# Level Ladders - Heathcote School ```Overview: consolidating level 5 and introducing level 6
Unit
Hours
Beyond the Classroom
Integers, powers and roots
6
Sequences, functions and graphs
3
Geometrical reasoning: lines, angles and shapes
5
Construction and loci
3
Probability
4
L5HD2, L5HD3 and L5HD5
Ratio and proportion
7
L5NNS6 and L5CALC5
Equations, formulae, identities and expressions
6
L5ALG1
Measures and mensuration; area
5
L5SSM7
L5SSM2
Learning review 1
Sequences, functions and graphs
6
Mental calculations and checking
4
L5CALC1, and L5NNS4
Written calculations and checking
5
L5CALC3 and L5CALC6
Transformations and coordinates
7
L5ALG2 and L5SSM3
Processing and representing data; Interpreting and discussing results
7
L5HD4 and L5HD6
Equations, formulae, identities and expressions
6
Learning review 2
Fractions, decimals and percentages
6
L5CALC2 and L6NNS1
Measures and mensuration
5
L5SSM5 and L5SSM6
Equations, formulae, identities and expressions
4
Sequences, functions and graphs: using ICT
4
Calculations and checking
5
L5NNS2
Geometrical reasoning: coordinates and construction
3
L5SSM1
Measures and mensuration; volume
5
Statistical enquiry
6
L5HD1
Learning review 3
11-16 Pathway
Stage
Page 1
3 to D+
4 to C+
5 to A+
1
2
3
2
3
4
3
4
5
4
5
6
5*
6*
7
Autumn Term 6 hours
Integers, powers and roots
48-59
Previously…
• Understand the significance of a counter-example
• Recognise and use multiples, factors, primes (less than 100), common factors, highest common factors and lowest common multiples in simple cases; use simple tests of divisibility
• Understand negative numbers as positions on a number line; order, add and subtract positive and negative integers in context.
• Recognise the first few triangular numbers, squares of numbers to at least 12  12 and the corresponding roots
Progression map
• Identify exceptional cases or counter-examples
• Use multiples, factors, common factors, highest common factors, lowest common multiples and primes
• Find the prime factor decomposition of a number (e.g. 8000) using index notation for small positive integer powers
• Add, subtract, multiply and divide integers
• Use squares, positive and negative square roots, cubes and cube roots
Progression map
Next…
• Use the prime factor decomposition of a number (to find highest common factors and lowest common multiples)
• Use ICT to estimate square roots and cube roots
• Use index notation for integer powers; know and use the index laws for multiplication and division of positive integer powers
Progression map
Suggested Activities
Maths Apprentice
 Exploring primes activities: Numbers of factors; factors of square numbers;
Mersenne primes; LCM sequence; Goldbach's theorem; n&sup2; and (n + 1)&sup2;; n&sup2; and n&sup2;
+ n; n&sup2; + 1; n! + 1; n! – 1; KPO: x2 + x +41
 Summing Up – add/subtract integers
 Developing Negatives – multiply/divide integers
 The root of the problem
 Consecutive products
 Cuisenaire factors
 Handling rules for primes, e.g. using the 4n-1 rule and comparing this to the list
of square numbers; use a spreadsheet to find difference between consecutive
cubes. This difference is always prime – true? What do you notice about the
second difference?
 Use a spreadsheet to show Eratosthenes' sieve in 6 columns - what do you
notice when you highlight the primes?
KS3 Top-up Bring on the Maths
 Problem Solving: v1, v2
Y8 Bring on the Maths
 Integers: v1, v2, v3
 Logic: v1, v2, v3
Criteria for Success
Standards Unit
 N9 Evaluating directed
number statements
NCETM Departmental
Workshops
 Operations with directed
numbers
What patterns arise when you multiply consecutive pairs
/ triples?
 Powers, integers, roots
182 is the product of 2 consecutive integers, but the
answer is not unique. Find both products
Beyond the Classroom
 Negative numbers
Are the prime factors of a number unique?
APP
Look for learners doing:
 L5NNS3
 L5CALC4*
 L5UA1
Is the prime factor decomposition of a number unique?
NRICH
 Playing Connect Three
 Weights
 Consecutive Negative
Numbers
 Stars
 14 Divisors
 Take Three from Five
 Differences
 Sissa's Reward
When finding prime factor decompositions using the
‘tree’ method, does it matter how you break down the
starting number?
What happens when you raise a number to a negative /
fractional power?
Can every cube of a number be written as the difference
of two squares?
Multiply the triangular numbers by 8 and add 1. What
numbers do you get? Why?
Resources
 Number line - extend to negative number line; consider negative movement along
number line
 Powers - HTU chart
 Venn diagrams for HCF/LCM
Is there a pattern in the prime numbers?
Page 2
Autumn Term 3 hours
Sequences, functions and graphs
144-157
Previously…
• Use appropriate procedures and tools, including ICT
• Describe integer sequences; generate terms of a simple sequence, given a rule (e.g. finding a term from the previous term, finding a term given its position in the sequence)
• Generate sequences from patterns or practical contexts and describe the general term in simple cases
Progression map
• Select appropriate procedures and tools, including ICT
• Use correct syntax when using ICT
• Generate terms of a linear sequence using term-to-term and position-to-term definitions of the sequence, on paper and using a spreadsheet or graphical calculator
• Use linear expressions to describe the nth term of a simple arithmetic sequence, justifying its form by referring to the activity or practical context from which it was generated
Progression map
Next…
• Generate terms of a sequence using term-to-term and position-to-term rules, on paper and using ICT
• Generate sequences from practical contexts and write and justify an expression to describe the nth term of an arithmetic sequence
Progression map
Suggested Activities
Criteria for Success
Maths Apprentice
 KPO: Generating Sequences
NCETM Departmental
Workshops
 Sequences
HORN, Cornwall
 Generating sequences 2
 Can you cut it?
 Checking temperatures
 Which way sequences 2
 Function machines
NRICH
 1 Step 2 Step
 Coordinate Patterns
 Seven Squares
Create a sequence of shapes / diagrams that have a
feature that can be described by 4n + 1, 3x – 2, etc.
Show me:
 a sequence that has the term-to-term rule of +2.
 the sequence that has the position-to-term rule of
+2.
 the sequence that has the nth term of i) n+2 ii) 3n+2
Convince me that:
 the nth term of the sequence 5, 8, 11, 14, … is 3n+ 2
 that the nth term of the sequence 15, 11, 7, 3, … is 19
– 4n
Y8 Bring on the Maths
 Problem Solving: v1, v2, v3
KS3 Top-up Bring on the Maths
 Sequences: v1, v2, v3
Resources
Physical equipment - multilink, matchsticks, counters, pattern blocks etc. so that the
shape can illustrate the rules generated.
Page 3
 Sequences, functions and
graphs
APP
Look for learners doing:
 L5NNS3
 L6ALG3
Autumn Term 5 hours
Geometrical reasoning: lines, angles and shapes
178-189
Previously…
• Recognise equivalent approaches
• Use correctly the vocabulary, notation and labelling conventions for lines, angles and shapes
• Identify parallel and perpendicular lines; know the sum of angles at a point, on a straight line and in a triangle; recognise vertically opposite angles
• Identify and use angle, side and symmetry properties of triangles and quadrilaterals; explore geometrical problems involving these properties, explaining reasoning orally, using step-bystep deduction supported by diagrams
Progression map
• Recognise efficiency in an approach
• Record solutions and conclusions
• Identify alternate angles and corresponding angles; understand a proof that:
(i) the sum of the angles of a triangle is 180&ordm; and of a quadrilateral is 360&ordm;;
(ii) the exterior angle of a triangle is equal to the sum of the two interior opposite angles.
• Solve geometrical problems using side and angle properties of equilateral, isosceles and right-angled triangles and special quadrilaterals, explaining reasoning with diagrams and text;
classify quadrilaterals by their geometrical properties
Progression map
Next…
• Look for and reflect on other approaches
• Distinguish between conventions, definitions and derived properties
• Explain how to find, calculate and use:
(i) the sums of the interior and exterior angles of quadrilaterals, pentagons and hexagons;
(ii) the interior and exterior angles of regular polygons
• Solve problems using properties of angles, of parallel and intersecting lines, and of triangles and other polygons, justifying inferences and explaining reasoning with diagrams and text
Progression map
Suggested Activities
Maths Apprentice
 Cut a parallelogram into two trapezia
 3x3, 4x4, 5x5 dotty paper activities: KPO: Shapes on dotty paper
 Perplexing Parallels
 Angles in polygons
HORN, Cornwall
 Alternate and corresponding angles
 Angles in a triangle
 Geometrical visualisations 2
Criteria for Success
Standards Unit
 SS1 Classifying Shapes
Find 2 shapes with an area of ___ but with different
perimeters. Extend to volume and surface area.
 Geometrical reasoning
NCETM Departmental
Workshops
 Angle Properties
Why are these called alternate angles?
Beyond the Classroom
 Angles
NRICH
 Eight Hidden Squares
 Square Coordinates
 Square It
How do you know this is true?
Y8 Bring on the Maths
 Lines and Angles: v1, v2, v3
Why are these called corresponding angles?
Can you cut a triangle into 4 quadrilaterals and a
triangle?
Can you cut a triangle into quadrilaterals only, without
putting new corners on the sides of the triangle?
KS3 Top-up Bring on the Maths
 Lines and Angles: v1, v2, v3
Level 5 Bring on the Maths
 Shape, Space and Measures: Using angle facts
Resources
 Spokes OHTs: clock (30&deg;), compass rose (45&deg;), 90&deg; spray
 Pattern Blocks
 3x3, 4x4, 5x5 dotty paper
Page 4
APP
Look for learners doing:
 L5SSM2*
 L6SSM1
 L6SSM3
Autumn Term 3 hours
Construction and loci
14-17, 220–223
Previously…
• Classify and visualise properties and patterns
• Use a ruler and protractor to:
(i) measure and draw lines to the nearest millimetre and angles, including reflex angles, to the nearest degree;
(ii) construct a triangle given two sides and the included angle (SAS) or two angles and the included side (ASA)
• Use ICT to explore constructions
Progression map
• Visualise and manipulate dynamic images
• Make accurate mathematical constructions on paper and on screen
• Find simple loci, both by reasoning and by using ICT, to produce shapes and paths, e.g. an equilateral triangle
• Use straight edge and compasses to construct;
(i) the mid-point and perpendicular bisector of a line segment;
(ii) the bisector of an angle;
(iii) the perpendicular from a point to a line;
(iv) the perpendicular from a point on a line
(v) a triangle, given three sides (SSS)
• Use ICT to explore these constructions
Progression map
Next…
• Find the locus of a point that moves according to a simple rule, both by reasoning and by using ICT
• Use straight edge and compasses to construct a triangle, given right angle, hypotenuse and side (RHS)
• Use ICT to explore constructions of triangles and other 2-D shapes
Suggested Activities
Maths Apprentice
 Demonstrate by getting pupils outside and asking them, one at a time to go and
stand at a position equidistant from two fixed points of your choosing
 KPO: Use dynamic geometry software to construct and explore dynamic
versions of these constructions
 Demonstrate perpendicular bisector using a 'story' and bits of rope (eg 2 icecream sellers on a beach)
 Set the task backwards - 'using straight-edge and compasses construct …'
 Using the perpendicular bisector
 Sketch a triangle. Draw in the bisectors of the three angles. Is it possible to draw
a triangle where two of these angle bisectors are at right angles to each other?
Can you prove it? (Hint: Assume that two of the angle bisectors do meet at 90&ordm;.
Progression map
Criteria for Success
NCETM Departmental
Workshops
 Constructions
 Loci
How can we construct an angle of 45 / 30 / 120 etc.
How many different triangles can be made with SSS,
SAS, SSA?
Show me:
 a construction you can do using a straight edge and a
pair of compasses
 a construction where it is important to keep the same
compass arc.
Autograph Resources
 The Perpendicular Bisector
Y8 Bring on the Maths
 Construction: v1, v2, v3
Page 5
 Construction, loci
APP
Look for learners doing:
 L5SSM4
 L6SSM4
 L6SSM8
Autumn Term 5 hours
Probability
276--283
Previously…
• Draw simple conclusions and explain reasoning
• Use vocabulary and ideas of probability, drawing on experience
• Understand and use the probability scale from 0 to 1; find and justify probabilities based on equally likely outcomes in simple contexts; identify all the possible mutually exclusive outcomes
of a single event
• Estimate probabilities by collecting data from a simple experiment and recording it in a frequency table; compare experimental and theoretical probabilities in simple contexts
Progression map
• Move between the general and the particular to test the logic of an argument
• Interpret the results of an experiment using the language of probability; appreciate that random processes are unpredictable
• Know that if the probability of an event occurring is p, then the probability of it not occurring is 1-p; use diagrams and tables to record in a systematic way all possible mutually exclusive
outcomes for single events and for two successive events
• Compare estimated experimental probabilities with theoretical probabilities, recognising that:
(i) if an experiment is repeated the outcome may, and usually will, be different
(ii) increasing the number of times an experiment is repeated generally leads to better estimates of probability
Progression map
Next…
• Pose questions and make convincing arguments to justify generalisations or solutions
• Interpret results involving uncertainty and prediction
• Identify all the mutually exclusive outcomes of an experiment; know that the sum of probabilities of all mutually exclusive outcomes is 1 and use this when solving problems
• Compare experimental and theoretical probabilities in a range of contexts; appreciate the difference between mathematical explanation and experimental evidence
Progression map
Suggested Activities
Maths Apprentice
 Set up a spreadsheet to automatically construct sample space diagrams for 2
dice etc
 KPO: Race game and marking guidance (can use Standards Unit S3 to extend
this)
 Probability pots
Criteria for Success
Standards Unit
 S3 Playing probability
computer games
The probability it will rain tomorrow is &frac12; - True or False?
Why?
How can you decide how many outcomes there will be?
NRICH
 Interactive Spinners
 Flippin' Discs
If I flip a coin 1000 times will I get 500 heads?
Y8 Bring on the Maths
 Probability: v1, v2, v3
If you repeat an experiment, will you always / sometimes
/ never get the same result?
KS3 Top-up Bring on the Maths
 Probability: v1, v2, v3
Design an experiment that will give probabilities of 1/3,
1/2, 2/5 etc.
Level 5 Bring on the Maths
 Handling Data: Using equally likely outcomes in probability, Using the
probability scale, Experimenting with probability
Selection (say 10) of different coloured counters in a
bag. Pick and replace several times. At each pick, what
do you think the colours of the 10 counters are? How
can we be even more sure?
HORN, Cornwall
 How many times? 2
How can you make a game fair?
Resources
Probability scale, Probability recording sheets, Probability pots, Possibility space
diagrams, Tree diagrams
A coin is flipped 10 times and you get 2H and 8T, is this
coin biased?
Page 6

http://www.kangaro
omaths.com/free_re
sources/ks3/resourc
es/other/cornwall_p
rogression/sow_pro
bability.docProbability
Beyond the Classroom
 Probability
 The probability scale
 Experiments
APP
Look for learners doing:
 L5HD2*
 L5HD3*
 L5HD5*
 L5UA5
 L6HD3
Ratio and proportion
2-35, 78-81
Previously…
• Interpret information from a mathematical representation or context
• Justify the mathematical features drawn from a context and the choice of approach
• Understand the relationship between ratio and proportion; use direct proportion in simple contexts; use ratio notation, simplify ratios and divide a quantity into two parts in a given ratio;
solve simple problems involving ratio and proportion using informal strategies
Progression map
• Use logical argument to interpret the mathematics in a given context or to establish the truth of a statement *
• Evaluate the efficiency of alternative strategies and approaches **
• Apply understanding of the relationship between ratio and proportion; simplify ratios, including those expressed in different units, recognising links with fraction notation; divide a quantity
into two or more parts in a given ratio; use the unitary method to solve simple problems involving ratio and direct proportion
Progression map
Next…
• Compare and evaluate approaches
• Use proportional reasoning to solve problems, choosing the correct numbers to take as 100%, or as a whole; compare two ratios; interpret and use ratio in a range of contexts
Progression map
Autumn Term 7 hours
Suggested Activities
Maths Apprentice
 Design a spreadsheet that someone could use to transfer cost price to selling
price, using 30% mark-up. How could they use this to find the cost price from
the selling price?
 Find the exchange rates for converting &pound;400 to euros from banks / PO/ bureaux
de change. (No coins). (Compare this with buying travellers’ cheques and
exchanging in Europe on three separate days, what was the overall exchange
rate on these days?)
 Fractions OHT
 Equivalents jigsaw (new Frac Dec.ppt)
 Play your cards right (comparing FDP)
 If we could shrink the world
Criteria for Success
Standards Unit
 KPO**: N6 Developing
proportional reasoning
The answer is ‘&pound;350 and &pound;450’. What is the question?
Draw a golden rectangle (sides are in the ratio 1:1.618)
Divide into the largest square possible and a rectangle.
What do you notice about the resulting rectangle?
Is a 10% increase on &pound;300 the same as 30% increase on
&pound;100? Why?
If you increase by e.g. 20% then by a further 20% is this
Ratio of squash to water is 1:3. Is 2:5 stronger or
weaker?
HORN, Cornwall
 Paper proportionality
Are these number sets in proportion?
Y8 Bring on the Maths
 Ratio: v1, v2, v3, v4, v5, v6
KS3 Top-up Bring on the Maths
 Ratio and Proportion 1: v1, v2, v3
 Ratio and Proportion 2: v1, v2, v3
 KPO*: Problem Solving: v3
Level 5 Bring on the Maths
 Numbers and the Number System: Simplifying ratios
 Calculating: Solving simple ratio problems
Resources
Fractions images / OHTs, Proportional sets 1, Proportional sets 2
Page 7
 Fractions
 Percentages
Beyond the Classroom
 Ratio
 Ratio and proportion
APP
Look for learners doing:
 L5NNS6*
 L5CALC5*
 L6CALC2
Autumn Term 6 hours
Equations, formulae, identities and expressions
112–119, 138–143
Previously…
• Explain and justify methods and conclusions
• Use letter symbols to represent unknown numbers or variables; know the meanings of the words term, expression and equation
• Understand that algebraic operations follow the rules of arithmetic
• Simplify linear algebraic expressions by collecting like terms; multiply a single term over a bracket (integer coefficients)
• Substitute positive integers into linear expressions
Progression map
• Use logical argument to establish the truth of a statement
• Manipulate algebraic expressions
• Recognise that letter symbols play different roles in equations, formulae and functions; know the meanings of the words formula and function
• Understand that algebraic operations, including the use of brackets, follow the rules of arithmetic; use index notation for small positive integer powers
• Simplify or transform linear expressions by collecting like terms; multiply a single term over a bracket
• Substitute integers into simple formulae
Progression map
Next…
• Generate fuller solutions by presenting a concise, reasoned argument using symbols, diagrams, graphs and related explanations
• Distinguish the different roles played by letter symbols in equations, identities, formulae and functions
• Use index notation for integer powers and simple instances of the index laws
• Simplify or transform algebraic expressions by taking out single-term common factors
• Substitute numbers into expressions and formulae
Suggested Activities
Maths Apprentice
 Compare expressions 1 &amp; Compare expressions 2
 Algebra ordering cards
 Farmer Fisher
Y8 Bring on the Maths
 Simplifying expressions: v1, v2, v3
 Substitution: v1, v2, v3
Progression map
Criteria for Success
Standards Unit
 KPO: A4 Evaluating
algebraic expressions
The answer is 2x+5y. What is the question?
NCETM Departmental
Workshops
 Constructing Equations
Find five expressions equivalent to 2y = 6x+4
The answer is 4n-12. What is the question?
Level 5 Bring on the Maths
 Algebra: Simplifying expressions, Substituting into simple formulae
Show me a formula involving a and b such that when
you substitute a = -2 and b = 3 into the formula you get
18.
HORN, Cornwall
 What's the difference?
 Inverse operations
 Finding equivalent expressions
 Substitution 2
 Deriving formulae 2
What is wrong:

3(b+1) = 3b + 1

10(p -4) = 10p - 6

-2 (3 - f) = -6 -2f

8 – (n – 1) = 7 – n
Resources
Equation snakes, Grid method of multiplying, extended to expanding brackets
Convince me that:

2(x+7) = 2x + 14

5(y -4) = 5y - 20
Page 8
Beyond the Classroom
 Simple formulae
Why are x, x2, x3 not like terms? Consider m, m2, m3.
Show me a formula involving a and b such that when
you substitute a = 2 and b = 7 into the formula you get
18.
KS3 Top-up Bring on the Maths
 Algebraic Expressions: v1, v2, v3
 Equations, formulae,
identities
APP
Look for learners doing:
 L5ALG1*
 L6UA4
Autumn Term 5 hours
Measures and mensuration; area
228–231, 234–241
Previously…
• Choose and use units of measurement to measure, estimate, calculate and solve problems in everyday contexts
• Know and use the formula for the area of a rectangle; calculate the perimeter and area of shapes made from rectangles
• Calculate the surface area of cubes and cuboids
Progression map
• Record methods, solutions and conclusions*
• Relate the current problem and structure to previous situations**
• Choose and use units of measurement to measure, estimate, calculate and solve problems in a range of contexts
• Derive and use formulae for the area of a triangle, parallelogram and trapezium; calculate areas of compound shapes
Progression map
Next…
• Solve problems involving measurements in a variety of contexts; convert between area measures (e.g. mm2 to cm2, cm2 to m2, and vice versa)
• Build on previous experience of similar situations and outcomes
• Know and use the formulae for the circumference and area of a circle
• Calculate the surface area of right prisms
Progression map
Suggested Activities
Maths Apprentice
 3x3, 4x4, 5x5 dotty paper activities - area and perimeter
 Dissection deductions
 Triangle Takeaway
 Disappearing squares
 KPO*: Equable shapes
Y8 Bring on the Maths
 Area and Volume: v1, v2, v3
Criteria for Success
Standards Unit
 KPO**: SS4 Evaluating
and area
How do you know which is the base and height?
Find six triangles with an area of 12cm2
Find six parallelograms with an area of 48cm2
NRICH
 All in a Jumble
 Isosceles Triangles
 Pick's Theorem
KS3 Top-up Bring on the Maths
 Area: v1, v2, v3
Level 5 Bring on the Maths
 Shape, Space and Measures: Area and perimeter
Resources
3x3, 4x4, 5x5 dotty paper
LEARNING REVIEW 1
Page 9
 Measures
Beyond the Classroom
 Area and perimeter
APP
Look for learners doing:
 L5SSM7*
 L6SSM9
 L6UA1
Spring Term 6 hours
Sequences, functions and graphs
6–13, 28–29, 160–177
Previously…
• Express simple functions in words, then using symbols; represent them in mappings
• Generate coordinate pairs that satisfy a simple linear rule; plot the graphs of simple linear functions, where y is given explicitly in terms of x, on paper and using ICT; recognise straight-line
graphs parallel to the x-axis or y-axis
• Plot and interpret the graphs of simple linear functions arising from real-life situations, e.g. conversion graphs
Progression map
• Express simple functions algebraically and represent them in mappings or on a spreadsheet
• Generate points in all four quadrants and plot the graphs of linear functions, where y is given explicitly in terms of x, on paper and using ICT; recognise that equations of the form y = mx + c
correspond to straight-line graphs
• Construct linear functions arising from real-life problems and plot their corresponding graphs; discuss and interpret graphs arising from real situations, e.g. distance–time graphs
Progression map
Next…
• Find the inverse of a linear function
• Generate points and plot graphs of linear functions, where y is given implicitly in terms of x (e.g. ay + bx = 0, y + bx + c = 0), on paper and using ICT; find the gradient of lines given by
equations of the form y = mx + c, given values for m and c
• Construct functions arising from real-life problems and plot their corresponding graphs; interpret graphs arising from real situations, e.g. time series graphs
Progression map
Suggested Activities
A second algebraic graphs unit in the summer term focuses on the use of ICT
Maths Apprentice
 Fascinating food
 Hare and Tortoise
 Use a graphic calculator with data logger to investigate time-distance graphs
Criteria for Success
NRICH
 How Steep Is the Slope?
 Parallel Lines
Find another line that is parallel to this one. How do you
know they are parallel?
 Sequences, functions, graphs
Find another line with the same y-intercept
APP
Look for learners doing:
 L6ALG4
 L6ALG5
Which of these are parallel: y = 2x+1, y = x+2, 2y = 4x –
10?
Autograph Resources
 Matching Graphs Quiz 3
Y8 Bring on the Maths
 Plotting graphs: v1, v2, v3
HORN, Cornwall
 Function machines
Resources
Page 10
Spring Term 4 hours
Mental calculations and checking
82–87, 92–107, 110–111, 108--109
Previously…
• Understand and use the rules of arithmetic and inverse operations in the context of positive integers and decimals
• Use the order of operations, including brackets
• Strengthen and extend mental methods of calculation to include decimals, fractions and percentages, accompanied where appropriate by suitable jottings; solve simple problems mentally
• Recall number facts, including positive integer complements to 100 and multiplication facts to 10 &times; 10, and quickly derive associated division facts
• Check results by considering whether they are of the right order of magnitude and by working problems backwards
Progression map
• Try out and compare mathematical representations
• Understand and use the rules of arithmetic and inverse operations in the context of integers and fractions
• Use the order of operations, including brackets, with more complex calculations
• Strengthen and extend mental methods of calculation, working with decimals, fractions, percentages, squares and square roots, and cubes and cube roots; solve problems mentally
• Recall equivalent fractions, decimals and percentages; use known facts to derive unknown facts, including products involving numbers such as 0.7 and 6, and 0.03 and 8
• Make and justify estimates and approximations of calculations
• Select from a range of checking methods, including estimating in context and using inverse operations
Progression map
Next…
• Recognise the impact of constraints or assumptions
• Understand the effects of multiplying and dividing by numbers between 0 and 1; consolidate use of the rules of arithmetic and inverse operations
• Understand the order of precedence of operations, including powers
• Use known facts to derive unknown facts; extend mental methods of calculation, working with decimals, fractions, percentages, factors, powers and roots; solve problems mentally
• Check results using appropriate methods
Progression map
Suggested Activities
Maths Apprentice
 Where’s the point?
 Misplaced Points
KS3 Top-up Bring on the Maths
 Using a Calculator: v1
 Solving Word Problems: v1
Criteria for Success
NRICH
 The Greedy Algorithm
 Thousands and Millions
 Keep it Simple
 Egyptian Fractions
Level 5 Bring on the Maths
 Calculating: Order of operations
 Numbers and the Number System: Equivalence between fractions, Ordering
fractions and decimals
Can division ever make a number larger?
 Mental calculations
Can multiplication ever make a number smaller?
digits / estimating]
Beyond the Classroom
 Known facts, place value and
order of operations
 Equivalence between
fractions
APP
Look for learners doing:
 L5CALC1*
 L5UA2
 L5NNS4*
Standards Unit
KPO: N5 Understanding the laws of arithmetic
Resources
Place value chart, Number lines, HTU chart, Gattegno charts
Page 11
Spring Term 5 hours
Written calculations and checking
82–87, 92–107, 110–111, 108--109
Previously…
• Generalise in simple cases by working logically
• Understand and use the rules of arithmetic and inverse operations in the context of positive integers and decimals
• Use the order of operations, including brackets
• Use efficient written methods to add and subtract whole numbers and decimals with up to two places
• Multiply and divide three-digit by two-digit whole numbers; extend to multiplying and dividing decimals with one or two places by single-digit whole numbers
• Carry out calculations with more than one step using brackets and the memory; use the square root and sign change keys
• Enter numbers and interpret the display in different contexts (decimals, percentages, money, metric measures)
• Check results by considering whether they are of the right order of magnitude and by working problems backwards
Progression map
• Manipulate numbers and apply routine algorithms
• Conjecture and generalise
• Understand and use the rules of arithmetic and inverse operations in the context of integers and fractions
• Use the order of operations, including brackets, with more complex calculations
• Use efficient written methods to add and subtract integers and decimals of any size, including numbers with differing numbers of decimal places
• Use efficient written methods for multiplication &amp; division of integers &amp; decimals, including by decimals such as 0.6 or 0.06; understand where to position the decimal point by considering
equivalent calculations
• Carry out more difficult calculations effectively and efficiently using the function keys for sign change, powers, roots and fractions; use brackets and the memory
• Enter numbers and interpret the display in different contexts (extend to negative numbers, fractions, time)
• Make and justify estimates and approximations of calculations
• Select from a range of checking methods, including estimating in context and using inverse operations
Progression map
Next…
• Recognise the impact of constraints or assumptions
• Understand the effects of multiplying and dividing by numbers between 0 and 1; consolidate use of the rules of arithmetic and inverse operations
• Understand the order of precedence of operations, including powers
• Use efficient written methods to add and subtract integers and decimals of any size; multiply by decimals; divide by decimals by transforming to division by an integer
• Use a calculator efficiently and appropriately to perform complex calculations with numbers of any size, knowing not to round during intermediate steps of a calculation; use the constant, 
and sign change keys; use the function keys for powers, roots and fractions; use brackets and the memory
• Check results using appropriate methods
Progression map
Suggested Activities
Maths Apprentice
 KPO: Number problems to explore
 Where’s the point?
 Misplaced Points
Criteria for Success
NRICH
 Largest Product
 Legs Eleven
Can division ever make a number larger?
Can multiplication ever make a number smaller?
digits / estimating]
KS3 Top-up Bring on the Maths
 Using a Calculator: v2, v3
 Written calculations
Beyond the Classroom
 Multiplying and dividing
 Checking solutions
APP
Look for learners doing:
 L5CALC3*
 L5CALC6*
Level 5 Bring on the Maths
 Calculating: Multiplying and dividing, Approximating and checking
Resources
Place value chart, Number lines, HTU chart, Gattegno charts
Page 12
Spring Term 7 hours
Transformations and coordinates
190–191, 202–215, 78–81
Previously…
• Communicate own findings effectively, orally and in writing, and discuss and compare approaches and results with others
• Understand and use the language and notation associated with reflections, translations and rotations
• Recognise and visualise the symmetries of a 2-D shape
• Transform 2-D shapes by:
(i) reflecting in given mirror lines;
(ii) rotating about a given point;
(iii) translating.
• Explore these transformations and symmetries using ICT
Progression map
• Refine own findings and approaches on the basis of discussions with others
• Identify all the symmetries of 2-D shapes
• Transform 2-D shapes by rotation, reflection and translation, on paper and using ICT
• Try out mathematical representations of simple combinations of these transformations
• Understand and use the language and notation associated with enlargement; enlarge 2-D shapes, given a centre of enlargement and a positive integer scale factor; explore enlargement
using ICT
• Know that if two 2-D shapes are congruent, corresponding sides and angles are equal
Progression map
Next…
• Review and refine own findings and approaches on the basis of discussions with others
• Identify reflection symmetry in 3-D shapes
• Recognise that translations, rotations and reflections preserve length and angle, and map objects on to congruent images
• Devise instructions for a computer to generate and transform shapes
• Explore and compare mathematical representations of combinations of translations, rotations and reflections of 2-D shapes, on paper and using ICT
• Enlarge 2-D shapes, given a centre of enlargement and a positive integer scale factor, on paper and using ICT; identify the scale factor of an enlargement as the ratio of the lengths of any
two corresponding line segments; recognise that enlargements preserve angle but not length, and understand the implications of enlargement for perimeter
• Understand congruence and explore similarity
Progression map
Suggested Activities
Maths Apprentice
 KPO: Two investigations
 Measuring Enlargements
 Moving House
 Diagonal Reflections
 Congruent halves
 Transformations template
 Tessellating Tess
Criteria for Success
NRICH
 Transformation Game
Where do you think the enlargement will be?
What is wrong with this enlargement?
What does the centre of enlargement mean?
What is the scale factor of this enlargement?
 Transformations
 Geometrical reasoning
Beyond the Classroom
 Coordinates in four
 Transforming shapes
Does a rectangle have four lines of symmetry?
Y8 Bring on the Maths
 Enlargement: v1, v2, v3
When enlarging on a coordinate grid: What connections
are there between the coordinates of corresponding
vertices?
Level 5 Bring on the Maths
 Algebra: Coordinates in four quadrants
 Shape, Space and Measures: Reflecting shapes, Rotational symmetry
Resources
3x3, 4x4, 5x5 dotty paper
Page 13
APP
Look for learners doing:
 L5ALG2*
 L5SSM3*
 L6SSM6
 L6SSM7
Spring Term 7 hours
Processing and representing data; Interpreting and discussing results
248–273
Previously…
• Represent problems, making correct use of symbols, words, diagrams, tables and graphs
• Calculate statistics for small sets of discrete data:
(i) find the mode, median and range, and the modal class for grouped data
(ii) calculate the mean, including from a simple frequency table, using a calculator for a larger number of items
• Construct, on paper and using ICT, graphs and diagrams to represent data, including:
(i) bar-line graphs
(ii) frequency diagrams for grouped discrete data
(iii) simple pie charts
• Interpret diagrams and graphs (including pie charts), and draw simple conclusions based on the shape of graphs and simple statistics for a single distribution
Progression map
• Try out and compare mathematical representations
• Calculate statistics for sets of discrete and continuous data, including with a calculator and spreadsheet; recognise when it is appropriate to use the range, mean, median and mode and, for
grouped data, the modal class
• Construct graphical representations, on paper and using ICT, and identify which are most useful in the context of the problem. Include:
(i) pie charts for categorical data
(ii) bar charts and frequency diagrams for discrete and continuous data
(iii) simple line graphs for time series
(iv) simple scatter graphs
(v) stem-and-leaf diagrams
• Interpret tables, graphs and diagrams for discrete and continuous data, relating summary statistics and findings to the questions being explored
Progression map
Next…
• Represent problems and synthesise information in algebraic, geometrical or graphical form; move from one form to another to gain a different perspective on the problem
• Calculate statistics and select those most appropriate to the problem or which address the questions posed
• Select, construct and modify, on paper and using ICT, suitable graphical representations to progress an enquiry and identify key features present in the data. Include:
(i) line graphs for time series
(ii) scatter graphs to develop further understanding of correlation
• Interpret graphs and diagrams and make inferences to support or cast doubt on initial conjectures; have a basic understanding of correlation
Progression map
Suggested Activities
Maths Apprentice
 Millions of sweets?
 Shoe shop manager: find the shoe sizes of pupils in the class. Which average should we use?
 Underwater experts? Repeat for a ‘holding your breath’ experiment (probable skewed data)
 Scatter diagram buddies
Autograph Resources
 Scatter Graphs
Y8 Bring on the Maths
 Graphs and Charts: v1, v2, v3
Criteria for Success
Standards Unit
KPO: S4 Understanding
mean, median, mode
and range
What is an appropriate graph / chart for this?
Why?
NCETM Departmental
Workshops
 Statistical Data
 Stem-and-leaf
diagrams
What do the scales mean?
KS3 Top-up Bring on the Maths
 Handling Data: v1, v2, v3
Can the mean = median = mode?
Beyond the Classroom
 Averages
 Graphs and diagrams
Is it more efficient to use ICT here?
Are scatter diagrams appropriate for these
data?
What does the graph tell you?
Why would you choose to use ICT here?
Level 5 Bring on the Maths
 Handling Data: Averages and the range, Interpreting pie charts
HORN, Cornwall
 Finding the mean
Does this piece of data fit the trend. If not,
can you think of a reason why it doesn’t?
Page 14
 Processing, representing and
interpreting data
APP
Look for learners doing:
 L5HD4*
 L5HD6*
 L6HD2
 L6UA2
Spring Term 6 hours
Equations, formulae, identities and expressions
112–113, 122–125
Previously…
• Know the meanings of the words term, expression and equation
• Construct and solve simple linear equations with integer coefficients (unknown on one side only) using an appropriate method (e.g. inverse operations)
Progression map
• Manipulate algebraic equations
• Recognise that letter symbols play different roles in equations, formulae and functions
• Construct and solve linear equations with integer coefficients (unknown on either or both sides, without and with brackets) using appropriate methods (e.g. inverse operations,
transforming both sides in same way)
• Use graphs and set up equations to solve simple problems involving direct proportion
• Substitute integers into simple formulae, including examples that lead to an equation to solve
Progression map
Next…
• Distinguish the different roles played by letter symbols in equations, identities, formulae and functions
• Construct and solve linear equations with integer coefficients (with and without brackets, negative signs anywhere in the equation, positive or negative solution)
• Use systematic trial and improvement methods and ICT tools to find approximate solutions to equations such as x 2 + x = 20
• Use algebraic methods to solve problems involving direct proportion; relate algebraic solutions to graphs of the equations; use ICT as appropriate
Progression map
Suggested Activities
Maths Apprentice
 KPO: Equation snakes: For example – give 5x – 13 = 2x + 5 in the first section of
the snake, and ask pupils to create a route through the snake where the same
operation is applied to both sides of the equation at each move, justifying as
they go. What is the most efficient way to arrive at x = 6? Alternatively, ‘cloud
the picture’ by starting with x = 7 (for example), and building up an increasingly
complicated equation, which can then be given to a friend to solve.
 Arithmagons
 Equal, balanced, same both sides
 Piles of stones
 Equation Balancer
 Cuisenaire simultaneous equations
Criteria for Success
NCETM Departmental
Workshops
 Constructing Equations
NRICH
 Think of Two Numbers
 Number Tricks
HORN, Cornwall
 Deriving formulae 2
KS3 Top-up Bring on the Maths
 Algebraic Equations: v1, v2, v3
Resources
Flowcharts for equations, Equal balanced, same both sides, Piles of stones
LEARNING REVIEW 2
Page 15
Model incorrect solutions to equation solving: What is
wrong with this?
Give me six equations with the same solution? How do
you work this out?
 Equations, formulae,
identities
APP
Look for learners doing:
 L6ALG2
Summer Term 6 hours
Fractions, decimals and percentages
60–77, 82–85, 88–101
Previously…
• Take account of feedback and learn from mistakes
• Express a smaller whole number as a fraction of a larger one; simplify fractions by cancelling all common factors and identify equivalent fractions; convert terminating decimals to fractions,
e.g. 0.23 = 23/100; use diagrams to compare two or more simple fractions
• Add and subtract simple fractions and those with common denominators; calculate simple fractions of quantities and measurements (whole-number answers); multiply a fraction by an
integer
• Understand percentage as the ‘number of parts per 100’; calculate simple percentages and use percentages to compare simple proportions
• Recognise the equivalence of percentages, fractions and decimals
Progression map
• Make connections with related contexts
• Estimate, approximate and check working
• Recognise that a recurring decimal is a fraction; use division to convert a fraction to a decimal; order fractions by writing them with a common denominator or by converting them to
decimals
• Add and subtract fractions by writing them with a common denominator; calculate fractions of quantities (fraction answers); multiply and divide an integer by a fraction
• Interpret percentage as the operator ‘so many hundredths of’ and express one given number as a percentage of another; calculate percentages and find the outcome of a given percentage
increase or decrease
• Use the equivalence of fractions, decimals and percentages to compare proportions
Progression map
Next…
• Use connections with related contexts to improve the analysis of a situation or problem
• Understand the equivalence of simple algebraic fractions; know that a recurring decimal is an exact fraction
• Use efficient methods to add, subtract, multiply and divide fractions, interpreting division as a multiplicative inverse; cancel common factors before multiplying or dividing
• Recognise when fractions or percentages are needed to compare proportions; solve problems involving percentage changes
Progression map
Suggested Activities
Maths Apprentice
 Which Fraction Charts
 Cuisenaire ratio and proportion
 If we could shrink the world
 Fractions poem
 Design a spreadsheet that someone could use to transfer cost price to selling
price, using 30% mark-up. How could they use this to find the cost price from
the selling price?
Y8 Bring on the Maths
 Fractions, Decimals &amp; Percentages: v1, v2, v3, v4, v5, v6
KS3 Top-up Bring on the Maths
 Fractions, Decimals &amp; Percentages 1: v1, v2, v3
 Fractions, Decimals &amp; Percentages 2: v1, v2, v3
Criteria for Success
Standards Unit
KPO: N7 Using percentages
to increase quantities
If you know that 1/5 = 0.2, what else can you deduce?
NCETM Departmental
Workshops
 Fractions
Find a unit fraction that is the sum of two other unit
fractions. How many can you find
NRICH
 Fair Shares
 Ben's Game
 Farey Sequences
 Round and Round and
Round
If you know that 1/8 = 0.125, what else can you deduce?
Find a fraction that is between &frac12;, and &frac34;. How did you
find this?
Camel problem - 3 sons to have 1/2, 1/3 and 1/9 of dad’s
17 camels. Bloke offers 1 camel, splits them and bloke
gets camel back. How? (nice plenary!)
Extend sequences of + and - fractions from Y7 – e.g. 1/2
+1/3 + 1/6 = 1. Can you predict the result of 1/4 + 1/6 +
1/12? Further predictions?
Level 5 Bring on the Maths
 Calculating: Fractions and percentages of quantities
Level 6 Bring on the Maths
 Numbers and the Number System: Comparing Proportions
HORN, Cornwall
 Paper proportionality
Resources
Fractions images, Proportional sets 1, Proportional sets 2
Page 16
 Fractions
 Percentages
Beyond the Classroom
 Use of a calculator
 Comparing proportions
APP
Look for learners doing:
 L5CALC2*
 L6NNS1*
 L6CALC1
 L6CALC4
Summer Term 5 hours
Measures and mensuration
228–231
Previously…
• Convert one metric unit to another, e.g. grams to kilograms; read and interpret scales on a range of measuring instruments
• Distinguish between and estimate the size of acute, obtuse and reflex angles
Progression map
• Choose and use units of measurement to measure, estimate, calculate and solve problems in a range of contexts; know rough metric equivalents of imperial measures in common use, such
as miles, pounds (lb) and pints
• Use bearings to specify direction
Progression map
Next…
• Solve problems involving measurements in a variety of contexts; convert between area measures (e.g. mm2 to cm2, cm2 to m2, and vice versa) and between volume measures (e.g. mm3 to
cm3, cm3 to m3, and vice versa)
Progression map
Suggested Activities
Maths Apprentice
 Cooking times, Cooking times – meat, Cooking times - beef
 How much are you worth?
 Weigh up the options
Criteria for Success
NCETM Departmental
Workshops
 Converting Units
What unit would you use to measure______?
Does every year have a Friday the 13th?
What is the greatest number of Friday the 13th’s you can
have in a single year?
Level 5 Bring on the Maths
 Shape, Space and Measures: Interpreting scales, Converting units
What is the difference between bearings and angles?
Resources
 Clocks, Sets of scales, Measuring jugs
 Spokes OHTs: clock (30&deg;), compass rose (45&deg;)
 360&ordm; overlay and Malverns topograph
 People bearings - pupils turn from North to East, or by given amounts in degrees.
Page 17
 Measures
APP
Look for learners doing:
 L5SSM5*
 L5SSM6*
Summer Term 4 hours
Equations, formulae, identities and expressions
116–137, 138–143
Previously…
• Use simple formulae from mathematics and other subjects; substitute positive integers into linear expressions and formulae and, in simple cases, derive a formula
• Simplify linear algebraic expressions by collecting like terms; multiply a single term over a bracket (integer coefficients)
• Construct and solve simple linear equations with integer coefficients (unknown on one side only) using an appropriate method (e.g. inverse operations)
Progression map
• Use formulae from mathematics and other subjects; substitute integers into simple formulae, including examples that lead to an equation to solve; substitute positive integers into
expressions involving small powers, e.g. 3x2 + 4 or 2x3; derive simple formulae
• Simplify or transform linear expressions by collecting like terms; multiply a single term over a bracket
• Construct &amp; solve linear equations with integer coefficients (unknown on either or both sides) using appropriate methods (e.g. inverse operations, transforming both sides in the same way)
Progression map
Next…
• Use formulae from mathematics and other subjects; substitute numbers into expressions and formulae; derive a formula and, in simple cases, change its subject
• Simplify or transform algebraic expressions by taking out single-term common factors
• Construct and solve linear equations with integer coefficients (with and without brackets, negative signs anywhere in the equation, positive or negative solution)
Progression map
Suggested Activities
Criteria for Success
Maths Apprentice
 Fibonacci biography
 Fibonacci Fun – to generate instant solutions to the questions in missing 3 step /
4 step activities - pupils could generate this themselves
 Generate Fibonacci using spreadsheets; explore ratios between terms. Graph
results.
What happens to the ratio of a Fibonacci number to its
previous number, as the size of the numbers increases?
 Equations, formulae,
identities
What happens with starting numbers other than 1, 1, …?
APP
Look for learners doing:
 L5ALG1
 L6ALG2
Resources
Generic assessment criteria
Page 18
Summer Term 4 hours
Sequences, functions and graphs: using ICT
6–13, 28–29, 164–177
Previously…
• Identify the necessary information to understand or simplify a context or problem
• Express simple functions in words, then using symbols; represent them in mappings
• Generate coordinate pairs that satisfy a simple linear rule; plot the graphs of simple linear functions, where y is given explicitly in terms of x, on paper and using ICT; recognise straight-line
graphs parallel to the x-axis or y-axis
• Plot and interpret the graphs of simple linear functions arising from real-life situations, e.g. conversion graphs
Progression map
• Identify the mathematical features of a context or problem
• Express simple functions algebraically and represent them in mappings or on a spreadsheet
• Generate points in all four quadrants and plot the graphs of linear functions, where y is given explicitly in terms of x, on paper and using ICT; recognise that equations of the form y = mx + c
correspond to straight-line graphs
• Construct linear functions arising from real-life problems and plot their corresponding graphs; discuss and interpret graphs arising from real situations, e.g. distance–time graphs
Progression map
Next…
• Break down substantial tasks to make them more manageable
• Find the inverse of a linear function
• Generate points and plot graphs of linear functions, where y is given implicitly in terms of x (e.g. ay + bx = 0, y + bx + c = 0), on paper and using ICT; find the gradient of lines given by
equations of the form y = mx + c, given values for m and c
• Construct functions arising from real-life problems and plot their corresponding graphs; interpret graphs arising from real situations, e.g. time series graphs
Progression map
Suggested Activities
The graph-plotting in this unit is intended to focus on the use of ICT. The real-life
graphs should focus on cross-curricular links with other subjects such as Science
and Geography
Maths Apprentice
 Using Autograph 1
 Using Autograph 2
Criteria for Success
Standards Unit
KPO: A5 Interpreting
distance-time graphs with a
computer
Find another line that is parallel to this one. How do you
know they are parallel?
NRICH
 Walk and Ride
 Buses
Which of these are parallel: y = 2x+1, y = x+2, 2y = 4x –
10?
HORN, Cornwall
 Checking temperatures
Y8 Bring on the Maths
 Plotting Graphs: v1, v2, v3
Page 19
 Sequences, functions and
graphs
Find another line with the same y-intercept
APP
Look for learners doing:
 L5UA4
 L6ALG4
 L6ALG5
Summer Term 5 hours
Calculations and checking
36–47, 92–111
Previously…
• Check the accuracy of the solution
• Relate findings to the original context
• Understand and use decimal notation and place value; multiply and divide integers and decimals by 10, 100, 1000, and explain the effect
• Compare and order decimals in different contexts; know that when comparing measurements the units must be the same
• Round positive whole numbers to the nearest 10, 100 or 1000, and decimals to the nearest whole number or one decimal place
• Strengthen and extend mental methods of calculation to include decimals, fractions and percentages, accompanied where appropriate by suitable jottings; solve simple problems mentally
• Multiply and divide three-digit by two-digit whole numbers; extend to multiplying and dividing decimals with one or two places by single-digit whole numbers
• Check results by considering whether they are of the right order of magnitude and by working problems backwards
Progression map
• Give accurate solutions appropriate to the context or problem
• Read and write positive integer powers of 10; multiply and divide integers and decimals by 0.1, 0.01
• Order decimals
• Round positive numbers to any given power of 10; round decimals to the nearest whole number or to one or two decimal places
• Make and justify estimates and approximations of calculations
• Strengthen and extend mental methods of calculation, working with decimals, fractions, percentages, squares and square roots, and cubes and cube roots; solve problems mentally
• Use efficient written methods for multiplication and division of integers and decimals, including by decimals such as 0.6 or 0.06; understand where to position the decimal point by
considering equivalent calculations
• Select from a range of checking methods, including estimating in context and using inverse operations
Progression map
Next…
• Extend knowledge of integer powers of 10; recognise the equivalence of 0.1, 1/10 and 10−1; multiply and divide by any integer power of 10
• Use rounding to make estimates and to give solutions to problems to an appropriate degree of accuracy
• Check results using appropriate methods
Progression map
Suggested Activities
Ensure progression from the previous ‘calculation and checking’ unit in the spring
term
Maths Apprentice
 Alfred and the prize money
 Decimal ordering cards
Criteria for Success
Standards Unit
 N1 Ordering fractions and
decimals
NCETM Departmental
Workshops
 Place Value
KS3 Top-up Bring on the Maths
 Place Value: v1, v2, v3
Find a number between 4.35 and 4.362. And another,
and another…
KPO: Is there a connection between a fraction with a
prime number denominator and recurring decimal
equivalent?
4x6 = 24. What else does this tell you?
 Place value, rounding
 Mental calculations
 Written calculations
Beyond the Classroom
 Decimals and negative
numbers
APP
Look for learners doing:
 L5NNS2*
 L6CALC1
 L6CALC4
Level 5 Bring on the Maths
 Numbers and the Number System: Rounding decimals, Ordering negative
numbers
Resources
Place value chart, Number lines, Decimal ordering cards, HTU chart, Gattegno charts,
Diene's blocks
Page 20
198–201, 216–
227
Summer Term 3 hours
Geometrical reasoning: coordinates and construction
Previously…
• Use 2-D representations to visualise 3-D shapes and deduce some of their properties
• Use conventions and notation for 2-D coordinates in all four quadrants; find coordinates of points determined by geometric information
Progression map
• Visualise 3-D shapes from their nets; use geometric properties of cuboids and shapes made from cuboids; use simple plans and elevations
• Make scale drawings
• Find the midpoint of the line segment AB, given the coordinates of points A and B
Progression map
Next…
• Visualise and use 2-D representations of 3-D objects; analyse 3-D shapes through 2-D projections, including plans and elevations
• Use and interpret maps and scale drawings in the context of mathematics and other subjects
• Use the coordinate grid to solve problems involving translations, rotations, reflections and enlargements
Suggested Activities
Progression map
Criteria for Success
Maths Apprentice
 What Shape?
 Visualisation
 Plans and elevations multilink groupwork activity
 Plans and elevations - another multilink activity
Given 2 elevations (or 1 elevation and a plan) of a 3-D
shape, what could the shape be?
 Construction, loci
 Transformations
 Geometric reasoning
Beyond the Classroom
 Properties of shapes
HORN, Cornwall
 Geometrical visualisations 2
APP
Look for learners doing:
 L5SSM1*
 L6SSM5
Resources
Maps, Architect’s plans
Page 21
Summer Term 5 hours
Measures and mensuration; volume
232–233, 238–241
Previously…
• Calculate the surface area of cubes and cuboids
Progression map
• Use accurate notation
• Calculate accurately, selecting mental methods or calculating devices as appropriate
• Know and use the formula for the volume of a cuboid; calculate volumes and surface areas of cuboids and shapes made from cuboids
Progression map
Next…
• Calculate the volume of right prisms
Progression map
Suggested Activities
Maths Apprentice
 Practical work - finding volumes of boxes
 KPO: Volume of classroom if modelled as a cuboid - extend to 'how many grains
of rice will fill the room' or equivalent.
 Make a cubic metre (out of tightly rolled newspaper perhaps)
 Maxbox
 What is the box with the greatest volume that can be made out of a sheet of A4
paper?
Criteria for Success
NRICH
 Sending a Parcel
 Painted Cube
 Cuboid Challenge
Draw a solid with an volume of 12cm&sup3; and another… and
another
 Measures
Find a cuboid with a surface area of 24cm&sup3; and another…
and another
APP
Look for learners doing:
 L6SSM9
What unit would you choose to measure ______? Why?
Y8 Bring on the Maths
 Area and Volume: v4, v5, v6
Page 22
Summer Term 6 hours
Statistical enquiry
248–273
Previously…
• Suggest possible answers, given a question that can be addressed by statistical methods
• Decide which data would be relevant to an enquiry and possible sources
• Plan how to collect and organise small sets of data from surveys and experiments:
(i) design data collection sheets or questionnaires to use in a simple survey
(ii) construct frequency tables for gathering discrete data, grouped where appropriate in equal class intervals
• Collect small sets of data from surveys and experiments, as planned
• Compare two simple distributions using the range and one of the mode, median or mean
• Write a short report of a statistical enquiry, including appropriate diagrams, graphs and charts, using ICT as appropriate; justify the choice of presentation
Progression map
• Make accurate mathematical graphs on paper and on screen
• Discuss a problem that can be addressed by statistical methods and identify related questions to explore
• Decide which data to collect to answer a question, and the degree of accuracy needed; identify possible sources; consider appropriate sample size
• Plan how to collect the data; construct frequency tables with equal class intervals for gathering continuous data and two-way tables for recording discrete data
• Collect data using a suitable method (e.g. observation, controlled experiment, data logging using ICT)
• Compare two distributions using the range and one or more of the mode, median and mean
• Write about and discuss the results of a statistical enquiry using ICT as appropriate; justify the methods used
Progression map
Next…
• Suggest a problem to explore using statistical methods, frame questions and raise conjectures
• Discuss how different sets of data relate to the problem; identify possible primary or secondary sources; determine the sample size and most appropriate degree of accuracy
• Design a survey or experiment to capture the necessary data from one or more sources; design, trial and if necessary refine data collection sheets; construct tables for gathering large
discrete and continuous sets of raw data, choosing suitable class intervals; design and use two-way tables
• Gather data from specified secondary sources, including printed tables and lists, and ICT-based sources, including the internet
• Compare two or more distributions and make inferences, using the shape of the distributions and appropriate statistics
• Review interpretations and results of a statistical enquiry on the basis of discussions; communicate these interpretations and results using selected tables, graphs and diagrams
Progression map
Suggested Activities
The aim of the project is: Write a short report of a statistical enquiry and illustrate
with appropriate diagrams, graphs and charts, using ICT as appropriate; justify the
choice of what is presented. However, it is essential that there is some progression
from the previous project completed. Refer to the previous handling data unit for
guidance on appropriate techniques.
Criteria for Success
NCETM Departmental
Workshops
 Data Collection
NRICH
 Reaction Timer
Maths Apprentice
 Exploring hypotheses
 KPO: National Lottery data project and winning numbers
 Discussing Statistics (Framework)
 Comparing Statistics
Concentrate on interpretation as a key element of the
project; relating outcomes directly with the hypothesis
being tested.
 Processing, representing and
interpreting data
Beyond the Classroom
 Collecting data
APP
Look for learners doing:
 L5HD1*
 L6HD1
 L6HD5
Y8 Bring on the Maths