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,

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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.

Solving a System of Equations by Substitution and Elimination

A system of equations involves trying to solve for more than one variable. What this means is that a system of equations helps you to see how to different equations relate or where they intersect if you were to graph them.

There are several different ways to solve a system of equations. In this post, we will solve y using the substitution and the elimination methods.

Substitution

Substitution involves choosing one of the two equations and solving for one variable. Once this is done we substitute the expression into the equation for which we did not solve a variable for. When this is done the second equation only has one unknown variable and this is basic algebra to solve.

The explanation above is abstract so here is a mathematical example

1.png

We are not done. We now need to use are x value to find our y value. We will use the first equation and replace x to find y.

1

This means that our ordered pair is (4, -1) and this is the solution to the system. You can check this answer by plugging both numbers into the x and y variable in both equations.

Elimination

Elimination begins with two equations and two variables but eliminates one variable to have one equation with one variable. This is done through the use of the addition property of equality which states when you add the same quantity to both sides of an equation you still have equality. For example 2+2 = 2 and if at 5 to both sides I get 7 + 7 = 7. The equality remains.

Therefore, we can change one equation using the addition property of equality until one of the variables has the same absolute value for both equations. Then we add across to eliminate one of the variables. If one variable is positive in one equation and negative in the other and has the same absolute value they will eliminate each other. Below is an example using the same system of equations as the previous example.

.1.png

You can take the x value and plug it into y. We already know y =1 from the previous example so we will skip this.

There are also times when you need to multiply both equations by a constant so that you can eliminate one of the variables

1.png

We now replace x with 0 in the second equation

1

Our ordered pair is (0, -3) which also means this is where the two lines intersect if they were graphed.

Conclusion

Solving a system of equations allows you to handle two variables (or more) simultaneously. In terms of what method to use it really boils down to personal choice as all methods should work. Generally, the best method is the one with the least amount of calculation.

Review “Pompeii…Buried Alive!”

This post is a review of the children’s book  Pompeii — Buried Alive! (Step into Reading) by Edith Kunhardt (pp. 48).

The Summary

This text provides the story of the eruption of Mount Vesuvius which destroyed the town of Pompeii at the base of the mountain in AD 79. The first part of the book seems to emphasize how the day of the eruption was like any other day. People were buying and selling at the market, going to the spas, visiting the temples, etc.

When the volcano initially erupts people are shocked and confused. Many people choose to flee by boarding ships to sail away from the place. For those who stayed the volcano dropped huge amounts of ash that buried almost everybody, If this did not finish someone off then the volcano dumped a huge volume of poisonous gas in the form of a pyroclastic eruption.

The disaster was seen from a distance and a boy who later became known as Pliny the Younger witnessed the events. Pliny would later write about these events which would provide historical evidence for the existence of Pompeii.

After several centuries, the original town of Pompeii is buried and forgotten and a new town was built on it. Eventually, construction workers uncover the city and archeologist descend on the site.

After doing some excavations the archeologist noticed something strange. There are many empty holes in the soil. Eventually, someone came up with the idea of dumping plaster inside them. The results were shocking. When the plaster hardens it left the impression of bodies of people in the position in which they died. It was a sobering reminder of the gruesome destruction of the volcano. Other artifacts were found such as jewelry, mosaics, and even food.

Today the original site of Pompeii is a tourist attraction.

The Good

The text is designed for young readers and it is truly simple in its writing. The illustrations are ok but a little dated.

The Bad

The author took some creative liberty in the development of the text. There is an unnamed family that provides a vechilce for depicting daily life. Since the family is unnamed it is hard to tell if they truly existed or not. This is probably not a problem for child but if they share this as if it was true it could reflect poorly on them.

The Recommendation

This is not the greatest text. There are better choices out there for explaining the destruction of Pompeii to kids. However, the price is great and you truly get what you pay fofr in this situation.

Making Lists in LaTeX

Lists are frequently used in communication in order to provide information succinctly. Often, the rules of grammar can be suspended because of the need for the list to communicate information in an unadorned way. In this post, we will learn how to make lists in LaTeX.

Basic List

For a simple list, you need to make an environment using the “itemize” declaration. Inside this environment, you must use the “\item” declaration for each bullet in the list. Below is the code and printout of a basic list in LaTeX.

\documentclass{article}
\begin{document}
   Shopping list
      \begin{itemize}
         \item lettace
         \item mango
         \item toothpaste
      \end{itemize}
\end{document}

0.jpg

As you can see this is fairly simple. THere is no need for any packages to complete this. If you want a number list instead of creating a “itemize” environment you would create an “enumerate” environment as shown below.

\documentclass{article}
\begin{document}
   Shopping list
   \begin{enumerate}
      \item lettace
      \item mango
      \item toothpaste
   \end{enumerate}
\end{document}

0

Nested List

It is possible to have lists within lists. To do this, you simply create an environment within an environment. LaTeX will automatically change the bullet type for you to enhance readability. Below is an example.

\documentclass{article}
\begin{document}
    Shopping list
    \begin{itemize}
       \item fruits
       \begin{itemize}
          \item lettace
          \item mango
        \end{itemize}
      \item other
      \begin{itemize}
         \item toothpaste
      \end{itemize}
    \end{itemize}
\end{document}

0

The example above has two levels in the list. LaTeX can go up to four levels.

Compact List

Generally, list by default in LaTeX are double-spaced. To reduce this you need to use the “paralist” package with either the “\compactitem” declaration and or the “\compactenum” declaration. Below is the same example but using the paralist features and also blending the use of bullets and numbers.

\documentclass{article}
\usepackage{paralist}
\begin{document}
   Shopping list
      \begin{compactitem}
         \item fruits
         \begin{compactenum}
             \item lettace
             \item mango
         \end{compactenum}
        \item other
        \begin{compactenum}
           \item toothpaste
        \end{compactenum}
      \end{compactitem}
\end{document}

0.jpg

Definition List

It is also possible to make a list of definition. This is useful for a glossary. In order to do this, you create a “description” environment. When you use the “\item” declaration you need to place the definition word in brackets. There are no packages needed for this. Below is the code.

\documentclass{article}
\usepackage{paralist}
\begin{document}
   \begin{description}
      \item[convoluted] complex and hard to understand
      \item [obtuse] slow to understand
   \end{description}
\end{document}

0.jpg

Conclusion

This post provided insights into the use of lists in LaTeX.

Relations and Functions

In mathematics, a relation is a connection between two distinct pieces of data or variables. For example, student name and ID number would be a relation commonly found at a school. What this means is that you can refer to a student by there name and get their ID number and vice versa.  These two pieces of information are connected and refer to each other. Another term for relation is ordered pair, however, this is more commonly use for coordinate graphing. Below is an example of several student names and ID numbers

Student Name (x values) ID Number (y values)
Jill Smith 12345
Eve Jackson 54321
John Doe 24681

Table 1

Two other pieces of information to know are domain and range. The domain represents all x values. In our table above the student names are the x values (Jill Smith, Eve Jackson, John Doe). The range is all of the y-values, THese are represented by ID number in the table above (12345, 54321, 24681).

The table above is nice and neat. However, sometimes the information is not organized into neat rows but is scrambled with the names and ID numbers not lining up. Below is the same information as the table 1 but the ID numbers are scrambled. The arrows tell who the ID number belongs to who.  This is known as mapping.

Student Name ID Number
Jill Smith ↘ 24681
Eve Jackson→ 54321
John Doe↗ 12345

If we find the ordered pair, domain and range it would be as follows.

  • Ordered pair = {(Jill Smith, 123450, (Eva Jackson, 54321), (John Doe,  24681)}
  • domain = {Jill Smith, Eva Jackson, John Doe}
  • Range = {24681, 54321, 12345}

Understanding Functions

A function is a specific type of relation. What a function does is assigns to each element in a domain. Below is an example of a function

f(x) = 2x + 7

Functions are frequently written to look the same as an equation  as shown below

y = 2x + 7

PLugging in different values of x in your function will provide you with a y as shown below

1

Here our x-value is 2 and the y-value is 11.

Of course, you can graph function as any other linear equation. Below is a visual.

save.png

Conclusion

This post explained the power of relations and functions. Relations are critical in computer science in particular relational databases. In addition., Functions are a bedrock in statistics and other forms of math. Therefore it is critical to understand these basic concepts of algebra.

Review of “Michelangelo”

This post is a review of the book Michelangelo by Diane Stanley (pp. 40).

The Summary

This book addresses the life of Michelangelo di Lodovico Buonarroti Simoni perhaps one of the greatest artists of all time. Michelangelo was born in the 15th century (1475) to a middle-class family in Italy during the Renaissance.

As as a small boy, Michelangelo was trained in stonecutting. This stokes a fire within him and he asked his father if he could be an artist apprentice. Initially, his father was angry about this as this was not an occupation for a gentleman. However, eventual the father relented and Michelangelo began his training.

With time Michelangelo learned painting and sculpting among other things and was eventually sponsored by the famous Medici family, living with them. After several years he would leave the family as politics became tense when there was a change of leadership within the Medici family who ruled Florence.

Over the next few years, Michelangelo sculpted many of his great masterpieces usually sponsored by the Catholic church. Examples include Pieta and David. The realistic nature of the statutes is due to Michelangelo’s talent as well as his knowledge of anatomy through the study of cadavers.

Michelangelo’s next project was to build a tomb for Pope Julius II. However, there was some misunderstanding and arguments over money that hounded this project. After fleeing and the returning to the Pope, Michelangelo was given the task of painting the ceiling of the Sistine Chapel. This was a monumental task as the ceiling was almost 6,000 square feet. It was all done by hand over the course of four years.

Michelangelo never married and he struggled to maintain social relationships. His work was his life and the excellence speaks for its self. He finally died at almost 90 years of age in 1564. A remarkable long life in an age of little health care and the plague.

The Good

This is an excellent text. The strong point is the pictures. The visuals are developed in a renaissance style and also include pictures of the various works Michelangelo made. The actual sculptures and paintings that he made are breath-taking. It almost appears as if Michelangelo was not even human.

The visuals also show Michelangelo as he progressed from small boy to old man. This supports the chronological nature which not all books do when sharing a biographical story.

Kids will love the pictures while older students will be able to appreciate the text. This book also provides exposure to some aspects of European and church history.

The Bad

The text is too complicated for anyone below 5th grade. Besides this, there is little to complain about in this text. In addition, a lot of background information may need to be provided in order for students to understand what is taking place in the story.

The Recommendation

This is a good book and perhaps should be a part of a teacher’s library if they want to expose kids to Renaissance art. However, it might be too detailed oriented and a more general book on art would provide the exposure kids may need

Web Scraping with R

In this post we are going to learn how to do web scrapping with R.Web scraping is a process for extracting data from a website. We have all done web scraping before. For example, whenever you copy and paste something from a website into another document such as Word this is an example of web scraping. Technically, this is an example of manual web scraping. The main problem with manual web scraping is that it is labor intensive and takes a great deal of time.

Another problem with web scraping is that the data can come in an unstructured manner. This means that you have to organize it in some way in order to conduct a meaningful analysis. This also means that you must have a clear purpose for what you are scraping along with answerable questions. Otherwise, it is easy to become confused quickly when web scraping

Therefore, we will learn how to automate this process using R. We will need the help of the “rest” and “xml2” packages to do this. Below is some initial code

library(rvest);library(xml2)

For our example, we are going to scrape the titles and prices of books from a webpage on Amazon. When simply want to make an organized data frame. The first thing we need to do is load the URL into R and have R read the website using the “read_html” function. The code is below.

url<-'https://www.amazon.com/s/ref=nb_sb_noss?url=search-alias%3Daps&field-keywords=books'
webpage<-read_html(url)

We now need to specifically harvest the titles from the webpage as this is one of our goals. There are at least two ways to do this. If you are an expert in HTML you can find the information by inspecting the page’s HTML. Another way is to the selectorGadget extension available in Chrome. When using this extension you simply click on the information you want to inspect and it gives you the CSS selector for that particular element. This is shown below

1.png

The green highlight is the CSS selector that you clicked on. The yellow represents all other elements that have the same CSS selector. The red represents what you do not want to be included. In this picture, I do not want the price because I want to scrape this separately.

Once you find your information you want to copy the CSS element information in the bar at the bottom of the picture. This information is then pasted into R and use the “html_nodes” function to pull this specific information from the webpage.

bookTitle<- html_nodes(webpage,'.a-link-normal .a-size-base')

We now need to convert this information to text and we are done.

title <- html_text(bookTitle, trim = TRUE) 

Next, we repeat this process for the price.

bookPrice<- html_nodes(webpage,'.acs_product-price__buying')
price <- html_text(bookPrice, trim = TRUE) 

Lastly, we make our data frame with all of our information.

books<-as.data.frame(title)
books$price<-price

With this done we can do basic statistical analysis such as the mean, standard deviation, histogram, etc. This was not a complex example but the basics of pulling data was provided. Below is what the first few entries of the data frame look like.

head(books)
##                                   title  price
## 1                          Silent Child $17.95
## 2 Say You're Sorry (Morgan Dane Book 1)  $4.99
## 3                     A Wrinkle in Time $19.95
## 4                       The Whiskey Sea  $3.99
## 5            Speaking in Bones: A Novel  $2.99
## 6 Harry Potter and the Sorcerer's Stone  $8.99

Conclusion

Web scraping using automated tools saves time and increases the possibilities of data analysis. The most important thing to remember is to understand what exactly it is you want to know. Otherwise, you will quickly become lost due to the overwhelming amounts of available information.

Line Breaks and Justification in LaTeX

In this post, we will look at line breaks and justification in LaTeX. These tools will provide a user with more nuanced command of their document.

Paragraph Break

By leaving a space between paragraphs in your document LaTeX will start a new paragraph. Below is the code followed by the output.

\documentclass{article}
\begin{document}
Hello, here is some text without a meaning. This text should show what a printed text will look like at this place. If you read this text, you will get no information.

Really?

Is there no information?

Is there a difference between this text and some nonsense like “Huardest gefburn”?

Kjift – not at all!

A blind text like this gives you information about the selected font.
\end{document}

1.png

Notice how each paragraph is indented. This is the default setting in LaTex. To remove indentation you need to use the “\noindet” declaration as shown below.

\documentclass{article}
\begin{document}
Hello, here is some text without a meaning. This text should show what a printed text will look like at this place. If you read this text, you will get no information.

\noindent

Really?
\noindent

Is there no information?
\noindent

Is there a difference between this text and some nonsense like “Huardest gefburn”?
\noindent

Kjift – not at all!
\noindent

A blind text like this gives you information about the selected font.
\end{document}

1.png

In this example, only the first paragraph is indented.

A simpler way to do this is with the short command line break \\. Below is what it looks like

\documentclass{article}
\begin{document}
Hello, here is some text without a meaning. This text should show what a printed text will look like at this place. If you read this text, you will get no information.\\
Really?\\
Is there no information? \\
Is there a difference between this text and some nonsense like “Huardest gefburn”? \\
Kjift – not at all! \\
A blind text like this gives you information about the selected font.
\end{document}

1

You can see that both “\noindent” and the short command \\ get the same results. However, the latter is probably more efficient and perhaps easier to read.

Justification

There are also ways to remove the default setting for justification. The three declaration are “\raggedright”, “\raggedleft”, and “\centering”. The “\raggedright” declaration makes the right side of the page ragged while the left side of the page is justified as shown below.

\documentclass{article}
\usepackage[english]{babel}
\usepackage{blindtext}
\begin{document}
{\raggedright
\Blindtext}
\end{document}

1.png

You can clearly see how the right side is truly ragged. The other packages in the code create the demo paragraph automatically for us.

The “\raggedleft” declaration does the opposite. See below

\documentclass{article}
\usepackage[english]{babel}
\usepackage{blindtext}
\begin{document}
{\raggedleft
\Blindtext}
\end{document}

1.png

I think we already know what centering does.

\documentclass{article} 
\usepackage[english]{babel} 
\usepackage{blindtext} 
\begin{document} 
{\centering 
\Blindtext} 
\end{document}

1.png

Conclusion

This post provided a demonstration of line breaks and justification in LaTeX.

Review of “The Titanic: Lost…and Found”

This post is a review of the book The Titanic: Lost and Found (Step-Into-Reading, Step 4) by Judy Donnelly (pp. 48).

The Summary

This text covers the classic story of the sinking of the Titanic in the early part of the 20th century. Originally build as unsinkable the Titanic collided with an iceberg and sank on its first voyage from Europe to America.

The text describes the accommodations and size of the ship. Such amenities as a pool and dining halls are depicted. At the time, the Titanic was also the largest passenger ship ever built.

When the ship had its incident with the iceberg people were supposedly laughing and joking as they were called to the deck for evacuation. This is actually an emotionally poweful moment in the text that a small child will miss. The people actually believed the foolish claim that a ship was unsinkable. To make matters worse, there were not enough lifeboats as even the builders of the ship arrogantly believed this as well.

Adding to the discouragement was the fact that a nearby ship ignored the radio calls of the sinking Titanic because their radio was turned off. When the people finally began to realize the danger they were in fear quickly set in. For whatever reason, the musicians continue to play music to try and keep the people calm and even played a hymn right before the final sinking of the ship. A somewhat chilling ending.

The book then concludes with the people in the lifeboat being rescued, it mentions changes to laws to prevent this disaster from happening again, and the final section of the text shares the story of how the Titanic was found in the 1980’s by researchers.

The Good

This book is written in simple language for small children. It can be read by early primary students. This text also provides a good introduction into one of the great tragedies of modern western history.

The illustrations also help to describe what is happening in the text. Lastly, the text is not that long and probably can be read in a few days by a child alone.

The Bad

There is little to complain about with this text. It should be in any primary teacher’s library. The only problem may be that it is a paperback book so it will not last long enduring the wear and tear that comes from small children.

The Recommendation

There will be no regrets if you purchase this book for your classroom or home.

Absolute Value Equations & Inequalities

The absolute value of a number is its distance from 0.  For example, 5 and -5 both have an absolute value of 5 because both are 5 units from 0. The symbols used for absolute value are |  | with a number or variable placed inside the vertical bars. With this knowledge lets look at an example of an absolute value.

1.png

The answer is +5 because both 5 and -5 are 5 units from 0.

In this post, we will look at equations and inequalities that use absolute values.

Solving one Absolute Value Equations

It is also possible to have inequalities with absolute values. To solve these you want to isolate the absolute value and solve the positive and also the negative version of the answer. Lastly, you never manipulate anything inside the absolute value brackets. you only manipulate and simplify values outside of the brackets. Below is an example.

1.png

As you can see absolute value inequalities involves solving two equations. Below is an example involving multiplication.

1

Notice again how the values inside the absolute value were never changed. This is important when solving absolute value inequalities.

Solving Two Absolute Values Equations

Solving two absolute values is not that difficult. You simply make one of the absolute values negative for one equation and positive for another. Below is an example.

1.png

Absolute Value Inequalities

Absolute value inequalities require a slightly different approach. You can rewrite the inequality in double inequality form and solve appropriately when the inequality is “less than.” Below is an example.

1.png

You can see that we put the absolute value in the middle and simply solved for x. you can even write this using interval notation as shown below.

1

“Greater than” inequalities are solved the same as inequalities with equal signs. You use the “or” concept to solve both inequalities.

1.png

The interval notation is as follows

1

We use the union sign in the middle is used in place of the word “or”.

Conclusion 

This post provided a brief overview of how to deal with absolute values in both equations and inequalities.

Modifying Text and Creating Commands in LaTeX

In this post, we are going to explore to separate features available in LaTeX. These two features are modifying the text size and creating custom commands.

Modifying Text

You can change the size and shape of text using many different declarations/environments in LaTeX. Declarations and environments serve the same purpose the difference is in the readability of the code. In the example below, we use an environment to make the text bigger than normal. The code is first followed by the example

\documentclass{article}
\usepackage[english]{babel}
\usepackage{blindtext}
\begin{document}
\begin{huge}
\blindtext
\end{huge}
\end{document}

1.png

Here is what we did.

  1. We create a document with the class of article
  2. We used the “babel” and “blindtext” packages to create some filler text.
  3. Next, we began the document
  4. We create the environment “huge” for enlarging the text.
  5. We used the declaration  “\blindtext” to create the paragraph
  6. We closed the “huge” environment with the “end” declaration
  7. We end the document

If you ran this code you will notice the size of the text is larger than normal. Of course, you can bold and do many more complex things to the text simultaneously. Below is the same example but with the text bold and in italics

\documentclass{article}
\usepackage[english]{babel}
\usepackage{blindtext}
\begin{document}
\begin{huge}
\bfseries
\textit
\blindtext
\end{huge}
\end{document}

1.png

The code is mostly the same with the addition of “\bfseries” for bold and  “\texit” for italics.

Making Commands

It is also possible to make custom commands in LaTeX. This can save a lot of time for repetitive practices. In the example below, we create a command to automatically print the name of this blog’s web address.

\documentclass{article}
\newcommand{\ert}{\bfseries{educationalresearchtechniques}}
\begin{document}
The coolest blog on the web is \ert
\end{document}

1.png

In the code, we use the declaration “\newcommand” in the preamble. This declaration had the command “\ert” which is the shorthand for the code to the right which is “\bfseries{educationalresearchtechniques}. This code tells LaTeX to bold the contents inside the brackets.

The next step was to begin the document. Notice how we used the “\ert” declaration and the entire word educationalresearchtechniques was printed in bold in the actual pdf.

It is also possible to make commands that format text. Below is an example.

\documentclass{article}
\newcommand{\mod}[1]{\textbf{\textit{#1}}}
\begin{document}
The is an example of modified \mod{text}
\end{document}

1.png

What is new is in line 2. Here we use the “\newcommand” declaration again but this time we create a command call “\mode” and give it an argument of 1 (see [1]) this is more important when you have more than one argument. Next, we put in curly brackets what we want to be done to the text. Here we want the text to be bold “\textbf” and in italics “\textit”. Lastly, we set the definition {#1}. Definition works with arguments in that argument 1 uses definition 1, argument 2 uses definition 2, etc.  Having more than one argument and definition can be confusing for beginners so this will not be explored for now.

Conclusion

This post provided assistance in understanding LaTeX’s font size capabilities as well as ways to make new commands.

Review of “Peter the Great”

In this post, we will take a look at the book Peter the Great by Diane Stanley (32 pp).

The Summary
This book covers the life and death of Peter the Great (1672-1725) one of the most influential Tsars of Russia. The book begins by showing Peter as small boy play war games with his friends. What is unique is that Peter is not the leader, despite his status, but is rather one of the junior soldiers taking orders from the other boys. This points already to an everyman personality of Peter.

The story does not neglect that Peter was royalty and shows some of the luxuries Peter enjoyed such as dancing animals and his own horses. Peter even designed and sailed his own ships.

As a student, Peter was educated by Europeans. He saw how they lived compared to how people in his country lived and it planted a seed for reformation in Peter’s heart.

I his early twenties Peter travels to Europe. While there he absorbs as much culture about Europe as possible and focuses heavily on learning various trades such as shipbuilding. His status as a King made it difficult to learn trades as people found this strange of someone of his rank.

Upon returning home, Peter began immediately to reform Russia. Immediately the long beards that Russian men favored were removed at least among the elite. In addition, the long robes were shortened. Men and women were encouraged to mingle at social settings and arranged marriages were discouraged.

Peter also built schools and canals. His greatest achievement may have been the founding and building of St. Petersburg. Today St. Petersburg is one of the largest cities in Russia.

Of course, all of these reforms had drawbacks. The poor were tax practically to death. Everything was taxed from candle to beards. Peasant young men had to spend as much as 25 years in the military. Lastly, thousands perhaps tens of thousands lost there lives in wars and building projects push by Peter.

Peter died in 1725 of a fever. He was 53 years old at the time of his death.

The Good
This book was extremely interesting. It captures your attention by giving with the rich illustration that has a renaissance feel to it. The illustrations always depict Peter as a man of action. The text is well-written and simple enough for an upper elementary student to understand and appreciate by themselves.

The Bad
There is little to complain about. This text is well-balanced between picture and text. Younger students (below grade 4) may need help with the text. Otherwise, this book is great for all kids and provides some understanding of the history of Russia.

The Recommendation
This is an excellent book to add ou your library as teacher or parent. Younger kids can enjoy the pictures while older kids can enjoy the text. Even an adult can benefit from reading this book if they have not been exposed to Russian history.

Solving Compound Inequalities

Compound inequalities are two inequalities that are joined by the word “and” or the word “or”. Solving a compound inequality means finding all values that make the compound inequality true.

For compound inequalities join ed by the word “and” we look for solutions that are true for both inequalities. Fo compound inequalities joined by the word “or” we look for solutions that work for either inequality.

It is also possible to graph compound inequalities on a number line as well as indicate the final answer using interval notation. Below is a compound inequality with the line graph solution

1.png

Solving the answer is the same as a regular equation. Below is the number line for this answer.

1.png

The empty circle at -8 means that -8 is not part of the solution. This means all values less than -8 are acceptable answers. This is why the line moves from right to left. All values less than -8 until infinity are acceptable answers. Below is the interval notation.

1

The parentheses mean that the value next to it is not included as a solution. This corresponds to the empty circle over the -8 in the lin graph. If the value should be included such as with a less/greater than sign you would use a bracket.

Double Inequality

A double inequality is a more concise version of a compound inequality. The goal is to isolate the variable in the middle. Below is an example

1.png

This is not complex. We simply isolate x in the middle using appropriate steps. The number line and interval notation or as follows

1.png

[-4, 2/3)

This time there is a bracket next to -4 which means that -4 is also a potential solution. In addition, notice how the -4 has a filled circle on the number line. This is another indication that -4 is a solution.

Practical Application

You have signed up for internet access through your cell phone. Your bill is a flat $49.00 per month please $0.05 per minute for internet use. How many minutes can you use internet per month if you want to keep your bill somewhere between $54-$74 per month?

Below is the solution using a double inequality

1

The answer indicates that you can spend anywhere from 100 to 500 minutes on the internet through your phone per month to stay within the budget. You can make the number line and develop the interval notation yourself.

Conclusion

Compound inequalities are useful for not only as an intellectual exercise. They can also be used to determine practical solutions that include more than one specific answer.

Benefits of Coding in Schools

There is a push in education to have more students learn to code. In fact, some schools are considering having computer coding count as the foreign language requirement to graduate from high school. This in many ways almost singles that coding has “arrived” and is not a legitimate subject not just for the computer nerd but for everybody.

In this post, we will look at several benefits of learning to code while in school.

Problem-Solving

People often code to solve a problem. It can be something as small as making an entertaining game or to try and get the computer to do something.  Generally, the process of developing code leads to all kinds of small problems that have to be solved along the way. For example, you want the code to A but instead, it does B. This leads to all kinds of google searches and asking around to try and get the code to do what you want.

All this is happening in the context in which the student is truly motivated to learn. This is perhaps no better situation in which problem-solving skills are developed.

Attention to Detail

Coding involves the ability to see the smallest details. I cannot remember how many times my code would not run because I forgot a comma or a semicolon or perhaps I misspelled a variable. These problems are small but they must be noticed in order to get the code to run.

When students develop code they must write the code perfect (not necessarily efficiently) in order for it to work. This attention to the small things helps in developing students who are not careless.

Computational Thinking

Computational thinking is the skill of being able to explain systematically what you are doing. When developing code students must be able to capture every step needed to execute an action in their code. It is not possible to skip steps. Everything must be planned for in order to have success.

This type of thinking carries over into the real world when communicating with people. The computational thinking comes out when presenting information, teaching, etc. This logical thinking is a key skill in today’s world where miscommunication is becoming so common.

Employment Opportunities

Naturally, learning to code can lead to employment opportunities. There is a growing demand for people with coding skills. Some of the strongest demands are in fields such as Data Science in which people need a blend of coding and domain expertise to develop powerful insights. In other words, it is better to be well-rounded rather than a super coder for the average person as the domain knowledge is useful in interpreting whatever results the coding helped to produce.

Conclusion

There are naturally other benefits of coding as well. The purpose here was just to consider a few reasons. As a minimum, learning to code should be experienced by most students just as they are exposed to music appreciation, art appreciation, and other subjects for the sake of exposure.

Writing Discussion & Conclusions in Research

The Discussion & Conclusion section of a research article/thesis/dissertation is probably the trickiest part of a project to write. Unlike the other parts of a paper, the Discussion & Conclusions are hard to plan in advance as it depends on the results. In addition, since this is the end of a paper the writer is often excited and wants to finish it quickly, which can lead to superficial analysis.

This post will discuss common components of the Discussion & Conclusion section of a paper. Not all disciplines have all of these components nor do they use the same terms as the ones mentioned below.

Discussion

The discussion is often a summary of the findings of a paper. For a thesis/dissertation, you would provide the purpose of the study again but you probably would not need to share this in a short article. In addition, you also provide highlights of what you learn with interpretation. In the results section of a paper, you simply state the statistical results. In the discussion section, you can now explain what those results mean for the average person.

The ordering of the summary matters as well. Some recommend that you go from the most important finding to the least important. Personally, I prefer to share the findings by the order in which the research questions are presented. This maintains a cohesiveness across sections of a paper that a reader can appreciate. However, there is nothing superior to either approach. Just remember to connect the findings with the purpose of the study as this helps to connect the themes of the paper together.

What really makes this a discussion is to compare/contrast your results with the results of other studies and to explain why the results are similar and or different. You also can consider how your results extend the works of other writers. This takes a great deal of critical thinking and familiarity with the relevant literature.

Recommendation/Implications

The next component of this final section of the paper is either recommendations or implications but almost never both. Recommendations are practical ways to apply the results of this study through action. For example, if your study finds that sleeping 8 hours a night improves test scores then the recommendation would be that students should sleep 8 hours a night to improve their test scores. This is not an amazing insight but the recommendations must be grounded in the results and not just opinion.

Implications, on the other hand, explain why the results are important. Implications are often more theoretical in nature and lack the application of recommendations. Often implications are used when it is not possible to provide a strong recommendation.

The terms conclusion and implications are often used interchangeably in different disciplines and this is highly confusing. Therefore, keep in mind your own academic background when considering what these terms mean.

There is one type of recommendation that is almost always present in a study and that is recommendations for further study. This is self-explanatory but recommendations for further study are especially important if the results are preliminary in nature. A common way to recommend further studies is to deal with inconclusive results in the current study. In other words, if something weird happened in your current paper or if something surprised you this could be studied in the future. Another term for this is “suggestions for further research.”

Limitations

Limitations involve discussing some of the weaknesses of your paper. There is always some sort of weakness with a sampling method, statistical analysis, measurement, data collection etc. This section is an opportunity to confess these problems in a transparent matter that further researchers may want to control for.

Conclusion

Finally, the conclusion of the Discussion & Conclusion is where you try to summarize the results in a sentence or two and connect them with the purpose of the study. In other words, trying to shrink the study down to a one-liner. If this sounds repetitive it is and often the conclusion just repeats parts of the discussion.

Blog Conclusion

This post provides an overview of writing the final section of a research paper. The explanation here provides just one view on how to do this. Every discipline and every researcher has there own view on how to construct this section of a paper.

Time Mangagement and E-Learning

Time management is a critical component of having success when studying online. However, many people struggle with the discipline to manage their time so that they can complete an online learning experience. In this post, we will look at several strategies that can help a student to complete an online course.

Routine is King

Perhaps the single most valuable piece of advice that can be given is the benefit of routine. Routine is as simple as dedicating a certain time of each day to the class. Routine can also manifest itself through setting aside a designated place for studying as well.  The beauty of a schedule is that you have taken control of what you will do during a given day.

It is important that a student has a dedicated place and time for e-learning studies. Even though e-learning can happen anywhere few people can learn anywhere as they must be in a learning mindset first. Often, the mindset to learn is dependent on the environment which the student needs to control.

One Thing at a Time

Multi-tasking is tempting but does not work. This is an extra special problem for e-learning because a student is on their computer. Being on the computer can allow the student to check emails, chat on facebook, listen to music, while they are also supposed to be learning.

These other activities are only distractions when trying to learning content online. As such, it is important to close these other applications and turn off notifications from various websites when really trying to learn online.

Ask for Help

Elearning can be an isolating experience. A student is all alone trying to maneuver the complexities of a subject. At times, a student may even get stuck and not know what to do.

In such situations, it is important that a student reaches out to a peer or the teacher for support. The feedback that is received can make a difference in completing a course or not.

Something that was alluded to in this section is the benefit of taking an online course with a friend. Having a friend in a course can be a source of encouragement and also a way to share the burden of larger assignments.

Conclusion

The anytime anywhere freedom of e-learning is perhaps the greatest blessing and also the greatest curse of this platform. The flexibility gives people the impression they can study whenever. However, when there is no structure to the learning experience there is usually no progress made either. Therefore, each student must put in constraints in order to function at a high level academically when studying online.

Major Challenges of Teachers

This post will provide some examples of common problems teachers face. Although the post may seem overwhelmingly negative the purpose here is to provide insight into the actual realities of teaching rather than the romantic experience portrayed in many venues.

Adminstration

Administrators are in charge of the “big picture” of guiding a school towards particular goals that are often laid out by local laws and the results of the prior accreditation visit. This focus on large institutional goals can often cause the administrator to lose sight of the needs of the teachers (unless this was a recommendation from the last accreditation visit),

What results is a task-oriented leadership that is focused on attaining goals or at least showing progress towards goals. This can lead administrators to step on, overwork, and even mistreat teachers. It is hard to blame administrators because if they do not meet specific targets they could lose their own employment.

The constant meetings and incredulous policies that are derived to “help the students” can become exceedingly frustrating for any teacher. Rest assure that few administrators just randomly think up bad ideas. Often the inspiration is from a higher source that is abusing the local administrator.

Co-Workers

There is a surprising amount of petty bickering and fighting among teachers that can become Machevellini in nature. Gossiping backbiting and of course backstabbing all take place. A teacher A confides in teacher B there having problems handling their students and teacher B spreads this to everyone on-campus that teacher A is a terrible teacher who cannot handle her duties.

I’ve heard of teachers complaining that other teachers do not collaborate during lunch with them as though lunchtime is meant to be a meeting that has required attendance. In another setting, I’ve seen teachers slander another teacher in order to help a friend get the job. Petty jealousy can lead teachers to isolate themselves to avoid political attacks which makes it harder to support students.

Parents & Students

Perhaps the biggest problem facing teachers is not necessarily students but parents. If a child is out of line it should only take a simple phone call home to resolve the problem. However, this is almost never the case. Today many parents are indifferent to the behavior of their children. This leaves the teacher only to provide intervention towards a wayward student.

The other extreme is the parent who overly protects and defends everything their child does. This undercuts the teacher’s authority in the same way as a parent who does not provide any sort of behavioral support. The same parents are often quick to get the attention of the administration which is always.

Class Administration

There are a bevy of things that a teacher must do in their own classroom such as

  • Class preparation
  • Marking assignments
  • Decorating
  • Meetings
  • Communicating with parents/students
  • Professional development

This all requires serious time management. It is hard to stay on top of all of these expectations if you are laid back and easy going. It requires strict discipline in order to keep some sort of sanity.

Conclusion

Teaching is tough. However, it is not all bad. There are many rewarding moments in being a teacher. Yet to be successful a teacher must be aware of the common problems that will face so that they are able to weather them.

Gamification

The concept of gamification has really picked up steam over the past few years in education. Gamification is the use design elements from traditional games within a curriculum. Examples of gamification include the use of badges, leveling up, progress bars, hit points, etc. As students focus on acquiring the various rewards in the gamification experience they also learn the content. In other words, gamification is primarily behavioral in nature.

The idea of gamification is not completely new. Many adults can remember earning stickers and or points for excellent behavior as a child. Gamification, however, tends to be focused on the online/ technology context. Instead of earning stickers like students in the 1980’s and 90’s today’s students can earn badges online in their schools learning management system as an example.

It may be clear that there are some pros and cons to gamification and these will be addressed below.

Pros

  • Engagement-Nothing motivates a student like playing a game. The badges and points in gamification often heighten engagement at least temporarily for many students.
  • Feedback-Through leveling up and earning badges students are provided with instant feedback. If they are unsuccessful it is readily apparent and there is no need to wait for teacher feedback.
  • Technology exposure-Gamification is focused in a technology domain. As such, it is a great way to help students to develop their technology skills.

Cons

  • Attention span-Games are often fast-paced. However, the real world is not often moving at the same speed. This can lead kids to struggle with everyday tasks that have not been gamified.
  • Assessment-Often the game serves as a platform to master a skill. However, the nuanced nature of grading can be difficult to apply to a gamified learning experience. In addition, the game may not always transfer to actual real-world skill, which further impairs grading. The focus on gaming often makes the learning take a backseat.
  • Cost/logistics-The cost of using software and other materials can be high. Even if you use a free system, such as Moodle, there is the logistics of setting up the badge system in your online platform.
  • Work ethic-A critical skill that students need to acquire is how to do something they don’t like to do. Gamification can make almost anything fun. However, in the actual world, there are a little of boring things that people have to do. Students must develop the discipline to engage in an activity because it needs to be done rather than because it’s fun.

Conclusion

The appropriate use of gamification is dependent on the extent to which it is used. Having a progress bar in a course probably will not influence attention spans detrimentally. However, more complex gamification is probably where you will start to see problems as shared in the con section.

Therefore, an appropriate analogy would be to compare gamification to salt. A little bit of salt makes food taste better. However, too much salt can ruin the food and impact the health of the eater. As such, a little gamification can enrich a learning experience but heavy doses could harm learning and perhaps character development.

Shaping the Results of a Research Paper

Writing the results of a research paper is difficult. As a researcher, you have to try and figure out if you answered the question. In addition, you have to figure out what information is important enough to share. As such it is easy to get stuck at this stage of the research experience. Below are some ideas to help with speeding up this process.

Consider the Order of the Answers

This may seem obvious but probably the best advice I could give a student when writing their results section is to be sure to answer their questions in the order they presented them in the introduction of their study. This helps with cohesion and coherency. The reader is anticipating answers to these questions and they often subconsciously remember the order the questions came in.

If a student answers the questions out of order it can be jarring for the reader. When this happens the reader starts to double check what the questions were and they begin to second-guess their understanding of the paper which reflects poorly on the writer. An analogy would be that if you introduce three of your friends to your parents you might share each person’s name and then you might go back and share a little bit of personal information about each friend. When we do this we often go in order 1st 2nd 3rd friend and then going back and talking about the 1st friend. The same courtesy should apply when answering research questions in the results section. Whoever was first is shared first etc.

Consider how to Represent the Answers

Another aspect to consider is the presentation of the answers. Should everything be in text? What about the use of visuals and tables?  The answers depend on several factors

  • If you have a small amount of information to share writing in paragraphs is practical. Defining small depends on how much space you have to write as well but generally anything more than five ideas should be placed in a table and referred too.
  • Tables are for sharing large amounts of information. If an answer to a research question requires more than five different pieces of information a table may be best. You can extract really useful information from a table and place it directly in paragraphs while referring the reader to the table for even more information.
  • Visuals such as graphs and plots are not used as frequently in research papers as I would have thought. This may be because they take up so much space in articles that usually have page limits. In addition, readers of an academic journal are pretty good at visually results mentally based on numbers that can be placed in a table. Therefore, visuals are most appropriate for presentations and writing situations in which there are fewer constraints on the length of the document such as a thesis or dissertation.

Know when to Interpret

Sometimes I have had students try to explain the results while presenting them. I cannot say this is wrong, however, it can be confusing. The reason it is so confusing is that the student is trying to do two things at the same time which are present the results and interpret them. This would be ok in a presentation and even expected but when someone is reading a paper it is difficult to keep two separate threads of thought going at the same time.  Therefore, the meaning or interpretation of the results should be saved for the Discussion Conclusion section.

Conclusion

Presenting the results is in many ways the high point of a research experience. It is not easy to take numerical results and try to capture the useful information clearly. As such, the advice given here is intending to help support this experience

Purpose of a Quantitative Methodology

Students often struggle with shaping their methodology section in their paper. The problem is often that students do not see the connection between the different sections of a research paper. This inability to connect the dots leads to isolated thinking on the topic and inability to move forward.

The methodology section of a research paper plays a critical role. In brief, the purpose of a methodology is to explain to your readers how you will answer your research questions. In the strictest sense, this is important for reproducing a study. Therefore, what is really important when writing a methodology is the research questions of the study. The research questions determine the following of a methodology.

What this means is that a student must know what they want to know in order to explain how they will find the answers. Below is a description of these sections along with one section that is not often influenced by the research questions.

Sample & Setting

In the sample section of the methodology, it is common or the student to explain the setting of the study, provide some demographics, and explain the sampling method. In this section of the methodology, the goal is to describe what the reader needs to know about the participants in order to understand the context from which the results were derived.

Research Design & Scales

The research design explains specifically how the data was collected. There are several standard ways to do this in the social sciences such.

  • Survey design
  • experimental design
  • correlational design

Within this section, some academic disciplines also explain the scales or the tool used to measure the variable(s) of the study. Again, it is impossible to develop this section of the research questions are unclear or unknown.

Data Analysis

The data analysis section provides an explanation of how the answers were calculated in a study. Success in this section requires a knowledge of the various statistical tools that are available. However, understanding the research questions is key to articulating this section clearly.

Ethics

A final section in many methodologies is ethics. The ethical section is a place where the student can explain how the protected participant’s anonymity, made sure to get the permission and other aspects of morals. This section is required by most universities in order to gain permission to do research. However, it is often missing from journals.

Conclusion

The methodology is part of the larger picture of communicating one’s research. It is important that a research paper is not seen as isolated parts but rather as a whole. The reason for this position is that a paper cannot make sense on its own if any of these aspects are missing.

Solving Inequalities

Inequalities are equations that use symbols related to less than, greater than, etc. This allows for the solution to be a range of values rather than only one specific one as in many standard equations where you solve for x.

Unique Property of Inequalities

The rules for solving inequalities are mostly the same as for solving a regular equation with one exception. If you multiply or divide both sides of an inequality by a negative number you need to flip the inequality sign.  Below is an example of the sign flipping

1

If you look at the final answer you can see that the x must be greater than -2. This makes sense as -5 * -2 would come to 10 which is not less than 10. Naturally,  any number that is larger than -2 would only be worst. Below is a word problem that employs an inequality.

Single Inequality

You have $8,000 to buy math textbooks for your classroom. Each math book cost $127.06. What is the maximum number of math books you can buy?

In the problem above, the keyword is maximum. In other words, there is a range of potential answers from 1 book to whatever the max is. This indicates that this problem is an inequality. Therefore,

  • Let 127.06 be the price of a math book
  • x the number of math books we can buy
  • < use less than because we do not want to exceed our budget of 8000

Below is the solution to the problem.

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You can buy up to 62 books and be less than or equal to 8000. We round down to 62 because we must stay under $8,000 in spending.

Below is another example but slightly more complex as it contains additional information.

Complex Single Inequality

You are planning a three-day camping trip for your students. Currently, there is $420 of money available. The students can earn $22.50 per hour through tutoring. The trip will cost $525 for transportation,  $390 for food, and $47.50 per night for the campground.  How many hours do the students need to tutor in order to have enough money for the trip?

This problem has three pieces of information on the left of the inequality

  • Transportation (525)
  • Food (390)
  • campground per night (47.5 * 3)

The information to the right is the following

  • The money available (420)
  • The earning rate per hour (22.50)
  • The variable for the hours to tutor (x)

We use the less than or equal to inequality <

Below is the solution

1.png

The students need to tutor for at least 28hours and 20 minutes in order to meet the expenses for the trip.

Conclusion

Inequalities are another useful tool taught in algebra. The applications are limitless. The key to appreciating inequalities is being able to determine when they can be used to solve real-world problems.

Tips for Writing a Quantitative Review of Literature

Writing a review of literature can be challenging for students. The purpose here is to try and synthesize a huge amount of information and to try and communicate it clearly to someone who has not read what you have read.

From my experience working with students, I have developed several tips that help them to make faster decisions and to develop their writing as well.

Remember the  Purpose

Often a student will collect as many articles as possible and try to throw them all together to make a review of the literature. This naturally leads to problems of the paper sounded like a shopping list of various articles. Neither interesting nor coherent.

Instead, when writing a review of literature a student should keep in mind the question

What do my readers need to know in order to understand my study?

This is a foundational principle when writing. Readers don’t need to know everything only what they need to know to appreciate the study they are ready. An extension of this is that different readers need to know different things. As such, there is always a contextual element to framing a review of the literature.

Consider the Format

When working with a student, I always recommend the following format to get there writing started.

For each major variable in your study do the following…

  1. Define it
  2. Provide examples or explain theories about it
  3. Go through relevant studies thematically

Definition

There first thing that needs to be done is to provide a definition of the construct. This is important because many constructs are defined many different ways. This can lead to confusion if the reader is thinking one definition and the writer is thinking another.

Examples and Theories

Step 2 is more complex. After a definition is provided the student can either provide an example of what this looks like in the real world and or provide more information in regards to theories related to the construct.

Sometimes examples are useful. For example, if writing a paper on addiction it would be useful to not only define it but also to provide examples of the symptoms of addiction. The examples help the reader to see what used to be an abstract definition in the real world.

Theories are important for providing a deeper explanation of a construct. Theories tend to be highly abstract and often do not help a reader to understand the construct better. One benefit of theories is that they provide a historical background of where the construct came from and can be used to develop the significance of the study as the student tries to find some sort of gap to explore in their own paper.

Often it can be beneficial to include both examples and theories as this demonstrates stronger expertise in the subject matter. In theses and dissertations, both are expected whenever possible. However, for articles space limitations and knowing the audience affects the inclusion of both.

Relevant Studies

The relevant studies section is similar breaking news on CNN. The relevant studies should generally be newer. In the social sciences, we are often encouraged to look at literature from the last five years, perhaps ten years in some cases. Generally, readers want to know what has happened recently as experience experts are familiar with older papers. This rule does not apply as strictly to theses and dissertations.

Once recent literature has been found the student needs to organize it thematically. The reason for a thematic organization is that the theme serves as the main idea of the section and the studies themselves serve as the supporting details. This structure is surprisingly clear for many readers as the predictable nature allows the reader to focus on content rather than on trying to figure out what the author is tiring to say. Below is an example

There are several challenges with using technology in class(ref, 2003; ref 2010). For example, Doe (2009) found that technology can be unpredictable in the classroom. James (2010) found that like of training can lead some teachers to resent having to use new technology

The main idea here is “challenges with technology.” The supporting details are Doe (2009) and James (2010). This concept of themes is much more complex than this and can include several paragraphs and or pages.

Conclusion

This process really cuts down on the confusion of students writing. For stronger students, they can be free to do what they want. However, many students require structure and guidance when the first begin writing research papers

Uniform Motion Equations

A uniform motion equation involves trying to make calculations when an object(s) is moving at a constant rate. The formula for this type of equation is below.

rate * time = distance

Generally, you want to make a table that includes all of the known information. This allows you to determine what the unknown information is that needs to be solved. Below is a table that you can use.

Rate            * Time            = Distance

Let’s go through some examples

Example 1

Dan and William are riding bicycles. Dan’s speed is 4 kph faster than William’s speed. It takes William 1.5 hours to reach the beach while it takes Dan 1 hour. Find the speed of both bicyclists.

Here is what we know

  • Dan is 4 kph faster than William
  • It takes Dan 1 hour to get to the beach
  • William is 4 kph slower than Dan
  • It takes William 1.5 hours to get to the beach

We will now setup our table

Rate            * Time            = Distance
 Dan  r + 4  1  1(r + 4)
 William  r  1.5  1.5r

We will now solve this equation by placing Dan’s information on one side of the equation and William’s information on the other side of the equation. Below is the solution

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We now know what r is so we need to plug this into the table to get the answers

Rate            * Time            = Distance
 Dan  8 + 4 = 12  1  1(8 + 4) = 12
 William  8  1.5  1.5(8) = 12

The speed of Dan was 12kph while the speed of William was 8kph. This first example was two people traveling the same distance. The next example will be two people travel a different distance.

Example 2

Jenny is traveling to meet her brother. She travels from Saraburi to Chang Mai while her brother travels from Chang Mai to Saraburi. They meet in Bangkok. The distance from Saraburi to Chang Mai is 620km. It takes Jenny 2 hours to get to Bangkok while it takes the brother 7.5 hours to get there. Jenny’s brother’s average speed is 30kph faster than hers. Find the average speed for both people.

The table below captures all of our information

Rate            * Time            = Distance
 Jenny r  2  2r
 Brother  r + 30  7.5  7.5(r + 30)
 620

To solve this problem we combine the information about Jenny and her brother and set this information to equal 620 which is the total distance. Below is the solved equation.

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We can now place this information in our table.

Rate            * Time            = Distance
 Jenny 41.57  2  2(41.57) = 83.14
 Brother  41.57 + 30 = 71.57  7.5  7.5(41.57 + 30) = 536.78
 620

Jenny average speed was 41.57kph while her brother’s speed was 71.57kph. If you add up the distance traveled it will sum to 620.

Our final example will look at determining the time travel when we know the rate of the two objects.

Example 3

A husband and wife both leave their home. The wife travels east and the husband travels west. Wife travels 80kph while the husband travels 100kph. How long will they travel before they are 360km apart?

Below is what we know

Rate            * Time            = Distance
Husband 100  t 100t
Wife 80 t  80t
 360

To solve this we combine the wife and husband information on one side of the equation and put the total distance traveled on the other side. The solution is below.

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We place our answer inside our table

Rate            * Time            = Distance
Husband 100  2 100(2) = 200
Wife 80 2  80(2) = 160
 360

It takes two hours for the wife and husband to be 360km apart.

Conclusion

Understanding uniform equations involve determining first what you know and then determining what the problem wants you to figure out. If you follow this simple process and are able to identify when an equation involves a uniform application it should not be difficult to find the solution.