Part 1:

- Evaluate the determinants using the coefficients aka D
- Evaluate the determinants using the constants in place of x aka Dx
- Evaluate the determinant using the constants in place of y aka Dy

Part 2:

- Find x by Dx / D
- Find y by Dy / D

This is modified if the system is 3 variables. Below we will go through an example with 2 variables.

**Example**

Here is our problem

3x – 2y = -6

Below is the matrix of the system of equations

We will first evaluate the Determinant D using the coefficients. In other words, we are going to calculate the determinant for the first two columns of the matrix. Below is the answer.

What we have just done we will do two more times. Once two find the determinant of x and once to find the determinant of y. When we say determinant x or y we are excluding that column from the 2×2 matrix. In other words, if I want to find the determinant of x I would exclude the x values from the 2×2 matrix when calculating. Below is the determinant of x.

Lastly, here is the determinant of y

We no have all the information we need to solve for x and y. To find the answers we do the following

- Dx / D = x
- Dy / D = y

We know these value already so we plug them in as shown below.

You can plug in these values into the original equation for verification.

The steps we took here can also be applied to a 3 variable system of equation. In such a situation you would solve one additional determinant for z.

**Conclusion**

Solving a system of equations using Cranmer’s rule is much faster and efficient than other methods. It also requires some additional knowledge of rows and matrices but the benefits far outweigh the challenge of learning some basic rules of row operations.

]]>Discussion is simply the flow of ideas between individuals and or groups. It is a two-way street in that both sides are actively expressing their ideas. This is how discussion varies from a lecture which is one-sided and most question and answers learning. In a discussion, people are sharing their thoughts almost in a democratic-like style.

Classroom discussion, of course, is focused specifically on helping students learn through interacting with each other and the teacher using this two-way form of communication.

Discussion can aid in the development of both thinking and affective skills. In terms, of thinking, classroom discussion helps students to use thinking skills from the various levels of Bloom’s taxonomy. Recalling, comparing, contrasting, evaluating, etc are all needed when sharing and defending ideas.

The affective domain relates to an individual’s attitude and morals. Discussion supports affective development through strengthing or changing a students attitude towards something. For example, it is common for students to hold strong opinions with little evidence. Through discussion, the matter and actually thinking it through critical students can realize that even if their position is not wrong it is not sufficiently supported.

**Barriers and Solutions to Classroom Discussion**

There are many common problems with leading discussions such as not understanding or failing to explain how a discussion should be conducted, focus on lower level questions, using the textbook for the content of a discussion, and the experience and attitude of the teacher.

Discussion is something everybody has done but may not exactly know how to do well. Teachers often do not understand exactly how to conduct a classroom discussion or, if they do understand it, they sometimes fail to explain it to their students. How to discuss should be at a minimum demonstrated before attempting to do it

Another problem is poor discussion questions. The goal of a discussion is to have questions in which there are several potential responses. If the question has one answer, there is not much to discuss. Many teachers mistakenly believe that single answer questions constitute a discussion.

The expertise of the teacher and the textbook can also be problems. Students often believe that the teacher and the textbook are always right. This can stifle discussion in which the students need to share contrasting opinions. Students may be worried about looking silly if disagreeing, One way to deal with this is to encourage openness and trying to make content relevant to something it the students lives rather than abstract and objecive.

In addition, students need to know there are no right or wrong answers just answers that are carefully thought out or not thought out. This means that the teacher must restrain themselves from correcting ridiculous ideas if they are supported adequately and show careful thought.

**Conclusion**

Discussion happens first through example. As the teacher show how this can be done the students develop an understanding of the norms for this activity. The ultimate goal should always be for students to lead discussion independent of the teacher. This is consistent with autonomous learning which is the end goal of education for many teachers.

]]>Determinants are calculated from a square matrix, such as 2×2 or 3×3. In a 2×2 matrix, the determinant is calculating by taking the product of the diagonal and finding the difference.

Here is how this look with real numbers

**Determinant for 3×3 Matrix**

To find the determinants of a 3×3 matrix it takes more work. By address, it is meant the row number and column letter. To calculate the determinant you must remove the row and column of that contains the variable you want to know the determinant too. Doing this creates what is called the minor. Below is an example with variables.

As you can see, to find the determinant of a1 we remove the row and the column that contains a1. From there, you do the same math as in a 2×2 matrix. When using real numbers you may need to add the row letter and column number to figure out what you are solving for. Below is an example with real numbers.

Find the determinant of c2

The number at c2 is -3. Therefore, we remove the row and the column that contains -3 and we are left with the minor of c2 shown below.

IF we follow the steps for a 2×2 matrix we can calculate the determinant of c2 as follows.

4

The answer is 4.

**Expand by Minors**

Knowing the minor is not useful alone, The minor of different columns can be added together to find the determinant for a 3×3 matrix. Below is the expression for finding the determinant of 3×3 matrix.

What is happening here is that you find the determinant of a1 and multiply it by the value in a1. You do this again for b1 and c1. Lastly, you find the sum of this process to evaluate the determinant of the 3×3 matrix. Below is another matrix this time with actual numbers. We are going to expand from the first row and first column

All we do not is obtain the determinant of each 2×2 matrix and multiply it by the outside value before adding it all together. Below is the math.

-25

**Conclusion**

This information is not as useful on its own as it is as a precursor to something else. The knowledge acquired here for finding determinants provides us with another way to approach a system of equations using matrices.

]]>For a beginner, the coding for developing a table is somewhat complex. Below is the code followed by the actual table. We will examine the code after you see it.

\documentclass{article} \begin{document} \begin{tabular}{ccc} \hline Vegetables & Fruits & Nuts\\ \hline lettace & mango & almond\\ spinach & apple & cashews\\ \hline \end{tabular} \end{document}

We will now go through the code.

- Line 1 is the preamble and tells LaTeX that we are making an article document class.
- Line 2 is the declaration to begin the document environment
- Line 3 is where the table begins. We create a tabular environment. IN the second set of curly braces we used the argument “ccc” this tells LaTeX to create 3 columns and each column should center the text. IF you wan left justification to use “l” and “r” for right justification
- Line 4 uses the “\hline” declaration this draws the top horizontal line
- Line 5 includes information for the first row of the columns. The information in the columns is separated by an ampersand ( & ) at the end of this information you use a double forward slash ( \\ ) to make the next row
- Line 6 is a second “\hline” to close the header of the table
- Line 7 & 8 are additional rows of information
- Line 9 is the final “\hline” this is for the bottom of the table
- Lines 10 & 11 close the tabular environment and the document

This is an absolute basic table. We made three columns with centered text with three rows as well.

**Table with Caption**

A table almost always has a caption in academics. The caption describes the contents of the table. We will use the example above but we need to add several lines of code. This is described below

- We need to create a “table” environment. We will wrap this around the “tabular” environment
- We need to use the “\caption” declaration with the name of the table inside curly braces after we end the “tabular” environment but before we end the table environment.
- We will also add the “\centering” declaration near the top of the code so the caption is directly under the table

Below is the code followed by the example.

\documentclass{article} \begin{document} \begin{table} \centering \begin{tabular}{ccc} \hline Vegetables & Fruits & Nuts\\ \hline lettace & mango & almond\\ spinach & apple & cashews\\ \hline \end{tabular} \caption{Example Table} \end{table} \end{document}

**Conclusion**

We explored how to develop a basic table in LaTeX. There are many more features and variations to how to do this. This post just provides some basic ideas on how to approach this.

]]>It is not necessary to have ones in the diagonal it simply preferred when possible. However, you must have the zeros underneath the diagonal in order to solve the system. Every zero represents a variable that was eliminated which helps in solving for the other variables.

**Two-Variable System of Equations**

Our first system is as follows

x + 2y = 1

Here is our system

Generally, for a 2X3 matrix, you start in the top left corner with the goal of converting this number into a 1.Then move to the second row of the first column and try to make this number a 0. Next, you move to the second column second row and try to make this a 1.

With this knowledge, the first-row operation we will do is flip the 2nd and 1st row. Doing this will give us a 1 in the upper left spot.

Now we want in the bottom left column where the 3 is currently at. To do this we need to multiply row 1 by -3 and then add row 1 to row 2. This will give us a 0.

We now need to deal with the middle row, bottom number, which is -2. To change this into a 1 we need to multiple rows to by the reciprocal of this which is -1/2.

If you look closely you will see that we have achieved row-echelon form. We have all 1s in the diagonal and only 0s under the diagonal.

Our new system of equations looks like the following

If we substitute -1 for y in our top equation we can solove for x.

We now know that x = 3 and y = -1. This indicates that we have solved our system of equations using matrices and row operations.

**Conclusion**

Using matrices to solve a system of equations can be cumbersome. However, once this is mastered it can often be faster than other means. In addition, understanding matrices is critical to being able to appreciate complex machine learning algorithms that almost exclusively use matrices.

]]>The religion of Persia was Zoroastrianism. The priestly class of Persia were called Magi. They responsible for sacred duties as well as the education of princes.

These are the same Magi that are found in the Bible in reference to the birth of Jesus. Due to their priestly responsibilities and knowledge of astronomy, this information merged to compel the Magi to head to Jerusalem to see Christ as a small child.

Teachers for the commoners were normally retired soliders. Exemption from the military began at the age of 50. At this age, if a male was able to live this long, he would turn his attention the education of the next generation.

**What was Taught**

The emphasis in Persian education was gymnastics, moral, and military training. The physical training was arduous, to say the least. Boys were pushed well nigh to their physical limits.

The moral training was also vigourously instilled. Boys were taught to have a strong understanding of right and wrong as well as a sense of justice. Cyrus the Great shared a story about how, as a boy, he was called to judge a case about coats. Apparently, a large student had a small coat and a small student had a large coat. The large student forced the small student to switch coats with him.

When Cyrus heard this story he decided that the large boy was right because both boys now had a coat that fitted him. The large boy had a large coat and the small boy had a small coat. However, Cyrus’ teacher was disappointed and beat him. Apparently, the question was not which coat fit which boy but rather which coat belonged to which boy.

Something that was neglected in ancient Persian education was basic literacy. The reading, writing, and arithmetic were taught at a minimal level. These skills were left for the Magi to learn almost exclusively.

**How Was the Curriculum Organized**

From the age of 0-7 education was in the home with the mother. From 7-15 boys were educated by the state and were even considered state property. After the age of 15, students spent time learning about justice in the marketplace.

Girls did not receive much of an education. Rather, they focused primarily on life in the home. This included raising small children and other domestic duties.

**Conclusion**

Persia education was one strongly dominated by the state. The purpose was primarily to mold boys into just, moral soldiers who could serve to defend and expand the empire. This system is not without merit as it held an empire together for several centuries. The saddest part may be the loss of individual freedom and expression at the expense of government will.

]]>This post will provide an example of the development of a tree diagram used the tikz package. Specifically, we will make a vertical and a horizontal tree.

**Vertical Tree**

First I want you to see what our final product looks like before we go through each step to make it.

As you can see it is a simple tree.

To develop the tree you need to setup the preamble with the following.

\documentclass{article} \usepackage{tikz} \begin{document} \end{document}

There is nothing to see yet. All we did was set the documentclass to an article and load the tikz package which is the package we will use to make the tree.

The next step will be to make a tikzpicture environment. We also need to set some options for what we want our nodes to look like. A node is a created unit in a picture. In our completed example above there are 5 nodes or rectangles there. We have to set up how we want these nodes to look. You can set them individually or apply the same look for all nodes. We will apply the same look for all of our nodes using the every node/.style feature. Below is the initial setup for the nodes. Remeber this code goes after \begin{document} and before \end{document}

\begin{tikzpicture} [sibling distance=10em,level distance=6em, every node/.style={shape=rectangle,draw,align=center}] \end{tikpicture}

The options we set are as follows

- sibling distance = how far apart nodes on the same level are
- level distance = how far apart nodes on different adjacent levels are
- every node/.style = sets the shape and text alignment of all nodes

We are now ready to draw our tree. The first step is to draw the root branch below is the code. This code goes after the tikzpicture options but before \end{tikzpicture}.

\node{root branch};

We will now draw our 1^{st} child and grandchild. This can be somewhat complicated. You have to do the following

- Remove the semicolon after {root branch}
- Press enter and type child
- make a curly brace and type node
- make another curly brace and type 1
^{st}child and close this with a second curly brace - press enter and type child
- type node and then a curly brace
- type grandchild and close the curly braces three times
- end with a semicolon

Below is the code followed by a picture

\node{root branch} child{node{1st child} child{node{grandchild}}};

We now repeat this process for the second child and grandson. The key to success is keeping track of the curly braces and the semicolon. A Child node is always within another node with the exception of the root. The semicolon is always at the end of the code. Below is the code for the final vertical tree.

\node{root branch} child{node{1st child} child{node{grandchild}}} child{node{2nd child } child{node{grandchild}}};

**Horizontal tree**

Horizontal trees follow all the same steps. To make a horizontal tree you need to add the argument “grow=right” to the options inside the brackets. Doing so and you will see the following.

**Conclusion**

As you can see, make diagram trees is not overly complicated in LaTeX. The flexibility of the tikz package is truly amazing and it seems there are no limits to what you can develop with it for visual representations.

]]>