© 2017 by Jerry L. Croasmun
"Why do we always have to keep trying to find X in Algebra?" Ben asked.
"I don't know but I once heard a joke," I said. "Dear Algebra, stop asking us to find your X because she is never coming back and please do not ask me Y."
"That is funny," chuckled Herr Specht. "Math is like many relationships; they don't necessary make sense, but they work out for the most part."
"But math is hard," exclaimed Ben.
"Ben, you are correct, but no harder than learning another subject or a foreign language. Math is its own language and as in life, opposites attract. Knowing and learning the vocabulary is the first step," said the little man.
Herr Specht continued, "Likewise in math, a positive five is the opposite of a negative five. They are both five steps or spaces from zero. The goal in solving for X is to have everything except X, zero-out or disappear from the equation. X is a variable and represents an unknown number. A variable changes or varies and it can be any letter and does not have to be X. We could use the letter n to represent the number we are looking for if we wanted to."
"Life is full of opposites. Opposites help keep things in balance. The German word Gegensätze means opposites. Here are some examples. Hot-cold ( heiß–kalt), strong-weak (stark–schwach), young-old (jung–alt), boy-girl (junge–mädchen), rich-poor (reich–arm), happy-sad/unhappy (glücklich–traurig/(unglücklich) and there are many more," said our German tutor. "Opposites complement each other and add spice to life. Just like a peanut butter and jelly sandwich only exists if we have both peanut butter and jelly."
"And don't forget the bread," Ben added. "I think I understand. Things like... you are the twinkle to my star, macaroni to my cheese, milk to my shake, flip to my flop."
I could not let Ben out do me so I interjected, "Sun to my shine, water to my ocean, yin to my yang, and Gomez to my Morticia."
Herr Specht had a wide grin on his face as he said, "I've never heard it put quite like that, but I think you kids are beginning to understand. The first thing to remember when dealing with an equation is the equal sign is what makes it an equation. Both sides are equal and must remain in balance. What is done on one side must also be performed to the other."
"Before we attempt to solve or look for X we need to understand a few of the math vocabulary words. First, who can tell me what a term is?" he asked.
Ben quickly answered, "We have trimesters or three terms at my school."
"And we have four quarters or terms at the high school. Is a term like a fraction or part?" I asked.
"Yes, in math the different terms are the parts of an equation and are separated by a plus or minus sign. Can I borrow your notepad?" he asked.
x2 + 5x - 2
x3 - 9x + 12
Writing on the notepad he explained, "Both have three terms and are a special type of polynomial called trinomials. Whereas, 7x + 21 and 36y - 18 are binomials each having two terms. I have something in the truck which will help us understand variables a little better. Ben could you give me a hand?" asked Herr Specht as he walked to the back of his 1963 Unimog.
Setting two baskets of fruit containing bananas, apples, peaches, pears, and cherries on the picnic table Herr Specht gave us each a basket and asked us to sort them. I counted 6 peaches, 10 cherries, 3 pears, 1 apple, and 8 bananas. Ben had 2 peaches, 1 pear, 5 apples, 13 cherries, and 4 bananas.
"How many total pieces of fruit are there?" Herr Specht asked.
Ben and I began counting and said we had 53 pieces of fruit with 8 peaches, 23 cherries, 4 pears, 6 apples, and 12 bananas. I wrote this on the spiral notepad which I had taken out of my backpack.
"Very good. Now comes the fun part where we get to assign a variable to each type of fruit. Let's say p = peaches, a = apples, c = cherries, etc. Next write how many of each in front of the letter or variable. In Ben's basket there are 2p + 5a + 13c, etc. Now you finish," said the little old man.
Ben started to write, "1p" indicating he had one pear, but he was quickly interrupted by Herr Specht who gently asked, "What are you doing?"
"I have one pear and four bananas. 1p + 4b," Ben answered.
"Good job, Ben. However, we cannot mix fruit in our equation. Bananas, apples, peaches, and pears might make a fruit salad, but we must keep them separate. We used the letter p to represent peaches so we must pick another variable. I think the letter r would be a good variable for pears," said Herr Specht as he watched Ben add 1r to the equation.
2p + 1r + 5a + 13c + 4b + 6p + 3r + 1a + 10c + 8b = 53
Herr Specht seemed pleased with our equation and said, "Now combine like terms to simplify your equation by counting the pears, apples, bananas, etc. Remember like terms are things which are the same and are separated by a plus or minus sign. The co-efficient is the number in front of the variable and indicates how many of each there are. In Algebra, we usually arrange our variables in alphabetical order."
Ben and I, working together and talking through the exercise, came up with:
6a + 12b + 23c + 8p + 4r = 53
"Algebra and math can be easy once you learn and understand the vocabulary. Finding the additive inverse might sound scary until you realize you are looking for a number which can be added to the original number and equals zero. The additive inverse of a +2 is a -2; they are opposites. It is like taking two steps forward and two steps back. Opposites attract and cancel one another out. When solving for a variable such as X, first find the X and then look at each term surrounding the variable and do the opposite. Subtraction is the opposite of addition, division is the opposite of multiplication, and squaring a number is the opposite of its square root. If it has a plus sign, subtract or add a negative number to both sides. If it is multiplied, divide," instructed Herr Specht.
He had a way of making Algebra and math seem easy and I secretly wished I had had a teacher like him. However, I was grateful for his lessons and knew I would be able to use his tips in college. He kept repeating that we need to keep everything in balance. He reminded us fractions are simple division problems and to get rid of a fraction we need to multiply by its reciprocal or inverse.
"I think you kids are ready to solve some real problems and find the value for X," said Herr Specht as he wrote two problems in my notebook.
7x = 21 18y = -36
"The first thing I do is find the variable and its co-efficient and read aloud the problem. Seven times a number is twenty-one. We know we have 7 Xs so the first step is to divide both sides of the equation by 7; which will leave us with 7/7. Any number over itself is equal to one. X therefore is equal to 21/7 or 21 divided by 7. So X is 3. Now you kids try eighteen times a number is negative thirty-six," said our math tutor.
"Did you get y is equal to a negative two?" Herr Specht asked.
Ben excitedly answered, "Yes, we did."
The little man was also excited and said, "I think you are ready to move on to something a little more difficult. Let's try solving some two-step equations. I like to find the variable and its co-efficient and place my finger over them, next I look at what is left. In our problem of 2x + 4 = 10, when we cover the 2x we are left with + 4 = 10. Remember opposites attract, so we need to subtract or add a negative four to both sides of the equation so 2x can be left alone. We now have 2x = 6. The final step in solving for X is to realize 2x means two Xs or the X is multiplied by 2. To get X by itself, we divide each side of the equation by 2. This gives us X = 6/2 or X is equal to 3. Now you try 4x - 6 = 30."
2x + 4 = 10 4x - 6 = 30
"What did you get for X?" asked Herr Specht.
Ben answered, "X is nine."
I do not know who was more excited Ben or Herr Specht, but I knew real learning had taken place and felt the joy in the air as Herr Specht said, "You two are the salt to my pepper. The ping to my pong and I think it is time we celebrated by helping me eat some of this fruit."
Read Chapter 1--Fraction Bar
Read Chapter 2--Pizza Pi
Read Chapter 3--Helen Marie
Read Chapter 4--The Wall Of Division
Read Chapter 5--Calendar Magic
Read Chapter 6--Opposites Attract
(NIV) Scripture taken from the HOLY BIBLE, NEW
Copyright (C) 1973, 1978, 1984 International Bible Society.
Used by permission of Zondervan Bible Publishers.