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A Go Figure Debut for A Piece of Pi Who Is New!

Anna is a fellow mathematician and the creator of the store Piece of Pi. She has been teaching 8th grade math for 19 years, and she can’t imagine doing anything else! She is married to her best friend, Jeff. They met in algebra class their freshman year of high school (of course it would be math class!) They have two beautiful children, Matthew (16) and Elise (13). Her hobbies (besides creating teaching materials) include reading, playing the piano and attending concerts. Four months ago, their family adopted an adorable beagle from animal rescue named Jinx.

This year, Anna is teaching algebra and geometry to 8th graders. She describes her classroom as a positive, structured environment. Although the classes she teaches are rigorous, plenty of support is offered. She teaches with guided notes, then lots of practice activities to help her students retain concepts. Anna has created many “looping” resources (another word for spiraling) for her classroom. She finds that these practice resources have really improved her students’ retention. Her color-coded note-taking techniques have also helped them learn new concepts and math standards…especially for those students who have special needs. She finds it rewarding when they become successful learners while mastering challenging content!

Her TPT store, Piece of Pi, contains 150 math resources
(14 free and 136 paid products). Her mathematics
$17.75 - Save 30%
curriculum materials are perfect for grades 7-9. The resource “Light Up Learning Guided Notes” are guided notes for Pre-Algebra using 8th grade common core math standards. Students color code with highlighters as a note-taking technique. This helps them to organize and match key pieces of information. They explore rate of change, use the Pythagorean Theorem, identify functions, and explore slope using similar triangles. In addition, they work with exterior angles of triangles, explore translations, reflections, rotations, dilations, distributive property, combine like terms, and calculate volume. In other words, they light up learning! When completed, these pages make great study guides and are a useful addition to interactive math notebooks.

Anna’s featured free resource is entitled, “Which One Doesn’t Belong: Math Journaling Activity.” Anna always wants her students to explain their mathematical reasoning, and this resource provides much-needed writing in math class! Furthermore, it is ideal for critical thinking! 

Many of her other resources are leveled, including several math “hurdle” activities…perfect for practice and student competition! You’ll find a variety of “looping” pages, which offer valuable practice for students’ retention of concepts. Creating resources that can help teachers save time and help their students become successful learners is what Anna loves to do. In fact, she is proud of the fact that her resources have been approved of by her own 8th graders!

So take some time to check out Anna’s store. If you like what you see, you might also want to visit her blog, Have a Piece of Pi. And no, you won’t find any pie recipes there!


Free Holiday Resource

Christmas is upon us, and I know as a teacher, we are always looking for fun, engaging activities for our students. Check out these free holiday lessons by The Best of Teacher Entrepreneurs Marketing Cooperative! This 20 page E-book includes activities for kindergarten through 12th grade from experienced Teachers Pay Teachers sellers. All you have to do is download it! 

Wishing you the very best during the holiday season.

The Members of the Best of Teacher Entrepreneurs Marketing Cooperative.

The Best Laid Plans. . .Writing Lesson Plans

Lesson plans have always been an Achilles heel for me.  I have taught for so-o-o long, that how to teach the lesson as well as knowing the content is not an issue.  I always have a Plan B, C, and D ready - just in case.  I now teach on the college level where no one checks my plans; however, I still write an outline for the day so I know that I have covered the important points. 

My first job, when I retired from our local school system, was teaching math at a private school.  Mind you, I had been teaching math for over twenty years; yet, the administrator wanted me to do detailed plans which had to be turned in every Friday. I grudgingly did them, but would add little comments in the comment section. That space became my way of quietly venting; so, I would write such things as:  "So many lesson plans; so little time. Writing detailed plans is not time well spent.  To plan or to grade, that is the question.  I am aging quickly; so, I need to make succinct plans."

My supervisor finally relented and allowed me to do an outline form of plans. However, he visited often to observe my teaching, which I didn't mind.  At least he knew what was happening in my classroom.  I have learned from teaching and observing student teachers that anyone can come up with dynamite plans, but the question is: "Do the plans match what the teacher is doing in the classroom?"  Remember Madelyn Hunter?  Oh, how my student teachers hated her lesson plan design, but they did learn how to make a good plan. To this day, I still do many of the items such as a focus activity and a lesson reflection at the end.

My Husband's Lesson Plans for a Week in October
As many of you know, my husband is a middle school science teacher. He is the "Sci" part of my name. Anyway, he is in his 43rd year of teaching, and he still does lesson plans - not the detailed ones we did our first couple of years of teaching, but plans he does have. He divides one of his white boards into sections using colored electrical tape as seen in the illustration on the left.  He then writes what each class is doing for the week in a designated square. In this way, the principal, parents, and students know the content that will be covered. Even the substitute (he is rarely sick) has a general idea of the day's activities. If plans change, he simply erases and makes the necessary corrections.

So what kind of plans are you required to do?  Maybe there are no requirements for you, but do you still write plans?  Are they in outline form or just brief notes to yourself?  I am interested in knowing what you do; so, please participate in the this conversation by leaving a comment.

Only $3.00
By the way, do you need a lesson plan that is easy to use, and yet is acceptable to turn into your supervisor or principal?   Check out my three lesson plan templates. One is a generic lesson plan; whereas, the other two are specifically designed for mathematics (elementary or secondary) and reading.  Checklists are featured on all three plans; hence, there is little writing for you to do. These lists include Bloom’s Taxonomy, multiple intelligences, lesson types, objectives, and cooperative learning structures. Just click under the resource cover.


Getting A Grip on Gratitude

We live in a nation where we have so much to be thankful for. We enjoy a measure of wealth that billions in this world can only dream of and previous generations could not have even imagined. Is it possible that we have grown so accustomed to our affluence that we have lost the wonder of it? Is it possible that our affluence is harming us even as it blesses us?

Unfortunately, I think many in America are infected with the contagious and dangerous disease of "affluenza". How do I know? Because daily, I see people exhibiting the symptoms of the disease. One of the first symptoms is discontentment with what they have. As we possess more things, satisfaction and contentment declines. Many times wealth doesn't deliver joy, only emptiness.

Secondly, obsession is a symptom of affluenza.  I want more; I need more; I deserve more is advertised everyday on T.V.  If we already have a product, we are enticed to upgrade to the latest and newest version or to replace it altogether.

Ingratitude is another indicator of affluenza.  We have so much that we have no needs, just wants, and as we acquire those desires, we tend to forget the words, "Thank you." Finally "affluenza" results in a non-giving spirit.  We grudgingly give or give a meager amount to satisfy our conscious. Shouldn't our giving reflect our abundant blessings?

This Thanksgiving, take time to be thankful.  Share with those you love why you are thankful for them. Call someone you haven't seen for a while and tell them you are thankful for their love and friendship.  Invite someone who has no family to have dinner with your family. And don't forget to give thanks to God who gives us eternal life through His Son, Jesus Christ.



It Depends on the Angle - Complimentary and Supplementary Angles

My Basic Algebra Concepts class just started a brief chapter on geometry...my favorite to teach! We are currently working on angles, and as we went through the definitions, I noticed my students were having difficulty distinguishing complimentary from supplementary angles. Since most of my students are visual learners, I had to come up with something that would jog their memory.


The definition states that complementary angles are any two angles whose sum is 90°. (The angles do not have to be next to each other to be complementary.) As seen in the diagram on the left, a 30° angle + a 60° angle = 90° so they are complementary angles. Notice that the two angles form a right angle or 1/4 of a circle.

If I write the word complementary and change the first letter "C" into the number nine and I think of the letter "O" as the number zero, I have a memory trick my mathematical brain can remember.


Supplementary Angles are two angles whose sum is 180°. Again, the two angles do not have to be together to be supplementary, just so long as the total is 180 degrees. In the illustration on your right, a 110° angle + a 70° angle = 180°; so, they are supplementary angles. Together, they form a straight angle or 1/2 of a circle.

If I write the word supplementary and alter the "S" so it looks like an 8, I can mentally imagine 180°.


Since there are so many puns for geometric terms. I have to share a bit of geometry humor. (My students endure many geometry jokes!)



Only $3.25
You might be interested in a variety of hands-on ideas on how to introduce angles to your students. Check out this resource.  It explains how to construct different kinds of angles (acute, obtuse, right, straight) using items such as coffee filters, plastic plates, and your fingers. Each item or manipulative is inexpensive, easy to make, and simple for students to use. All of the activities are hands-on and work well for kinesthetic, logical, spatial, and/or visual learners.


Making Parent Teacher Conferences Worthwhile


Are You….....
  • Tired of always talking about grades at parent/teacher conferences? 
  • Tired of feeling like nothing is ever accomplished during the allotted time? 
  • Are you having problems with a student, but don’t know how to tell the parents? 
  • Do you want to be specific and to-the-point? 
When I taught middle school and/or high school, these were the items that really discouraged me. I knew I had to come up with a better plan if I wanted parent/teacher conferences to be worthwhile and effective for both the student and the parents. I created a a checklist that I could follow, use during conferences, and then give a copy to the parents at the end of the conference.  It contained nine, brief, succinct checklists which were written as a guide so that during conferences I could have specific items to talk about besides grades. I found it easy to complete and straight forward plus it provided me with a simple outline to use as I talked and shared with parents.

Since other teachers were able to use it successfully, I took that checklist and turned it into a resource called Parent/Teacher Conference Checklist, Based on Student Characteristics and Not Grades. Nine different categories are listed for discussion.  They include:
  1. Study Skills and Organization 
  2. Response to Assignments 
  3. In Class Discussion 
  4. Class Attitude 
  5. Reaction to Setbacks 
  6. Accountability 
  7. Written Work 
  8. Inquiry Skills 
  9. Evidence of Intellectual Ability 
To get ready for conferences, all you have to do is place a check mark by each item within the category that applies to the student. Then circle the word that best describes the student in that category such as "always, usually, seldom". (See example above.)


Finally, make a copy of the checklist so that the parent(s) or the guardian(s) will have something to review with their student when they return home.

Now you are ready for a meaningful and significant conference.




Spiders Are Your Friends - Learning about Spiders

Spiders! We see pretend ones in the store as Halloween decorations (some are pretty terrifying) or real ones outside in a web they have created.  For some reason, these creatures are always something that students want to learn about. How are spiders different than insects? What is an orb web? Are all spiders poisonous? How does the spider not get stuck in her own web? These are questions that students will ask because they are curious and inquisitive.

Did you know spiders are really useful animals and serve mankind well? They eat mosquitoes, grasshoppers, locusts and other insects that are harmful to man. A single spider may kill about two thousand insects in its lifetime. Even though you may be afraid of spiders, very few are dangerous. The black widow and the brown house (recluse) spider do have poisonous bites, but there are no other common house spiders known to be dangerous.

Only $5.25
Spiders are not insects, and insects are not spiders. Spiders are arthropods because they have spinning glands used to create silken threads. Sometimes spiders are called arachnids because of their eight legs. Spiders and insects have different attributes. All insects have six legs, but all spiders have eight legs. An insect has a three-part body, but a spider has only two parts to its body. Insects have antennae or feelers, and spiders do not. Spiders can usually be found in basements, barns, garages, or attics. In warm weather, you can find them under rocks or logs, sitting on fences, or in the grass and flowers. There are about forty thousand different species of spiders.

Interested in learning more?  Check out a ten page short mini reading/science unit  about spiders. First, the students read a short passage about spiders. Then they answer several questions about the reading based on Bloom's Taxonomy, or they do an activity related to the reading passage. Activities include dictionary work, spider math problems, labeling the parts of a spider, and completing a spider web. This mini unit is appropriate for grades 3-5 and will take about five days to complete. An answer key is included.

Want more spider activities? Check out the two word search puzzles and the two crossword puzzles available in time for Halloween! 

A Go Figure Debut for a Board Certified Teacher Who Is New!

Our newest Go Figure Debut teacher is from Arizona. Elizabeth has been in the teaching profession for 12 years, with the last seven being a Master Teacher (teacher coach). In her position, she gets to work with hundreds of different kids and dozens of teachers, and she absolutely LOVES it!

Elizabeth is also a National Board Certified Teacher who loves working with children. Every time she leaves a class, she leaves with a cute or funny story to tell! Building relationships with her students is a vital belief system of hers.

Her personal teaching style might be described as goofy, fun, and entertaining but structured. She loves to joke and have fun with her students, as long as they are behaved and getting their work done. She loves creating resources that bring that love of teaching out in all teachers and students. In addition, she tries to make resources that are colorful, cute and purposeful without being overwhelming.

When Elizabeth is not teaching, she enjoys making all sorts of crafts, spending time with her family and pets, going on camping trips and catching up on movies and TV.

Elizabeth’s TPT store is called “Balke’s Resources”. Currently, her store contains 121 products, 18 of which are free. They vary from back to school activities, positive praise, songs, and get to know you activities. One of her freebies is a Grammar Center Game to practice verb tenses. Included in this printable file are:
  1. The verb cards which are balls of yarn
    Free Resource
  2. A verb sort page
  3.  An instruction sheet with directions written in sequence, using sequencing words 
  4. A special tent CCSS sheet that includes: objective, Common Core Standards and directions to set up at the center
  5.  Two independent student worksheets with the answer sheet 
This free resource is appropriate for students up through the fourth grade.

One of her paid resources is a Hygiene Sequencing Activity Bundle for $4.00. It centers around hygiene, tying reading and science together. This unit is appropriate for Kindergarten through 6th Grade. Included are:
$4.00
  • Objectives for all the activities 
  • Healthy hygiene and poor hygiene cards. 
  • Color coded hygiene cards for easier sorting. 
  • A sorting activity to distinguish healthy hygiene from poor hygiene with directions.
  • An activity to order a morning routine to get ready, with directions on how to sequence. 
  • A transition word poster. 
  • A comic strip for students to draw their morning routine based on their sequenced cards. 
I believe your students will enjoy Elizabeth's fun filled activities because she takes the time to make them simple for you to use. Take a few moments to check out her store. You won't be disappointed!

Using the Periodic Table to Create Bulletin Boards

As many of you know, my husband teachers middle school science.  Together, this is our 86th year of teaching; so, you can tell that we both still love what we do. In fact, we can't imagine doing anything else.

Only $4.00
My husband isn't one to do bulletin boards, never has been and never will be. My daughter (also a teacher) and I usually construct them for him. For many months now, I have been looking for individual tiles of the periodic table.  I saw a bulletin board on Pinterest (one of my favorite places to gather ideas) that I wanted to recreate for my husband's science lab.  I finally turned to Teachers Pay Teachers (where I should have gone in the first place) and asked in the Forum if anyone had such an item. I found that The Triple Point had just what I was looking for. It was a set containing 118 images (png) of Periodic Table tiles, one for each of the 118 elements. Since the resource was only $4.00, I purchased and downloaded it immediately.

After copy the individual tiles onto card stock and laminating them for durability, I laid out the bulletin board (see below). To be honest, my husband did staple everything onto the board as well as arrange the other items. Didn't he do a great job?


In case you can't read the meme in the middle, it says, "That will be $5.00 for the Electrons; the Neutrons are Free of Charge." After all, every classroom needs a bit of humor!

Be-Leaf Me! Fall is Great! Using Leaves in Science Investigations


When Aunt Sue moved to Florida, she would send home some strange requests.  One year, she wanted us to send her a box of fall leaves.  Since Florida lacks deciduous trees, her students were unaware of the gorgeous colors produced by the trees up north.  The only problem with her request was that the leaves we sent would be dry and crumbling by the time she received them. What to do?

I solved the problem by ironing the leaves between two sheets of wax paper.  It was something I had learned in elementary school many, many years ago (back when the earth was cooling).  My granddaughters still collect leaves so we can do the activity together.  Here is how you do it.
  1. Find different sizes and colors of leaves.
  2. Tear off two sheets of waxed paper - about the same size.
  3. Set the iron on "dry".  No water or steam here!
  4. The heat level of the iron should be medium.
  5. Place leaves on one piece of the waxed paper.
  6. Lay the other piece on top.
  7. Iron away!
Up above, on the right, you will see what ours looked like when we were finished.

Only $5.25
You can also use this activity to identify leaves.  According to my husband who knows trees, leaves and birds from his college studies, we "waxed" a maple leaf, sweet gum leaf, elm leaf, cottonwood leaf (the state tree of Kansas - they are everywhere), and two he doesn't recognize because they are some kind of ornamentals. So my suggestion is to get out there and start gathering leaves because your students, children and grandchildren will love it....be-leaf me!

Do you want your students to have fun with leaves? Check out  a six lesson science performance demonstration for the primary grades which utilizes leaves. This inquiry guides the primary student through the scientific method – 1) exploration time, 2) writing a good investigative question, 3) making a prediction, 4) designing a plan, 5) gathering the data, and 6) writing a conclusion based on the data. A preview of the investigation is available. Just click on the title. After all you might have an unbe-leaf-able time!

Dots Lots of Fun - Using Dominoes in Math

I am always looking for ordinary items that can be used in the classroom as manipulatives. I'm a firm believer in the Conceptual Development Model which advocates teaching the concrete (using manipulatives) prior to moving to the pictorial before even thinking about the abstract. When I was at the Dollar Tree (a great, inexpensive place to purchase school stuff) I saw sets of dominoes for $1.00 each. Since they were inexpensive and readily available, I decided to create several math activities and games to introduce, reinforce, or reteach math concepts.

The Number 52
Think about it; if you lay a domino horizontally, you have a two digit number. Put two dominoes side-by-side, and a four digit number is created. Now you can work with place value, estimation, or rounding.  How about lining up dominoes in a column, and working on addition (with or without regrouping) or subtraction (with or without renaming)? 

Another perfect domino activity is practicing addition or multiplication facts.  How about adding the two sides of the domino or multiplying the two sides together?

The Fraction 1/4
If a domino is placed vertically, you immediately have a fraction.  Placed one way it is a proper fraction, but rotated around, it is an improper fraction which can then be reduced.  A fraction can also be changed into a division problem, a ratio, a decimal, or a percent.

So think outside that box of dominoes and use them as an inexpensive math manipulative because Dots Lots of Fun!

Check out all my Domino Resources available on Teachers Pay Teachers.
The first two are absolutely FREE!
  1. Dots Fun for Everyone - FREE  Three math activities and one game for the intermediate grades.
  2. Dots Fun - FREE  Three math activities and one game for the primary grades.
  3. Dots Fun   A 24 page resource for grades 1-3 that includes 13 math activities and four games.
  4. Dots Fun for Everyone  A 29 page resource that features 15 math activities and three games for grades 3-6.
  5. Dots Lots of Fun  Seven math games that use dominoes for grades 2-5.

A Go Figure Debut for an Illinoisan who is new!

My latest Go Figure Debut is an elementary teacher from Illinois. Jenny believes in making math fun for all students by building their confidence through math center games and small group lessons because confident students are successful students!

Jenny is in her 9th year of teaching. She currently teaches first and second grade math, science and social studies. What she loves most about teaching is sharing her love of math with her students. She loves seeing their light bulbs go off as well as seeing her students enjoy math more as they become confident in their math abilities. Her classroom is a safe place where students know that it is okay to make mistakes or to not know how to do something, but it is not okay to ever give up. Her students learn math through centers, games and small guided math groups. Learning is a team effort in her classroom, and she enjoys hearing her students help each other.

In addition, Jenny likes playing games with her family. She spends lots of time with her girls, ages three and two, playing Candy Land and Soggy Doggy. She also loves traveling anywhere and everywhere. In June, her girls will be surprised with a trip to Disney, and Jenny cannot wait to see their faces when they meet their favorite princesses in real life.

Jenny currently has 102 products in her store called Foreman Fun. Twelve of the resources are free. Most of her resources are for 1st, 2nd and 4th grade math and are aligned to the CCSS.

Only $5.00
One of her resources is called Dice Math Center Games! ~ NO PREP! ~ Add, Subtract, Place Value, and Time. The games are for Kindergarten, first or second grade, and there is NO CUTTING! Just print, laminate, and play! All you need is dice! The games are perfect for centers on addition, subtraction, place value and telling time. You could also use them for independent work or as a fun assessment! Want to challenge your students? Just give them more than a six sided die!
FREE

Jenny’s free item is a quick, half page, no prep geometry assessments on 2-D and 3-D shapes and fractions. They are ideal for exit slips, as pre/post tests before and after a unit, or to use as evidence in a portfolio for standards based report cards. Each assessment is aligned to one specific common core standard making it easy to assess each individual standard.

I believe students will enjoy Jenny's center activities and games because she takes the time to make them simple for you to use.  Take a few moments to check out her store and use the custom categories on the left of her store's home page to make your search easier.

FOIL - It Doesn't Always Work!

Using FOIL
In more advanced math classes, many instructors happen to hate "FOIL" (including me) because it only provides confusion for the students. Unfortunately, FOIL (an acronym for first, outer, inner and last) tends to be taught as THE way to multiply all polynomials, which is certainly not true. As soon as either one of the polynomials has more than a "first" and "last" term in its parentheses, the students are puzzled as well as off course if they attempt to use FOIL. If students want to use FOIL, they need to be forewarned: You can ONLY use it for the specific case of multiplying two binomials. You can NOT use it at ANY other time!

When multiplying larger polynomials, most students switch
to vertical multiplication, because it is much easier to use, but there is another way. It is called the clam method. (An instructor at the college where I teach says that each set of arcs reminds her of a clam. She’s even named the clam Clarence; so, at our college, this is the Clarence the Clam method.)

Let’s say we have the following problem:

(x + 2) (2x + 3x – 4)

Simply multiply each term in the second parenthesis by the first term in the first parenthesis. Then multiply each term in the second parenthesis by the second term in the first parenthesis.

I have my students draw arcs as they multiply. Notice below that the arcs are drawn so they connect to one another to designate that this is a continuous process. Begin with the first term and times each term in the second parenthesis by that first term until each term has been multiplied.
When they are ready to work with the second term, I have the students use a different color.  This time they multiply each term in the second parenthesis by the second term in the first while drawing an arc below each term just as they did before.  The different colors help to distinguish which terms have been multiplied, and they serve as a check point to make sure no term has been missed in the process.

As they multiply, I have my students write the answers horizontally, lining up the like terms and placing them one under the other as seen below. This makes it so much easier for them to add the like terms:

This "clam" method works every time a student multiplies polynomials, no matter how many terms are involved.

Let me restate what I said at the start of this post: "FOIL" only works for the special case of a two-term polynomial multiplied by another two-term polynomial. It does NOT apply to in ANY other case; therefore, students should not depend on FOIL for general multiplication. In addition, they should never assume it will "work" for every multiplication of polynomials or even for most multiplications. If math students only know FOIL, they have not learned all they need to know, and this will cause them great difficulties and heartaches as they move up in math.

Personally, I have observed too many students who are greatly hindered in mathematics by an over reliance on the FOIL method. Often their instructors have been guilty of never teaching or introducing any other method other than FOIL for multiplying polynomials. Take the time to show your students how to multiply polynomials properly, avoid FOIL, if possible, and consider Clarence the Clam as one of the methods to teach. 


Sock It Away! What To Do With Those Annoying Cell Phones in the Classroom

Most of us can't live without our cell phones.  Unfortunately, neither can our students.  I teach on the college level, and my syllabus states that all cell phones are to be put on "silent", "vibrate", or turned off when class is in session.  Sounds good, doesn't it?  Yet, one of the most common sounds in today's classrooms is the ringing of a cell phone, often accompanied by some ridiculous tune or sound effect that broadcasts to everyone a call is coming in.  It’s like “technological terror" has entered the classroom uninvited.  Inevitably, this happens during an important part of a lesson or discussion, just when a significant point is being made, and suddenly that "teachable moment" is gone forever.

What are teachers to do?  Some instructors stare at the offender while others try to use humor to diffuse the tension. Some collect the phone, returning it to the student later.  A few have gone so far as to ask the student to leave class.

In my opinion the use of cell phones during class time is rude and a serious interruption to the learning environment. What is worse is the use of the cell phone as a cheating device.  The college where I teach has seen students take a picture of the test to send to their friends, use the Internet on the phone to look up answers, or have answers on the phone just-in-case.  At our college, this is cause for immediate expulsion without a second chance.  To avoid this problem, I used to have my students turn their cell phones off and place them in a specific spot in the classroom before the test was passed out.  Unfortunately, the students’ major concern during the test was that someone would walk off with their phone.  Not exactly what I had planned!

It's a CUTE sock and perfect for a cell phone!
A couple of years ago, a few of us in our department tried something new.  Each of us has purchased those long, brightly colored socks that seem to be the current fashion statement.  (I purchased mine at the Dollar Tree for $1.00 a pair.)  Before the test, each student had to turn off their cell phone, place it in the sock, tie the sock into a knot and place the sock in front of them. This way, the student still had control over their cell phone and could concentrate on doing well on the test, and I did not have to constantly monitor for cheating.

At the end of the semester, we compared notes.  Overall, we found that the students LOVED this idea.  Many said their students were laughing and comparing their stylish sock with their neighbor's.  I was surprised that a few of the students even wanted to take their sock home with the matching one – of course.  So here is a possible side benefit....maybe socking that cell phone away caused my students to TOE the line and study!
Just $2.00

If you want some additional help with other irritations that might "drive you up the wall", try the resource Seven Greatest Irritations for Teachers with Possible Solutions. You can find it in my store for only $2.00.



Fibonacci Numbers and The Golden Ratio

Fibonacci
Even if you were taught about the Fibonacci number sequence in school, you probably don’t remember much about it. As with other higher levels of math, many aren’t sure how Fibonacci could possibly be relevant to their real lives; so, why should they even attempt to remember him or his sequence? In reality, Fibonacci numbers are something you come across practically every day. Even so, let’s go back and start at the beginning.

The Fibonacci number sequence is named after Leonardo of Pisa (1175-1240), who was known as Fibonacci. (I love to say that name because it sounds like I know a foreign language.) In mathematics, Fibonacci numbers are this sequence of numbers:
As you can see, it is a pattern, (all math is based on patterns). Can you figure out the number that follows 89? Okay, let's pretend I waited for at least 60 seconds before giving you the answer….144. By definition, the first two numbers in the Fibonacci sequence are 0 and 1, and each subsequent number is the sum of the previous two. For those who are still having difficulty (like my daughter who is sitting here), it is like this.
  

The next number is found by adding up the two numbers that precede it.
  • The 8 is found by adding the two numbers before it (3 + 5)
  • Similarly, 13 is found by adding the two numbers before it (5 + 8),
  • And the 21 is (8 + 13), and so on!
It is that simple! For those who just love patterns, here is a longer list:

 

Can you figure out the next few numbers?


The Fibonacci sequence can be written as a "Rule “which is:   xn = xn-1 + xn-2   The terms are numbered from 0 forwards as seen in the chart below.   xn is the term number n.   xn-1 is the previous term (n-1) and xn-2 is the term before that (n-2)

Sometimes scientists and mathematicians enjoy studying patterns and relationships because they are interesting, but frequently it's because they help to solve practical problems. Number patterns are regularly studied in connection to the world we live in so we can better understand it. As mathematical connections are uncovered, math ideas are developed to help us be aware of the relationship between math and the natural world. 

As stated previously, we come across Fibonacci numbers almost every day in real life. For instance, my husband and I were at the Wonders of Wildlife Aquarium in Springfield, Missouri. (If you haven't been, you should go because it is spectacular.) He was noticing how the herrings were swimming counter clockwise and discussing the Coriolis effect with the guide. When we got to the lower levels, where the sharks were, they were all swimming in a counterclockwise direction as well. I asked my rocket scientist husband why this was and again he said, with a straight face, "The Coriolis Effect."

Inside of a Nautilus Shell
I then spied seashells and started talking about Fibonacci numbers and the Golden Ratio. (I know the visitors around us were wondering just who we were!) On the right, you will see a picture of the inside of a Nautilus Shell taken by me! It clearly shows the Golden Ratio. (The Golden Ratio is a special number equal to about 1.6180339887498948482. The Greek letter Phi is used to refer to this ratio. Like Pi, the digits of the Golden Ratio go on forever without repeating.) Many shells, including snail shells and nautilus shells, are perfect examples of the Golden spiral.

Are you still not sure what I am talking about? Have you ever watched the Disney movie entitled Donald in Mathmagic Land? (It's an old one that
The Golden Ratio
you can find on You Tube.) Well, in the movie they talk about the Golden ratio. This is a proportion that is found in nature and in architecture. The proportion creates beauty. And that proportion is the Fibonacci sequence! If you divide consecutive Fibonacci numbers you will always get the Golden ratio. Try it! Start with the big numbers. If you divide 89 by 55, you get 1.61. If you divide 55 by 34, you get 1.61. If you divide 34 by 21, you get 1.61, and so on. You can look up the Golden Ratio and explore it more. It’s fun!

As I close, here are four questions to think about:
  1. How might knowing this number pattern be useful? 
  2. What kinds of things can the numbers in the Fibonacci sequence represent?
  3. Where is the Golden Ratio found in the human body?
  4. Why is the golden rectangle important in architecture and art?