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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
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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
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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
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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
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KM: Swillions
KM: Lottery project
NRICH: Half a Minute
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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
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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
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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
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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
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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
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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
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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
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Learning review
GLOWMaths/JustMaths: Sample Questions Both Tiers
GLOWMaths/JustMaths: Sample Questions Higher Tiers
KM: 10M13 BAM Task
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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
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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
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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
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KM: Stick on the Maths HD2: Frequency polygons and scatter diagrams
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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
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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
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(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
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KM: Stick on the Maths: Inequalities
KM: Convinced?: Inequalities in one variable
NRICH: Inequalities
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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.
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Learning review
www.diagnosticquestions.com
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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
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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
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(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
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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
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The unshaded region represents the
solution set for the inequalities:
Learning review
GLOWMaths/JustMaths: Sample Questions Both Tiers
GLOWMaths/JustMaths: Sample Questions Higher Tiers
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x < 1, y ≥ 0 and x + y > 6
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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
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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
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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
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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