meiassets.blob.core.windows.net · Web view2021. 4. 12. · AMSP Core Maths platform content: Drag and drop these shapes into the blank planner above, then stretch them to fit the

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OCR Core Maths A- PlannerHalf term

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AMSP Core Maths platform content: Drag and drop these shapes into the blank planner above, then stretch them to fit the number of weeks you require. The shapes contain links, which can be activated in the usual way (Ctrl and click), to navigate to areas of the scheme with more details on the activities and the specification content covered. You can return to this page by pressing simultaneously the Ctrl and Home keys.

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Percentage 3Statistics 2

Estimation

Correlation & causation

Exchanging Currency

Risk Percentage 1 Percentage

2Statistics

1

Comparing and deciding

Standard form

Probability & frequencytrees

Fallacies in probability

Interpreting & communicating

Algebra, graphs and gradients

Exponentials and logs

RCT, bias & sampling

Modelling

Shape and measures

Normal Distributions

Decisions business & risk

Approximately Normal?

Budgeting, inflation, saving

& borrowing

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Risk – Teacher overview

Summary of content: This resource deals with the probability of adverse events – often these probabilities are approximate or based on data from the (recent) past.

Comparison of risks in the form 1 in n can be difficult for students; they need to understand that 1 in 10 is a higher risk than 1 in 50, for example. A common misconception is to assume that the higher number means a higher risk.

There is an opportunity here to check student understanding of percentages; can they convert between risks expressed as 1 in n, a percentage and a decimal?

Understanding risk is important in business, science and in careers which involve looking after other people such as teaching and health.

Pre-requisites: GCSE probability and ratio

Approximate guided learning hours: 3-4 guided learning hours can be used a rough guide, however the amount of time required for this section will vary depending on the needs of individual students.

Auto-marked online assessment: Section test – Probability and risk 1

(Solutions available in the teachers section only)

Teacher notes, solutions and any other support materials are contained in the teacher folders

Folder name: Probability and risk 1

Activity title Description Specification learning outcomes

Worksheet on dangerous sports

Worksheet on dangerous sports black and white

Checking risk spreadsheet

An introductory card sort activityIQRu3 Be able to estimate probability from long-run relative frequency

IQRu7 Understand risk given as a probability or as 1 in n or as a description such as “once in n

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years”.

IQRu8 Be able to interpret a risk assessment, understanding that it involves measures of both likelihood and impact.

Assessing risk

Risk diagram

Risk diagram worksheet

A powerpoint leading students through assessing risk

A diagram on risk and impact.

A worksheet on risk and impact.

Probability starter

Probability and risk

Lottery numbers

Risk matching cards

Risk matching cards solutions

Worksheet on lottery numbers

Worksheet on accidents

Worksheet on accidents teacher solutions

Road accidents 2012

Vehicle miles

Worksheet on cause of death

Cause of death table

Worksheet on cause of death teacher solutions

A starter to remind students about probability.

A teacher powerpoint about working out risk.

An Excel file to produce lottery numbers.

A card sorting exercise.

Solutions to the card sorting exercise.

A worksheet to go with the Excel file

A worksheet for students to work out risk.

Solutions for teachers to the worksheet.

An Excel file to use with the worksheet.


A worksheet for students to work out risk.


Solutions for teachers to the worksheet.

Questions on risk Questions to check students’ understanding of

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risk.

Solutions to questions on risk Full solutions to the questions.

Student interactive study packages

Basic probability A recap on GCSE probability but involving the sort of contexts applicable to Core Maths

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Percentages 1 Change and comparisons – Teacher overview

Summary of content

This section reviews percentages starting from the context of finance and extending to other contexts. Some students will already have a good understanding of percentages from GCSE and will need to spend very little time on this section. The use of percentages in the risk section will indicate how much work students are likely to need on percentages.

In the context of finance, linking percentage to number of pence per pound may help students. Students may be quite good at working out simple percentages of an amount, such as 10% and 50% of £25, but find it much more difficult to work out what

percentage something has changed by; e.g. a leaf grows from 2.8 cm to 3.1 cm, what is the percentage change? Using a spreadsheet will require them to think about what steps the calculation needs and so help them work with harder numbers.

Some students may not have used multipliers for percentages at GCSE; they will need to do so in order to work with interest rates for loans and savings. For students with a good understanding of percentages, this section can be extended to including repeated percentage change, appreciation and depreciation.

For students who need quite a bit of revision of percentages, that can be left till later. A good understanding of percentages is important for everyday life, understanding the news and in business, science and social science. Many careers will include the use of spreadsheets; this section is an opportunity to use basic formulas and graphs in a spreadsheet to work with and compare

percentage change.

Pre-requisites: GCSE percentage workApproximate guided learning hours: 4-5 guided learning hours can be used a rough guide, however the amount of time required for this section will vary depending on the needs of individual students.Auto-marked online assessment: Percentages (this section test is currently not available to students as Q8 contains content from percentage 3 a new test is being prepared)(Solutions available in the teachers section only)Teacher notes, solutions and any other support materials are contained in the teacher folders

Folder name: Percentage for comparison Note: This resource has two supporting video clips


Percent starter A question to start students thinking about how to compare quantities – see below.

IQRq2 Be able to read information from a standard spreadsheet.IQRq1 Be able to use a standard calculator (scientific or graphical).IQR f2 Know how to obtain the necessary information.

Using percentages for comparison A powerpoint to introduce the idea of using percentages for comparison – some students will already be familiar with these ideas.

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IQRf3 Be able to process the information to provide one or more possible solutions.

IQRf5 Be able to do calculations involving percentages in context; the use of an index number to compare a number or value to that in a base year

Questions percent for comparison Questions to practise using percentages for comparison

Driver data A spreadsheet with the data for question 5

Folder name: Percentage Change


Percentage change A powerpoint to introduce students to percentage increase and decrease mentally and on a calculator. Notice the suggested spreadsheet exercise below.

IQRq2 Be able to read information from a standard spreadsheet.IQRq1 Be able to use a standard calculator (scientific or graphical).IQRf2 Know how to obtain the necessary information.IQRf3 Be able to process the information to provide one or more possible solutions.IQRf5 Be able to do calculations involving percentages in context; the use of an index number to compare a number or value to that in a base year

Multipliers A simple spreadsheet to let students practise getting the correct multiplier for a percentage change.

Percentage change questions Questions to practice percentage change.

Discussion better offer A question to promote discussion and problem solving – students should be able to justify their solutions to convince other

Reverse percentages A powerpoint to introduce students to the idea of using multipliers to find a quantity before a percentage change.

Multipliers for percentages This powerpoint can be used to show students how to use multipliers to find what the percentage is – some students will prefer to use the methods they are already familiar with and you may choose not to use this.

Mixed percent questions These mixed questions include forward and reverse percentage change as well as finding the percentage

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Folder name: The richest country in the world

Document title Description Specification learning outcomes

The richest country in the world .PPT This resource introduces students to Gross Domestic Product (GDP) and calculations related to GDP, including GDP per capita and GDP per capita, Purchasing power parity (PPP)

F2.2 expressing one quantity as a percentage of another

F2.3 comparing two quantities using percentages

F2.4 working with percentages over 100%The richest country in the world worksheet Students explore further about GDP and

interpret two sets of data related to GDP and GDP per capita.

Folder name: How much does a Big Mac cost?


Big Mac .PPT

Big Mac worksheet

Big Mac fill the gaps

Introduction to how the cost of a Big mac is used across the world as a simple measure of the cost of living.

Starts with an example of how the Big Mac index is calculated and moves on to investigate how long people would need to work in different countries to buy a Big Mac.

A table of calculations with some calculations and key information missing. Students use problem solving skills to insert the correct calculations and information that is missing.

F2.5 solving problems involving percentage change, including percentage increase/decrease and original value problems, including simple and compound interest

Folder name: Income, wealth and inequality; Autonomous task


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TBC

Folder name: The Gini coefficient


Gini Coefficient Students explore inequality and look at a measure of inequality, the Gini Coefficient, that is used internationally to compare levels of inequality. They will use percentages, interpreting graphs and plotting using 2 axes that are built up using cumulative percentages.

Use with The Gini PPT

F2.2 expressing one quantity as a percentage of another

F2.3 comparing two quantities using percentages

F2.5 solving problems involving percentage change, including percentage increase/decrease and original value problems, including simple and compound interest

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Percentage 2- Use and abuse – Teacher overview

Summary of content This section deals with some of the ways in which percentages are used and misused in order to support claims. Students should be

encouraged not to accept figures at face value and to examine the narrative behind the figures. Recognising that a percentage is a measure of proportion, so when making decisions it can be important to know the numbers behind the

percentages because a small percentage of a large figure may still be a large amount and a large percentage of a small amount is actually still a small amount. The difference between relative and absolute risk is a specific illustration of this point.

Examining the wording of particular claims to ensure the correct figures are used for the numerator and denominator in order to calculate the correct percentage.

Techniques to help visualise percentages; one good way of addressing this is to get them to think of 2% as 2 in 100, 0.2% as 2 in 1000 and 0.02% as 2 in 10 000 etc. Encouraging students to visualise these representations with reference to something from their own personal experience for example :

100 – a very full railway carriage1000 – size of a particular school perhaps, or the population of a particular (small) village10 000 – size of Cambridge United Stadium, or particular big village100 000 capacity of Wembley (not quite, but near enough)1 000 000 population of Birmingham

In this way 2% becomes two people in a busy railway carriage, 0.2% becomes 2 people in a school etc.

Pre-requisites: Students who do not have a good understanding of percentages might benefit from doing Percentages 1


Auto-marked online assessment: Section test - Percentage use and abuse section test



Folder name: Percentage use and abuse

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Percentage use and abuse student’s sheetV9.1 This resource is aimed at highlighting some of the ways in which percentages are used and misused in order to support claims. Students should be encouraged not to accept figures at face value and to examine the narrative behind the figures.

CMn1Learners should have a correct and critical understanding of the use of percentages including use of percentage change in a probability.

CMp8 Be able to interpret the solution to a problem effectively in terms of the original problem.

CMp10, Be able to evaluate critically information in public statements such as news reports and political comments. e.g. Proportions and percentages.

CMp 14 Be able to use numerical values to decide whether a general statement is realistic. Includes the consideration of extreme values.

Student’s clip percentages 1 A series of problems posed on the video

Percentage helpful hints Some hints to answering the problems

Teacher’s clip - percentages Answers


H5P Proportions and amounts

H5P Using proportions to compare risks

H5P Percentage – Preparing for exam questions

(Follow up questions to above H5p)

CMn1,p8,10,14 see above

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Shape and Measures – Teacher overview

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Summary of content If also done after the standard form section, applications to biological contexts of magnification with microscopes can be used. The idea of scale factors is basic to many areas of uses of mathematics such as maps, scale drawings, photographs from microscopes or space. Some students will need to interpret scale drawings and photographs as part of future work and study but most students will use online or paper maps when

planning a journey or researching a place they intend to visit. Relationships between lengths, areas, weights and volumes of similar figures are especially important in biology but also in manufacturing when considering how

to scale up a container to a larger size.

Pre-requisites: GCSE work on ratio

Approximate guided learning hours: 4 - 5 guided learning hours can be used a rough guide, however the amount of time required for this section will vary depending on the needs of individual students.

Auto-marked online assessment: Section test - Measures and scaling



Folder name: 2D and 3D shapes


How many coins starter A starter to introduce the topic. IQR m1 Be able to recognise and use relationships between lengths, areas, weights and volumes of similar figures to model real-world situations.

IQRm2Be able to work with time, length, area and volume to meet given regulations. e.g. In the context of meeting health and safety requirements. Regulations to be met will be given to learners.

IQRm3Be able to work with commonly used units and know that quantities being compared should have the same units; this includes compound units. e.g. Units of time, speed. e.g. Units of speed are units of distance divided by units of time. e.g. Understanding that the units of quantities arise from the way they are calculated. Where appropriate, conversion factors between metric and imperial units will be given.

IQRm4Be able to interpret diagrams, maps and scale drawings and be able to use them in problem solving.

IQRm5Be able to work with representations of

Worksheet for similar shapes powerpoint A worksheet to be used with the powerpoint

Changing between metric units A spreadsheet to practice changing between units

Questions on area and volume Questions for students to practice.

Worksheet for regulations A worksheet to be used with the card matching activity on regulations.

Regulation sorting card activity A card sorting activity to check students’ understanding of regulations..

Three-dimensional shapes A powerpoint with descriptions of three- dimensional shapes.

Worksheet for three-dimensional representations A worksheet to check students’ understanding of drawing three-dimensional shapes in two dimensions.

Cards for three-dimensional shapes A card matching activity to use with the

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H5P Ball pools

H5P Regulation cards

H5P 2 dimensional representations

H5P Scale drawings

H5P Speed

Geogebra applet–speed graphs

H5P Fastest Speeds

H5P Proportion practice

Geogebra applet – ratio practice

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Statistics 1 – Teacher overview

Summary of content This section provides an overview of the statistical problem-solving cycle, the collection of data and the use of statistical diagrams and summary

statistics to make sense of the data. Students should be encouraged to see statistics as a means of problem solving rather than as a collection of techniques. The focus of work on statistical diagrams should be on their use to make sense of data, using a spreadsheet to draw and label appropriate

diagrams and interpretation of diagrams. The collection and interpretation of data is important in science and social science and is also used in many careers, in politics, business and in

understanding the news. The ability to spot errors and rogue figures is part of the CBI’s definition of functional numeracy.

Pre-requisites: GCSE work on statistics


Auto-marked online assessment: Section test - Data types and sources



Folder name: Introducing the statistical cycle


Introduction to cycle A PowerPoint presentation for use in the classroom. This is a short introductory session, so there are no questions for students. The rest of the Statistics section is about considering each part of the cycle in turn.

Contains link to YouTube clips. Please check links still work.

IQRs1 Be able to decide what data need to be collected in order to answer a question requiring statistical evidence.

IQRs3 Be able to process and present the data and so provide an answer to the original question.

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Folder name: Asking the right questions


Asking the right question and collecting data Asking the right question and collecting data IQRs2 Be able to use a suitable method for collecting data, taking ethical considerations into account, and judge whether data are of sufficient quality. The data may be primary or secondary, and may be read off a graph or diagram

A PowerPoint presentation for use in the classroom. It covers all the key points of this section.

A PowerPoint presentation for use in the classroom. It covers all the key points of this section.

IQRs5 Understand and use the language describing types of data. Primary, secondary; categorical, numerical; continuous, discrete.

Folder name: Interpreting data shown in bar charts and population pyramids


Students sheet – UK babies 2016 This session is designed to encourage students to think more deeply about data presented in diagrams.

A good starting point would be a diagram which you suspect will get students talking and thinking about the data. The teacher’s powerpoint has two slides as an example.

Show the first slide and give the students some time to think. Ask them:

What do you notice?

What do you want to ask?

This sheet contains further diagrams and population pyramids to encourage students to think more deeply about data.

IQRs11 Be able to use and interpret statistical diagrams appropriate to a variety of types of data. Diagrams include: box and whisker plots, dot plots, scatter diagrams, bar charts, pie charts, histograms, frequency charts, cumulative frequency diagrams. Learners may be asked to complete these diagrams in the examination.

IQRs12 Be able to identify when a statistical diagram is misleading and explain how it could be improved. e.g. Improvement by clearer labelling or a better scale.

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Folder name: Social media influencers


Student sheet – influencers V5 Students explore data on social media influences.

Includes interpreting data displayed in tables, and considering how to measure the effectiveness of influencer advertising.



Folder name: Mean and standard deviation


Student sheet standard deviation Students learn about the mean and standard deviation from the introductory problem on the ppt, then progress onto a number of examination style questions involving calculating and interpreting the mean and standard deviation in a variety of contexts.

IQRs15 Be able to select and calculate appropriate measures of central tendency and to interpret them. Mean, median, mode. Includes grouped data and calculation or estimation for data in a statistical diagram. Number of data items = n Sample mean =

IQRs16 Be able to use appropriate measures of spread and to interpret them. Calculate range, inter-quartile range, semi inter-quartile range. Includes grouped data and calculation or estimation for data in a statistical diagram. Know that standard deviation is a measure of spread

Students sheet – DigitalBreathTestData2013 Spreadsheet task related to Q5 on student sheet

Folder name: Dot plots – Box plots - Stem and leaf diagrams

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Student’s sheet – Dot plots- Stem and leaf - Bar charts

(includes a matching task)

The students examine data from a variety of contexts.

The sheet provides guidance on how to calculate medians, Q1, Q3 and interquartile ranges. Students use these measures to compare samples.


IQRs16 Be able to use appropriate measures of spread and to interpret them. Calculate range, inter-quartile range, semi inter-quartile range. Includes grouped data and calculation or estimation for data in a statistical diagram. Know that standard deviation is a measure of spread


Folder name: Interpreting data using iNZight – Note this also has an H5P interactive video


Student sheet interpreting data (word version) Question shown at the end of the clip, plus another.


IQRs15 Be able to select and calculate appropriate measures of central tendency and to Student help sheet – using insight (word version) Instruction sheet about how to use the online

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data tool to obtain diagrams and summary statistics, plus interpretation questions

interpret them. Mean, median, mode. Includes grouped data and calculation or estimation for data in a statistical diagram. Number of data items = n Sample mean =

IQRs16 Be able to use appropriate measures of spread and to interpret them. Calculate range, inter-quartile range, semi inter-quartile range. Includes grouped data and calculation or estimation for data in a statistical diagram. Know that standard deviation is a measure of spread.

Data from CIA world factbook Data set as an Excel file.

File Name – Reaction time investigation


Reaction time investigation Core A version Idea for a teacher led project IQRs1-s17, CMp1-4, IQRp7, CMp7,8,9,11,14Student interactive study packages

H5P - Quantitative and qualitative data Online auto marked activity, with video content. IQRs5 Understand and use the language describing types of data. Primary, secondary; categorical, numerical; continuous, discrete.

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Estimation– Teacher overview

Summary of content Encourage students to practice estimation skills throughout the course; being able to recognise when an answer is reasonable is an important skill

for life. Estimation is included in the requirements for biology, chemistry, psychology, geology and environmental science A levels. The ability to spot

errors is included in the CBI’s definition of functional numeracy. When making an estimate, students need to make sensible assumptions about the sizes of numbers that will make the working easier; This is a

good opportunity to follow up by introducing the modelling cycle and making explicit what students have done. Students need to be able to communicate their method of estimation clearly.

Students should develop a quantitative understanding of the world – having a rough idea of what order of magnitude of answer is reasonable. The skills developed in this section are highly valued in industry or further study because they enable students to develop a sense of whether the

answer to a calculation is about right and enable them to get a rough answer with little time spent. Fermi estimates are named in honour of physicist Enrico Fermi who was skilled in such estimation and used questions like this with his students.

Questions like this are sometimes used in job interviews to test candidates’ thinking skills. Rather than doing all this section at once, teachers may prefer to develop and maintain students’ skills by using Fermi estimation from time to time

during the course.

Pre-requisites: This section links with modelling and standard form but students need not have done these before tackling this section.


Auto-marked online assessment: Section test errors and estimates , section test - Making estimates , section test Fermi estimates



Folder name: Roughly speaking

This folder contains teacher’s notes and a range of resources. The teacher notes are available in written form, or as two presentations with an audio commentary. There is also a mathematical background for teachers, again available in written form or as a presentation with audio commentary.

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6. Estimation starter Opening estimation task IQRe1 Be able to make a rough estimate of a quantity from available information.

IQRe2 Be able to use estimates when checking calculations.

IQRe3 Be able to make and justify upper and lower bounds for a calculation. Includes selecting and/or justifying an appropriate level of accuracy for an answer to a calculation. Maximum, minimum, upper bound, lower bound.

IQRe4 Be able to interpret and present error bounds or tolerances on diagrams and in writing, understanding that different levels of tolerance are appropriate in different situations. Error bounds may be required in percentage form. 12 ± 0 5. 340 ± 10% 8 5cm ≤ D ≤ 9 5. cm

7. Dealing with errors ppt Teacher’s powerpoint containing questions

8. Tiger population in India Spreadsheet which can be adapted for use with students

9. Calculations with error bounds Student’s worksheet

10 Calculations with error bounds solutions Solutions to student’s questions

11 Sf rule of thumb Spreadsheet with teacher’s note tab, and activity for use with students

12 Estimates of calculations Teacher powerpoint

13 Estimates of calculations Student’s worksheet building on activity introduced via the spreadsheet

14 Estimates of calculations (solutions) Answers to student’s worksheet of same name

Folder name: Tolerances


Worksheet for auction A worksheet to go with the powerpoint IQRe4 Be able to interpret and present error bounds or tolerances on diagrams and in writing, understanding that different levels of tolerance are appropriate in different situations. Error bounds may be required in percentage form.

Tolerances A powerpoint to introduce tolerance

Airplane A GeoGebra file to use with the powerpoint

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Notation

12 ± 0.5

340 ± 10%

8.5cm ≤ D ≤ 9 5 cm

Tolerances A GeoGebra file to show how tolerances can be presented

Cookies A GeoGebra file on tolerance

Worksheet for Cookies A worksheet to use with the GeoGebra file

Questions on tolerance Questions for students to discuss

Folder name: Getting started with estimates


Getting started with estimates Students sheet CMe1 Be able to make reasonable estimate of quantities typically encountered in everyday life. Quantities such as the following: • lengths, areas, volumes • estimates of adult height, weight and other body measurements • the time an adult would take to perform an everyday task.

CMe2 Know the rough size of the population of a large UK city, the UK and the world. About a million, about 65 million, 7 billion respectively.

CMe3 Know the rough size and seating capacity of common means of transport such as cars, buses, trains, and know typical speed limits on UK roads. Speed limits in miles per hour.


H5P Learning activity: Accuracy Online auto marked activity with video content IQRe3 Be able to make and justify upper and lower bounds for a calculation. Includes selecting and/or justifying an appropriate level of accuracy for an answer to a calculation.

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Maximum, minimum, upper bound, lower bound.

IQRe4 Be able to interpret and present error bounds or tolerances on diagrams and in writing, understanding that different levels of tolerance are appropriate in different situations. Error bounds may be required in percentage form. 12 ± 0 5. 340 ± 10% 8 5cm ≤ D ≤ 9 5. cm

H5P estimation Online auto marked activity with video content Provides an introduction to estimation comparing/linking GCSE to Core Maths

Folder name: Making estimates with limited data


Students sheet A series of problems involving estimation CMe4 Be able to calculate an estimate with limited information (a Fermi estimate) using quantitative understanding of the world. e.g. Estimate how many dentists there are in Birmingham. Assumptions and reasoning should be communicated clearly.

CMe5 Recognise when the order of magnitude of an answer is reasonable.

CMp12 Recognise that a “good enough” solution to a problem can save time and money compared to a more accurate solution.

CMp13 Be able to criticise or refine a proposed solution to a problem.

CMp14 Be able to use numerical values to decide whether a general statement is realistic.

Making estimates part1.mp4 A video which poses an estimation problem CMe4 Be able to calculate an estimate with

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related to a business start up limited information (a Fermi estimate) using quantitative understanding of the world. e.g. Estimate how many dentists there are in Birmingham. Assumptions and reasoning should be communicated clearly.





Making estimates part2.mp4 Some hints – with learning points

Making estimates teachers clip1.mp4 A possible solution

Folder name: Estimations


Teacher notes and student sheet Fermi estimates

Provided to support the teaching of this activity.

Includes answers to the questions on the student sheet.

Fermi estimates or “back of an envelope” calculations can be used in a variety of situations. These calculations can often save time and money for both individuals and also for

CMe1 Be able to make reasonable estimate of quantities typically encountered in everyday life. Quantities such as the following: • lengths, areas, volumes • estimates of adult height, weight and other body measurements • the time an adult would take to perform an everyday task.

CMe2 Know the rough size of the population of a large UK city, the UK and the world. About a

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companies. They can also be used to test quickly and informally the accuracy of statements which are made in the press or in ‘too good to be true’ business offers.

The idea is not to calculate something precisely, nor to spend hours searching the internet for information, but to get a quick rough idea.

Fermi estimates get their name from the physicist Enrico Fermi, who liked to discuss questions which drew upon everyday knowledge and experience of the world.

million, about 65 million, 7 billion respectively.


CMe4 Be able to calculate an estimate with limited information (a Fermi estimate) using quantitative understanding of the world. e.g. Estimate how many dentists there are in Birmingham. Assumptions and reasoning should be communicated clearly.





Folder name: Further estimates

These resources cover the same learning points as the previous resources. They have been included to allow their use as:

An alternative context/approach to some of the previous content A revision or consolidation resource


Student sheet - Put these in order Quick sorting activity to get students thinking about estimation and quantifying

CMe1 Be able to make reasonable estimate of quantities typically encountered in everyday life. Quantities such as the following: • lengths, areas, volumes • estimates of adult height, Student sheet – Estimation questions Some estimation questions – linked to

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powerpoint weight and other body measurements • the time an adult would take to perform an everyday task.

CMe2 Know the rough size of the population of a large UK city, the UK and the world. About a million, about 65 million, 7 billion respectively.


CMe4 Be able to calculate an estimate with limited information (a Fermi estimate) using quantitative understanding of the world. e.g. Estimate how many dentists there are in Birmingham. Assumptions and reasoning should be communicated clearly.





Student sheet – Loch Ness map Needed for the long question on the powerpoint

Student sheet – Match these facts A matching activity

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Statistics 2 – Teacher overview

Summary of content Students should be encouraged to see statistics as a means of problem solving rather than as a collection of techniques. The focus of work on statistical diagrams should be on their use to make sense of data, using a spreadsheet to draw and label appropriate

diagrams and interpretation of diagrams. The collection and interpretation of data is important in science and social science and is also used in many careers, in politics, business and in

understanding the news. The ability to spot errors and rogue figures is part of the CBI’s definition of functional numeracy.

Pre-requisites: This section builds on the content of Statistics 1


Auto-marked online assessment: Section test - Representing data OCR



Folder name: Percentiles and cumulative frequency Note - this activity has supporting videos


Percentiles

Student worksheet Introduces the idea of percentiles using baby/child growth curves. Leads into work on cumulative frequency curves


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Cumulative frequency

Student worksheet – Cumulative Frequency

Plus

Demo video – Autograph cumulative frequency

Demo video – GeoGebra cumulative frequency

Using GeoGebra to draw cumulative graphs (instruction sheet)

GeoGebra applet – Cumulative frequency practice

GeoGebra applet – Cumulative frequency practice

Using GeoGebra to draw cumulative graphs (instruction sheet)

Work on cumulative frequency curves starting with the context of baby growth charts.

A set of instruction explaining how to draw a cumulative frequency curve using GeoGebra


IQRs16 Be able to use appropriate measures of spread and to interpret them. Calculate range, inter-quartile range, semi inter-quartile range. Includes grouped data and calculation or estimation for data in a statistical diagram. Know that standard deviation is a measure of spread.

Folder name: Weighted mean


Weighted mean questions Some questions on weighted means IQRs17 Be able to calculate a weighted mean and recognise when it is appropriate to do so.

Folder name: Histograms


Students- Histograms resourceV2 Students interpret and construct histograms from data covering a variety of context including real data on road traffic accidents, and roadside breath tests.

IQRs11 Be able to use and interpret statistical diagrams appropriate to a variety of types of data. Diagrams include: box and whisker plots, dot plots, scatter diagrams, bar

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charts, pie charts, histograms, frequency charts, cumulative frequency diagrams. Learners may be asked to complete these diagrams in the examination.


IQRs13 Be able to identify skewness from a histogram, frequency chart or box and whisker plot. In appropriate contexts. Positive and negative skewness.

Folder name: Which diagram?


Students sheet – which diagram activity See comments on related teacher sheet IQR s7 Be able to read information from a table and to construct a table to present information.Includes grouping data using suitableclass intervals.

IQRs11 Be able to use and interpret statistical diagrams appropriate to a variety of types of data. Diagrams include: box and whiskerplots, dot plots, scatter diagrams,bar charts, pie charts, histograms,frequency charts, cumulativefrequency diagrams.

IQRs12 Be able to identify when a statistical diagram is misleading and explain how it could be improved.

Student sheet – Which diagram data set sheet See comments on related teacher sheet

Which diagram - data sets Excel file of data sets to support the activity

6 Student sheet – Which diagram data set sheet

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Folder name: Summary measures


Summary measures and diagrams

A powerpoint presentation for use in the classroom. This is a substantial presentation which contains many of the key ideas. You will want to split up this work.

This set of resources provides an opportunity to revisit aspects of work from statistics 1 and 2.

Air pollution data

An Excel file with data on air pollution which you may wish to use, if you are following the ongoing investigation, instead of asking students to download it.

Newspaper articleA task which invites students to write a short piece for a newspaper which uses a diagram to illustrate the thrust of the article.

Questions: Understanding statistical diagrams This needs the following resource.

Diagrams for Questions: Understanding statistical diagrams

Needed for the questions. It is a PowerPoint presentation best viewed on individual screens, saving the need for printouts.

Solutions for Questions: Understanding statistical diagrams Solutions for the questions above

Questions: using a spreadsheet to explore data

Question 1 guides students through some practice of using a spreadsheet to answer some statistical questions. Question 2 asks students to choose an aspect of a dataset which they will illustrate using summary measures and diagrams. Both questions need the following datafile.

Data for Questions: using a spreadsheet to An Excel file with several sets of data. These are

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explore data primarily for use with the Questions, but can be used in other ways.

Solutions to Questions: using a spreadsheet to explore data

Solutions to question 1; question 2 is open-ended.

Folder name: Autonomous learning taskThis task is designed to give students the opportunity to bring together the knowledge, skills and understanding they have developed in this section, in a realistic, work related context.


ALT – Student’s task sheet Encourages the students to think of three questions they would like to answer using the data set. Then think about how they will display the data and what calculations to use.

Students are asked to produce a poster or a powerpoint and present their findings.

IQR s15 Be able to select and calculate appropriate measures of central tendency and to interpret them. Mean, median, mode. Includes grouped data and calculation or estimation for data in a statistical diagram. Number of data items = n Sample mean = IQR s16 Be able to use appropriate measures of spread and to interpret them. Calculate range, inter-quartile range, semi inter-quartile range. Includes grouped data and calculation or estimation for data in a statistical diagram. Know that standard deviation is a measure of spread.

ALT – Student’s sheet – DigitalBreathTest2013.xlx

Large data set on Excel spreadsheet

File name: Exam style questions – Histograms


Exam style questions- Histograms A selection of questions based on previous examination questions

Examinations practice/awareness


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Weighted mean An auto-marked online activity which supports the learning outcomes for weighted mean.

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Correlation and causation – Teacher overview

Summary of content

Correlation is where the scatter diagram shows an (approximate) straight line relationship between the two variables being graphed. There is association between two variables if knowing the value of one provides information about the likely value of the other (the relationship

does not need to be linear for association); association is a more general term than correlation. News reports and political statements sometimes identify an association between two variables and imply that one is causing the other; in this

section students should develop a clear understanding of the difference between some form of association/correlation and causation. They should be able to recognise and explain some of the reasons for incorrectly assuming causation, and identify where further information is

required before a decision can be reached. The material in this section is relevant for careers in journalism and politics as well as for informed citizenship.

Pre-requisites: Familiarity with scatter diagrams from GCSE


Auto-marked online assessment: Section test – Is this a valid argument?



Folder name: Correlation – diagrams for bivariate data


2.14 Bivariate data powerpoint

2.15 Bivariate data ggb

A presentation to support the teaching of this section

A GeoGebra file which can be used with the presentation.

IQRs14 Be able to interpret a scatter diagram for bivariate data, draw a line of best fit by eye when it is appropriate to do so and understand that extrapolation might not be justified. Including the terms association, correlation, line of best fit.

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2.16 introduction to scatter diagrams Student’s question sheet IQR s11 Be able to use and interpret statistical diagrams appropriate to a variety of types of data. Diagrams include: box and whisker plots, dot plots, scatter diagrams, bar charts, pie charts, histograms, frequency charts, cumulative frequency diagrams. Learners may be asked to complete these diagrams in the examination.

IQRs14 Be able to interpret a scatter diagram for bivariate data, draw a line of best fit by eye when it is appropriate to do so and understand that extrapolation might not be justified. Including the terms association, correlation, line of best fit.

2.18 corr coeff An interactive Geogebra file for investigating how to interpret correlation on scatter diagrams.

Knowing how to calculate the pmcc is not in the IQR subject content, but meeting it like this is valuable.

2.19 corr coeff 2 An interactive activity for students, based on a Geogebra file. Students have to estimate which of two scatter diagrams shows the stronger positive correlation.

Correlation and causation (updated 2020)

H5P – Correlation and causation Interactive video with pause point reflection questions for use in teacher led session.

CMs4 Understand that neither correlation nor association implies causation.

Including being able to suggest possible alternative explanations.H5P – Correlation and causation without pause

pointsAbove video but without pause points

Students sheet - Correlation sort Activity linked to the video clip.

Students sheet – independent and dependent variables

Activity linked to the video clip

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Students sheet – Is this a valid argument? Activity investigating false/spurious correlations

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Exchanging currency – Teacher overview

Summary of content In this section, students will use multipliers to work out equivalent amounts in different foreign currencies and also estimate equivalent amounts in

different foreign currencies. In this section, students learn to interpret and work with foreign exchange information presented in a variety of ways – it would be useful to ask

them to bring examples of information about foreign currency exchange which they see advertised or in their area. Nearly all students will go abroad at some time and need to be able to decide where to change their money and to calculate rough equivalent

prices in pounds when shopping abroad.

Pre-requisites: It will be helpful if students have done the work on multipliers in Percentages 1 and the introduction to estimation before doing this section.


Auto-marked online assessment: Section test: Exchanging currency



Folder name: Exchanging Currency


Student’s sheet – Currency exchange investigation

Investigating the price of the same article on different Amazon sites – (Amazon.uk, Amazon.us, Amazon.de ….)

IQRf8 Be able to use foreign exchange rate information to make calculations, including calculations for currency exchange with commission or a fee. In the UK, “sell at 1.54, buy at 1.69” means that when converting from pounds to the currency, a customer gets 1.54 of the currency for £1, but when changing the currency to pounds, 1.69 of the currency is needed for £1. “buy at”; “we buy”; “buy rate” are

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different ways of saying the same thing; similarly, “sell at”; “we sell”; “sell rate”.

IQRf9 Be able to decide which foreign exchange rate is most advantageous for a particular exchange without doing the calculations. Deciding and justifying which exchange rate is most advantageous to the customer.

Student’s spreadsheet – Comparing prices in different countries

Spreadsheet to help with above task. (this can be adapted)

IQRq2 Be able to read information from a standard spreadsheet.

IQRq3 Be able to enter formulae and data into a spreadsheet, knowing that a standard spreadsheet formula starts with =. Formulae based on the 4 rules of arithmetic and other standard functions required by the rest of this specification, e.g. to the power of, square root. e.g. =B2*(C2+D2) =C3^4 =SQRT(A10)

IQRq4 Be able to interpret simple formulae on a spreadsheet given in terms of cell references. Examples include money, number patterns and simple sequences. e.g. =AVERAGE(A1:A9)

IQRq5 Be able to copy a formula and to ensure that only the required cell addresses increment. e.g. $A1, $A$1, A$1

Student’s sheet – Four currencies sorting puzzle Mixing percentage and currency exchange IQR f8, f9 shown above

Student’s sheet – NO commission Currency exchange without commission IQR f8, f9 shown above

Student’s sheet – Currency exchange with commission

Currency exchange with commission IQR f8, f9 shown above

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H5P Exchanging currency Online auto marked activity with video content. Useful as a homework or to consolidate classwork.

IQR f8, f9 shown above



Big Mac .PPT

Big Mac worksheet





IQRf8 Be able to use foreign exchange rate information to make calculations, including calculations for currency exchange with commission or a fee. In the UK, “sell at 1.54, buy at 1.69” means that when converting from pounds to the currency, a customer gets 1.54 of the currency for £1, but when changing the currency to pounds, 1.69 of the currency is needed for £1. “buy at”; “we buy”; “buy rate” are different ways of saying the same thing; similarly, “sell at”; “we sell”; “sell rate”.

IQRf9 Be able to decide which foreign exchange rate is most advantageous for a particular exchange without doing the calculations. Deciding and justifying which exchange rate is most advantageous to the customer.

.

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Comparing and deciding – Teacher overview

Summary of content This section focuses on ways to decide whether a general statement is true, whether one quantity is bigger than another or which course of

action is better. The emphasis is on looking at the information presented in a way which makes a decision easy to make. A number of strategies should be encouraged.

o Using a graph and looking at the message it shows rather than reading off individual values.o Using a large or small number to test a general statement.o Thinking about which calculation will give a larger answer without actually doing the calculation.

The skills which this section focuses on are useful in testing statements made in the press or in advertising and quickly seeing whether they are likely to be correct.

Teachers may prefer to come back to this section from time to time rather than doing it all at once.

Pre-requisites: Some of the skills in this section are used in the IQR section on foreign exchange which can either be done before or after this section.

Approximate guided learning hours: 4 – 5 guided learning hours can be used a rough guide, however the amount of time required for this section will vary depending on the needs of individual students.

Auto-marked online assessment: Section test – Comparing quantities



Folder name: Comparing quantities


Students sheet - Comparing quantities finalStudent problems/tasks

Currency game - Spreadsheet to support introductory activity

The questions contained in this resource are designed to develop a student’s ability to compare quantities using reasoning rather than calculations. In addition, this skill can be developed further throughout the course, for example when checking the reasonableness of solutions to other questions.

CMp1 Be able to decide what information is needed to address a problem.

CMp2 Be able to make, and justify, simplifying assumptions in order to solve a problem.

CMp5 Be able to compare two quantities by reasoning, without working out the value of

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either.

Contexts include finance, area, volume, speed, probability. e.g. Recognising that the probability of a car breaking down on a Monday is less than the probability of the car breaking down sometime during that week because that latter event includes the former event.




Folder name: Exam style question


Available to teachers only an exam style questions which can be used either as is, with low or high level hints.

Other resources

These resources are external to the Integral websiteBuying Cars from the MAP resources.


Under developmentPercentages 3 - Repeated change – Teacher overview

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Summary of content

Percentages are often used in the news, not always correctly – students should be encouraged to look out for examples of the use of percentages and to explain what the numbers are saying in simple terms – this is a useful skill for life, work and study.

Appreciation means growth, depreciation means getting smaller – the words are usually applied to increase and decrease in value. Savings increase in value, by the percentage shown by the interest rate – this can be either constant or changing but it is often easier to

assume it is constant to give an idea of how the savings might grow over time. This is a form of modelling. This section needs to be done after the earlier percentage work on multipliers. This section makes use of mathematical modelling, considering growth or decrease by a constant percentage rate. It would be useful for

students to connect this to the modelling cycle. Ideas of appreciation and depreciation are important when thinking about investments (including pension funds), the value of cars, houses and

other major purchases including those made by businesses. The effect of inflation on prices should also be considered here.

Pre-requisites: Percentage 1 and 2Approximate guided learning hours: 5-6 guided learning hours can be used a rough guide, however the amount of time required for this section will vary depending on the needs of individual students.Auto-marked online assessment: Section test – Percentages 1-3(Solutions available in the teachers section only)Teacher notes, solutions and any other support materials are contained in the teacher folders

Folder name: Repeated percentage change. Note: This resource has a supporting video clip


Depreciation worksheet A simple worksheet to promote discussion comparing constant percentage decrease in value with the way car values actually decrease.

IQRf5 Be able to do calculations involving percentages in context; the use of an index number to compare a number or value to that in a base year.

IQRf6 Know how to use percentages to work with appreciation or depreciation

IQRf7 Be able to work out an average annual

Depreciation spreadsheet A spreadsheet which allows students to compare a constant annual percentage decrease with decreasing values from a website.

Repeated percentage change A powerpoint introducing repeated percentage

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change with a spreadsheet and with a calculator – the latter using powers of multipliers.

percentage growth (or reduction) rate for a given change over a period.

IQRf14 Understand and use the language of finance. Words such as income, expenditure, budget, profit, loss, investment, tax, revenue, inflation, APR and AER.

IQRq3 Be able to enter formulae and data in to a spreadsheet, knowing that a standard spreadsheet formula starts with =. (Formulae based on the 4 rules of arithmetic and other standard functions required by the rest of this specification, e.g. to the power of, square root.)

IQRq4 Be able to interpret simple formulae on a spreadsheet given in terms of cell references. (Examples include money, number patterns and simple sequences.)

IQRq5 Be able to copy a formula and to ensure that only the required cell addresses increment.

IQRp4 Be able to compare the outcomes from a model with actual data, information, experience or common sense and comment on the appropriateness of the model. (For example, compare an exponential growth model with actual population figures. The information may be given in diagrammatic or graphical form.)

Rising prices questions An introduction to the effects of inflation over time and the idea of an average annual percentage increase.

Growing money spreadsheet A spreadsheet showing the effect of a repeated percentage increase on money – it can be used for inflation or interest calculations.

Rising prices solutions Full written solutions to the rising prices questions

Average percentage change A powerpoint showing how to find the average annual percentage change over a period of time.

AER and APR questions Information and questions about interest on savings and loans

Repaying loans spreadsheet A spreadsheet which allows students to explore loan repayments; amount borrowed, interest rate and repayment can all be varied.

Rule of 72 A spreadsheet exploration of an approximating rule used in finance calculations. Students should write their own spreadsheet to explore this.

Compare House Price A starter question to encourage students to think about the need for index numbers – available in pdf or powerpoint format.

Index powerpoint A powerpoint which goes through the process of calculating an index.

Average House Index spreadsheet This allows students to practice calculating an index and gives feedback on whether they are

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right. Answers are also available.

Index Compare questions Questions for students about using an index to make comparisons.

UK House price spreadsheet Data for students to use when comparing house prices (see last question of Index Compare).

Index percent change powerpoint A powerpoint which goes through the process of finding a percentage change from an index number.

House building spreadsheet Data used in the powerpoint.

Index percent change questions Questions about finding percentage change from an index number.

Calculating Inflation spreadsheet Data needed for the final Index percent change question.

Index percent change solutions Full written solutions to the questions.

Questions: repeated change Mixed questions about repeated percentage change, includes questions about the average percentage change over a period of years.


Under development

Standard index form – Teacher overview

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Summary of content Some students will have used standard form to express very large or very small numbers at GCSE. Others may not be familiar with this. They are

likely to encounter numbers expressed in standard form when working with repeated percentage change on either a calculator or a spreadsheet and need to be able to understand the way that the calculator or spreadsheet shows the number.

Standard form is used in science to write very large or small numbers; it is also called scientific notation.

Pre-requisites: It would be useful to do this section soon after the section on Percentages 3


Auto-marked online assessment: Section test – Standard form



Folder name: Standard index form


Order of size A video to look at how big objects are. Students could look at this video answer the questions.

IQRl1 Be able to interpret large or small numbers in standard form, including the use of a spreadsheet or calculator. Standard form is sometimes called scientific notation. 3.1 × 108

3.1 E +08 3.1 E +008 3.1 EXP 08

IQRl2 Be able to calculate with numbers in standard form. e.g. Contexts such as astronomy, wavelengths, atoms or cells.

IQRq1 Be able to use a standard calculator (scientific or graphical).

Ordering numbers A spreadsheet for students to check they have ordered numbers correctly. Starter begins with large numbers and what they mean which will help to find out how much students know about large numbers. Film Link

Film Link

https://www.youtube.com/watch?v=0fKBhvDjuy0Powers of ten - this show the size of the universe and atoms and is an introduction to the standard form notation.

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IQRq2 Be able to read information from a standard spreadsheet.

IQRq3 Be able to enter formulae and data into a spreadsheet, knowing that a standard spreadsheet formula starts with =. Formulae based on the 4 rules of arithmetic and other standard functions required by the rest of this specification, e.g. to the power of, square root.

Standard form powerpoint Shows how to covert numbers to and from standard form.

Large numbers video A video for students to work through themselves before the lesson, looking at changing large numbers to and from standard form

Small numbers video A video for students to work through themselves before the lesson, looking at changing small numbers to and from standard form

Checking standard form An Excel spreadsheet for students to use to check they can put numbers into standard form

Excel spreadsheet An Excel spreadsheet for students to work to check they can successfully enter numbers into Excel spreadsheets

Worksheet on how to use standard form on Excel

A worksheet to lead students through the Excel spreadsheet

Converting to and from standard form questions Questions for students to practise skills

Card Activity A matching activity for students to check they are able to convert between numbers in ordinary form and standard form.

The solar system A video for students to work through to see if they can work with numbers in standard form

Questions on planets Questions for students to attempt by themselves to check understanding of standard form.

Calculations using standard from powerpoint Shows how to tackle problems with large or small numbers on a calculator

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True or false questions Questions for students to check their understanding of standard form

Calculations using standard form questions Questions for students to practise skills


H5P Writing numbers using standard index form

H5P Match the cards activity

Online auto marked student activity to support the teaching of this section

IQRl1 Be able to interpret large or small numbers in standard form, including the use of a spreadsheet or calculator. Standard form is sometimes called scientific notation. 3.1 × 108 3.1 E +08 3.1 E +008 3.1 EXP 08

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Probability, frequency trees – Teacher overview

Summary of content An understanding of conditional probability is important in both medicine and law so this section is especially relevant to students who intend to

work in health, law and social work as well as for sport and for every citizen who may have to interpret medical or legal information one day either as a patient or on a jury.

When working with conditional probabilities, either in a two way table or in a tree diagram, students may find that they are better able to understand the situation if they work with representative frequencies than if they work with probabilities. Of course, if they are happier working with probabilities and able to use them successfully then that is also fine.

This section explores conditional probability in contexts such as understanding the risk surrounding a positive result from a screening test. These tests occur in a wide variety of situations such as pre-natal screening, cancer screening, DNA testing, airport security, lie detection and random drugs tests. Research shows that people find it difficult to understand the probabilities involved in such a test. In particular, if testing is taking place for some rare condition (that most people being tested don’t have), then with any test that is less than 100% accurate there is always a possibility of error that can result in a high proportion of the people testing positive not actually having the condition (false positives).

The students should be encouraged to investigate the different outcomes from these tests, particularly the ones which result in a false indication by the test – both false positive and false negative (people who do have the condition but it does not show up on the test). It is suggested that this is best achieved by examining the expected frequencies of each of the outcomes and then using these figures to make judgements about the likelihood of a false indication and the possible consequences.

An understanding of conditional probability is important in both medicine and law so this section is especially relevant to students who intend to work in health, law and social work as well as for sport and for every citizen who may have to interpret medical or legal information one day either as a patient or on a jury.

Pre-requisites: It will be best to do this section after the review of percentages; it follows up on the section on risk to look at more complex probabilities.


Auto-marked online assessment: Section test - Probability, section test - Probability and risk 2, section test - Probability and risk 3

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Deal or no deal game A game to start students thinking about probability and associated instructions

IQRu1 Be able to identify relevant equally likely outcomes in appropriate contexts. Includes understanding when outcomes are not equally likely.

IQRu2 Be able to count equally likely outcomes in appropriate contexts and hence estimate a probability. Includes listing and use of tree diagrams to find number of outcomes.

IQRu3 Be able to estimate probability from long-run relative frequency.

IQRu4 Be able to interpret two-way tables and use them to calculate or estimate probability.

IQRu5 Understand the difference between dependent and independent events and be able to calculate probability in simple cases. Contexts include games of chance and risk of suffering from diseases.

IQRu6 Be able to work with a tree diagram when calculating or estimating a probability, including conditional probability. Learners can choose to work with either frequencies or probabilities in tree diagrams.

Coins spreadsheet An Excel file to help student think about probability.

Worksheet for coins Excel file A worksheet to give students some ideas about using the spreadsheet

Dice Simulation worksheet A worksheet to guide students through designing a spreadsheet.

Excel file for the game PIG A spreadsheet to allow students to investigate the game PIG

2 Dice An Excel file which students can use to simulate throwing dice with different number of faces

Card activity A card activity to match bar charts with dice.

Outcome questions Questions to practise writing outcomes

Biased coin An Excel file for students to use to estimate probabilities

Estimating probability An Excel file for students to use to estimate probabilities

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Worksheet for estimating probabilities A worksheet to lead students through the Excel files

Relative frequency A powerpoint to lead students through relative frequency.

Relative frequency problems Questions to practise finding relative frequencies.

Two way tables questions Questions to check students’ understanding of two way tables.



Introduction to frequency trees A video to introduce frequency trees. IQRu2 Be able to count equally likely outcomes in appropriate contexts and hence estimate a probability. Includes listing and use of tree diagrams to find number of outcomes.

IQRu3 Be able to estimate probability from long-run relative frequency. u4 Be able to interpret two-way tables and use them to calculate or estimate probability.

IQRu5 Understand the difference between dependent and independent events and be able to calculate probability in simple cases. Contexts include games of chance and risk of suffering from diseases.

IQRu6 Be able to work with a tree diagram when

Galton grid A spreadsheet to accompany the video

Probability tree starter To start students thinking about probability trees

Worksheet driving tests A worksheet for students to draw a probability tree and answer questions on it

Initial representative frequency An Excel spreadsheet to help students work out a good starting number

Worksheet to use with initial representative frequencies

A worksheet to help understand the spreadsheet

Independent and dependent events questions Questions to practise on independent and dependent events

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calculating or estimating a probability, including conditional probability. Learners can choose to work with either frequencies or probabilities in tree diagrams.

Conditional probability starter To start students thinking about conditional probability

Tree diagram template A template to use with the activity.

Ideas to use with the tree diagram template A card activity to use with the template

Worksheet for theory tests A worksheet for students to draw a probability tree with conditional probability

Probability tree questions Questions on probability trees for students to practise.

Folder name: Screening


Students sheet screening V8.1 This activity is designed to make students aware of some of the issues coming from a positive result from a screening test. These tests occur in a wide variety of situations such as cancer screening, DNA testing, airport security, lie detection and drugs tests. Research shows that people find it difficult to understand the probabilities involved in such a test. In particular, if testing is taking place for some rare condition (that most people being tested don’t have) then any test that is less than 100% accurate (and there is always a possibility of error) can result in a high proportion of the people testing positive not actually having the condition (false positives).

A spreadsheet tool to support the teaching of this activity has been provided

CMu1 Know that a conditional probability of event A given event B is different from the conditional probability of event B given event A in relevant contexts.

Questions will be asked in words in context. e.g. Recognising that the probability of testing positive for a disease given that the patient has the disease may not be the same as the probability of the patient having the disease given that he/she tests positive.

CMu2 Interpret statements about population proportions in terms of probability and use them to make further deductions.

CM s2 Recognise common examples of incorrect reasoning in probability and be able to explain the errors. (partial)Recognise and explain the following: • the prosecutor’s fallacy and the defendant’s fallacy •

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the gambler’s fallacy


Setting up and using frequency trees

Conditional probability

Online auto-marked activities with video clips, to support the students learning in this area.

Fallacies in probability and statistics – Teacher overview

Summary of content This section includes regression to the mean, a statistical phenomenon that occurs when values or measurements of the values of a property

have variability. Sometimes the random variation of this property can produce what may appear to be a causal change. The aim of this section is not to encourage students to be cynical about all statistics quoted at them, but rather that students can recognise

situations where regression to the mean may form part of the explanation for an observation, then be able to check to see whether this has been taken into account in explaining the results.

Students should recognise situations where regression to the mean is likely to be most noticeable, for example an exceptionally good performance is often followed by a performance which is not as good, whilst an unusually poor performance is often followed by a better one.

Scams is intended to consolidate the students understanding of some aspects of probability such as expected frequencies. Students should develop an understanding of expected return and use this and other factors to enable them to form opinions about the validity of offers which

sound too good to be true. This is one of a number of sections aimed at demonstrating how mathematics can be helpful when making decisions. Being able to evaluate offers and opportunities and avoid being taken in is an important skill for future life.

Pre-requisites: There is no pre-requisite knowledge for regression to the mean. Scams is related to the section on probability but may be done before or after it. It is also related to the business and risk section -

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Auto-marked online assessment: Section test – Regression to the mean, section test - Scams



Folder name: Regression to the mean


Student sheets – Regression to the mean A series of questions related to regression to the mean

CMs2 Recognise common examples of incorrect reasoning in probability and be able to explain the errors

CMs5 Recognise situations where regression to the mean might occur.

Student sheet – Galton surveyV8.xlsm Spreadsheet for students to build on

Folder name: Scams


Student sheet – ScamsV3.1

Other resources

These resources are external to the Integral website.

Derren Brown Coin flipping Scam nrich

Scams Advice from the Financial Conduct Authority

By using this resource, the students should consolidate their understanding of some aspects of probability such as expected frequencies and develop an understanding of expected return.

The students should use mathematics to enable them to form opinions about the validity of offers which sound too good to be true.

CMs1 Understand that an event with small probability is not surprising in a sufficiently large population



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H5P regression to the mean Online activity to introduce regression to the mean. Auto marked with video content.


CMs5 Recognise situations where regression to the mean might occur.

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Randomised controlled trials, sampling and bias Teacher overview

Summary of content Randomised controlled trials are used in medicine to decide whether a proposed new medical treatment is effective. They have also been used by charities

trying to decide how best to help people and in education. Understanding how randomised controlled trials work is becoming increasingly important for informed decision making as a citizen as well as for careers in medicine and research.

Pre-requisites: This section follows on from the section Regression to the mean and is best done after that section.


Auto-marked online assessment: Section test – Randomised controlled trials



Folder name: Comparing sampling techniques


Student’s card sort – Statistical terms/definitions

Student’s card sort – Variance and bias

Two card sort activities have been provided. One which is designed to examine the difference between variance and bias and a second which looks at the definitions of some of the terms used in the section.

These card sorts could be used as a formative assessment/introductory task at the beginning of the session, or as a summative assessment/exit task, or one of each.

Although this content is not assessed it has been included to provide a better understanding of the overall theme of this section which is understanding potential sources of error in experiments and studies.

This activity relates to

IQR s10 Understand about the variability of data and be able to describe the main features of a distribution.

Sampling – Students sheet 1 Excel sheet (data file for use in the task)

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Instruction sheets

Random sample Stratified sample Systematic sample

These sheet are intended for students. They provide guidance on how to use Excel to take a sample from a large data set.

The intention is that students then analyse the distribution of the sample means, and examine the variation of the samples obtained using each technique.

Further guidance on using the sheets can be found in 1.1 Teacher’s notes sampling

Although this content is not assessed it has been included to provide a better understanding of the overall theme of this section which is understanding potential sources of error in experiments and studies.

These activities relate to

IQRs8 Understand the meaning of the terms sample and population. (The names of particular sampling methods will not be examined.)

IQRs10Understand about the variability of data and be able to describe the main features of a distribution.

CMp9 Be able to identify and comment on possible sources of bias or error when solving a problem.

Students sheet – Personal research task Students are encouraged to do some personal research. Some structure is provided in order to guide the project and summarise their findings.

This could be done either after the spreadsheet tasks detailed above or a pre-session task.

Folder name: Randomised controlled trials


Students sheet – RCT version4 Student’s worksheet to accompany this activity.

This resource aims to examine some of the aspects involved in setting up a statistical trial. The students are introduced to the idea of random variation by examining the way in which speed cameras are evaluated.

Objectives. Students should :

• Recognise what information is necessary in order to address a problem.

• Understand and explain some of the possible sources of error/bias when choosing a sample,

CMs6 Know what randomised controlled trials are and understand their importance in trying to establish causation.

CMs7 Understand the concepts and importance of “blind” and “double blind” in statistical trials.

CMs10 Understand that proportions from small samples are more variable than those from large samples.

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collecting data or conducting an experiment. For example, sample size being too small for confident conclusions, selection method or survey questions being biased.

• Evaluate the accuracy of results obtained when deciding on a model or experiment.

• Criticise or refine a proposed solution to a problem.

• Recognise ways in which statistics are misused and be aware of common misconceptions.

• Understand the importance of randomised controlled trials when trying to establish causation.


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Algebra, graphs and gradients – Teacher overview

Summary of content The main ways that all students are likely to encounter algebra in future life is when using a spreadsheet or when using a formula related to their future work or

study. It will be helpful if students have already used spreadsheets for calculations before doing this section. In some careers, formulas are given in words rather than in terms of algebraic symbols – this is typical in nursing, for example. For scientists who want to communicate their work internationally, algebraic symbolism is understood in the same way all over the world. We live in a scientific

age and it is important for all informed citizens to appreciate the power of algebraic notation. Interpreting graphs is important in science, business, economics, psychology and social science; it is also important for future life and work – graphs are used in magazines and reports.

Pre-requisites: This section is largely a review of GCSE work on formulas and equations.


Auto-marked online assessment: Section test – Algebra, section test - Graphs and gradients



Folder name: Algebra resources


Setting up and solving equations A powerpoint to lead students through setting up and solving equations.

IQRa1 Be able to represent a situation mathematically using a formula or equation. Using both traditional algebra and spreadsheet notation.

IQRa2 Be able to substitute values into a formula given in symbols, words or as a flow chart. Formulae will be confined to the following cases (or simple combinations of these): • polynomials • simple rational expressions • exponential growth and decay • trigonometric

Equations spreadsheet A spreadsheet to check answers.

Setting up and solving equations questions Questions for students to practise.

Setting up and solving equations solutions Full written solutions to the questions.

Worksheet for trial and improvement A worksheet to lead students through trial and

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improvement. functions (sin and cos).

IQRa3 Be able to solve equations and change the subject of a formula. In simple cases using the four operations, powers and roots. Solve more complex equations using trial and improvement or a graphical method.

Worksheet for trial and improvement teacher’s solutions

Solutions to the worksheet for the teacher.

Area of garden spreadsheet A spreadsheet to use for trial and improvement.

Rearranging equations A powerpoint to lead students through one way of rearranging equations.

Converting temperature worksheet A worksheet for students to rearrange equations using Excel.

Converting temperature spreadsheet A completed spreadsheet for teachers to use to help students.

Rearranging formulas questions Questions for students to practise.

Folder name: Graphs


Examples of graphs A matching activity to match graphs with words. IQRg1 Know that the independent variable is plotted on the horizontal axis of a graph. Dependent variable, response variable, independent variable, explanatory variable.

IQRg2 Be able to construct a table of values for a graph from a simple formula and use it to plot the graph. Includes trigonometric graphs (sin and cos) for angles in degrees.

IQRg3 Be able to use a graph to construct a table of values.

IQRg4 Be able to work with graphs drawn from

Examples of graphs solutions Solutions to the activity.

Worksheet for motion graphs A worksheet to fill in to check understanding of motion graphs.

Worksheet for motion graphs solutions Solutions to the worksheet.

Motion graphs questions Questions on motion graphs.

Motion graphs solutions Full solutions to the questions.

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a variety of contexts. Includes graphs representing motion along a straight line, time series graphs, step graphs, periodic graphs, graphs of exponential growth and decay and piecewise graphs.

IQRg7 Understand the relationship between a straight line graph and the formula connecting the variables graphed.

Worksheet for drawing straight line graphs A worksheet to lead students through drawing a straight line graph.

Worksheet for drawing straight line graphs solutions

Solutions for the teacher.

Worksheet for drawing curved graphs A worksheet to lead students through drawing curved graphs.

Worksheet for drawing curved graphs solutions . Solutions for the teacher

Drawing graphs questions Questions on drawing graphs to check understanding.

H5P Motion

H5P Motion graphs

Student resources on motion graphs

Student resource to consolidate motion graphs

Folder name: Gradients


Order of steepness A video to introduce gradient. IQRg7 Understand the relationship between a straight line graph and the formula connecting the variables graphed.

IQRg8 Be able to find the gradient of a straight line graph and interpret it in context, taking account of the scales on the axes and using appropriate units. Includes finding units for the gradient from units on the axes.

IQRg9 Be able to estimate the gradient of a curve at a point by drawing a tangent and interpret it as a rate of change. Includes e.g.

Steepness worksheet

Steepness worksheet black and white

A worksheet to be used with the video.

Checker An Excel spreadsheet to check the order of steepness and gradients.

Gradient of a straight line video A video revising gradients for those who have some knowledge of it from GCSE.

Gradients A powerpoint to introduce gradients of straight

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line graphs kinematics graphs and growth curves.

What does gradient mean video A video for students to consider the meaning of the gradient of a straight line.

What does gradient mean answers Answers for the teacher only

Gradient of curves A powerpoint to introduce gradients of curves.

Gradient of curves A Geogebra file to investigate gradients at different points on curves.

Worksheet for gradients on curves A worksheet to be used with the Geogebra file.

Changing units An Excel spreadsheet to help change units

Gradients questions Questions on gradients to check understanding.

H5P Gradients

H5P Gradients on graphs

H5P Which is the steepest

Student resource on gradients

Student resources to check understanding of gradients

Folder name: Direct and inverse proportion


Direct and inverse proportion questions Questions for students to tackle IQRg5 Recognise graphs of direct and inverse proportion.

IQRg6 Be able to linearise the graph of a relationship where the dependent variable is directly proportional to some function of the independent variable. e.g. Plot y against x2 to investigate a relationship of the form y = ax2

Proportional graphs ppt To support teaching of this content

Direct proportion questions Questions for students to tackle

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H5P Equations An online auto marked activity recapping on GCSE work and building understanding for Core Maths work.

H5P Gradients

H5P Gradients on graphs

H5P Put in order of steepness

Student resource on gradients

Student resources to check understanding of gradients

H5P Motion

H5P Motion graphs

Student resources on motion graphs

Student resource to consolidate motion graphs

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Interpreting problems and communicating solutions – Teacher overview

Summary of content This section contains activities which aim to promote the usefulness of mathematics when making informed choices. such as deciding a price

for a new product to be sold by a business and an understanding of some of the systems used to decide on the winner of a contest. Teachers may wish to use individual activities periodically throughout the course rather than doing the whole section at once. In product pricing students should consider issues such as :

o The price of the product is likely to affect the number of sales for example they are likely to sell more if the price is lower. o If they produce more of the product, this might also make the cost for each one lower, particularly when considering fixed costs.o Selling more is not always more profitable, and that there are some other advantages to selling fewer items at a higher cost.

In voting systems, the students should be able to comment on their interpretation of the fairness of the system using relevant points as justification.

The knowledge and skills gained from working on this resource could be used in other contexts such as an examination of the voting systems used by governments to pass laws, or in general elections.

This section encourages the students to aim to find an optimum solution, hopefully revealing that mathematics can be useful in understanding this process.

This section connects to further study and careers in business.

Pre-requisites: A spreadsheet may help with some task.


Auto-marked online assessment: Section test – Voting systems, section test – Product prices



Folder name: Investigating voting systems


The cutest animal photo competition blank Template for use with optional introductory activity


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Students sheet Voting systems

Resources for students covering content in learning outcomes.


CMp3 Be able to recognise when a problem is similar to one which has already been solved. Learners may be asked to give examples related to problems which they have solved during the course.

CMp4 Evaluate different strategies for problem solving.

CMp6 Be able to use data to make and justify a decision. e.g. Contexts include using data to produce a rank order.

CMp7 Be able to communicate the solution to someone who understands the problem.



Folder name: Investigating the proposed Olympic climbing event (Toyko 2020)


Students Sheet – Climbing – points system

Simulation Olympic Climbing final A spreadsheet tool



CMp3 Be able to recognise when a problem is similar to one which has already been solved. Learners may be asked to give examples related to problems which they have solved during the

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course.

CMp4 Evaluate different strategies for problem solving.





Folder name: Pricing products


Product pricing template A template for a spreadsheet with the questions from the teachers powerpoint in a text box.



CMp3 Be able to recognise when a problem is similar to one which has already been solved. Learners may be asked to give examples related to problems which they have solved during the course.

CMp4 Evaluate different strategies for problem

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solving.






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Exponentials and logarithmic scales – Teacher overview

Summary of content Appreciation by a constant percentage is a form of exponential growth. Depreciation by a constant percentage is a form of exponential decay. The modelling cycle can be introduced here in the context of the use of exponential growth as a model. Exponential growth is used as a model for population growth and so has applications in all the social sciences as well as business, economics and

science. Exponential decay is important in business when modelling depreciation and also in science for radioactive decay. Radioactive decay is important in

carbon dating in archaeology and also in considering issues to do with nuclear power so a basic understanding of it is important for all citizens. Examples of logarithmic scales are pH in chemistry, decibels and the Richter scale for earthquakes. Logarithmic scales are sometimes used in graphs

showing economic data to make the scale more manageable.

Pre-requisites: It will be helpful if students have done both the Percentages 3 and the algebra and graphs sections before this section.


Auto-marked online assessment: Section test – Exponential growth and decay, section test – Working with exponentials, section test – Log scales



Folder name: Exponential graphs


Bacteria graph

Depreciation graph

Half life

Savings

GeoGebra files to support the session – see teachers’ notes for further guidance

IQRl3 Be able to explore exponential growth and decay, including interpreting output from a spreadsheet. Contexts include borrowing and saving money, bacterial growth and radioactive decay.

IQRl4 Be able to represent and interpret Simulating dice to use Excel files to support the session – see

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Simulating dice example teachers’ notes for further guidance exponential growth or decay in a graph. Learners may be asked to plot or sketch exponential graphs. Half-life. y = kax with k and a constant.

Student worksheets Materials for students.

Folder name: Logarithmic scales


Changing to Joules A spreadsheet activity to use with video on Richter scale

IQRq3 Be able to enter formulae and data into a spreadsheet, knowing that a standard spreadsheet formula starts with =.

IQRq5 Be able to copy a formula and to ensure that only the required cell addresses increment.

IQRq7 Be able to use a spreadsheet to draw graphs and standard statistical diagrams and interpret graphs produced on spreadsheets.

IQRp2 Be able to develop or choose a simple mathematical model for a real-world situation. Model in words, numerically, algebraically, diagrammatically or in a spreadsheet.

IQRp3 Be able to use a model to make predictions or get information about a situation..

Worksheet for Richter scale

https://www.youtube.com/watch?v=pUSzVL0kGEA&feature=youtu.be

A worksheet for use with video

Worksheet for log graphs A worksheet to use with log graph video

How many cells starter A revision starter to check understanding from exponential growth. Related teacher ppt provided “Looking back on exponential graphs”

Worksheet for Excel files A worksheet to give students some ideas about using the spreadsheet.

Worksheet for logarithmic scales A worksheet for students to use to see the relationship between linear scales and logarithmic scales. Related powerpoint for teachers provided “Logarithmic scales”

Follow me card activity A card matching activity to consolidate understanding of powers of ten.

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IQRp4 Be able to compare the outcomes from a model with actual data, information, experience or common sense and comment on the appropriateness of the model. For example, compare an exponential growth model with actual population figures. The information may be given in diagrammatic or graphical form.

IQRl6 Be able to use and interpret a logarithmic scale on a graph. Learners should know that the value equidistant between a and b on a linear scale is the arithmetic mean (a+b)/ 2 but for a logarithmic scale it is the geometric mean √(ab).

Logarithmic scales problems Questions to practise solving problems using logarithmic scales. Related powerpoint supplied “Using logarithmic scales to solve problems”


H5P - Exponential growth and decay

H5P – Logarithmic scales

Geogebra applet – logarithmic scales – finding values

H5P – Match the cards – powers of 10

H5P – Logarithmic scales problems

Interactive online activities linked to the content of this section. Auto marked with video content.

IQRl3 Be able to explore exponential growth and decay, including interpreting output from a spreadsheet. Contexts include borrowing and saving money, bacterial growth and radioactive decay.

IQRl4 Be able to represent and interpret exponential growth or decay in a graph. Learners may be asked to plot or sketch exponential graphs. Half-life. y = kax with k and a constant.

IQRl6 Be able to use and interpret a logarithmic scale on a graph. Learners

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should know that the value equidistant between a and b on a linear scale is the arithmetic mean (a+b)/ 2 but for a logarithmic scale it is the geometric mean √(ab).

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Modelling – Teacher overview

Summary of content This section makes the modelling process more explicit and allows a number of models to be developed in different contexts. It will be best if this section is done after the algebra, exponentials and graphs sections. Modelling is intrinsic to the use of mathematics in real life situations, especially in science and economics. Due to the easy availability of

computers, mathematical models are used to predict how an epidemic might unfold, the possible effects of climate change and the long term effects of national economic strategy. It is helpful for every informed citizen to have a general understanding of mathematical modelling.

Pre-requisites: It will be helpful if students have already encountered the modelling cycle, when doing estimation or when working with exponentials


Auto-marked online assessment: Section test - Modelling



Folder name: Modelling


The modelling cycle Information sheet

Graph for CO2 A graph to use with the powerpoint. IQRp1 Be able to identify simplifying assumptions that allow a situation to be modelled.

IQRp2 Be able to develop or choose a simple mathematical model for a real-world situation. Model in words, numerically, algebraically,

Worksheet for 100m record

Excel file 100m

Worksheet for demand curves

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Graph for demand curves diagrammatically or in a spreadsheet.

IQRf13 Be able to use a demand curve as a model for the relationship between price and demand.

IQRp3 Be able to use a model to make predictions or get information about a situation. For example, use a simple demand curve (e.g. a linear model) to predict the change in revenue following a given change in price.

IQRp4 Be able to compare the outcomes from a model with actual data, information, experience or common sense and comment on the appropriateness of the model. For example, compare an exponential growth model with actual population figures. The information may be given in diagrammatic or graphical form.

IQRp5 Be able to appraise the assumptions underlying a model critically.

Worksheet for vehicle miles

Excel file Cars

Projects

Mobile phones

Hooke’s law

High jump records

Modelling the population

Spreadsheet exercise pension

A modelling exercise to predict number of phones.

A modelling exercise to find a relationship between weight an extension on a spring.

A modelling exercise to predict how high a man can jump

A modelling exercise to see how the population is growing

A modelling exercise about pensions


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Normal distributions – Teacher overview

Summary of content This section begins with an activity which encourages students to think about the source of errors in measurements. A normal distribution is introduced using the context of the e-mark. Finally, more formal use of z scores is introduced.

Pre-requisites: This is better done after statistics 2


Auto-marked online assessment: Section test – Normal distribution MEI OCR



Folder name: Variation in measurements


Students sheet – Variation in measurement This activity can be used to recap on the mean and standard deviation leading to the introduction to the Normal distribution.

The distribution of measurements is often close to a Normal distribution due to variation in measurements. The students examine how sources of error in measurements can lead to a Normal distribution.

IQR s15 Be able to select and calculate appropriate measures of central tendency and to interpret them. Mean, median, mode. Includes grouped data and calculation or estimation for data in a statistical diagram. Number of data items = n Sample mean =

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Folder name: The e mark


Students sheet emark The students are introduced to the e-mark and related regulations by the teacher’s powerpoint, before starting on the sheet.

Students are encouraged to think about the shape and properties of a Normal distribution by examining a scenario involving quality control and weight/measures regulations.

The students start examining the proportion of a product within certain weight limits, then use these proportions to estimate probabilities, leading into the idea of a probability distribution.

This idea is then used in reverse to estimate the expected proportion and then number of items within a given range, for example between µ and µ+σ

The teacher’s powerpoint uses images from the probability calculator in GeoGebra to help illustrate proportions.

The student interactive package “Exploring the shape of a Normal distribution” compliments this lesson.

IQR s19 Know that, for a Normal distribution, values more than three standard deviations from the mean are very unusual; know that approximately 95% of the data lie within two standard deviations of the mean and that 68% (just over two thirds) lie within one standard deviation of the mean. Learners may be asked to estimate mean and standard deviation from a Normal curve. n for population mean. σ for population standard deviation.

The Normal Distribution Summary A fact sheet to help with some of the questions.

Folder name: Modelling with a Normal distribution


Data files GeoGebra files containing the data used in the

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student sheet. Data is in rank order to help when answering questions at the board.

The Normal distribution summary sheet A summary of properties of a Normal distribution provided to help students.

Student sheet - Modelling with a Normal distribution

To access this work students should be able to find the standard deviation of a set of data.

The students examine the distribution of two data sets, particularly the proportion of data in the interval µ±σ and µ±2σ

Although the proportion in these intervals for both data sets are close to those seen with a Normal distribution, the distance data is closer to the symmetrical bell shape of the Normal distribution, the rainfall data has a positive skew.

IQRs18 Know that the Normal distribution is a model which can be used for real data and recognise a Normal curve. Know that the distribution is symmetrical about the mean for the population but understand that histograms for samples will usually not be exactly symmetrical.

Student sheet – Distribution matching puzzle Students are asked to match a set of Normal distributions to their descriptions.

use of the notation N(μ,σ2) to describe a normal distribution in terms of mean and standard deviationuse of the notation N(0, 1) for the standardised normal distribution with mean = 0 and standard deviation = 1

Folder name: Normal distributions and z scores


Introduction to the Normal distribution This is a PowerPoint presentation for students to watch. It has an audio commentary and runs automatically. It shows students what they will learn in this section. It might be suitable for students to watch this before the lesson.

IQRs18 Know that the Normal distribution is a model which can be used for real data and recognise a Normal curve. Know that the distribution is symmetrical about the mean for the population but understand that histograms

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for samples will usually not be exactly symmetrical.

IQRs19 Know that, for a Normal distribution, values more than three standard deviations from the mean are very unusual; know that approximately 95% of the data lie within two standard deviations of the mean and that 68% (just over two thirds) lie within one standard deviation of the mean. Learners may be asked to estimate mean and standard deviation from a Normal curve. μ for population mean σ for population standard deviation

IQRs20 Be able to use mean and standard deviation to calculate a z-score and use z-scores for comparison or quality control. Includes interpreting z-score as number of standard deviations away from the mean. Standardised score; z-score; z-value

IQRs21 Be able to interpret a Normal probability plot from statistical software. A straight line indicates a Normal distribution.

Normal1 A GeoGebra file, needed for the investigations.

Investigations: The Normal distribution These investigations, based on using the GeoGebra file ‘Normal 1’ introduce all the key ideas in this section.

Summary of the Normal distribution This is a PowerPoint presentation (no audio) which summarises what students may have learned from the investigations and what they need to know

Questions: The Normal distribution Some questions to check that students have learned what they should.

Introduction to z-scores This is a powerpoint which will lead you through how to calculate and use z-scores.

z-scores

z-scores investigations

A GeoGebra file to use for investigating z-scores

A video to accompany the GeoGebra file

Questions for z-scores Some questions on z-scores to check understanding


Exploring the shape of a normal distribution Online auto marked activities covering most aspects of IQR 18-21

Is the normal distribution a good fit?

Decisions, business and risk– Teacher overview

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Summary of content This section illustrates how insurance companies, banks, services, agriculture and the manufacturing industry all make key decisions based on

what they predict is going to happen in a given market. In some businesses, an understanding of probability, in particular risk and expectation, linked to an ability to gather and evaluate data, is crucial.

Students should also see that, from a personal point of view, the same skills can allow you to assess the value for money of things like loans and insurance; decisions which most of them will need to make.

Pre-requisites: This section is linked to the IQR section on risk; it would be useful to have done that section first as well as the Critical Maths section scams. Students should be able to work out percentages fairly confidently before doing this section.


Auto-marked online assessment: Section test – Business and risk, section test – Making decisions involving risk



Folder name: Business and risk


Student sheet – Business and risk Insurance companies, banks, service industries, agriculture and manufacturing all make key decisions based on what they predict is going to happen in a given market. In some businesses, an understanding of probability, in particular risk and expectation, linked to an ability to gather and evaluate data is crucial. From a personal point of view, the same skills can allow you to evaluate the value for money of things like loans and insurance. This resource is an attempt to develop these skills.

CMe6 Be able to estimate the probability of an event.

CMe7Understand expected number as the average number of occurrences in the long run.

Includes being able to estimate expected return from a game or investment.



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CMp8 Be able to interpret the solution to a problem effectively in terms of the original problem


Folder name: Making decisions involving risk


Student sheet Making decisions involving risk The ideas in this activity came from “Thinking, Fast and Slow” by Daniel Kahneman. In this book the author explains how emotions play a large part in our decision-making process, particularly how the fear of a loss balances with the joy of a gain.

CMs1 Understand that an event with small probability is not surprising in a sufficiently large population.

CMs2 Recognise common examples of incorrect reasoning in probability and be able to explain the errors.

CMe6 Be able to estimate the probability of an event.

CMe7 Understand expected number as the average number of occurrences in the long run.

Includes being able to estimate expected return from a game or investment.


Approximately Normal? – Teacher overview

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Summary of content This section examines some of the ideas related to selecting an appropriately sized sample on which to make decisions. This decision-making leads students into exploration of ideas related to probability distributions, including the Normal distribution, mean, spread

and gives them an informal, intuitive idea of the principles underlying statistical hypothesis testing. Ideas of hypothesis testing are important in many social sciences and in medicine; it is helpful for students to understand that such decisions are

made on the basis of probability. The essential idea in this section is to give some examples of when the Normal distribution is a suitable approximate distribution even when it is

not an exact match for the data. For situations which can be modelled by the toss of a fair coin, both the mean and standard deviation can be simply calculated and used to find

values that are unusually large or small. This links to ideas of statistical hypothesis testing. Many students going on to further study in science or social science will encounter the Normal distribution so it is helpful for them to understand

that it can occur approximately in many contexts. The “bell-shaped curve” is sometimes referred to in the news so a general understanding of places where the Normal distribution can occur is also useful for the informed citizen.

Pre-requisites: This section follows on from the Statistics sections and the Normal distribution section. It should be done after them


Auto-marked online assessment: Section test – How convincing is the information? Section test – Setting a pass mark



Folder name: Setting a pass mark


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Setting a pass mark – sample generator

Student sheet – setting a pass mark

A spreadsheet tool which generates random samples of different sizes. Can be use with students to demonstrate variability of samples.

Student’s worksheets to accompany the session.

This resource starts with the context of a multiple choice test and asks students to consider the problem of setting a pass mark which will not allow guessers to pass easily. This context leads students into exploration of ideas related to probability distributions, including the Normal distribution, mean, spread and gives them an informal, intuitive idea of the principles underlying statistical hypothesis testing. Students look at multiple choice tests with two answers per question.


CMs8 Know that, for large samples of a fixed size, sums of independent observations are distributed approximately Normally.

Folder name: Spreadsheet modelling


Spreadsheet modelling Presentation to introduce the model (Also see video clip Spreadsheet modelling – building a model)

This activity is intended as a lead in to the How convincing? Activity which is next in this document.

The students learn some spreadsheet and modelling skills related to the component objectives shown below

• engage in solving authentic problems appropriate to this level

• evaluate quantitative information

• recognise when mathematical and statistical analysis will be helpful

• improve their understanding of the applicability

Basic 50 percent sheet The completed first spreadsheet

Basic 40 percent sheet The completed second spreadsheet

Spreadsheet 200 juries Combined spreadsheet with charts

Instructions for building Students instruction sheet (there is a PDF version in the student folder)

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of mathematics

• develop skills of representing new situations mathematically

• think flexibly in problem solving

Folder name: How convincing is the information?


How convincing is the information? V5.1 student sheet.

How convincing is the information? - support spreadsheet. (A Spreadsheet tool to aid the teaching of this section.)

This resource is designed to examine the issues surrounding sample size. In particular, how large does a sample need to be in order for the evidence to be convincing and how representative are the figures in opinion polls and survey questions that have two possible responses, for example agree or disagree, yes or no.

The students start by examining three sets of results. They are asked to place these results in order of how convincing they find the evidence. This task is designed to illustrate the connection between sample size and variability.

The resource then looks at the expected value for the mean and spread of a probability distribution which can be modelled by the toss of a fair coin as the number of trials (n) increases.

This leads the students to use an estimate of

the mean and as an estimate of the

CMs8 Know that, for large samples of a fixed size, sums of independent observations are distributed approximately Normally.

e.g. The distribution of the number of successes in 100 Bernouilli trials. e.g. The distribution of the total score obtained when 50 dice are thrown.

CMs9 Know that, for an experiment which can be modelled by the toss of a fair coin, for n repetitions of the experiment, on average n/2 occurrences will happen and that the standard

deviation is √n2

.

Learners should recognise when this is an appropriate model and be able to use it to decide what kinds of results are unusual in the context.


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standard deviation in examples where there are two equally likely outcomes.


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Budgeting, inflation, saving and borrowing – Teacher overview

Summary of content

This section brings together earlier work on finance and introduces the financial problem solving cycle as a way of organising thinking when addressing a financial situation arising in either business or in personal life.

This section should be done after the other financial sections. There is also the opportunity here for students to tackle different problems, present solutions to the rest of the class and discuss and refine solutions.

The ability to apply financial thinking to real life is important for everyone. Costing and being able to monitor a budget has been identified as an important skill for work by ACME’s Mathematical Needs in the Workplace

report (2011). It can also be included in business qualifications at level 3.

Pre-requisites:


Auto-marked online assessment: Section test – Saving and borrowing, section test - Costing



Folder name: Budgeting


1a Lifestyle calculations exemplar.docx

1b Exemplar answers making decisions scenarios

1c Further Costing exercises exemplar answers

Exemplar answers to the activities. IQRf1 Be able to decide what information is needed to address a financial situation.

IQRf2 Know how to obtain the necessary information. The information may be presented

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in a graph or diagram.


IQRf4 Be able to decide which, if any, of the solutions are appropriate.

IQRf10 Be able to work out the regular outgoing cost for a financial decision. Examples include the monthly cost of buying and running a car or renting and running a home.

IQRf12 Be able to use a spreadsheet to cost a project or business proposal, recognising that some costs are more predictable than others over time. Learners may be asked to monitor whether a budget is being followed over time, and to calculate projected costs from the budget.

IQ f14 Understand and use the language of finance. Words such as income, expenditure, budget, profit, loss, investment, tax, revenue, inflation, APR and AER.

2 Budgeting teach presentation Powerpoint presentation to introduce and support the teaching of the activities.

3a Card set 1 Employment

3b Card set 2 Accommodation

3c Card set 3 Cars

3d Card set 4 Money

4 Lifestyle questions

Lifestyle activity and scenario cards.

5 Worksheet 1 Job offer

6 Worksheet 2 Windfall

Two worksheet with decision making scenarios.

Folder name: Football tickets - inflation


Spreadsheet Football tickets Spreadsheet for student tasks IQRf12 Be able to use a spreadsheet to cost a project or business proposal, recognising that some costs are more predictable than others over time.

IQRf14Understand and use the language of finance. Words such as income, expenditure, budget, profit, loss, investment, tax, revenue, inflation, APR and AER.

Students sheet - Football tickets Worksheets containing tasks and information

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Folder name: Student Loans


Student sheet – Initial questions Starter activity – questions IQR f10Be able to work out the regular outgoing cost for a financial decision. Examples include the monthly cost of buying and running a car or renting and running a home.


Students question sheet Follow on questions

Spreadsheet for last question To support the final task on the spreadsheet


H5P - Interest rates

H5P Loans and interest rates


Folder name: The richest country in the world


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The richest country in the world .PPT

The richest country in the world worksheet

This resource introduces students to Gross Domestic Product (GDP) and calculations related to GDP, including GDP per capita and GDP per capita, Purchasing power parity (PPP)

Students explore further about GDP and interpret two sets of data related to GDP and GDP per capita.

IQRf1 Be able to decide what information is needed to address a financial situation.

IQR f2 Know how to obtain the necessary information. The information may be presented in a graph or diagram.




Big Mac .PPT

Big Mac worksheet








.

Folder name: Income, wealth and inequality; Autonomous task


TBC

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Folder name: The Gini coefficient


Gini .PPT This resource starts looking at inequality, then some measures of inequality and finally focussing on a specific measure, the Gini Coefficient. Following on from this students can explore the Gini Coefficient online interactive activity.




Gini Coefficient Students explore inequality and look at a measure of inequality, the Gini Coefficient, that is used internationally to compare levels of inequality. They will use percentages, interpreting graphs and plotting using 2 axes that are built up using cumulative percentages.




IQRf5 Be able to do calculations involving percentages in context; the use of an index number to compare a number or value to that in a base year. Contexts include those outside finance. Examples of financial contexts include VAT, inflation and compound interest for savings or loans. Expected calculations include forward and reverse percentage increase and decrease, repeated and combined percentage change and finding a percentage change.

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meiassets.blob.core.windows.net · Web view2021. 4. 12. · AMSP Core Maths platform content: Drag and drop these shapes into the blank planner above, then stretch them to fit the