School level

# 8th May

## 8th May: Leaving Cert Higher level Proofs and Derivations

### Learning Outcomes

2.1 Synthetic Geometry

• prove theorems 11,12,13, concerning ratios (see Geometry for Post-primary School Mathematics), which lay the proper foundation for the proof of the theorem of Pythagoras studied at junior cycle

2.3 Trigonometry

• derive the trigonometric formulae 1, 2, 3, 4, 5, 6, 7, 9

3.1 Number systems

• prove that √2 is not rational
• prove by induction

• simple inequalities such as n! > 2n, 2n ≥ n2 (n ≥ 4) (1+ x)n ≥ 1+nx (x > –1)
• simple identities such as the sum of the first n natural numbers and the sum of a finite geometric series
• factorisation results such as 3 is a factor of 4n –1

• solve problems involving finite and infinite geometric series including applications such as recurring decimals and financial applications, e.g. deriving the formula for a mortgage repayment
• derive the formula for the sum to infinity of geometric series by considering the limit of a sequence of partial sums

4.4 Complex Numbers

• prove De Moivre’s Theorem by induction for n ∈ N

Resources and Presentations
SEC Exam Questions
Explanatory Videos

### Theorem 11 proof

Proof of theorem 11 from the Geometry for Post-primary school Mathematics

SC Mathematics

#### How it maps to the curriculum

SC
Mathematics

Strand: 2. Geometry & Trigonometry

Strand unit: 1. Synthetic Geometry

Suggestions for use: Useful demonstration for the Proof of theorem 11 from the Geometry for Post-primary school Mathematics

Asset type: Presentation
Language: English
Special Needs:

### Theorem 12 Proof

https://www.scoilnet.ieuploads/resources/32902/32646.pptx Added: 08 May 2020 Contributor: PDST Resource type: Demonstration / Presentation

Proof of theorem 12 from the Geometry for Post-primary school Mathematics

SC Mathematics

#### How it maps to the curriculum

SC
Mathematics

Strand: 2. Geometry & Trigonometry

Strand unit: 1. Synthetic Geometry

Suggestions for use: Demonstration for the Proof of theorem 11 from the Geometry for Post-primary school Mathematics

Asset type: Presentation
Language: English
Special Needs:

### Proof of theorem 13

https://www.scoilnet.ieuploads/resources/32903/32647.pptx Added: 08 May 2020 Contributor: PDST Resource type: Demonstration / Presentation

Proof of theorem 13 from the Geometry for Post-primary school Mathematics

SC Mathematics

#### How it maps to the curriculum

SC
Mathematics

Strand: 2. Geometry & Trigonometry

Strand unit: 1. Synthetic Geometry

Suggestions for use: Demonstration for the Proof of theorem 11 from the Geometry for Post-primary school Mathematics

Asset type: Presentation
Language: English
Special Needs:

### Proof by contradiction that the square root of 2 is irrational

Demonstration of the proof that the square root of 2 is irrational

SC Mathematics

#### How it maps to the curriculum

SC
Mathematics

Strand: 3. Number

Strand unit: 1. Number Systems (Complex Numbers,Sequences)

Suggestions for use: Presentation of the proof that the square root of two is irrational.

Asset type: Presentation
Language: English
Special Needs:

### Presentation on Financial Mathematics from Module 4

https://www.scoilnet.ieuploads/resources/32905/32649.pdf Added: 08 May 2020 Contributor: PDST Resource type: Demonstration / Guide/notes / Presentation

Presentation on financial Mathematics from module 4. Derivation for the formula of a mortgage repayment on slide 44.

SC Mathematics

#### How it maps to the curriculum

SC
Mathematics

Strand: 3. Number

Strand unit: 1. Number Systems (Complex Numbers,Sequences)

Suggestions for use: Presentation on financial Mathematics from module 4. Derivation for the formula of a mortgage repayment on slide 44.

Asset type: Document
Language: English
Special Needs:

### Derivation of the Compound Angle Formulae

https://www.scoilnet.ieuploads/resources/32604/32346.docx Added: 20 Apr 2020 Contributor: PDST Resource type: Guide/notes / Reference

These are the formal derivations of formula 4, 5, 7 and 9, as required on the LC higher level syllabus. These derivations have been taken from the Senior Cycle Teachers Handbook, Projectmaths.ie

SC Mathematics

#### How it maps to the curriculum

SC
Mathematics

Strand: 2. Geometry & Trigonometry

Strand unit: 3. Trigonometry

Suggestions for use: This resource gives the derivations of the compound angle formulae 4, 5, 7 and 9 required by SEC for the HL leaving certificate examination.

Asset type: Document
Language: English
Special Needs:

### Prove by Induction

https://www.scoilnet.ie/pdst/maths/student/lc/lc-numbers/induction/ Added: 09 Nov 2019 Contributor: PDST Resource type: Presentation

Powerpoint Examples of Proof by Induction created by the Project Maths Development Team.

SC Mathematics

#### How it maps to the curriculum

SC
Mathematics

Strand: 3. Number

Strand unit: 1. Number Systems (Complex Numbers,Sequences)

Asset type: Online Resource
Language: English
Special Needs:

### Complex Number Proofs

https://www.scoilnet.ie/pdst/maths/...ent/lc/lc-algebra/demoivrestheorem/ Added: 09 Nov 2019 Contributor: PDST Resource type: Presentation

Powerpoints of the Proof of De Moivre's Theorem along with the proofs of the sin3x and cos3x formulae.

SC Mathematics

#### How it maps to the curriculum

SC
Mathematics

Strand: 4. Algebra

Strand unit: 4. Complex Numbers(Conjugate Root+De Moivre)

Asset type: Online Resource
Language: English
Special Needs:

### 2019 LCHL P1 Q6 Proof by contradiction

https://www.scoilnet.ieuploads/resources/32906/32650.pdf Added: 08 May 2020 Contributor: PDST Resource type: Assessment / Exam Papers

2019 leaving cert higher level paper 1 exam question on proof by contradiction

SC Mathematics

#### How it maps to the curriculum

SC
Mathematics

Strand: 3. Number

Strand unit: 1. Number Systems (Complex Numbers,Sequences)

Suggestions for use: Could be used as an assessment to test prior knowledge

Asset type: Document
Language: English
Special Needs:

### 2019 LCHL P1 Q2 Proof by induction

https://www.scoilnet.ieuploads/resources/32907/32651.pdf Added: 08 May 2020 Contributor: PDST Resource type: Assessment / Exam Papers

Question from the 2019 Leaving Certificate Higher Level Paper 1. Question 2 Proof by induction

SC Mathematics

#### How it maps to the curriculum

SC
Mathematics

Strand: 3. Number

Strand unit: 1. Number Systems (Complex Numbers,Sequences)

Suggestions for use: assessment on proof by induction

Asset type: Document
Language: English
Special Needs:

### 2018 LCHL P2 Q6 Theorem 12

https://www.scoilnet.ieuploads/resources/32908/32652.pdf Added: 08 May 2020 Contributor: PDST Resource type: Assessment / Exam Papers

Leaving Certificate exam paper 2 Question 6. Proof of theorem 12

SC Mathematics

#### How it maps to the curriculum

SC
Mathematics

Strand: 2. Geometry & Trigonometry

Strand unit: 1. Synthetic Geometry

Suggestions for use: assessment of theorem 12

Asset type: Document
Language: English
Special Needs:

### 2018 LCHL P1 Q4 Prove De Moivre's theorem by induction

https://www.scoilnet.ieuploads/resources/32909/32653.pdf Added: 08 May 2020 Contributor: PDST Resource type: Assessment / Exam Papers

2018 Leaving Certificate Higher level Paper 1 Question 4. Prove De Moivre's theorem by induction

SC Mathematics

#### How it maps to the curriculum

SC
Mathematics

Strand: 3. Number

Strand unit: 1. Number Systems (Complex Numbers,Sequences)

Suggestions for use: Assessment for proof of De Moivre's theorem by induction

Asset type: Document
Language: English
Special Needs:

### 2017 LCHL P2 Q5 Mini Geometry proof

https://www.scoilnet.ieuploads/resources/32910/32654.pdf Added: 08 May 2020 Contributor: PDST Resource type: Assessment / Exam Papers

2017 Leaving Certificate Higher level exam paper 2. Question 5 mini geometry proof.

SC Mathematics

#### How it maps to the curriculum

SC
Mathematics

Strand: 2. Geometry & Trigonometry

Strand unit: 1. Synthetic Geometry

Suggestions for use: Assessment of a mini geometry proof.

Asset type: Document
Language: English
Special Needs:

### 2017 LCHL P2 Q3 Geometry proof on Orthocentre

https://www.scoilnet.ieuploads/resources/32911/32655.pdf Added: 08 May 2020 Contributor: PDST Resource type: Assessment / Exam Papers

2017 Leaving Certificate Higher level exam paper 2 Question 3. Mini geometry proof on orthocentre.

SC Mathematics

#### How it maps to the curriculum

SC
Mathematics

Strand: 2. Geometry & Trigonometry

Strand unit: 1. Synthetic Geometry

Suggestions for use: assessment on a mini proof of the orthocentre

Asset type: Document
Language: English
Special Needs:

### 2017 LCHL P1 Q8 Derivation of formula for a mortgage repayment

https://www.scoilnet.ieuploads/resources/32912/32656.pdf Added: 08 May 2020 Contributor: PDST Resource type: Assessment / Exam Papers

2017 Leaving Certificate exam paper 1. Question 8 Derivation of the formula for a mortgage repayment.

SC Mathematics

#### How it maps to the curriculum

SC
Mathematics

Strand: 3. Number

Strand unit: 1. Number Systems (Complex Numbers,Sequences)

Suggestions for use: Assessment on the derivation of the formula for a mortgage repayment

Asset type: Document
Language: English
Special Needs:

https://www.youtube.com/watch?v=jJXVarQANNU Added: 17 Jul 2018 Contributor: UCD MSC Resource type: Audio / Video

Higher level students need to know the concept 'proof by contradiction'. This video explores the concept along with illustrating an example. The example given requires students to prove by contradiction that the square root of two is an irrational number.

SC Mathematics

#### How it maps to the curriculum

SC
Mathematics

Strand: 2. Geometry & Trigonometry

Strand unit: 1. Synthetic Geometry

Asset type: Video
Special Needs:

### Mini Proofs - Useful Terms

https://www.youtube.com/watch?v=rwl6ITIkv0Q Added: 17 Jul 2018 Contributor: UCD MSC Resource type: Audio / Video

This video introduces the concept of mini-proofs by detailing useful terms and identifying the steps used in proofs. This video was produced by the UCD Maths Support Centre in conjunction with UCD access centre.

SC Mathematics

#### How it maps to the curriculum

SC
Mathematics

Strand: 2. Geometry & Trigonometry

Strand unit: 1. Synthetic Geometry

Asset type: Video
Language: English
Special Needs:

### Mini Proof - Example 1

https://www.youtube.com/watch?v=X0RJsWWyxpA Added: 17 Jul 2018 Contributor: UCD MSC Resource type: Audio / Video

Video demonstrates an example of a mini proof by following the steps of a formal proof. In this example, the question asks student to proof that |<1| +|<2| +|<3| = 360 Degrees from a given diagram.

SC Mathematics

#### How it maps to the curriculum

SC
Mathematics

Strand: 2. Geometry & Trigonometry

Strand unit: 1. Synthetic Geometry

Asset type: Video
Special Needs:

### Mini Proof - Example 2

https://www.youtube.com/watch?v=kEFfcjHMJ5M Added: 17 Jul 2018 Contributor: UCD MSC Resource type: Audio / Video

Video demonstrates an example of a mini proof by following the steps of a formal proof. In this example, the question asks student to proof that XZ ||BD from the given diagram, knowing that XY||BC and YZ||CD.

SC Mathematics

#### How it maps to the curriculum

SC
Mathematics

Strand: 2. Geometry & Trigonometry

Strand unit: 1. Synthetic Geometry

Asset type: Video
Language: English
Special Needs:

### Mini Proof - Example 3

https://www.youtube.com/watch?v=IckvJk974kg Added: 17 Jul 2018 Contributor: UCD MSC Resource type: Audio / Video

Video demonstrates an example of a mini proof by following the steps of a formal proof. In this example, the question asks student to proof that |AC|.|FG| = |AG|.|BC| in the given diagram.

SC Mathematics

#### How it maps to the curriculum

SC
Mathematics

Strand: 2. Geometry & Trigonometry

Strand unit: 1. Synthetic Geometry

Asset type: Video
Language: English
Special Needs:

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