Mochi does Maths!

Ms Carolina's Math class

Topic 2 - Functions

Office Hours

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

Week A - Tuesday 7:00 - 8:00 PM JST
Week B - Wednesday 7:00 - 8:00 PM JST

Video Lessons

The structure of the lessons is as follows:

1. Assigned reading.
2. Video explanation
3. Video and classroom examples
4. Class work
Suggested concepts embedded in this topic: 
Representation, relationships, space, quantity, equivalence. AHL: Systems, patterns. 

Content-specific conceptual understandings: 
• Different representations of functions, symbolically and visually as graphs, equations and tables provide different ways to communicate mathematical relationships. 
• The parameters in a function or equation correspond to geometrical features of a graph and can represent physical quantities in spatial dimensions.
 • Moving between different forms to represent functions allows for deeper understanding and provides different approaches to problem solving. 
• Our spatial frame of reference affects the visible part of a function and by changing this “window” can show more or less of the function to best suit our needs. 
• Equivalent representations of quadratic functions can reveal different characteristics of the same relationship. 
• Functions represent mappings that assign to each value of the independent variable (input) one and only one dependent variable (output). AHL 
• Extending results from a specific case to a general form can allow us to apply them to a larger system. 
• Patterns can be identified in behaviours which can give us insight into appropriate strategies to model or solve them. • The intersection of a system of equations may be represented graphically and algebraically and represents the solution that satisfies the equations.
                                                                                                                                    MAA HL Subject guide IBO 2019

This unit will focus on understanding and recognizing the behaviour of the different families of functions, their domain and range, and what changes when functions are combined. The Topic we will be discussing are:

1. Domain and Range

2. Linear functions

3. Quadratic functions

4. Rational functions

5. Radical functions

6. Partial fractions and other rational functions

7. Absolute value functions and equations

8. Piecewise functions

9. Classification of functions according to their mapping

10. Classification of functions according to their parity

11. Operations with functions 

12. Inverse of a function

13. Transformations of functions

14. Properties of exponents

15. Exponential functions and equations

16. Properties of Logarithms

17. Logarithmic functions and equations

18. Trigonometric functions

19. Applications and modelling

Unit Introduction

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

Lesson 1: Domain and Range/Linear functions AA SL 2.1, AA SL 2.2

Period 2

Approx. time 80 minutes


In this lesson you will learn: (This is usually assumed known for HL)

- the concepts of domain and range.

- how to distinguish a functions from a relation using the vertical line test.

- to identify the graphs, end behaviour, domain and range of Linear functions.


INSTRUCTIONS

Before class:

- Read: Mathematics Analysis and Approaches HL by Oxford, pages 72 - 85

- Watch: Video lessons part I on domain and range.

- Watch: Video Lesson  part II on the Vertical Line Test


During Class:

- Investigation: Please work on this investigation regarding the linear function. Solutions

- If you have any questions at any point, please ask them at this time.


Bonus materials

-  part III On linear and quadratic functions

- Do: work along part IV.


Homework: Create a free account at inThinking at the link for the course Mathematics AA HL. Go to MAA then functions, then Equation of a straight line and do the two quizzes and the four ESQ.

Week 18: January 25th - January 29th (Week B)

Lesson 2: The graph of a function AA SL 2.3/2.4

Period 1 and 2

Approx. time 80 minutes


In this lesson you will learn: (This is usually assumed known for HL)

- to understand the difference between graph and sketch.

- to identify important features on the graph of a function, such as x-intercepts, y-intercept, maxima, minima, asymptotes.

- to identify the graphs, end behaviour, domain and range of linear, quadratic, exponential, logarithmic, rational, radical and trigonometric functions


INSTRUCTIONS

Before class:

- Read: Mathematics Analysis and Approaches HL by Oxford, pages 72 - 85

-  part III On linear and quadratic functions


During Class:

- Investigation: Please work on this investigation regarding the graphs of functions and the quadratic function. 

- Submit: Look at the picture on Investigation 5, on page 83. Use it as inspiration to create your own original character. Please use Geogebra or Desmos to create your own character, playing with quadratics, and lines. Submit your picture here.

- Past paper questions using quadratics (Assigned problems 4~13). Solutions

- If you have any questions at any point, please ask them at this time.

- Mini IA presentation. 

               Task 1. 

               Task 2. 

               Task 3 prompt: Analysis of population rates of decrease and increase for various endangered species. How long will it take to bring them back to healthy levels?

                Task 4: Choose your own topic.


Bonus materials

- Do: work along part IV.


Homework

Finish the exercises on quadratic functions.



Week 20: February 8th - February 12th (Week C)

Lesson 3: Rational functions and Radical functions AA SL 2.8,  HL 2.13

Period 1 HL 2

Approx. time 80 minutes


In this lesson you will learn:

- determine the domain and range of rational and radical functions.

- the concept of asymptote.

- to find the vertical and horizontal asymptotes of rational functions.


INSTRUCTIONS


Before class:

- Read: Mathematics Analysis and Approaches HL by Oxford, pages 86 - 90.

- Watch: Video lessons part I, part II, Examples (optional) part III.


During class:

- Questions about homework

- Do: Mathematics Analysis and Approaches HL by Oxford, Exercise 2C (page 89) and Exercise 2D (page 90).

Week 21: February 15th - 19th (Week B)

Lesson 4: Asymptotes and partial fractions AA HL 2.8, 2.13

Period 1:

In this lesson you will learn:
- How to express rational functions as partial fractions. (Part II)
- To solve problems of applications of rational functions.

INSTRUCTIONS

Before class:
- Read: Mathematics Analysis and Approaches HL by Oxford, pages 90 - 93
- Watch: Video lessons part I, part II, Bonus

During class:
- Homework check: You were asked to use your knowledge of important features of functions to graph 1~3 rational functions. How did you do?
- Do: Mathematics Analysis and approaches by Oxford, Exercise 2E, questions 3 and 7.
- Do: Mathematics Analysis and approaches by Oxford, Exercise 2F, questions 3 and 7.
- Do: Additional practice: Revision village - Rational functions questions 1~6.
- Submit: Do and submit your answers to the questions included in this form.


Week 22: February 22nd - 26th (Week C)

Lesson 5: Hybrid functions

Period 1

Approx. time 80 minutes


In this lesson you will learn:

- determine the domain and range of rational and radical functions.

- interpret and graph piece-wise functions manually and using technology.


INSTRUCTIONS


Before class:

- Read: Mathematics Analysis and Approaches HL by Oxford, pages 86 - 90.

- Watch: Video lessons part I


During class:

- Homework check: Last lesson you were asked to complete the questions about application of rational functions. How did you do? See the solutions here.

- Investigation Piece-wise Functions. You will have only one lesson to complete this investigation.

Lesson 6: Absolute value functions AA HL 2.16

Period 2 and 3
Approx. time 80 minutes

In this lesson you will learn:
- The concept of absolute value. (part I)
- The graph, domain and range of the function f(x)=|x|. (part I)
- How absolute value changes graphs of other functions, |f(x)|. (part I)
- How to solve equations and inequalities involving absolute value. (part II and part III)

INSTRUCTIONS
Before class:
- Read: Mathematics Analysis and approaches HL by Oxford, page 93 - 99.
- Watch: Video lessons part I (absolute value functions), part II (Equations), part III (inequalities)

During class:
- Practice: Equations with absolute value
- Do: Mathematics analysis and approaches HL by Oxford, Exercise 2H, questions e and g, page 98.
- Do: Mathematics analysis and approaches HL by Oxford, Exercise 2I, questions 1a, 1d, 1e, 2a, 2d. page 100.

Week 23: March 1st - 6th (Week A) 

Lesson 5 - continued - Equations and inequalities using absolute value

Period 1 


In this lesson you will learn:
- The concept of absolute value. (part I)
- The graph, domain and range of the function f(x)=|x|. (part I)
- How absolute value changes graphs of other functions, |f(x)|. (part I)
- How to solve equations and inequalities involving absolute value. (part II and part III)

INSTRUCTIONS
Before class:
- Read: Mathematics Analysis and approaches HL by Oxford, page 93 - 99.
- Watch: Video lessons part II (Equations), part III (inequalities)

During class:
- Question: How did you do with the introductory paper 3 question? Is there anything you would like to revisit about it?
- Practice: Equations with absolute value
- Do: Mathematics analysis and approaches HL by Oxford, Exercise 2H, questions e and g, page 98.
- Do: Mathematics analysis and approaches HL by Oxford, Exercise 2I, questions 1a, 1d, 1e, 2a, 2d. page 100.

Paper 2 (2)

March 4th

This session you will sit a paper 2.

Please be aware that you are allowed a calculator and a formula booklet.

Week 24: March 8th - March 12th (Week B)

Lesson 6: Classification of functions AA SL 2.2

Period 1

Approx. time 80 minutes


In this lesson you will learn:

- to classify functions according to their mapping. (Part I)

- to classify functions according to their parity. (Part II)


INSTRUCTIONS

Before class:

- Read: Mathematics analysis and approaches by Oxford pages 102 - 108.

- Watch: Video lessons Part I, Part II.


During class:

- Do: Mathematics Analysis and approaches by oxford, exercise 2K, page 105.

- Do: Mathematics Analysis and approaches by oxford, exercise 2L, page 108.

- Submit: Consider the information that you have learnt up to this point about functions. Please submit the answers to the exercises included in the following form.

Lesson 7: Operations with functions AA SL 2.3

Period 2

Approx. time 80 minutes


In this lesson you will learn:

- how to add, subtract, multiply and divide functions. (Part I)

- to identify the domain and range of a resulting functions. (Part I)

- to do the composition of functions (Part II)

- to identify the domain and range of a composition. (Part II)


INSTRUCTIONS

- Read: Mathematics analysis and approaches by Oxford pages 108 - 111.

- Watch: Video lessons Part I, Part II, Part III (examples 1, 2), Part IV (Examples 3, 4, 5)

- Do: Mathematics Analysis and approaches by oxford, exercise 2M, page 111.

- Do: Write a comment in the stream for this lesson giving your own example of a function that is the composition of three functions, f(g(h(x))) and state the individual functions f, g, and h. (example: f(g(h(x)))=1-(x+1)^2, where f(x)=1-x, g(x)=x^2, and h(x)=x+1)

Week 25: March 29th - April 3rd (Week C)

Lesson 8: Inverse functions AA SL 2.2, AA SL 2.5

Period 1

Approx. time 80 minutes


In this lesson you will learn:

- The concept of identity function. (Part II)

- the concept of inverse function and how to find them. (Part I)

- the concept of self inverse. (Part II)

- how to find the domain and range of an inverse functions. (Part III)


INSTRUCTIONS

- Read: Mathematics analysis and approaches by Oxford pages 112 - 116.

- Watch: Video lessons Part I, Part II, Part III.

- Do: Mathematics Analysis and approaches by oxford, exercise 2N, page 116.

- Submit: Please submit the answers to the exercises included in the following form.

Lesson 9: Transformations of functions I AA SL 2.11, AA HL 2.16

Period 2

Approx. time 80 minutes


In this lesson you will learn:

- to transform functions using vertical and horizontal displacements. (Part I)

- to transform functions using magnifications. (Part II)

- to transform functions using reflections. (Part II)


INSTRUCTIONS

- Read: Mathematics analysis and approaches by Oxford pages 117 - 127.

- Watch: Video lessons Part I, Part II.

- Do: Mathematics Analysis and approaches by oxford, exercise 2O, page 118, questions 3, 5, and 6.

- Do: Mathematics Analysis and approaches by oxford, exercise 2P, page 120, question 2.

- Do: Mathematics Analysis and approaches by oxford, exercise 2Q, page 126, questions 1 and 3.

- Do: play this kahoot! (Link expires May 15th) Game PIN: 05886179

Lesson 10: Transformations of functions II (Further examples)

Period 3

Approx. time 80 minutes


This is a practice lesson. In this lesson you will continue to learn:

- How to graph functions with multiple transformations

- How to write a function's equation undergoing multiple transformations


INSTRUCTIONS

- Read: Mathematics analysis and approaches by Oxford pages 127 - 139.

- Watch: Video lessons Part I (Further Examples).

- Do: Mathematics Analysis and approaches by oxford, exercise 2R, page 134, question 2.

- Do: Mathematics Analysis and approaches by oxford, exercise 2S, page 139.

- Submit: Please submit the  answers to the exercises included in the following form.

Week 26th: April 5h - April 9th (Week A)

Lesson 11: Properties of Exponents and Exponential equations AA SL 2.9, AA SL 2.10

Period 1

Approx. time 80 minutes


For this lesson you need to review beforehand:

- Properties of exponents (Bonus video)

- Factorization


In this lesson you will learn:

- How to solve equations with exponentials algebraically. (Part I)

- How to solve equations with exponentials using technology. (Part II)

- Graphs of exponential functions. (Part I)


INSTRUCTIONS

- Read: Mathematics Analysis and approaches HL by Oxford, pages 460 - 464

- Read: Mathematics analysis and approaches HL by Oxford, pages 473 - 475 (476* Read only up to the yellow information box and recommend doing investigation 11 (to understand number e), you do not need to be concerned about the text that follows this box yet AND investigation 12 question 6.)

- Watch: Video lessons part I, Part II, Bonus (optional - watch first)

- Do: Mathematics Analysis and approaches HL by Oxford, exercise 7C questions 2, 6, 8, page 464.
-Do: Mathematics Analysis and Approaches HL by Oxford, exercise 7E questions 3, 4, 5, page 481-482.

- Do: Answer the question in the classroom for the corresponding lesson 19.

Paper 1 (3)

April 8th


You will sit a paper 1 test. 

This test will include everything in topics 1 and 2, including and up to Lesson 10 on functions.

This is a non calculator test.

Bring a printed copy of your formula booklet.

Week 27: April 12th - April 16th (Week B)

Lesson 12: Properties of logarithms and logarithmic equations I AA SL 2.9

Period 1

Approx. time 80 minutes


For this lesson you need to review beforehand:

- The concept of logarithm (Bonus video by Vi Hart)

- Factorization


In this lesson you will learn:

- The definition of logarithm (Part I)

- The properties of logarithms (Part I)

- The graph of logarithmic functions (Part I)

- The connection between logarithms and exponential functions (Part I)


INSTRUCTIONS

- Read: Mathematics Analysis and approaches by Oxford, pages 465 - 472

- Read: Mathematics Analysis and approaches by Oxford, pages 477** - 482 (477 from Investigation 13)

- Watch: Video Lessons part I, Bonus (Bi Vi Hart - Optional)

- Do: Mathematics Analysis and Approaches HL by Oxford, Exercise 7E questions 6, 7, 8, 10

- Submit: Form

Week 28th: April 19th - April 23rd (Week C)

Lesson 13: Properties of logarithms and logarithmic equations II AA SL 2.10

Period 1

Approx. time 80 minutes


For this lesson you need to review beforehand:

- The concept of logarithm (Bonus video by Vi Hart)

- Factorization


In this lesson you will learn:

- To solve equations with Logarithms (Part I)

- The laws of logarithms (Part I and III)


INSTRUCTIONS

- Read: Mathematics Analysis and approaches by Oxford, pages 465 - 472

- Read: Mathematics Analysis and approaches by Oxford, pages 477** - 482 (477 from Investigation 13)

- Watch: Video Lessons part I, part III, and part II (examples), Bonus (Bi Vi Hart - Optional)

- Do: Mathematics Analysis and approaches by Oxford, exercise 7D, odd questions, page 472.

- Do: Answer in classroom the question corresponding to Lesson 21

Lesson 14: Trigonometric functions Part I AA SL 3.7

Period 3

Approx. time 80 minutes


For this lesson you need to review beforehand:

- Trigonometric ratios

- Trigonometric identities


In this lesson you will learn:

- The graphs of Sine, Cosine and Tangent as functions of x (Investigation and Part I)

- How to graphs Sine and cosine (Part II and III)

- How to transform the graphs of sine and cosine (Part III)


INSTRUCTIONS

- Do and Submit: Trigonometric functions activity. You will need: blank paper, ruler, protractor, thread, compass (or any circular object of about 6 cm in diameter). Submit your work in the corresponding classroom assignment for lesson 22.

- Read: Mathematics Analysis and approaches by Oxford, pages 410 - 417

- Watch: Video Lessons part I, part II, and part III.

- Do: Mathematics Analysis and approaches by Oxford, exercise 6N, questions 1, 2, and 4. 

- Do: Mathematics Analysis and approaches by Oxford, exercise 6O, questions 1 and 4 (a, b, c, d).

Week 29th: April 19th - April 23rd

Lesson 15: Trigonometric functions part II AA SL 3.7

Period 2

Approx. time 80 minutes


For this lesson you need to review beforehand:

- Trigonometric ratios

- Trigonometric identities


In this lesson you will learn:

- To model data using trigonometric functions (Part I)

- To solve word problems involving trigonometric functions (Part II)


INSTRUCTIONS

- Read: Mathematics Analysis and approaches by Oxford, pages 410 - 417

- Watch: video lessons Part I and Part II

- Do: Mathematics Analysis and approaches by Oxford, exercise 6O, questions 2 and 3.

- Do: In the stream for this lesson mention 3 things, places, etc you have seen in your culture/city/etc that can be modelled through a sine or cosine curve.

Lesson 16: Trigonometric functions Part III AA HL 3.9

Period 3

Approx. time 80 minutes


For this lesson you need to review beforehand:

- Trigonometric ratios

- Trigonometric identities


In this lesson you will learn:

- To graph calculate inverse trigonometric functions (Part I)

- To solve trigonometric equations by hand and using technology (Part II)


INSTRUCTIONS

- Read: Mathematics Analysis and approaches by Oxford, pages 418 - 421

- Watch: video lessons Part I, Part II.

- Do: Mathematics analysis and approaches HL by Oxford, exercise 6P, questions 2, 4, 5.

- Submit: Assignment sent via Kognity: Functions review.

Review session

Period 4


Please be aware this is a support period. 


I highly recommend that you spend time working on either of the following:


1. Asking for support in the areas that you have identified you need help with.

2. working on your semester 2 essay.

3. Working on past exam questions or exercise sets from the different textbooks.

4. reviewing concepts and quizzing your knowledge against a friend.

Week 29th: April 26th - April 30th (Week A)

Lesson 17: Graphs of tangent and reciprocal trigonometric functions  AA HL 3.9

Sep 8th, 2020

Approx. time 80 minutes


For this lesson you need to review beforehand:

- Trigonometric ratios

- Trigonometric identities

- Graphs of Sine, Cosine and Tangent.


In this lesson you will learn:

- To graph calculate reciprocal trigonometric functions Secant, Cosecant and Cotangent.

- To solve trigonometric equations by hand and using technology involving reciprocal trigonometric functions.


INSTRUCTIONS

Before Class:

Please make notes of any questions that show up and have them ready when you come to class.

- Read: Mathematics Analysis and approaches by Oxford, pages 410 - 413

             - Supporting reading: Kognity. Lesson 18 - Graphs of Tangent and reciprocal trigonometric functions

- Watch: video lessons 

During Class:

At any point during our lesson you can come to me for additional support.

- Do: Share your questions with the class (20 minutes)

- Do: Mathematics: Analysis and approaches HL by Oxford, page 414, exercise 6N, q. 1, 2. (10 minutes)

- Do: From this set of graphing tasks, do 1, 3, 4, 5, 7, 9, 12, 15, 17, 21, 23, 24. Please check your work with your Calculator. If there are any differences check what went wrong and how your graph can be improved. Check these guidelines for good graphing. (50 minutes)

Stewart. PreCalculus Mathematics for Calculus. 5th Edition. Page 255.

 
 

Though these guidelines are for polynomials, they apply as well for any other type of function

Lesson 18: Graphs of inverse trigonometric functions AA HL 3.9

Sep 9th, 2020

Approx. time 80 minutes


For this lesson you need to review beforehand:

- Trigonometric ratios

- Trigonometric identities

- Graphs of Sine, Cosine and Tangent.

- Graphs of reciprocal functions


In this lesson you will learn:

- To graph the inverse trigonometric functions for Sine, Cosine and Tangent.

- To solve trigonometric equations by hand and using technology involving inverse trigonometric functions.


INSTRUCTIONS

Before Class:

Please make notes of any questions that show up and have them ready when you come to class.

- Read: Mathematics Analysis and approaches by Oxford, pages 418 - 421

             - Supporting reading: Kognity. Lesson 19: Graphs of inverse trigonometric functions

- Watch: video lessons 

During Class:

At any point during our lesson you can come to me for additional support.

- Example questions using Reciprocal and inverse trigonometric functions. (15 minutes)

- Do: Share your questions with the class (20 minutes)

- Do: Mathematics: Analysis and approaches HL by Oxford, page 420, exercise 6P, q. 1, 2, 3, 4. (20 minutes) (Optional)

- Do: Kognity. Lesson 19: Graphs of inverse trigonometric functions Task. If possible, please check your work with your Calculator. If there are any differences check what went wrong and how your work can be improved. (20 minutes)

- Do: past exam questions (Priority)

Lesson 19: Trigonometric equations revisited

Sep 10th (HL1) / 11th (HL2), 2020

Approx. time 80 minutes


For this lesson you need to review beforehand:

- Trigonometric ratios

- Trigonometric identities

- Graphs of all trigonometric functions and their inverses.


In this lesson you will Practice:

- Graphing trigonometric functions

- Solving trigonometric equations analytically using trigonometric identities and using technology.


INSTRUCTIONS

Before Class:

There is no reading assign nor videos to watch for this lesson.

Please read your notes and the mind-map you created for the unit functions. If there are any questions make note of them and prepare to ask them during class.

During Class:

If there are any questions regarding Functions please ask them at this time.

Work on this set of question to practice the whole of the unit Functions. Solutions

End Of Unit