Math in a Box with Susan Johnsey

Is our world Hyperbolic or Euclidean? Math helps us explain our universe.

2016-02-02T10:26:00.001-06:00

Corals, crochet and the cosmos: how hyperbolic geometry pervades the universe

Margaret Wertheim, University of Melbourne

We have built a world of largely straight lines – the houses we live in, the skyscrapers we work in and the streets we drive on our daily commutes. Yet outside our boxes, nature teams with frilly, crenellated forms, from the fluted surfaces of lettuces and fungi to the frilled skirts of sea slugs and the gorgeous undulations of corals.

These organisms are biological manifestations of what we call hyperbolic geometry, an alternative to the Euclidean geometry we learn about in school that involves lines, shapes and angles on a flat surface or plane. In hyperbolic geometry the plane is not necessarily so flat.

Yet while nature has been playing with hyperbolic forms for hundreds of millions of years, mathematicians spent hundreds of years trying to prove that such structures were impossible.

But these efforts led to a realisation that hyperbolic geometry is logically legitimate. And that, in turn, led to the revolution that produced the kind of maths now underlying general relativity, and thus the structure of the universe.

Non-Euclidean clause

Hyperbolic geometry is radical because it violates one of the axioms of Euclidean geometry, which long stood as a model for reason itself.

The fifth and final axiom of Euclid’s system – the so-called parallel postulate – turns out not to be correct. Or at least not necessarily so. If we accept it, we get Euclidean geometry, but if we abandon it, other geometries become possible, most famously the hyperbolic variety.

Here’s how the parallel postulate works. Consider a simple question: if I have a straight line, and a point outside the line, how many straight lines can I draw through the point that never meet the original line? Euclid said the answer is one and there couldn’t be any more, which feels intuitively right.

Euclid could only see one possible straight line through a point that does not meet the original line. Margaret Wertheim, Author provided

Mathematicians, being sticklers, wanted to prove this was true, but in the end such efforts led them to see that there is a logically consistent geometric system in which the answer is infinity. We can represent the situation as follows.

What if the straight lines look curved? Margaret Wertheim, Author provided

This seems impossible and a first reaction is to say it’s cheating because the lines look curved. But they only look curved because we’re trying to project an image of a curved surface onto a flat plane.

It’s the same as when we’re trying to project an image of the surface of the Earth onto a flat map; the relationships get distorted. To really see countries relative to one another we have to look at a globe.

How to turn our home planet into a flat Earth.

So also with hyperbolic geometry. To really see what’s going on we have to look at the curved surface itself, and here the lines are straight.

This image shows straight lines drawn on a paper model of a hyperbolic plane. All the pencil lines that appear to be curved were drawn with a ruler so they are actually straight. Margaret Cagyle, Institute For Figuring, Author provided

One way of understanding different geometries is in terms of their curvature. A flat, or Euclidean plane has zero curvature. The surface of a sphere (like a beach ball) has positive curvature, and a hyperbolic plane has negative curvature. It’s a geometric analogue of a negative number.

When mathematicians discovered this aberrant geometry in the early 19th century they were nearly driven mad. “For God’s sake please give it up,” said the Hungarian mathematician Wolfgang Bolyai to his son János Bolyai, urging him abandon to work on hyperbolic geometry.

Nature’s work

Yet critters who’d never studied non-Euclidean geometry had meanwhile just been doing it. Along with corals, many other species of reef organisms have hyperbolic forms, including sponges and kelps.

Wherever there is an advantage to maximising surface area – such as for filter feeding animals – hyperbolic shapes are an excellent solution. There are hyperbolic structures in cells, hyperbolic cacti and hyperbolic flowers, such as calla lilies. In the film Avatar, there is a fabulous CGI grove of giant hyperbolic blooms that curl up when touched.

Hyperbolic surfaces can also be built at the molecular scale from carbon atoms. These carbon nano-foams were discovered in 1997 by physicist Andrei Rode and his colleagues at the Australian National University.

That year Cornell mathematician Daina Taimina also worked out how to model such surfaces using crochet, which was a big deal because it’s actually hard for humans to construct these forms.

For the past 10 years, I’ve been spearheading a project where we use hyperbolic crochet to make woolly simulations of coral reefs. Our Crochet Coral Reefs are an artistic response to the devastation of living reefs due to global warming and have been exhibited at art galleries and science museums around the world, including the Smithsonian.

A crochet coral reef based on hyperbolic geometry. Institute For Figuring, Author provided

Here, a ball of wool and a crochet hook become pedagogical tools bringing mathematics out of textbooks, and taking it to people as a living tactile experience.

More than 8,000 women in a dozen countries (including Australia, the United States of America, and the United Arab Emirates) have participated in making these installations, which reside at the intersection of mathematics, marine biology, community art practice and environmentalism.

The shape of the universe

Once mathematicians realised that different geometrical spaces are possible, a question arose as to which one is realised in physical space. What is the shape of our universe?

Galileo Galilei and Isaac Newton founded modern physics on the assumption that space is Euclidean, but Albert Einstein’s equations of general relativity describe a universe that can have complex curved forms.

One of the major questions astronomers are trying to resolve, with instruments such as the Hubble Space Telescope, is what shape our universe has. While most of the large-scale evidence points to a Euclidean structure, there is some tantalising evidence that we might just live in a hyperbolic world.

Margaret Wertheim, Vice-Chancellor’s Fellow in Science Communication, University of Melbourne

This article was originally published on The Conversation. Read the original article.

2015-08-19T11:13:00.000-05:00

Quick Review Solving Equations in Algebra 1 or 2

If you need a quick thorough review of solving LINEAR and QUADRATIC equations then follow the link below to my classroom.   There are numerous examples, diagrams and videos to help you.
Take a quick look or stay for an hour and relearn several chapters on solving equations -ALL in one Lesson!

Solving Equations
Easy:    2x/3 =   44       to more challenging:    2x³- 8x²+ 8x = 0

Questions?   Email me. Susan Johnsey   my contact info is somewhere on these web pages!

3 Million Dollars for Math homework!

2014-06-24T16:27:00.003-05:00

If you are interested in math study the LINK below will introduce you to some of the great math minds of our decade. These individuals are interested in many areas of math and physics. You can find something that perhaps interests you here.

Dr. Maxim Kontsevich who works at the Institute of Advanced Scientific Studies outside Paris was awarded 3 million dollars by Yuri Milner a few years back. Yuri Milner is a Russian who dropped out of graduate studies in physics and became a successful investor in Internet companies like Facebook.
Just recently, Mr. Milner told him Dr. Kontsevich that he was one of five inaugural winners of the Breakthrough Prize in Mathematics. The awards are financed by Mr. Milner and Mark Zuckerberg, the founder of Facebook; this prize also comes with $3 million. Mr. Milner officially announced the winners on Monday, June 23, 2014.

"The Breakthrough Prize in Mathematics is the latest effort in Mr. Milner's crusade to make science lucrative and COOL in a society that much more often celebrates athletes, entertainers, politicians and business tycoons. 'It is really out of balance', he said. 'This is really to emphasize the importance of fundamental science in our world today'."

READ more here

Summer Ideas for Math

2013-07-08T10:31:00.001-05:00

Do you need some clear goals for the summer?
Should you review last years math problems?
Improve your calculating skills?
Look to your new subject that will begin in the fall?

Math is an awesome skill to have. Your skills and knowledge and even speed can improve each day.
What do you want to learn this summer?

It can be fun! It can be interesting! Are you ready for math this summer.

Here is a link for fun ideas in improving your math skills this summer.
Summer Ideas for Math!

Need more Algebra or Geometry? Then try this link:
Math classes for review or for new skills

Factoring Class offered by Math in a Box dot com -- FREE

2013-06-14T15:07:00.002-05:00

Factoring Polynomials

Factoring and working with POLYNOMIALS is a very important skill for Algebra and Calculus students.
If you have studied Algebra 1 and are not sure you can still factor (there are at least 5 types) polynomials then consider completing this class before you start your next math class.

You do need to know some algebra: exponents, variables, order of operations, signed numbers, and all of these combined into 1 problem!

Here is the link to the free class. Factoring Polynomials.

2013-06-14T14:53:00.001-05:00

What is Stealth? Make your choice:

1. it is teaching a child a topic or concept without them know it!
2. an object that is not detectable, but it really is there!
3. both of the above.

1.
Can we really teach our children without them realizing they are learning a "school topic" or life lesson?
Sure. Read this person's blog to see the fun they had learning- build a pendulum in your living room.
Stealth learning may not always appear orderly, but it’s natural and it can work! It doesn’t have to be used exclusively, but it’s worth trying now and then and seeing how it might work for you and your child.
Stealth Learning

2.
As you might expect, hiding a giant plane is no easy task. Northrop Grumman, the defense firm that won the bomber contract, spent billions of dollars and nearly 10 years developing the top secret project. The finished product is a revolutionary machine -- a 172-foot wide flying wing that looks like an insect to radar scanners! The craft is also revolutionary from an aeronautics perspective: It doesn't have any of the standard stabilizing systems you find on a conventional airplane, but pilots say it flies as smoothly as a fighter jet. The B-2 bomber has a completely different design: It's one big wing; it is like a boomerang.
BUT how does the B-2 fly and how does it vanish? LOTS of MATH and science learning needed for this job.

Shall we make it stealth learning or book learning?

Wonderful article that does not support Math education!

2013-05-30T12:36:00.000-05:00

Wonderful article that does not support Math education!

View this article about the need for algebra in the life of Liberal art students.

The article is written by Barbara Ellis in support of the book Shakespeare Didn't Need College Algebra, author Barbara G. Lenmark.

Read her article here.

You may also want to review the book; but, I could not find it even with Google search and Amazon! If you do find it then let me know!

Transitional Students need your help for their schooling

2012-12-23T18:18:00.000-06:00

Transitional Students Need Your Help!

My friend Cheri Thompson has a farm in Kenya. The farm helps street kids make a new life for themselves.

Education is, of course, a part of that new life.

Will you help? Read the December 2012 post by Amy at Challenge Farm to see the present news and needs of their children.

Education is such an important part of what we do at Challenge Farm. This week standard 8 students are taking exams to see if they qualify for high school and we very much appreciate your prayers for them. For some students it doesn’t matter how much they study and how many times they take a test – they are just not going to qualify for high school. These are our transitional students and this is the new CHALLENGE for Challenge Farm. Read more and accept the challenge: Click here to visit the full blog post.

New approach to developmental college math

2012-11-12T11:09:00.000-06:00

I can give a definite "YES" to these comments about math curriculum changes proposed in California by Katie Hern of Chabot College, director of the California Acceleration Project.

" A growing number of community colleges have developed an innovative new approach for students who are under-prepared for college math. It is less expensive than the traditional curriculum and significantly more effective. The innovation has been spotlighted by several national organizations focused on college completion."

Please read the complete article here:
New approach to remedial math challenged

The need for math is present in almost all areas of study, but more so in sciences, engineering, business, and mathematics. Algebra, geometry and trig skills need to be very strong for these majors.
But for English (and other languages), elementary education, art or music majors, they usually need only a couple of college math courses. Those are usually finite math and statistics. Algebra maybe helpful, but not a must for finite math and statistics. If students are very well prepared with today's definition of pre-algebra skills then these students can be successful without college algebra, precalculus or calculus.

New Factoring Polynomials lessons- free math class

2012-06-01T23:58:00.000-05:00

Please visit my new class.   Do you need to learn or review Factoring Polynomials?
The class is free, but you must register first.   No strings attached. I have other classes that require a fee, but this one is free.

My Factoring Polynomials Class    A new window will open for you.

Susan Johnsey

Experts: Remedial college classes need fixing

2012-05-31T11:35:00.000-05:00

Remedial math college classes

The student debt for taking remedial classes is growing and it discourages students from completing their degree. Read the full report at the Gainesville Sun:

http://www.gainesville.com/article/20120528/WIRE/120529609/-1/entertainment?p=1&tc=pg or go to www.gainesville.com and search for remedial college classes.

You can complete my online math courses and have the knowledge you need to make higher scores on the college placement tests. Prices range from $80 to $180. Many adults have taken my courses and then moved onto college at a higher level.

Visit www.mathinabox.com for more info about myself and my Algebra and Geometry courses.

This was posted by the Gainesville Sun on Gainseville.com :

"The Associated Press

Published: Monday, May 28, 2012 at 8:07 p.m.

Last Modified: Monday, May 28, 2012 at 8:07 p.m.

OVERLAND PARK, Kan. — Each year, an estimated 1.7 million U.S. college students are steered to remedial classes to catch them up and prepare them for regular coursework. But a growing body of research shows the courses are eating up time and money, often leading not to degrees but student loan hangovers."

Math Resource links for middle school students

2012-04-23T18:37:00.001-05:00

A few Math resource links that are helpful for middle school students:

Math Dictionary

Triple A Math

Algebra Practice

Cool Math - Pre-Algebra

Pre-Algebra Textbook for Alabama
You can find your state too.

That Quiz
Try the Algebra (under Integers column) and Geometry levels 1 to 4.

2012-03-22T13:52:00.001-05:00

Need help finding the right apps for your child's iPad™?

eSpark creates personalized curriculum on iPads™ to meet student needs!!

eSpark uses your child's student data to determine the child's learning needs and then helps you set academic goals for them. You or your child are guided through a tailored set of excellent 3rd party apps, games and instructional content.

View there entertaining visual map to see what they offer: eSPARK apps for iPAD

One of the apps suggested by eSparks is Motion Math.

eSparks promotional video:

Need INSPIRATION for TEACHING~ create "BLUE SKY" in your classroom!

2012-01-31T00:18:00.000-06:00

Teaching students can be exhilarating or draining. If you are in need of some new motivation then read the book published just a few months ago by Walter Lewin (with Warren Goldstein): For the LOVE of PHYSICS. Lewin is the wonderfully inspiring MIT professor of PHYSICS. Watch his videos on YouTube; I guarantee that you will be inspired to improve your class presentations and enjoy them. You do not have to be a physics teacher.

One of the many hundreds of his activities for teaching is to create a "blue sky" and then "white clouds" inside his MIT classroom! Do you think his students will remember why our skies are blue and clouds are white?

Even Bill Gates has watched many of his YouTube videos.

Purchase the book at AMAZON or watch his videos at YouTube. See his enthusiasm in his writing and his classroom presentation.

Walter Lewin writes: "I've always tried to make physics come alive for my students. I believe it's much more important for them to remember the beauty of the discoveries than to focus on the complicated math. My goal is to impart enthusiasm to my students, to help them see the beauty of the world all around them in a NEW way. What counts, I found, is not what you cover, but what you uncover. Students love being apart of that discovery."


Math in a Box appreciates your support and comments.

Top 100 Tools for Learning

2011-12-31T13:53:00.001-06:00

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.

Top 100 Tools for Learning 2011

View more presentations from Jane Hart

Free Web whiteboard - Tutor students online - includes audio

2011-11-07T10:45:00.003-06:00

Scribblar
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 Scribblar.com. 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.

Math and Tornadoes

2011-04-29T01:50:00.001-05:00

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.

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

2011-04-20T00:02:00.014-05:00

Quaternions
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.

Super Egg in Geometry!

2011-04-15T17:23:00.003-05:00

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

2011-04-15T16:27:00.002-05:00

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.

Graph Sine and Cosine using GEOGEBRA

2011-02-17T23:17:00.008-06:00

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

Email request for solving linear equations

2011-02-06T17:25:00.000-06:00

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.

Geometry SSS of congruent triangles Illustration

2011-01-28T12:47:00.000-06:00

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.

The Twelve Days of a Large Family Christmas

2011-01-17T23:10:00.001-06:00

I know this is a bit late, but you will enjoy.
Counting to 12!

Exponents and the Order of Operations

2011-01-14T14:09:00.005-06:00

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.

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.

- 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)²is +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.

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.