Quadratic Equations and the Square Root Property

A quadratic equation is an equation that includes a variable raised to the second power. Below is a common format for a quadratic equation.

1

This characteristic makes it difficult to rely on linear equation tricks of addition, subtraction, and multiplying to isolate the variable. One trick that we often have to use now is factoring as shown below.

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An alternative way to solve quadratic functions is through having knowledge of the square root property which is shown below.

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Below is the same example as our first example but this time we use the square root property.

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This trick works for numbers that cannot be factored.

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This leads us to the point that the square root property is used for speed or when factoring is not an option.

With this knowledge, all the other possible ways to solve a linear equation can be used to solve a quadratic equation

Division

In the example below is a quadratic formula in which you have to divide to isolate the variable. From there you solve like always1.pngFraction

To remove a fraction you must multiply both sides by the reciprocal as shown below.

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When we got the square root of 18 we had to further simplify the radical by finding the factors of 18. In the second to last line if you multiply these numbers together you will get 18 because  9 * 2 = 18. Furthermore, if you square root 9 you get 3 but you cannot square root 2 and get a whole number. This is why the final answer is 3 * the square root of 2.

Conclusion

 

Quadratic formulas are common in algebra and as such there are many different ways to solve them. In this post, we looked at an alternative to factoring called the square root property. Understanding this approach is valuable as you can often solve quadratic equations faster and or they can be used when factoring is not possible.

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The Life of Pythagoras

 

Pythagoras was a highly influential educator during the time of ancient Greece. In this post, we will take a brief look at his life and impact on education.

Early Life

Pythagoras was born around 570BC on the island of Samos. His early life was spent in private study. However, as a young man, Pythagoras traveled to Egypt to acquire additional education.

Down To Egypt

While in Egypt, Pythagoras studied with the Egyptian priest. The Egyptian priest were the masters of education in Egypt and was the only class in Egypt that received an advanced education. Under their tutelage, Pythagoras was exposed to various math and science subjects as well as some of the religious practices of Egypt. He was particularly touched by their way of life and it led him to develop his own style of living that would eventually be called Pythagoreanism.

After completing additional studies in Egpyt, Pythagoras moved to Italy and founded his own school. The school had essentially two levels which were the exoteric and esoteric. Students began in the exoteric studies and stayed there for at least 3 years. After completing exoteric studies a student would begin esoteric studies with Pythagoras himself.

In Italy

The subjects taught at Pythagoras’ school includes physics, geography, medicine, math and even metaphysics. In terms of math, it was Pythagoras who gave algebra students the Pythagorean theorem which states that the square of a  hypotenuse of a right triangle is equal to the sum of the square of the base and the square of the height as shown in the expression below

 base2 + height2 = hypotenuse2

Pythagoras also had distinct metaphysical views. He believed in one true called as a form of monotheism. This was in stark contrast to the commonly held beliefs of Greece at the time. Christianity was a rather strong religion at this time and it is possible that Pythagoras may have come into contact with this religion.

Pythagoras also believed in the transmigration of the soul. This essentially means that when an animal died they would come back as a lower animal. This is in many ways a form of reincarnation. It was simply another way of saying “You shall not really die” which was an idea shared in a garden by a snake to a woman.

Pythagoras’ school was known for being authoritative, strict, and even have a habit of being aristocratic. This along with other ideas made Pythagoras school unpopular. So unpopular that a mob would eventually burn his school down to the ground.

Conclusion

It is not clear if Pythagoras died in the flames or lived on as scholars are still debating this. What can be seen is that Pythagoras view of education has continued to live on to this day. His way of life had an influence on many people and his contribution to mathematics has touched the life of practical every algebra student on the planet.

Education in Ancient Sparta

With Ancient Greece there was a small city-state called Sparta. MAny today know of Sparta because of the movies that have been made of this war-like people. Spartan education was primarily a one about military training.

The reason for this emphasis on developing soldiers was due in part to the context in which the Spartans lived. In their own country, they were a minority with a large population of neighboring freeman and an even larger population of slaves. The only way in the Spartans minds to maintain power was through the use of strength. As such this was the focus of their education.

Background

The founder of the government of what makes up classical Sparta was Lycurgus. After spending time in Egypt Lycurgus came to Sparta and developed their constitution. Some of the practices he made lawe included the making all money out of iron to discourage greed and to require men to live in barracks together to encourage unity towards the military and state over the family.

By discouraging greed and familial affections Spartan men were focused on developing strength and military prowess almost to the exclusion of anything else. What else is there for a man to do when he cannot acquire wealth or enjoy his family?

One last point to mention is that children were seen as the property of the state. In a rather cruel way, weak children were eliminated at birth and only the strong were allowed to live. This further strengthens the idea of the state over family.

What They Learned

The training was primarily physical in nature. Young boys were taken from their homes at the age of 7 to live in the state barracks. Once there, they were given a minimum amount of clothes and food. The cold and hunger often compelled the boys to steal. Stealing was actually encourage as it taught stealth. However, being caught was punished severely because it indicated carelessness, which could prove deadly on the battlefield.

Gymnastics, wrestling, and the use of weapons were also emphasized. Despite the contradiction in encouraging stealing the Spartan education also strongly inculcated moral training as well. Boys were to control their appetites, respect the aged as well as their parents, and to be indifferent to suffering. It was considered shameful to lose control of one’s behavior in any way. This naturally discourages such behaviors as drunkness.

Unlike other ancient cultures, the Spartans loved music and spent a large of amount of free time developing this skill. Songs were frequently about war and brave acts.

Women also received an education and the focus was on the development of the physical nature.

How Were They Taught

Spartan boys were taught primarily by the senior citizens or the aged of the society. The old would spend time with the young boys. The common forms of instruction involved a question and answer format. This instilled a great deal of practical wisdom in the youth.

Another primary method of learning was imitation. Young people would learn simply through copying the actions and behaviors of the aged. This imitation of the aged rather than of other young people help Spartans to mature and develop a seriousness to them that would be hard to find in young people today.

COnclusion

The Spartans were a military culture with a strong state apparatus. Their educational system was developed to suppress the people around them in an attempt to maintain their own safety. This desire to survive contributed to a highly oppressive system from the viewpoint of an outsider but perhaps a saving grace for the Spartan.

Solving Single Radical Equations

This post will look at how to solve radical equations. The concepts are mostly similar to solving any other equation in terms of isolating terms etc. However, for people who are new to this, it may still be confusing. Therefore, we will go through several examples.

Example 1

Our first example is a basic radical equation that includes a constant outside the radical. Below is the equation.

1

Solving this problem requires to main steps.

  1. Isolate the radical
  2. Remove the radical by squaring it

Doing these two steps will lead to our answer. We will have two answers but the reason for this will become clear as we solve the equation.

First, we will isolate the radical by subtracting 1 from both sides

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Now, to remove the radical we will square both sides. This new equation will need to be simplified and will become a quadratic equation.

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With our new quadratic equation we will factor this and as expected get two answers.

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Index Other than 2

For a radical that has an index other than  2, the process involves raising the radical to whatever power will cancel out the radical. Below is an example that has an index of 3. We will first subtract the constant from both sides.

1

In order to remove the index of 3, we need to raise each side of the equation to the power of 3. After doing this, we solve a simple equation.

1

Radicals as Fractions

One a number is a raised to a power that is a fraction it is the same as a radical. Below is an example.

1

This means that the steps we took to solve equations with radicals can be mostly used to deal with equations with powers that are a fraction.

below is an equation. To solve this equation you must raise each side of the equation to the power of the denominator in the fraction.

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As you can see both sides were raised to the 4th power because that is the number in the denominator of the fraction. On the left side of the equation, the 4th power cancels out the fraction. Now you can simply solve the equation like any other equation.

1

Hopefully, this is clear.

Conclusion

Solving radical equatons in not that diffcult. Usually, the ultimate goal is to remove the radical. The difference between this and solving for other equations is that with radical equations you want to first isolate the radical, remove the radical, and then solve for the unknown variable.

Education in Ancient Egypt

Ancient Egypt is perhaps one of the oldest if not the oldest civilization on the planet. With a rich history dating several thousand years Egypt also had a reputation for education as well. This post will discuss education in Egypt with a focus on training by caste.

Background

Egypt was famous for their wisdom and architectural work. In terms of architecture, we are probably all familiar with the pyramids that are still standing after several thousand years. In terms of wisdom, Egypt was so highly regarded in the past that the Greeks sent several future philosophers and leaders to Egypt to study. Among those who went include Plato perhaps the greatest philosopher of all time, Lycurgus, the founder of Sparta, and Solon, the famous Athenian Statesman.

Egypt also had a strong caste system similar to India’s. There were essentially three classes. At the top were the priests, second, was the military, and the lowest classes was everybody else. The lowest class was also sub-divided into three subclasses of farmers/boatmen, then mechanics and tradesman, and lastly the herdsman, fishermen, and laborers. A person was born into their class and it was almost impossible to move from one to the other.

The priestly class was also exempt from taxes and owned as much as 1/3 of the land in Egypt. Their skills and training also commanded high salaries. Egypt was essentially a priests’ country in terms of status and privileges.

What They Study

The education an Egyptian received was heavily influenced by the caste they came from. The priest received the most extensive training. They studied philosophy, natural history, medicine, math, history, law, etc. With this training, a person from the priestly caste could be a physician, historian, surveyor, customs inspector, judge, counselor, etc.

Everyone else received a basic education depending on their occupation. Merchants learned how to read, write, and perform simple math. Tradesman only learned their trade from their parents.

The writing was also divided along class lines. THere were two types of writing systems. The Demotic style was for the masses while the Hieratic style was for the priestly class. The main difference between these two styles is the proportion of hieroglyphics used.

Two subjects commonly ignored in Egyptian education was gymnastics and music. Gymnastics was considered dangerous due to the risk of bodily harm. Music was considered to have an effminate influence on a man if studied to excess.

Conclusion

Ancient Egyptian was unique in terms of the dominance of the priestly class. The priest was allowed to study extensively while everyone else did not seem to enjoy the same access to education. This allowed the priest to wield tremendous informal power within Egypt and to quietly work behind the scenes to achieve goals

Roots, Radicands, and Radicals

Roots, radicands, and radicals are yet another way to express numbers in algebra. In this post, we will go over some basic terms to know.

Roots

A square root is a number that is multiplied by itself to get a new number. Below is an example

1

In the example above 5 is the square root of 25. This means that if you multiply 5 by its self you would get 25.

Another term to know is the square. The square is the result of multiplying a number by its self. In the expression above 25 is the square of 5 because you get 25 by multiplying 5 by its self.

Square Roots

Square roots, in particular, have a lot of other ways to be expressed. To understand square roots you need to know what roots, radicands, and radical sign are. Below is a picture of these three parts.

1

The radical sign is simply a sign like multiplication and division are. The radicand is the number you want to simplify by finding a number that when multiplied by itself would equal the value in the radicand. We also call this new number the square root. For example,

1

What the example above means is that the number you can multiply by itself to get 100 is 10.

The index is trickier to understand. It tells you how many times to multiply the number by its self to get the radicand. If no number is there you assume the index is 2. Below is an expression with an index that is not 2.

1

What this expression is saying is that you can multiply 2 by its self 3 times to get eight as you can see below.

2 * 2  = 4 * 2 =  8

Additional Terms

There are some basic terms that are needed to understand using radicals. Generally, when every we are speaking of multiplying two times we call it square. Multiplying three times is referred to as cub or cubic. Anything beo=yond 3 is called to the nth powered. For example, multiplying a number by its self 4 times would be called to the 4th power, 5 times to the 5th power etc. However, some people referred to the square as the 2nd  power and the cube as the 3rd power if this is not already confusing. Below is a table that clarifies things

Number Power Example
2 square n2
3 cube n3
4 4th power n4
5 5th power n5

Conclusion

There are many more complex ideas and operations that can be performed with radicands and radicals. One of the primary benefits is that you can avoid dealing with decimals for many calculations when you understand how to manipulates these terms. As such, there actual are some benefits in understanding radicands and radicals use.

Education in Ancient China

As one of the oldest civilizations in the world, China has a rich past when it comes to education. This post will explore education in Ancient China by providing a brief overview of it. The following topics

  1. Background
  2. What was Taught
  3. How was it Taught
  4. The Organization of what was Taught
  5. The Evidence Students Provided of their Learning

Background

Ancient Chinese education is an interesting contrast. On the one hand, they were major innovators of some of the greatest invention of mankind which includes paper, printing, gunpowder, and the compass. On the other hand, Chinese education in the past was strongly collective in nature with heavy governmental control. There was extreme pressure to conform to ancient customs and independent deviate behavior was looked down upon.  Despite this, there as still innovation.

Most communities had a primary school and most major cities had a college. Completing university study was a great way to achieve a government position in ancient China.

What Did they Teach

Ancient Chinese education focused almost exclusively on Chinese Classics. By classics, it is meant the writings of mainly Confucius. Confucius emphasized strict obedience in a hierarchical setting. The order was loosely King, Father, Mother, then the child. Deference to authority was the ultimate duty of everyone. There is little surprise that the government support such an education that demanded obedience to them.

Another aspect of Confucius writings that was stressed was the Five Cardinal Virtues which were charity, justice, righteousness, sincerity, and conformity to tradition. This was the heart of the moral training that young people received. Even leaders needed to demonstrate these traits which limited abuses of power at times.

What China is also famous for in their ancient curriculum is what they did not teach.  Supposedly, they did not cover in great detail geography, history, math, science, or language. The focus was on Confucius apparently almost exclusively.

How Did they Teach

Ancient Chinese education was taught almost exclusively by rote memory. Students were expected to memorized large amounts of information.  This contributed to a focus on the conservation of knowledge rather than the expansion of it. If something new or unusual happened it was difficult to deal with since there was no prior way already developed to address it.

How was Learning Organized

School began at around 6-7 years of age in the local school. After completing studies at the local school. Some students went to the academy for additional studies.  From Academy, some students would go to university with the hopes of completing their studies to obtain a government position.

Generally,  the education was for male students as it was considered shameful to not educate a boy. Girls often did not go to school and often handle traditional roles in the home.

Evidence of Learning

Evidence of learning in the Chinese system was almost strictly through examinations. The examinations were exceedingly demanding and stressful. If a student was able to pass the gauntlet of rot memory exams he would achieve his dream of completing college and joining the prestigious Imperial Academy as a Mandarin.

Conclusion

Education in Ancient China was focused on memorization, tradition,  and examination. Even with this focus, Ancient China developed several inventions that have had a significant influence on the world. Explaining this will only lead to speculation but what can be said is that progress happens whether it is encouraged or not.

Polynomials

Polynomial is an expression that has more than one algebraic term. Below is an example,

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Of course, there is much more to polynomials then this simple definition. This post will explain how to deal with polynomials in various situations.

Add & Subtract

To add and subtract polynomials you must combine like terms. Below is an example1

All we did was combine the terms that had the y^2 in them. This, of course, applies to subtraction as well.

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In the example above, the terms with “a” in them are combined and the terms with “n” in then are combined.

Exponents

When dealing with exponents when multiplication is involved you add the exponents together.

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Notice how like terms were dealt with separately.

Since we add exponents during multiplication we subtract them during division.

1

The 2 in the numerator and denominator cancel out and  7 – 5 = 2.

When parentheses are involved it is a little more complicated. For example, when the exponent is negative you multiply the exponents below

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For negative exponents, you fill the numerator and denominator around and make the negative exponent positive.

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There are other concepts involving polynomials not covered here. Examples include long division with polynomials and synthetic division. These are fascinating concepts however in the books I have consulted, once these concepts are taught they are never used again in future chapters. Therefore, perhaps they are simply interesting but not commonly used in practice.

Conclusion

Understanding polynomials is critical to future success in algebra. As concepts become more advanced, it will seem as if you are always trying to simplify terms using concepts learned in relation to polynomials.

Education in Ancient Israel

The Nation of Israel as described in the Bible has a rich and long history of several thousand years. This particular group of people believed that they are the keepers of the knowledge of the true God. Their influence in religion is remarkable in that a large part of the theology of Christianity is derived from Hebrew writings.

In this post, we will only look at a cross-section of Hebrew education around the time of the time of the monarchy period of David and Solomon.

What Did they Teach

The goal of Hebrew education was to produce people who obeyed God. This is in stark contrast to other educational systems that emphasized obeying earthly rulers. The Hebrew system stress first allegiance to God and then allegiance to man when this did not conflict with the will of God. When there was a disagreement in terms of what man and God commanded the Hebrew was taught to obey  God. This thinking can be traced even in Christianity with the death of martyrs throughout Church history.

The educational system was heavily inspired by their sacred writings. At the time we are looking at, the majority of the writings were by Moses. The writings of Moses provide a detailed education of health principles, morality, and precise explanation of performing the rites of the sacrificial system.

The sacrificial system in the Hebrew economy is particularly impressive in that the ceremonies performed were all meant to help the Israelites remember what God had done for them and to be shadows of the life, death, and resurrection of Christ as understood by some Christian theologians.

In order for children to learn all of the laws and sacred writings of their nation, it required that almost everyone learn to read. People were held personally responsible to understand their role in society as well as in how to treat others and God’s will for them. Again this is in contrast to other religions in which people simply obeyed the religious leaders. The Jew was expected to know for themselves what their religion was about.

How Did They Teach

Despite the theocratic nature of the government and the details of the religious system, the educational system in Ancient Israel was highly decentralized. The school was the home and the teachers were the parents. Most nations that reached the strength and level of the Monarchy of Israel had a state ran educational system. However, the Hebrews never had this.

The decentralized nature of education is unusual because secular leaders normally want to mold the people to follow and obey them. In Israel, this never happens because of the focus on serving God. The personalized education allowed children to grow as individuals rather than as cogs in a nation-state machine. The idea of allowing parents to all educate their children as they decide would seem chaotic in today’s standardized world. Yet the Israel monarchy lasted as long as any other kingdom in the world.

How Did They Organize 

Once a child completed their studies they would learn a trade and begin working. Higher education was not focused on secular matters and was often reserved for the priestly class to learn skills related to their office. Example include law, sacred writings weights and measures, and astronomy to determine when the various feast days would be.

Another form of additional education was the Schools of the Prophets. Apparently, these were independent institutions that provided training in the scriptures, medicine, and law. At least one author claims that the Schools of the Prophets were established because Hebrew parents were neglecting the education of their children.  In other words, when the parents began to neglect the education of their children is when the nation begin to decline as well.

Conclusion

The Israelite educational system during the early monarchy period was an interesting example of contrasts. Highly detailed yet decentralized in execution, focused on obeying God yet having a monarchy that probably wanted to keep power, and little regard for higher education while producing some of the most profound theological works of all-time.  The strength of this system would be considered a weakness in many others.

Cramer’s Rule

Cramer’s rule is a method for solving a system of equations using the determinants. In order to do this, you must be familiar with matrices and row operations. Generally, it is really difficult to explain that is a simple matter but there are two main parts to completing this

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

2x + y = -4
3x – 2y = -6

Below is the matrix of the system of equations

1

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.

1
1

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.

1

1

Lastly, here is the determinant of y

1

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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.

1

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 Cramer’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.

Classroom Discussion

Classroom discussion is a common yet critical aspect of the educational experience. For many, learning happenings not necessarily when students listen but also when students express their thoughts and opinions regarding a matter. This post will look at reasons for discussion, challenges, and ways to foster more discussion in the classroom.

Reasons for Discussion

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.

Evaluating the Determinant of a Matrix

There are several different ways to solve a system of equations. Another common method is Cranmer’s Rule. However, we cannot learn about Cranmer’s rule until we understand determinants.

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.

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Here is how this look with real numbers

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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.

1

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

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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.

1

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

4(4) – (-2)(-2) =
16 – 12 =
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.

1

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

1

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.

 2(-4 – 0) + 3(-6 – 0) – 1(-3 – (2-2)) = 
-18 – 18 + 1= 
-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.

Making Tables with LaTeX

Tables are used to display information visually for a reader. They provide structure for the text in order to guide the comprehension of the reader. In this post, we will learn how to make basic tables.

Basic Table

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}

1

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}

1.png

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.

Solving a System of Equations with Matrices: 2 Variables

This post will provide examples of solving a system of equations with 2 variables. The primary objective of using a matrix is to perform enough row operations until you achieve what is called row-echelon form. Row-echelon form is simply having ones all across the diagonal from the top left to the bottom right with zeros underneath the dia. Below is a picture of what this looks like

1.png

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

3x + 4y = 5
x + 2y = 1

Here is our system

1

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.

1.png

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.

1

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.

1

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

1x + 2y = 1
0x +1y = -1 or y = -1

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

1

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.

Education in Ancient Persia

The Persian Empire was one of the great empires of ancient civilization. It was this Empire that defeated the Babylonians. This post will provide a brief examination of the educational system of Persia.

Background

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.

Making Diagram Trees with LaTeX

There are times when we want to depict hierarchical relationships in a diagram. Examples include workflow chart, organizational chart, or even a family tree. In such situations, the tree diagram feature in LaTeX can meet the need.

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.

Screenshot 2018-04-17 09:02:18

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};

Screenshot 2018-04-17 09:17:18

We will now draw our 1st 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 1st 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}}};

Screenshot 2018-04-17 09:24:14

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}}};

Screenshot 2018-04-17 09:02:18.png

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.

Screenshot 2018-04-17 09:29:52

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.

Augmented Matrix for a System of Equations

Matrices are a common tool used in algebra. They provide a way to deal with equations that have commonly held variables. In this post, we learn some of the basics of developing matrices.

From Equation to Matrix

Using a matrix involves making sure that the same variables and constants are all in the same column in the matrix. This will allow you to do any elimination or substitution you may want to do in the future. Below is an example

11

Above we have a system of equations to the left and an augmented matrix to the right. If you look at the first column in the matrix it has the same values as the x variables in the system of equations (2 & 3). This is repeated for the y variable (-1 & 3) and the constant (-3 & 6).

The number of variables that can be included in a matrix is unlimited. Generally,  when learning algebra, you will commonly see 2 & 3 variable matrices. The example above is a 2 variable matrix below is a three-variable matrix.

11

If you look closely you can see there is nothing here new except the z variable with its own column in the matrix.

Row Operations 

When a system of equations is in an augmented matrix we can perform calculations on the rows to achieve an answer. You can switch the order of rows as in the following.

1.png

You can multiply a row by a constant of your choice. Below we multiple all values in row 2 by 2. Notice the notation in the middle as it indicates the action performed.

1

You can also add rows together. In the example below row 1 and row 2, are summed to create a new row 1.

1

You can even multiply a row by a constant and then sum it with another row to make a new row. Below we multiply row 2 by 2 and then sum it with row 1 to make a new row 1.

1

The purpose of row operations is to provide a way to solve a system of equations in a matrix. In addition, writing out the matrices provides a way to track the work that was done. It is easy to get confused even the actual math is simple

Conclusion

System of equations can be difficult to solve. However, the use of matrices can reduce the computational load needed to solve them. You do need to be careful with how you modify the rows and columns and this is where the use of row operations can be beneficial.

Drawing Diagrams in LaTeX

There is an old saying that most of us are familiar with that says that “a picture is worth a thousand words.” Knowing means that a communicating cannot only include text but most also incorporate visuals as well. LaTeX allows you to develop visuals and diagrams using various packages for this purpose.

The visuals we will make are similar to those found in Microsoft Word Smart Graphics. One of the main advantages of using code to make diagrams is that they are within the document and you do not need to import images every single time you compile the document. If the image disappears it will not work but as long as the code is where you can always regenerate it.

In this post, we will use the “smartdiagram” package to make several different visuals that can be used in LaTeX. The types we will make are as follows…

  • Flow diagram
  • Circular diagram
  • Bubble diagram
  • Constellation diagram
  • Priority diagram
  • Descriptive diagram

The code for each individual diagram is almost the same as you will see. The preamble will only include the document class of “article” as well as the package “smartdiagram”. After this, we will create are document environment. Below is the preamble and the empty document environment.

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

Flow Diagram

The flow diagram is a diagram using boxes with arrows pointing from left to right in-between each box until the last box has an arrow that points back to the first bo indicating a cyclical nature. Below is  the code followed by the diagram

\smartdiagram[flow diagram:horizontal]{Step 1,Step2,Step3,Step4}

1

The syntax is simple.

  1. Call the declaration “\smartdiagram”
  2. Inside the brackets, you indicate the type of diagram which was “flow diagram: horizontal” for us.
  3. Next, you indicate how many boxes by typing the text and separating them by commas inside the curly braces.

This pattern holds for most of the examples in this post.

Circular Diagram

Below is a circular diagram. The syntax for the code is the same. Therefore, the code is below followed by the diagram

\smartdiagram[circular diagram:clockwise]{Step 1,Step2,Step3,Step4}

1.png

Bubble Diagram

The same syntax as before. Below is the code and diagram.

\smartdiagram[bubble diagram]{Step 1,Step2,Step3,Step4}

1.png

Constellation Diagram

This diagram looks similar to the bubble diagram but has arrows jutting out of the center. The syntax is mostly the same.

\smartdiagram[constellation diagram]{Step 1,Step2,Step3,Step4}

1.png

Priority Descriptive Diagram

This diagram is useful if the order matters

\smartdiagram[priority descriptive diagram]{Step1, Step2, Step3,Step4}

1.png

Descriptive Diagram

The coding for the descriptive diagram is slightly different. Instead of one set of curly braces, you have a set of curly braces within a set of curly braces within a final set of curly braces. The outer layer wraps the entire thing. The second layer is for circles in the diagram and the inner curly braces are for adding text to the rectangle. Each double set of curly braces are separated by a comma. Below is the code followed by the diagram.

\smartdiagram[descriptive diagram]{
{Step 1,{Sample text, Sample text}},
{Step2,{More text, more text}},
{Step3,{Text again, text again}},
{Step4,{Even more text}}
}

1.png

Hopefully, you can see the formatting of the code and see how everything lines up.

Conclusion

Developing diagrams for instructional purposes is common in many forms of writing. Here, we simply look at creating diagrams using LaTeX. The power of this software is the ability to create almost whatever you need for communication.

System of Equations and Mixture Application

Solving a system of equations with a mixture application involves combining two or more quantities. The general setup for the equations is as follows

Quantity * value = total

This equation is used for both equations. You simply read the problem and plug in the information. The examples in this post are primarily related to business as this is one of the more practical applications of solving a system of equations for the average person. However, a system of equations for mixtures can also be used for determining solutions but this is more common in chemistry.

Example 1: Making Food 

John wants to make 20 lbs of granola using nuts and raisins. His budget requires that the granola cost $3.80 per pound. Nuts are $4.50 per pound and raisins are $1.00 per pound. How many pounds of nuts and raisins can he use?

The first thing we need to determine what we know

  • cost of the raisins
  • cost of the nuts
  • total cost of the granola
  • number of pounds of granola to make

Below is all of our information in a table

Pounds * Price Total
Nuts n 4.50 4.5n
Raisins r 1 r
Granola 20 3.80 3.8(20) = 76

What we need to know is how many pounds of nuts and raisins can we use to have the total price per pound be $3.80.

With this information, we can set up our system of equations. We take the pounds column and create the first equation and the total column to create the second equation.

1

We will use elimination to solve this system. We will multiply the first equation by -1 and combine them. Then we solve for n as in the steps below

1.png

We know n = 16 or that we can have 16 pounds of nuts. To determine the amount of raisins we use our first equation in the system.

1.png

You can check this yourself if you desire.

Example 2: Interests

Below is an example that involves two loans with different interest rates. Our job will be to determine the principal amount of the loan.

Tom owes $43,080 on two student loans. The bank’s interest rate is 5.25% and the federal loan rate is 2.95%. The total amount of interest he paid last two years was 6678.72. What was the principal for each loan

The first thing we need to determine what we know

  • bank interest rate
  • Federal interest rate
  • time of repayment
  • Amount of loan
  • Interest paid so far

Below is all of our information in a table

Principal * Rate Time Total
Bank b 0.0525 1 0.0525b
Federal f 0.0295 1 0.0295f
Total 43080 1752.45

Below is our system of equation

1.png

To solve the system of equations we will use substitution. First, we need to solve for b as shown below

1.png

We now substitute  and solve

1

We know the federal loan is $22,141.30 we can use this information to find the bank loan amount using the first equation.

1.png

The bank loan was $20,938.70

Conclusion

Hopefully, it is clear by now that solving a system of equations can have real-world significance. Applications of this concept can be useful in the context of business as shown here.

Education in Ancient India

In this post, we take a look at India education in the ancient past. The sub-continent of India has one of the oldest civilizations in the world. Their culture has had a strong influence on both the East and West.

Background

One unique characteristic of ancient education in India is the influence of religion. The effect of Hinduism is strong. The idea of the caste system is derived from Hinduism with people being divided primarily into four groups

  1. Brahmins-teachers/religious leaders
  2. Kshatriyas-soldiers kings
  3. Vaisyas-farmers/merchants
  4. Sudras-slaves

This system was ridged. There was no moving between caste and marriages between castes was generally forbidden. The Brahmins were the only teachers as it was embarrassing to allow one’s children to be taught by another class. They received no salary but rather received gifts from their students

What Did they Teach

The Brahmins served as the teachers and made it their life work to reinforce the caste system through education. It was taught to all children to understand the importance of this system as well as the role of the  Brahmin at the top of it.

Other subjects taught at the elementary level include the 3 r’s. At the university level, the subjects included grammar, math, history, poetry, philosophy, law, medicine, and astronomy. Only the Brahmins completed formal universities studies so that they could become teachers. Other classes may receive practical technical training to work in the government, serve in the military, or manage a business.

Something that was missing from education in ancient India was physical education. For whatever reason, this was not normally considered important and was rarely emphasized.

How Did they Teach

The teaching style was almost exclusively rote memorization. Students would daily recite mathematical tables and the alphabet. It would take a great deal of time to learn to read and write through this system.

There was also the assistance of an older student to help the younger ones to learn. In a way, this could be considered as a form of tutoring.

How was Learning Organized

School began at 6-7. The next stage of learning was university 12 years later. Women did not go to school beyond the cultural training everyone received in early childhood.

Evidence of Learning

Learning mastery was demonstrated through the ability to memorize. Other forms of thought and effort were not the main criteria for demonstrating mastery.


Conclusion

Education in India serves a purpose that is familiar to many parts of the world. That purpose was social stability. With the focus on the caste system before other forms of education, India was seeking stability before knowledge expansion and personal development. This can be seen in many ways but can be agreed upon is that the country is still mostly intact after several thousand years and few can make such a claim even if their style of education is superior to India’s.

Making Presentations in LaTeX

One of the more interesting abilities of LaTeX is the ability to make presentations similar to those that are commonly made with PowerPoint. In this post, we will explore this capability of generating presentations with LaTeX

Setting Up The Preamble

The document class used for making presentations is called “beamer”. With this, you also need to set the theme of the presentation. The theme is the equivalent of a template in powerpoint. For our purposes, we will use the Singapore theme. After doing this the preamble is complete for our example

\documentclass{beamer}
\usetheme{Singapore}

Title Page

We will create the title page of the document. This involves using the “frame” environment. The code is below followed by the actual example.

\documentclass{beamer}
\usetheme{Singapore}
\begin{document}
 \title{Example Project}
  \subtitle{For Blog}
  \author{Yours Truly}
  \date{June 20, 2099}
  \begin{frame}
   \titlepage
  \end{frame}
\end{document}

1.png

Overview of Presentation

Another section that you can include is a table of contents. This will allow you to provide a big picture of what to expect in the presentation. The beamer class does not include animation so to make bullets appear LaTeX will create several slides and each slide will include one additional piece of information. This gives the appearance of animation when in fact it is a new slide. Below is the code with the additional information in bold. Unfortunately, you will not be able to see anything after this step because our example is incomplete.

\documentclass{beamer}
\usetheme{Singapore}
\begin{document}
\title{Example Project}
\subtitle{For Blog}
\author{Yours Truly}
\date{June 20, 2099}
\begin{frame}
\titlepage
\end{frame}
\begin{frame}{Outline}
\tableofcontents[pausesections]
\end{frame}
\section{Beginning}
\section{Middle}
\section{End}

The “\section” declarations tell LaTeX what is in the table of contents.

Completed Presentation

We will now make several different slides that have some sample text. On each slide, we will use an “itemize” environment in order to create bullets. The bullets help to organize the text visually.  Below is the final code followed by several pictures of what it should look like.

\documentclass{beamer}
\usetheme{Singapore}
\begin{document}
 \title{Example Project}
 \subtitle{For Blog}
 \author{Yours Truly}
 \date{June 20, 2099}
 \begin{frame}
  \titlepage
 \end{frame}
 \begin{frame}{Outline}
  \tableofcontents[pausesections]
 \end{frame}
 \section{Beginning}
 \section{Middle}
 \section{End}
 \begin{frame}{Beginning} 
  \begin{itemize} 
   \item first point 
   \item Second point 
  \end{itemize}
 \end{frame}
 \begin{frame}{Middle} 
  \begin{itemize} 
   \item first point 
   \item Second point 
  \end{itemize}
 \end{frame}
 \begin{frame}{End} 
  \begin{itemize} 
   \item first point 
   \item Second point 
  \end{itemize}
 \end{frame}
\end{document}

1.png

1.png

1.png

Conclusion

The beamer class allows a person to develop simple efficient presentations using LaTeX. The main advantage may be speed. As you can type and add content fast simply with a few keystrokes rather than with mouse clicks. However, many people would find this cumbersome and you can do a great deal of typing using the outline view in Powerpoint. Despite this, it is good to know that LaTeX provides this feature.

Designing a Poster in LaTeX

LaTeX provides the option of being able to make posters. This can be useful for academics who often present posters at conferences in order to share their research. In this post, we will go through the development of a poster using LaTeX.

First, we will setup the preamble and title of the poster. The document class is “tikzposter” with a1paper and size 25pt font. We will use the “graphicx” package for inserting images, the “lipsum” package for dummy text, and the “multicol” package to divide the poster into columns. The theme we are using is “Rays” and is one of many available themes.

Before we begin the coding it will be beneficial if you see what the final product looks like.

1

This poster has two columns and a block along the bottom. The column 1 to the left has two blocks A and B. Block A has an inner block. In column 2, we have a block with a picture in it.

Coding

Next, we begin the document and insert the code for the title. Below is the code followed by what are document look likes so far.

\documentclass[25pt,a1paper]{tikzposter} %size is 84cmX120cm
\usepackage{graphicx}
\graphicspath{{YOUR DIRECTORY HERE}}
\usetheme{Rays} %theme of poster design
\usepackage{lipsum} %for dummy text
\usepackage{multicol}
\begin{document}
\title{This is Amazing}
\author{ERT Blogger}
\maketitle
\end{document}

1.png

Creating Columns

How you design the poster is up to you but in our example, we are going to create two columns with a horizontal block across the bottom. Column one will use 65% of the available space while column 2 will use the remaining 35%. We will not include the code for the horizontal block along the bottom yet.  For now, just look at the code and after the next step, you can copy it if you want.

\begin{columns}
%COLUMN 1
\column{.65}%use 65% of the available space for this column
%COLUMN 2
\column{.35} %last 1/3 of space
\end{columns}

Making Blocks

Inside column one, we are going to place two blocks of text called Block A and Block B. Inside each block you can design it however you want. You can even have blocks within blocks.

For Block A we will make a title, a small bit of text, a colored box with some bullets, more text, and lastly an inner-block with some math text. The \bigskip declaration provides spacing and the \lipsum declaration provides dummy text. The code is below followed by a screenshot of the current poster. This code must be placed before the \end{column} command.

\begin{columns}
%COLUMN 1
\column{.65}%use 65% of the available space for this column 
%Block A
\block{More Examples of LaTeX}{
\bigskip
You can even put stuff in colored boxes.\\
\coloredbox{\begin{itemize}
  \item Point 1
  \item point 2
\end{itemize}}
\lipsum[2]
\bigskip
\innerblock{here is some math for fun}
{\begin{center}
  $2^5+5x-\frac{2}{x} * 3= y$
  \end{center}
  }
}

1.png

Block B is much simpler and includes a title with some dummy text. The code is below.

%Block B 
\block{More Text}{\lipsum[1]}

Column 2

We will now turn our attention to column 2. This column uses the last 35% of remaining space and has a picture in it with some dummy text. Inside the column, we set up a block and include the graphic followed by the dummy text. The code is below with the image afterward. This code must be placed before the \end{column} command.

%COLUMN 2
\column{.35} %last 1/3 of space
\block{More Pictures}{
\includegraphics[width=\linewidth]{{"1"}.jpg}
\lipsum[4]
}

1.png

Final Block

We will now put a block that runs along the bottom of the poster. Just a title with some text. This code must be placed above the \end{document} command.

\block{The End}{
\lipsum[10-11]
}

1.png

Conclusion

Perhaps, you can see how cool and versatile LaTeX can be. You can make a poster for presentations that are rather beautiful and much more symmetrical than trying to draw boxes by hand using powerpoint.

Solving a System of Equations with Three Variables

A system of equations can be solved involving three variables. There are several different ways to accomplish this when three variables are involved. In this post, we will focus on the use of the elimination method.

Our initial system of equations is below

1.png

The values eq1,eq2 and eq3 just mean equation 1, 2, 3

To solve this system we need to first solve two equations as a system and create a fourth equation we will call eq4. We then take eq1 and eq3 to create a new system of equations that creates eq5.

It is important to note that for the first two two-variable system of equations you create that you eliminate the same variable it both systems. So fare our example when we take equation 1 and 2 to create equation 4 and then take equation 1 and 3 create equation 5 we must solve for y in both situations or else we will have problems. In addition, you must make sure that all three equations appear at least once in the two two-variable systems of equations. For our purpose, we will use eq1 twice and eq2 and eq3 once.

Eq4 and eq5 are used to find the actual values we need for all three variables. This will make more sense as we go through the example. Therefore, we are going to solve first for y for eq1 and eq2.
1.png

To eliminate y we need to multiple eq2 by 2 and then combine the equations. Below is the process and the new eq 4

1.png

We will come back to eq4. For now, we will create eq 5 by eliminating y from eq1 and eq3.

1

We are essentially done using equations 1, 2, and 3. They will not reappear until the end. We will now use equations 4 and 5 to find our answers for two of the three variables.

We now will use eq 4 and 5 to eliminate the variable x. Eliminating x will allow us to solve for z. Doing means we will multiply eq4 by -1.

1.png

We know z = -3 we can plug this value into either eq4 or 5 to find the answer for x.

1.png

Now that we know x and z we can plug the two numbers into one of the three original equations to find the value for y. Notice how the first variable we eliminated becomes the last one we solve for.

1.png

We now know all three values which are

(4, -1, -3)

What this means is that if we were to graph this three equations they would intersect at (4, -1, -3). A solving a system of equations is simply telling us where the lines of the equations intersect.

Conclusion

Solving a system of equations involving three variable is an extension of the two variable system that has already been covered. It provides a mathematician with a tool for solving for more unknown variables. There are practical applications of this as we shall see in the future,

Cross-Referencing with LaTeX

Cross-referencing allows you to refer to almost anything in your document automatically through the use of several LaTeX commands. This can become extremely valuable if you have to edit your document and things change. With whatever updates you make the cross-referencing is update automatically.

There are many different ways to cross-reference in LaTeX but we will look at the following.

  • Tables
  • Items in a list
  • Pages

When using cross-referencing you must compile the document twice in order for the numbers to show up. The first time you will only see question marks.

Tables

Below is an example of LaTeX referring to a table.

\documentclass{article}
\begin{document}
   In Table~\ref{Example} we have an example of a table
   \begin{table}[h]
   \begin{center}
   \begin{tabular}{ll}
      \hline
         Fruits&Vegetables\\
      \hline
         Mango&Lettuce\\
         Papaya&Kale\\
      \hline
   \end{tabular}
      \caption{Example} \label{Example}
   \end{center}
   \end{table}
\end{document}

1.png

How a table is created has been discussed previously, what is new here are two pieces of code.

  • ~\ref{  }
  • \label{ }

The “\label” declaration gives the table a label that can be used in the text. In the example, we labeled the table “Example”. The “~\ref” declaration is used in the text and you put the label name on the table inside the curly braces. If you look at the text we never use the number 1 in the text. LaTeX inserts this for us automatically.

The same process can be used to label images as well.

Item in List

Cross-referencing an item on a list is not that complex either. Below is an example.

\documentclass{article}
\begin{document}
   Simple list
   \begin{enumerate}
      \item Mango
      \item Papaya \label{fruit2}
      \item Apple
   \end{enumerate}
   Number \ref{fruit2} is a common fruit in tropical countries.
\end{document}

1.png

As you can see, you can label almost anything anywhere.

Referring to Pages

It is also possible to refer to pages. This can save a lot of time if you update a document and page numbers change. Below is the code and example.

\documentclass{article}
\begin{document}
 Simple list \label{list}
 \begin{enumerate}
    \item Mango
    \item Papaya \label{fruit2}
    \item Apple
 \end{enumerate}
Number \ref{fruit2} in the list on page~\pageref{list} 
is a common fruit in tropical countries.
\end{document}

1.png

We made a label right above our list and then we used the “~\pageref” declaration with the name of the label inside. This provides us with the page number automatically.

Conclusion

There are more complex ways to cross-reference. However, unless you are developing a really complex document they are not really necessary for most practical applications. The ideas presented here will work in most instances as they are.

System of Equations and Uniform Motion

This post will provide examples of the use of a system of equations to solve uniform motion applications. A system of equations is used to solve for more than one variable. In the context of uniform motion, the basic equation is as follows

distance  = rate * time

We will look at the following examples

  • Two objectives moving in the same direction
  • Affect of a headwind/tailwind

Objects Moving in the Same Directions

Below is the problem followed by the solution.

Dan leaves home and travels to Springfield at 100 kph. About 30 minutes later Sue leaves the house and also travels the same way to Springfield driving 125 kph. How long will it take Sue to catch Dan?

The easiest way to solve this is to create a table with all of the information we have. The table is below.

Names Rate * Time = Distance
Dan 100 j 100j
Sue 125 k 125k

We first need to recognize that they will drive the same distance this leads to one of our equations

1.png

However, we are not done. We also need to realize that Sue leaves half an hour later, which leads to the second equation

1.png

We can now solve our system of equations

1

We know Dan travels for 2.5 hours before Sue catches him but we need to determine how long Sue drives before she catches Dan. We will take our answer J and plug it into the original equation for k.

1

It will take Sue 2 hours to catch up with Dan.

Affect of a Headwind/Tailwind

In transportation, it is common for a plan or ship to be able to travel faster with a tailwind or downstream than with a headwind or upstream The example below shows you how to determine the speed needed to travel a certain distance in the same amount of time as well as the speed of the wind/current.

A plane can travel 548 miles in 1.5 hours with a tailwind but only 494 hours when flying into a headwind. Find the speed of the plane and the wind.

We will have two variables because there are two things we want to know

  • p = the speed of the plane
  • w = the speed of the wind

The tailwind makes the plane go faster, therefore, the speed of the plane will be the plane speed + the wind speed

The tailwind slows the plane down. Therefore, the tailwind will be the speed of the plane minus the windspeed.

Below is a table with all of the available information

Rate * Time = Distance
Tailwind p + w 1.5 548
Headwind p – w 1.5 494

The initial system of equations is as follows

1

To solve this system of equations we will use the elimination method as shown below.

1.png

The plane travels 347.33 mph. We now take the value of p plug it into one of our equations to find the speed of the wind.

1.png

The speed of the wind is 18 mph. We know the plane travels 347 + 18 = 365 mph with a tailwind and 347-18 = 329mph with a headwind.

Conclusion

A system of equations is proven to have a practical application. The assumption of a uniform speed is somewhat unrealistic in most instances. However, this assumption simplifies the calculation and prepares us for more complex models in the future.

Review of “Marie Curie’s: Search for Radium”

This post is a review of the children’s book Marie Curie’s Search for Radium (Science Stories) by Beverly Birch and Christian Birmingham (pp. 40).

The Summary

As you can surmise from the title, this book focuses specifically on Marie Curie’s discovery of Radium. As such, the text skips most of the life of Marie such as her childhood, early education, and even any insight into her marriage and children.

The book begins with Marie being interested in X-rays. Through her study of X-rays Marie finds out about rays that come from uranium. This led Marie to wonder if other elements emit electricity. She decides to test this with the help of her husband’s electrometer.

She soon begins to find other elements that emit electricity in the air. She calls this rays radiation or radioactive rays. Eventually, she discovers two new elements polonium and radium. To find these elements she had to sift through huge amounts of pitchblende a mineral in order to concentrate the radium or polonium. Radium is million times more radioactive than uranium. As such, Marie was actually slowly poisoning herself through her research.

After years of work, Marie had a thimble size amount of radium to share with the world. The blue liquid actually glows in the dark. Another sign of how dangerous it was without Marie knowing.

The Good

The visuals have an impressionistic feel to them. In many ways, a younger child can determine what is happening just from looking at the pictures.

The Bad

The book seems to narrowly focus. Marie’s husband comes out of nowhere as if she was magically married somehow. In addition, the book leaves out some of Marie’s most impressive achievements such as the fact that she won two Nobel Prizes. In fact, Marie first Noble Prize was shared with her husband and Antoine Becquerel. It was the research done with Becquerel that led to Marie’s future work with radium and a second Nobel Prize. This is never stated in the text. A passing reference is enough for such monumental achievements

The Recommendation

This book would be a reasonable read for older elementary students. However, the book need will require supplemental materials and or instruction in order for the students to truly understand the impact and influence Marie Curie had in science.

Insert Images into a LaTeX Document

We have all heard that a picture is worth a thousand words. Images help people to understand concretely what a writer is trying to communicate with text. In this post, we will look at how to include images inside documents prepared with LaTeX.

Basic Example

One way to include an image is to use the “graphicx” package and to set the path for where the image is using the “\garphicspath” declaration in the preamble of a LaTeX document. Below is an example. Included in the example is the “babel” and “blindtext” packages to create some filler text.

\documentclass{article}
\usepackage[english]{babel}
\usepackage{blindtext}
\usepackage{graphicx}
\graphicspath{ {PUT THE PATH HERE} }
\begin{document}
\blindtext

\includegraphics[scale=.1]{1.jpg}

\blindtext
\end{document}

1

Inside the actual document we use the following declaration

\includegraphics[scale=.1]{1.jpg}

“\includegraphics” is the declaration. The “scale” argument reduces the size of the image. The information in the curly braces is the name of the actual file. You can see that our print out is rather ugly and needs refinement.

Adding a Caption

One thing our picture needs is a caption that describes what it is. This can be done by first creating a figure environment, placing the “\includegraphics” declaration inside it, and using the “\caption” declration. Below is an example. We will also center the image for aesthetic reasons as well.

\documentclass{article}
\usepackage[english]{babel}
\usepackage{blindtext}
\usepackage{graphicx}
\graphicspath{ {PUT THE PATH HERE} }
\begin{document}
\blindtext

\begin{figure}
\centering
\includegraphics[scale=.1]{1.jpg}
\caption{Using Images in \LaTeX}
\end{figure}

\blindtext
\end{document}

1.png

We created a figure environment added our image and type a caption. LaTeX automatically added “Figure 1” to the image. In addition,  you can see that the picture moved to the top of the page. This is because environments are able to float to the best position on a page as determined by calculations made by LaTeX.

If you want the image to appear in a particular place you can add the optional arguments h,t,b,p next to the “\begin{figure}” declaration. h =  here, t = top, b = bottom, and p = separate page.

To get rid of floating use the package called “capt-of”  and the declaration “\captionof{figure or table}{name here}}”. This will freeze the image in place so that it does not move all over the place as you add content to your document. Below is the same example but using the “capt-of” package.

\documentclass{article}
\usepackage[english]{babel}
\usepackage{blindtext}
\usepackage{graphicx}
\graphicspath{ {PUT PATH HERE} }
\usepackage{capt-of}
\begin{document}
\blindtext

\begin{center}
\includegraphics[scale=.1]{1.jpg}
\captionof{figure}{Using Images in \LaTeX}
\end{center}

\blindtext
\end{document}

1

This is almost like our first example except now we have a caption. We did have to create a center environment but this type of environment does not float.

Wrapping Figures

The last example is wrapping text around a figure. For this, you need the “wrapfig” package and you need to create an environment with the “Wrapfigure” command. You also must indicate where the figure should be to the left (l),  center (c), or to the right (r). Lastly, you need to indicate the width of the figure. Below is the code followed by the results.

\documentclass{article}
\usepackage[english]{babel}
\usepackage{blindtext}
\usepackage{graphicx}
\graphicspath{ {/home/darrin/Downloads/} }
\usepackage{wrapfig}
\begin{document}
\blindtext

\begin{wrapfigure}{r}{7.8cm}
\includegraphics[scale=.1]{1.jpg}
\caption{Using Images in \LaTeX}
\end{wrapfigure}

\blindtext
\end{document}

1.png

In the example above, we moved the image to the left. For the width, you have to guess several times so that all of the text appears next to the figure rather than behind it.

Conclusion

This post provided several practical ways to include images in a LaTeX document. With this amount of control, you are able to make sophisticated documents that are consistently reproduced.

Solving a System of Equations with Direct Translation

In this post, we will look at two simple problems that require us to solve for a system of equations. Recall that a system of equations involves two or more variables that must be solved. With each problem, we will use the direct translation to set up the problem so that it can be solved.

Direct Translation 

Direct translation involves reading a problem and translating it into a system of equations. In order to do this, you must consider the following steps

  1. Determine what you want to know
  2. Assigned variables to what you want to know
  3. Setup the system of equations
  4. Solve the system

Example `1

Below is an example  followed by a step-by-step breakdown

The sum of two numbers is zero. One number is 18 less than the other. Find the numbers.

Step 1: We want to know what the two numbers are

Step 2: n = first number & m =  second number

Step 3: Set up system

1

Solving this is simple we know n = m – 18 so we plug this into the first equation n + m = 0  and solve for m.

1.png

Now that we now m we can solve for n in the second equation

1.png

The answer is m = 9 and n = -9. If you add these together they would come to zero and meet the criteria for the problem.

Example 2

Below is a second example involving a decision for salary options.

Dan has been offered two options for his salary as a salesman. Option A would pay him $50,000 plus $30 for each sale he closes. Option B would pay him $35,000 plus $80 for each sale he closes. How many sales before the salaries are equal

Step 1: We want to know when the salaries are equal based on sales

Step 2: d =  Dan’s salary & s = number of sales

Step 3: Set up system

1.png

To solve this problem we can simply substitute d  for one of the salaries as shown below

1

You can check to see if this answer is correct yourself. In order for the two salaries to equal each other Dan would need to sale 300 units. After 300 units option B is more lucrative. Deciding which salary option to take would probably depend on how many sales Dan expects to make in a year.

Conclusion

Algebraic concepts can move beyond theoretical ideas and rearrange numbers to practical applications. This post showed how even something as obscure as a system of equations can actually be used to make financial decisions.

Review of “Eric the Red & Leif the Lucky”

This post is a review of the book Eric the Red and Leif the Lucky by Barbara Schiller (pp. 48).

The Summary

This book covers the lives of Eric the Red and his son Leif the Lucky. Eric was a hot-tempered Viking who was banished from Iceland for murdering a man. Since he had to leave Eric decided to explore a mysterious land to the west of Iceland.

Upon his arrival, Eric explores this new land and see that this could be a place to live. After several years of exploration, Eric gives the land a name. For marketing purposes, Eric calls the place Greenland and returns to Iceland to try and convince people to come to the new country. With famine and poverty afflicting many people it was not hard to get some people to come.

From here, the book moves to focus on Eric the Red’s son Leif the Lucky. Leif was also an explorer like his father. One day, Leif hears of a strange land further to the west of Greenland. Leif decides to go and find this land for himself.

After several days of travel, Leif and his team find the new land. He arrives at the beginning of the 11th century 500 years before Columbus came to America. The men landed somewhere in what is today Canada and set up temporary living quarters and began exploring the land.

Leif decided to call this land Vinland. Vin means grapes and he named the country this because they discovered grapes in the area. After filling their boat with cargo to sail, Leif returns to Greenland to tell others and his father about Vinland.

The Vikings tried to return to Vinland (America). However, the Indians were waiting for them and fighting between the two groups made it impossible for the Vikings to stay on a permanent basis. Leif never returned to America as he became the leader of Greenland when his father Eric the Red died.

The Good

This text is highly informative and provides students with some basic understanding of the men who came to America so long ago. The black and white illustrations are also interesting as they portray Vikings in a highly traditional manner which is in a position of strength and dominance.

The Bad

The text is tough for a child to read. Therefore, they would probably need help with the reading. There also might be issues with relevance as a child would try to figure out how to connect with a story of Vikings finding America.

The Recommendation

This book would be great for older children. In addition, if it can be integrated into the learning of the students it could help with the relevancy issue. It would be somewhat unusual for a kid to pick this book up and read it for its own value but as part of an assignment/project, this text is excellent.

Prerequistes to Conducting Research

Some of the biggest challenges in helping students with research is their lack of preparation. The problem is not an ignorance of statistics or research design as that takes only a little bit of support. The real problem is that students want to do research without hardly reading any research and lacking knowledge of how research writing is communicated. This post will share some prerequisites to performing research.

Read Extensively

Extensive reading means reading broadly about a topic and not focusing too much on specifics. Therefore, you read indiscriminately perhaps limited yourself only to your general discipline.

In order to communicate research, you must first be familiar with the vocabulary and norms of research. This can be learned to a great extent through reading academic empirical articles.

The ananoloy I like to use is how a baby learns. By spends large amounts of time being exposed to the words and actions of others. The baby has no real idea in terms of what is going on at first. However, after continuous exposure, the child begins to understand the words and actions fo those around them and even begins to mimic the behaviors.

In many ways, this is the purpose of reading a great deal before even attempting to do any research. Just as the baby, a writer needs to observe how others do things, continue this process even if they do not understand, and attempt to imitate the desired behaviors. You must understand the forms of communication as well as the cultural expectations of research writing and this can only happen through direct observation.

At the end of this experience, you begin to notice a pattern in terms of the structure of research writing. The style is highly ridge with litter variation.

It is hard to say how much extensive reading a person needs. Generally, the more reading that was done in the past the less reading needed to understand the structure of research writing. If you hate to read and did little reading in the past you will need to read a lot more to understand research writing then someone with an extensive background in reading. In addition, if you are trying to write in a second language you will need to read much more than someone writing in their native language.

If you are still desirous of a hard number of articles to read I would say aim for the following

  • Native who loves to read-at least 25 articles
  • Native who hates to read-at least 40 articles
  • Non-native reader-60 articles or more

Extensive reading is just reading. There is no notetaking or even highlighting. You are focusing on exposure only. Just as the observant baby so you are living in the moment trying to determine what is the appropriate behavior. If you don’t understand you need to keep going anyway as the purpose is quantity and not quality. Generally, when the structure of the writing begins to become redundant ad you can tell what the author is doing without having to read too closely you are ready to move on.

Read Intensively

Intensive reading is reading more for understanding. This involves slows with the goal of deeper understanding. Now you select something, in particular, you want to know. Perhaps you want to become more familiar with the writing of one excellent author or maybe there is one topic in particular that you are interested in. With intensive writing, you want to know everything that is happening in the text. To achieve this you read fewer articles and focus much more on quality over quantity.

By the end of the extensive and intensive reading, you should be familiar with the following.

  • The basic structure of research writing even if you don’t understand why it is the way it is.
  • A more thorough understanding of something specific you read about during your intensive reading.
  • Some sense of purpose in terms of what you need to do for your own writing.
  • A richer vocabulary and content knowledge related to your field.

Write Academicly

Once a student has read a lot of research there is some hope that they can now attempt to write in this style. As the teacher, it is my responsibility to point out the structure of research writing which involves such as ideas as the 5 sections and the parts of each section.

Students grasp this but they often cannot build paragraphs. In order to write academic research, you must know the purpose of main ideas, supporting details, and writing patterns. If these terms are unknown to you it will be difficult to write research that is communicated clearly.

The main idea is almost always the first sentence of a paragraph and writing patterns provide different ways to organize the supporting details. This involves understanding the purpose of each paragraph that is written which is a task that many students could not explain. This is looking at writing from a communicative or discourse perspective and not at a minute detail or grammar one.

The only way to do this is to practice writing. I often will have students develop several different reviews of literature. During this experience, they learn how to share the ideas of others. The next step is developing a proposal in which the student shares their ideas and someone else’s. The final step is writing a formal research paper.

Conclusion

To write you must first observe how others write. Then you need to imitate what you saw. Once you can do it what others have done it will allow you to ask questions about why things are this way. Too often, people just want to write without even understanding what they are trying to do. This leads to paralysis at best (I don’t know what to do) to a disaster at worst (spending hours confidently writing garbage). The enemy to research is not methodology as many people write a lot without knowledge of stats or research design because they collaborate. The real enemy of research is neglecting the preparation of reading and the practicing of writing.