School level

# Daily Maths - PDST

Welcome one and all.

We hope these resources may be of some use to you and your students in the coming days.

We've picked materials from projectmaths.ie which we think are most useful and have added in some videos and materials from other sites which may help you when planning for supporting your students in their learning. As teachers you're in the best place to judge which materials are most useful for your students and the design of activities around these resources.

If you have any questions email us at postprimarymaths@pdst.ie.

## 3rd April: Complex Numbers

### Learning Outcomes

Number systems:
Students working at OL should be able to:

• investigate the operations of addition, multiplication, subtraction and division with complex numbers C in rectangular form a+ib
• illustrate complex numbers on an Argand diagram
• interpret the modulus as distance from the origin on an Argand diagram and calculate the complex conjugate

Students working at HL should be able to:

• calculate conjugates of sums and products of complex numbers

Algebra:
Students working at HL should be able to:

• use the Conjugate Root Theorem to find the roots of polynomials
• work with complex numbers in rectangular and polar form to solve quadratic and other equations including those in the form zn = a, where n ∈ Z and z = r (Cos θ + iSin θ )
• use De Moivre’s Theorem
Teaching & Learning Plans
Workshop Materials
Operations - Geogebra Files
Polar Form & DeMoivre’s Theorem
SEC Exam Questions

### Complex Number Operations - Plan

To familiarise students with operations on Complex Numbers and to give an algebraic
and geometric interpretation to these operations

SC Mathematics

#### How it maps to the curriculum

SC
Mathematics

Strand: 3. Number

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

Suggestions for use: To introduce students to the operations of complex numbers. To show that multiplication of a Complex Number by another Complex Number corresponds to a rotation and a scaling of the Complex Number.

Asset type: Document
Language: English
Rights: Attribution-Noncommercial Share Alike Creative Commons
Special Needs:

### Complex Number Operations - Plan

https://www.scoilnet.ieuploads/resources/29638/29374.ppt Added: 16 May 2019 Contributor: PDST Resource type: Lesson Plan / Presentation

To familiarise students with operations on Complex Numbers and to give an algebraic
and geometric interpretation to these operations

SC Mathematics

#### How it maps to the curriculum

SC
Mathematics

Strand: 3. Number

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

Suggestions for use: To give students a deeper understanding of the operations of complex numbers. show that multiplication of a Complex Number by another Complex Number corresponds to a rotation and a
scaling of the Complex Number

Asset type: Presentation
Language: English
Rights: Attribution-Noncommercial Share Alike Creative Commons
Special Needs:

### Complex Numbers Workshop

https://www.scoilnet.ieuploads/resources/32449/32191.pptx Added: 02 Apr 2020 Contributor: PDST Resource type: Demonstration / Guide/notes / Presentation

This presentation is the Complex Numbers sections of a Numbers workshop presented by the PDST. It focusses on the geometric view of complex numbers.

SC Mathematics

#### How it maps to the curriculum

SC
Mathematics

Strand: 3. Number

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

Suggestions for use: This can be used to introduce a geometric view of complex numbers, building on a students' prior knowledge of number to give them an understanding of complex numbers.

Asset type: Presentation
Language: English
Special Needs:

### Complex Numbers Workshop booklet

https://www.scoilnet.ieuploads/resources/32450/32192.docx Added: 02 Apr 2020 Contributor: PDST Resource type: Worksheet / Help Sheets / Activity

This booklet is best used with the Complex Numbers Workshop presentation and contains activities to work through to help gain an understanding of the geometric view of complex numbers.

SC Mathematics

#### How it maps to the curriculum

SC
Mathematics

Strand: 3. Number

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

Suggestions for use: Use with the Complex Numbers Workshop presentation to complete activities and gain a better understanding of the geometric view of complex numbers.

Asset type: Document
Language: English
Special Needs:

### The Argand Diagram and the Modulus of a Complex Number

https://www.scoilnet.ie/pdst/maths/...lc/lc-numbers/complexnumbersargand/ Added: 09 Nov 2019 Contributor: PDST Resource type: Guide/notes / Simulation

A GeoGebra file and student worksheet to investigate the Argand diagram and modulus of complex numbers.

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:

### Multiplying a complex number by a constant

https://www.geogebra.org/classic/hqc5kfft Added: 02 Apr 2020 Contributor: PDST Resource type: Demonstration / Simulation / Tool

This GeoGebra file invites students to explore what happens when a complex number is multiplied by a constant.

SC Mathematics

#### How it maps to the curriculum

SC
Mathematics

Strand: 4. Algebra

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

Suggestions for use: GeoGebra file to explore the multiplication of a complex number by a constant

Asset type: Online Resource
Language: English
Special Needs:

### Multiplying a complex number by i

https://www.geogebra.org/classic/q8mj3eym Added: 02 Apr 2020 Contributor: PDST Resource type: Demonstration / Simulation / Tool

This GeoGebra file allows the exploration of multiplying a complex number by i and by exponents of i.

SC Mathematics

#### How it maps to the curriculum

SC
Mathematics

Strand: 4. Algebra

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

Suggestions for use: This GeoGebra file should be used to explore what happens to a complex number when it is multiplied by i or an exponent of i.

Asset type: Online Resource
Language: English
Special Needs:

https://tinyurl.com/WS4task2demo Added: 02 Apr 2020 Contributor: PDST Resource type: Demonstration / Simulation / Tool

GeoGebra file to allow the exploration of addition of complex numbers

SC Mathematics

#### How it maps to the curriculum

SC
Mathematics

Strand: 4. Algebra

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

Suggestions for use: This GeoGebra file can be used to explore what happens when two complex numbers are added together, students will see a geometric view of the addition of complex numbers.

Asset type: Online Resource
Language: English
Special Needs:

### Complex conjugate

https://www.geogebra.org/classic/wrxxyrsf Added: 02 Apr 2020 Contributor: PDST Resource type: Demonstration / Simulation / Tool

GeoGebra file to explore the conjugate of a complex number

SC Mathematics

#### How it maps to the curriculum

SC
Mathematics

Strand: 4. Algebra

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

Suggestions for use: Use as a demonstration or an exploration tool for the geometric view of the complex conjugate.

Asset type: Online Resource
Language: English
Special Needs:

### Multiplication of complex numbers

https://www.geogebra.org/classic/bmedez74 Added: 02 Apr 2020 Contributor: PDST Resource type: Demonstration / Simulation / Tool

The GeoGebra file can be used to explore the geometric view of a complex number multiplied by another complex number

SC Mathematics

#### How it maps to the curriculum

SC
Mathematics

Strand: 4. Algebra

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

Suggestions for use: This file can be used to explore the multiplication of complex numbers.
It could also be used when introducing students to DeMoivre's theorem.

Asset type: Online Resource
Language: English
Special Needs:

### Polar form & DeMoivre's theorem

This presentation is created to guide students through the polar form of a complex number to introduce DeMoivre's theorem

SC Mathematics

#### How it maps to the curriculum

SC
Mathematics

Strand: 4. Algebra

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

Suggestions for use: Use with the GeoGebra file (attached in presentation) 'multiplication of a complex number' to use the argument and modulus to guide students to get the polar form of a complex number and then to introduce DeMoivre's theorem.

Asset type: Presentation
Language: English
Special Needs:

### Student activity - Argument & Modulus

https://www.scoilnet.ieuploads/resources/32469/32211.pdf Added: 02 Apr 2020 Contributor: PDST Resource type: Worksheet / Activity

This worksheet should be used with the GeoGebra file (link in document) to encourage students to explore the relationship between the argument and the modulus when multiplying complex numbers.

SC Mathematics

#### How it maps to the curriculum

SC
Mathematics

Strand: 4. Algebra

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

Suggestions for use: Worksheet to explore the relationship between the argument and the modulus when multiplying complex numbers. This could also be used to introduce DeMoivre's theorem.

Asset type: Document
Language: English
Special Needs:

### SEC 2019 HL p1 Q5 Complex Numbers

SEC exam question on complex numbers

SC Mathematics

#### How it maps to the curriculum

SC
Mathematics

Strand: 4. Algebra

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

Suggestions for use: Past exam question to assess understanding of complex numbers.

Asset type: Document
Language: English
Special Needs:

### SEC 2019 OL p1 Q2 Complex Numbers

SEC past exam question on complex numbers

SC Mathematics

#### How it maps to the curriculum

SC
Mathematics

Strand: 3. Number

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

Suggestions for use: SEC past exam question to assess understanding of complex numbers

Asset type: Document
Language: English
Special Needs:

Daily Banks
Date Topic
2nd AprilLCA Maths Applications (House & Home Mathematics)
1st AprilExpected Value
31st MarchCo-Ordinate Geometry, Parallel and Perpendicular Lines
30th MarchInvestigating the Various Types of Functions
27th MarchNets
26th MarchInferential Statistics - Normal Distribution
25th MarchGeometry - Circle, Tangents, Cords
24th MarchCalculus
23rd March Geometric Series Financial Applications
20th MarchIntegral Calculus
19th MarchSine and Cosine Rules & 3D Applications
18th March Correlations Coefficient

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