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AQA Modular GCSE Two Year Scheme of Work 2010 4360 Specification Foundation – Unit 1 OVERVIEW for Foundation Tier Notes: 1 hour calculator exam 54 marks – 26.7% Topic Y10 AUTUMN TERM Y9 SUMMER TERM 1. Data collection 30 -40% Functional Elements 14 marks Number, 40 marks Statistics Teaching hours 6 2. Interpreting and representing data 12 3. Range and averages 4. Probability 6 5. Fractions, decimals and percentages 6. Ratio and proportion 12 8 6 AQA Modular specification reference The Data Handling Cycle: S1 Data Collection: S2.1, S2.2, S2.3, S2.4, S2.5 Data presentation and analysis: S3.1 Data presentation and analysis: S3.2 Data Interpretation: S4.2, S4.3 Data presentation and analysis: S3.3 Data Interpretation: S4.1 Data presentation and analysis: S3.1 Probability: S5.1, S5.2, S5.3, S5.4, S5.7, S5.8, S5.9 Working with numbers and the number system: N1.14 Fractions, Decimals and Percentages: N2.1, N2.5,N2.6, N2.7 Ratio and Proportion: N3.1, N3.3 AQA Modular GCSE Two Year Scheme of Work 2010 4360 Specification Topic 1 Data collection Time: 6 hours S1 Understand and use the statistical problem solving process which involves specifying the problem and planning collecting data processing and presenting the data interpreting and discussing the results. S2.1 Types of data: qualitative, discrete, continuous. Use of grouped and ungrouped data. S2.2 Identify possible sources of bias. S2.3 Design an experiment or survey. S2.4 Design data collection sheets distinguishing between different types of data. S2.5 Extract data from printed tables and lists. S3.1 Design and use two-way tables for grouped and ungrouped data. AQA Spec ref S1 Learning objectives Grade Common mistakes and misconceptions Discuss all aspects of the data handling cycle within one situation Know the meaning of the term hypothesis Write a hypothesis to investigate a given situation D S2.3, S2.4 Know where to look for information Distinguish between primary and secondary data D S2.1 Decide whether data is qualitative, discrete or continuous – use this to choose suitable diagrams Understand the difference between grouped and ungrouped data including advantages and disadvantages D S2.4 E S2.4, S2.5 Understand methods of observation, controlled experiment, questionnaire, survey, data logging Know where and why each might be used Design and use data collection sheets. Work out methods for gathering data that can take a wide range of values Tabulate ungrouped data into grouped data. Formulating a hypothesis that cannot be tested. Thinking that a hypothesis is not valuable if it is eventually proved false. Not realising that data collected by a third party (even if the results of a survey or experiment) is classed as secondary data. Not appreciating that some data can be treated as either discrete or continuous depending on the context (e.g. age – this is really continuous, but is often treated as discrete, such as when buying child or adult tickets). Using shortcuts in the tallying process – counting up the items in each class, rather than tallying items one by one. D Using overlapping class intervals. Recording data which is on the boundary of a class interval in the wrong class. S2.5 S3.1 Interrogate tables or lists Design and use two-way tables D Not checking that the totals in two-way tables add up. AQA Modular GCSE Two Year Scheme of Work 2010 4360 Specification S2.3, S2.4 S2.2, S2.3, S2.4 Write or criticise questions and response questions for a questionnaire Suggest how a simple experiment may be carried out Understand how to collect survey data Know the techniques to use to get a reliable sample. Understand how bias may arise and offer ways to minimise bias for a data collection method Resources: C C Giving an answer that is not requested. Reading the data from the table incorrectly Using overlapping classes, or gaps between classes, for response options. Mistaking biased samples for random samples. AQA GCSE Maths Middle sets Book Sections 1.1 to 1.8, 5.3 AQA Modular GCSE Mathematics Foundation Tier Tally charts P1 Two way tables P104 Questionnaires & surveys P147 Observation and data logging P151 Foundation Practice Book sections 1.1, 5.1, 7.1, 7.2 www.aqa.org.uk/ 2010 ready/GCSE Maths/Free online resources/Foundation Tier/Unit 1/Collecting Data/Teaching Resources Questionnaire activity/powerpoint/worksheet Body correlation lesson plan/teacher handout/worksheet Collecting data homework sheet 2010 ready/GCSE Maths/Free online resources/Foundation Tier/Unit 1/Representing Data/Teaching Resources Two way tables lesson plan/worksheet Functional skills activities 2.1, 13.1 Notes: Questions may be set that require candidates to go through the stages of the Data Handling cycle without individual prompts Questions may explicitly test knowledge of the words discrete, continuous, qualitative, grouped etc but it is recognition of the type of data that will be more important – e.g. ‘Draw a suitable diagram for the data’. An understanding of the terms ‘primary data’ and ‘secondary data’ is expected. Causes of bias will be clear for candidates to identify Real data may be used in examination questions. Including knowing and using the term ‘hypothesis’ for a general prediction which is to be tested. Closely related - N6.12 Discuss, plot and interpret graphs (which may be non-linear) modelling real situations S5.9 Understand that increasing sample size generally leads to better estimates of probability and population characteristics AQA Modular GCSE Two Year Scheme of Work 2010 4360 Specification Topic 2 Interpreting and representing data Time: 12 hours S3.2 Produce charts and diagrams for various data types. Scatter graphs, stem-and-leaf, tally charts, pictograms, bar charts, dual bar charts, pie charts, line graphs, frequency polygons, histograms with equal class intervals. S4.2 Look at data to find patterns and exceptions. S4.3 Recognise correlation and draw and/or use lines of best fit by eye, understanding what they represent. AQA Mod spec ref S3.2 Learning objectives Grade Common mistakes and misconceptions Draw bar charts, dual bar charts and composite bar charts E, F, G Bar widths or gaps between bars not equal. Incorrect scale S3.2 S3.2 S3.2 Produce a tally chart Draw or complete a pictogram Draw a pie chart. F S3.2 Complete an ordered stem-and-leaf diagram D S3.2, S4.2 Draw a scatter diagram on a given grid Find patterns in data that may lead to a conclusion being drawn e.g. Interpret points on a scatter diagram Look for unusual data values (outliers/anomalies) D S4.3 D, C Trying to make the line of best fit go through the origin, rather than drawing it appropriately. Not appreciating correlation in terms of ‘positive’ and ‘negative’. S3.2 Draw a line of best fit on a scatter diagram or know that a line of best fit is not justified due to lack of correlation Recognise and name types of correlation and strength of correlation – none, positive, negative, weak, strong, moderate Understand that if correlation exists, this does not mean causality exists Use the line of best fit to estimate unknown values when appropriate Produce histograms with equal class intervals D S3.2 Draw frequency polygons for grouped data C Using grouped labels on the data axes (e.g. 15–20, rather than the ends of the bar being clearly marked with a 15 at one end and a 20 at the other end). Using a grouped label on the horizontal axis rather than a continuous scale. E Not drawing the angles in the pie chart accurately or using the appropriate scale on the protractor. Measuring each angle from the same starting point Forgetting to put a key and order the leaves. Forgetting to recombine the stem and leaf and just giving the leaf as the value. Assuming that all the plotted points must be joined with a line. Drawing the diagram without spending time working out the best scale. AQA Modular GCSE Two Year Scheme of Work 2010 4360 Specification Plotting the upper bound instead of the midpoint. S3.2 Produce line graphs Resources: AQA GCSE Maths Middle sets Book Sections 3.1 to 3.6 AQA Modular GCSE Mathematics Foundation Tier Tally charts P1 Pictograms P8 Bar charts P13 Pie charts P75 Stem and leaf diagrams P81 Scattergraphs P86 Time series (line graphs) P93 Frequency polygons P173 Histograms P178 Foundation Practice Book 1.1, 1.2, 1.3, 4.1, 4.2, 4.3, 4.4, 8.3, 8.4 www.aqa.org.uk/ 2010 ready/GCSE Maths/Free online resources/Foundation Tier/Unit 1/Representing Data/Teaching Resources Stem and leaf lesson plan/worksheet/homework sheet 2010 ready/GCSE Maths/Free online resources/Foundation Tier/Unit 1/Scattergraphs/Teaching Resources Scattergraphs lesson plans/homeworksheet/powerpoint Functional skills activities 2.2, 2.3, 2.4, 5.1, 13.1, 13.2, 13.3, 16.1, 16.2, 16.3 Notes: Candidates may be asked to draw a suitable diagram for data. An understanding of the type and nature of the data is expected from the candidate in order to make a choice. Axes and scales may or may not be given. Though the words interpolation and extrapolation will not be used in the examination, the idea that finding estimates outside of the data range is less reliable than finding estimates from within the data range is expected to be understood by candidates. Closely related – N6.12 Discuss, plot and interpret graphs (which may be non-linear) modelling real situations S4.1 Interpret a wide range of graphs and diagrams and draw conclusions. AQA Modular GCSE Two Year Scheme of Work 2010 4360 Specification Topic 3 Range and averages Time: 6 hours S3.3 Calculate median, mean, range, mode and modal class. S4.1 Interpret a wide range of graphs and diagrams and draw conclusions. AQA Mod spec ref S3.3, S4.1 Learning objectives Grade Common mistakes and misconceptions Find the mean, median and range from a set of data, including data given in a stem and leaf diagram E, D S3.3, S4.1 Calculate the mode, median and range from an ungrouped frequency table Use lists, tables or diagrams to find values for the above measures Find the median for a discrete frequency distribution Find the mode for frequency distributions Find the interval containing the median for a grouped frequency distribution Choose an appropriate measure according to the nature of the data to be the ‘average’ Calculate the mean from an ungrouped frequency table Find the mean for a discrete frequency distribution Omitting units when writing averages or range. Forgetting to include the stem when reading the median from a stem and leaf diagram. Confusing the frequencies and the data values. D, C Dividing by the number of rows in the frequency table (i.e. the number of different data values), not by the sum of the frequencies. Not appreciating that the statistics calculated from grouped frequency tables are estimates. Not understanding that the estimate for the range is an upper limit. Incorrectly calculating the mid-points of class intervals for grouped discrete data (e.g. the mid-point of the class interval 10–19 is 14.5, not 15). Interpreting ‘find an estimate for the mean’ as ‘guess the mean’. S3.3, S4.1 S3.3, S4.1 Find the modal class from a grouped frequency table Estimate the range from a grouped frequency table Work out the class interval which contains the median from data given in a grouped frequency table D, C S3.3, S4.1 Estimate the mean of data given in a grouped frequency table, knowing why it is an estimate C AQA Modular GCSE Two Year Scheme of Work 2010 4360 Specification Resources: AQA GCSE Maths Middle sets Book Sections 4.1 to 4.5 AQA Modular GCSE Mathematics Foundation Tier Mode P25 Mean P 29 Median P34 Range P39 Mean of a discrete frequency distribution P109 Comparing distributions P117 Mean for grouped data P162 Finding the median for frequency distributions P168 Foundation Practice Book 2.1 – 2.4, 5.2, 5.3, 8.1, 8.2 www.aqa.org.uk/ 2010 ready/GCSE Maths/Free online resources/Foundation Tier/Unit 1/Statistical measures/Teaching Resources lesson plans/worksheets/homework sheet/problem sheet Functional skills activities 6.1, 6.2, 17.1 – 17.3 Notes: Candidates may be asked to draw a suitable diagram for data. An understanding of the type and nature of the data is expected from the candidate in order to make a choice, Axes and scales may or may not be given. Closely related – N6.12 Discuss, plot and interpret graphs (which may be non-linear) modelling real situations S4.4 Compare distributions and make inferences – includes comparisons of average and range at tier F S3.2 Produce charts and diagrams for various data types AQA Modular GCSE Two Year Scheme of Work 2010 4360 Specification Topic 4 Probability Time: 8 hours S5.1 Understand and use the vocabulary of probability and the probability scale. S5.2 Understand and use estimates or measures of probability from theoretical models (including equally likely outcomes), or from relative frequency. S5.3 List all outcomes for single events, and for two successive events, in a systematic way and derive related probabilities. S5.4 Identify different mutually exclusive outcomes and know that the sum of the probabilities of all these outcomes is 1. S5.7 Compare experimental data and theoretical probabilities. S5.8 Understand that if an experiment is repeated, this may – and usually will – result in different outcomes. S5.9 Understand that increasing sample size generally leads to better estimates of probability and population characteristics. AQA Mod spec ref S5.1, S5.4 S5.4 S5.2 S5.2, S5.7, S5.8, S5.9 5.3 Learning objectives Grade Common mistakes and misconceptions Use words to indicate the chances for an outcome of an event Use fractions, decimals or percentages to put values to probabilities Place probabilties or outcomes to events on a probability scale Work out the probability of an event not happening when you know the probability that it does happen Understand when outcomes can or cannot happen at the same time. Use this understanding to calculate probabilities Understand and use the fact that the sum of the probabilities of all mutually exclusive outcomes is 1 Find the probability of a single outcome from knowing the probabilities of all other outcomes Work out probabilities by counting or listing equally likely outcomes Estimate probabilities by considering relative frequency Predict the likely number of successful events given the probability of any outcome and the number of trials or experiments Estimate probabilities from experimental data Understand and use the term relative frequency Consider the differences between theoretical probability and relative frequency from a practical situation Understand that experiments rarely give the same results when a random process is involved Appreciate the lack of memory in a random situation e.g. tails or heads are equally likely even after 5 heads in a row Understand the greater the number of trials the more the reliable the results Understand how relative frequency diagram may settle as sample size increases List all outcomes for a single event in a systematic way List all outcomes for two events in a systematic way G F Not remembering that probabilities can be written as fractions, decimals and percentages. Incorrectly subtracting decimals from 1. E D Adding or subtracting the incorrect values due to misreading the question. G, F E D Incorrectly finding fractions of an amount. Not cancelling a fraction to its simplest form. C Trying to plot decimals worked out to three decimal places or more. Comparing theoretical probability with relative frequency without taking into account the number of trials carried out. G,F E AQA Modular GCSE Two Year Scheme of Work 2010 4360 Specification Use two-way tables to list outcomes Use lists or tables to find probabilities Resources: D AQA GCSE Maths Middle sets Book Sections 5.1 to 5.6 AQA Modular GCSE Mathematics Foundation Tier Chance P48 Probability scales and calculations P51 Listing outcomes P56 Calculating probabilities P61 Probability for mutually exclusive outcomes P66 Relative frequency P128 Using relative frequency P135 Foundation Practice Book 3.1 – 3.5, 6.1, 6.2 www.aqa.org.uk/ 2010 ready/GCSE Maths/Free online resources/Foundation Tier/Unit 1/Probability/Teaching Resources lesson plans & starters/worksheets/homework sheets/problem sheets/powerpoints/interactive activities Functional skills activities 7.1, 7.2, 18.1, 18.2 Notes: The words candidates should be familiar with will be limited to impossible, (very) unlikely, evens or even chance, (very) likely and certain. Candidates should not use word forms or ratio for numerical probabilities such as 1 out of 2 or 1 : 2. Situations will be familiar, such as dice or bags containing numbered counters. Probabilities and relative frequencies should be written using fractions, decimals or percentages. Cancelling a fraction to its simplest form may be required. If not directed, listing can be done using lists, tables or sample space diagrams. The term sample space will not be tested. The term mutually exclusive will not be tested though the principle will. Closely related S3.1 Design and use two-way tables for grouped and ungrouped data. Work from N2.1 may be assessed with this specification reference. AQA Modular GCSE Two Year Scheme of Work 2010 4360 Specification Topic 5 Fractions, decimals and percentages Time: 12 hours N1.14 Use calculators effectively and efficiently, including statistical functions. N2.1 Understand equivalent fractions, simplify by cancelling all common factors N2.5 Understand that percentage means parts per hundred and use this to compare proportions N2.6 Interpret fractions, decimals and percentages as operators. N2.7 Calculate with fractions, decimals and percentages. AQA Mod spec ref N2.7 Learning objectives Grade Common mistakes and misconceptions Find a fraction of a quantity with a calculator Find a fraction of an amount with a calculator in more complex situations Use a calculator for the four rules for fractions Calculate with fractions, decimals or percentages in a variety of contexts including statistics and probability Write one quantity as a fraction of another E, D Incorrectly inputting numbers on the calculator. Being unsure of what to work out when a fraction calculation is set in context. D N1.14 Use the fraction key on a calculator Use the fraction key on a calculator with mixed numbers E, D N2.7 Find a percentage of a quantity with a calculator Find percentages of amounts in more complex situations - Perform calculations involving repeated percentage changes E, D Not making the denominator the total in questions involving a number of quantities. Working with quantities in different units. Incorrectly cancelling down. Not recognising or know how to use the fraction key on a calculator. Misinterpreting a mixed number on a calculator display. Thinking that percentages over 100% cannot exist. Treating a percentage such as 0.05% as though it were 5%. Adding the percentage to the cost when finding a percentage increase (e.g. £315 + 15% VAT = £330). Leaving the multiplier as a percentage, instead of converting to a decimal. Inaccurately converting to a decimal. Not understanding compounding (e.g. treating compound interest as simple). N2.7 AQA Modular GCSE Two Year Scheme of Work 2010 4360 Specification N2.7 Write one quantity as a percentage of another Write one quantity as a percentage of another in more complex situations – work what percentage one is of another D, C N2.6, N2.7 Calculate a percentage increase or decrease Use fractions, decimals or percentages to compare or interpret data sets or statistical diagrams Interpret a fraction, decimal or percentage as a multiplier when solving problems Convert between fractions, decimals and percentages Understand and use a retail prices index Understand and use a retail prices index in more complex situations Identify equivalent fractions Simplify a fraction using a calculator Understand whether a value is a percentage, decimal or fraction Convert values between fractions, decimals or percentages in order to compare them D C N2.7 N2.1 N2.5 D, C E, D G, F E Not using the original amount as the denominator, when finding a percentage difference. Working with quantities in different units. Giving the actual increase/decrease as the answer when the amount after the increase/decrease is what is required. Using the multiplier as 1.5 rather than 1.05 for an increase of 5%. Using a previously found price instead of the base year price. AQA Modular GCSE Two Year Scheme of Work 2010 4360 Specification Resources: AQA GCSE Maths Middle sets Book Sections 2.1 to 2.7 AQA Modular GCSE Mathematics Foundation Tier Social Statistics P156 Conversions of fractions, decimals and percentages P253 Fraction of a quantity P258 Percentage of a quantity P261 Ordering decimals, fractions and percentages P264 Expressing one quantity as a fraction of another P268 Addition and subtraction of fractions P273 Multiplication and division of fractions P279 Addition and subtraction of decimals P285 Multiplication and division of decimals P288 Expressing one quantity as a percentage of another P314 Increasing and decreasing by a percentage P318 Finding a percentage change P321 Percentages in real life P324 Foundation Practice Book 7.3, 13.1 – 13.5, 14.1 – 14.4, 16.1 – 16.4 www.aqa.org.uk/ 2010 ready/GCSE Maths/Free online resources/Foundation Tier/Unit 1/Fractions/Teaching Resources lesson plans/worksheets/homework sheets 2010 ready/GCSE Maths/Free online resources/Foundation Tier/Unit 1/Decimals/Teaching Resources Lesson plans & starters/worksheets/homework sheets 2010 ready/GCSE Maths/Free online resources/Foundation Tier/Unit 1/Percentages/Teaching Resources Lesson plans/matching cards/powerpoints/homework sheets/plenary questions Functional skills activities 3.1 – 3.4, 14.1 -14.4 Notes: This is part of the core number work required across all units. The term common denominator will not be used. Candidates should be able to interpret percentage problems using a multiplier. Closely related N1.3 Understand and use number operations and the relationships between them, including inverse operations and the hierarchy of operations N1.4 Approximate to a given power of 10 (nearest 10,100, 1000), up to 3 decimal places and 1 significant figure N1.14 Use a calculator effectively and efficiently, including statistical functions AQA Modular GCSE Two Year Scheme of Work 2010 4360 Specification Topic 6 Ratio and proportion Time: 6 hours N3.1 Use ratio notation, including reduction to its simplest form and its various links to fraction notation. N3.3 Solve problems involving ratio and proportion, including the unitary method of solution. AQA Mod spec ref N3.1 Learning objectives Grade Common mistakes and misconceptions Understand the meaning of ratio notation Simplify a ratio to its lowest terms a:b where a and b integers Use a ratio in practical situations F, E, D N3.1, N3.3 Write a ratio as a fraction Use a ratio to find one quantity when the other is known Use a ratio and proportion to solve statistical and number problems E, D, C Swapping over the numbers in the ratio (e.g. 2 : 5 becomes 5 : 2). Simplifying ratios without ensuring the quantities are in the same units. Turning a ratio into a fraction (e.g. the ratio 4 4 : 5 becomes 5). Failing to find the value of the unit fraction in more complex problems. N3.3 Write a ratio in the form 1 : n or n : 1 C N3.3 Solve word problems involving ratio C N3.3 Understand direct proportion Solve proportion problems, including using the unitary method D N3.3 Work out which product is the better buy D Ignoring different units in a ratio (e.g. simplifying 2 days : 15 hours to 1 : 7½) . Not multiplying both sides of the ratio by the same number. Giving an answer without considering the context. Not always seeing the relationships between numbers (e.g. if the cost of 4 items is given, and the price of 8 is asked for). Not making the units the same for each item. Comparing unlike unit rates (e.g. price per gram for one item but amount for 1p for the other). AQA Modular GCSE Two Year Scheme of Work 2010 4360 Specification Resources: AQA GCSE Maths Middle sets Book Sections 7.1 to 7.8 (7.8 higher) AQA Modular GCSE Mathematics Foundation Tier Ratio P331 Proportion and best value P336 Foundation Practice Book 17.1, 17.2 www.aqa.org.uk/ 2010 ready/GCSE Maths/Free online resources/Foundation Tier/Unit 1/Ratio/Teaching Resources lesson plans & starters/sorting activity/problem sheets/homework sheets Functional skills activities 4.1, 4.2, 15.1 – 15.3 Notes: This is part of the core number work required across all units. Ratio may be linked to a probability. Division in a given ratio is only tested in Unit 2.