HIGHER UNIT 3 (OLD UNIT 4) SCHEMES OF WORK
HIGHER CHAPTER 1
Time: 3 – 5 hours
CALCULATIONS
SPECIFICATION REFERENCE
Finding reciprocals of whole numbers, fractions and decimals
Using the rules with negative numbers
Using the rules of indices to multiply and divide numbers in index form
Using the 4 rules with fractions
Converting fractions to decimals and decimals to fractions; recurring decimals
NA3a
NA3a
NA2b/3a
NA3c/d
NA3c
PRIOR KNOWLEDGE
Heinemann GCSE Modular Mathematics Unit 2 (old Unit 3) Higher Chapter 1: Integers and powers
Heinemann GCSE Modular Mathematics Unit 2 (old Unit 3) Higher Chapter 2: Fractions, decimals and percentages
Heinemann GCSE Modular Mathematics Unit 2 (old Unit 3) Higher Chapter 3: Index notation and standard form
OBJECTIVES
By the end of the unit the student should be able to:
Perform multiplications and divisions of whole numbers and decimal numbers without a calculator
4 2
,
7 x
-
Find the reciprocal of, e.g. 0.75,
-
Multiply and divide positive and negative integers

-
Use the three index laws to evaluate numbers, e.g. find the value of
Add, subtract, multiply and divide fractions
Convert fractions to decimals (including recurring fractions).
81
3
4
DIFFERENTIATION AND EXTENSION
Use the three index laws with fractions and/or fractional powers.
Change a recurring decimal to an exact fraction.
Multiply/divide whole numbers (no calculator) where a conversion is required, e.g. speed.
RESOURCES
Heinemann GCSE Modular Mathematics Unit 3 (old Unit 4) Higher Chapter/section: 1.1 – 1.6
See the Teaching and Learning Software for activities linking to this chapter.
ASSESSMENT ISSUES
Written tests and homework to assess knowledge of content
Mental tests using simple fraction, e.g. the reciprocal of 2 times three quarters
Extra examples given regularly for revision purposes
HOMEWORK
Homework at each stage could comprise 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 work clearly with decimals points in line and/or 0 as a place holder.
Make sure remainders/carries are shown clearly and in the right place.
Use Exercises 1A – 1F for practice. Use Mixed Exercise 1 for consolidation.
HIGHER UNIT 3 (OLD UNIT 4) SCHEMES OF WORK
HIGHER CHAPTER 2
PERCENTAGES
SPECIFICATION REFERENCE
Converting between percentages, fractions and decimals
Percentage of amounts
Finding increases and decreases as a percentage
Finding the original amount
Finding one number as a percentage of another
Calculating compound interest and depreciation
Time: 3 – 5 hours
NA3c
NA2e/3e
NA2e/3f
NA3f
NA2e/3j
NA3k/4a
PRIOR KNOWLEDGE
Heinemann GCSE Modular Mathematics Unit 2 (old Unit 3) Higher Chapter 2: Fractions, decimals and percentages
OBJECTIVES
By the end of the unit the student should be able to:
Convert between percentages, fractions and decimals
Find the percentage of an amount, e.g. 34% of £120
Increase/decrease an amount by a given percentage (including multipliers), e.g. 5% increase =
Find the original amount, given the amount after a percentage increase/decrease
Find one number as a percentage of another number, e.g. percentage profit
Calculate compound interest and depreciation.
1.05
DIFFERENTIATION AND EXTENSION
Fractional percentages of amounts (non-calculator), e.g. VAT
Combine multipliers to simplify a series of percentage changes.
-
Percentages which convert to recurring decimals (e.g. 33 13 %), and situations which lead to percentages of more than 100%.
-
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
RESOURCES
Heinemann GCSE Modular Mathematics Unit 3 (old Unit 4) Higher Chapter/section: 2.1 – 2.6
See the Teaching and Learning Software for activities linking to this chapter.
ASSESSMENT ISSUES
Written tests and homework to assess knowledge of content
Mental tests using simple percentages, e.g. increase £21 by 33 13 %
Extra examples given regularly for revision purposes
HOMEWORK
Homework at each stage could comprise consolidation of work in class by completion of exercises set, additional work of a similar
nature, or extension work detailed above.
HINTS AND TIPS
Amounts of money should always be rounded to the nearest penny where necessary, except where such rounding is
premature (e.g. in successive calculations of compound interest).
In preparation for this chapter students should be reminded of basic percentages and recognise their fraction and decimal
equivalents.
Use Exercises 2A – 2E for practice. Use Mixed Exercise 2 for consolidation.
HIGHER UNIT 3 (OLD UNIT 4) SCHEMES OF WORK
HIGHER CHAPTER 3
RATIO AND PROPORTION
Time: 2 – 4 hours
SPECIFICATION REFERENCE
Writing ratios in their lowest terms by canceling
Changing a ratio into a fraction
Sharing quantities in a given ratio
Solving problems using direct and inverse proportion
Using a multiplier raised to a power
Finding the rule connecting quantities using ratio
NA2f/3f/n
NA2f/3e/f/n
NA2f/3f/n/4a
NA3l/4a/5h
NA3k
NA5h
PRIOR KNOWLEDGE
Heinemann GCSE Modular Mathematics Unit 2 (old Unit 3) Higher Chapter 2: Fractions decimals percentages
OBJECTIVES
By the end of the unit the student should be able to:
Simplify a ratio by dividing common factors, e.g. simplify 3: 27
Express a ratio as a fraction and divide an amount in a given ratio (includes a : b : c)
Solve problems involving direct proportion and indirect proportion
Solve problems involving repeated proportional change (constant changes)
Find the rule for direct/indirect proportional changes, e.g. a  b, so a = kb.
DIFFERENTIATION AND EXTENSION
Problems involving other types of proportionality, e.g. a  b2
Treat direct/indirect proportions as functions. Draw/interpret graphs.
Use fractions in ratios.
RESOURCES
Heinemann GCSE Modular Mathematics Unit 3 (old Unit 4) Higher Chapter/section: 3.1 – 3.7
See the Teaching and Learning Software for activities linking to this chapter.
ASSESSMENT ISSUES
Written tests and homework to assess knowledge of content
Mental tests using simple percentages, e.g. 6 books cost £30, how much will 8 cost?
HOMEWORK
Homework at each stage could comprise consolidation of work in class by completion of exercises set, additional work of a similar
nature, or extension work detailed above.
HINTS AND TIPS
Work involving a = kb should be set out clearly.
Some students may find using a : b : c difficult.
Use Exercises 3A – 3G for practice. Use Mixed Exercise 3 for consolidation.
HIGHER UNIT 3 (OLD UNIT 4) SCHEMES OF WORK
HIGHER CHAPTER 4
Time: 3 – 5 hours
POWERS AND SURDS
SPECIFICATION REFERENCE
Calculating in standard form
Understanding and using yx and x√
Exploring growth and decay
Using surds and π to give exact answers
Manipulating expressions with surds
Finding bounds of numbers expressed to a given degree of accuracy
Calculating the bounds in problems
Na3m
NA2b
NA3t
NA3n
NA3n
NA3q
NA3q
PRIOR KNOWLEDGE
Heinemann GCSE Modular Mathematics Unit 2 (old Unit 3) Higher Chapter 3: Index notation and standard form
OBJECTIVES
By the end of the unit the student should be able to:
Multiply and divide numbers in standard form
2.32  4.5
-
Use a calculator to evaluate powers and roots of numbers, e.g. Work out the value of
-
Solve problems involving exponential growth/decay, e.g. growth of bacteria
-
Multiply and divide surds, e.g.
-
Find the area/perimeter of circle, or parts of circles, leaving the answer in terms of 
Understand upper and lower bounds of measurements made in real-life situations
Calculate the bounds in problems involving addition, multiplication, subtraction or division.

3  5  15 ; express 2  3
 in the form a  b
100  3.13
2
3
DIFFERENTIATION AND EXTENSION
Use standard form in real-life contexts, e.g. distances of planets in astronomical units.
-
1
More complex use of calculators with multi-stage calculations, e.g. evaluate
1
-
1
1  ...
Draw graphs of exponential growth/decay. Discuss the limitations of exponential growth/decay models.
Calculate bounds for more complex problems, e.g. find bounds for the volume if the surface area is to given accuracy.
RESOURCES
Heinemann GCSE Modular Mathematics Unit 3 (old Unit 4) Higher Chapter/section: 4.1 – 4.7
See the Teaching and Learning Software for activities linking to this chapter.
ASSESSMENT ISSUES
Written tests and homework to assess knowledge of content
HOMEWORK
Homework at each stage could comprise consolidation of work in class by completion of exercises set, additional work of a similar nature, or extension work detailed above.
HINTS AND TIPS
Some makes of calculator may require the extensive use of brackets in calculations.
Use Exercises 4A – 4G for practice. Use Mixed Exercise 4 for consolidation.
HIGHER UNIT 3 (OLD UNIT 4) SCHEMES OF WORK
HIGHER CHAPTER 5
SIMPLE LINEAR EQUATIONS AND ALGEBRA
Time: 4 – 6 hours
SPECIFICATION REFERENCE
Solving linear equations by balancing
Using the terms expressions, equation and identity
Setting up and solving equations
Solving equations expressed as algebraic fractions
Using indices notation
Substituting values into expressions
NA5e/f
NA5c
NA5a/e/f
NA5f
NA 3g/5d
NA5g
PRIOR KNOWLEDGE
Heinemann GCSE Modular Mathematics Unit 2 (old Unit 3) Higher Chapter 4: Essential algebra
OBJECTIVES
By the end of the unit the student should be able to:
1  3x   x  5
-
Solve linear equations both with and without brackets, e.g. 2
-
Understand the difference between expression, equation and identity
Generate equations from practical situations and solve them
-
Solve harder equations involving fractions, e.g.
-
Use the 3 index laws with algebra, e.g. simplify
-
Substitute positive and negative numbers into algebraic expressions (including fractions).
x 1 2x 1
2

 1,
3
2
3
3 x
 2x 
3 4
(including zero and negative indices)
DIFFERENTIATION AND EXTENSION
2
 9  7  x2
-
Solve simple quadratic equations, e.g. solve 2 x
-
Use the index laws with fractions, e.g. simplify
-
Substitute positive and negative numbers into practical functions, e.g. s  ut  0.5at
Investigate the range of values that may be substituted into calculator functions, e.g. yx, sin x, ln x.
Investigate the formula E = mc2.
 9x 
4
1
2
2
RESOURCES
Heinemann GCSE Modular Mathematics Unit 3 (old Unit 4) Higher Chapter/section: 5.1 – 5.10
See the Teaching and Learning Software for activities linking to this chapter.
ASSESSMENT ISSUES
Written tests and homework to assess knowledge of content
Mental tests using index laws
HOMEWORK
Homework at each stage could comprise consolidation of work in class by completion of exercises set, additional work of a similar
nature, or extension work detailed above.
HINTS AND TIPS
Revise basic work with fractions/indices before dealing with algebraic fractions/indices.
Show all stages when solving equations.
Use Exercises 5A – 5J for practice. Use Mixed Exercise 5 for consolidation.
HIGHER UNIT 3 (OLD UNIT 4) SCHEMES OF WORK
HIGHER CHAPTER 6
Time: 4 – 6 hours
FORMULAE
SPECIFICATION REFERENCE
Deriving a formula
Evaluating formulae
Finding the value of a variable that is not the subject of the formula
Rearranging a formula to change the subject
Combining formulae
NA5g
NA5g
NA5g
NA5g
NA5g
PRIOR KNOWLEDGE
Heinemann GCSE Modular Mathematics Unit 3 (old Unit 4) Higher Chapter 5: Simple linear equations and algebra
OBJECTIVES
By the end of the unit the student should be able to:
Write a formula in words and/or symbols, e.g. A = x, where A is the area of a square of side x
Substitute positive and negative numbers into formulae (including fractions)
Find the value of a variable that is not the subject of the formula
-
Change the subject of a formula (several steps may be required), e.g. make x the subject of g
-
Substitute one formula into another, e.g. x = 2  3t, y = 2x + 1, find y in term of t.
DIFFERENTIATION AND EXTENSION
vu
t
in
s  ut  0.5at 2
 2t  x 2
-
Harder examples, e.g. replace a by
and simplify.
-
Draw graphs of curves with simple parametric equations, e.g. x = t2, y = t – 2, plot (x, y) for different t.
RESOURCES
Heinemann GCSE Modular Mathematics Unit 3 (old Unit 4) Higher Chapter/section: 6.1 – 6.5
See the Teaching and Learning Software for activities linking to this chapter.
ASSESSMENT ISSUES
Written tests and homework to assess knowledge of content
HOMEWORK
Homework at each stage could comprise consolidation of work in class by completion of exercises set, additional work of a similar
nature, or extension work detailed above.
HINTS AND TIPS
Students often find substituting negative numbers difficult.
-
Some calculators evaluate 2 as 4, brackets may be required.
Use Exercises 6A – 6E for practice. Use Mixed Exercise 6 for consolidation.
2
HIGHER UNIT 3 (OLD UNIT 4) SCHEMES OF WORK
HIGHER CHAPTER 7
LINEAR AND REAL-LIFE GRAPHS
SPECIFICATION REFERENCE
Understanding gradient m and y intercept c
Finding the gradient and equation of a line perpendicular to a given line
Using linear graphs in practical applications
Drawing and interpreting graphs, both accurate and sketched
Time: 3 – 5 hours
NA6c/d
NA6c
NA6c/d
NA6d
PRIOR KNOWLEDGE
Heinemann GCSE Modular Mathematics Unit 2 (old Unit 3) Higher Chapter 5: Coordinates and graphs
OBJECTIVES
By the end of the unit the student should be able to:
Find the equation of a straight line from its graph
Rearrange an equation of a straight line, e.g. 3 x  2 y  7 to find the gradient and intercept
Understand that the graphs of linear functions are parallel if they have the same value of m
Know that the line perpendicular to y = mx + c has gradient 1/m
Find an equation for a line perpendicular to a given line and through a given point
Find an equation for the line of best fit
Interpret real-life graphs, e.g. find the speed from a distance-time graph.
DIFFERENTIATION AND EXTENSION
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.
Find the area under a simple velocity-time graph (i.e. distance traveled).
y  y1  m  x  x1 
-
Derive general results, e.g.
-
Investigate relationships of the form
y  ax 2  b
RESOURCES
Heinemann GCSE Modular Mathematics Unit 3 (old Unit 4) Higher Chapter/section: 7.1 – 7.4
See the Teaching and Learning Software for activities linking to this chapter.
ASSESSMENT ISSUES
Written tests and homework to assess knowledge of content
HOMEWORK
Homework at each stage could comprise consolidation of work in class by completion of exercises set, additional work of a similar
nature, or extension work detailed above.
HINTS AND TIPS
The values of m and c will be given.
Careful annotation should be encouraged. Label the coordinate axes and write the equation of the line.
Use Exercises 7A – 7D for practice. Use Mixed Exercise 7 for consolidation.
HIGHER UNIT 3 (OLD UNIT 4) SCHEMES OF WORK
Time: 4 – 6 hours
HIGHER CHAPTER 8 SOLVING EQUATIONS AND INEQUALITIES
SPECIFICATION REFERENCE
Solving simultaneous equations by finding the point of intersection of their graphs
Solving simultaneous equations using algebraic methods
Solving quadratic equations by rearranging and factorising
Completing the square for a quadratic expression
Using the quadratic formula to solve quadratic equations
Representing inequalities on a number line
Solving inequalities in one variable
Using trial and improvement to solve problems
NA5i
NA5i
NA5k
NA5k
NA5k
NA5j
NA5j
NA5m
PRIOR KNOWLEDGE
Heinemann GCSE Modular Mathematics Unit 2 (old Unit 3) Higher Chapter 5: Coordinates and graphs
Heinemann GCSE Modular Mathematics Unit 3 (old Unit 4) Higher Chapter 7: Linear and real-life graphs
OBJECTIVES
By the end of the unit the student should be able to:
Solve linear simultaneous equations graphically and algebraically
Factorise and solve a quadratic equation
Complete the square of a quadratic equation and solve it
Solve a quadratic equation by using the formula
Solve linear inequalities in one variable, e.g. solve 5x  2  3x  5 , write down all the integer values of x which
-
satisfy 1  2 x  9
Represent inequalities in two variables on a coordinate grid (including dotted lines)
-
Solve equations by trial and improvement to a given degree of accuracy, e.g. solve x
2

1
 10
x
DIFFERENTIATION AND EXTENSION
Find the coordinates of the point of intersection of the medians of a triangle and explore further.
Find a graphical solution to problems involving lines and quadratic functions.
-
 bx  c  0
x  y2  4 , y2  x
Derive the quadratic formula by completing the square of ax
2
2
-
Represent non-linear regions on a coordinate grid, e.g.
-
Solve harder problems by trial and improvement using calculator functions, e.g. solve e
x
x2
RESOURCES
Heinemann GCSE Modular Mathematics Unit 3 (old Unit 4) Higher Chapter/section: 8.1 – 8.13
See the Teaching and Learning Software for activities linking to this chapter.
ASSESSMENT ISSUES
Written tests and homework to assess knowledge of content
HOMEWORK
Homework at each stage could comprise consolidation of work in class by completion of exercises set, additional work of a similar
nature, or extension work detailed above.
HINTS AND TIPS
Show all the figures on the calculator display when solving by trial and improvement.
Correct use of notation is essential.
Use Exercises 8A – 8M for practice. Use Mixed Exercise 8 for consolidation.
HIGHER UNIT 3 (OLD UNIT 4) SCHEMES OF WORK
HIGHER CHAPTER 9
Time: 4 – 6 hours
QUADRATIC GRAPHS
SPECIFICATION REFERENCE
Drawing graphs of quadratic functions
Rearranging equations and solving graphically
Solving linear equation and a quadratic equation functions graphically
Using algebra to find the points of intersection of a line and a circle
NA6e
NA6e/f
NA6e
NA5l/6e/h
PRIOR KNOWLEDGE
Heinemann GCSE Modular Mathematics Unit 3 (old Unit 4) Higher Chapter 8: Solving equations and inequalities
OBJECTIVES
By the end of the unit the student should be able to:
Plot and draw graphs of quadratic functions
Solve quadratic functions graphically
Solve linear and quadratic simultaneous equations graphically and algebraically
Solve simultaneous equations involving lines and circles graphically and algebraically.
DIFFERENTIATION AND EXTENSION
Solve simultaneous equations involving circles and quadratic functions.
Use a graphic calculator to find the points of intersection of curves and lines.
-
Solve simultaneous equations involving other standard functions, e.g.
y
1
, y  x 1
x
RESOURCES
Heinemann GCSE Modular Mathematics Unit 3 (old Unit 4) Higher Chapter/section: 9.1 – 9.6
See the Teaching and Learning Software for activities linking to this chapter.
ASSESSMENT ISSUES
Written tests and homework to assess knowledge of content
HOMEWORK
Homework at each stage could comprise consolidation of work in class by completion of exercises set, additional work of a similar
nature, or extension work detailed above.
HINTS AND TIPS
Factorisation should always be attempted before using the quadratic formula.
The accuracy of graphical solutions will be affected by the quality of the graphs.
Algebraic solutions should be set out clearly.
Use Exercises 9A – 9F for practice. Use Mixed Exercise 9 for consolidation.
HIGHER UNIT 3 (OLD UNIT 4) SCHEMES OF WORK
Time: 3 – 5 hours
HIGHER CHAPTER 10 GRAPHS AND TRANSFORMATIONS OF GRAPHS
SPECIFICATION REFERENCE
Drawing graphs of cubic and reciprocal functions
Setting up equations to solve problems involving direct/inverse proportion
Drawing graphs of exponential and trigonometric functions
Transforming graphs of functions
NA6f
NA5h
NA6f
NA6g
PRIOR KNOWLEDGE
Heinemann GCSE Modular Mathematics Unit 3 (old Unit 4) Higher Chapter 9: Quadratic equations
OBJECTIVES
By the end of the unit the student should be able to:
Plot and draw graphs of cubic and reciprocal functions
-
Use given information to solve problems involving direct/inverse proportion, e.g.
-
Plot and draw graphs of exponential and trigonometric functions
-
Find the value of p and the value of q in y  pq given two points on the curve
Translate and reflect curves, e.g. given y = f(x), sketch y = f(x – 2) + 3
Understand the definitions for odd and even functions.
y  x, y  x , y 
1
x2
x
DIFFERENTIATION AND EXTENSION
Investigate transformations of the form y = f(ax) and y = af(x)
Investigate the transformation of a circle.
Complete the square of a quadratic function and relate this to transformations.
Investigate curves which are unaffected by particular transformations.
Use a graphic calculator to investigate transformations.
Investigate simple trigonometric relationships, e.g. sin(180 – x) = sinx, sin(90 – x) = cosx
RESOURCES
Heinemann GCSE Modular Mathematics Unit 3 (old Unit 4) Higher Chapter/section: 10.1 – 10.4
See the Teaching and Learning Software for activities linking to this chapter.
ASSESSMENT ISSUES
Written tests and homework to assess knowledge of content
Written investigation
HOMEWORK
Homework at each stage could comprise consolidation of work in class by completion of exercises set, additional work of a similar
nature, or extension work detailed above.
HINTS AND TIPS
Graphic calculators and/or dedicated computer software will underpin the main ideas
Use Exercises 10A – 10D for practice. Use Mixed Exercise 10 for consolidation.
HIGHER UNIT 3 (OLD UNIT 4) SCHEMES OF WORK
HIGHER CHAPTER 11
TRANSFORMATION
Time: 3 – 5 hours
SPECIFICATION REFERENCE
Translating, reflecting, rotating and enlarging shapes
Combining two transformations
SSM3a/b/c/d/f
SSM3b
PRIOR KNOWLEDGE
Heinemann GCSE Modular Mathematics Unit 2 (old Unit 3) Higher Chapter 7: Properties of shapes
OBJECTIVES
By the end of the unit the student should be able to:
Translate shapes on a 2-D grid (including translation vector)
Reflect shapes on a 2-D grid, e.g. reflect a triangle in the line y = x (includes planes of symmetry)
Rotate shapes on a 2-D grid, e.g. rotate trapezium by +90, centre (2, 1)
Enlarge shapes on a 2-D grid, e.g. enlarge rectangle scale factor 1.5, centre (3, 0)
Describe the single transformation that replaces two others.
DIFFERENTIATION AND EXTENSION
Relate enlargement scale factors to area/volume scale factors.
Classify 3-D shapes by their planes of symmetry.
Design a wall paper pattern. Use transformations to describe the patterns used.
Introduce vector addition by combining translation vectors.
RESOURCES
Heinemann GCSE Modular Mathematics Unit 3 (old Unit 4) Higher Chapter/section: 11.1 – 11.5
See the Teaching and Learning Software for activities linking to this chapter.
ASSESSMENT ISSUES
Written tests and homework to assess knowledge of content
HOMEWORK
Homework at each stage could comprise consolidation of work in class by completion of exercises set, additional work of a similar
nature, or extension work detailed above.
Hints and Tips
Transformations need to be described fully, e.g. rotation by –90 centre (0, 0).
Students may use tracing paper in the examination.
Use Exercises 11A – 11E for practice. Use Mixed Exercise 11 for consolidation.
HIGHER UNIT 3 (OLD UNIT 4) SCHEMES OF WORK
HIGHER CHAPTER 12 BEARINGS, CONSTRUCTIONS AND PROPERTIES OF SHAPES
SPECIFICATION REFERENCE
Drawing diagrams and calculating bearings
Drawing and interpreting scale diagrams and maps
Using compasses to construct triangles, perpendiculars and bisectors
Constructing loci
Using nets of 3-D shapes
Representing shapes on an isometric grid
Using plans and elevations of 3-D shapes
Using angle properties of polygons
Identifying similar shapes; scale factor
Identifying congruent triangles
Proving congruence using formal arguments
Time: 6 – 8 hours
SSM4a
SSM3d
SSM2h/4c/d
SSM4e
SSM2i
SSM2i
SSM2i
SSM2a/b/c/d
SSM2g/3c/d
SSM2e/4b
SSM2e
PRIOR KNOWLEDGE
Heinemann GCSE Modular Mathematics Unit 2 (old Unit 3) Higher Chapter 7: Properties of shapes
OBJECTIVES
By the end of the unit the student should be able to:
Measure and write down a bearing
Use a scale to find the distance on/from a map, e.g. 1:50 000
Use ruler and compass to construct an accurate triangle from given information
Construct: the perpendicular bisector; the perpendicular from a point to a line; the perpendicular from a point on a line; the
angle bisector
Find the locus of points equidistant from: one point; two points; a line segments
Construct/draw/sketch accurate nets for 3-D shapes
Represent a 3-D shape on an isometric grid
Draw/use plans and elevations of 3-D shapes
Find the interior/exterior angle of a regular polygon
 n  2 180
-
Find a missing angle in a polygon, e.g. by using
-
Use scale factor to find missing lengths in congruent shapes (2-D shapes)
Identify congruent triangles formally (using SAS, ASA, SSS and RHS)
Use formal arguments to prove congruence.
DIFFERENTIATION AND EXTENSION
Solve loci problem that require a combination of loci.
Draw complex 3-D shapes on isometric grids.
Use accurate diagrams to solve practical problems, e.g. lighthouse, port and ship.
More complex loci, e.g. the locus of a point which is equidistant from a point and a line
Use scale factors to find missing lengths in 3-D shapes.
Further problems involving formal arguments to prove congruence.
RESOURCES
Heinemann GCSE Modular Mathematics Unit 3 (old Unit 4) Higher Chapter/section: 12.1 – 12.11
See the Teaching and Learning Software for activities linking to this chapter.
ASSESSMENT ISSUES
Written tests and homework to assess knowledge of content
Mental tests using congruent triangles/polygons, e.g. sketch and label two triangles that show SAS congruence; find/write down the
interior angle of a pentagon
HOMEWORK
Homework at each stage could comprise consolidation of work in class by completion of exercises set, additional work of a similar
nature, or extension work detailed above.
HINTS AND TIPS
Sturdy compasses are essential.
Construction lines should not be erased.
Use real maps for interest.
Students often find the presentation of formal arguments difficult.
Use Exercises 12A – 12K for practice. Use Mixed Exercise 12 for consolidation.
HIGHER UNIT 3 (OLD UNIT 4) SCHEMES OF WORK
HIGHER CHAPTER 13 PYTHAGORAS’ THEOREM IN 2-D AND 3-D
Time: 8 – 10 hours
SPECIFICATION REFERENCE
Using Pythagoras’ theorem to find the third side of a right-angled triangle
Finding lengths and angles in 3-D graphs
Using tangents in triangles
Using sine, cosine and tangent to solve problems
Trigonometry in 3-D
Calculating area of a triangle using A = ½ ab sinC
Using the sine and cosine rules to solve problems
Trigonometric graphs
Solving trigonometric equations
SSM2f
SSM2g
SSM2g
SSM2g
SSM2g
SSM2g
SSM2g
NA6f
SSM2g
PRIOR KNOWLEDGE
Heinemann GCSE Modular Mathematics Unit 3 (old Unit 4) Higher Chapter 11: Transformations
OBJECTIVES
By the end of the unit the student should be able to:
Use Pythagoras’ theorem to find missing lengths (including 3-D applications)
Use the trigonometric ratios to find missing lengths/angles (including practical applications)
Find the angle between a line and a plane
Use A = ½ ab sinC to find the area of a triangle
Use the sine and cosine rules to find missing lengths/angles
-
Draw/sketch the graphs of trigonometric functions, e.g. sketch 3cos2x for 0 
x  360
Solve simple trigonometric equations, e.g. solve 3tanx  1 = 0 for 180  x  180
Know/use elementary trigonometric identities, e.g. sin   x   sin x, cos 180  x    cos x .
DIFFERENTIATION AND EXTENSION
Harder problems involving multi-stage calculations
Use a spreadsheet to find Pythagorean triples.
Appreciate/reproduce derivations of: the cosine rule; the sine rule; the area formula.
Find the angle between two planes.
Investigate the graphs of combinations of trigonometric functions, e.g. 3sinx + 4cosx.
RESOURCES
Heinemann GCSE Modular Mathematics Unit 3 (old Unit 4) Higher Chapter/section: 13.1 – 13.12
See the Teaching and Learning Software for activities linking to this chapter.
ASSESSMENT ISSUES
Written tests and homework to assess knowledge of content
Mental tests using trigonometric ratios, e.g. here is a right angle triangle, write down the value of cosA
HOMEWORK
Homework at each stage could comprise consolidation of work in class by completion of exercises set, additional work of a similar
nature, or extension work detailed above.
HINTS AND TIPS
Graphic calculators and/or dedicated computer software will underpin the main ideas.
Use Exercises 13A – 13L for practice. Use Mixed Exercise 13 for consolidation.
HIGHER UNIT 3 (OLD UNIT 4) SCHEMES OF WORK
HIGHER CHAPTER 14
CIRCLE THEOREMS
Time: 2 – 4 hours
SPECIFICATION REFERENCE
Using the circle theorems to find missing angles
Using the theorems for cyclic quadrilaterals
Using the alternate segment theorem
Knowing the proof of circle and geometric theorems
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PRIOR KNOWLEDGE
Heinemann GCSE Modular Mathematics Unit 2 (old Unit 3) Higher Chapter 8: Properties of circles
OBJECTIVES
By the end of the unit the student should be able to:
Know/use the circle theorems involving radii, chords and tangents
Know/use the circle theorems involving angles at the centre/circumference
Know/use opposite angles of a cyclic quadrilateral add to 180
Know/use the alternate segment theorem
Understand/reproduce the proofs of circle theorems.
DIFFERENTIATION AND EXTENSION
Harder problems involving multi-stage calculations
Investigate which theorems are used to prove the circle theorems, e.g. angles at a point add up to 360, the angle subtended
at the centre is twice the angle at the circumference (opposite angles of a cyclic quadrilateral add up to 180).
Investigate the historical proof of circle theorems (Euclid’s Elements).
RESOURCES
Heinemann GCSE Modular Mathematics Unit 3 (old Unit 4) chapter/section: 14.1 – 14.4
See the Teaching and Learning Software for activities linking to this chapter.
ASSESSMENT ISSUES
Written tests and homework to assess knowledge of content
Mental tests using circle theorems, e.g. write down the alternate segment theorem; in the diagram, explain why angle ACB equals
90 (the angle in a semicircle is a right angle)
HOMEWORK
Homework at each stage could comprise consolidation of work in class by completion of exercises set, additional work of a similar
nature, or extension work detailed above.
HINTS AND TIPS
Students may be required to write down an appropriate circle theorem to explain a result.
Use Exercises 14A – 14D for practice. Use Mixed Exercise 14 for consolidation.
HIGHER UNIT 3 (OLD UNIT 4) SCHEMES OF WORK
HIGHER CHAPTER 15
2-D AND 3-D SHAPES
Time: 4 – 6 hours
SPECIFICATION REFERENCE
Calculating the area and circumference of circles
Calculating the volume and surface area of prisms
Converting units of area and volume
Calculating the arc length and sector area of a circle
Calculating the area of a segment
Calculating volumes of pyramids, cones and spheres
Using area and volume scale factors
Calculating volumes of combined shapes, including truncated solids
Recognising formulae for length, area and volume from the dimensions
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SSM2i
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SSM2i
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PRIOR KNOWLEDGE
Heinemann GCSE Modular Mathematics Unit 2 (old Unit 3) Higher Chapter 10: Perimeter, area and volume
OBJECTIVES
By the end of the unit the student should be able to:
Find the area/circumference of circles and parts of circles
Find the surface area/volume of prisms (including cylinders)
Convert between units of area/volume, e.g. change 2m2 to cm2
Find the arc length/sector area of circles
 r 2
1
 r 2 sin 
360 2
-
Find the segment area, e.g. by using
-
Find the surface area of regular 3-D shapes (including spheres and cones and combinations of these)
Find the volume of regular shapes (including right pyramids, cones and spheres)
Find the surface area and volume of similar shapes, e.g. using k, k2 and k3
Find the volume of compound shapes (including truncated cones/pyramids)
Use dimensions to identify formulas for length, area and volume.
DIFFERENTIATION AND EXTENSION
Multi-step problems, e.g. find the volume of a cylinder given its surface area (leaving answer in terms of l)
Express the volume of a sphere in terms of its surface area.
Find the new depth of a liquid in a container when some of the liquid is removed, e.g. conical container.
Use Index notation when converting between units.
Estimate volumes/surface areas in real-life situations, e.g. surface area of the Moon (sphere), volume of a mountain (cone).
-
Use
22
7
as an approximation for .
RESOURCES
Heinemann GCSE Modular Mathematics Unit 3 (old Unit 4) Higher Chapter/section: 15.1 – 15.11
See the Teaching and Learning Software for activities linking to this chapter.
ASSESSMENT ISSUES
Written tests and homework to assess knowledge of content
Mental tests using formulae, e.g. write down the formula for the volume of a sphere
HOMEWORK
Homework at each stage could comprise consolidation of work in class by completion of exercises set, additional work of a similar
nature, or extension work detailed above.
HINTS AND TIPS
Students may be asked to leave their answer in terms of .
Use Exercises 15A – 15K for practice. Use Mixed Exercise 15 for consolidation.
HIGHER UNIT 3 (OLD UNIT 4) SCHEMES OF WORK
HIGHER CHAPTER 16
VECTORS
SPECIFICATION REFERENCE
Understanding the properties of vectors
Combining vectors in the form pa + qb
Using vectors related to the origin
Using the properties of parallel vectors
Time: 4 – 6 hours
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PRIOR KNOWLEDGE
Heinemann GCSE Modular Mathematics Unit 3 (old Unit 4) Higher Chapter 11: Transformation
OBJECTIVES
By the end of the unit the student should be able to:
Understand the properties of a vector, e.g. length and direct, and use vector notation
Add/subtract column vectors; multiply a column vector by a number
Understand/use the parallelogram rule for adding/subtracting vectors
Represent vectors as a linear combination of two other vectors, e.g. pa + qb, where a, b are given
Understand position and displacement (free) vectors
Use the triangle rule for vectors
Find the position vector of the point which divides a line in a given ratio
Use vectors to prove geometric results.
DIFFERENTIATION AND EXTENSION
Harder geometric proofs, e.g. show that the medians of a triangle intersect in the ratio 1:2
Vector problems in 3-D
Use i and j (and k) notation.
RESOURCES
Heinemann GCSE Modular Mathematics Unit 3 (old Unit 4) Higher Chapter/section: 16.1 – 16.4
See the Teaching and Learning Software for activities linking to this chapter.
ASSESSMENT ISSUES
Written tests and homework to assess knowledge of content
Mental tests using properties of vectors, e.g. write down a vector parallel to 2a + 3b
HOMEWORK
Homework at each stage could comprise consolidation of work in class by completion of exercises set, additional work of a similar
nature, or extension work detailed above.
HINTS AND TIPS
Draw diagrams to illustrate vector algebra.
Use Exercises 16A – 16C for practice. Use Mixed Exercise 16 for consolidation.