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“Straight-Line-Graph-Through-The-Origin”

The words of Mr Michael Twomey, physics teacher, in Coláiste an Spioraid Naoimh, I can still hear them.

There were two main reasons to produce this straight-line-graph-through-the-origin:

• to measure some quantity (e.g. acceleration due to gravity, speed of sound, etc.)
• to demonstrate some law of nature (e.g. Newton’s Second Law, Ohm’s Law, etc.)

We were correct to draw this straight-line-graph-through-the origin for measurement, but not always, perhaps, in my opinion, for the demonstration of laws of nature.

The purpose of this piece is to explore this in detail.

Direct Proportion

Two variables $P$ and $Q$ are in direct proportion when there is some (real number) constant $k$ such that $P=k\cdot Q$.

Quadratics are ubiquitous in mathematics. For the purposes of this piece a quadratic is a real-valued function $q:\mathbb{R}\rightarrow \mathbb{R}$ of the form

$q(x)=ax^2+bx+c$,

where $a,\,b,\,c\in \mathbb{R}$ such that $a\neq 0$. There is a little bit more to be said — particularly about the differences between a quadratic and a quadratic function but for those this piece is addressed to (third level: non-maths; all second level), the distinction is unimportant.

Geometry

The basic object we study is the square function, $s:\mathbb{R}\rightarrow \mathbb{R}$, $x\mapsto x^2$:

All quadratics look similar to $x^2$. If $a>0$ then the quadratic has this $\bigcup$ geometry. Otherwise it looks like $y=-x^2$ and has $\bigcap$ geometry

The geometry dictates that quadratics can have either zero, one or two real roots. A root of a function is an input $x$ such that $f(x)=0$. As the graph of a function is of the form $y=f(x)$, roots are such that $y=f(x)=0\Rightarrow y=0$, that is where the graph cuts the $x$-axis. With the geometry of quadratics they can cut the $x$-axis no times, once (like $s(x)=x^2$), or twice.

There are a number of ways of explaining why you cannot divide by zero. Here are my two favourites.

Any Set of Numbers Collapses to a Single Number

How old are you? Zero years old.

How tall are you? Zero metres old.

How many teeth do you have? Zero.

How many Superbowls has Tom Brady won? Zero

Yep, if you allow division by zero you only end up with one number to measure everything with.

Assessment 5

Assessment 5 will be on Tuesday, 6 December 2016.

The following  group will sit the assessment at 17:15 (arrive around 17:05):

• COMP1B-X

The following group will sit the assessment at 18:15 (arrive around 18:05)

• COMP1B-Y

Keep an eye on your CIT Blackboard and email for the latest and definitive assessment information.

Week 12

We looked at graphing and mathematical modelling.

Week 13

Two tutorials:

• COMP1B-Y: Monday 10:00 in F1.3
• COMP1B-X: Monday 15:00 in B251
• COMP1B-X and 1B-Y: Tuesday 14:00 in B225

Study

Please feel free to ask me questions about the exercises via email or even better on this webpage.

Student Resources

You will need graph paper for the remaining MATH6000 classes. I advise that you gang up and buy graph paper together in the shop downstairs.

Assessment 4 Results

If you require a result of 40% or more on Assessment 5, you will receive an email regarding your results on or before Wednesday. Otherwise you must wait until your results are released, in January.

Assessment 5

Assessment 5 will be on Tuesday, 6 December 2016.

The following  group will sit the assessment at 17:15 (arrive around 17:05):

• COMP1B-X

The following group will sit the assessment at 18:15 (arrive around 18:05)

• COMP1B-Y

Keep an eye on your CIT Blackboard and email for the latest and definitive assessment information.

Week 11

We began our study of coordinate geometry.

Week 12

We will look at graphing and mathematical modelling.

Assessment 4

Assessment 4 will be on this Wednesday, 23 November 2016.

The following  group will sit the assessment at 17:15 (arrive around 17:05):

• COMP1B-X

The following group will sit the assessment at 18:15 (arrive around 18:05)

• COMP1B-Y

Keep an eye on your CIT Blackboard and email for the latest and definitive assessment information.

Week 10

We looked at scientific notation, unit conversion and logarithms.

Week 11

We will finish our study of logarithms and begin our study of coordinate geometry.

Study

Please feel free to ask me questions about the exercises via email or even better on this webpage.

Assessment 4

Assessment 4 will be on this Wednesday, 23 November 2016.

The following  group will sit the assessment at 17:15 (arrive around 17:05):

• COMP1B-X

The following group will sit the assessment at 18:15 (arrive around 18:05)

• COMP1B-Y

Keep an eye on your CIT Blackboard and email for the latest and definitive assessment information.

Week 9

We began looking at indices.

Week 10

We will look at scientific notation, unit conversion, functions and logarithms.

Assessment 3

Assessment 3 will be on the Wednesday of Week 9, 9 November 2016.

The following  group will sit the assessment at 17:15 (arrive around 17:05):

• COMP1B-X

The following group will sit the assessment at 18:15 (arrive around 18:05)

• COMP1B-Y

Keep an eye on your CIT Blackboard and email for the latest and definitive assessment information.

Week 8

We looked at quadratic and simultaneous equations.

Week 9

We will look at functions and begin looking at indices and scientific notation.

This post follows on from this post where the logic for the below is discussed. I am not going to define here what easy means!

Here is the strategy/guiding principle:

Fundamental Principle of Solving ‘Easy’ Equations

Identify what is difficult or troublesome about the equation and get rid of it. As long as you do the same thing to both numbers (the “Lhs” and the “Rhs”), the equation will be replaced by a simpler equation with the same solution.

I am emailing a link of this to everyone on the class list every week. If you are not receiving these emails or want to have them sent to another email address feel free to email me at jpmccarthymaths@gmail.com and I will add you to the mailing list.

Assessment 3

Assessment 3 will be on the Wednesday of Week 9, 9 November 2016.

The following  group will sit the assessment at 17:15 (arrive around 17:05):

• COMP1B-X

The following group will sit the assessment at 18:15 (arrive around 18:05)

• COMP1B-Y

Keep an eye on your CIT Blackboard and email for the latest and definitive assessment information.

Week 7

We looked at equations.

Week 8

We will look at quadratic and simultaneous equations.