No.48

Properties of sine and cosine functions

—  1. The domain is the set of real numbers.

—  2. The range is the set of y values such that . 

—  3. The maximum value is 1 and the minimum value is –1.

—  5. Each function cycles through all the values of the range
 over an x-interval of      .

—  6. The cycle repeats itself indefinitely in both directions of the x-axis.

—  y = a sin b (x - c)           y = a cos b (x – c)

—  |a| is the amplitude of the sine or cosine graph.

—  The amplitude describes the height of the graph.

—  “b” affects the period of the sine or cosine graph. 

—  “c” indicates the phase shift of the sine graph or of the cosine graph.  The x-coordinate of the key point is c.

No.47

Key Definitions:

• Parameter - The third variable (usually t or Θ).
• t represents time; Θ represents angle
• Parametric Equation - An equation that equates x or y to the third variable (ex: y=2t)
• Plane Curve - An interval in which both x and y are continuous functions of t. Often represented by C.

Main Purpose - to give a more realistic illustration of a graph by including the time or angle at which any given point occurs.

Steps to Solving Parametric Equations:

1. Eliminate the parameter to find a rectangular equation
2. Sketch the graph of basic x,y equation once parameter is eliminated.
3. Use parametric equations to find (x,y) at point t.
1. Usually use t = -2,-1,0,1,2.

Credit by Joe Laski

No. 46

Ax²+Bxy+Cy²+Dx+Ey+F=0 A’(x’)²+C’(y’)²+D’x’+E’y’+F’=0 cos2θ=A-C/B   x=x’cosθ-y’sinθ   y=x’sinθ+y’cosθ

Steps to Rotate a Conic:

1. Find the angle using the 3rd equation above

2. Plug angle measure from unit circle into last 2 equations to get new x’ and y’

3. Plug back into the original equation and make sure the xy-term cancels out

  

Credit by Alec Ainger

No.45

Mean Deviation

The mean of the distances of each value from their mean.

Yes, we use "mean" twice: Find the mean ... use it to work out distances ... then find the mean of those!

Three steps:

• 1. Find the mean of all values
• 2. Find the distance of each value from that mean (subtract the mean from each value, ignore minus signs)
• 3. Then find the mean of those distances

Like this:

Example: the Mean Deviation of 3, 6, 6, 7, 8, 11, 15, 16

Step 1: Find the mean:

Mean =	3 + 6 + 6 + 7 + 8 + 11 + 15 + 16	=	72	= 9
	8		8

Step 2: Find the distance of each value from that mean:

Value	Distance from 9
3	6
6	3
6	3
7	2
8	1
11	2
15	6
16	7

Which looks like this:

Step 3. Find the mean of those distances:

Mean Deviation =	6 + 3 + 3 + 2 + 1 + 2 + 6 + 7	=	30	= 3.75
	8		8

So, the mean = 9, and the mean deviation = 3.75

It tells us how far, on average, all values are from the middle.

In that example the values are, on average, 3.75 away from the middle.

For deviation just think distance

Formula

The formula is:

Mean Deviation =	Σ|x - μ|
	N

Let's learn more about those symbols!

Firstly:

• μ is the mean (in our example μ = 9)
• x is each value (such as 3 or 16)
• N is the number of values (in our example N = 8)

Absolute Deviation

Each distance we calculated is called an Absolute Deviation, because it is the Absolute Value of the deviation (how far from the mean).

To show "Absolute Value" we put "|" marks either side like this: |-3| = 3 For any value x: Absolute Deviation = |x - μ|

From our example, the value 16 has Absolute Deviation = |x - μ| = |16 - 9| = |7| = 7

And now let's add them all up ...

No.44

Inverse of a Matrix

Please read our Introduction to Matrices first.

What is the Inverse of a Matrix?

The Inverse of a Matrix is the same idea as the reciprocal of a number:

Reciprocal of a Number

But we don't write 1/A (because we don't divide by a Matrix!), instead we write A^-1 for the inverse:

(In fact 1/8 can also be written as 8^-1)

And there are other similarities:

When you multiply a number by its reciprocal you get 1

8 × (1/8) = 1

When you multiply a Matrix by its Inverse you get the Identity Matrix (which is like "1" for Matrices):

A × A^-1 = I

It also works when the inverse comes first: (1/8) × 8 = 1 and A^-1 × A = I

Identity Matrix

Note: the "Identity Matrix" is the matrix equivalent of the number "1":

A 3x3 Identity Matrix

• It is "square" (has same number of rows as columns),
• It has 1s on the diagonal and 0s everywhere else.
• It's symbol is the capital letter I.

The Identity Matrix can be 2×2 in size, or 3×3, 4×4, etc ...

Definition

So we have a definition of a Matrix Inverse ...

The Inverse of A is A^-1 only when:

A × A^-1 = A^-1 × A = I

Sometimes there is no Inverse at all.

2x2 Matrix

OK, how do we calculate the Inverse?

Well, for a 2x2 Matrix the Inverse is:

In other words: swap the positions of a and d, put negatives in front of b and c, and divideeverything by the determinant (ad-bc).

Let us try an example:

How do we know this is the right answer?

Remember it must be true that: A × A^-1 = I

So, let us check to see what happens when we multiply the matrix by its inverse:

And, hey!, we end up with the Identity Matrix! So it must be right.

No.43

Plane Geometry

If you like drawing, Geometry is for you!

Plane geometry is all about shapes like lines, circles and triangles ... shapes that can be drawn on a flat surface called a Plane (it is like on an endless piece of paper).

Point, Line, Plane and Solid

A Point has no dimensions, only position
A Line is one-dimensional
A Plane is two dimensional (2D)
A Solid is three-dimensional (3D)

Hint: Try drawing some of the shapes and angles as you learn ... it helps.

Shapes

	2D Shapes
	Activity: Sorting Shapes
	Triangles
	Right Angled Triangles
	Interactive Triangles
	Quadrilaterals (Rhombus, Parallelogram, etc)
	· Rectangle, Rhombus, Square, Parallelogram, Trapezoid, Kite
	Interactive Quadrilaterals
	Shapes Freeplay
	Perimeter
	Area
	Area of Plane Shapes
	Area Calculation Tool
	Area of Polygon by Drawing
	Activity: Garden Area
	General Drawing Tool

Polygons

A Polygon is a 2-dimensional shape made of straight lines. Triangles and Rectangles are polygons.

Here are some more:

Pentagon

Pentagram

Hexagon

	Properties of Regular Polygons
	Diagonals of Polygons
	Interactive Polygons

The Circle

	Circle
	Pi
	Circle Sector and Segment
	Circle Area by Sectors
	Annulus
	(Advanced) Circle Theorems
	Activity: Dropping a Coin onto a Grid

Symbols

There are many special symbols used in Geometry. Here is a short reference for you: Geometric Symbols

No.42

Why learn math?

When are we ever going to use this? I’m sure other teachers hear this but it seems to be especially prevalent in math classes.

This question is frustrating for two reasons.  1) There is an implied “if you can’t answer then I don’t have to learn it.”   2) I could come up with an infinite number of answers and they, probably, would not be adequate.  Students invariably say yeah yeah, but I don’t planning on doing that.

But it is a fair question.  I’ve created this blog, in part, to hone my answers.  Here are my current top ten reasons.  (They are not in any particular order except, perhaps, #1 and 10.)

Learning math can help you….

10.  Prepare for a career.

In my humble opinion, this is the least important reason to learn math….

9.  Develop problem solving skills.

You can only develop problem solving skills by …wait for it… solving problems.  I don’t mean problems where the teacher shows you an example and you do 20 problems just like it.  I am referring to problems you’ve never seen before.  Math gives you practice in organizing what you know, rearranging information, testing hypotheses, etc.  These are the same skills you can use in everyday life.  Unfortunately, most teachers (including myself…  There’s that stupid test each year) take the problem solving element out of math class.

8.  Increase your capacity to think analytically

Without props, without manipulatives….  Just with your brain…

7.  Be able to learn science

6.  Argue better

5.  Be less gullible

Astrology is completely bogus.  This can be demonstrated.  If you don’t understand or believe the reasons (or can’t sit through an explanation), you’re missing some of the skills mentioned above and your math teachers should be fired, (except for the fact that learning is, ultimately, the students’ responsibility.)

4.  Distinguish us from animals 

What makes humans different?  Build tools?  Nope, certain apes do it, even ants. Develop language? Nope, dolphins, birds, and primates beat us to it.  Use money?  Nope.  One reason is our ability to think and communicate abstract ideas.  One language for doing so is mathematics.  (To be fair there are many others: art, music, literature, etc.)

3.  Better appreciate the history of our civilization

Wait, what?

The industrial revolution was made possible because of thermodynamics which require an understanding of differential equations which is the main sequel to Newton’s calculus.  The exploration of the world happened because the Egyptians knew the earth was round. etc.

2.  So you don’t look like an idiot

And to show off at parties…

1.  If you understand math….  And can read,  you can teach yourself anything…. 

So when I ask you, “why is math important.” if you say, “to count my money.” I will start banging my head against the wall.

information fromhttp://wildmath.org/2012/07/18/top-10-reasons-to-learn-math-a-hed-manifesto/

No.41

Slope  =

Change in Y Change in X

You can find an average slope between two points.
But how do you find the slope at a point? There is nothing to measure!
But with derivatives you use a small difference ... ... then have it shrink towards zero.

No.40

Sometimes you can't work something out directly ... but you can see what it should be as you get closer and closer!

Let's use this function as an example:

(x²-1)/(x-1)

And let's work it out for x=1:

(1²-1)/(1-1) = (1-1)/(1-1) = 0/0

Now 0/0 is a difficulty! We don't really know the value of 0/0, so we need another way of answering this.

So instead of trying to work it out for x=1 let's try approaching it closer and closer:

x	(x2-1)/(x-1)
0.5	1.50000
0.9	1.90000
0.99	1.99000
0.999	1.99900
0.9999	1.99990
0.99999	1.99999
...	...

Now we can see that as x gets close to 1, then (x²-1)/(x-1) gets close to 2

We are now faced with an interesting situation:

• When x=1 we don't know the answer (it is indeterminate)
• But we can see that it is going to be 2

We want to give the answer "2" but can't, so instead mathematicians say exactly what is going on by using the special word "limit"

The limit of (x²-1)/(x-1) as x approaches 1 is 2

And it is written in symbols as:

So it is a special way of saying, "ignoring what happens when you get there, but as you get closer and closer the answer gets closer and closer to 2"

As a graph it looks like this: So, in truth, you cannot say what the value at x=1 is. But you can say that as you approach 1, the limit is 2.