Math tutoring, advising, and online courses taught by Susan Johnsey. www.MathinABox.com
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Friday, January 28, 2011
Monday, January 17, 2011
The Twelve Days of a Large Family Christmas
I know this is a bit late, but you will enjoy.
Counting to 12!
Friday, January 14, 2011
Exponents and the Order of Operations
Exponents are a shorthand for writing a multiply that uses only one number several times.
5^{2} is 5 with an exponent of 2 means 5 times 5 = 25
6 with an exponent of 3 or 6^{3} means 6 times 6 times 6 = 216. I wrote the 6 down three times.
But notice the differences:
6^{2} = 36, but (6)^{2}= +36 and (6^{2})= 36 One of these is tricky!!
The parenthesis around the 6 tells me the 6 is squared (another word for exponent of 2). We write the 6 down 2 times and then we multiply.
SEE:
(6)^{2}= (6)(6) = +36 .We write the 6 down two times and multiply. Recall two negatives multiplied yields a positive answer.
Without the parenthesis only the 6 is squared. We write the 6 down two times and multiply.
 6^{2}=  (6)(6) =  36.
With the parenthesis BUT the 2 is inside: (6^{2}) =  36 This is really the same as 6^{2}. tricky one!!
Do you see that the exponent 2 is inside the parenthesis?
That means only the 6 is squared not the whole parenthesis, not the 6.
Thus the 6^{2} and the (6^{2}) are really the same. They equal  36.
And ( 6)^{2 }is +36.
And ( 6)^{2 }is +36.
Write these down on a note card or place a sticky note.
You will see this on the QUIZ below. .
We have studied the FOUR operations of math:
addition, subtraction, multiplication and division.
And, you have studied exponents.
You know how to do each of these, but can you do them if I mix 3 or 4 of them together?
Study these carefully.
A. 60 divided by 10 * 2
B. 60 divided by 5 * 4
C. 60 divided by (5 * 4)
D. 60 divided by 2 divided by 5 * 2
Which of the above means  3, and which means 12?
Which one has another number for its answer?
You must understand the above problems and these below about exponents before leaving this lesson.
Do not jump into these without studying the rules.
Most people remember this somewhat, but there are pitfalls that many always seem to fall into.
When the operations and exponents are MIXED together which do we do first?
THE ORDER is important. An agreement has been made.
All calculators and teachers and computers know it.
It is time for you to learn it too.
The Order of Operations AGREEMENT:
You must follow the order.
Step 1 requires you to SEARCH for the grouping symbols and then complete the work INSIDE them.
Grouping symbols are the parenthesis ( ), brackets[ ], bars , braces{ } and even the long fraction bar. While working "inside" you follow these 4 steps.
Step 2 requires you to SEARCH for the exponents in the expression and evaluate them.
Step 3 states that you are to do the multiply and the divides but not necessarily in that order. Many students think that you do the multiplies and then do the divides. That is WRONG.
You do NOT search for the multiplies and then search for the divides!!!!! They MUST be worked from left to right. Before beginning step 3 be sure you have returned to the left side of the problem. Then do the divide or multiply as you move across the problem going toward the right. If the divide is left most then divide first, but if the multiply is left most then you multiply first.
Step 4 states that you are to do the add and subtracts, but not necessarily in that order.
You do NOT search for the adds and then search for the subtracts!!!!! Many students think that you do the adds and then do the subtracts. That is WRONG. They MUST be worked from left to right. Before beginning step 4 be sure you have returned to the left side of the problem. Then do the subtracts or adds as you move across the problem going toward the right.
ORDER of OPERATIONS EXAMPLES
Example 1 :
 3 + 36 ÷ (3)( 2) = There are no grouping symbols nor exponents.
 3 + 36 ÷ (3)( 2) = There are no grouping symbols nor exponents.
Do we do the division or the multiply first? WRITE these on your paper going down the page as I have done.
You MUST learn to write them this way. Otherwise you will only be able to do the simple ones.
 3 + 36 ÷ (3)( 2) =
3 + 12( 2) =
3 + 24=21
I started with step 3 since there are no grouping symbols and no exponents. Begin on the left and do multiply or divide as you come to them. Many people will do the (3)( 2) before the divide; that is wrong.
But you must begin on the left.
Multiply is done before division only if it is left most.
Then begin again on the left and do add or subtract as you come to them.
Try this problem:
Example 2:
45 divided by (3)( 6  3) =
45 divided by (3)( 6  3) =
we have 45 ÷ (3)( 3)
We did the work inside the parenthesis. Now do we multiply the threes or divide the 45 by 3?
= 15(3)= 45.
Do not be tempted to multiply to get 9. That would mean that you did not return to the left to begin step 3.
Example 3 :
3  ( 5 + 8*2÷4  1)
3  ( 5 + 8*2÷4  1)
The (5+8*2÷41) is done first becauseof the parenthesis, but the Steps 2 through 4 must be followed
as you work inside the parenthesis. Step 2 (exponents, not needed). Step 3 says do multiply or divide ( start on the left).
So do 8*2 then divide by the 4. Then step 4 says do add or subtract( start on the left).
5+4  1=8. We have finished inside the parenthesis. 3(5+8*2÷4 1)= 3(5+4 1) =
I hope you are writing these on your paper as I have them typed. 3  (8) = 5
Example 4:
5 + ( 6^2) NOTE: the ^2 means exponent of 2. We use that on computers sometimes.
5 + ( 6^2) NOTE: the ^2 means exponent of 2. We use that on computers sometimes.
5  36 = 31 Do you know why this has  36 and not +36? We discussed that above.
Sometimes we use the "/" to mean "divide by".
Example 5 :
60/3 + 2(2) ^2 = What is the first thing to do?
60/3 + 2(2) ^2 = What is the first thing to do?
There is NO Step 1. We begin with Step 2, SEARCH for the exponents.
Then we go to Step 3, NO SEARCHING just start on the left.
We have one division and then one multiply for Step 3. And, we have one addition to do for Step 4.
60/3 + 2(2)^2= The ^2 means exponent of 2. We use that on computers sometimes.
60/3 + 2(4) =
20 + 8 = 28
Example 6:
35  ( 4 2)^2  8^2 = Step 1: Search for the grouping symbols and
35  ( 4 2)^2  8^2 = Step 1: Search for the grouping symbols and
work INSIDE the ( ).
35  ( 4 2)^2  8^2=
35  ( 6)^2  8^2= Search for the exponents for Step 2 and evaluate those.
35  (+36)  64 = Now what? Step 3. START on the LEFT.
Begin on the left and do what? The subtract or the multiply or the add?
Well , of course we multiply before we subtract. I hope you know (+36) is same as 36.
35  36  64 =
1  64.
The final answer is 65. Did you get that?
You really should copy some of the examples above into your notebook and use them to help you.
Wednesday, January 12, 2011
Tutoring by email or ?
Math in a Box www.mathinabox.com
email Susan O. Johnsey
sjohnsey at bellsouth dot net
sjohnsey at bellsouth dot net
Online Math assistance: some free and some for a small fee. I give free advice (I will gladly help you with 2 or 3 problems for free) and I tutor online for a reasonable rate. You may pay me by using the Google Shopping cart below. $13 for 30 minutes of work or $25 for one hour.  Online Math Courses You may find the prices and registration information for my online math classes (Algebra 1 and 2, Geometry)here: Online Math Courses by Susan O. Johnsey Prices range from $85 to $165 
Please email me first or send the problems and discuss what you need. I answer email at least 350 days of the year, several times a day.

Multiplying and Dividing Numbers
Post 2 Multiplying and Dividing Numbers and special note on ZERO.
 4 times 5 = 20 50 divided by 10 =  5 ( 8) 9 =72 If multiplying or dividing by one positive number and one negative then the answer is negative. 
 4( 8) = +32 or 32 (7)( 6) = +42 or 42 18 divided by  3 = 6 If multiplying or dividing by a negative number and another negative then the answer is positive. 
+4(+ 8) =+32 You know a positive number multiplied or divided by another positive number yields a positive answer. 
HOW to write a Multiply in Algebra!In algebra we do NOT use the x or the dot for multiply.We write the numbers next to each other; some times we use parentheses. 3(6) or (3)(6) means multiply 3 and 6. 
You must use only two of the numbers at a time to determine their sign.Try these examples:A. 6(7)(4) = +42(4) =  168. The 6 times the 7 gives + 42. Then multiply by the 4. B. (4)6(2) = 24(2) = +48 or 48. The 4 times the 6 gives 24. Then multiply by the 2. The rules for deciding the sign of the answer for multiplying and dividing are the same. The / symbol means divide by or is the fraction bar. It is also the divide key on the calculator. 
FRACTIONS:which ones are less than 1 whole? 5/8 or 8/5. And 9/11 or 11/9?And 14/7 or 7/14?DO YOU KNOW?5/8 means 5 divided by 8.This should give you a decimal number that is less than 1 whole if you complete the division.Do NOT confuse it with 8 divided by 5 or also called 8fifths.This is more than 1 whole. It is 1 and 3/5.Be sure you know which "way" to divide.Do the division for the 5/8 and send me your answer in an email Click: Susan O. Johnsey . 
WHAT ABOUT ZERO in division? Zero is well understood with adding, subtracting and multiplying. But it is a bit peculiar, some would say, when it comes to division.Look carefully:0/4 = 0 4/4= 1 4/(4)= 1 0/4 = 0 4 / 4 = 1 4/(4) = 1 4/0 is undefined. Division by zero and Division into zero are quite different. 
A common fraction that most students know is 4/5, that is, 4fifths. 4/5 means 4 divided by 5 . Will the division give you a number larger than 1 or smaller than 1?? Can you read these fractions? Watch out!! 6/5 is read "6fifths" 5/5 is read "5fifths" and equals 1 whole 5/4 is read "5fourths". OK? Keep going please: 5/3 is read_______ 5/2 is read_________ 5/1 is read__________ 5/3 is read 5thirds 5/2 is read 5halves 5/1 is read 5 wholes 5/0 is read_______ oops what did you say here? 5 zeroths!!! what is that? there is no such fraction! And what about the 5/0? 5/0 is undefined (have you ever heard of 5zeroths? I hope not! ) 5/0 means 5 divided by 0. There is not a simple number answer that can be given. This is studied in Calculus so I will tell you for now to just know division BY zero is undefined. Do not confuse it with division into zero. 0/4 = 0 
All division BY zero is undefined:6 /0 or 6 divided by 0, and even, 0/0 are undefined.0 divided by 3 (zerothirds) is 0, but 3 divided by 0 (that is 3zeroths) is undefined. 
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