4. Functional Skills: Applying Lowest Common Multiple to a practical problem

5. Surface Area of a Triangular Prism

6. Interior and Exterior Angles of Polygons

7. Area of Trapezium

8. Functional Skills: Best buy comparison of holidays

9. Speed and conversions from miles to kilometres

10. 3D coordinates and midpoints

11. Expanding, Factorising and Simplifying Algebraic Expressions

12. Indices

13. Equations of Lines and Perpendicular Lines

14. Simplifying Algebraic Fractions by Factorising

15. Expanding Brackets to find a Formula for the Volume of a Prism

16. Tangents to Circles and Proof

A detailed breakdown:

MODULE

CONTENTS

PRIOR KNOWLEDGE

OBJECTIVES

DIFFERENTIATION & EXTENSION

NOTES

2-1

Integers and decimals*

Understand place value and round to a given power of 10
Understand and use negative integers both as positions and translations on a number line
Order integers
Multiply and divide by negative numbers
Multiply or divide any number by powers of 10, and any positive number by a number between 0 and 1
Check and 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 equivalent fractions are multiplied or divided
Check and estimate answers to problems, by approximations or inverse operations (Decimals)

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)
• Add, subtract, multiply and divide integers (BIDMAS) (1.5)
• Round whole numbers to the nearest, 10, 100, 1000, … (Revision of
prior knowledge)
• Multiply and divide whole numbers by a given multiple
of 10 (1.1, 1.3)
• Check their calculations by rounding, eg 29 ´ 31 » 30 ´ 30 (3.5)
• Put digits in the correct place in a decimal number (3)
• Write decimals in ascending order of size (3)
• Approximate decimals to a given number of decimal places or significant figures (3.3, 3.4)
• Multiply and divide decimal numbers by whole numbers and decimal numbers (up to 2 d.p.) eg 266.22 ¸ 0.34 (3.2)
• Know that eg 13.5 ¸ 0.5 = 135 ¸ 5 (3.2)
• Check their answer by rounding, eg 2.9 ´ 3.1 » 3.0 ´ 3.0 (3.6)

More work on long multiplication and division without using a calculator
Estimating 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
Introduce standard form for very large/small numbers
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; emphasise that all working is to be shown
For non-calculator methods make sure that ‘remainders’ and ‘carrying’ are shown
Amounts of money should always be rounded to the nearest penny where necessary
It is essential to ensure that 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
Use Functional Elements problems as additional practice for the topic area

2-2

Factors and multiples

Understand 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) (1.4)
• Understand odd and even numbers, and prime numbers (1.2)
• Find the HCF and the LCM of numbers (1.2)
• Write a number as a product of its prime factors, eg 108 = 22 ´ 33 (1.2)

Calculator exercise to check factors of larger numbers
Further work on indices to include negative and/or fractional indices (intro to 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)

2-3

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, fractional and negative powers
Use inverse operations involving

Recall the fact that n0 = 1 and n–1 = for positive integers n, the corresponding rule for negative integers, = square root n and = cube root n for any positive number n

The ability to order numbers
Appreciation of place value
Experience of the four operations using whole numbers
Knowledge of integer complements to 10
Knowledge of multiplication facts to 10 ´ 10
Knowledge of strategies for multiplying and dividing whole numbers by 10

By the end of the module the student should be able to:
• Understand and order integers (2.1)
• Add, subtract, multiply and divide integers (2.1)
• Understand simple instances of BIDMAS, eg work out 12 ´ 5 – 24 ¸ 8 (2.1)
• Round whole numbers to the nearest, 10, 100, 1000, … (1.6)
• Multiply and divide whole numbers by a given multiple of 10 (2.1)
• Check their calculations by rounding, eg 29 ´ 31 » 30 ´ 30 (2.6)
• Check answers by reverse calculation, eg if 9 ´ 23 = 207 then
207 ¸ 9 = 23

Estimating answers to calculations involving the four rules
Directed number work with two or more operations, or with decimals
Encourage effective use of a calculator

Present all working clearly
Show what is entered into your calculator, not just the answer

2-4

Standard form

Use standard form, expressed in standard notation and on a calculator display
Calculate with standard form
Convert between ordinary and standard form representations
Convert to standard form to make sensible estimates for calculations involving multiplication and/or division
Use standard form display and know how to enter numbers in standard index form

Rounding decimals to a given number of decimal places or significant figure
Multiplying decimal numbers with, and without, a calculator
Some experience with powers of 10, eg know that 102 = 100, 103 = 1000, 10–1 = 0.1
Negative indices and laws of indices

By the end of the module the student should be able to:
• Understand the standard form convention (5.2)
• Convert numbers to, and from, standard form (5.2)
• Calculate with numbers given in standard form with, and without, a calculator (5.2)
• Round numbers given in standard form to a given number of significant figures (5.2)

Use standard index form in real-life situations, eg stellar distances, sizes of populations and atomic distances for small numbers

This work can be enriched by using examples drawn from the sciences, eg Avogadro’s constant 6.02 x 10-23

2-5

Fractions*

Express a given number as a fraction of another
Add and subtracting 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, subtract, multiply and divide 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:
• Visualise a fraction diagrammatically (2.1)
• Understand a fraction as part of a whole (2.1)
• Recognise and write fractions in everyday situations (2.1)
• Write a fraction in its simplest form and recognise equivalent fractions (2.1, 3.1)
• Compare the sizes of fractions using a common denominator (Revise
prior knowledge)
• Add and subtract fractions by using a common denominator (2.2 - 2.5)
• Write an improper fraction as a mixed fraction (2.1 – 2.5)
• Recognise common recurring decimals can be written as exact fractions, eg (3.1)
• Converting a recurring decimal into a fraction (3.1)

Careful differentiation is essential for this topic dependent upon the student’s ability
Relating simple fractions to remembered percentages and vice-versa
Using a calculator to change fractions into decimals and looking for patterns
Working with improper fractions and mixed numbers
Multiplication and division of fractions to link 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
Introduction to algebraic fractions

Understanding of equivalent fractions is the key issue in order to be able to tackle the other content
Constant revision of this topic is needed
Students need to learn how to identify and use the fraction button on their calculators to check solutions
Link with probability calculations using AND and OR Laws (Unit 1)
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

2-6

Percentages*

Understand that ‘percentage’ means ‘number of parts per 100’
Interpret percentage as the operator 'so many hundredths of'
Use multiplier to increase or decrease a given amount
Convert between fractions, decimals and percentages

4 operations of number, particularly for decimals.
The concepts of a fraction and a decimal
An 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 (4.1)
• Write a percentage as a decimal; or as a fraction in its simplest terms (4.1)
• Write one number as a percentage of another number (4.1)
• Calculate the percentage (or fraction) of a given amount (4.1, 2.5)
• Find a percentage increase/decrease of an amount (4.2, 4.3)
• Use a multiplier to increase by a given percent, eg 1.10 ´ 64 increases 64 by 10% (4.3)
• Calculate simple and compound interest for two, or more, periods of time (4.3)

Fractional percentages of amounts (non-calculator)
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 Functional Elements contexts
Comparisons between simple and compound interest calculations
Formulae in simple interest/compound interest methods

Show how the multiplier for a 10% increase (1.10) is most useful for working out a value before the increase, by using division
Show that finding 10% of the new price and subtracting does not work. (This is a useful introduction to Unit 3 contents)

2-7

Ratio and scale*

Use ratio notation, including reduction to its simplest form and its 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, eg 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
Harder problems involving multi-stage calculations
Relate ratios to Functional Elements contexts, eg investigate the proportions of the different metals in alloys, and work out the new amounts of ingredients for a recipe for a different number of guests

Students often find three-part ratios difficult
Link ratios to metric and imperial units module

2-8

Introduction to algebra*

Distinguish the different roles played by letter symbols in algebra
Distinguish the meaning between the words ‘equation’, ‘formula’, ‘identity’ 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 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)
• Understand the meaning between the words ‘equation’, ‘formula’, ‘identity’ and expression (10.1)

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)

2-9

Algebraic manipulation

Simplify terms, products and sums
Multiply a single term over a bracket
Take out common factors
Expand the product of two linear expressions
Factorise quadratic expressions

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 (7.2)
• Expand and factorise expressions with one pair of brackets,
eg expand x(2x + 3y); factorise 3xy2 - 6x2y (8.1, 8.2)
• Expand and simplify expressions involving more than one pair of brackets, eg 3(x + 4) – 2(x – 3); (2x + 3)(3x – 4) (8.3, 8.4)
• Factorise quadratic expression (including the difference of two squares) (8.4)
• Simplify algebraic fractions, eg (11.1)

Expand algebraic expressions involving three pairs of brackets
Further examples in factorising quadratic expression with non-unitary values of a (including fractional values)
Simplification of algebraic fractions which involve the addition of fractions

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
Link the difference of two squares with the rationalisation of surds

2-10

Formulae

Substitute numbers into formulae
Use formulae from mathematics and other subjects that require simplification of brackets, including those that have negative signs occurring anywhere in the equation, and those with a negative solution
Change the subject of a formula including where the subject occurs once
Generate a formula

Understanding of the mathematical meaning of the words expression, simplifying, formulae and equation
Experience of using letters to represent quantities
Substituting into simple expressions using words
Using brackets in numerical calculations and removing 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 (10.2)
• Substitute positive and negative numbers into simple algebraic formulae (10.2)
• Substitute positive and negative numbers into algebraic formulae involving powers (10.2)
• Find the solution to a problem by writing an equation and solving it (10.3)
• 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 (10.3)

Use negative numbers in formulae involving indices
Various investigations leading to generalisations
Further problems in generating formulae form given information
Apply changing the subject to y = mx + c

Emphasise good use of notation, eg 3ab means 3 ´ a ´ b
Students need to be clear on the meanings of the words expression, equation, formula and identity
Changing the subject is covered fully in Unit 3, use this Unit as an introduction to this concept
Use Functional Elements problems as a rich source of formulae given in everyday contexts
Link with formulae for area, volume, surface area, etc

2-11

Exact answers and surds

Simplify surds and rationalise denominators
Write answers in exact form

Knowing the common square numbers (for simplifying surds)
Knowledge of square roots and π
Collecting together like terms
Removing brackets in simple algebraic expressions
Difference of two squares

By the end of the module the student should be able to:
• Simplifying surds (5.4)
• Use surds and π in exact calculations, without a calculator (5.4)
• Write (3 – √3)2 in the form a + b√3 (5.4)
• Rationalise a denominator (5.4)
• Rationalise the denominator of fractions, eg = , and
eg write (Ö18 +10) ¸ Ö2 in the form p + qÖ2 (5.4)
• Give an answer to a question involving the area of a circle as 25π (link with Unit 3) (5.4)

Explain the different between rational and Irrational numbers as an introduction to surds
Stress it is better to write answers in exact form, eg is better than 0.333333…..
Prove that √2 is irrational
Revise the difference of two squares to show why we use, for example (√3 – 2) as the multiplier to rationalise (√3 + 2)
Link to Unit 3 work on circle measures (involving π) and Pythagoras calculations in exact form.
Give an answer to use of Pythagoras’ Theorem as √13 (link with Unit 3)

Link simplifying surds to collecting together like terms. eg 3x + 2x = 5x,
so therefore 3√5 + 2√5 = 5√5
Stress it is better to write answers in exact form (eg is better than 0.333333…..)
A-Level textbooks (Core 1) are a good source of questions on surd manipulation, some of which are algebraic.
A useful generalisation to learn is

2-12

Patterns and sequences

Generate 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 (7.6, 7.7)
• Find the nth term of a number sequence as an algebraic expression (7.7)
• Explain why a number is, or is not, a member of a given sequence (7.6, 7.7)
• Produce a sequence of numbers from a given nth term formula (7.7)

Match-stick problems
Sequences and nth term formula for triangle numbers, Fibonacci numbers etc
Prove a sequence cannot have odd numbers for all values of n
Extend to quadratic sequences whose nth term is an2 + bn + c

Emphasise the use of appropriate algebraic notation, eg 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

2-13

Linear graphs*

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
Use the formula to find the mid-point of line segments
Identify 2-D and 3-D shapes; 3-D coordinates
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
Explore the gradients of parallel lines and lines perpendicular to each other
Construct linear functions and plot the corresponding graphs arising from real-life problems
Discuss, plot and interpreting graphs modelling real situations (including travel & conversion graphs)

Substitute positive and negative numbers into algebraic expressions
Plot coordinates in the first quadrant
Calculating the mean of two numbers
Knowledge of basic shapes
Rearrange to change the subject of a formula

By the end of the module the student should be able to:
• Add a point to a coordinate grid to complete a given shape (parallelogram; rhombus; trapezium; square) (9.1)
• Use the formula to calculate the midpoint of a line segment (9.2)
• Understand how to represent points in 1-D, 2-D and 3-D (9.1, 14.9)
• Substitute values of x into linear functions to find corresponding
values of y (9.1, 9.3 – 9.5)
• Plot points for linear functions on a coordinate grid and draw the
corresponding straight lines (9.1, 9.3 – 9.5)
• Interpret m and c as gradient and y-intercept in linear functions (9.3)

• Understand that the graphs of linear functions are parallel if they have the same value of m (9.5)
• Know that the line perpendicular to y = mx + c has gradient (9.5)
• Understand linear functions in practical problems, eg distance-time graphs (9.6)

Find the equation of the line through two given point
Find the equation of the perpendicular bisector of the line segment joining two given points
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

With the exception of parallel and perpendicular lines, this topic is also contained in Unit 1
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
Link to graphs and relationships in other subject areas, ie science, geography etc
Interpret straight line graphs for Functional Elements problems
– Ready reckoner graphs
– Conversion graphs
– Fuel bills & Mobile Phone tariffs
– Fixed charge (standing charge) and cost per unit
Link conversion graphs to converting metric & imperial units and equivalents (Unit 2)
A-Level Text books (C1) are a good source of extension questions on this topic
Cover horizontal and vertical lines (x = c and y = c), as students often forget these

2-14

Lines and angles*

Measure lines and using 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°

An understanding of angle as an amount of rotation or a measure of turning
Experience of using a ruler and a protractor

By the end of this module the student should be able to:
• Distinguish between acute, obtuse, reflex and right angles (12.6)
• Estimate the size of an angle in degrees (revision of
prior knowledge)
• Measure and draw angle to the nearest degree (Chapter 12
Introduction)
• Measure and draw line to the nearest mm (Chapter 12
Introduction)
• Use angle properties at a point to calculate unknown angles (12.6)

Extend to other angle facts in triangles, parallel lines and/or quadrilaterals (prep for next topic)

Make sure that all pencils are sharp and drawings are neat and accurate
Angles should be correct to within 2°, lengths correct to the nearest mm
Apply skills to constructing pie charts (Unit 1)

2-15

Angle and shapes

Definitions and names of polygons
Properties of triangles and quadrilaterals
Geometric proof
Angles associated with parallel lines
Calculate and use the sums of the interior angles of quadrilaterals, pentagons and hexagons
Calculate and use the angles of regular polygons
Understand that inscribed regular polygons can be constructed by equal divisions of a circle

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:
• Name a polygon with 3, 4, ..., 10 sides (13.6)
• Identify triangles by their properties (scalene, isosceles, equilateral, right-angled, obtuse, and acute) (12.8)
• Prove the angle sum in a triangle is 180° (13.2)
• Use the angle properties of triangle to find missing angles (12.7, 12.8)
• Prove the exterior angle of a triangle is equal to the sum of the two opposite interior angles (13.2)
• Identify quadrilaterals by their properties (trapezium, parallelogram, rhombus, rectangle, square, kite and arrowhead) (12.2)
• Use alternate and corresponding angles in parallel lines to find missing angles (13.1)
• Calculate and use the sums of the interior angles of convex polygons of sides 3, 4, 5, 6, 8, 10 (13.6)
• Know, or work out, the relationship between the number of sides of a polygon and the sum of its interior angles (13.6)
• Know that the sum of the exterior angles of any polygon is 360° (13.6)
• Find the size of each exterior/interior angle of a regular polygon (13.6)

Use triangles to find the angle sums of polygons
Use the angle properties of triangles to find missing angles in combinations of triangles
Harder problems involving multi-step calculations
Extend to bearings (Unit 3)
Link with symmetry and tessellations

Use plenty of practical drawing examples to help illustrate properties of various shapes – group/displays
Diagrams used in examinations are seldom drawn accurately
Use tracing paper to show which angles in parallel lines are equal
Encourage students to always write down the reasons and ‘quote’ the angle fact/theorem used (this is important for the new assessment objectives)

2-16

Circle geometry

Use names and definitions of parts of a circle
Understand and use the fact that tangents from an external point are equal in length
Explain why the perpendicular from the centre to a chord bisect the chord
Understand that the tangent at any point on a circle is perpendicular to the radius at that point

Properties of shapes and polygons
Basic angle facts
Experience of drawing a circle with a pair of compasses
Some experience of algebraic manipulation

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) (13.8)
• Use the angle properties of tangents to find missing angles (tangent at a point, tangents from a point) (13.10)
• Understand, prove and use circle theorems (see above) (13.9, 13.10)
• Use circle theorems to find unknown angles and explain their method - quoting the appropriate theorem(s) (13.9, 13.10)

Investigate other circle theorems by accurately measuring angles (link with Unit 3)
Harder problems involving multi-stage calculations
Algebraic solutions or general proofs

All working should be presented clearly, and accurately
Encourage students to always put the reasons and ‘quote’ the angle fact or theorem used
Draw lines using an HB pencil
A sturdy pair of compasses are essential - spare equipment is advisable
For weaker students a good idea for algebraic solutions or proofs is to replace the ‘x’ with a number, say 60o (in pencil). The students work through the question and then repeat it with the ‘x’, but using the same processes
Encourage students to always put the reasons and ‘quote’ the angle fact/theorem used

2-17

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 calculate 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, weights (12.4)
• Decide on the appropriate units to use in real life problems (12.3)
• Read measurements from instruments: scales, analogue and digital clocks, thermometers (Assumed
prior knowledge)
• 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 (12.5)
• Knowledge and use of common imperial units and time conversions and units (12.5)

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 make the lesson more real
Use Functional Elements questions as a source of practice questions

2-18

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 Module 17 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 (12.3, 12.5)
• Use common metric/imperial equivalents to convert between units* (12.5)
• Round measurements to 1 s.f. to find an estimate for a calculation (12.4)
• Understand that measurements can not be precise, and write down the maximum and minimum possible values (12.3)
• Work out the maximum/minimum possible error in a calculation involving measures (12.3)
• Use speed = distance/time to work out speed, distance or time (9.8)
• Change the units of speed between metric/metric units or imperial/imperial units (12. 5)
• Compare quantities by converting to the same units, eg km/h and m/s (12. 5)

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
30 cm 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 with more difficult examples, eg with quantities in standard form
Use uncommon units, eg astronomical units, speed of light, etc

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 questions as a source of practice questions, eg Best Buys, Travel Graphs leading to speed calculations

2-19

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 (14.2)
• Work out the surface area of 3-D shapes based on rectangles and triangles (by working out the area of each face) (14.8)
• Use v = l ´ w ´ h to solve problems involving the volume and dimensions of a cuboid (14.6)
• Work out how many small boxes fit into a large box (14.6)
• Use volume = cross-section ´ length to find the volume of a regular prism, eg with trapezium cross-section (14.7)

Further problems involving combinations of shapes
Using 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
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 concept
There are many Functional Elements skills which can be applied to this topic area, eg floor tiles, optimization type questions

2-20

3-D shapes, symmetry and tessellations

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 visualise reflection and rotational symmetry of 2D shapes
Construct cubes, tetrahedra, square-based pyramids and other 3-D shapes from given information
Tessellate combinations of polygons
Understand tessellations of regular and irregular polygons

The names of standard 3-D shapes
The unit on 2-D shapes

By the end of the module the student should be able to:
• Count the vertices, faces and edges of 3-D shapes (14.4)
• Draw nets of solids and recognise solids from their nets (14.4)
• Draw and interpret plans and elevations (14.5)
• Recognise line and rotational symmetry in 2-D shapes (12.1)
• 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 (12.1)
• Understand tessellations and explain why some shapes tessellate and why other shapes do not (13.7)
• Recognise and name examples of solids, including prisms, in the real world (14.4)

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
Discover Euler’s Formula for solids
Work out how many small boxes can be packed into a larger box (Functional type example)

Accurate drawing skills need to be reinforced
Some students find visualising 3-D objects difficult and simple models will assist
Use tracing paper/mirrors to help with symmetry questions
Link tessellations with the polygons

Unit 2 Higher## Past Papers

Latest Paper:Unit 2 Higher | Past Papers | Revision resources: | Unit 2 Topic List: | March 2011 Solutions: | A detailed breakdown:

Revision resources:Some Unit 2 revision questions (47 pages worth!):

The answers so you can check your work:

Mrs Walters list of MyMaths links related to the topic list:

Unit 2 Higher | Past Papers | Revision resources: | Unit 2 Topic List: | March 2011 Solutions: | A detailed breakdown:

## Unit 2 Topic List:

Factorise

Solving Equations

Solving All types of Equations

Selection of Questions

Algebraic Proof

Rearranging to get y=mx+c

Creating Formulae

Surds - Rationalising denominator

Straight Line Graphs

Tangents

Circle Theorems - All

Unit 2 Higher | Past Papers | Revision resources: | Unit 2 Topic List: | March 2011 Solutions: | A detailed breakdown:

## March 2011 Solutions:

1. Arithmetic Sequences, finding the nth term

2. Sharing in a Ratio

3. Plotting Equations of Straight Lines

4. Functional Skills: Applying Lowest Common Multiple to a practical problem

5. Surface Area of a Triangular Prism

6. Interior and Exterior Angles of Polygons

7. Area of Trapezium

8. Functional Skills: Best buy comparison of holidays

9. Speed and conversions from miles to kilometres

10. 3D coordinates and midpoints

11. Expanding, Factorising and Simplifying Algebraic Expressions

12. Indices

13. Equations of Lines and Perpendicular Lines

14. Simplifying Algebraic Fractions by Factorising

15. Expanding Brackets to find a Formula for the Volume of a Prism

16. Tangents to Circles and Proof

## A detailed breakdown:

Understand and use negative integers both as positions and translations on a number line

Order integers

Multiply and divide by negative numbers

Multiply or divide any number by powers of 10, and any positive number by a number between 0 and 1

Check and 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 equivalent fractions are multiplied or divided

Check and estimate answers to problems, by approximations or inverse operations (Decimals)

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

• Understand and order integers (1.1)

• Add, subtract, multiply and divide integers (BIDMAS) (1.5)

• Round whole numbers to the nearest, 10, 100, 1000, … (Revision of

prior knowledge)

• Multiply and divide whole numbers by a given multiple

of 10 (1.1, 1.3)

• Check their calculations by rounding, eg 29 ´ 31 » 30 ´ 30 (3.5)

• Put digits in the correct place in a decimal number (3)

• Write decimals in ascending order of size (3)

• Approximate decimals to a given number of decimal places or significant figures (3.3, 3.4)

• Multiply and divide decimal numbers by whole numbers and decimal numbers (up to 2 d.p.) eg 266.22 ¸ 0.34 (3.2)

• Know that eg 13.5 ¸ 0.5 = 135 ¸ 5 (3.2)

• Check their answer by rounding, eg 2.9 ´ 3.1 » 3.0 ´ 3.0 (3.6)

Estimating 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

Introduce standard form for very large/small numbers

Money calculations that require rounding answers to the nearest penny

Multiply and divide decimals by decimals (more than 2 decimal places)

For non-calculator methods make sure that ‘remainders’ and ‘carrying’ are shown

Amounts of money should always be rounded to the nearest penny where necessary

It is essential to ensure that 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

Use Functional Elements problems as additional practice for the topic area

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

Use a number line to show how numbers relate to each other

Recognise basic number patterns

Experience of classifying integers

• Find: squares, cubes, square roots, cube roots of numbers, with and without a calculator (including the use of trial and improvement) (1.4)

• Understand odd and even numbers, and prime numbers (1.2)

• Find the HCF and the LCM of numbers (1.2)

• Write a number as a product of its prime factors, eg 108 = 22 ´ 33 (1.2)

Further work on indices to include negative and/or fractional indices (intro to 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)

Use index laws to simplify and calculate the value of numerical expressions involving multiplication and division of integers, fractional and negative powers

Use inverse operations involving

Recall the fact that n0 = 1 and n–1 = for positive integers n, the corresponding rule for negative integers, = square root n and = cube root n for any positive number n

Appreciation of place value

Experience of the four operations using whole numbers

Knowledge of integer complements to 10

Knowledge of multiplication facts to 10 ´ 10

Knowledge of strategies for multiplying and dividing whole numbers by 10

• Understand and order integers (2.1)

• Add, subtract, multiply and divide integers (2.1)

• Understand simple instances of BIDMAS, eg work out 12 ´ 5 – 24 ¸ 8 (2.1)

• Round whole numbers to the nearest, 10, 100, 1000, … (1.6)

• Multiply and divide whole numbers by a given multiple of 10 (2.1)

• Check their calculations by rounding, eg 29 ´ 31 » 30 ´ 30 (2.6)

• Check answers by reverse calculation, eg if 9 ´ 23 = 207 then

207 ¸ 9 = 23

Directed number work with two or more operations, or with decimals

Encourage effective use of a calculator

Show what is entered into your calculator, not just the answer

Calculate with standard form

Convert between ordinary and standard form representations

Convert to standard form to make sensible estimates for calculations involving multiplication and/or division

Use standard form display and know how to enter numbers in standard index form

Multiplying decimal numbers with, and without, a calculator

Some experience with powers of 10, eg know that 102 = 100, 103 = 1000, 10–1 = 0.1

Negative indices and laws of indices

• Understand the standard form convention (5.2)

• Convert numbers to, and from, standard form (5.2)

• Calculate with numbers given in standard form with, and without, a calculator (5.2)

• Round numbers given in standard form to a given number of significant figures (5.2)

Add and subtracting 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, subtract, multiply and divide fractions (without a calculator)

Recognise that recurring decimals are exact fractions, and that some exact fractions are recurring decimals

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)

• Visualise a fraction diagrammatically (2.1)

• Understand a fraction as part of a whole (2.1)

• Recognise and write fractions in everyday situations (2.1)

• Write a fraction in its simplest form and recognise equivalent fractions (2.1, 3.1)

• Compare the sizes of fractions using a common denominator (Revise

prior knowledge)

• Add and subtract fractions by using a common denominator (2.2 - 2.5)

• Write an improper fraction as a mixed fraction (2.1 – 2.5)

• Recognise common recurring decimals can be written as exact fractions, eg (3.1)

• Converting a recurring decimal into a fraction (3.1)

Relating simple fractions to remembered percentages and vice-versa

Using a calculator to change fractions into decimals and looking for patterns

Working with improper fractions and mixed numbers

Multiplication and division of fractions to link 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

Introduction to algebraic fractions

Constant revision of this topic is needed

Students need to learn how to identify and use the fraction button on their calculators to check solutions

Link with probability calculations using AND and OR Laws (Unit 1)

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

Interpret percentage as the operator 'so many hundredths of'

Use multiplier to increase or decrease a given amount

Convert between fractions, decimals and percentages

The concepts of a fraction and a decimal

An awareness that percentages are used in everyday life

• Understand that a percentage is a fraction in hundredths (4.1)

• Write a percentage as a decimal; or as a fraction in its simplest terms (4.1)

• Write one number as a percentage of another number (4.1)

• Calculate the percentage (or fraction) of a given amount (4.1, 2.5)

• Find a percentage increase/decrease of an amount (4.2, 4.3)

• Use a multiplier to increase by a given percent, eg 1.10 ´ 64 increases 64 by 10% (4.3)

• Calculate simple and compound interest for two, or more, periods of time (4.3)

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 Functional Elements contexts

Comparisons between simple and compound interest calculations

Formulae in simple interest/compound interest methods

Show that finding 10% of the new price and subtracting does not work. (This is a useful introduction to Unit 3 contents)

Divide a quantity in a given ratio

Solve word problems about ratio, including using informal strategies and the unitary method of solution

• Appreciate that, eg 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)

Harder problems involving multi-stage calculations

Relate ratios to Functional Elements contexts, eg investigate the proportions of the different metals in alloys, and work out the new amounts of ingredients for a recipe for a different number of guests

Link ratios to metric and imperial units module

Distinguish the meaning between the words ‘equation’, ‘formula’, ‘identity’ and expression

Word formulae or rules to describe everyday situations, eg time to cook a nut roast linked to weight of nut roast

• Distinguish the different roles played by letter symbols in algebra (7.1)

• Understand the meaning between the words ‘equation’, ‘formula’, ‘identity’ and expression (10.1)

Look at word equations written in symbolic form, eg F = 2C + 30 to roughly convert temperature and compare with F = + 32

(See Emporium website)

Multiply a single term over a bracket

Take out common factors

Expand the product of two linear expressions

Factorise quadratic expressions

Ability to use negative numbers with the four operations

Experience of using BIDMAS in calculations without a calculator

• Simplify expressions with like terms, eg x2 + 3x2; 3ab + 5ab +2c2 (7.2)

• Expand and factorise expressions with one pair of brackets,

eg expand x(2x + 3y); factorise 3xy2 - 6x2y (8.1, 8.2)

• Expand and simplify expressions involving more than one pair of brackets, eg 3(x + 4) – 2(x – 3); (2x + 3)(3x – 4) (8.3, 8.4)

• Factorise quadratic expression (including the difference of two squares) (8.4)

• Simplify algebraic fractions, eg (11.1)

Further examples in factorising quadratic expression with non-unitary values of a (including fractional values)

Simplification of algebraic fractions which involve the addition of fractions

Present all work neatly, writing out the questions with the answers to aid revision at a later stage

Link the difference of two squares with the rationalisation of surds

Use formulae from mathematics and other subjects that require simplification of brackets, including those that have negative signs occurring anywhere in the equation, and those with a negative solution

Change the subject of a formula including where the subject occurs once

Generate a formula

Experience of using letters to represent quantities

Substituting into simple expressions using words

Using brackets in numerical calculations and removing brackets in simple algebraic expressions

• Use letters or words to state the relationship between different quantities (10.2)

• Substitute positive and negative numbers into simple algebraic formulae (10.2)

• Substitute positive and negative numbers into algebraic formulae involving powers (10.2)

• Find the solution to a problem by writing an equation and solving it (10.3)

• 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 (10.3)

Various investigations leading to generalisations

Further problems in generating formulae form given information

Apply changing the subject to y = mx + c

Students need to be clear on the meanings of the words expression, equation, formula and identity

Changing the subject is covered fully in Unit 3, use this Unit as an introduction to this concept

Use Functional Elements problems as a rich source of formulae given in everyday contexts

Link with formulae for area, volume, surface area, etc

Write answers in exact form

Knowledge of square roots and π

Collecting together like terms

Removing brackets in simple algebraic expressions

Difference of two squares

• Simplifying surds (5.4)

• Use surds and π in exact calculations, without a calculator (5.4)

• Write (3 – √3)2 in the form a + b√3 (5.4)

• Rationalise a denominator (5.4)

• Rationalise the denominator of fractions, eg = , and

eg write (Ö18 +10) ¸ Ö2 in the form p + qÖ2 (5.4)

• Give an answer to a question involving the area of a circle as 25π (link with Unit 3) (5.4)

Stress it is better to write answers in exact form, eg is better than 0.333333…..

Prove that √2 is irrational

Revise the difference of two squares to show why we use, for example (√3 – 2) as the multiplier to rationalise (√3 + 2)

Link to Unit 3 work on circle measures (involving π) and Pythagoras calculations in exact form.

Give an answer to use of Pythagoras’ Theorem as √13 (link with Unit 3)

so therefore 3√5 + 2√5 = 5√5

Stress it is better to write answers in exact form (eg is better than 0.333333…..)

A-Level textbooks (Core 1) are a good source of questions on surd manipulation, some of which are algebraic.

A useful generalisation to learn is

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

Recognise simple number patterns eg 1, 3, 5, ...

Writing simple rules algebraically

Raise numbers to positive whole number powers

• Find the missing numbers in a number pattern or sequence (7.6, 7.7)

• Find the nth term of a number sequence as an algebraic expression (7.7)

• Explain why a number is, or is not, a member of a given sequence (7.6, 7.7)

• Produce a sequence of numbers from a given nth term formula (7.7)

Sequences and nth term formula for triangle numbers, Fibonacci numbers etc

Prove a sequence cannot have odd numbers for all values of n

Extend to quadratic sequences whose nth term is an2 + bn + c

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

Plot points in all four quadrants

Find the coordinates of the final point to make a given shape

Use the formula to find the mid-point of line segments

Identify 2-D and 3-D shapes; 3-D coordinates

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

Explore the gradients of parallel lines and lines perpendicular to each other

Construct linear functions and plot the corresponding graphs arising from real-life problems

Discuss, plot and interpreting graphs modelling real situations (including travel & conversion graphs)

Plot coordinates in the first quadrant

Calculating the mean of two numbers

Knowledge of basic shapes

Rearrange to change the subject of a formula

• Add a point to a coordinate grid to complete a given shape (parallelogram; rhombus; trapezium; square) (9.1)

• Use the formula to calculate the midpoint of a line segment (9.2)

• Understand how to represent points in 1-D, 2-D and 3-D (9.1, 14.9)

• Substitute values of x into linear functions to find corresponding

values of y (9.1, 9.3 – 9.5)

• Plot points for linear functions on a coordinate grid and draw the

corresponding straight lines (9.1, 9.3 – 9.5)

• Interpret m and c as gradient and y-intercept in linear functions (9.3)

• Understand that the graphs of linear functions are parallel if they have the same value of m (9.5)

• Know that the line perpendicular to y = mx + c has gradient (9.5)

• Understand linear functions in practical problems, eg distance-time graphs (9.6)

Find the equation of the perpendicular bisector of the line segment joining two given points

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

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

Link to graphs and relationships in other subject areas, ie science, geography etc

Interpret straight line graphs for Functional Elements problems

– Ready reckoner graphs

– Conversion graphs

– Fuel bills & Mobile Phone tariffs

– Fixed charge (standing charge) and cost per unit

Link conversion graphs to converting metric & imperial units and equivalents (Unit 2)

A-Level Text books (C1) are a good source of extension questions on this topic

Cover horizontal and vertical lines (x = c and y = c), as students often forget these

Use a protractor to draw angles accurately

Use the fact that angles at a point add to 360°

Experience of using a ruler and a protractor

• Distinguish between acute, obtuse, reflex and right angles (12.6)

• Estimate the size of an angle in degrees (revision of

prior knowledge)

• Measure and draw angle to the nearest degree (Chapter 12

Introduction)

• Measure and draw line to the nearest mm (Chapter 12

Introduction)

• Use angle properties at a point to calculate unknown angles (12.6)

Angles should be correct to within 2°, lengths correct to the nearest mm

Apply skills to constructing pie charts (Unit 1)

Properties of triangles and quadrilaterals

Geometric proof

Angles associated with parallel lines

Calculate and use the sums of the interior angles of quadrilaterals, pentagons and hexagons

Calculate and use the angles of regular polygons

Understand that inscribed regular polygons can be constructed by equal divisions of a circle

The concept of vertical and horizontal

The concept of an angle between two lines

Experience in drawing triangles, quadrilaterals and circles

• Name a polygon with 3, 4, ..., 10 sides (13.6)

• Identify triangles by their properties (scalene, isosceles, equilateral, right-angled, obtuse, and acute) (12.8)

• Prove the angle sum in a triangle is 180° (13.2)

• Use the angle properties of triangle to find missing angles (12.7, 12.8)

• Prove the exterior angle of a triangle is equal to the sum of the two opposite interior angles (13.2)

• Identify quadrilaterals by their properties (trapezium, parallelogram, rhombus, rectangle, square, kite and arrowhead) (12.2)

• Use alternate and corresponding angles in parallel lines to find missing angles (13.1)

• Calculate and use the sums of the interior angles of convex polygons of sides 3, 4, 5, 6, 8, 10 (13.6)

• Know, or work out, the relationship between the number of sides of a polygon and the sum of its interior angles (13.6)

• Know that the sum of the exterior angles of any polygon is 360° (13.6)

• Find the size of each exterior/interior angle of a regular polygon (13.6)

Use the angle properties of triangles to find missing angles in combinations of triangles

Harder problems involving multi-step calculations

Extend to bearings (Unit 3)

Link with symmetry and tessellations

Diagrams used in examinations are seldom drawn accurately

Use tracing paper to show which angles in parallel lines are equal

Encourage students to always write down the reasons and ‘quote’ the angle fact/theorem used (this is important for the new assessment objectives)

Understand and use the fact that tangents from an external point are equal in length

Explain why the perpendicular from the centre to a chord bisect the chord

Understand that the tangent at any point on a circle is perpendicular to the radius at that point

Basic angle facts

Experience of drawing a circle with a pair of compasses

Some experience of algebraic manipulation

• Identify and name the various parts of a circle (centre, radius, diameter, circumference, sector, segment, arc and chord) (13.8)

• Use the angle properties of tangents to find missing angles (tangent at a point, tangents from a point) (13.10)

• Understand, prove and use circle theorems (see above) (13.9, 13.10)

• Use circle theorems to find unknown angles and explain their method - quoting the appropriate theorem(s) (13.9, 13.10)

Harder problems involving multi-stage calculations

Algebraic solutions or general proofs

Encourage students to always put the reasons and ‘quote’ the angle fact or theorem used

Draw lines using an HB pencil

A sturdy pair of compasses are essential - spare equipment is advisable

For weaker students a good idea for algebraic solutions or proofs is to replace the ‘x’ with a number, say 60o (in pencil). The students work through the question and then repeat it with the ‘x’, but using the same processes

Encourage students to always put the reasons and ‘quote’ the angle fact/theorem used

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 calculate time intervals

Convert from one metric unit to another

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

• Make estimates of: length, volume and capacity, weights (12.4)

• Decide on the appropriate units to use in real life problems (12.3)

• Read measurements from instruments: scales, analogue and digital clocks, thermometers (Assumed

prior knowledge)

• 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 (12.5)

• Knowledge and use of common imperial units and time conversions and units (12.5)

Link with compound units such as speed for travel graphs (m/s or km/h or mph) and Best Buys (g/penny)

Use a range of everyday objects to make the lesson more real

Use Functional Elements questions as a source of practice questions

Estimate answers

Use the formulae for speed

Change units in compound measures

Some experience of Metric/Imperial measures from Module 17 or Unit 1

• Convert measurements to the same unit before doing a calculation,

eg 3 m ´ 12 cm ´ 10 mm (12.3, 12.5)

• Use common metric/imperial equivalents to convert between units* (12.5)

• Round measurements to 1 s.f. to find an estimate for a calculation (12.4)

• Understand that measurements can not be precise, and write down the maximum and minimum possible values (12.3)

• Work out the maximum/minimum possible error in a calculation involving measures (12.3)

• Use speed = distance/time to work out speed, distance or time (9.8)

• Change the units of speed between metric/metric units or imperial/imperial units (12. 5)

• Compare quantities by converting to the same units, eg km/h and m/s (12. 5)

- These are the Metric to Imperial equivalents expected

Metric Imperial1 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

30 cm 1 foot

Work with more difficult examples, eg with quantities in standard form

Use uncommon units, eg astronomical units, speed of light, etc

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 questions as a source of practice questions, eg Best Buys, Travel Graphs leading to speed calculations

Calculate the surface area and volume of cuboids

Use the formula to calculate the volume of a prism

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

• Use the area formulae for triangles, parallelograms and trapeziums (14.2)

• Work out the surface area of 3-D shapes based on rectangles and triangles (by working out the area of each face) (14.8)

• Use v = l ´ w ´ h to solve problems involving the volume and dimensions of a cuboid (14.6)

• Work out how many small boxes fit into a large box (14.6)

• Use volume = cross-section ´ length to find the volume of a regular prism, eg with trapezium cross-section (14.7)

Using 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

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 concept

There are many Functional Elements skills which can be applied to this topic area, eg floor tiles, optimization type questions

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 visualise reflection and rotational symmetry of 2D shapes

Construct cubes, tetrahedra, square-based pyramids and other 3-D shapes from given information

Tessellate combinations of polygons

Understand tessellations of regular and irregular polygons

The unit on 2-D shapes

• Count the vertices, faces and edges of 3-D shapes (14.4)

• Draw nets of solids and recognise solids from their nets (14.4)

• Draw and interpret plans and elevations (14.5)

• Recognise line and rotational symmetry in 2-D shapes (12.1)

• 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 (12.1)

• Understand tessellations and explain why some shapes tessellate and why other shapes do not (13.7)

• Recognise and name examples of solids, including prisms, in the real world (14.4)

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

Discover Euler’s Formula for solids

Work out how many small boxes can be packed into a larger box (Functional type example)

Some students find visualising 3-D objects difficult and simple models will assist

Use tracing paper/mirrors to help with symmetry questions

Link tessellations with the polygons