Unit 2 Foundation

A list of Unit 2 topics:
Module number
Integers and decimals
Factors and multiples
Laws of indices
Ratio and scale
Introduction to algebra
Algebraic manipulation
Patterns and squares
Linear graphs y = mx + c
Lines and angles
Shapes & angles
Estimates & reading scales
Metric, imperial & compound units
Area, perimeter & volume
3-D Shapes, symmetry and similarity

A detailed breakdown:
Integers and decimals
Add, subtract, multiply and divide any number
Order integers
Multiply and dividing by negative numbers
Multiply or divide any number by powers of 10, and any positive number by a number between 0 and 1
Estimate answers to problems
Write decimal numbers in order of size
Round to a given number of significant figures (and decimal places)
Add and subtract decimal numbers
Understand where to position the decimal point by considering what happens when multiplying or dividing equivalent fractions
The ability to order numbers
Appreciation of place value
Experience of the four operations using whole numbers
Knowledge of integer complements to 10 and 100
Knowledge of multiplication facts to 10 ´ 10
Knowledge of strategies for multiplying and dividing whole numbers by 10
The concepts of a fraction and a decimal
By the end of the module the student should be able to:
• Understand and order integers (1.1, 1.2)
• Add, subtract, multiply and divide integers (BIDMAS) (1.4, 1.5)
• Round whole numbers to the nearest, 10, 100, 1000, … (1.6)
• Multiply and divide whole numbers by a given multiple of 10 (1.5)
• Check their calculations by rounding, eg 29 ´ 31 » 30 ´ 30 (3.10)
• Put digits in the correct place in a decimal number (3.1)
• Write decimals in ascending order of size (3.2)
• Approximate decimals to a given number of decimal places or significant figures (12.3)
• Multiply and divide decimal numbers by whole numbers and decimal
numbers (up to 2 d.p.) eg 266.22 ¸ 0.34 (3.4, 3.6)
• Know that eg 13.5 ¸ 0.5 = 135 ¸ 5 (3.6, 3.11)
• Check answers by rounding and know that, eg 2.9 ´ 3.1 » 3.0 ´ 3.0
More work on long multiplication and division without using a calculator
Estimate answers to calculations involving the four rules
Consideration of mental maths problems with negative powers of 10: 2.5 ´ 0.01, 0.001
Directed number work with two or more operations, or with decimals
Use decimals in Functional Elements problems
Use BIDMAS to establish the order of operations
Money calculations that require rounding answers to the nearest penny
Multiply and divide decimals by decimals (more than 2 decimal places)
Present all working clearly with decimal points in line and emphasise that all working is to be shown
For non-calculator methods, make sure that ‘remainders’ and ‘carrying’ are shown as evidence of working
Amounts of money should always be rounded to the nearest penny where necessary
It is essential to ensure the students are absolutely clear about the difference between significant figures and decimal places
Extend to multiplication of decimals and/or long division of integers
Try different methods from the traditional ones, eg Russian or Chinese methods for multiplication
Always round measures to an appropriate degree of accuracy
Incorporate Functional Elements where appropriate
Factors and multiples
Recognise and identify even, odd and prime numbers
Find factors and multiples of numbers
Find squares and cubes of numbers; and finding square roots and cube roots of numbers
Find HCF, LCM and prime factor decomposition
Number complements to 10 and multiplication/division facts
Use a number line to show how numbers relate to each other
Recognise basic number patterns
Experience of classifying integers
By the end of the module the student should be able to:
• Find: squares; cubes; square roots; cube roots of numbers, with and
without a calculator (including the use of trial and improvement) (2.3, 3.5)
• Understand odd and even numbers, and prime numbers (2.1)
• Find the HCF and the LCM of numbers (2.2)
• Write a number as a product of its prime factors, eg 108 = 22 ´ 33 (2.1, 2.2)
Calculator exercise to check factors of larger numbers
Further work on indices to include negative and/or fractional indices (introduction to the next section)
Use prime factors to find LCM
Use a number square to find primes (sieve of Eratosthenes)
Calculator exercise to find squares, cubes and square roots of larger numbers (using trial and improvement)
All of the work in this unit is easily reinforced by starters and plenaries
Calculators are only to be used when appropriate
Encourage students to learn square, cube, prime and common roots as Unit 2 is a
non-calculator examination
Laws of indices
Use index notation and index laws for multiplication and division of integer powers
Use index laws to simplify and calculate the value of numerical expressions involving multiplication and division of integers.
Knowledge of squares, square roots, cubes and cube roots
By the end of the module the student should be able to:
• Understand that Index notation can be used to represent repeated multiplication (of the same base) (8.1)
• Use index rules to simplify and calculate numerical expressions involving powers, eg (23 ´ 25) ¸ 24 (8.2, 13.2)
Use index rules to simplify algebraic expressions. eg y y y y = y4
Treat index rules as formulae (state which rule is being at each stage in a calculation)
Use a simple division example to illustrate how zero and negative indices occur, eg 33 ÷ 33 for zero index as extension of work at this level
Show that x3 y4 cannot be combined
Cover examples like (4x3 3x5) ÷ 2x2
Explain that a reciprocal will be useful for other topics, like fractions
Express a given number as a fraction of another
Add and subtract fractions by writing them with a common denominator
Understand equivalent fractions; simplifying a fraction by cancelling all common factors
Order fractions by rewriting them with a common denominator
Add and subtract fractions (without a calculator)
Recognise that recurring decimals are exact fractions, and that some exact fractions are recurring decimals
Multiplication facts
Ability to find common factors
A basic understanding of fractions as being ‘parts of a whole unit’
Use of a calculator with fractions (from Unit 1)
By the end of the module the student should be able to:
• Recognise and write fractions in everyday situations (prior knowledge)
• Write a fraction in its simplest form and recognise equivalent fractions (4.2)
• Compare the sizes of fractions using a common denominator (4.1)
• Add and subtract fractions by using a common denominator (4.7)
• Write an improper fraction as a mixed fraction and visa versa (4.4)
• Recognise common recurring decimals can be written as exact fractions eg (4.8)
• Convert common recurring decimals into fractions eg 0.6666666…. (4.8)
Careful differentiation is essential as this topic dependent upon the student’s ability
Relate simple fractions to percentages and visa versa
Using a calculator to change fractions into decimals and looking for patterns
Working with improper fractions and mixed numbers, eg divide 5 pizzas between 3 people
Link fractions with probability calculations in Unit 1
Recognising that every terminating decimal has its fraction with 2 and/or 5 as a common factor in the denominator
Solve word problems involving fractions and in real-life problems, eg find perimeter using fractional values
Regular revision of this topic is needed
Students need to learn how to identify and use the fraction button on their calculators in order to check solutions
Use of fractions for calculations involving compound units (Unit 1 and 2)
Use Functional Elements questions and examples using fractions, eg off the list price when comparing different sale prices
Understand that ‘percentage’ means ‘number of parts per 100’
Interpret percentage as the operator 'so many hundredths of'
Four operations of number, particularly for decimals
The concepts of a fraction and a decimal
Awareness that percentages are used in everyday life
By the end of the module the student should be able to:
• Understand that a percentage is a fraction in hundredths (5.1)
• Write a percentage as a decimal; or as a fraction in its simplest terms (5.1)
• Write one number as a percentage of another number (2.3 in Unit 3)
• Calculate the percentage (or fraction) of a given amount (4.5, 5.2)
• Find a percentage increase/decrease of an amount (2.2 in Unit 3)
• Use a multiplier to increase by a given percent, eg 1.10 ´ 64
increases 64 by 10% (2.2 in Unit 3)
• Use of 90% to represent a 10% decrease (2.2 in Unit 3)
Fractional percentages of amounts (non-calculator methods)
Combine multipliers to simplify a series of percentage changes
Percentages which convert to recurring decimals, eg 33 % and situations which lead to percentages of more than 100%
Problems which lead to the necessity of rounding to the nearest penny, eg using Functional Elements contexts
Keep using non-calculator methods, eg Start with 10%, then 1% in order to find other required percentage
Use plenty of practical examples linked to Functional Elements, eg VAT calculations (17 %)
Ratio and scale
Use ratio notation, including reduction to its simplest form and it various links to fractions notation
Divide a quantity in a given ratio
Solve word problems about ratio, including using informal strategies and the unitary method of solution
Fractions and Decimals
By the end of the module the student should be able to:
• Appreciate that the ratio 1:2 represents and of a quantity (6.1)
• Divide quantities in a given ratio, eg divide £20 in the ratio 2:3 (6.3)
• Solve word problems involving ratios, eg find the cost of 8 pencils given that 6 pencils cost 78p (6.2)
Currency calculations using currency exchange rates
Use harder problems involving multi-stage calculations
Relate ratios to Functional Elements contexts, eg investigate the proportions of the different metals in alloys and the new amounts of ingredients needed for a recipe used for different numbers of guests
Students often find three-part ratios difficult
Link ratios given in different units to metric and imperial units topic
Introduction to algebra
Distinguish the different roles played by letter symbols in algebra
Distinguish the meaning between the words ‘equation’, ‘formula’ and expression
Experience of using a letter to represent a number
Word formulae or rules to describe everyday situations eg time to cook a nut roast linked to weight of the nut roast
By the end of the module the student should be able to:
• Distinguish the different roles played by letter symbols in algebra (7.1, 7.4)
• Understand the meaning between the words ‘equation’, ‘formula’, and
expression (7.2, 7.8)
Extend the above ideas to the ‘equation’ of the straight line, y = mx + c
Look at word equations written in symbolic form, eg F = 2C + 30 to roughly convert temperature and compare with F = + 32
There are plenty of old exam papers with matching tables testing knowledge of the ‘Vocabulary of Algebra’
(See Emporium website)
Present all work neatly
Use the appropriate algebraic vocabulary
Emphasise the correct use of symbolic notation, eg 3x rather than 3 x
Algebraic manipulation
Simplify terms and expressions
Multiply a single term over a bracket
Take out common factors
Know that a letter can be used to represent a number
Ability to use negative numbers with the four operations
Experience of using BIDMAS in calculations without a calculator
By the end of this Unit the student should be able to:
• Simplify expressions with like terms, eg x2 + 3x2; 3ab + 5ab +2c2 (7.3)
• Expand and factorise expressions with one pair of brackets,
eg expand x(2x + 3y); factorise 3xy2 - 6x2y (7.7, 8.6)
Expand and simplify expressions involving more than one pair of brackets,
eg 3(x + 4) – 2(x – 3); (2x + 3) (3x – 4)
Multiply out two (linear) brackets, eg (2x – 4) (3x + 2)
Expand algebraic expressions like (x – 2) (x + 2) to illustrate how the ‘middle’ term cancels
Emphasise correct use of symbolic notation, eg 3x2 rather than 3 ´ x2
Present all work neatly, writing out the questions with the answers to aid revision at a later stage
Substitute numbers into formulae
Use formulae from mathematics and other subjects that require prior simplification of brackets, including those that have negative signs occurring anywhere in the equation, and those with a negative solution
Generate a formula
Understand of the mathematical meaning of the words expression, simplifying, formulae and equation
Experience of using letters to represent quantities
Substitute into simple expressions using words
Use brackets in numerical calculations and remove brackets in simple algebraic expressions
By the end of the module the student should be able to:
• Use letters or words to state the relationship between different quantities (13.3)
• Substitute positive and negative numbers into simple algebraic formulae (13.2)
• Substitute positive and negative numbers into algebraic formulae involving powers (7.9, 13.2)
• Find the solution to a problem by writing an equation and solving it (13.1)
• Generate a formula from given information, eg find the formula for the perimeter of a rectangle given its area A and the length of one side (13.3)
Use negative numbers in formulae involving indices
Use various investigations leading to generalisations
Use further problems in generating formulae from given information
Simple changing the subject to prepare students for y = mx + c and Unit 3
Emphasis on good use of notation 3ab means 3 ´ a ´ b
Students need to be clear on the meanings of the words; expression, equation, formula and identity
Use a formula in words and put in letters to represent the words, eg cooking time for a joint of meat
Changing the subject is covered again in Unit 3, for brighter students, use this as an introduction to these manipulations
Use Functional Elements questions given in everyday contexts as a rich source of formulae
Link with formulae for area, volume, surface area etc
Patterns and squares
Generate common integer sequences (including: sequences of odd or even integers; squared integers; powers of 2; powers of 10; triangle numbers)
Generate terms of a sequence using term-to-term and position-to-term definitions of the sequence
Use linear expressions to describe the nth term of an arithmetic sequence
Know about odd and even numbers
Recognise simple number patterns eg 1, 3, 5, ...
Writing simple rules algebraically
Raise numbers to positive whole number powers
By the end of the module the student should be able to:
• Find the missing numbers in a number pattern or sequence (9.1)
• Find the nth term of a number sequence as an algebraic expression (9.3)
• Explain why a number is, or is not, a term of a given sequence (9.4)
• Produce a sequence of numbers from a given nth term formula (9.3)
Match-stick problems
Sequences and nth term formula for triangle numbers (for top students)
Fibonacci numbers etc
Extend to quadratic sequences whose nth term is an2 + b and link to square numbers
Show different ways to get the constant term, e.g. zero term or working backwards one term
Emphasise good use of notation 3n means 3 ´ n
When investigating linear sequences, students should be clear on the description of the pattern in words, the difference between the terms and the algebraic description of the nth term
Stress that the difference is always the coefficient of n, as students often put n + difference
Linear graphs y = mx + c
Use the conventions for coordinates in the plane
Plot points in all four quadrants
Find the coordinates of the final point to make a given shape
Recognise (when values are given for m and c) that equations of the
form y = mx + c correspond to straight-line graphs in the coordinate plane
Plot graphs of functions in which y is given explicitly in terms of x, or implicitly
Find the gradient of lines given by equations of the form y = mx + c (when values are given for m and c)
Understand that the form y = mx + c represents a straight line and that m is the gradient of the line and c is the value of the y-intercept
Construct linear functions and plotting the corresponding graphs arising from real-life problems
Discuss and interpret graphs modelling real situations (including Travel & Conversion Graphs)
Being able to:
Substitute positive and negative numbers into algebraic expressions
Plot coordinates in the first quadrant
Calculate the mean of two numbers
Knowledge of basic shapes
By the end of the module the student should be able to:
• Add a point to a coordinate grid to complete a given shape,
eg (parallelogram; rhombus; trapezium; square) (10.1)
• Use the formula to calculate the midpoint of a line segment (10.3)
• Understand how to represent points in 2-D (10.2)
• Substitute values of x into linear functions to find corresponding values of y (11.2, 11.3)
• Plot points for linear functions on a coordinate grid and draw the
corresponding straight lines (11.2, 11.3)
• Interpret m and c as gradient and y-intercept in linear functions (11.2, 11.3)
• Understand that the graphs of linear functions are parallel if they have the same value of m (11.2, 11.3)
• Understand linear functions in practical problems, eg distance-time graphs (12.3)
Find the equation of the line through two given points (extension task for brightest)
Use a spreadsheet to generate straight-line graphs, posing questions about the gradient of lines
Use a graphical calculator or graphical ICT package to draw straight-line graphs
Link to scatter graphs and correlation from Unit 1
Cover horizontal and vertical lines (x = c and y = c), as students often forget these
Careful annotation should be encouraged. Label the coordinate axes and write down the equation of the line
Recognise linear graphs and hence when data may be incorrect
The midpoint can be found by drawing and/or using the mean of the end points (rather than by using the formula)
Link to graphs and relationships in other subject areas, eg science, geography etc
Interpret straight line graphs for Functional Elements
– Ready reckoner graphs
– Conversion graphs
– Fuel bills & mobile phone tariffs
– Fixed charge (standing charge) and cost per unit
Link conversion graphs to converting metric and imperial units and equivalents (Unit 2)
Lines and angles
Measure lines and use a protractor to measure angles of all sizes
Use a protractor to draw angles accurately
Use the fact that angles at a point add to 360°
Label angles and lines
Recall and use properties of perpendicular lines
An understanding of angle as a measure of turning
The ability to use a ruler and protractor
By the end of this module the student should be able to:
• Distinguish between acute, obtuse, reflex and right angles (14.2)
• Estimate the size of an angle in degrees (14.4)
• Measure and draw angle to the nearest degree (14.5, 14.6)
• Measure and draw line to the nearest mm (14.6)
• Use two letter notation for a line and three letter notation for an angle (14.3)
• Recall and use properties of perpendicular lines (15.2)
• Mark perpendicular lines on a diagram (15.2)
• Identify a line perpendicular to a given line (15.2)
• Use geometric language appropriately (14, 15)
• Use letters to identify points, lines and angles (14.3)
• Use angle properties at a point to calculate unknown angles (14.1)
Extend to other angle facts in triangles, parallel lines and/or quadrilaterals (prep for next topic)
Perpendicular bisector construction (Intro to Unit 3)
Make sure that all pencils are sharp and drawings are neat and accurate
Use a lot of drawing practice and encourage students to check each others diagrams
Pass around a mark scheme on tracing paper to save time and to give instant feedback
Angles should be correct to within 2°, lengths correct to the nearest mm
Apply skills to constructing pie charts (Unit 1)
Shapes & angles
Recall properties of triangles and quadrilaterals
Use geometric proof of angle sum of triangles
Use facts associated with parallel lines
Calculate and use the sums of the interior angles of triangles & quadrilaterals
The concept of parallel lines
The concept of vertical and horizontal
The concept of an angle between two lines
Experience in drawing triangles, quadrilaterals and circles
By the end of the module the student should be able to:
• Identify triangles by their properties (scalene, isosceles, equilateral,
right-angled, obtuse, and acute) (15)
• Prove the angle sum in a triangle is 180° (15.4)
• Use the angle properties of triangle to find missing angles (15)
• Prove the exterior angle of a triangle is equal to the sum of the two opposite interior angles (15.4)
• Identify quadrilaterals by their properties (trapezium, parallelogram, rhombus, rectangle, square, kite and arrowhead) (16.2)
• Use alternate and corresponding angles in parallel lines to find missing angles (15.2)
Use the angle properties of triangles to find missing angles in combinations of triangles
Use harder problems involving multi-step calculations
Link with symmetry
A lot of practical drawing examples helps to illustrate properties of various shapes, eg group work or displays
Diagrams used in examinations are often not drawn accurately
Use tracing paper to show which angles in parallel lines are equal
Encourage students to always write down the reasons and to ‘quote’ the angle fact used
Understand the related parts of a circle
Properties of shapes and polygons
Basic angle facts
Ability to draw a circle with compasses
By the end of the module the student should be able to:
• Identify and name the various parts of a circle (centre, radius, diameter, circumference, sector, segment, arc and chord) (16.4)
• Use the angle properties of two radii to make an isosceles triangle within a circle. (This is a good application of the triangle angle properties from the previous section and links well with Unit 3 (Pythagoras’ Theorem) (16.5)
Use harder problems involving multi-stage calculations
Define a circle by using the language of loci (introduction to Unit 3 topic)
All working should be clearly and accurately presented
Draw lines using an HB pencil
A sturdy pair of compasses is essential
Estimates & reading scales
Make estimates in everyday life
Make estimates of length using metric and imperial units
Make estimates of weight, volume and capacity using metric and imperial units
Use sensible units for measuring
Read analogue and digital clocks
Read measurements on different types of scales
Read timetables and calculating time intervals.
Convert from one metric unit to another
An awareness of the metric and imperial system of measures
Strategies for multiplying and dividing by 10 (for converting metric units)
Knowledge of the conversion facts for metric lengths, mass and capacity
Knowledge of the conversion facts between seconds, minutes and hours
By the end of this module the student should be able to:
• Make estimates of: length, volume and capacity, weight (revision of
prior knowledge)
• Decide on the appropriate units to use in real life problems (revision of
prior knowledge)
• Read measurements from instruments: scales; analogue and digital clocks; thermometers, etc (17.1, 17.2)
• Recognise that measurements given to the nearest whole unit may be inaccurate by up to one half in either direction
• Convert from one metric unit to another metric unit (17.3)
• Knowledge and use of common imperial units and time conversions (17.4)
This could be made a practical activity by collecting assorted everyday items for weighing and measuring to check the estimates of their lengths, weights and volumes
Link with compound units such as speed for travel graphs (m/s or km/h or mph) and Best Buys (g/penny)
Measurement is essentially a practical activity
Use a range of everyday objects to bring more meaning to lessons
Use Functional Elements as practice questions for this topic area
Metric, imperial & compound units
Convert units
Estimate answers
Use the formulae for speed
Change units in compound measures
Know that: 1 hour = 60 minutes; 1 day = 24 hours
Some experience of Metric/Imperial measures from Topic 15 (above) or Unit 1
By the end of the module the student should be able to:
• Convert measurements to the same unit before doing a calculation,
eg 3 m ´ 12 cm ´ 10 mm (17.3)
• Use common metric/imperial equivalents to convert between units* (17.4)
• Round measurements to 1s.f. to find an estimate for a calculation (17.4)
• Use speed = distance/time to work out speed, distance or time (17.5)
• Change the units of speed between metric/metric units or imperial/imperial units (17.5)
• Compare quantities by converting to the same units, eg km/h and m/s (17.3, 17.4)

*These are the metric to imperial equivalents expected
Metric Imperial
1 kg 2.2 pounds
1 litre 1¾ pints
4.5 litre 1 gallon
8 km 5 miles
1 m 39 inches
2.5 cm 1 inch
30cm 1 foot
Use the internet and/or reference books to find the weights, volumes and heights of large structures such as buildings, aeroplanes and ships
Work out the maximum/minimum possible error in a calculation involving measures
Measurement is essentially a practical activity
All working should be shown with multiplication or division by powers of 10
Use a distance/speed/time triangle (ie Drink Some Tea)
Use Functional Elements as practice questions for this topic area ie Best Buys, travel graphs leading to speed calculations, etc
Area, perimeter & volume
Use formulae to find the area of triangles, parallelograms, and trapeziums
Calculate the surface area and volume of cuboids
Use the formula to calculate the volume of a prism
The names of quadrilaterals
Ability to substitute numbers into a formula
Some notion of the difference between length, area and volume
Properties of cubes, cuboids and other common 3-D objects
By the end of the module the student should be able to:
• Use the area formulae for triangles, parallelograms and trapeziums (18.3)
• Work out the surface area of 3-D shapes based on rectangles and triangles (by working out the area of each face) (19.4)
• Use v = l ´ w ´ h to solve problems involving the volume and dimensions of a cuboid (19.3)
• Work out how many small boxes fit into a large box (19.3)
• Use volume = cross-section ´ length to find the volume of a regular prism, eg with trapezium cross-section (19.3)
Further problems involving combinations of shapes
Use compound shape methods to investigate the areas of other standard shapes, eg kites
Practical activities, eg using estimation and accurate measuring to calculate perimeters and areas of classroom/corridor floors
Discuss the correct use of language and units [Sometimes one mark is for putting in the correct unit]
Ensure that students can distinguish between perimeter, area and volume
Many students have little real understanding of perimeter, area and volume
Practical experience is essential to clarify these concepts, bring in cornflake boxes
There are many Functional Elements skills which can be applied to this topic area, eg floor tiles, optimization type questions etc
3-D Shapes, symmetry and similarity
Identify and name common solids.
Explore the geometry of cuboids (including cubes), and shapes from cuboids
Use 2-D representations of 3-D shapes and analysing 3-D shapes through 2-D projections and cross-sections, including plan and elevation
Recognise and visualising reflection and rotational symmetry of 2-D shapes
Construct cubes, tetrahedra, square-based pyramids and other 3-D shapes from given information
Understand congruent and similar shapes
Names of common 2-D shapes
By the end of the module the student should be able to:
• Identify and name common solids: cube, cuboid, cylinder, prism, pyramid, sphere and cone (19.2)
• Count the vertices, faces and edges of 3-D shapes (19.1, 19.2)
• Draw nets of solids and recognise solids from their nets (Isometric drawing) (9.1 in Unit 3)
• Draw and interpret plans and elevations (9.2 in Unit 3)
• Recognise line and rotational symmetry in 2-D shapes (16.6, 16.7)
• Draw in the line of symmetry (or state the equation if the shape is on a coordinate grid) and state the order of rotational symmetry (16.6, 16.7)
• Recognise and name examples of solids, including prisms, in the real world (19.1)
Make solids using equipment such as clixi or multi-link
Draw shapes made from multi-link on isometric paper
Build shapes from cubes that are represented in 2-D
Extend to planes of symmetry for 3-D solids
Work out how many small boxes can be packed into a larger box (Functional Elements example)
Accurate drawing skills need to be reinforced
Some students find visualising 3-D objects difficult, simple models will assist
Use tracing paper/mirrors to help with symmetry questions
Introduce the concept of congruency