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Foundation 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.
Collecting and representing data
6 hours
Key concepts
The Big Picture: Statistics progression map
 infer properties of populations or distributions from a sample, whilst knowing the limitations of sampling
 interpret and construct tables, charts and diagrams, including frequency tables, bar charts, pie charts and pictograms for categorical data, vertical line charts for ungrouped discrete numerical data and know their appropriate use
Possible learning intentions
Possible success criteria
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Explore types of data
Construct and interpret graphs
Select appropriate graphs and charts
Understand the limitations of sampling
Use a sample to infer properties of a population
Know the meaning of categorical data
Know the meaning of discrete data
Interpret and construct frequency tables
Construct and interpret pictograms (bar charts, tables) and know their appropriate use
Construct and interpret comparative bar charts
Interpret pie charts and know their appropriate use
Construct pie charts when the total frequency is not a factor of 360
Choose appropriate graphs or charts to represent data
Construct and interpret vertical line charts
Prerequisites
Mathematical language
Pedagogical notes
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Data, Categorical data, Discrete data
Sample, Population
Pictogram, Symbol, Key
Frequency
Table, Frequency table
Tally
Bar chart
Time graph, Time series
Bar-line graph, Vertical line chart
Scale, Graph
Axis, axes
Line graph
Pie chart
Sector
Angle
Maximum, minimum
In stage 6 pupils constructed pie charts when the total of frequencies is a factor of 360.
More complex cases can now be introduced.
Much of the content of this unit has been covered previously in different stages. This is
an opportunity to bring together the full range of skills encountered up to this point,
and to develop a more refined understanding of usage and vocabulary.
Construct and interpret a pictogram
Construct and interpret a bar chart
Construct and interpret a line graph
Understand that pie charts are used to show proportions
Use a template to construct a pie chart by scaling frequencies
Bring on the Maths+: Moving on up!
Statistics: #1, #2, #3
William Playfair, a Scottish engineer and economist, introduced the bar chart and line
graph in 1786. He also introduced the pie chart in 1801.
NCETM: Glossary
Common approaches
Pie charts are constructed by calculating the angle for each section by dividing 360 by
the total frequency and not using percentages.
The angle for the first section is measured from a vertical radius. Subsequent sections
are measured using the boundary line of the previous section.
Notation
When tallying, groups of five are created by striking through each group
of four
Reasoning opportunities and probing questions
Suggested activities
Possible misconceptions
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KM: Constructing pie charts
KM: Maths to Infinity: Averages, Charts and Tables
NRICH: Picturing the World
NRICH: Charting Success
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Show me a pie chart representing the following information:
Blue (30%), Red (50%), Yellow (the rest). And another. And
another.
Always / Sometimes / Never: Bar charts are vertical
Always / Sometimes / Never: Bar charts, pie charts,
pictograms and vertical line charts can be used to represent
any data
Kenny says ‘If two pie charts have the same section then the
amount of data the section represents is the same in each pie
chart.’ Do you agree with Kenny? Explain your answer.
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Learning review
www.diagnosticquestions.com
Some pupils may think that the lines on a line graph are always meaningful
Some pupils may think that each square on the grid used represents one unit
Some pupils may confuse the fact that the sections of the pie chart total 100% and
360°
Some pupils may not leave gaps between the bars of a bar chart
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 (positive, negative, strong, weak, no 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
Pythagoras’ theorem
5 hours
Key concepts
The Big Picture: Measurement and mensuration progression map
 know the formulae for: Pythagoras’ theorem, a² + b² = c², and apply it to find lengths in right-angled triangles in two dimensional figures
Possible learning intentions
Possible success criteria
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Investigate right-angled triangles
Solve problems involving Pythagoras’ theorem
Know Pythagoras’ theorem
Identify the hypotenuse in a right-angled triangle
Know when to apply Pythagoras’ theorem
Calculate the hypotenuse of a right-angled triangle using Pythagoras’ theorem
Calculate one of the shorter sides in a right-angled triangle using Pythagoras’ theorem
Prerequisites
Mathematical language
Pedagogical notes
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Hypotenuse
Pythagoras’ theorem
Pupils must experience right-angled triangles in different orientations to appreciate
the hypotenuse is always opposite the right angle.
NCETM: Glossary
Common approaches
Pythagoras’ theorem is stated as ‘the square of the hypotenuse is equal to the sum of
the squares of the other two sides’ not just a² + b² = c².
Reasoning opportunities and probing questions
Suggested activities
Possible misconceptions
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KM: One old Greek (geometrical derivation of Pythagoras’ theorem. This 
is explored further in the next unit)
KM: Stick on the Maths: Pythagoras’ Theorem
KM: Stick on the Maths: Right Prisms
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Find the squares and square roots of numbers
Simple rearranging to find the subject of the formula
Always/ Sometimes/ Never: If a² + b² = c², a triangle with sides a,
b and c is right angled.
Kenny thinks it is possible to use Pythagoras’ theorem to find the
height of isosceles triangles that are not right- angled. Do you
agree with Kenny? Explain your answer.
Convince me the hypotenuse can be represented as a horizontal
line.
Learning review
KM:, 9M11 BAM Task
Some pupils may use Pythagoras’ theorem as though the missing side is always
the hypotenuse
Trigonometry (Additional foundation content)
10 hours
Key concepts
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The Big Picture: Properties of Shape progression map
make links to similarity (including trigonometric ratios) and scale factors
know the exact values of sinθ and cosθ for θ = 0°, 30°, 45°, 60° and 90°; know the exact value of tanθ for θ = 0°, 30°, 45° and 60°
know the trigonometric ratios, sinθ = opposite/hypotenuse, cosθ = adjacent/hypotenuse, tanθ = opposite/adjacent
apply it to find angles and lengths in right-angled triangles in two dimensional figures
Possible learning intentions
Possible success criteria
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Investigate similar triangles
Explore trigonometry in right-angled triangles
Set up and solve trigonometric equations
Use trigonometry to solve practical problems
Bring on the Maths: GCSE Higher Shape
Investigating angles: #5, #6, #7, #8, #9
Appreciate that the ratio of corresponding sides in similar triangles is constant
Label the sides of a right-angled triangle using a given angle
Choose an appropriate trigonometric ratio that can be used in a given situation
Understand that sine, cosine and tangent are functions of an angle
Establish the exact values of sinθ and cosθ for θ = 0°, 30°, 45°, 60° and 90°
Establish the exact value of tanθ for θ = 0°, 30°, 45° and 60°
Know how to select the correct mode on a scientific calculator
Use a calculator to find the sine, cosine and tangent of an angle
Know the trigonometric ratios, sinθ = opp/hyp, cosθ = adj/hyp, tanθ = opp/adj
Set up and solve a trigonometric equation to find a missing side in a right-angled triangle
Set up and solve a trigonometric equation when the unknown is in the denominator of a fraction
Set up and solve a trigonometric equation to find a missing angle in a right-angled triangle
Use trigonometry to solve problems involving bearings
Use trigonometry to solve problems involving an angle of depression or an angle of elevation
Prerequisites
Mathematical language
Pedagogical notes
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Similar
Opposite
Adjacent
Hypotenuse
Trigonometry
Function
Ratio
Sine
Cosine
Tangent
Angle of elevation, angle of depression
Ensure that all students are aware of the importance of their scientific
calculator being in degrees mode.
Ensure that students do not round until the end of a multi-step calculation
This unit of trigonometry should focus only on right-angled triangles in two
dimensions. The sine rule, cosine rule, and applications in three dimensions
are covered in Stage 11.
NRICH: History of Trigonometry
NCETM: Glossary
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Understand and work with similar shapes
Solve linear equations, including those with the unknown in the
denominator of a fraction
Understand and use Pythagoras’ theorem
Notation
sinθ stands for the ‘sine of θ’
sin-1 is the inverse sine function, and not 1÷ sin
Common approaches
All students explore sets of similar triangles with angles of (at least) 30°, 45°
and 60° as an introduction to the three trigonometric ratios
The mnemonic ‘Some Of Harry’s Cats Are Heavier Than Other Animals’ is used
to help students remember the trigonometric ratios
Reasoning opportunities and probing questions
Suggested activities
Possible misconceptions
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KM: From set squares to trigonometry
KM: Trigonometry flowchart
NRICH: Trigonometric protractor
NRICH: Sine and cosine
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Learning review
GLOWMaths/JustMaths: Sample Questions Both Tiers
GLOWMaths/JustMaths: Sample Questions Higher Tiers
KM: 10M10 BAM Task
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Show me an angle and its exact sine (cosine / tangent). And another
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Convince me that you have chosen the correct trigonometric
function
(When exploring sets of similar triangles and working out ratios in
corresponding cases) why do you think that the results are all similar,
but not the same? Could we do anything differently to get results
that are closer? How could we make a final conclusion for each
ratio?
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Some students may not appreciate the fact that adjacent and opposite
labels are not fixed, and are only relevant to a particular acute angle. In
situations where both angles are given this can cause difficulties.
Some students may not balance an equation such as sin35 = 4/x correctly,
believing that the next step is (sin35)/4 = x
Some students may think that sin-1θ = 1 ÷ sinθ
Some students may think that sinθ means sin × θ