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
Theorem 11 proof
https://www.scoilnet.ieuploads/resources/32900/32644.pptx Added: 08 May 2020 Contributor: PDST Resource type: DemonstrationProof of theorem 11 from the Geometry for Post-primary school Mathematics
View full descriptionHow 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
Theorem 12 Proof
https://www.scoilnet.ieuploads/resources/32902/32646.pptx Added: 08 May 2020 Contributor: PDST Resource type: Demonstration / PresentationProof of theorem 12 from the Geometry for Post-primary school Mathematics
View full descriptionHow 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
Proof of theorem 13
https://www.scoilnet.ieuploads/resources/32903/32647.pptx Added: 08 May 2020 Contributor: PDST Resource type: Demonstration / PresentationProof of theorem 13 from the Geometry for Post-primary school Mathematics
View full descriptionHow 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
Proof by contradiction that the square root of 2 is irrational
https://www.scoilnet.ieuploads/resources/32904/32648.pptx Added: 08 May 2020 Contributor: PDST Resource type:Demonstration of the proof that the square root of 2 is irrational
View full descriptionHow 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.
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 / PresentationPresentation on financial Mathematics from module 4. Derivation for the formula of a mortgage repayment on slide 44.
View full descriptionHow 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.
Prove by Induction
https://www.scoilnet.ie/pdst/maths/student/lc/lc-numbers/induction/ Added: 09 Nov 2019 Contributor: PDST Resource type: PresentationPowerpoint Examples of Proof by Induction created by the Project Maths Development Team.
View full descriptionHow it maps to the curriculum
- SC
- Mathematics
Strand: 3. Number
Strand unit: 1. Number Systems (Complex Numbers,Sequences)
Complex Number Proofs
https://www.scoilnet.ie/pdst/maths/...ent/lc/lc-algebra/demoivrestheorem/ Added: 09 Nov 2019 Contributor: PDST Resource type: PresentationPowerpoints of the Proof of De Moivre's Theorem along with the proofs of the sin3x and cos3x formulae.
View full descriptionHow it maps to the curriculum
- SC
- Mathematics
Strand: 4. Algebra
Strand unit: 4. Complex Numbers(Conjugate Root+De Moivre)
Proof by Contradiction
https://www.youtube.com/watch?v=jJXVarQANNU Added: 17 Jul 2018 Contributor: UCD MSC Resource type: Audio / VideoHigher 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.
View full descriptionHow it maps to the curriculum
- SC
- Mathematics
Strand: 2. Geometry & Trigonometry
Strand unit: 1. Synthetic Geometry
Mini Proofs - Useful Terms
https://www.youtube.com/watch?v=rwl6ITIkv0Q Added: 17 Jul 2018 Contributor: UCD MSC Resource type: Audio / VideoThis 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.
View full descriptionHow it maps to the curriculum
- SC
- Mathematics
Strand: 2. Geometry & Trigonometry
Strand unit: 1. Synthetic Geometry
Mini Proof - Example 1
https://www.youtube.com/watch?v=X0RJsWWyxpA Added: 17 Jul 2018 Contributor: UCD MSC Resource type: Audio / VideoVideo 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.
View full descriptionHow it maps to the curriculum
- SC
- Mathematics
Strand: 2. Geometry & Trigonometry
Strand unit: 1. Synthetic Geometry
Mini Proof - Example 2
https://www.youtube.com/watch?v=kEFfcjHMJ5M Added: 17 Jul 2018 Contributor: UCD MSC Resource type: Audio / VideoVideo 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.
View full descriptionHow it maps to the curriculum
- SC
- Mathematics
Strand: 2. Geometry & Trigonometry
Strand unit: 1. Synthetic Geometry
Mini Proof - Example 3
https://www.youtube.com/watch?v=IckvJk974kg Added: 17 Jul 2018 Contributor: UCD MSC Resource type: Audio / VideoVideo 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.
View full descriptionHow it maps to the curriculum
- SC
- Mathematics
Strand: 2. Geometry & Trigonometry
Strand unit: 1. Synthetic Geometry
