually exclusive outcomes and know that the sum of the probabilities of all these outcomes is 1

Know when to add or multiply two probabilities

Use tree diagrams to represent outcomes of compound events, recognising when events are independent

Compare experimental data and theoretical probabilities

Understand that if they repeat an experiment they may, and usually will, get different outcomes, and that increasing sample size generally leads to better estimates of probability and population parameters

Know how to add, and multiplying fractions and decimals (link with Unit 2)

Experience of expressing one number as a fraction of another number (link with Unit 2)

Recognise the language of statistics, eg words such as likely, certain, impossible, etc

• List all the outcomes from mutually exclusive events, eg from two coins, and sample space diagrams (5.3)

• Write down the probability associated with equally likely events, eg the probability of drawing an ace from a pack of cards (5.1, 5.3)

• Know that if the probability of an event occurring is p than the probability of it not occurring is 1 – p (5.3)

• Find the missing probability from a list or table (5.3)

• Know that the probability of A or B is P(A) + P(B) (5.3)

• Know that the probability of A and B is P(A) ´ P(B) (5.3)

• Draw and use tree diagrams to solve probability problems (including examples of non-replacement) (5.7, 5.8)

• Find estimates of probabilities by considering relative frequency in experimental results (including two-way tables) (5.4, 5.5)

• Know that the more an experiment is repeated the better the estimate of probability (5.4)

Show sample space for outcomes of throwing two dice

Stress that there are 36 outcomes, as students will initially guess it’s 12 outcomes for two dice

Binomial probabilities (H or T)

Do a question ‘with’, and then repeat it ‘without’ replacement. It is a good idea to show the students the contents of the bag and then physically remove the object to illustrate the change of probability fraction for the second selection

Fractions needed not be cancelled to their lowest terms. This makes it easier to calculate tree diagram probabilities ie easier to add like denominators

Show that each cluster of branches adds up to 1, and that all the outcomes add up to 1 too

Explain that if two objects are picked, then this is the same as one followed by another without replacement

Show that it is often easier to do a question involving multiple outcomes by considering the opposite event and subtracting from 1, eg at least 2 reds, at least 2 beads of a different colour etc

Understand and use positive numbers and negative integers both as positions and translations on a number line

Order integers

Add, subtract, multiply and divide integers and then by any number

Use standard column procedures for addition and subtraction of integers

Use standard column procedures for multiplication of integers

Use a variety of checking procedures, including the problem backwards, and considering whether a result is of the right magnitude

Use brackets and the hierarchy of operations ie BIDMAS

Appreciation of place value

Experience of the four operations using whole numbers

Knowledge of integer complements to 10

Knowledge of multiplication facts to 10 ´ 10

Knowledge of strategies for multiplying and dividing whole numbers by 10

• Understand and order integers (2.1)

• Add, subtract, multiply and divide integers (2.1)

• Understand simple instances of BIDMAS, eg work out 12 ´ 5 – 24 ¸ 8 (2.1)

• Round whole numbers to the nearest, 10, 100, 1000, … (1.6)

• Multiply and divide whole numbers by a given multiple of 10 (2.1)

• Check their calculations by rounding, eg 29 ´ 31 » 30 ´ 30 (2.6)

• Check answers by reverse calculation, eg if 9 ´ 23 = 207 then

207 ¸ 9 = 23

Directed number work with two or more operations, or with decimals

Encourage effective use of a calculator

Show what is entered into your calculator, not just the answer

Round to a given number of significant figures (and decimal places)

Estimate answers to problems

The concepts of a fraction and a decimal

• Put digits in the correct place in a decimal number (2.1)

• Write decimals in ascending order of size (2.2)

• Approximate decimals to a given number of decimal places or significant figures (1.6)

Money calculations that require rounding answers to the nearest penny

Multiply and divide decimals by decimals (more than 2 decimal places)

Round answers to appropriate degrees of accuracy to suit a particular everyday unit

Link decimals to statistics and probability, eg the mean should not be rounded, the probabilities of all events add up to 1

Also link decimals to converting units and compound measures (Unit 1 and 2)

Order fractions by rewriting them with a common denominator

Ability to find common factors

A basic understanding of fractions as being ‘parts of a whole unit’

Use of a calculator with fractions

• Visualise a fraction diagrammatically (5.1)

• Understand a fraction as part of a whole (5.1)

• Recognise and write fractions in everyday situations (Throughout Ch 5)

• Write a fraction in its simplest form and recognise equivalent fractions (5.1)

• Compare the sizes of fractions using a common denominator (5.1)

• Add and subtract fractions by using a common denominator (5.1)

• Write an improper fraction as a mixed fraction

• Recognising common recurring decimals can be written as exact (revision of

fractions, eg prior knowledge)

Relating simple fractions to remembered percentages and vice-versa

Using a calculator to change fractions into decimals and looking for patterns

Working with improper fractions and mixed numbers

The four rules of number applied to fractions with a calculator

Solve word problems involving fractions and in real-life problems, eg find perimeter using fractional values

Constant revision of this topic is needed

Students should learn how to identify and use the fraction button on their calculators

Link with probability calculations using AND and OR Laws

Use of fractions for calculations involving compound units (Unit 1 and 2)

Use Functional Elements questions and examples using fractions, eg off the list price when comparing different sale prices

Interpret percentage as the operator 'so many hundredths of'

Use percentages (and fractions) in real-life situations

Solve percentage problems, including increase and decrease.

Represent repeated proportional change using a multiplier raised to a power

Use calculators to explore exponential growth and decay, using a multiplier and power key

The concepts of a fraction and a decimal

An awareness that percentages are used in everyday life

• Understand that a percentage is a fraction in hundredths (5.1)

• Write a percentage as a decimal; or as a fraction in its simplest terms (5.1)

• Write one number as a percentage of another number (5.2)

• Calculate the percentage (or fraction) of a given amount (5.2)

• Find a percentage increase/decrease of an amount (5.2)

• Use a multiplier to increase by a given percent, eg 1.10 ´ 64 increases 64 by 10% (5.2)

• Calculate simple and compound interest for two, or more, periods of time (5.2)

Combine multipliers to simplify a series of percentage changes

Percentages which convert to recurring decimals, eg 33 %, and situations which lead to percentages of more than 100%

Problems which lead to the necessity of rounding to the nearest penny, eg Functional Elements contexts

Comparisons between simple and compound interest calculations

Formulae in simple interest/compound interest methods

Divide a quantity in a given ratio

Solve word problems about ratio, including using informal strategies and the unitary method of solution

• Appreciate that, eg the ratio 1:2 represents and of a quantity (5.9)

• Divide quantities in a given ratio, eg divide £20 in the ratio 2:3 (5.9)

• Solve word problems involving ratios, eg find the cost of 8 pencils given that 6 pencils cost 78p

Harder problems involving multi-stage calculations

Relate ratios to Functional Elements contexts, eg investigate the proportions of the different metals in alloys or the new amounts of ingredients needed for a recipe for different numbers of guests

Also link ratios given in different units to metric and imperial units (Unit 2)

Distinguish the meaning between the words ‘equation’, ‘formula’, ‘identity’ and expression

Word formulae or rules to describe everyday situations, eg time to cook a nut roast linked to weight of the nut roast

• Distinguish the different roles played by letter symbols in algebra (2.3)

• Understand the meaning between the words ‘equation’, ‘formula’, ‘identity’ and expression (2.3)

Look at word equations written in symbolic form, eg F = 2C + 30 to roughly convert temperature and compare with F = + 32

(See Emporium website)

Discuss and interpreting graphs modelling real-life situations

Interpret information presented in a range of linear graphs

Use the conventions for coordinates in the plane

Plot points in all four quadrants

Recognise (when values are given for m and c) that equations of the

form y = mx + c correspond to straight-line graphs in the coordinate plane

Plot graphs of functions in which y is given explicitly in terms of x, or implicitly

Find the gradient of lines given by equations of the form y = mx + c, when (a) values are given for m and c and (b) the line has been plotted

Linear Sequences and basic number patterns

• Draw linear graphs from tabulated data, including real-world examples (4.1 – 4.3)

• Interpret linear graphs, including conversion graphs and distance-time graphs (4.2 – 4.3)

• Draw and interpret graphs in the form y = mx + c (when values for m and c are given) (4.2)

• Understand that lines are parallel when they have the same value of m (4.2)

• Find the gradient and intercept of a straight line graph (4.2)

Use a spreadsheet to generate straight-line graphs, posing questions about the gradient of lines

Use a graphical calculator or graphical ICT package to draw straight-line graphs

Link to scatter graphs and correlation

Recognise linear graphs and hence when data may be incorrect

Link to graphs and relationships in other subject areas, eg science, geography etc

Interpret straight line graphs for Functional Elements contexts

– Ready reckoner graphs

– Conversion graphs

– Fuel bills

– Fixed charge (standing charge) and cost per unit

Also link conversion graphs to converting metric and imperial units and equivalents (Unit 2)

Plot graphs of simple cubic functions, the reciprocal function y = with x 0, the exponential function y = kx for integer values of x and simple positive values of k, the circular functions y = sin x and y = cos x, using a spreadsheet or graph plotter as well as pencil and paper

Recognise the characteristic shapes of all these functions

BIDMAS

• Plot and recognise quadratic, cubic, reciprocal, exponential and circular functions

• Use the graphs of these functions to find approximate solutions to equations, eg given x find y (and vice versa)

• Match equations with their graphs

• Sketch graphs of given functions

Explore the function y = tan x (Unit 3)

Find solutions to equations of the circular functions y = sin x and y = cos x over more than one cycle (and generalise)

Group work with each group assigned a different type of graph is an effective way to explore the graphical properties

There are plenty of past exam papers with matching tables testing knowledge of the ‘Shapes of Graphs’

(See Emporium website)

Use a protractor to draw angles accurately

Use the fact that angles at a point add to 360°

The ability to use a ruler and protractor

• Distinguish between acute, obtuse, reflex and right angles (3.1)

• Estimate the size of an angle in degrees (revision of

prior knowledge)

• Measure and draw angle to the nearest degree (3.1)

• Measure and draw line to the nearest mm (revision of

prior knowledge)

• Use angle properties ‘at a point’ to calculate unknown angles (3.1)

Angles should be correct to within 2°, lengths correct to the nearest mm

Apply skills to constructing pie charts

Make estimates of length using metric and imperial units

Make estimates of weight, volume and capacity using metric and imperial units

Use sensible units for measuring

Read analogue and digital clocks

Read measurements on different types of scales

Read timetables and calculating time intervals

Convert from one metric unit to another

Strategies for multiplying and dividing by 10 (for converting Metric Units)

Knowledge of the conversion facts for metric lengths, mass and capacity

Knowledge of the conversion facts between seconds, minutes and hours

• Make estimates of: length, volume and capacity, weights (Throughout Ch 1)

• Make approximate conversions between metric and imperial units (revision of

prior knowledge)

• Decide on the appropriate units to use in real life problems (1.1)

• Read measurements from instruments: scales, analogue and

digital clocks, thermometers etc (Throughout

Chapter 1)

• Recognise that measurements given to the nearest whole unit may be inaccurate by up to one half in either direction (revision of

prior knowledge)

• Convert from one metric unit to another metric unit (1.1)

Link with compound units such as speed for travel graphs (m/s or km/h or mph) and Best Buys (g/penny)

Use a range of everyday objects to make the lesson more realistic

Use Functional Elements problems as a rich source of practice questions for this topic area