Year: 2009

Free Mathematics Tutorial Videos

Audio, Video and Animation, Resources and Freebies, Web 2.0 Technology

One of the best videos about mathematics I have seen in the internet are from Khan Academy.  Khan Academy offers more than 1100 downloadable videos for free in Algebra, Geometry, Calculus, Differential Equations, Physics, Linear Algebra and other fields. Shown below is one of their mathematics tutorial videos discussing the concept of slope.

If you want to see the list of all their free videos, click here.

If you are a teacher, you  can watch the video above and refer it to your students.  But did you know that it is quite easy to make such video?

If you want to learn how to make a video such as the one you have seen above, I have posted a tutorial about it here.

Update:

Here are some other sites offering math videos for free.

  1. Bright Storm
  2. FreeVideoLectures
  3. Just Math Tutoring
  4. Math Playground
  5. Math Videos Online
  6. Math Vids Website
  7. MIT Blossoms Library
  8. MIT Open Courseware
  9. Visual Math Learning
  10. YouTube – patrickJMT’s Channel
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Latex Tutorial: Commonly Used Latex Commands

Latex Typesetting, Software Tutorials

INTRODUCTION

In my previous latex post, we have seen that Latex is capable of displaying complex mathematical expressions in blogs and forums. In this post, we will discuss the very basics Latex – the most commonly used symbols in high school mathematics and how to code them.

Almost all Latex commands or codes begin with the \ symbol. For example, we want to write a fraction, we must use the \frac command. If we want to use Latex in a WordPress blog, we should enclose it with two dollar signs (see Figure 2) with the word “latex” after the first dollar sign. The dollar signs and the word “latex” is not a latex command, but it tells WordPress (or other applications) that the enclosed text is a mathematical expression and should be displayed in Latex form.  Blogs and forums have different ways of embedding Latex commands (see my previous latex blog for further explanation), so you must know how they work.  In the following discussion, we will use the WordPress format.

If we want to copy the text writtenbelow (without the drawing, of course), we can use a combination of text and Latex in writing the solution.

Figure 1 – A forum post embedded with Latex

The solution was written as follows:

Figure 2 – The Latex code of the text in Figure 1

You should notice that the only latex command above is the \rightarrow command, but the equations are all written in Latex. This is because most of the time, it is advisable to write entire mathematical expressions in Latex to make it look better in web pages.

You should also be careful about the spaces between in your latex code. No spaces before and after the dollar sign, and 1 space after the word “latex”.

BASIC COMMANDS

Below are the basic commands commonly used in high school mathematics. If you want to learn more about the other symbols, a list of symbols can be viewed here and a comprehensive list can be viewed here.

Exponents and Subscripts

In Latex, the symbol ^ is used for exponents and the symbol _ is used for subscript. The {} symbol is used for grouping.

Expression Command Notes
2^3 2^3
2^{10} 2^{10} Note how {} is used. 2^10 will yield
p_k p_k
x^{3}_1 x^{3}_1
2^{a_k} 2^{a_k}
t_{n - 1} t_{n – 1} Try omitting the {} and see what happens.
c^2 = a^2 + b^2 c^2 = a^2 + b^2

 

Fractions

The command \frac is used to type fractions in Latex. The syntax is \frac{numerator}{denominator}.

Expression Command Notes
\frac{2}{3} \frac{2}{3}
\frac{3}{x + 5} \frac{3}{x + 5}
\frac{5}{x + \frac{1}{x}} \frac{5}{x + \frac{1}{x}} Notice that enclosed by the blue  braces are the numerator and denominator of a fraction in the denominator, there is another fraction.

 

Radicals

Radicals are written using the the \sqrt{expression} command. If the index, however, is greater than 2, the syntax is \sqrt[index]{expression}. (See examples 4 and 5}

Expression Command Notes
\sqrt{5} \sqrt{5}
\sqrt{a + b} \sqrt{a + b}
\sqrt{a + \frac{1}{a}} \sqrt{a + \frac{1}{a}}
\sqrt[3]{x + 3} \sqrt[3]{x + 3} You can write the index of the radical before the radicand expression.
\sqrt[5]{(32)^2} \sqrt[5]{(32)^2}

Geometry

Expression Command Notes
\overline{AB} \bot \overline{CD} \overline{AB} \bot \overline{CD} \botis the symbol for perpendicular
\angle {ABC} \cong \angle {PQR} \angle {ABC} \cong \angle {PQR}
\Delta{PQR} \approx \Delta{XYZ} \Delta{PQR} \approx \Delta{XYZ} Delta, the fourth letter of the Greek alphabet, is the symbol for triangle

 

With the Latex commands you already know, you will be surprised that you can already code a lot of expressions. Now try coding the quadratic formula. The code form \pm (plus-minus) is \pm.

The fraction \frac{3}{x+5} above is somewhat small. If you want it to appear in its full size, just add the \displastyle command before the \frac command and do not leave any white spaces. The command

\displaystyle\frac{3}{x+5}

will display \displaystyle\frac{3}{x+5}.

GOING FURTHER

You can do a lot more things using Latex. You can create documents, articles, create cross references among your documents and so on.  In fact, many books and ebooks nowadays are coded in Latex. In creating documents, however, we will need a Latex editor and a compiler.  I am planning to make a tutorial on how to make Latex documents using a Latex editor (not very soon though), so you may want to subscribe to my blog for later updates if you are interested.

Mr. Kogler has also a latex tutorial for  for advanced users here.

Note: If you want to test some of the latex commands, feel free to use the comment box.

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Introduction to Permutations

High School Algebra, High School Mathematics, High School Probability, Questions and Quandaries

Problem: In how many ways can Anna, Brenda and Connie stand in a single line for picture taking?

Intuitively, we can count the number of ways by listing. We can list randomly as shown below.

Anna, Brenda, Connie

Brenda, Connie, Anna

Connie, Anna, Brenda

and so on.

Q1: Do you think that listing randomly is a good idea? What are its advantages and its disadvantages?

Listing randomly can solve our problem, if there are only a few things, or in our case persons, to be arranged; however, we can do better than that.  Learning mathematics has taught us to be organized, and has taught us to do things systematically.  Besides, if there are many persons to be arranged, it is hard to keep track if we have listed all possible arrangements. For example, what if David joins the group? Try to list randomly and determine how many possible arrangements are there.

Q2: Before proceeding, can you think of a way to come up with an organized way to list all the possible arrangements?

One possible strategy is to list in alphabetical order. Let us represent Anna, Brenda and Connie by the first letter of their names. If we choose A to occupy the leftmost position, then there are two possible choices for the middle position, namely B and C. That means have AB and AC as all possible arrangements if A is chosen to occupy the leftmost position. Now, in each of the cases, we only have one person left to occupy the rightmost position.  This gives as ABC and ACB as all possible arrangements of the three girls if A were to occupy the leftmost position.

We can also use a tree diagram as shown in Figure 1.  If we choose A to be the person in leftmost position, then the branches B and C mean our possible choices for the middle position. If we have chosen a person who will occupy the middle position, then we are left with only one person to occupy the rightmost position.  Hence, if we choose A to occupy the first position, the only possible arrangements for picture taking are ABC and ACB.

Figure 1- The tree diagram of all the possible arrangements of A, B and C.

But we know that we can also choose B or C as the person who will occupy the leftmost position. This means, that there are 3 possible choices for the first position, 2 possible choices  for the second position and 1 possible choice for the third position (see Figure 1). Hence, there are 3 x 2 x 1 possible arrangements for 3 persons in a single line.

We will denote 3 x 2 x 1 as 3! (3 factorial).  This implies that if we say 5!, we mean 5 x 4 x 3 x 2 x 1 = 120.  In general, n! means n(n-1)(n-2)…(3)(2)(1). Note that the ellipsis symbol … denotes that there are numbers in the sequence that are not shown.  For example, 100(99)(98)…(3)(2)(1), means the product of all integers from 100 all the way down to 1.

Q3: If David joins the group, how many possible arrangements are there?

You may want to solve this problem first before proceeding.

Figure 2 – The tree diagram of all possible arrangments of A, B, C and D.

Looking at the tree diagram, there are four possible choices to occupy the leftmost position, 3 possible choices to occupy the second position, 2 possible choices to occupy  the third position and 1 possible choice to occupy the rightmost position.  Hence there are 4! = 4 x 3 x 2 x 1 = 24 possible arrangements.

By now, you would have realized that the number of arrangements or the number of permutations of n persons on a single line for picture taking is n!. We will denote it as P(n,n) orthe permutations of n objects taken n at a time. We sayn objects taken n at a time because we have the choice to choose numbers less than n to be arranged.  For example, we can choose A and C from A, B, C and D .  This means that we a permutation of 4 objects taken 2 at a time. In general, we describe this type of permutation as permutations of n objects taken k at a time and write P(n,k).

Let us see what happens to our computation with P(4,2). Since there are 4 possible choices for the first choice, and 3 choices for the second position, therefore, there are 4 x 3 possible permutations.  This is the same as removing the smallest two factors by division. If we do this, we come up with the following computation:

If we list the elements of P(4,2), we have the following: AB, BA, AC, CA, AD, DA, BC, CB, BD, DB, CD, and DC. Indeed, we have 12 possible arrangements.

With our findings above, let us try to perform a few more computations and see if we can find a pattern.

Therefore, by looking at the pattern, we can conclude that the number of permutations of n things taken k at a time described by the formula

You may want to read  Introduction to Combinations, the continuation of this post.

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