Susan Johnsey - online math courses and math tutoring

Saturday, December 31, 2011

Top 100 Tools for Learning

Jane Hart offers valuable info for educators, students, and entrepreneurs with her top 100 Tools for Learning. At her web site she also gives the next top 50 - the 101st to 150th web software/apps/sites- that are quite helpful for learning, teaching, creating and communicating with your students or clients.

Most of the tools are free!

  • Find tools for quick communication or for producing your own movie or screen cast.
  • Geogebra will help you teach Geometry or Algebra online or in your classroom.
  • Free tools  for drawing, painting, editing pics and docs.
  • Watch Science and Math videos.
  • Read research papers.
  • Build visual maps
  • Store your files in safe place with easy network or internet access
  • Create a Slidecast (narrated presentation)
 And remember these are the top of the line software, apps and web sites. The offerings will produce quality goods for you or allow you to experience excellent goods.   I use some of these in my work and will probably  try some others in the new year.

Monday, November 7, 2011

Free Web whiteboard - Tutor students online - includes audio

Scribblar is a Flash-based hosted web site that is ideal for real-time tutoring and web collaboration. Built with simplicity in mind, Scribblar is
  • Easy to use – no training required
  • Browser based and cross-platform
  • Customizable and brandable to fit into your existing website
Try it now  for free at The application requires no large downloads.  It has a clear, user-friendly interface so it’s easy to get up and running quickly.

Send your students a link to your whiteboards and then they can join you.     You can speak with them by turning on the microphone icon (one click), but of course, you must have a mic on you computer.  If you do not have a mic then you can still chat with the student in the CHAT box beside the Whiteboard.

You can write, type, write math equations with LATEX, draw geometric shapes,   use different colors for the text or background or shapes.   You can even upload documents or pictures and edit these together.     SAVE the EDITS or new Creations on the whiteboard and then send to your students
I have used this with several students.   

Friday, April 29, 2011

Math and Tornadoes

Scientists still do not understand the formations,  movements and growth of tornadoes.  Tornadoes  strike sometimes before a tornado warning can even be issued.  

Many warnings (sirens and radio-TV broadcasts) were issued and heeded by Alabama residents this week and lives were saved.  Please pray for the citizens of my home state. At least 200 people lost their lives.  Thousands have lost friends, family, pets, homes, businesses, and jobs. 

Our state is blessed to have brilliant and compassionate weather forecasters and broadcasters, as well as, brave reporters. Many reporters were in the field chasing and sometimes running from the tornadoes in order to help the weather broadcasters better warn the people of our state.

So what about the math?    MORE is needed.    Numerical modeling has provided us with some insights as to how a tornado occurs.  Video observations and numerical modeling has helped us  discover and better understand the formations,  movements and growth of tornadoes. The numerical modeling allows researchers to create computer simulations which can validate their ideas, as well as, uncover new information about tornadoes.   

Can tornadoes be slowed down or detoured? Are you ready to study more math so you can help in future years?   Think about it, and please pray for guidance.

Wednesday, April 20, 2011

Quaternions: Are they from outer space, depths of the sea or are they surreal?

Would you believe that they are NUMBERS, that is 4-dimensional numbers used to describe dynamics of motion in 3-D.  Although created in 1800's they are today used for virtual reality games, robotics and geometry of space-time.  Quaternions are implemented in flight software for the NASA Space Shuttles.  If you have heard of complex numbers, a+bi where bi is the imaginary part, then think of quaternions as complex triple imaginary numbers.   a+bi+cj+dk.

 Irishman William Hamilton devoted his life to studying and teaching about his invention, quaternions. He founded a school of "quaternionists" and popularized them in several books.  He was from Dublin and belong to the Irish Royal Academy.  

In the 1900's the MATHEMATICS of quaternions was replaced with VECTOR analysis.  Who would have thought that 100 years later they would be revived for NASA and  for virtual reality games of the 21st century.


Friday, April 15, 2011

Super Egg in Geometry!

Super Egg - 
The upright egg was created in 1965 by a Dane, Piet Hein. 
 The 3-D  shape is a superellipse defined by the graph of the relation:


The ellipse is then revolved about the z-axis.    In most super eggs that you see a/b = 4/3 !  Or have you not seen one yet!.

Hein's super eggs were popular as toys in the 1960's.  The world's largest super egg can be seen outside Kelvin Hall in Glascow, Scotland, UK.  The super egg with a/b= 6/5 was used as a round-about road in Stockholm, Sweden.  Traffic flowed more fluidly with the super egg design.

You can purchase a set of super eggs on Amazon!   They were designed by Hein as salt and pepper shakers.    They will wobble a bit but do not fall over!

I gleaned this from a book you may want to consider reading at your public library.
The Math Book : from Pythagoras to the 57th dimension, 250 milestones in the history of mathematics by  Clifford A. Pickover.


Need more information about the SUPER ELLIPSE?

Learn to create your own here:
Math World at Wolfram

iPad 2 and its Classroom possibilities

Do you enjoy learning the newest technology?  Have you tried out the new iPad 2?

Houghton Mifflin this month, April 2011, introduced new apps for their Algebra 1 curriculum.

Houghton Mifflin Harcourt, a Boston company known for such educational products as textbooks, said it is launching HMH Fuse: Algebra 1, the first core K-12 education solution developed exclusively for the iPad. 

"The portability of a complete Algebra 1 course on an iPad enables students to learn in the classroom, on the bus, or at home — anytime, anywhere," the company said.

These are current prices, updated each day:

With iPad, the classroom is always at your fingertips.

At the iTunes App Store, there are thousands of apps available to download.
Apple’s iTunes U is home to more than 350,000 free lectures, videos, readings, and podcasts.  Universities such as Yale, Stanford, UC Berkeley, Oxford, Cambridge, MIT, Beijing Open University and The University of Tokyo, as well as broadcasters such as PBS, offer free content on iTunes U.

Thursday, February 17, 2011

Graph Sine and Cosine using GEOGEBRA

You should print this worksheet, see below,  or you can make up your own functions to study.  
Then click link to go to the Graphing ACTIVITY.

Click here for the Geogebra activity: Graphing Sine and Cosine Functions with Variations

Worksheet for Graphing Sine and Cosine:
 Use pencil and paper to copy each of these graphs that Geogebra will create for you.
You should have 12 separate graphs.
Can you predict the graph changes?
Try some yourself.     If you use  y=2sin(x)  what graph will you have?
Change the 2 to 3 or -3, what happens to the graph.  The amplitude changes.
Try other combinations.     y=sin(2x). This 2 will affect the period of the function rather than the amplitude.

1. SINE function
A. period=2pi amplitude = 3 Equation_________
B. period=2pi amplitude = 3 up 3 Equation_________
C. period=2pi amplitude = 3 up 3 left pi/4 Equation_________
D. Complete each graph and also state the "5 points" for B. and C. on the graph.
Does your graph match the info I gave you for C? I hope so HOW can you know for sure?
2. SINE function
A. period=pi amplitude = 3 Equation_________
B. period=pi amplitude = 3 down 1 Equation_________
C. period=pi amplitude = 3 down 1 right pi/2 Equation_________
D. Complete each graph and also state the "5 points" for B. and C. on the graph.
3. COSINE function
A. period=2pi amplitude = -2 Equation_________
B. period=2pi amplitude = -2 up 1 Equation_________
C. period=2pi amplitude = -2 up 1 right pi/4 Equation_________
D. Complete each graph and also state the "5 points" for B. and C. on the graph.
4. COSINE function
A. period= 4pi amplitude = 3 Equation_________
B. period=4pi amplitude = 3 down 2 Equation_________
C. period=4pi amplitude = 3 down 2 right pi/3 Equation_________
D. Complete each graph and also state the "5 points" for B. and C. on the graph.

Let me know if you need help.       If you need the equations then email me.  Be specific in you request as I have many online students studying different topics.
Susan Johnsey  sjohnsey at bellsouth dot net

Sunday, February 6, 2011

Email request for solving linear equations

Hi   johnsey, I have a hard time figuring out solving 2 math problems.
Here they are:
5y - 3 = 8 + 4(y - 2) and   if 7x + 3 = 5x - 5, evaluate 2x - 1.

5y - 3 = 8 + 4(y - 2)  Multiply on right side and add LIKE terms  then solve.
5y-3 = 8+4y-8       the 8-8 is 0 so we get
5y-3 =4y      Since the only number term, the -3, is on the left side I will move the y-terms to the right.   Subtract 5y from both sides.
 5y-3 -5y =4y -5y

-3  = -y   Now we want to finish with y or +1y  so divide both sides  by -1.
-3/-1 = -y/-1   thus we  get y = 3.


If 7x + 3 = 5x - 5, evaluate 2x - 1.

Solve for x first then we evaluate.
7x + 3 = 5x - 5 I will move the variable terms to the left side and the number terms to the right.
subtract 5x from both sides.
7x + 3 -5x = 5x - 5 -5x

2x+3 = -5   Now subtract 3 from both sides.
2x+3-3 = -5-3
2x=-8   so x =  - 4.

now evaluate 2x - 1. let x=-4 we get 2(-4) -1 -8-1 = -9.   The -9 is the answer.

Friday, January 28, 2011

Geometry SSS of congruent triangles Illustration

Here is an illustration  for explaining congruent triangles when we know the lengths of the sides of the triangles. I would suggest using pipe cleaners, or strips of paper to make this a hands-on activity.

Monday, January 17, 2011

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.

52  is 5 with an exponent of 2 means 5 times 5 = 25

6 with an exponent of 3 or 63 means 6 times 6 times 6 = 216.   I wrote the 6 down three times.

But notice the differences:

-62 = -36, but (-6)2= +36 and (-62)= -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.


(-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.

- 62= - (6)(6) = - 36.

With the parenthesis BUT the 2 is inside: (-62) = - 36 This is really the same as -62. 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 -62 and the (-62) are really the same. They equal - 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.


Example 1 : 
    - 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) =

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)

The (5+8*2÷4-1) is done first becauseof 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 - 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?

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

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.

Order of Operations Game.      Try to get a score of 3000.

Wednesday, January 12, 2011

Tutoring by email or ?

Math in a Box   
email Susan O. Johnsey
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.

  sjohnsey at bellsouth dot net


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.



    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 8-fifths.  

       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, 4-fifths.   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 "6-fifths"    5/5 is read "5-fifths" and equals 1 whole    5/4 is read "5-fourths". 

    OK?    Keep going please:

             5/3 is read_______    5/2 is read_________    5/1  is read__________   
             5/3 is read 5-thirds   5/2 is read 5-halves   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 5-zeroths?   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 (zero-thirds) is 0,  
        but 3 divided by 0 (that is 3-zeroths) is undefined.