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Higher 2 Year Scheme: Year 11 Autumn 1 Statistical Measures (1 of 2) 6 hours Key concepts The Big Picture: Statistics progression map interpret, analyse and compare the distributions of data sets from univariate empirical distributions through appropriate measures of central tendency (median, mean, mode and modal class) and spread (range, including consideration of outliers) apply statistics to describe a population Possible learning intentions Possible success criteria Investigate averages Explore ways of summarising data Analyse and compare sets of data Find the modal class of set of grouped data Find the class containing the median of a set of data Find the midpoint of a class Calculate an estimate of the mean from a grouped frequency table Estimate the range from a grouped frequency table Analyse and compare sets of data Appreciate the limitations of different statistics (mean, median, mode, range) Choose appropriate statistics to describe a set of data Justify choice of statistics to describe a set of data Prerequisites Mathematical language Pedagogical notes Average Spread Consistency Mean Median Mode Range Statistic Statistics Approximate, Round Calculate an estimate Grouped frequency Midpoint The word ‘average’ is often used synonymously with the mean, but it is only one type of average. In fact, there are several different types of mean (the one in this unit properly being named as the ‘arithmetic mean’). NCETM: Glossary Understand the mean, mode and median as measures of typicality (or location) Find the mean, median, mode and range of a set of data Find the mean, median, mode and range from a frequency table Common approaches Every classroom has a set of statistics posters on the wall All students are taught to use mathematical presentation correctly when calculating and rounding solutions, e.g. (21 + 56 + 35 + 12) ÷ 30 = 124 ÷ 30 = 41.3 to 1 d.p. Notation Correct use of inequality symbols when labeling groups in a frequency table Reasoning opportunities and probing questions Suggested activities Possible misconceptions KM: Swillions KM: Lottery project NRICH: Half a Minute Show me an example of an outlier. And another. And another. Convince me why the mean from a grouped set of data is only an estimate. What’s the same and what’s different: mean, modal class, median, range? Always/Sometimes/Never: A set of grouped data will have one modal class Convince me how to estimate the range for grouped data. Learning review www.diagnosticquestions.com Some pupils may incorrectly estimate the mean by dividing the total by the numbers of groups rather than the total frequency. Some pupils may incorrectly think that there can only be one model class. Some pupils may incorrectly estimate the range of grouped data by subtracting the upper bound of the first group from the lower bound of the last group. Statistical Measures (2 of 2) and Collecting and representing data 12 hours Key concepts The Big Picture: Statistics progression map infer properties of populations or distributions from a sample, whilst knowing the limitations of sampling construct and interpret diagrams for grouped discrete data and continuous data, i.e. cumulative frequency graphs and histograms, and know their appropriate use interpret, analyse and compare the distributions of data sets from univariate empirical distributions through appropriate graphical representation involving discrete, continuous and grouped data, including box plots interpret, analyse and compare the distributions of data sets from univariate empirical distributions through appropriate measures of central tendency including quartiles and inter-quartile range Possible learning intentions Possible success criteria Construct and interpret cumulative frequency graphs Construct and interpret box plots Analyse distributions of data sets Construct and interpret histograms Understand the limitations of sampling Use a sample to infer properties of a population Know the meaning of the lower quartile and upper quartile Find the quartiles for discrete data sets Calculate and interpret the interquartile range Construct and interpret a box plot for discrete data Use box plots to compare distributions Understand the meaning of cumulative frequency Complete a cumulative frequency table Construct a cumulative frequency curve Use a cumulative frequency curve to estimate the quartiles for grouped continuous data sets Use a cumulative frequency curve to estimate properties of grouped continuous data sets Use histograms with equal and unequal class intervals and know their appropriate use Prerequisites Mathematical language Pedagogical notes Categorical data, Discrete data Continuous data, Grouped data Axis, axes Population Sample Cumulative frequency Box plot, box-and-whisker diagram Central tendency Mean, median, mode Spread, dispersion, consistency Range, Interquartile range Skewness Histogram In Stage 8, pupils explore how to find the modal class of set of grouped data, the class containing the median of a set of data, the midpoint of a class, an estimate of the mean from a grouped frequency table and an estimate of the range from a grouped frequency table This unit builds on the knowledge by exploring measures of central tendency using quartiles and inter-quartile range. Cumulative frequency curves are usually S-shaped, known as an ogive. Box plots are also known as ‘box and whisker’ plots. Know the meaning of discrete and continuous data Interpret and construct frequency tables Analyse data using measures of central tendency Notation Correct use of inequality symbols when labeling groups in a frequency table Reasoning opportunities and probing questions Suggested activities NCETM: Glossary Common approaches The median is calculated by finding the (n+1)/2 th item and the lower quartile by finding the (n+1)/4 th item unless n is large (n>30). In the case when n>30, n/2 and n/4 can be used to find the median and lower quartile. Possible misconceptions Show me a box plot with a large/small interquartile range. And another. And another. What’s the same and what’s different: inter-quartile range, median, mean, mode Convince me how to construct a cumulative frequency curve Always/Sometimes/Never: The median is greater than the interquartile range KM: Stick on the Maths HD1: Statistics, HD2: Comparing Distributions KM: Cumulative Frequency and Box Plots NRICH: The Live of Presidents NRICH: Olympic Triathlon NRICH: Box Plot Match OCR: Sampling, Analysing Data Learning review GLOWMaths/JustMaths: Sample Questions Both Tiers GLOWMaths/JustMaths: Sample Questions Higher Tiers KM: 10M13 BAM Task Some pupils may plot the cumulative frequencies against the midpoints or lower bounds of grouped data Some pupils may try to construct a cumulative frequency curve by plotting the frequencies against the upper bound of grouped data Some pupils may try to construct a cumulative frequency curve by joining the points with straight lines rather than a smooth curve Some pupils may forget to add the ‘whiskers’ when constructing a ‘box and whisker’ plot. Some pupils may forget to plot frequency density on the y-axis of a histogram. Scatter graphs 3 hours Key concepts The Big Picture: Statistics progression map interpret and construct tables, charts and diagrams draw estimated lines of best fit; make predictions know correlation does not indicate causation; interpolate and extrapolate apparent trends whilst knowing the dangers of so doing Possible learning intentions Possible success criteria Interpret a range of charts and graphs Interpret scatter diagrams Explore correlation Interpret a wider range of non-standard graphs and charts Understand that correlation does not indicate causation Interpret a scatter diagram using understanding of correlation Construct a line of best fit on a scatter diagram Use a line of best fit to estimate values Know when it is appropriate to use a line of best fit to estimate values Prerequisites Mathematical language Pedagogical notes Categorical data, Discrete data Continuous data, Grouped data Axis, axes Scatter graph (scatter diagram, scattergram, scatter plot) Bivariate data (Linear) Correlation Positive correlation, Negative correlation Line of best fit Interpolate Extrapolate Trend Lines of best fit on scatter diagrams are first introduced in Stage 9, although pupils may well have encountered both lines and curves of best fit in science by this time. William Playfair, a Scottish engineer and economist, introduced the line graph for time series data in 1786. NCETM: Glossary Know the meaning of discrete and continuous data Interpret and construct frequency tables Construct and interpret pictograms, bar charts, pie charts, tables, vertical line charts, histograms (equal class widths) and scatter diagrams Common approaches As a way of recording their thinking, all students construct the appropriate horizontal and vertical line when using a line of best fit to make estimates. In simple cases, students plot the ‘mean of x’ against the ‘mean of y’ to help locate a line of best fit. Notation Correct use of inequality symbols when labeling groups in a frequency table Reasoning opportunities and probing questions Suggested activities Possible misconceptions KM: Stick on the Maths HD2: Frequency polygons and scatter diagrams What’s the same and what’s different: correlation, causation? What’s the same and what’s different: scatter diagram, time series, line graph, compound bar chart? Convince me how to construct a line of best fit. Always/Sometimes/Never: A line of best fit passes through the origin Learning review www.diagnosticquestions.com Some pupils may think that correlation implies causation Some pupils may think that a line of best fit always has to pass through the origin Some pupils may misuse the inequality symbols when working with a grouped frequency table Inequalities (Solving - 1 of 2) 5 hours Key concepts The Big Picture: Algebra progression map understand and use the concepts and vocabulary of inequalities solve linear inequalities in one variable represent the solution set to an inequality on a number line Possible learning intentions Possible success criteria Explore the meaning of an inequality Solve linear inequalities Understand the meaning of the four inequality symbols Choose the correct inequality symbol for a particular situation Represent practical situations as inequalities Recognise a simple linear inequality Find the set of integers that are solutions to an inequality Use set notation to list a set of integers Use a formal method to solve an inequality Use a formal method to solve an inequality with unknowns on both sides Use a formal method to solve an inequality involving brackets Know how to deal with negative number terms in an inequality Know how to show a range of values that solve an inequality on a number line Know when to use an open circle at the end of a range of values shown on a number line Know when to use an filled circle at the end of a range of values shown on a number line Use a number line to find the set of values that are true for two inequalities Prerequisites Mathematical language Pedagogical notes (Linear) inequality Unknown Manipulate Solve Solution set Integer The mathematical process of solving a linear inequality is identical to that of solving linear equations. The only exception is knowing how to deal with situations when multiplication or division by a negative number is a possibility. Therefore, take time to ensure pupils understand the concept and vocabulary of inequalities. NCETM: Departmental workshops: Inequalities NCETM: Glossary Notation The inequality symbols: < (less than), > (greater than), ≤ (less than or equal to), ≥ (more than or equal to) The number line to represent solutions to inequalities. An open circle represents a boundary that is not included. A filled circle represents a boundary that is included. Set notation; e.g. {-2, -1, 0, 1, 2, 3, 4} Common approaches Pupils are taught to manipulate algebraically rather than be taught ‘tricks’. For example, in the case of -2x > 8, pupils should not be taught to flip the inequality when dividing by -2. They should be taught to add 2x to both sides. Many pupils themselves will later generalise. Reasoning opportunities and probing questions Suggested activities Possible misconceptions KM: Stick on the Maths: Inequalities KM: Convinced?: Inequalities in one variable NRICH: Inequalities Understand the meaning of the four inequality symbols Solve linear equations including those with unknowns on both sides Show me an inequality (with unknowns on both sides) with the solution x ≥ 5. And another. And another … Convince me that there are only 5 common integer solutions to the inequalities 4x < 28 and 2x + 3 ≥ 7. What is wrong with this statement? How can you correct it? 1 – 5x ≥ 8x – 15 so 1 ≥ 3x – 15. Learning review www.diagnosticquestions.com Some pupils may think that it is possible to multiply or divide both sides of an inequality by a negative number with no impact on the inequality (e.g. if -2x > 12 then x > -6) Some pupils may think that a negative x term can be eliminated by subtracting that term (e.g. if 2 – 3x ≥ 5x + 7, then 2 ≥ 2x + 7) Some pupils may know that a useful strategy is to multiply out any brackets, but apply incorrect thinking to this process (e.g. if 2(3x – 3) < 4x + 5, then 6x – 3 < 4x + 5) Inequalities (Graphs - 2 of 2) 6 hours Key concepts The Big Picture: Algebra progression map solve linear inequalities in two variables represent the solution set to an inequality using set notation and on a graph Possible learning intentions Possible success criteria Understand and use set notation Solve inequalities Represent inequalities on a graph Understand the use of a graph to represent an inequality in two variables State the (simple) inequality represented by a shaded region on a graph Know when to use a dotted line as a boundary for an inequality on a graph Know when to use a solid line as a boundary for an inequality on a graph Construct and shade a graph to show a linear inequality of the form y > ax + b, y < ax + b, y ≥ ax + b or y ≤ ax + b Construct and shade a graph to show a linear inequality in two variables stated implicitly Construct and shade a graph to represent a set of linear inequalities in two variables Find the set of integer coordinates that are solutions to a set of inequalities in two variables Use set notation to represent the solution set to an inequality Solve quadratic inequalities in one variable Prerequisites Mathematical language Pedagogical notes (Linear) inequality Variable Manipulate Solve Solution set Integer Set notation Region Pupils have explored the meaning of an inequality and solved linear inequalities in one variable in Stage 9. This unit focuses on solving linear equalities in two variables, representing the solution set using set notation and on a graph Therefore, it is important that pupils can plot the graphs of linear functions, including x = a and y = b. NCETM: Departmental workshops: Inequalities NCETM: Glossary Understand the meaning of the four inequality symbols Find the set of integers that are solutions to an inequality Use set notation to list a set of integers Use a formal method to solve an inequality in one variable Plot graphs of linear functions stated explicitly Plot graphs of linear functions stated implicitly Notation The inequality symbols: < (less than), > (greater than), ≤ (less than or equal to), ≥ (more than or equal to) A graph to represent solutions to inequalities in two variables. A dotted line represents a boundary that is not included. A solid line represents a boundary that is included. Set notation; e.g. {-2, -1, 0, 1, 2, 3, 4} Common approaches All students experience the use of dynamic graphing software, such as Autograph, to represent the solution sets of inequalities in two variables Reasoning opportunities and probing questions Suggested activities Possible misconceptions Show me a pair of integers that satisfy x + 2y < 6. And another. And another … Convince me that the set of inequalities x > 0, y > 0 and x + y < 2 has no positive integer solutions. Convince me that the set of inequalities x ≥ 0, y > 0 and x + 2y < 6 has 6 pairs of positive integer solutions. What is wrong with this statement? How can you correct it? KM: Stick on the Maths 8: Inequalities KM: Convinced?: Inequalities in two variables KM: Linear Programming NRICH: Which is bigger? Hwb: How do we know? MAP: Defining regions using inequalities CIMT: Inequalities The unshaded region represents the solution set for the inequalities: Learning review GLOWMaths/JustMaths: Sample Questions Both Tiers GLOWMaths/JustMaths: Sample Questions Higher Tiers x < 1, y ≥ 0 and x + y > 6 Some pupils may think that it is possible to multiply or divide both sides of an inequality by a negative number with no impact on the inequality (e.g. if -2x > 12 then x > -6) Some pupils may think that strict inequalities, such as y < 2x + 3, are represented by a solid, rather than dashed, line on a graph Some pupils may shade the incorrect region Numerical methods (iteration/trial & improvement) 3 hours Key concepts The Big Picture: Algebra progression map find approximate solutions to equations numerically using iteration Possible learning intentions Possible success criteria Find approximate solutions to complex equations Solve simultaneous equations Understand the concept of decimal search to solve a complex equation Use decimal search to solve a complex equation Understand the process of interval bisection to locate an approximate solution for a complex equation Use interval bisection to locate an approximate solution for a complex equation Rearrange an equation to form an iterative formula Use an iterative formula to find approximate solutions to equations Prerequisites Mathematical language Pedagogical notes Unknown Solve Solution set Interval Decimal search Iteration Iteration is introduced as a process for finding approximate solutions to nonlinear equations. GCSE examples can be found here. NCETM: Glossary Common approaches Pupils use the ‘ANS’ key on their calculators when finding an approximate solution using iteration Notation (a, b) for an open interval [a, b] for a closed interval Reasoning opportunities and probing questions Suggested activities AQA: Bridging Units Resource Pocket 4 Learning review GLOWMaths/JustMaths: Sample Questions Both Tiers GLOWMaths/JustMaths: Sample Questions Higher Tiers KM: 10M4 BAM Task Possible misconceptions