# Mochi does Maths!

## Ms Carolina's Math class

### Office Hours

Office hours will happen on a rotation basis during a cycle.

Tuesdays: 14:30 ~ 16:30
Thursdays: 14:30 ~ 16:30

### Video Lessons

The structure of the lessons is as follows:

2. Video
3. Video and classroom examples
4. Class work.

Videos may  contain additional examples or content that may not be in the textbook. However, if you can't watch the videos, you should not worry, you still have access to all the textbooks and tasks via our classroom and mochidoesmaths.com.

In this section you will find all the video lessons regarding our first unit: Algebra.

Essential understandings:

Number and algebra allow us to represent patterns, show equivalencies and make generalizations which enable us to model real-world situations. Algebra is an abstraction of numerical concepts and employs variables which allow us to solve mathematical problems.

Suggested concepts embedded in this topic: Generalization, representation, modelling, equivalence, patterns, quantity AHL: Validity, systems.

Content-specific conceptual understandings:

• Modelling real-life situations with the structure of arithmetic and geometric sequences and series allows for prediction, analysis and interpretation.

• Different representations of numbers enable equivalent quantities to be compared and used in calculations with ease to an appropriate degree of accuracy.

• Numbers and formulae can appear in different, but equivalent, forms, or representations, which can help us to establish identities.

• Formulae are a generalization made on the basis of specific examples, which can then be extended to new examples.

• Logarithm laws provide the means to find inverses of exponential functions which model real-life situations.

• Patterns in numbers inform the development of algebraic tools that can be applied to find unknowns.

• The binomial theorem is a generalization which provides an efficient method for expanding binomial expressions.

AHL

• Proof serves to validate mathematical formulae and the equivalence of identities.

• Representing partial fractions and complex numbers in different forms allows us to easily carry out seemingly difficult calculations.

• The solution for systems of equations can be carried out by a variety of equivalent algebraic and graphical methods.

MAA HL Subject guide IBO 2019

### Lesson 0 and Lesson 1 - Presumed knowledge

Monday

In this lesson you will learn or revise:

- the use of scientific notation.

- to operate numbers in scientific notation (+ - x /).

- To apply exponents to numbers written in scientific notation.

INSTRUCTIONS

Before you come to class

- Read from the subject guide the pages 20 - 25.

- Watch the video lesson: Lesson 0 (Note please ignore the comments regarding Kognity and Managebac)

In class:

- Introduction to the course

- Introduction to assessment

- First mock essay

- Self taught topics

### Lesson 2 - Arithmetic Sequences

Wednesday

In this lesson you will learn:

- the concept of sequence.

- to identify an arithmetic sequence.

- to find a general formula for an arithmetic sequence.

- to find the number of terms in an arithmetic sequence.

- to predict the nth term of an arithmetic sequence.

- to find the number of terms in an arithmetic series.

- to find the partial sum of an arithmetic series.

INSTRUCTIONS

Before Class:

- Watch: Video lesson for this session

Lesson 3 - Part II    Series and Sigma notation

Lesson 3 - Part III   Arithmetic Series

During class:

- Do: If there are any questions regarding the materials, please ask them during the first 10 minutes.

- Pay close attention to Further examples.

﻿- Homework: Watch next lesson's video and read the assigned sections in preparation for next class. ### Lesson 3: Geometric sequences

Friday

In this lesson you will learn:

- to identify a geometric sequence.

- to find a general formula for a geometric sequence.

- the concept of series.

- to read and write using appropriate notation (sigma notation).

- to identify an arithmetic series.

- to find a general formula for an arithmetic series.

INSTRUCTIONS

Before Class:

- Watch: Video lessons for this day:

Lesson 3 - Part I     Geometric Sequences;

Please make sure to take a 5 minute break between videos. Stretch your arms, legs and torso!

During class:

- Do: If there are any questions regarding the materials, please ask them at this time.

- Homework: Watch next lesson's video and read the assigned sections in preparation for next class. ### Lesson 4: Geometric Series and Convergence

Wednesday

In this lesson you will learn:

- to find the number of terms in a geometric sequence.

- to predict the nth term of a geometric sequence.

- to identify a geometric series.

- to find a general formula for a geometric series.

- to find the number of terms in a geometric series.

- to find the partial sum of a geometric series.

- applications of geometric sequences and series.

INSTRUCTIONS

Before Class:

- Watch: Video lessons for this day:

Lesson 4 - Part I         Geometric Series

Lesson 4 - Part II        Infinite Geometric Series and convergence.

During class:

﻿﻿- Worked examples.

### Applications

Friday

In this lesson you will learn:

- Applications of sequences and series to real life situations.

INSTRUCTIONS

During class:

- If you have any questions please address them at this time.

- Further examples on Applications: compound interest, appreciation, depreciation, etc.

### Test 1

Wednesday

In this lesson you will reinforce your knowledge of sequences and series.

INSTRUCTIONS

Before Class:

- Put your observations and results from the exploration of the Koch snowflake into this shared document: Link

- Work on the study guide, share dwith you in the Google classroom.

- Review your notes on compound interest

During class:

- Test I: This test will be a mix of Paper 1 and Paper 2 style questions, that means you must bring your calculator to sit the test as well as a printed copy of the formula booklet.

- We will continue to develop pending and further examples on applications of sequences and series

### Lesson Mini IA I

Friday

INSTRUCTIONS

Before Class:

- Read the subject guide's section on the toolkit i.e the internal assessment, paying special attention to the assessment criteria.

During class:

- Ask any relevant questions regarding the exploration/Toolkit.

- Read and mark the sample essay provided and compare your notes to the original scoring.

- Start to work on your first semester essay: Lacsap's fractions.

Homework:

- You will continue to work on this task at home. The expectation is that you will submit an essay. Please don't leave this task for last minute. Next week you will be given one more block to work on it.

### Lesson 5 - Proof I

Monday

In this lesson you will learn:

- the basics of Logic

- to prove simple statements using direct proof

INSTRUCTIONS

Before Class:

- Watch: Video lessons for this day:

Lesson 5 - Part I         Basic Laws and simple proofs (Logic) (Optional)

Lesson 5 - Part II.       Direct Proof

During class:

﻿﻿- Worked examples (Supplied in class)

### Lesson 6 - Proof II

Wednesday

In this lesson you will learn:

- to prove simple statements using proof by contradiction

- to prove statements using the contrapositive

INSTRUCTIONS

Before Class:

- Watch: Video lessons for this day:

Lesson 6 - Part III         Contradiction and Contrapositive

During class:

﻿﻿- Worked examples (Supplied in class)

### Lesson 7: Proof by Mathematical induction I

Friday

In this lesson you will learn:

- to prove simple statements using proof by contradiction

- to prove statements using the contrapositive

INSTRUCTIONS

Before Class:

- Watch: Video lessons for this day:

Lesson 7 - Part I         The process of Mathematical Induction

During class:

﻿﻿- Worked examples (Supplied in class)

### Lesson 8: Proof by Mathematical Induction II

October 29th

Approx. time 80 minutes

In this lesson you will learn:

- to prove statements using Mathematical induction: divisibility and inequalities

INSTRUCTIONS

Before Class:

- Read: Kognity reading Assignment assignment Lesson 8 - Mathematical Induction (Reading assignment shared with Lesson 9)

- Read: Complementary reading from Pearson. pages 226 - 234. (Pearson textbook available via Managebac.)

- Watch: Video lessons for this day:

Lesson 8 - Part II.       Mathematical Induction to prove inequalities

During class:

﻿﻿- Worked examples (Supplied in class)

- Do: Kognity. Lesson 9 - Mathematical Induction II Problem Set

### Pre test Practice

HL 1 Nov 3rd / HL2 Nov 5th

I will provide exercises for you to work on during class, as practice for the test Paper 1 (1)

### Paper 1 (1)

HL1 Nov 4th / Nov 6th

This is our first summative test of the year. Please make note of the following information:

- This test included everything we have studied so far: sequences and series, direct proof, contradicion, proofby contrapositive and mathematical induction.

- This is a non-calculator test.

- If you feel you need language support, you are allowed to bring a your-language/English dictionary, that does not contain definitions, only translations.

- Bring pen black or blue. Do not answer any questions with pencil.

- Do not use whiteout or any king of correction tapes.

### Study Guide

HL1: November 10th

HL2: November 9th

In this lesson you will work on the study guide for paper 2.

### Paper 2

November 11th

This day you will sit Paper 2.

You will be required to:

- Bring a clean printed copy of the MAAHL information booklet.

- Pens with black or blue ink.

### Mathematical induction - continued

HL1: November 12th

HL2: November 10th

In this lesson you will learn:

- To use Mathematical induction to propositions including inequalities.

INSTRUCTIONS

Before Class:

- Read: Kognity reading Assignment assignment Lesson 8 - Mathematical Induction (Reading assignment shared with Lesson 9)

- Read: Complementary reading from Pearson. pages 226 - 234. (Pearson textbook available via Managebac.)

- Watch: Video lessons for this day:

Lesson 8 - Part II.       Mathematical Induction to prove inequalities

During class:

﻿﻿- Worked examples (Supplied in class)

- Do: Kognity. Lesson 9 - Mathematical Induction II Problem Set

From IBID textbook (available for download on managebac) Exercise 12.2 Problems a through j, Page 432

You can complement the video lesson with examples 12.6 and 12.7 on pages 431 and 432.

### Lesson 9 - Binomial Theorem

HL1: November 13th

HL2: November 12th

In this lesson you will learn:

- the postulate of the binomial theorem

- to use the binomial theorem to expand binomials

- to use the binomial theorem to find a term in a binomial expansion

INSTRUCTIONS

Before Class:

- Watch: Video lessons for this day:

Lesson 9 - Part I

During class:

﻿﻿- Worked examples (Supplied in class)

- Do: Kognity. Lesson 9 - Binomial theorem Exercise Set

### Lesson 10 - Binomial Theorem II

HL1: November 19th

HL2: November 17th

In this lesson you will learn:

- to find coefficients of specific terms in a binomial expansion

- to find the value of an unknown given the value of a term

- to find terms in a polynomial expansion resulting from the product of two binomial polynomials

INSTRUCTIONS

Before Class:

- Watch: Video lessons for this day:

Lesson 10 - Part I

During class:

﻿﻿- Worked examples (Supplied in class)

- Do: Please finish the Type I task called Binomial coefficients supplied last week. This will be the last lesson you will have to complete the task. Don't forget to support each other.

### Lesson 11 - Polynomial division

HL1: November 19th

HL2: November 20th

In this lesson you will learn:

- To use long division to factorize polynomials

- to find the roots of a polynomial/solve polynomial equations

- To use Synthetic division to factorize polynomials

NOTE: This topic IS NOT included in the syllabus however it is a very important tool of Algebra that will be useful in our study of Polynomials, thus you are required to be familiar with either of the two methods presented.

INSTRUCTIONS

Before Class:

- Watch: Video lessons for this day:

Lesson 11 - Part I - Long Division

Part III - Systems of linear equations

During class:

﻿﻿- Worked examples (Supplied in class)

### Lesson 12 - Simultaneous linear equations

HL1: November 24th

HL2: November 26th

In this lesson you will learn:

- To solve systems of 3 equations and 3 unknowns by elimination

- To solve systems of 3 equations and 3 unknowns by substitution

- To solve systems of 3 equations and 3 unknowns using technology (GDC)

INSTRUCTIONS

Before Class:

- Read: MAA HL Oxford. Page 205-208.

- Watch: Video lessons for this day:

Lesson 12 - part I

During class:

﻿﻿- Worked examples (Supplied in class)

- Discuss: what are the conditions for a system to have one, none, or an infinite number of solutions.

### Lesson 13 - Complex Numbers: Introduction

HL1: November 25th

HL2: November 30th Week C - OC

In this lesson you will learn:

- The definition of a complex number

- to add, subtract and multiply complex numbers

- to find the complex conjugate of a complex number

- To divide two complex numbers

- the concept of modulus of a complex number

INSTRUCTIONS

Before Class:

- Read: MAA HL Oxford. Page 161-171

- Watch: Video lessons for this day:

Lesson 13 - Part I

During class:

﻿﻿- Worked examples (Supplied in class)

- Discuss: How are complex numbers different from real numbers? What rules do you foresee not applying to complex numbers? Where in real life do we use complex numbers?

- Task I: Oxford MAA HL. page 165. Exercise 3F, question 3.

- Task II: Oxford MAA HL. page 171. Exercise 3G, question 1-5.

### Study guide

December 1st

Today you will work on a study guide to prepare for tomorrow's test.

### Paper 1 (2)

HL1: December 2nd

HL2: December 2nd

In this lesson you will sit a paper 1 exam. Please note the following:

1. This is a non-calculator paper.

2. Please bring a printed copy of the formula booklet

3. The content is cumulative, in this test you may find questions on:

SL

Sequences and Series and applications

Direct proof

Number sets

Scientific notation

HL

Sequences and Series and applications

Direct proof, proof by contradiction, contrapositive, and induction

Number sets, Scientific notation

Systems of simultaneous equations 3x3

Binomial theorem

Operations with complex numbers, including addition, subtraction, multiplication, division and complex conjugate.

Simple equations

### Lesson 14 - Polar form (Cis form)

HL1: December 3rd

HL2: December 3rd - OC

In this lesson you will learn:

- The definition of modulus and argument

- How to write a complex number in Euler form

- How to write a number in Cis form

- How to find the principal argument

- the differences between principal argument and argument

- the properties of the modulus and the argument.

INSTRUCTIONS

Before Class:

- Read: MAA HL Oxford. Page 650-655

- Watch: Video lessons for this day:

Lesson 14 - Part I

Lesson 14 - Part II

During class:

﻿﻿- Worked examples (Supplied in class)

- Discuss: How are complex numbers different from real numbers? What rules do you foresee not applying to complex numbers? Where in real life do we use complex numbers?

- Task I: Oxford MAA HL. page 655. Exercise 10A.

- Task II: Cambridge MAA HL page 363 Exercise 8.2, questions 1, 2, 3.

### Lesson 15 - DeMoivre's Theorem

HL1: December 10th

HL2: December 8th

In this lesson you will learn:

- to apply DeMoivre's theorem to calculate powers of complex numbers

INSTRUCTIONS

Before Class:

- Read: MAA HL Oxford. Page 664-667

- Read: MAA HL Oxford. Page 672-674

- Watch: Video lessons for this day:

Lesson 15 - Part I

Lesson 15 - Part II

During class:

﻿﻿- Worked examples (Supplied in class)

- Discuss: What are the properties of Cis form? How are these properties connected to the properties of the argument?

- Task I: Oxford MAA HL. page 667. Exercise 10D.

- Task II: MAA HL Oxford. Page 674. Exercise 10F

### Lesson 16 - Roots of polynomials I

HL1: December 13th

HL2: December 12th

In this lesson you will learn:

- to apply DeMoivre's to the calculation of fractional roots of complex numbers

- to solve equations of the form z^n=a+ib

- to understand the roots of a complex number as the vertices of a regular polygon

- to graph the roots of complex numbers on the complex plane

INSTRUCTIONS

Before Class:

- Read: MAA HL Oxford. Page 664-671

- Watch: Video lessons for this day:

Lesson 16 - Part I

During class:

﻿﻿- Worked examples (Supplied in class)

- Discuss: What relationships exists between pairs of complex roots?

- Task I: Oxford MAA HL. page 671. Exercise 10E.

### Lesson 17 - Roots of polynomials II

HL1: December 13th

HL2: December 12th

In this lesson you will learn:

- the Remainder theorem

- the factor theorem

- How to use the remainder and factor theorems to find the roots of polynomials.

INSTRUCTIONS

Before Class:

- Read: MAA HL Oxford. Page 178-190

- Watch: Video lessons for this day:

Lesson 17 - Part I

During class:

﻿﻿- Worked examples (Supplied in class)

- Discuss: What relationships exists between pairs of complex roots?

- Task I: Oxford MAA HL. page 181. Exercise 3J. question 2.

- Task II: Oxford MAA HL. page 184. Exercise 3K. questions 6-10.

- Task III: Oxford MAA HL. page 190. Exercise 3L. questions 1-6, (a, b, and c only from each numeral)

### Week 17: January 18 - January 22 (Week A)

We are changing to a counter system rather than dates. For example this week we will meet twice. You have tour schedules and you know what date each period will happen. From here onwards lessons will me noted by the number of the period in the week. For easier planning.

Period 1:

In this lesson you will review:

HL

- the Remainder theorem

- the factor theorem

- Algebra of complex numbers

INSTRUCTIONS

During class:

- Kahoot HL (we will play a kahoot Precalc Complex numbers review). (Game Pin: 09334462)

- Kahoot SL (we will play a kahoot IB Math Number and Algebra). (Game Pin: 0150606)

- Formative test HL SL

Note Period 2 will be at the start of the Functions unit.