Monthly Archives: June 2018

Polynomials

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

1.png

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

Add & Subtract

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

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

1

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

Exponents

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

1

Notice how like terms were dealt with separately.

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

1

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

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

1

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

1

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

Conclusion

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

Education in Ancient Israel

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

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

What Did they Teach

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

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

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

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

How Did They Teach

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

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

How Did They Organize 

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

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

Conclusion

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

Cramer’s Rule

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

Part 1:

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

Part 2:

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

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

Example

Here is our problem

2x + y = -4
3x – 2y = -6

Below is the matrix of the system of equations

1

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

1
1

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

1

1

Lastly, here is the determinant of y

1

1.png

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

  • Dx / D = x
  • Dy / D = y

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

1

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

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

Conclusion

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

Classroom Discussion

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

Reasons for Discussion

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

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

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

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

Barriers and Solutions to Classroom Discussion

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

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

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

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

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

Conclusion

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

Evaluating the Determinant of a Matrix

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

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

1

Here is how this look with real numbers

1.png

Determinant for 3×3 Matrix

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

1

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

Find the determinant of c2

1

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

1

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

4(4) – (-2)(-2) =
16 – 12 =
4

The answer is 4.

Expand by Minors

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

1

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

1

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

 2(-4 – 0) + 3(-6 – 0) – 1(-3 – (2-2)) = 
-18 – 18 + 1= 
-25

Conclusion

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

Making Tables with LaTeX

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

Basic Table

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

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

1

We will now go through the code.

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

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

Table with Caption

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

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

Below is the code followed by the example.

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

1.png

Conclusion

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

Solving a System of Equations with Matrices: 2 Variables

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

1.png

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

Two-Variable System of Equations

Our first system is as follows

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

Here is our system

1

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

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

1.png

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

1

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

1

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

Our new system of equations looks like the following

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

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

1

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

Conclusion

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

Education in Ancient Persia

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

Background

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

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

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

What was Taught

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

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

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

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

How Was the Curriculum Organized

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

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

Conclusion

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

Making Diagram Trees with LaTeX

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

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

Vertical Tree

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

Screenshot 2018-04-17 09:02:18

As you can see it is a simple tree.

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

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

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

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

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

The options we set are as follows

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

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

\node{root branch};

Screenshot 2018-04-17 09:17:18

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

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

Below is the code followed by a picture

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

Screenshot 2018-04-17 09:24:14

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

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

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

Horizontal tree

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

Screenshot 2018-04-17 09:29:52

Conclusion

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

Augmented Matrix for a System of Equations

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

From Equation to Matrix

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

11

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

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

11

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

Row Operations 

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

1.png

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

1

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

1

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

1

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

Conclusion

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

Drawing Diagrams in LaTeX

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

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

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

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

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

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

Flow Diagram

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

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

1

The syntax is simple.

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

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

Circular Diagram

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

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

1.png

Bubble Diagram

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

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

1.png

Constellation Diagram

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

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

1.png

Priority Descriptive Diagram

This diagram is useful if the order matters

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

1.png

Descriptive Diagram

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

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

1.png

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

Conclusion

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

System of Equations and Mixture Application

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

Quantity * value = total

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

Example 1: Making Food 

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

The first thing we need to determine what we know

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

Below is all of our information in a table

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

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

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

1

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

1.png

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

1.png

You can check this yourself if you desire.

Example 2: Interests

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

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

The first thing we need to determine what we know

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

Below is all of our information in a table

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

Below is our system of equation

1.png

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

1.png

We now substitute  and solve

1

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

1.png

The bank loan was $20,938.70

Conclusion

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

Education in Ancient India

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

Background

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

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

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

What Did they Teach

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

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

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

How Did they Teach

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

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

How was Learning Organized

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

Evidence of Learning

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


Conclusion

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