Unsure about Methods of Proof?

Unsure about Methods of Proof?

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Covers the whole of the AH Maths course

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Resources used with students in Scottish Secondary Schools

Methods of Proof

Welcome to advancedhighermaths.co.uk

A sound understanding of Methods of Proof  is essential to ensure exam success.

Study at Advanced Higher Maths level will provide excellent preparation for your studies when at university. Some universities may require you to gain a pass at AH Maths to be accepted onto the course of your choice. The AH Maths course is fast paced so please do your very best to keep on top of your studies.

For students looking for extra help with the AH Maths course in 2019/20 you may wish to consider subscribing to the fantastic additional exam focused resources available in the Online Study Pack.

We hope you find this website useful and wish you the very best with your studies in 2019/20.

Please find below:

1. AH Maths Past Exam Worksheets by Topic

2. Links and Exam Formulae

3. Methods of Proof Resources

4. Methods of Proof Worksheets and Solutions

5. About Methods of Proof

6. AH Maths Past & Practice Papers

7. AH Maths Past Paper Questions by Topic

8. AH Maths Recommended Text Book

9. Exam Focused Online Study Pack – For students looking for a ‘good’ Pass

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Advanced Higher Maths Resources

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1. AH Maths Past Paper Exam Worksheets by Topic

Thanks to the SQA for making these available. The worksheets by topic below are an excellent study resource since they are actual SQA past paper exam questions.

Clear, easy to follow, step-by-step worked solutions to all SQA Higher Maths Questions below are available in the Online Study Pack.

Number
______
Topic
____________________________________________
Answers
_________
1Binomial TheoremAnswers
2Complex NumbersAnswers
3DifferentiationAnswers
4Differentiation (Further)Answers
5Differential Equations - Variables SeparableAnswers
6Differential Equations (Further)Answers
7Functions & GraphsAnswers
8IntegrationAnswers
9Integration (Further)Answers
10MatricesAnswers
11Number Theory - Methods of ProofAnswers
12Number Theory (Further) - Euclidean & Number BasesAnswers
13Partial FractionsAnswers
14Sequences & SeriesAnswers
15 Sequences & Series - Maclaurin Answers
16Systems of EquationsAnswers
17VectorsAnswers

2. Links and Exam Formulae

AH Maths - Home Page LinkHERE
AH Maths - Whole Course LinkHERE
AH Maths - Exam Formula ListHERE
Higher Maths - Exam Formulae ListHERE
AH Maths - Complete Formula ListHERE
SQA Full Course Content and OutlineHERE

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3. Methods of Proof Resources

Thanks to the author(s) for making these available.

Methods of Proof Theory Guide 1HERE
Methods of Proof Theory Guide 2HERE
Methods of Proof HSN Summary NotesHERE

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4.  Methods of Proof Worksheets and Solutions

Thanks to the author(s) for making these available.

Topic
_______________________________
Lessons
__________
Questions
_________
Typed Solutions
_______________
Handwritten Solutions
______________________
Exam Questions - Worked Solutions in Online Study Pack
______________________________________________________
Direct ProofLesson 1Ex 1 & 2 Ex 1 & 2 Handwritten Solns2018-Q9,2015-Q12, 2010-Q8a
Proof by Counterexample Lesson 2Ex 3Ex 3 Typed SolnsEx 3 Handwritten Solns2016-Q10, 2013-Q12, 2008-Q11
Proof by Counterexample Ex 4Ex 4 Typed SolnsEx 4 Handwritten Solns2016-Q10, 2013-Q12, 2008-Q11
Proof by ContradictionLesson 3Ex 5Ex 5 Typed SolnsEx 5 Handwritten Solns2010-Q12
Proof by ContrapositiveLesson 4Ex 6Ex 6 Typed SolnsEx 6 Handwritten Solns2017-Q13
Proof by InductionLesson 5Ex 7Ex 7 Typed SolnsEx 7 Handwritten Solns2014-Q7,2013-Q9,2012-Q16a,2011-Q12,2010-Q8b,2009-Q4,2007-Q12
Proof by Induction - Sigma NotationLesson 6Ex 8Ex 8 Typed SolnsEx 8 Handwritten Solns2018-Q12,2016-Q5, 2013-Q9,2009-Q4

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5. About Methods of Proof

For a more detailed explanation, please first read the Theory Guides above.

At AH Maths, the following is covered in the topic ‘Methods of Proof’:

Direct Proof

No set rules, just make logical deductions

Proof by Counter Example

If you can find one example to disprove the statement, then it must be false

Proof by Contradiction

This method assumes that the statement is false and then shows that this leads to something we know to be false (a contradiction). Therefore the original assumption must be false, which means that the statement is true.

Proof by Contrapositive 

Pythagoras Theorem states “A triangle is right-angled at ‘A’ if a² = b² + c²”.

The contrapositive of this statement is “if a² ≠ b² + c² then the triangle in not right-angled at ‘A'”.

If you can prove that the contrapositive of a statement is true then the original statement must also be true.

Proof by Induction

This is a 4 step process:

  • STEP 1: Show conjecture is true for n = 1 (or the first value n can take)
  • STEP 2: Assume statement is true for n = k
  • STEP 3: Show conjecture is true for n = k + 1
  • STEP 4: Closing Statement – this is crucial in gaining all the marks

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Types of Number

Numbers are classified according to their type. The types of number at AH Maths you need to be familiar with are:

  • Natural Numbers – Counting numbers such as 1,2,3 …
  • Integers – Positive or negative whole number
  • Primes – Have only 2 factors – 2,3,5,7,11 etc
  • Rational – Are of the form m/n (n ≠ 0), m & n are integers
  • Irrational – Cannot be expressed on the form m/n eg √2
  • Real – These are either rational or irrational
  • Complex – In the form x + iy where x and y are real numbers and i = √-1

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6. AH Maths Past & Practice Papers

Thanks to the SQA for making these available.

Worked solutions to SQA Exam Questions available in the Online Study Pack.

Year
____
Paper Type
_________________
Exam Paper
______________
Marking Scheme
_______________________________________
2019AH SpecimenSpecimenMarking Scheme
2019Advanced HigherExam PaperMarking Scheme
2018Advanced HigherExam PaperMarking Scheme
2017Advanced HigherExam PaperMarking Scheme
2016Advanced HigherExam PaperMarking Scheme
2016AH SpecimenSpecimenMarking Scheme
2016AH ExemplarExemplarMarking Scheme
2015Advanced HigherExam PaperMarking Scheme
2014Advanced HigherExam PaperMarking Scheme
2013Advanced HigherExam PaperMarking Scheme
2012Advanced HigherExam PaperMarking Scheme
2011Advanced HigherExam PaperMarking Scheme
2010Advanced HigherExam PaperMarking Scheme
2009Advanced HigherExam PaperMarking Scheme
2008Advanced HigherExam PaperMarking Scheme
2007Advanced HigherExam PaperMarking Scheme
2006Advanced HigherExam PaperMarking Scheme
2005Advanced HigherExam PaperMarking Scheme
2004Advanced HigherExam PaperMarking Scheme
2003Advanced HigherExam PaperMarking Scheme
2002Advanced HigherExam PaperMarking Scheme
2001Advanced HigherExam Paper

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7. AH Maths Past Paper Questions by Topic

Thanks to the SQA for making these available.

Questions and answers have been split up by topic for ease of reference.

Worked solutions to all SQA questions below available in the Online Study Pack.

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Paper
___________
.
Marking
______
Binomial
Theorem
________
Partial
Fractions
________
.
Differentiation
___________
Further Differentiation
___________
.
Integration
___________
Further
Integration
____________
Functions
& Graphs
___________
Systems of
Equations
____________
Complex
Numbers
__________
Seq &
Series
_________
Further Seq
& Series
____________
.
Matrices
_________
.
Vectors
__________
Methods
of Proof
__________
Further No
Theory
___________
Differential
Equations
____________
Further
Differential Eqns
_________________
Specimen P1MarkingQ2Q4Q6Q8Q3Q5Q1Q7
Specimen P2MarkingQ3Q1Q2,4,8,10Q7Q11Q5Q13Q9Q6Q12
2019MarkingQ9Q4Q1a,b,6Q1c,5,10Q16bQ16aQ3Q18Q7,17Q2Q15Q11,14Q12Q13Q8
2018MarkingQ3Q2Q1bQ1a,c,6,13Q8Q15aQ16aQ4,10Q14Q17Q7,11Q16Q9,12Q5Q15b
2017MarkingQ1Q2Q3Q11,18Q16Q6Q12Q5Q17Q4,10Q7Q15Q13Q8Q9Q14
2016MarkingQ3Q13Q1a,bQ1c,11Q13Q9Q12Q4Q8Q2Q6Q7Q14Q5,10Q16Q15
2015MarkingQ1,9Q2Q4,6,8Q17Q10Q14Q13Q3Q5,11Q15Q12Q7Q18Q16
2014MarkingQ214bQ1,13Q1,4,6Q10,12Q15Q11Q3Q16Q14Q9Q7Q5Q7Q8
2013MarkingQ1Q2Q11Q4,6Q8Q13Q7,10Q17Q3Q15Q9,12Q5Q16Q14
2012MarkingQ415aQ1Q12,13Q8Q11Q7Q14Q3,16bQ2Q6Q9Q516aQ10Q15
2011MarkingQ2Q13b,73aQ1,11aQ1,11,16Q6Q10Q8,13Q5Q4Q15Q12Q9Q14
2010MarkingQ5Q1Q13Q15Q3,7Q10Q16Q2Q9Q4,14Q6Q8,12Q11
2009MarkingQ8Q14Q1aQ1b,11Q5,7Q9Q1316aQ6Q12Q14Q2Q16Q4Q10Q3Q15
2008MarkingQ8Q4Q10,15Q2,5Q4,9,10Q7Q3Q16Q1Q12Q6Q14Q11Q13
2007MarkingQ1Q4Q2Q13Q4,10Q4Q16Q3,11Q9Q6Q5Q15Q12Q7Q14Q8
MixedMixedMixedMixedMixedMixedMixedMixedMixedMixedMixedMixedMixedMixedMixedMixedMixed

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8. AH Maths Text Book – Maths In Action (2nd Edition) by Edward Mullan 

A fully revised course for the new Curriculum for Excellence examination that is designed to fully support the course’s new structure and unit assessment. A part of the highly regarded Maths in Action series, it provides students with a familiar, clear and carefully structured learning experience that encourages them to build confidence and understanding.

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9. Exam focused Online Study Pack

Through step-by-step worked solutions to exam questions available in the Online Study Pack we cover everything you need to know about Methods of Proof  to pass your final exam.

As well as students studying Advanced Higher Mathematics, the resources will benefit young adults studying A-Level Mathematics and undergraduates who need a little extra help. Particular benefit will be to students who have gained a ‘Conditional’ University place and are therefore required to pass in order to gain entry onto the course of their choice.

Subscribing to the Online Study Pack may therefore be one of your best ever investments. Please click Online Study Pack to view screenshots, examples and instructions on how to subscribe.

Please speak with your mum/dad and show them our fantastic website.

We hope the resources on this website prove useful and wish you the very best of success with your AH Maths course in 2019/20.

Get the Study Pack - just £9.99

Teachers click here >

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