foundation module 1 · web viewtime: 7–9 hours. specification reference. understanding place...

FOUNDATION CHAPTER 1 UNDERSTANDING WHOLE NUMBERS Time: 7–9 hours

SPECIFICATION REFERENCE Understanding place value in whole numbers NA2a Using figures and words for whole numbers and writing numbers in order of size NA2a Adding, subtracting, multiplying and dividing whole numbers NA3a/g/j/k Problems in selecting the correct operation from addition, subtraction, multiplication and division NA3a/g/4b Writing numbers to the nearest 10, 100 and 1000 and to one significant figure before

approximating and estimating calculations NA3h/4c

PRIOR KNOWLEDGEThe ability to order numbersAppreciation of place valueExperience of the four operations using whole numbersKnowledge of integer complements to 10Knowledge of multiplication facts to 10 10Knowledge of strategies for multiplying and dividing whole numbers by 10

OBJECTIVESBy the end of the chapter the student should be able to: Understand and order integers Add, subtract, multiply and divide integers Understand simple instances of BIDMAS, e.g. work out 12 5 – 24 8 Round whole numbers to the nearest 10, 100, 1000, … Multiply and divide whole numbers by a given multiple of 10 Check their calculations by rounding, e.g. 29 31 30 30 Check answers by reverse calculation, e.g. If 9 23 = 207 then 207 9 = 23

RESOURCESFoundation Student book Chapter/section: 1.1–1.5Foundation Practice book Chapter 1

DIFFERENTIATION AND EXTENSION 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

ASSESSMENTHeinemann online assessment tool will be available in 2007.

HOMEWORKThis could include consolidation of work in class by completion of exercises set, additional work of a similar nature, or extension work detailed above.

HINTS AND TIPS Present all working clearly with numbers in line; stress that all working is to be shown. For non-calculator methods make sure that remainders and carrying are shown. Use Exercises 1A–1L for practice. Use Mixed Exercise 1 for consolidation.

FOUNDATION CHAPTER 2 NUMBER FACTS Time: 6–8 hours

SPECIFICATION REFERENCE Understanding negative numbers in context NA2a Ordering, adding, subtracting, multiplying and dividing positive and negative numbers NA2a/3a Recognising even, odd and prime numbers and finding factors and multiples of numbers NA2a Finding squares and cubes of numbers NA2b Finding squares and square roots of numbers NA2b/3o/4d Finding cubes and cube roots of numbers NA2b/3o Finding square roots and cube roots by trial and improvement NA2b/3o Writing numbers as a product of their prime factors; HCF and LCM NA2a/3a

PRIOR KNOWLEDGENumber complements to 10 and multiplication/division factsUse a number line to show how numbers relate to each otherRecognise basic number patternsExperience of classifying integers

OBJECTIVESBy the end of the chapter the student should be able to: Understand and use negative numbers in context, e.g. thermometers Find: squares; cubes; square roots; cube roots of numbers, with and without a calculator (including the use of trial and

improvement) Understand odd and even numbers, and prime numbers Interpret and represent numbers expressed in index form on calculator displays (using the correct notation) Find the HCF and the LCM of numbers Write a number as a product of its prime factors, e.g. 108 = 2 2 3 3 3

RESOURCESFoundation Student book Chapter/section: 2.1–2.8Foundation Practice book Chapter 2Teaching and Learning software Chapter 2

DIFFERENTIATION AND EXTENSION Calculator exercise to check factors of larger numbers Further work on indices to include negative and/or fractional indices 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)

ASSESSMENT Heinemann online assessment tool will be available in 2007Regular oral work, e.g. a five-minute assessment at the beginning or end of a lessonMental test to check knowledge of squares and cubesMental test on the recognition of odd and even numbersTest on performance using a calculator to find squares, cubes and square rootsTest without a calculator on knowledge of squares, cubes and square numbers (keeping the numbers small)

HOMEWORKThis could include consolidation of work in class by completion of exercises set, additional work of a similar nature, or extension work detailed above.Investigational tasks leading to number patterns involving powers of numbers.GCSE past paper questions.

HINTS AND TIPS All of the work in this unit is easily reinforced by starter and end activities. Oral discussion should be used to ensure:

Recognition of odd and even numbersCalculators are only used when appropriate.

Do Exercises 2A–2L for practice. Do Mixed Exercise 2 for consolidation.

FOUNDATION CHAPTER 3A ESSENTIAL ALGEBRA Time: 4–6 hours

SPECIFICATION REFERENCE Introducing the use of letters NA5a The meaning of expressions such as 5x NA5a Simplifying expressions involving one letter only NA5a Simplifying expressions involving more than one letter NA5a Omitting the multiplication sign NA5a Simple algebraic multiplication NA5a/b Simple algebraic division NA5a/b

PRIOR KNOWLEDGEExperience of using a letter to represent a numberAbility to use negative numbers with the four operations

OBJECTIVESBy the end of the chapter the student should be able to: Simplify algebraic expressions in one, or more like terms, by adding and subtracting Multiply and divide with letters and numbers


DIFFERENTIATION AND EXTENSION Further work on collecting like terms, involving negative terms Collecting terms where each term may consist of more than one letter, e.g. 3ab + 4ab Examples where all the skills above are required

ASSESSMENT Heinemann online assessment tool will be available in 2007.


HINTS AND TIPS Explanations about mathematical terminology should be tailored to suit the ability of the group. Emphasise correct use of symbolic notation, e.g. 3x rather than 3 x. Present all work neatly, writing out the questions with the answers to aid revision at a later stage. Do Exercises 3A–3G for practice.

FOUNDATION CHAPTER 3B ESSENTIAL ALGEBRA 2 Time: 6–8 hours

SPECIFICATION REFERENCE Using index notation with numbers NA5c Finding the rules for adding and subtracting indices NA3a Using index notation in algebra NA5c Multiplying and dividing using indices NA5c Using brackets in numerical calculations (BIDMAS) NA3b Simplifying algebraic expressions involving brackets NA3b/5b Factorising using a single bracket NA5b

PRIOR KNOWLEDGEExperience of using a letter to represent a numberAbility to use negative numbers with the four operations

OBJECTIVESBy the end of the chapter the student should be able to: Multiply and divide powers of the same number Understand and use the index rules to simplify algebraic expressions, e.g. 55 52 = 53

Use brackets to expand and simplify simple algebraic expressions


DIFFERENTIATION AND EXTENSION Examples where all the skills above are required Factorising where the factor may involve more than one variable Use index rules with negative numbers (and fractions)



HINTS AND TIPS Emphasise correct use of symbolic notation, e.g. 3x rather than 3 x. Present all work neatly, writing out the questions with the answers to aid revision at a later stage. Do Exercises 3H–3T for practice. Do Mixed Exercise 3 for consolidation.

FOUNDATION CHAPTER 4 PATTERNS AND SEQUENCES Time: 5–7 hours

SPECIFICATION REFERENCE Finding the missing numbers in number patterns or sequences NA6a Investigating number patterns NA6a Finding the nth term of a number sequence NA6a Finding whether a number is part of a given sequence Na6a Using a calculator to produce and generate patterns and sequences NA3o/p/6a Using spreadsheets to investigate and to generate number patterns and sequences NA6a/5f

PRIOR KNOWLEDGEKnow about odd and even numbersRecognise simple number patterns, e.g. 1, 3, 5, ...Writing simple rules algebraicallyRaise numbers to positive whole number powers

OBJECTIVESBy the end of the chapter the student should be able to: Find the missing numbers in a number pattern or sequence Find the nth term of a number sequence Find whether a number is part of a given sequence Use a calculator to produce a sequence of numbers


DIFFERENTIATION AND EXTENSION Match-stick problems Sequences of triangle numbers, Fibonacci numbers, etc

ASSESSMENT Heinemann online assessment tool will be available in 2007.Simple investigation of a sequence, using diagrams and number patternsUse of mental maths in the substitution of simple numbers into expressionsGCSE A01 coursework tasks

HOMEWORKThis could include consolidation of work in class by completion of exercises set, additional work of a similar nature, or extension work detailed above.Fibonacci sequence, Pascal’s triangleUses of algebra to describe real situation, e.g. n quadrilaterals have 4n sides

HINTS AND TIPS Emphasis on good use of notation, e.g. 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. The cube (and cube root) function on a calculator may not be the same for all makes. Do Exercises 4A–4I for practice. Do Mixed Exercise 4 for consolidation.

FOUNDATION CHAPTER 5 DECIMALS Time: 5–7 hours

SPECIFICATION REFERENCE Putting digits in the correct places NA2a/d Writing decimals in ascending and descending order NA2d Rounding decimal numbers to the nearest whole number and to one decimal place NA2a/3h Approximating to a given number of decimal places NA3h Approximating to a given number of significant figures NA3h Pencil and paper methods for adding and subtracting decimal numbers NA3i/j Multiplying decimals by whole numbers and decimals NA3k/i Dividing decimals by whole numbers and decimals NA3k/i

PRIOR KNOWLEDGEChapter 1: Understanding whole numbersThe concepts of a fraction and a decimal

OBJECTIVESBy the end of the chapter the student should be able to: Put digits in the correct place in a decimal number Write decimals in ascending order of size Approximate decimals to a given number of decimal places or significant figures Multiply and divide decimal numbers by whole numbers and decimal numbers (up to two decimal points), e.g. 266.22 0.34 Know that, for instance 13.5 0.5 = 135 5 Check their answer by rounding, know that, for instance 2.9 3.1 3.0 3.0


DIFFERENTIATION AND EXTENSIONUse decimals in real-life problemsUse standard form for vary large/small numbersMoney calculations that require rounding answers to the nearest pennyMultiply and divide decimals by decimals (more than 2 dp)



HINTS AND TIPS Present all working clearly with decimal points in line; emphasising 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. Do Exercises 5A–5J for practice. Do Mixed Exercise 5 for consolidation.

FOUNDATION CHAPTER 6A ANGLES AND TURNING 1 Time: 4–6 hours

SPECIFICATION REFERENCE An introduction to angles SSM2a Types of angles, estimating the size of angles SSM2b Using vertices to name angles SSM2a Using a protractor to measure angles of all sizes SSM4a/d Using a protractor to draw angles accurately SSM4d Using the fact that angles on a straight line add to 180 SSM2a Using the fact that angles at a point add to 360 and vertically opposite angles are equal SSM2a

PRIOR KNOWLEDGEAn understanding of angle as a measure of turningThe ability to use a protractor to measure angles

OBJECTIVESBy the end of the chapter the student should be able to: Distinguish between acute, obtuse, reflex and right angles Estimate the size of an angle in degrees Measure and draw angle to the nearest degree Use angle properties on a line and at a point to calculate unknown angles


DIFFERENTIATION AND EXTENSION Measuring angles in polygons Investigate the angle sum of polygons


HOMEWORKThis could include consolidation of work in class by completion of exercises set, additional work of a similar nature, or extension work detailed above.Regular quick test-type homework on angle properties of various shapes.

HINTS AND TIPS Make sure that all pencils are sharp and drawings are neat and accurate Angles should be within 2 degrees Do Exercises 6A–6G for practice.

FOUNDATION CHAPTER 6B ANGLES AND TURNING 2 Time: 3–5 hours

SPECIFICATION REFERENCE Calculating unknown angles in triangles and quadrilaterals SSM2d Using parallel lines to identify alternate and corresponding angles SSM2c Understanding simple formal proofs in geometry SSM2c/d Calculating interior and exterior angles in polygons SSM2g Measuring and calculating bearings SSM4b

PRIOR KNOWLEDGEChapter 6A: Angles and turning 1Recall the names of special types of triangle, including equilateral, right-angled and isoscelesKnow that angles on a straight line sum to 180 degreesKnow that a right angle = 90 degreesUnderstand the concept of parallel lines

OBJECTIVESBy the end of the chapter the student should be able to: Mark parallel lines in a diagram Find missing angles using properties of corresponding angles and alternate angles, giving reasons Use angle properties of triangles and quadrilaterals to find missing angles Find the three missing angles in a parallelogram when one of them is given Calculate and use the sums of the interior angles of convex polygons of sides 3, 4, 5, 6, 8, 10 Prove that the angle sum of a triangle is 180 degrees Explain why the angle sum of a quadrilateral is 360 degrees Know, or work out, the relationship between the number of sides of a polygon and the sum of its interior angles Measure and calculate bearings


DIFFERENTIATION AND EXTENSIONHarder problems involving multi-step calculations



HINTS AND TIPS Generally the diagrams in examinations are not accurately drawn Do Exercises 6H–6M for practice. Do Mixed Exercise 6 for consolidation.

FOUNDATION CHAPTER 7 2-D SHAPES Time: 4–6 hours

SPECIFICATION REFERENCE Basic information about special triangles, quadrilaterals and polygons SSM2d/f Using square and isometric paper to draw polygons and make patterns SSM4d Recognising when shapes are congruent SSM2d Tessellating with assorted shapes SSM3b/4d Accurate constructions of triangles, quadrilaterals and polygons SSM4d/e Scale drawings and using scales in maps SSM3d Drawing angles using a ruler and compasses bisecting lines using a pair of compasses SSM4e Constructing loci and regions SSM4j/e

PRIOR KNOWLEDGEAn ability to use a pair of compassesUnderstanding of the terms perpendicular and parallel

OBJECTIVESBy the end of the chapter the student should be able to: Use a ruler and compass to draw accurate triangles, and other 2-D shapes, given information about their side lengths and

angles Use straight edge and compass to construct: an equilateral triangle; the midpoint and perpendicular bisector of a line

segment; the bisector of an angle Find the locus of points, such as the locus of points equidistant to two given points Understand, by their experience of constructing them, that triangles satisfying SSS, SAS, ASA and RHS are unique, but SSA

triangles are not Recall and use angle properties of equilateral, isosceles and right-angled triangles Recall and use the properties of squares, rectangles, parallelograms, trapeziums and rhombuses Recall and use properties of circles Appreciate why some shapes tessellate and why some shapes do not tessellate


DIFFERENTIATION AND EXTENSION Solve loci problems that require a combination of loci Construct combinations of 2-D shapes to make nets Investigate tessellation



HINTS AND TIPS All working should be presented clearly, and accurately. A sturdy pair of compasses are essential. Have some spare equipment available. Do Exercises 7A–7H for practice. Do Mixed Exercise 7 for consolidation.

FOUNDATION CHAPTER 8A FRACTIONS 1 Time: 5–7 hours

SPECIFICATION REFERENCE Understanding parts of a whole NA3c Writing fractions from everyday situations NA3c Converting between improper fractions and mixed numbers NA2c Finding fractions of different quantities NA3c Cancelling fractions into their lowest terms NA2c/3c Finding equivalent fractions by cancelling and multiplying NA2c Using common denominators to compare the size of fractions NA2c Using common denominators to add fractions NA3c Using common denominators to subtract fractions NA3c

PRIOR KNOWLEDGEMultiplication factsAbility to find common factorsA basic understanding of fractions as being ‘parts of a whole unit’Use of a calculator with fractions

OBJECTIVESBy the end of the chapter the student should be able to: Visualise a fraction diagrammatically Understand a fraction as part of a whole Recognise and write fractions in everyday situations Write a fraction in its simplest form and recognise equivalent fractions Compare the sizes of fractions using a common denominator Add and subtract fractions by using a common denominator Write an improper fraction as a mixed fraction


DIFFERENTIATION AND EXTENSION 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 Solve word problems involving fractions (and in real-life problems, e.g. find perimeter using fractional values)

ASSESSMENT Heinemann online assessment tool will be available in 2007.Testing the ability to perform calculations, using simple fractions, without a calculator.Mental arithmetic test involving simple fractions such as ½, ¼, ...

HOMEWORKThis could include consolidation of work in class by completion of exercises set, additional work of a similar nature, or extension work detailed above.An equivalent fractions worksheet as a preliminary, following on from the initial lesson.Use the worksheet for comparing fractions, ordering fractions, and adding and subtracting fractions.Extra examples on a regular basis for revision purposes.Other work given could have fractional answers as a part of the process.

HINTS AND TIPS Understanding of equivalent fractions is the key issue in order to be able to tackle the other content. Calculators should only be used when appropriate. Constant revision of this aspect is needed. All work needs to be presented clearly with the relevant stages of working shown. Do Exercises 8A–8I for practice.

FOUNDATION CHAPTER 8B FRACTIONS 2 Time: 3–5 hours

SPECIFICATION REFERENCE Multiplying fractions by cancelling and by using top-heavy fractions NA3d/l Dividing fractions by inverting the divisor and multiplying NA3d/l Assorted questions using fractions NA3c/q/4a Converting between fractions and decimals NA3c Changing fractions into recurring decimals NA3c/2d Finding reciprocals of whole numbers, fractions and decimals NA3a

PRIOR KNOWLEDGEChapterR 8A Fractions 1

OBJECTIVESBy the end of the chapter the student should be able to: Multiply and divide a number with a fraction, and a fraction with a fraction (expressing the answer in its simplest form) Simplify multiplication of fractions by first cancelling common factors Convert a fraction to a decimal, or a decimal to a fraction Convert a fraction to a recurring decimal Find the reciprocal of whole numbers, fractions, and decimals, e.g. find the reciprocal of 0.4 Know that 0 does not have reciprocal, and that a number multiplied by its reciprocal is 1 Use fractions in contextualised problems


DIFFERENTIATION AND EXTENSION Using a calculator to find fractions of given quantities Working with improper fractions and mixed numbers Using the four operations using fractions (and in real-life problems, e.g. find area using fractional values) For very able students, cancelling down of algebraic expressions could be considered

ASSESSMENT Mental testing on a regular basis, of the basic conversions of simple fraction into decimalsTesting the ability to perform calculations, using simple fractions, without a calculatorMental arithmetic test involving simple fractions such as ½, ¼ ...

HOMEWORKThis could include consolidation of work in class by completion of exercises set, additional work of a similar nature, or extension work detailed above.Extra examples on a regular basis for revision purposes.Other work given could have fractional answers as a part of the process.

HINTS AND TIPS Constant revision of this aspect is needed. All work needs to be presented clearly with the relevant stages of working shown. Non-calculator work with fractions is generally poorly attempted at GCSE. Students may have difficulty with the concept of

dividing by a fraction. Do Exercises 8J–8O for practice. Do Mixed Exercise 8 for consolidation.

FOUNDATION CHAPTER 9 ESTIMATING AND USING MEASURES Time: 7–9 hours

SPECIFICATION REFERENCE Making estimates in everyday life SSM4a Making estimates of length using metric and imperial units SSM4a Making estimates of volume and capacity using metric and imperial units SSM4a Making estimates of weights using metric and imperial units SSM4a Using sensible units for measuring SSM4a Reading analogue and digital clocks SSM4a Reading measurements on different types of scales SSM4a

PRIOR KNOWLEDGEAn awareness of the imperial system of measuresStrategies for multiplying and dividing by 10Knowledge of the conversion facts for metric lengths, mass and capacityKnowledge of the conversion facts between seconds, minutes and hours

OBJECTIVESBy the end of the chapter the student should be able to: Make estimates of: length; volume and capacity; weights Make approximate conversions between metric and imperial units Decide on the appropriate units to use in real life problems Read measurements from instruments: scales; analogue and digital clocks; thermometers, etc


DIFFERENTIATION AND EXTENSION 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 Use ICT and reference books to find the weights, volumes and heights of large structures such as buildings, aeroplanes and

ships Work with more difficult examples

ASSESSMENT Heinemann online assessment tool will be available in 2007.Mental testing to check for knowledge of everyday measures, and estimation.Mental test questions on changing units, changing between metric and imperial units.Mental questions to test knowledge of common conversion factors.Aural questions on dates, times and timetables.

HOMEWORKThis could include consolidation of work in class by completion of exercises set, additional work of a similar nature, or extension work detailed above.Plan a foreign holiday with dates, timetables to arrive at the airport etc.

HINTS AND TIPS Measurement is essentially a practical activity. Use a range of everyday objects to make the lesson more real. All working should be shown with multiplication or division by powers of 10. Do Exercises 9A–9M for practice. Do Mixed Exercise 9 for consolidation.

FOUNDATION CHAPTER 10 COLLECTING AND RECORDING DATA Time: 4–6 hours

SPECIFICATION REFERENCE Ways of collecting data HD3a Designing questions for a questionnaire HD3a Random samples and bias HD3a Designing a data capture sheet HD3a Using a data capture sheet to collect results from an experiment HD3a Secondary data and primary data HD3b Organised collections of information HD3b Data from the Internet HD3b Mileage charts HD3c

PRIOR KNOWLEDGEAn understanding of why data needs to be collectedSome idea about different types of graphs

OBJECTIVESBy the end of the chapter the student should be able to: Design a suitable question for a questionnaire Understand the difference between: primary and secondary data; discrete and continuous data Design suitable data capture sheets for surveys and experiments Understand about bias in sampling Use a mileage chart


DIFFERENTIATION AND EXTENSIONCarry out a statistical investigation of their own including- designing an appropriate means of gathering the data

ASSESSMENT Heinemann online assessment tool will be available in 2007.Their own statistical investigation.GCSE coursework – data handling project.

HOMEWORKThis could include consolidation of work in class by completion of exercises set, additional work of a similar nature, or extension work detailed above.Completion of data collection exercise and statistical project.

HINTS AND TIPS Students may need reminding about the correct use of tallies. Emphasis the differences between primary and secondary data. If students are collecting data as a group they should all use the same procedure. Emphasis that continuous data is data that is measured. Do Exercises 10A–10H for practice. Do Mixed Exercise 10 for consolidation.

FOUNDATION CHAPTER 11 LINEAR EQUATIONS Time: 6–8 hours

SPECIFICATION REFERENCE Solving equations with one operation NA5e Solving equations involving more than one stage NA5e Solving equations with brackets and more than one operation NA5e

PRIOR KNOWLEDGEExperience of finding missing numbers in calculationsThe idea that some operations are ‘opposite’ to each otherAn understanding of balancingExperience of using letters to represent quantitiesBe able to draw a number line

OBJECTIVESBy the end of the chapter the student should be able to: Solve linear equations with one, or more, operations Solve linear equations involving a single pair of brackets Solve simple equations using inverse operations

RESOURCESFoundation Student book Chapter/section: 11.1–11.3Foundation Practice book Chapter 11 Teaching and Learning software Chapter 11

DIFFERENTIATION AND EXTENSION Use of inverse operations and rounding to one significant figure could be applied to more complex calculations Derive equations from practical situations (such as angle calculations) Solve equations where manipulation of fractions (including the negative fractions) is required Solve linear inequalities where manipulation of fractions is required



HINTS AND TIPS Students need to realise that not all linear equations can easily be solved by either observation or trial and improvement, and

hence the use of a formal method is vital. Students can leave their answers in fractional form where appropriate. Interpreting the direction of an inequality is a problem for many. Do Exercises 11A–11O for practice. Do Mixed Exercise 11 for consolidation.

FOUNDATION CHAPTER 12 SORTING AND PRESENTING DATA Time: 5–7 hours

SPECIFICATION REFERENCE Using tally charts and bar charts to display data HD4a/5b Using class intervals to record data HD3a Representing sets of data in a dual bar chart HD4a/5b Representing information in a dual bar chart HD4a/5b Discrete and continuous data, and line graphs HD4a/5b/c Information that changes with time HD4a/5b/c/k Using a histogram to display continuous data HD4a/5b Showing the pattern of data in a histogram HD4a/5b/c

PRIOR KNOWLEDGEAn understanding of why data needs to be collected and some idea about different types of graphsExperience of collecting, interpreting, displaying and calculating with data

OBJECTIVESBy the end of the chapter the student should be able to represent data as: Bar charts (including dual bar charts) Pictograms Line graphs Histograms (intervals with equal width) Frequency polygons Time series graphs In addition, students should be able to: Choose an appropriate way to display discrete, continuous and categorical data Identify trends in data over time Identify exceptional periods by comparison with similar previous periods


DIFFERENTIATION AND EXTENSION Carry out a statistical investigation of their own and use an appropriate means of displaying the results Use a spreadsheet to draw different types of graphs Collect examples of charts and graphs in the media which have been misused, and discuss the implications Make predictions by considering trends of line graphs for time series Additional work on making predictions based on current trends, using time series and/or moving averages Collect data from the Internet (e.g. RPI) and analyse it for trend

ASSESSMENT Heinemann online assessment tool will be available in 2007.Their own statistical investigation.GCSE coursework – data handling project.

HOMEWORKThis could include consolidation of work in class by completion of exercises set, additional work of a similar nature, or extension work detailed above.Completion of a simple statistical project.

HINTS AND TIPS Clearly label all axes on graphs and use a ruler to draw straight lines. Many students enjoy drawing statistical graphs for classroom displays. Do Exercises 12A–12G for practice. Do Mixed Exercise 12 for consolidation.

FOUNDATION CHAPTER 13 3-D SHAPES Time: 4–6 hours

SPECIFICATION REFERENCE Recognising horizontal and vertical surfaces SSM2j Counting the faces, edges and vertices in three-dimensional solids SSM2j 3-D shapes in everyday life SSM2k Examples of prisms SSM2k Drawing nets of solids and recognising solids from their nets SSM2j/4d Drawing and interpreting plans and elevations SSM2k Identifying and drawing planes of symmetry in 3-D shapes SSM3b

PRIOR KNOWLEDGEThe names of standard 3-D shapesChapter 7: 2-D shapes

OBJECTIVESBy the end of the chapter the student should be able to: Count the vertices, faces and edges of 3-D shapes Draw nets of solids and recognise solids from their nets Draw and interpret plans and elevations Draw planes of symmetry in 3-D shapes Recognise and name examples of solids, including prisms, in the real world


DIFFERENTIATION AND EXTENSION 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 Work out how many small boxes can be packed into a larger box


HOMEWORKThis could include consolidation of work in class by completion of exercises set, additional work of a similar nature, or extension work detailed above.Sketch a plan view of your bedroom or an elevation of your house.

HINTS AND TIPS Accurate drawing skills need to be reinforced. Some students find visualising 3-D objects difficult – simple models will assist. Do Exercises 13A–13G for practice. Do Mixed Exercise 13 for consolidation.

FOUNDATION CHAPTER 14 UNITS OF MEASURE Time: 5–7 hours

SPECIFICATION REFERENCE Changing metric units of length, capacity and weight SSM4a Converting between metric and imperial units used in everyday situations SSM4a Questions involving the use of time in everyday situations SSM4a Questions involving the use of dates in everyday situations SSM4a Reading timetables and working out journey times SSM4a

PRIOR KNOWLEDGEChapter 9: Estimating and using measures

OBJECTIVESBy the end of the chapter the student should be able to: Change metric units of length, capacity and weight Convert between metric and imperial units used in everyday situations Do calculations involving time, including the use of time tables and calendars


DIFFERENTIATION AND EXTENSION Use ICT and reference books to find the weights, volumes and heights of large structures such as buildings, aeroplanes and

ships Work with more difficult examples Work with “real” timetables and “real” holiday brochures for working out holiday dates Use combinations of timetables for multi-stage trips

ASSESSMENT Heinemann online assessment tool will be available in 2007.Mental testing to check for knowledge of everyday measures, and estimation.Mental test questions on changing units, changing between metric and imperial units.Mental questions to test knowledge of common conversion factors.Aural questions on dates, times and timetables.

HOMEWORKThis could include consolidation of work in class by completion of exercises set, additional work of a similar nature, or extension work detailed above.Plan a foreign holiday with dates, timetables to arrive at the airport etc.

HINTS AND TIPS Measurement is essentially a practical activity. Use a range of everyday objects to make the lesson more real. All working should be shown with multiplication or division by powers of 10. Do Exercises 14A–14J for practice. Do Mixed Exercise 14 for consolidation.

FOUNDATION CHAPTER 15A PERCENTAGES 1 Time: 3–5 hours

SPECIFICATION REFERENCE Introducing percentages NA2e Writing percentages as decimals and fractions in their lowest terms NA2c/e/3c Calculating percentages of different amounts NA3c/m

PRIOR KNOWLEDGEFour operations of numberThe concepts of a fraction and a decimalNumber complements to 10 and multiplication tablesAwareness that percentages are used in everyday life

OBJECTIVESBy the end of the CHAPTER the student should be able to: Understand that a percentage is a fraction in hundredths Write a percentage as a decimal; or as a fraction in its simplest terms Write one number as a percentage of another number Calculate the percentage of a given amount


DIFFERENTIATION AND EXTENSION Fractional percentages of amounts Percentages which convert to recurring decimals, e.g. 33 %), and situations which lead to percentages of more than 100%

ASSESSMENT Heinemann online assessment tool will be available in 2007.Reinforce equivalence and the connection between percentage, fraction and decimal.Mental methods of calculating common percentages, e.g. 17.5% using 10%, 5%, 2.5%.

HOMEWORKThis could include consolidation of work in class by completion of exercises set, additional work of a similar nature, or extension work detailed above.Independent research into the many uses made of percentages, particularly in the media.

HINTS AND TIPS For non-calculator methods make sure that remainders and carrying are shown. In preparation for this unit students should be reminded of basic percentages and recognise their fraction and decimal

equivalents. Do Exercises 15A–15D for practice.

FOUNDATION CHAPTER 15B PERCENTAGES 2 Time: 4–6 hours

SPECIFICATION REFERENCE Adding a percentage of an amount NA3m Reducing an amount by a percentage NA3m Adding VAT on to an amount NA3m Using an index to look at increases and decreases over time NA2e/HD5k Writing one number as a percentage of another NA3e Using percentages to compare different fractions and decimals NA2e Solving numerical problems involving compound interest NA3m

PRIOR KNOWLEDGEFour operations of numberThe concepts of a fraction and a decimalNumber complements to 10 and multiplication tablesAwareness that percentages are used in everyday life

OBJECTIVESBy the end of the chapter the student should be able to: Find a percentage increase/decrease, of an amount Calculate simple and compound interest for two, or more, periods of time Calculate an index number


DIFFERENTIATION AND EXTENSION Combine multipliers to simplify a series of percentage changes Problems which lead to the necessity of rounding to the nearest penny, e.g. real-life contexts Comparisons between simple and compound interest calculations Formulae in simple interest/compound interest methods

ASSESSMENT Heinemann online assessment tool will be available in 2007.Reinforce equivalence and the connection between percentage, fraction and decimal.Mental methods of calculating common percentages, e.g. 17.5% using 10%, 5%, 2.5%.

HOMEWORKThis could include consolidation of work in class by completion of exercises set, additional work of a similar nature, or extension work detailed above.The construction of a VAT ready-reckoner table.

HINTS AND TIPS For non-calculator methods make sure that remainders and carrying are shown. In preparation for this unit students should be reminded of basic percentages and recognise their fraction and decimal

equivalents. Do Exercises 15E–15K for practice. Do Mixed Exercise 15 for consolidation.

FOUNDATION CHAPTER 16 COORDINATES AND GRAPHS Time: 5–7 hours

SPECIFICATION REFERENCE Using coordinates in the first quadrant to draw shapes and find the mid-point of a line segment NA6b/SSM3e Drawing linear graphs from tabulated data NA6c Using conversion graphs to convert from one measurement to another NA6c Reading distance–time graphs and calculating speed NA6c Drawing curved graphs for real-life situations NA6e Plotting coordinates that include negative numbers NA6b/SSM3e Drawing the graphs of straight line equations NA6b Using coordinates in 1-D, 2-D and 3-D SSM3e

PRIOR KNOWLEDGEExperience at plotting points in all quadrants

OBJECTIVESBy the end of the chapter the student should be able to: Draw linear graphs from tabulated data, including real-world examples Interpret linear graphs, including conversion graphs and distance-time graphs Understand the difference between a line and a line segment Solve graphically simultaneous equations, e.g. find when/where the car overtakes the bus Plot and reading coordinates on a coordinate grid (in all four quadrants) Understand that one coordinate identifies a point on a line, two coordinates identify a point in a plane and three coordinates

identify a point in space, and use the terms ‘1-D’, ‘2-D’ and ‘3-D’ Find the coordinates of the fourth vertex of a parallelogram Identify the coordinates of the vertex of a cuboid on a 3-D grid Writing down the coordinates of the midpoint of the line connecting two points


DIFFERENTIATION AND EXTENSION Plot graphs of the form y = mx + c where pupil has to generate their own table and set out their own axes Use a spreadsheet to generate straight-line graphs, posing questions about the gradient of lines More able students could extend to identifying regions relating to straight-line graphs Draw quadratic graphs Use a graphical calculator to draw straight-line graphs Find the coordinates of the point of intersection of two lines Shade regions in a linear inequality Find the coordinates of the point if intersection of the medians of a triangle, and explore further Identify the coordinates of the mid-point of a line segment in 3-D

ASSESSMENT Heinemann online assessment tool will be available in 2007.Test ability to join points up in consecutive order

HOMEWORKThis could include consolidation of work in class by completion of exercises set, additional work of a similar nature, or extension work detailed above.Drawing shapes using co-ordinates.Use data collected by students to draw linear graphs.Consolidation work using conversion graphs.

HINTS AND TIPS Clear presentation with axes labelled correctly is vital. Recognise linear graphs and hence when data may be incorrect. Link to graphs and relationships in other subject areas, i.e. science, geography etc. Coordinate axes should be labelled. Students often have difficulty visualising 3-D coordinates. Do Exercises 16A–16J for practice. Do Mixed Exercise 16 for consolidation.

FOUNDATION CHAPTER 17 RATIO AND PROPORTION Time: 4–6 hours

SPECIFICATION REFERENCE Examples of ratios used in everyday life NA2f Writing ratios in their lowest terms by cancelling NA2f Finding and using equivalent ratios NA2f/3n Ratios as 1:n or n:1 NA2f Sharing in a given ratio NA3f Examples of direct proportion NA3n Examples in which increasing one quantity causes another quantity to increase in the same ratio NA3n Using a scale to convert between map distances and ground distances NA2f

PRIOR KNOWLEDGEUsing the four operationsAbility to recognise common factorsKnowledge of fractions

OBJECTIVESBy the end of the chapter the student should be able to: Understand what is meant by ratio Write a ratio in its simplest form; and find an equivalent ratio Share a quantity in a given ratio Understand and use examples in direct proportion Interpret map/model scales as a ratio


DIFFERENTIATION AND EXTENSION Similar triangles Share quantities in a ratio involving fractions, and decimals Use a map to plan a journey, e.g. how long will the journey take travelling at average speed of 40 kmph (or mixed speeds for

different roads)


HOMEWORKThis could include consolidation of work in class by completion of exercises set, additional work of a similar nature, or extension work detailed above.Draw a map, e.g. of a room in your house.Use a map to work out a distance, e.g. the real distance between two exits on a motorway.Plan a journey on a map.

HINTS AND TIPS Students often cope well with ratios of two quantities. They have greater difficulty with ratios of three quantities and particular attention needs to be given to this. Do Exercises 17A–17H for practice. Do Mixed Exercise 17 for consolidation.

FOUNDATION CHAPTER 18 SYMMETRY Time: 2–4 hours

SPECIFICATION REFERENCE Finding lines of symmetry and completing shapes given their lines of symmetry SSM3a/3b Line symmetry in more complicated shapes SSM3a/3b Finding the order of rotational symmetry of various shapes SSM3a/3b Line symmetry and rotational symmetry of regular polygons SSM3a/3b

PRIOR KNOWLEDGEChapter 7: 2-D shapes

OBJECTIVES Identify reflection and rotation symmetry in 2-D shapes (including shapes with patterns) Complete a diagram given one (or two) lines of symmetry State the order of rotational symmetry For regular polygons, see the relationship between the order of rotational symmetry and the number of lines of symmetry


DIFFERENTIATION AND EXTENSION Generate snowflake patterns with paper and scissors Investigate symmetry in tessellations Investigate symmetry in the natural world



HINTS AND TIPSDo Exercises 18A–18D for practice. Do Mixed Exercise 18 for consolidation.

FOUNDATION CHAPTER 19 SIMPLE PERIMETER, AREA AND VOLUME Time: 7–9 hours

SPECIFICATION REFERENCE Perimeters of shapes made up of straight lines and circumferences of circles SSM4f Finding areas by counting squares SSM4f Finding volumes by counting cubes SSM4g Using formulae to find areas of rectangles, triangles and composite shapes SSM2e/4f Using the formulae for the volume of a cuboid SSM4g Finding the surface area of cuboids and prisms SSM4f Division of volumes SSM4g Solving problems involving area and volume SSM4f/g Changing from one unit to another SSM4a/4i Relationship between speed, time and distance and assorted questions SSM4c Maximising areas using a spreadsheet SSM4f/NA5f

PRIOR KNOWLEDGEChapter 7: 2-D shapesChapter 13: 3-D shapesChapter 14: Units of measureExperience of multiply by powers of 10, e.g. 100 100 = 10 000

OBJECTIVESBy the end of the chapter the student should be able to: Find the perimeters, areas and volumes of shapes made up from triangles and rectangles Find areas and volumes of shapes by counting squares Find the surface area of cuboids and prisms Solve a range of problems involving area and volume Convert between units of area Use the relationship between distance, speed and time to solve problems Convert between metric units of speed, e.g. km/h to m/s


DIFFERENTIATION AND EXTENSION Calculating areas and volumes using formulae Using compound shape methods to investigate areas of other standard shapes such as a trapezium or a kite Practical activities , e.g. using estimation and accurate measuring to calculate perimeters and areas of classroom/corridor

floors


HOMEWORKThis could include consolidation of work in class by completion of exercises set, additional work of a similar nature, or extension work detailed above.Find the perimeter and area of the floor of a room at home.A fencing problem, such as finding the smallest/largest area with a fixed perimeter.

HINTS AND TIPS Discuss the correct use of language and units. Ensure that pupils 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. Use a distance, speed, time triangle to help students see the relationship between the variables. Do Exercises 19A–19N for practice. Do Mixed Exercise 19 for consolidation.

FOUNDATION CHAPTER 20 PRESENTING AND ANALYSING DATA 1 Time: 4–6 hours

SPECIFICATION REFERENCE Finding the most common item in a set of data HD4b Finding the middle value in a set of data HD4b Finding the mean HD4b/j Using the range of a set of data to compare it with other sets of similar data HD4b/5ad The advantages/disadvantages of each kind of average HD4b/5a Drawing and using stem and leaf diagrams HD4a/5b Drawing and using pie charts HD4a/5b

PRIOR KNOWLEDGESome experience of the measures of averagesAbility to order numbersMeasuring and drawing anglesFractions of simple quantities

OBJECTIVESBy the end of the chapter the student should be able to: Find the mode, the median, the mean, and the range for (small) sets of data Use a stem and leaf diagram to sort data Know the advantages/disadvantages of using the different measure of average Represent categorical data in a pie chart Interpret categorical data in a pie chart


DIFFERENTIATION AND EXTENSION Collect data from class, e.g. children per family etc. Find measures of average for data collected in a frequency distribution Use stem and leaf diagrams with unusual stems, e.g. 234.1, 234.6, 235.1, ... Discuss occasions when one average is more appropriate, and the limitations of each average Compare distributions and making inferences, using the shapes of distributions and measures of average and spread, e.g.

‘boys are taller on average but there is a much greater spread in heights’

ASSESSMENT Heinemann online assessment tool will be available in 2007.A group work assessment through selected questions and mini-projects.Data handling project coursework task.

HOMEWORKThis could include consolidation of work in class by completion of exercises set, additional work of a similar nature, or extension work detailed above.Collect data at home for processing in class.

HINTS AND TIPS Students tend to select modal class but identify it by the frequency rather than the class description. Explain that the median of grouped data is not necessarily from the middle class interval. Accurate drawing skills need to be reinforced. Angles should be correct to within 2 degrees. Do Exercises 20A–20H for practice. Do Mixed Exercise 20 for consolidation.

FOUNDATION CHAPTER 21 FORMULAE AND INEQUALITES Time: 8–10 hours

SPECIFICATION REFERENCE Using words to state the relationship between different quantities NA5f Using letters to state the relationship between different quantities NA5f Substituting into algebraic formulae NA5f Addition, subtraction and multiplication using negative numbers NA3a Substituting into formulae which include powers of a variable NA5f Finding the solution to a problem by writing an equation and solving it NA5e Solving linear inequalities NA5d

PRIOR KNOWLEDGEUnderstanding of the mathematical meaning of the words: expression, simplifying, formulae and equationExperience of using letters to represent quantitiesSubstituting into simple expressions using wordsUsing brackets in numerical calculations and removing brackets in simple algebraic expressions

OBJECTIVESBy the end of the chapter the student should be able to: Use letters or words to state the relationship between different quantities Substitute positive and negative numbers into simple algebraic formulae Substitute positive and negative numbers into algebraic formulae involving powers Find the solution to a problem by writing an equation and solving it Solve linear inequalities in one variable and present the solution set on a number lineRESOURCESFoundation Student book Chapter/section: 21.1–21.7Foundation Practice book Chapter 21Teaching and Learning software Chapter 21

DIFFERENTIATION AND EXTENSION Use negative numbers in formulae involving indices Change the subject of a formula, e.g. given the formula to convert C to F, what is the formula to convert F to C? Develop algebraic skills in the Higher specification Various investigations leading to generalisations

ASSESSMENT Heinemann online assessment tool will be available in 2007.Discussion of situations that lead to formulae.Spreadsheet tasks such as ‘guess my rule’.

HOMEWORKThis could include consolidation of work in class by completion of exercises set, additional work of a similar nature, or extension work detailed above.Uses of algebra to describe real situation, e.g. n quadrilaterals have 4n sides.

HINTS AND TIPS Emphasis on good use of notation, e.g. 3ab means 3 a b. Students need to be clear on the meanings of the words expression, equation, word formulae and algebraic formulae as they

can find this confusing. Do Exercises 21A–21Q for practice. Do Mixed Exercise 21 for consolidation.

FOUNDATION CHAPTER 22 TRANSFORMATIONS Time: 2–4 hours

SPECIFICATION REFERENCE Understanding translation as a sliding movement with two components SSM3b Understanding that rotation is a turning motion about an angle and with a centre SSM3b Reflecting various shapes in mirror lines in different positions SSM3b Making various shapes larger using different centres of enlargement and scale factors SSM3c

PRIOR KNOWLEDGERecognition of basic shapesAn understanding of the concept of rotation, reflection and enlargement

OBJECTIVESBy the end of the chapter the student should be able to: Transform triangles and other shapes by translation, rotation and reflection Understand translation as a combination of a horizontal and vertical shift Understand rotation as a turn about a given origin Reflect shapes in a given mirror line Enlarge shapes by a given scale factor from a given point Distinguish properties that are preserved under transformations, e.g. write down the angles of a triangle that has been

enlarged


DIFFERENTIATION AND EXTENSIONThe tasks set can be extended to include combinations of transformations

ASSESSMENT Heinemann online assessment tool will be available in 2007.Practical poster work.


HINTS AND TIPS Diagrams should be drawn carefully. The use of tracing paper is allowed in the examination. Do Exercises 22A–22D for practice. Do Mixed Exercise 22 for consolidation.

FOUNDATION CHAPTER 23A PROBABILITY 1 Time: 2–4 hours

SPECIFICATION REFERENCE Comparing probabilities. How likely is it? HD4c/5g Finding a number to represent a probability HD4d Simple probability HD4c

PRIOR KNOWLEDGESome idea of chance and the likelihood of an event happening; and recognition that some events are more likely than othersExperience of using the language of likelihood

OBJECTIVESBy the end of the chapter the student should be able to: Use the language of probability to describe the likelihood of an event Represent and compare probabilities on a number scale List outcomes for single mutually exclusive events and write down their probability


DIFFERENTIATION AND EXTENSION Write down probabilities of events that may or may not happen Play simple probability games, predicting outcomes , horse race for sum of two dice



HINTS AND TIPS Where possible introduce practical work to support the theoretical work. Only fractions, decimals or percentages should be used for probability. Do Exercises 23A–23C for practice.

FOUNDATION CHAPTER 23B PROBABILITY 2 Time: 2–4 hours

SPECIFICATION REFERENCE Working out the probability that something will not happen if you know the probability that it will happen HD4f Estimating a probability from the results of an experiment HD4d/5i/j Using a diagram to record all the possibilities HD4e Using the information contained in a two-way table to find an estimate of probability HD3c/4d Estimating probability by relative frequency

HD4d/5g/h/i/j

PRIOR KNOWLEDGEChapter: 23A Probability 1Ability to read from a two-way table

OBJECTIVESBy the end of the chapter the student should be able to: Write down the theoretical probability for an equally likely event Estimate a probability by relative frequency Know that a better estimate for a probability is achieved by increasing the number of trials


DIFFERENTIATION AND EXTENSIONThe work can be extended to include that of the Higher specification



HINTS ANDTIPS Students can be unsure of the relationship P(not n) = 1 – P(n). Only fractions, decimals or percentages should be used for probability. Do Exercises 23D–23H for practice. Do Mixed Exercise 23 for consolidation.

FOUNDATION CHAPTER 24A PRESENTING AND ANALYSING DATA 2A Time: 2–4 hours

SPECIFICATION REFERENCEDrawing and using scatter diagrams and lines of best fit; correlation HD4a/h/5b/f

PRIOR KNOWLEDGEPlotting coordinatesAn understanding of the concept of a variableRecognition that a change in one variable can affect anotherLinear graphs

OBJECTIVESBy the end of the chapter the student should be able to: Draw and produce a scatter graph Appreciate that correlation is a measure of the strength of association between two variables Distinguish between positive, negative and zero correlation using a line of best fit Appreciate that zero correlation does not necessarily imply ‘no correlation’ but merely ‘no linear relationship’ Draw line of best fit by eye and understand what it represents Find the equation of the line of best fit and use it to interpolate/extrapolate

RESOURCESFoundation Student book Chapter/section: 24.1Foundation Practice book Chapter 24Teaching and Learning software Chapter 24

DIFFERENTIATION AND EXTENSION Vary the axes required on a scatter graph to suit the ability of the class. Carry out a statistical investigation of their own including; designing an appropriate means of gathering the data, and an

appropriate means of displaying the results. Use a spreadsheet, or other software, to produce scatter diagrams/lines of best fit. Investigate how the line of best fit is

affected by the choice of scales on the axes

ASSESSMENT Heinemann online assessment tool will be available in 2007.Test a given hypothesis either using data provided or by collecting data from the class.Their own statistical investigation.

HOMEWORKThis could include consolidation of work in class by completion of exercises set, additional work of a similar nature, or extension work detailed above.Completion of simple statistical project.

HINTS AND TIPS Students should realise that lines of best fit should have the same gradient as the correlation of the data. Clearly label all axes on graphs and use a ruler to draw straight lines. Do Exercises 24A for practice.

FOUNDATION CHAPTER 24B PRESENTING AND ANALYSING DATA 2B Time: 3–5 hours

SPECIFICATION REFERENCE Finding the mean, median and mode of a set of data; using appropriate averages HD4b Finding the mean, mode and median from frequency tables HD4g/j Finding the range and interquartile range from a frequency table HD4b Finding the mean, mode, median from grouped frequency tables HD4g

PRIOR KNOWLEDGEChapter 20: Presenting and analysing data 1Finding the average of two number, i.e. the midpoint

OBJECTIVESBy the end of the chapter the student should be able to: Identify the modal class interval in grouped and ungrouped frequency distributions Find the class interval containing the median value Find the mean of an ungrouped frequency distribution Find an estimate for the mean of a grouped frequency distribution by using the mid-interval value Use the statistical functions on a calculator or a spreadsheet to calculate the mean for discrete data


DIFFERENTIATION AND EXTENSION Find the mean for grouped and continuous data with unequal class intervals Collect continuous data and decide on appropriate (equal) class intervals; then find measures of average Find the median by cumulative frequency diagram Consider other measures of spread, e.g. interquartile range; appreciate advantages/limitations of the range Use the statistical functions on a calculator or a spreadsheet to calculate the mean for continuous data

ASSESSMENT Heinemann online assessment tool will be available in 2007.Test a given hypothesis either using data provided or by collecting data from the class.GCSE coursework.

HOMEWORKThis could include consolidation of work in class by completion of exercises set, additional work of a similar nature, or extension work detailed above.Completion of simple statistical project.

HINTS AND TIPS Students should be aware that the actual mean can not be calculated from a grouped frequency distribution; and that using

the midpoint of the class intervals gives the best estimate for the mean. The modal class is found for grouped frequency distributions in which the class intervals have an equal. Do Exercises 24B–24E for practice. Do Mixed Exercise 24 for consolidation.

FOUNDATION CHAPTER 25 PYTHAGORAS’ THEOREM Time: 3–5 hours

SPECIFICATION REFERENCE Using Pythagoras’ theorem to calculate the length of the hypotenuse SSM2h/NA4d Using Pythagoras’ theorem to find one of the shorter sides of a right-angle triangle SSM2h/NA4d

PRIOR KNOWLEDGERecognise right-angled triangles in different orientationsBe able to square a number with and without a calculatorBe able to rearrange an equationBe able to use a calculator to taker the square root of a numberGive an answer to one decimal place

OBJECTIVES Find the length of the missing side in a right-angled triangle (only 2-D examples) Recall and use Pythagoras’ theorem


DIFFERENTIATION AND EXTENSION Find the distance of the line segment between two given points on a coordinate grid Use Pythagoras’ theorem in multi-step questions involving two right-angled triangles



HINTS AND TIPS Students should be encouraged to learn both a linguistic and a mathematical statement of Pythagoras’ theorem. Do Exercises 25A–25C for practice. Do Mixed Exercise 25 for consolidation.

FOUNDATION CHAPTER 26A ADVANCED PERIMETER, AREA AND VOLUME 1 Time: 7–9 hours

SPECIFICATION REFERENCE Perimeter and area of triangles and quadrilaterals SSM4f Introducing pi and the formula for the circumference of a circle SSM2i/4h Developing the formula for the area of a circle SSM4h Compound shapes made from circle parts, e.g. semicircles, quadrants SSM2i/4h Surface areas and volumes of prisms SSM4f/g

PRIOR KNOWLEDGEThe ability to substitute numbers into formulae

OBJECTIVESBy the end of the chapter the student should be able to: Use formulae to find the surface area of shapes made up of rectangles and triangles Solve problems involving the circumference and area of a circle (and simple fractional parts of a circle) Solve problems involving the volume of a cylinder Find exact answers by leaving answers in terms of pi Understand a formula by considering its dimensions, e.g. identify formulae that represent area from a list


DIFFERENTIATION AND EXTENSION Use more complex 2-D shapes, e.g. (harder) sectors of circles Use more complex 3-D shapes, e.g. half-cylinders Surface area of cylinder/half-cylinder Approximate pi as 22

7 Consider the dimensions of harder formulae, e.g. the surface area of a cone (including base)


HOMEWORKThis could include consolidation of work in class by completion of exercises set, additional work of a similar nature, or extension work detailed above.Need to constantly revise the expressions for area/volume of shapes.

HINTS AND TIPS Need to constantly revise the expressions for area/volume of shapes. Do Exercises 26A–26M for practice.

FOUNDATION CHAPTER 26B ADVANCED PERIMETER, AREA AND VOLUME 2 Time: 3–5 hours

SPECIFICATION REFERENCE Degrees of accuracy of given measurements SSM4a Using sensible units for various measurements SSM4a Using speed and density formulae SSM4c Converting between units of speed SSM4a Changing units as necessary SSM4a

PRIOR KNOWLEDGEKnowledge of metric units, e.g. 1 m = 100 cm, etc.Know that 1 hour = 60 mins, 1 min = 60 secondsExperience of multiply by powers of 10, e.g.. 100 100 = 10 000

OBJECTIVESBy the end of the chapter the student should be able to: Use the relationship between distance, speed and time to solve problems Convert between metric units of speed, e.g. km/h to m/s Know that density is found by mass ÷ volume Use the relationship between density, mass and volume to solve problems Convert between metric units of density, e.g. kg/m to g/cm


DIFFERENTIATION AND EXTENSION Perform calculations on a calculator by using standard form Convert imperial units to metric units, e.g. mph into kmph



HINTS AND TIPS Use a distance, speed and time triangle to help students see the relationship between the variables. Do Exercises 26N–26S for practice. Do Mixed Exercise 26 for consolidation.

FOUNDATION CHAPTER 27 DESCRIBING TRANSFORMATIONS Time: 4 –6 hours

SPECIFICATION REFERENCE Describing translations using column vectors SSM3a/b/f Describing a reflection by giving the equation of a mirror line SSM3a/b Describing a rotation by giving the angle, the direction and the centre of rotation SSM3a/b Describing an enlargement by giving the scale factor and centre SSM3c/3d

PRIOR KNOWLEDGECoordinates in four quadrantsLinear equations parallel to the coordinate axes


OBJECTIVESBy the end of the chapter the student should be able to: Transform triangles and other shapes by translation, rotation and reflection including combinations of transformations Understand translation as a combination of a horizontal and vertical shift including vector notation Understand rotation as a clockwise turn about a given origin Reflect shapes in a given mirror line. Initially line parallel to the coordinate axes and then y = x or y = –x Enlarge shapes by a given scale factor from a given point; using positive whole number scale factors, then positive fractional

scale factors Distinguish properties that are preserved under transformations, e.g. write down the angles of a triangle that has been

enlarged

DIFFERENTIATION AND EXTENSIONThe tasks set can be extended to include combinations of transformations



HINTS AND TIPS Emphasis needs to be placed on ensuring that students do describe the given transformation fully. Do Exercises 27A–27F for practice. Do Mixed Exercise 27 for consolidation.

FOUNDATION CHAPTER 28 EXPRESSING, FORMULAE, EQUATIONS AND GRAPHS Time: 8–10 hours

SPECIFICATION REFERENCE Multiplying out brackets NA5b Graphs of functions y = ax2 + c where a and c can be positive or negative NA6e Graphs of functions y = ax2 + bx + c NA6e Using graphs to solve equations such as ax2 + bx + c = d NA6e Assorted situations treated graphically NA6d Changing the subject of a formula NA5f Solving equations of the form ax2 + bx = c or a/x = b NA5e Solving cubic equations using trial and improvement NA5m Using a spreadsheet to solve cubic equations by trail and improvement NA5m

PRIOR KNOWLEDGESubstituting numbers into algebraic expressionsPlotting points on a coordinate gridExperience of dealing with algebraic expression with one pair of bracketsKnow , where a and b are integersSubstituting numbers into algebraic expressionsDealing with decimals on a calculatorOrdering decimals

OBJECTIVESBy the end of the chapter the student should be able to: Substitute values of x into a quadratic function to find the corresponding values of y Draw graphs of quadratic functions Use quadratic graphs to solve quadratic equations Solve quadratic and reciprocal functions Expand and simplify expressions with two pairs of brackets Solve cubic functions by successive substitution of values of x


DIFFERENTIATION AND EXTENSION Draw simple cubic graphs Solve graphically simultaneous equations involving a quadratic graph and a line

Solve functions of the form



HINTS AND TIPS The graphs of quadratic functions should be drawn freehand; and in pencil. Turning the paper often helps. Squaring negative integers may be a problem for some. Students will often forget the middle term of the expansion- they will need to be reminded later. Students should be encouraged to use their calculators efficiently- by using the ‘replay’ function. The cube function on a calculator may not be the same for different makes. Students should write down all the digits on their calculator display. Do Exercises 28A–28M for practice. Do Mixed Exercise 28 for consolidation.

foundation module 1 · web viewtime: 7–9 hours. specification reference. understanding place...

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