Class Notes Variables and Expressions

Help with the homework, see example problems:
Class Notes Example Problems for C1 L6 Variables and Expressions

Variables and Expressions You Tube link for following video. Thank you Mrs. Griffin at www.nutshellmath.com

This video is a good introduction to algebra, but it goes far beyond this lesson. No need to purchase the full lesson.

1. Multiply the numbers together.
2. Multiply the variables (letters) together.
3. Multiply the signs together.

Sample Problem:

5(2 – 3)
5 squished up against the parenthesis means multiply 5 by everything in the parenthesis.
so multiply 5 x 2 (the first number inside the parenthesis) and then do 5 x -3 (the second thing inside the parenthesis)

Follow the steps.

1. Multiply the numbers. 5 x 2 = 10
2. Multiply the variables/letters. There are none, so skip this step.
3. Multiply the signs. The 5 has an invisible positive sign. The 2 has an invisible positive sign, so multiply + • + = +

So 5 x 2 = +10
Write +10 down as the first part of your answer.

Next, multiply 5 x -3

Follow the steps:

1. Multiply the numbers 5 x 3 = 15
2. Multiply the variables. There are none, so skip this step.
3. Multiply the signs. 5 has an invisible positive sign. 3 has a negative sign. So multiply + • – = –

So 5 x -3 = -15
Write this down as the next part of your answer.

The answer now looks like this: 10 -15.

Add the +10 and the -15 together.
10 + (- 15) =
Think of “poster number 2″. Add two integers with different signs.
Steps to do this. Take the numbers only (not the signs) and subtract (find the difference of) the numbers.
15-10 = 5
Now look at the two numbers 10 and 15. Choose the biggest number. 15. It wins. Put the sign in front of the biggest number( 15 has a negative sign in front of it), in front of your final answer, 5.

So, 10 + (- 15) = -5

Another Sample Problem:

7(-c + 4)
7 squished up against the parenthesis means multiply 7 by everything in the parenthesis.
so multiply 7 x -c (the first thing inside the parenthesis) and then do 7 x +4 (the second thing inside the parenthesis)

Follow the steps.

1. Multiply the numbers. 7 x invisible 1 in front of the c = 7
2. Multiply the variables/letters. c, so squish c next to the 7 = 7c.
3. Multiply the signs. The 7 has an invisible positive sign. The c has a negative sign,
so multiply + • – = -

So 7 x -c = -7c

Write -7c down as the first part of your answer.

Next, multiply 7 x + 4

Follow the steps:

1. Multiply the numbers 7 x 4 = 28
2. Multiply the variables. There are none, so skip this step.
3. Multiply the signs. 7 has an invisible positive sign. 4 has a visible positive sign. So multiply + • + = +

So 7 x + 4 = + 28
Write this down as the next part of your answer.

The answer now looks like this: – 7c + 28.

This means -7 • mystery number c (must multiply this first) AND THEN add +28.

I do not know what the variable c is yet. It’s a mystery. So I can’t multiply -7 • c until I do know.
c could turn out to be the number 8; then I would multiply -7 • 8= -56. Then I could add 28 to -56 and get my final answer which would be -28.
But c could turn out to be the number -11; then I would multiply -7 • -11 = +77. Then I could add 28 to +77 and get my final answer which would be 105.

As it is, I have no idea what number c will turn out to be. So I just have to be patient and let it be for now, until I do find out.

This means, for now, I have done as much as I can do. And my final answer is -7c + 28

My final answer IS DEFINITELY NOT 21c. I can not add the -7 and the 28 together; because that disobeys the order of operations. I have to be able to multiply -7 and c together first.

Here’s a really good YouTube video that should help you understand distributive property.

Clues to recognize
Commutative Property of Addition and Commutative Property of Multiplication:

1. The order of the numbers on one side of the equals sign is different than the order of the numbers on the other side of the equals sign.

The numbers are flip-flopped.

Clues to recognize
Associative Property of Addition and Associative Property of Multiplication:

1. There must be at least three numbers or letters being added or being multiplied together.

2. At least two of the numbers or letters must be in a group (inside parenthesis).

3. At least one of the numbers or letters inside the group (parenthesis) must be changed on one side of the equals sign from the other side of the equals sign.

4. The order of the numbers or letters stays the same on both sides of the equals sign.

Clues to recognize Identity Properties:
Identity Properties state that after you add (or multiply) one special number to any number in the world, the answer will be the same number you started with.

In addition, the only number you can add to any number in the world without changing it, is the amazing number zero.

In multiplication, the only number you can multiply any number in the world by without changing it, is the magic number one.

Additive Identity Property or the Identity Property of Addition

1. Zero was added to a number or letter.

Multiplicative Identity Property or the Identity Property of Multiplication

1. A number or letter was multiplied by 1.

Multiplicative Property of Zero

1. A number or letter was multiplied by 0.

No matter what number is multiplied by 0, the answer will always be 0.

That’s because 0 means nothing. When you have nothing, no matter how many times you have nothing, or how much of nothing you have; you will always have nothing.

1 pile of nothing = nothing. 27 piles of nothing = nothing. 500 piles of nothing = nothing.

So

1 X 0 = 0

27 x 0 = 0

500 x 0 = 0

You will need to know these properties by name. Recognize which property is which. And recognize when an equation uses one of the properties. Questions about properties appear on the benchmark and the FCAT tests.

Hello Oxbow Talons,

Please read this information carefully. It contains directions for your 1st 9 weeks project.

I AM…
You, one of the world’s famous mathematicians, will be writing your “auto”biography and publishing it on your own page in our wiki located here at http://soms.wikispaces.com.

You will also appear in person on a date yet to be selected in our classroom’s living history math museum. This will be the equivalent of a rock concert for all your adoring fans (and parents and community members) who will want to hear straight from the horse’s mouth all about your life and most important mathematical discovery(ies).

In order to be ready, here’s what you’ll need to do:

1. Select a famous mathematician using the following criteria:

• You are genuinely interested in this person.
• This person made an important contribution to the field of mathematics OR they used mathematics extensively AND critically (couldn’t have managed to do what they did without math) to accomplish their most well-known achievement(s).
• This person may be living or deceased (dead).
• You are able to locate at least three (3) different, reliable, trustworthy sources of information about this person. At least one source MUST be from a book. Other sources may include non-internet-based encyclopedias with the following exceptions.
  ONLINE encyclopedias you MAY use:

www.britannica.com
www.worldbookonline.com
Grolier’s online encyclopedia
MSN Encarta Encyclopedia
Smithsonian Encyclopedia
The MacTutor History of Mathematics Click on Biographies Index in left hand margin.

ONLINE encyclopedias you may NOT use:
www.wikipedia.com
or any online encyclopedia NOT included on the approved list above

2. On an index card, write and turn in to the appropriate homework basket in the back of the room:

• your name (first and last), (1 point)
• Period___, (1 point)
• the name of the mathematician you selected, (1 point)
• the titles, authors, publishers, and ISBN numbers of the 3 books or encyclopedias you found that reference this mathematician. (12 points)
• the reason you chose this person, (1 point)
• one interesting fact you’ve discovered about this mathematician so far. (1 point)

Complete Steps 1 and 2 by September 1, 2008.

3. Receive approval to go ahead with your project from Ms. Walsh. (3 points, if first submission approved)

Complete Step 3 by September 8, 2008.

(Students who do not receive approval on their first selection, may have until September 8, 2008, to select a mathematician and receive project approval.)

Need ideas? Google “famous mathematicians”, visit your local public library, browse the biographies and autobiographies sections and mathematics sections at a local bookstore, visit the school library.

Equations

Equations are math problems with an equals sign.

An equals sign means that everything on one side of the equals sign is the SAME as everything on the other side of the equals sign.

How to solve an addition or subtraction equation using the opposite operation
For problems that look like this:
a number (by itself with no variable attached) + or - a variable (letter of the alphabet) = a number

5 + x = 15
x + 7 = 12
k – 14 = 26
13 – z = 9

Thanks to Mr. Karaba (skyegentille) for posting this video on YouTube.

Variables and Expressions You Tube link for following video. Thank you Mrs. Griffin at www.nutshellmath.com

Expressions
Think of an expression as one side of a math problem.
If the math problem is 2 + 3 = 5. The math expression is 2 + 3.

Numerical expressions
Numerical expressions are expressions that contain only numbers and operation signs. NO variables (letters) are used.

Algebraic expressions
Algebraic expressions are expressions that contain variables and operation signs. They can contain numbers too.

Variables
Variables are used in math for two reasons:

1. There is a mystery number. Noone knows what the number is yet. We know it’s a number, but we don’t know which number. Your job is to find out the mystery number or “variable”.

2. There is a number which we want to be able to change or switch, so that we can see what the answer would be if that number were this number or that number.

In the above example of 2 + 3 = 5, we knew quite a few things.
a. The + sign, told us it was an addition problem.
b. The 2 and the 3, told us the exact two numbers (amounts) being added together.
c. The = sign told us we know the answer for the question. What is 2 + 3 added together? The = sign says, “Keep reading, I’m going to tell you right after this symbol for equals ‘=’ .”
d. The 5 tells us that 5 is the exact same amount (or you could say, the answer, or our book likes to call the answer, the value) as 2 + 3.

If I change that problem to this problem, 2 + ? = 5. Now all of a sudden, we don’t know what one number is in advance. We have to figure out what is added to 2, to become 5. When we figure that out, we figure out what the ? is. The ? can only be one number, the number 3.

The ? sign in that problem is called a variable. It is a symbol put in the place the number belongs. Once the number is figured out, we put the actual number in its place. We “solve for the variable”.

When the variable is defined
Sometimes you will be told what the variable mystery number is. Like this: x = 4.

Then all you need do to solve an expression, or find the value of an expression, or find the answer is substitute the known mystery number for the variable.
Like this: 7 + x
They told us what the variable (or mystery number) is x = 4, so we rewrite the expression like this: 7 + 4. We took out the variable x, and in its place substituted the 4.
Now, because we CAN, we solve the expression like this: 7 + 4 = 11.

The expression 7 + x can NOT be solved any further, so we keep it just that way, UNTIL we are told or can figure out what the variable (mystery number) is.

If all we are given is 7 + x we don’t have enough information to solve the problem any further. So we keep it this way.

If we are given an equation, 7 + x = 11; now we have enough information to figure out the one number x has to be. And we solve for x. x = 4.

You can use anything you want to show a variable. You can use an empty box, a blank line, a question mark, anything at all. And you have probably seen empty boxes and blank lines used in first and second grade a lot.
But, in math, the favorite choice to use for variables are letters of the alphabet. And the favorite letter to use in math and in algebra is x.

That creates a problem. From now on, you have to be careful.
Is the x a variable, or does it mean multiply?

Other ways to show the multiplication operation

Most mathematicians do not use × to show multiplication anymore.
Instead they choose one of these methods for showing the multiplication operation.

1. A raised dot. • Like this: 2 • 3 = 6 or like this 2 • x = 6
2. A parenthesis around one or both of the numbers or variables.
Like this 2(3) = 6 or (2)(3) = 6 or like this 2(a) = 6 or this (2)(a) = 6.
3.Or they leave out the parenthesis and just smush two or more things together. BUT not when ONLY numbers are involved.
Like this 2a = 6, BUT NOT like this 23 = 6, because 23 is the number 23, NOT 2 × 3, and 23 does NOT = 6.

IF mathematicians use this symbol × for multiplication, they usually write it like a lower case printed x.
Then, if they need to use the variable x, they make the x cursive or italicize (slant) it.

Expressions vs. equations

Expressions are math problems that do NOT have an equals sign. Like this: 2 + 3
Equations are math problems that DO have an equals sign. Like this 2 + 3 = 5

Do you understand everything? If not, what do you still have questions about? Ask (and answer!) in the comments section.

p. Red 24-27; Blue 14-17; Green 12-16

Math problems that involve more than one operation (+, –, × , ÷) must be solved in the same order by everyone in the world; or else, it’s possible to obtain several different answers for the same problem.

Mathematicians around the world agreed to ALWAYS solve EVERY math problem in the following order, known as “The Order of Operations.”

1. First, every problem (and each step in a problem) must be solved from left to right, just like you read a sentence in English.

If more than one set of brackets or parenthesis, exponents, multiplication or division signs, addition or subtraction signs are present; follow the order of operations, and solve the operation that is to the left, first, each time.

2. Second, solve everything inside outside brackets first [ ].

3. Third, solve everything inside parenthesis next ( ).

4. Fourth, solve exponents next. Exponents look like this 3 ⁴. This means multiply 3 by itself 4 times (3 × 3 × 3 × 3)

5. Fifth, solve all of the multiplication and division problems next, in the order they appear from left to right. (With fractions-multiply the numerator (top number) and divide the denominator (bottom number).

6. Sixth, solve all of the addition and subtraction problems next, in the order they appear from left to right.

Here is a well known mnemonic (saying to help you remember):
Please excuse my dear Aunt Sally.

The initial letter of each word can also stand for what you are supposed to do in the order it should be done.
parenthesis
exponents
multiplication and/or division
addition and/or subtraction

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Slide presentation credit: renegarmath