math2015Term1

Term 1 IB Mathematics Class of 2015
Here a log of our classes with a summary of what we covered and some additional material and points to reflect on.

Remember, there is a search function for you to use. Remember to check the Resources page before you ask me for material you think I may have already made available to you.

Success & enjoy, Arno.

=== //Thu Dec 5//

Well done to Asa, Michael, Richard to give the Unit 3 Test a go in class, and thanks to all who handed it in.

Here are the solutions the test:

Enjoy, Arno.

=== //Week Dec 2 - 6//

Revision week.

Here the practice exam papers(hard copy handed out in class) with their solutions:
 * paper 1: [[file:Semester One Exam Review Paper One.pdf]] & [[file:Semester One Exam Review Paper One - Solutions.pdf]]
 * paper 2: [[file:Semester One Exam Review Paper Two.pdf]] & [[file:Semester One Exam Review Paper Two - Solutions.pdf]].

Success & enjoy, Arno.

=== //Wed 27 & 28 Nov//

Trig compound angle formulae are a very useful property to obtain exact solutions, prove identities and re-writing trig expressions:.

Homework from the textbook: p. 359 Qs 7, 8, 11 p. 363 Qs 1, 3, 7 p. 368 Qs 2, 5 P. 374 Qs 4, 5

Monday for those who wish, they can sit for a grade the Unit 3 Test, otherwise can do it as practice.

The next week will all about revision for the examination papers, two of them, which will be held during the last week before the holidays, during all three lessons of the week.

Make sure you are familiar with your Formula Booklet for the exams.

Success & enjoy, Arno.

=== //Mon 25 Nov evening//

And here the solutions to the Problem Set for Self-assessment:

Ensure that you go through your answers with these solutions and grading yourself with a different colour pen.

Success & enjoy, Arno.

=== //Mon 25 Nov//

An introduction and exploration of the reciprocals of Sin, Cos and Tan, namely, in order, Cosec/Csc, Sec, and Cot. And then some very fine **trig identities** based on the Pythagorean we discovered earlier in our unit circle.

Remember how to use the graphical capabilities of your TI solve trig equations and find **all** the solutions.

Notes here:

You may find the following problems from the textbook of interest: page 302 Short Qs 1 - 6 and 7 & 8 for the brave, pages 302-303 Long Qs 2 & 3.

Pablo found a useful website with exact values for special angles: Exact Values Special Angles

Success & enjoy, Arno.

=== //Sun 24 Nov//

As promised a few extensions to the material discussed this Thursday. I am focussing here on the trigonometric objects Sin, Cos and Tan as functions. While you have everything you need for the self-assessment by now, you may still want to practice a little from the textbook. Hence, the self-assessment won't be due till Thursday of this week. In our textbook, the most pertinent information talked about in the screencasts is found in Chapter 9. In particular, Questions 4 - 7 on pages 260-261 are of interest, as are Questions 2 - 4 on page 264.

On an aside, this week is the last of new material until the holidays. The whole of the week of December 2nd will be devoted to revision of the material covered to date. Then during the last week, spread over all three classes, will be a set of exams looking at the material, divided up in a paper do be done without a GDC, and one with your trusty TI. Hence, the last exam paper will be during our Thursday December 12th class.

Finally, the two screencasts I talked about below.

Success & enjoy, Arno. media type="custom" key="24514264" media type="custom" key="24514268"

=== //Thu 21 Nov//

Angles beyond 90 degrees. So much fun to move around the unit circle! In the course of that exploration, we even discovered our first Trigonometric Identity math \sin^2(\theta) + \cos^2(\theta) = 1, \quad \forall \quad \theta math known as the Pythegorean identity.

Importantly from this class' first section is the realisation that the trig ratios repeat as we move through the 4 quadrants of the unit circle, e.g., cosine of 120 degrees is the same as the negative of cosine of 60 degrees. Details in the notes:.

This then let us to consider these trig rations as functions, namely periodic functions, which we require if ever we would like to describe, i.e., model, periodic, or cyclical, phenomena. What do periodic phenomena have: periods, amplitudes, vertical and horizontal off-sets, i.e., what do they oscillate about, the equilibrium y-value, and what is the value of the function at the "start" where the independent variable has the value 0. We explored this by looking at this GeoGebra file:

Lastly, I introduced a new type of assessment, namely self-assessment, in order to promote and encourage reflection in your learning also when you have finished an assessment. The criteria on which your self-assessment mark, i.e., how well did you perform assessing your own work, is based on these:.

I will need to share a screencast before I can let you loose on this week's assessment. Stay tuned. Regardless, I will handout the assessment tonight at 18:30 during prep sign in.

Success & enjoy, Arno.

=== //Mon 18 Nov//

A long and hard look at the purpose and rationale for mathematical investigations, as well as some in-class time to work on the investigation from last week. Due date this Friday, in PDF, by email.

Success & enjoy, Arno.

=== //Thu 13 Nov//

Importantly, we will often considers angles not in degrees but in **radians**, which is the natural measure of an angle, defined in a circle: when the arc length in a circle is equal to its radius, then its opening angle is defined as 1 radian. You must be able to convert between raidans and degrees, and know the exact trig ratios for Sin and Cos when the angles are given in radians, i.e. for the following fractions of pi: 2, 3, 4, 6.

Notes here:

Homework from the textbook:

This week is also the requisite assignment for this unit, which is an investigation. It is based on a real problem of being a piece of land and will make use of trigonometry studied this week. This is it: And it will be graded based on three aspects detailed in this rubric: Due date for the investigation is **by** next class, typed in and PDFed emailed to me.

Success & enjoy, Arno.

=== //Wed 12 Nov//

And our third unit will be trigonometry.

In this week we will consider method to deal with general triangles, i.e., once which are not right-angled triangles. For that we need the Cosine and Sine rule. And there is a nifty equation to give you the area of a triangle based on two sides and the angle in between. Notes here:

Make sure you know without reference the exact trig ratios of Sin and Cos for the following angles (in degrees): 90, 60, 45, 30, 0.

Homework is from the textbook.

Success & enjoy, Arno.

=== //Thu 7 Nov//

Test time. Unit 2 finishes

And here are the solutions:

=== //Wed 6 Nov//

Today you received your assessment form last week back. Note that only one of you! decided to impress me with doing the Investigation, rather than the Quiz -and impress me he did! You are still encouraged to work through that exercise as a preparation (hint) for the Unit Assessment tomorrow. The solutions are here:

And we continue with modelling today, but now using Euler's number and looking at the Gaussian distribution. For that, you need to use this GeoGebra files:
 * SAT math scores: [[file:SAT Math Scores.ggb]]
 * logistic model: [[file:Logistical Growth Model.ggb]]

And the Notes are here:

Success & enjoy, Arno.

=== //Mon 4 Nov//

A week of modelling ahead of us using GeoGebra extensively. Here is what we worked on today:

This is also the week of this test, i.e., your requisite major assessment will be this week, a pleasant little assessment on Thursday this week.

The solutions to last week's quiz, which even if you didn't decide to hand in, would be a great (hint) preparation for the test on Thursday, is here:.

In Question 2, using your TI would have been ideal. Here is a video showing you how, sorry no sound as I am not carrying my headphones with me and wanted to make this available to you folks: media type="custom" key="24323250".

Success & enjoy, Arno.

=== //Thu 31 Oct//

Exponential and logarithms are widely used in applied mathematics. Hence a look at how to use it. However, we spend first a little time on the homework (see yesterday's log entry).

Then we considered the Rule of 72 in some detail, and how the work of this week gives us more precise information on how to consider the doubling of your capital given an interest rate can be calculated; see these notes:

Then we looked at the IPO graph and watched the explanatory video considering what happens when you plot on log scales. This ties in with one of the two possible assignments for this week, and for this one you may find the following GeoGebra file useful:

So, the minor assessment for this week is a **choice** out of two, meaning you must do at least one, and as said in class, I will be more impressed if you tackle the second than if you complete the first one.
 * 1) [[file:Diagnostic Quiz Lessons 1, 2 & 3.pdf]]
 * 2) [[file:Week 7 Minor Summative Assessment.pdf]]

Due on Monday.

Success & enjoy, Arno.

=== //Wed 30 Oct//

//A very special base today:// **//e//** which is Euler's number math e = lim_{n->\infty} (1 + 1/n)^n math

Also how to change base on logs.

Here the notes:

Practice from your textbook: Exercise 3H, Pages 117—188: Work at least one from 11a, b and c 13a,b Exercise 4D.1, Pages 135—136, A small selection from 1 to 6 Exercise 4D.2, Pages 136—137, A small selection from 1 to 5 Exercise 4E, Page 138—139 A small selection from 2; At least one from 5d, e and f A small selection from 6 and 7 Exercise 4F, Page 140, A small selection from 1 to 7.

Success & enjoy, Arno. === //Mon 28 Oct//

Exponents & Logarithms is the flavour of this week. Here are the notes:

Today we looked at the link between exponents and logarithms, what they "do", look like and that they are each other's inverse. Importantly, we looked at the exponent and log laws, which are crucial, as they will be with us for many other sections in the course ... **know them and know how to apply them!**

To that affect, try your hand at the following problems from the digital textbook (I just ordered paper copies):

//Exponents:// Exercise 3B p. 99—101: choose a small selection from Problems 1—10 Exercise 3C p. 101-102: choose a small selection from Problems 1—5 Exercise 3D.1 p. 105: 2a, d, i, j and a few more in this section if you feel you need the practice Exercise 3D.2 p. 105—106: choose a small selection from Problems 1—6 Exercise 3E p. 107—108: choose a small selection from 1—5

//Logarithms:// Exercise 4B p. 128: choose a small selection from P. 1—6 Exercise 4C.1 p. 131—132: choose a small selection from Problems 1—6 Exercise 4C.2 p. 133—134: choose a small selection from Problems 1—4 Exercise 4E p. 138: choose a selection from Problem 1a—1f

Success & enjoy, Arno.

=== //Wed 23 Oct//

And back to functions. Today we looked at math \sqrt{f(x)} math
 * radical functions .... radical function is an expressions which looks like this
 * reciprocal functions and how to **sketch** them from an equation as well as a "parent" sketched function; stick to the recipe!

The notes for today are:

And for tomorrow, please take a good look and try your hand at the diagnostic quiz you were given in class.

The solutions to the quiz are here:

Success & enjoy, Arno.

=== //Mon 21 Oct//

A start to a our Unit 2 which is more Functions and Equations. This week we will studies some nifty functions, starting with Rational Functions and a fresh look at Self-inverse functions. Here are the notes:.

For this week you will need these two GeoGebra files:
 * [[file:Rational Analysis.ggb]]
 * [[file:Domain Restriction.ggb]]

As always, bring your laptop to class.

Success & enjoy, Arno.

=== //Thu 17 Oct//

Today we looked at the absolute value function, see these notes:.

And we did a GeoGebra exercise. Please have it finished by Monday for me to look at.

Success & enjoy, Arno.

=== //Wed 16 Oct//

And now we are with the HL students only!

Today we went over the test.

Bring your laptop with GeoGebra tomorrow.

Enjoy, Arno.

//===// //Thu 10 Oct//

Test day.

=== //Wed 9 Oct//

Today we didn't do any math.

But you may want to take a look at these notes which will not be on the test tomorrow.

Success & enjoy, Arno.

=== //Mon 7 Oct//

All the transformation
 * 1) translations 2x
 * 2) stretches 2x
 * 3) reflections 2x
 * 4) rotations by 180 degrees

Here are some notes:

This is the GeoGebra file I played with showing the transformations:

Keep an open mind and re-read my musings on mathematics and climbing here: home

Success & enjoy, Arno.

=== //Thu 3 Oct//

We finally got to explore inverse functions in all its glory ... glory of doing nothing, right? Notes are on the one uploaded for yesterday's class.

We started exploring transformations of functions by looking at the easiest one, namely translations, also known as shifts. These can be either up/down, vertical translations, or right/left, horizontal translations.

The solutions to last week's quiz will come soon, I will also upload a screencast soon to go over some of the common mistakes.

Take care, Arno.

=== //Wed 2 Oct//

A further look at composite functions, after a refresher on the various formal language surrounding functions.

We didn't quite make it inverse functions (though we did not do nothing!), but here are the notes for today and tomorrow:

The Unit 1 test will be next week Thursday!

Success & enjoy, Arno.

=== //Mon 30 Sept//

Functions, functions, and more functions. We will be studying functions for all their lovely functionality. Today we looked at some formal definitions surrounding functions, to wit
 * functions as a map
 * domain, co-domain and range
 * image under a mapping
 * even and odd function
 * mention of inverse function (more on that later in the week)

Most of the information can be found in these notes:

You were given a few short exercises to go with this work.

Success & enjoy, Arno.

=== //Saturday 28 Sept - New flash//

I have not been able to locate a downloadable copy of the software I need to show you how to use the TI via my computer, only a DVD which I have ordered but will take at least a week to make it to Dwight. So, keep on working with pencil and paper!

Note that your morning blocks have changed, and our morning Math class has moved to Thursday, meaning not two math classes on Monday.

Enjoy the weekend, Arno.

=== //Wed 25 Sept//

We finished our formal work on polynomials for now with a look at the Remainder Theorem and the Factor Theorem, including some examples. I encourage you to try the following questions from the textbook:

And remember, here are the notes on this, with some more examples as well:.

Then I handed out a Quiz for you to do in the study block in the library tomorrow. Keep in mind:
 * only do section 1 without a calculator
 * do it by yourself
 * hand it in at 3:30 sharp, i.e., after the library block, at the front desk and ask kindly for your quiz to be put in my mailbox.

Finally, I handed out the problem set with a promise of a little feedback. The solutions are here:. That feedback you can find in the embedded screencast below (Smarties for who tells me the error first). media type="custom" key="23924910"

Later this weekend, when all of you will have picked up a TI 84 GDC, I will upload a screencast on how to use your GDC for solving some of the things we have been doing.

Note that I have put a new resource in the ... Resources page, namely a link to videos for both Mathematics HL as well as SL stuff.

Success & enjoy, Arno.

=== //Mon 23 Sept - morning//

And a look at cubics, quartics and thus higher order "polynomial" expressions. The notes can be found here:. Pay particular attention to the features of these functions:
 * 1) general form
 * 2) x-intercepts - solutions to y=0
 * 3) y-intercepts - value at x=0
 * 4) turning points: (local) maxima, (local) minima

And some GeoGebra files to play with:

Success & enjoy, Arno.

=== //Mon 23 Sept - shortened afternoon class//

And now time to be a bit more formal about polynomial functions and then look at a couple of theorems. We didn't get through all the material, but **please read ahead over these notes for next class: .**

Success & enjoy, Arno.

=== //Mon 16 Sept - morning//

And our look at quadratics continues. As you saw in the notes which I shared last week, or should have seen in the notes (!), there are three forms of the quadratic function General form: math y = a x^2 + b x + c math

Vertex form: math y = k (x - p)+ q math with the vertex at (//p,q//)

Factorised form: math y = k ( x - m)(x-n) math with x-intercepts at //m,n//.

In terms of finding the number of x-intercepts, i.e., the number of solutions to the quadratic equation, the discriminant is a rather useful object, as it **discriminates** between the three options: two solutions, one solution, no solution. We define the discriminant, using the general form, as math \Delta = b^2 - 4 a c math

Then the following cases hold

With all these tools, you can accurately **sketch** quadratic functions and indicating the general form and interesting features:
 * 1) form: up or down concave
 * 2) position of vertex
 * 3) x-intercepts, if any
 * 4) y-intercept

Here are the two GeoGebra files I played with today in class

Success & enjoy, Arno.

=== //Thu 12 Sept//

And here are the solutions for the practice problems in
 * factorisation -[[file:Lesson 010201 Solution to Practice Problems on Solving Quadratic Equations that Factorize.pdf]]
 * completing the square -[[file:Lesson 010201 Solutions to Practice Problems on Completing the Square.pdf]].

For Monday, I expect you all to have read in detail, ensuring that you following each and every step, the handout I already made available yesterday, namely this one:.

By Monday, you should also all have a GDC! And have [|GeoGebra] installed on your laptop. Please bring your laptop to class on Monday, we may use it.

Success & enjoy, Arno.

===

//Wed 11 September//

A further look at quadratic equations and three methods to solve them We even proved the quadratic formula, **something that I expect you all to be able to do!**
 * 1) factorisation (if the coefficients lead to "nice" factors)
 * 2) completing the square
 * 3) quadratic formula

Here some notes on this:.

I then handed out some practice questions on factorisation:, and completing the square:.


 * There will be no class tomorrow.** You are expected to be in the library working on the practice problems, for which the solutions will be released tomorrow afternoon.


 * If you are done already with the practice problems, you can read the following handout: [[file:Notes 010202.pdf]].**

Success & enjoy, Arno.

===

//Mon 9 September//

Things to ensure you have and do sooner rather than later:
 * 1) graphing display calculator (hereafter referred to as GDC) preferably of the TI 84 type;
 * 2) check your Dwight email daily;
 * 3) dito for this wiki;
 * 4) install GeoGebra on your laptop, I have put an introduction to GeoGebra in the Resources page;
 * 5) make sure you know how to get to the WolframAlpha site and/or get the app for your smartphone or tablet; and for some more information, watch the Ted talk by Stephen Wolfram here: [|Stephen Wolfram Ted talk];
 * 6) once I have sent it around, get the digital Mathematics books on your laptop.

And then today we saw a few instances where we encountered quadratic expressions which shows us the need to study them in a bit more detail, both from an algebraic point of view as well as graphically.

We briefly looked at the importance of significant figures -they are important, one could say, they are significant, as they reflect your claim about precision- and you can find the presentation on that in the Resources section, found among the pages on the top left of the screen.

Enjoy, Arno.