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The Left Angle Mystery

Geometry is probably my favorite part of math to teach because it is so visual; plus the subject lends itself to doing many hands-on activities, even with my college students.  When our unit on points, lines and angles is finished, it is time for the unit test.  Almost every year I ask the following question:  What is a left angle?   Much to my chagrin, here are some of the responses I have received over the years.

1)   A left angle is the opposite of a right angle.

2)  On a clock, 3:00 o'clock is a right angle, but 9:00 o'clock is a left angle.

3)  A left angle is when the base ray is pointing left instead of right.

    4)      A left angle is 1/2 of a straight angle, like when it is cut into two pieces, only it is the part on the left, not the part on the right.
5)      A left angle is 1/4 of a circle, but just certain parts. Here is what I mean.


Now you know why math teachers, at times, want to pull their hair out!  Just to set the record straight, in case any of my students are reading this, there is no such thing as a left angle!  No matter which way the base ray is pointing, any angle that contains 90is called a right angle.


If you would like some different ways to teach angles, you might look at the resource entitled, Angles: Hands-on Activities.

A Go Figure Debut for A Texas Teacher Who Is New


Lauren's Store
Lauren is a Texas girl who has been a teacher for 12 years! She has taught a variety of 5th and 6th grade math and science classes, and this is her fourth year as an instructional math coach. She works at an intermediate (5th/6th grade) campus that serves a diverse group of students that includes many English language learners and at-risk students.

Lauren loves working with students in small group instruction for math. This was her favorite part of her day when she was in the classroom and one of the ways she still interacts with students in her job as an instructional coach. She also enjoys designing curriculum with her teachers and creating new lesson ideas to teach difficult topics. Her favorite topics to teach are fractions and proportionality. In science, she likes teaching about plate tectonics.

Lauren has three sons from 18 months of age to 11 years old. Together, they like to watch movies and play outside. Between her boys and her work, she stays really busy, but she still finds time to read science fiction books and create educational resources (Surprising, right?).

Her Teachers Pay Teachers store, Leaf and STEM Learning, focuses on resources for 4th - 7th grade. Teachers can use them for guided math instruction, including centers and stations materials, problem solving, utilizing manipulatives plus concept development. Her materials are aligned with Common Core and the Texas TEKS. Since she has used these materials in her own classroom or in classrooms throughout her school and school district, you can be confident in using them in your classroom, too!
Free Resource

Currently, her store contains just over 100 resources, with eight of those resources being free. One of those freebies is entitled Place Value & Powers of 10. Using an engaging low prep interactive notebook set, this resource introduces and practices using powers of ten for place value notation. All the included parts, an interactive mini book, practice activities and formative assessment, are designed to fit perfectly in your students' composition books or math journals.

I am really partial to her paid resource called Ratios & Proportions. These differentiated task cards have 132 total questions that go with 44 real world and model scenarios. Students write ratios in word form, colon form, fraction form and decimal form and solve proportional relationships and percent problems.

Only $3.00
Three levels of questions can be selected randomly by rolling dice or by the teacher with the differentiated dice cards to make the perfect practice for your students. The task cards can also be used as prompts for small group instruction or tutorials. Also included are a teacher guide to help you set up, student instructions, a student recording sheet, and a full answer key.

Her Blog
Lauren has a blog entitled Leaf and STEM Learning, just like her store. She not only shares interesting posts about teaching, but in addition she gives instructional tips and specific ideas for math teachers. Maybe you have heard the common adage that teachers must learn how to “beg, borrow, and steal” to find the best resources and practices for their classroom. Because this saying resonates true for most of us, you ought to take a moment to read her “Steal It” articles! I believe they will really “hit home” as they did with me!

Happy Earth Day

Happy Earth Day Everyone! 
Earth Day is observed each year on April 22nd. The purpose of the day is to encourage awareness of and appreciation for the earth's environment. It is usually celebrated with outdoor shows, where individuals or groups perform acts of service to the earth. Typical ways of observing Earth Day include planting trees, picking up roadside trash, and conducting various programs for recycling and conservation.


Symbols used by people to describe Earth Day include: an image or drawing of planet earth, a tree, a flower or leaves depicting growth or the recycling symbol. Colors used for Earth Day include natural colors such as green, brown, or blue. The universal recycling symbol as seen on your left is internationally recognized and used to designate recyclable materials. It is composed of three mutually chasing arrows that form a Mobius strip which, in math, is an unending single-sided looped surface. (And you wondered how I would get math in this article!?!) This symbol is found on products like plastics, paper, metals and other materials that can be recycled. It is also seen, in a variety of styles, on recycling containers, at recycling centers, or anywhere there is an emphasis on the smart use of materials and products.

Free 8 Page Resource

Inspired by Earth Day, Trash to Treasure is an eight page FREE handout. Discover how to take old, discarded materials and make them into new, useful, inexpensive products or tools for your classroom. To download the free version, just click under the cover page on your right.




Recycled Butterflies

Two of my grandchildren are in kindergarten and of course, everything is new and exciting to them.  They came home one day with egg carton caterpillars.  I know most of us have made one of these in our lifetime, but to these two, they were the best craft ever!

They told me that their teachers were raising butterflies in their classroom, and soon they would hatch.  Anticipation and excitement reigned until the day they came out of school telling everyone that one of the butterflies had hatched.  However, much to their chagrin, the teacher was going to let it go.  They just couldn't understand why or how their teacher could do that!


But, here is the good part!  They got to make a cocoon out of a toilet paper cylinder.  They covered it by gluing on white cotton balls.  Then the made a butterfly out of tissue paper and a small plastic bag tie.  They put the butterfly inside the cocoon and then pretended to have the butterfly hatch!  This was done over and over and over until the cocoon was no more.  Luckily, I was able to get pictures before both were literally destroyed!

Now, what does all of this have to do with math?  I contemplated all the ways to use recycled products to make items for the classroom.  Thus Trash to Treasure was created. It is 34 pages of art ideas, fun and engaging mini-lessons as well as cute and easy-to-construct crafts all made from recycled or common, everyday items.
Only $7.00

Find out more than 14 ways to use milk lids for math. Did you know that you can practice math facts using clear plastic containers? Learn how to take two plastic plates and turn them into angle makers. How about using two plastic beverage lids to make card holders for kindergartners or for those whose hands are disabled? Discover ten ways to use carpet squares as well as nine ways to use old calendars. How about playing hop scotch on old carpet squares? Were you aware that butter tubs can become an indoor recess game to practice addition or multiplication facts? These are just a few of the fun and exciting activities that use recycled items found in this 34 page resource entitled Trash to Treasure.

Because these numerous activities vary in difficulty and complexity, they are appropriate for any PreK - 3rd classroom, and the visual and/or kinesthetic learners will love them.

Yes or No? Stay or Go? Solving for "x".


My basic algebra classes have just begun solving equations containing one unknown. As I tell them, we are inquisitive detectives looking for the unknown.

My students' greatest difficulty is deciding what stays and what goes in an equation. In other words, which term should be cleared by using the inverse operation and which term should stay where it is?

Hands-On Equation
 Balance Beam
www.borenson.com
I always start this chapter using Hands-On Equations®. I have used them for years because it provides a visual for those concrete learners. I also refer to the written equation as a teeter-totter or a see-saw which must always stay balanced. In other words, the equal sign is the pivotal point and both sides of that = sign must be the same.  (Notice that Hands-On Equations® uses a balance beam.) We also discuss the importance of the"Whatsoever thou doest to one side of the equation, we must doest to the other". (Out of necessity, I admit that I was with Moses when he received the Ten Commandments, but it "fell upon me" to convey The First Commandment of Solving Equations to future mathematicians.)

One Unknown
After much practice with the Hands-On Equations®, we move to actual written equations such as: x + 9 = 12. Here's the rub; a few of my students know the answer and do not want to show any of their work. Maybe some of you have this type of student as well. Since, after 30+ years, I am still unable to grade what is in their minds, I insist that all steps are written down. I explain that it's like riding a tricycle to ride a bicycle to ride a unicycle.

First, I instruct the students to look at the equation and determine which terms are out of place. (Side note: Because my students are easily confused, at the present, we keep all of the unknowns on the left side and all of the numbers on the right side of the equal sign.) Let's go back to our sample of x + 9 = 12. Because the x is already on the left side of the equation, the students write a "Y" over it for the word, "Yes". The 9 is on the wrong side of the equal sign, so the students write a "N" over it for "No".  Finally, they write a "Y" over the 12 since it is the correct place. They now have exactly what they want, a Y and N on the right side and a Y on the left side. They now must clear anything that has a "N" over it.  The students recognize they if they use the inverse operation of addition, they can clear the 9. They therefore subtract 9 from each side of the equation resulting in an answer of 3.

Many algebra teachers will have the students write the step x + 0 = 9.  You may wish to include this step in the process, but since my college students readily see that +9 and -9 make zero, they put an X over the two opposites to show that they cancel each other out or when added together, they equal zero.

What if the equation is: 3 = y - 4? This always freaks my students out; yet, if they do the yes/no process, they will discover that they have two "no's" and one "yes", not a yes, no = yes.  This means they can rewrite the equation as y - 4 = 3 to get a yes, no = yes. The problem can now easily be solved like the one above.

Unknown on both sides
of the equation
The next step is what to do when an unknown appears on both sides of the equal sign.  Usually, my students are sure they are incapable of solving such a difficult problem, but let's use the yes/no method and see what it looks like. 

Notice in the sample on the left that we have a yes, no = no, yes. We start by clearing the "N" on the left hand side of the equation by using the inverse of -9. We then go to the right side and clear the y by using the inverse operation of addition. (Yes, I am aware both can be cleared at the same time, but again simple and methodical is what is best for my mathphobics.) We then divide each side by 4 resulting in the answer of 3. When the problem is completed, my students are amazed and proud that they could solve such a long equation. (You might notice in the illustration, a dotted line is drawn vertically where the equal sign is. This helps my visual students to separate the two sides of the equation.)

If any of you try this approach with your students or have a different method, I would love to hear from you. Just leave a comment and a short statement of how this process worked for you or what process you use that is even better. That way, we can learn from each other.

----------------------------------------------------------------------------
Hands-On Equations® is algebra for the visual and kinesthetic learner. This system, developed by Dr. Henry Borenson, enables students (even those in 4th or 5th grade) to easily learn essential algebraic concepts and skills. Dr. Borenson received a U.S. patent for his teaching invention.

Metrics - Not Going the Whole Nine Yards!

Did you know that there are only three nations which do not use the metric system: Myanmar, Liberia and the United States? The U.S. uses two systems of measurement, the customary and the metric. Yes, since our country does use the metric system, we have given more than an inch, but we haven't gone the whole nine yards.

Today, when we shop for groceries, soda is sold in liters. Medicine is sold in milligrams, food nutrition labels are metric, and what about a 100-meter sprint or a 5K race? Still, we are the only industrialized nation in the world that does not conduct business in metric weights and measures. To be or not to be a metric nation has been a question of great consternation for our country for many years.

Here are some reasons why I think our nation should go to the metric system.
  1. It's the measurement system 96% of the world uses. 
  2. It is much easier to do conversions since it is based on units of ten. Water freezes at zero, not 32°, and it boils at 100, not 212°. 
  3. Teaching two measurement systems to children is time consuming and confusing. 
  4. It is the "official" language of science and medicine. 
  5. Its use is necessary when you travel outside of the United States. 
  6. Conversion from customary to metric is often fraught with errors. Because the metric system is a decimal system of weights and measures, it is easy to convert between units. 
  7. There are fewer measures to learn. Once you learn the meaning of the prefixes, you can easily convert mass, volume and distance measurements. No further conversion factors need to be memorized except the specific power of 10. For the Customary System you have to remember 5280 feet = 1 mile, 4 quarts = 1 gallon, 3 feet = 1 yard, 16 oz. = 1 pound, etc. 
  8. And just think, I would have less clutter in my kitchen since I wouldn’t need liquid and dry measuring cups or teaspoons and tablespoons! All I would need is a scale and liquid measuring cups!
So, while most nations use the metric system, the United States still clings to pounds, inches, and feet. Why do you think Americans refuse to convert? I’d be interested in your perspective and ideas.

A Go Figure Debut for Amy Lynn who is new!

Amy Lynn's Store
Amy Lynn has been teaching for 12 years. For her first three years, she was a K-5 science teacher at a magnet school in Tampa, with a side role of being the drama teacher. From there, she and her husband moved to NYC so she could pursue acting, and he could pursue music. While in New York, she taught second grade for seven years at a Christian school while starting up a drama club on the side. (She studied and performed improv at night with the upright Citizens Brigade Theater).

Amy then discovered she was pregnant! A son was born, and the three of them lived in a tiny (280 sq. ft.) studio apartment. (Talk about tiny house living!) At that point, they were a bit tired, and having accomplished a great deal, they packed up and moved back to Tampa, where she is now an elementary special needs teacher. As you can see, her teaching experience is broad...from inner city science...to a private school...to now. Knowing that no day or year is the same is what she loves about teaching.

Amy claims to be eclectic, as is her classroom. It is colorful, with a side of the arts and a side of Zen. She incorporates skits into many lessons (Her drama background seems to keep rearing its head). She has a corner filled with tactile stress balls/tubes/instruments that students can visit when they just need a mental break. Other teachers stop in and use them too, which is always fun. She LOVES fun. She loves it when her students laugh and feel safe to be themselves.

Only $7.00
Amy married the boy next door whom she met as a seven year old and has loved ever since. (They are those kind of people.) They both work hard during the week; so, on the weekends, when not at church, they can be found on the beach, on a bike trail, or just at home, reading books together in a hammock.

Amy is a HUGE fan of creating materials that any teacher can use with as little prep as possible. One such resource is entitled Word Wall - 100 Printables for Any Word Wall. This product includes 100+ printables/activities/skits to use with any class word wall.  It can be used as centers, seatwork, early-finishers or more! Skit writing and performance pages, creative writing, seatwork for individual and for partner work, dictionary and Thesaurus skills, poetry, games, and much more are included in this activity pack of 106 different printables and games for ANY WORD WALL!

Amy currently has over 134 products in her store. Of these, 24 are free, many of which are science items.

One such freebie is on food chains. It contains four fun and simple activities to help engage students as they learn! Included is:
Free Resource
  • A graphic organizer for students to color and fill in their prior knowledge
  • Two "Create a Food Chain" pages
  • A full color page that can be used as seat-work, and then as a colorful wall display of what your students have learned!
Her store is as varied as her 12 years of teaching have been. She has many desk/center labeling tags to help teachers change the looks of their rooms, packets of 100 standards aligned anytime reading/math/writing/language printables per grade level, yearlong Dolch activity packs, and Christian resources as well. There seems to be something for everyone!

Her current goal, (You can see a sample by clicking on her "superhero theme" category at her storefront, as this category is almost complete) is to create grade level standards aligned themed printables that will assist teachers as they endeavor to incorporate their school’s yearly themes. These will include themed 100th day packs, 100 standards aligned printables, reward coupons, free "getting to know me" pages, as well as end of the year memory books/ writing sets. So if anyone knows their school’s upcoming themes and would love these types of resources, she would love all your suggestions! Just go to her store and click on “Ask A Question.”

Pi Day is on March 14!



Mathematicians love to celebrate anything related to math; consequently, Pi Day was invented by some anonymous mathematician.  It is observed every year on March 14th because if this date is written in a month - day format, we get 3-14 which is similar to 3.14, the estimate we use when working with Pi.

The symbol for Pi comes from the 16th letter of the Greek alphabet which is written as “π”.  When used in mathematics, this symbol stands for a constant which is the ratio of a circle’s circumference to its diameter.  This is approximately 3.14159...  Remember using the formula C = π d in high school ? This formula is used because the circumference of a circle is about three times the diameter.

We classify Pi as an irrational number because it cannot be written as a simple fraction. The decimal goes on forever without ending or repeating any numbers. In other words, it is infinite.  While only a few digits are needed for typical math calculations (we usually use 3.14) mathematicians (not me) have calculated Pi to over one trillion digits beyond its decimal point. Pi’s unlimited nature makes it a fun challenge to memorize, but I doubt if anyone has memorized a trillion digits!  Now there is a great bonus assignment for some math teacher to try! 


Since Pi Day is a reason to celebrate, let's have some fun with Pi.  On the left you will see a pumpkin "Pi".  (Was it baked by a mathematician?) On the right, is some "pi" in the sky!



Have a great and glorious "Pi" Day which, in my opinion, 
is just another way to celebrate math.


St. Patrick's Day Myths and Fun Facts

March 17th is St. Patrick’s Day; so, for fun, let’s explore some of the
myths surrounding this Irish holiday as well as a few fun facts.

Myths

1) St. Patrick was born in Ireland. Here is a surprise; St. Patrick isn’t Irish at all! He was really born in Britain, where as a teen, he was captured, sold into slavery, and shipped to Ireland.

2) St. Patrick drove all of the snakes out of Ireland. It’s
true there are none living in Ireland today, but according to scientists, none every did. You can’t chase something away that isn't there in the first place!

3) Since the leaves of a shamrock form a triad (a group of three), St. Patrick used it to describe the Trinity, the Father, the Son, and the Holy Spirit so that people could understand the Three in One. However, there is nothing in any literature or history to support this idea although it does make a great object lesson.

4) Legend says each of the four leaves of the clover means something. The first leaf is for hope; the second for faith; the third for love and the fourth leaf is for luck. Someone came up with this, but since a clover is just a plant, the leaves mean absolutely nothing.

5) Kissing the Blarney Stone will give you the eloquent power of winning or convincing talk. Once upon a time, visitors to this stone had to be held by the ankles and lowered head first over the wall surrounding the Blarney Stone to kiss it. Those attempting this were lucky not to receive the kiss of death.

Fun Facts

1) The tradition of wearing green originally was to promote Ireland otherwise known as "The Green Isle." After the British invasion of Ireland, few people wore green because it meant death. It would be like wearing red, white, and blue in the Middle East today. When the Irish immigrated to the U.S. because of the potato famine, few were accepted and most were scorned because of their Catholic beliefs. For fear of being ridiculed and mocked only a small number would wear green on St. Patrick’s Day. Those who didn't adorn green were pinched for their lack of Irish pride. This “pinching” tradition continues today.

2) Did you know that in 1962, Chicago, Illinois began dying the Chicago River green, using a vegetable dye? An environmentally safe dye is used in amounts that keep the river festively green for about four to five hours.

3) The Irish flag is green, white, and orange. The green 
represents the people of southern Ireland, and orange
signifies the people of the north. White is the symbol of peace that brings the two groups together as a nation.

4) Boston was the first city to celebrate Saint Patrick’s Day way back in 1737, but they did it in August, not March!

5) A famous Irish dish is cabbage and corned beef which I love to eat!

It is estimated that there are about 10,000 regular three-leaf clovers for every one lucky four-leaf clover you might find. Those aren’t very good mathematical odds whether you are Irish or not!

Want some activities for your classroom? Check out my St. Patrick's Day resources.

March Fraction Word Puzzles

Pot of Gold Glyph

St. Patrick's Day Crossword

Writing Papers - HELP!

I am currently teaching a new class called Conquering College which is a required class for all new in-coming freshmen. In the class, we learn about learning styles, AVID strategies, how to take notes, how to read a textbook, etc. Their final project is a poster with an accompanying paper.  Here are the guidelines I give my students
when it comes to writing the paper.

1) This paper should link and connect your ideas with any aspect of self, identity and personality concepts, mindset or learning styles we have discussed in class. In other words, use the class readings and discussions as a “lens” through which you view this person. Do this by using specific vocabulary used in class (e.g. conscious identity claims, growth or fixed mindset, grit, introvert or extrovert, learning style, soft and hard skills, etc.). 

2) Be sure to discuss how and what made this person successful. You might discuss their background, how and where they were raised, what challenges they overcame to succeed, how they reacted to failures and mistakes, what gave them the desire to succeed. 

3) This is not a facts paper about the person. This is about the character traits and attributes of the individual. Although facts can be included, most facts should be on the poster part of this project.

The first semester, the papers were just awful. I could use other words, but needless to say, they were painful to read. The next semester, I created A Graphic Organizer for Writing Papers. My students were amazed at how much easier writing a paper was. Many had never used a graphic organizer like this in English; so, this whole concept was new to them. (This was hard for me to believe, but I guess on the college level, such visuals are rarely used.) 

Only $2.75
This graphic organizer not only helped my students to arrange ideas thus communicating more effectively, but it also facilitated understanding of key concepts by allowing the students to visually identify key points and ideas more efficiently.

The blank graphic organizer found on Teachers Pay Teachers is divided into 11 sections, one for each paragraph. The students write the main idea followed by five details for each paragraph, not in sentence form but in a few words. Separate grids for the introduction and conclusion paragraphs are included. Even though there are 11 paragraphs, the organizer can be reduced to include as many paragraphs as you desire. My students were required to write a paper that was about two pages in length (500 words) when typed; so, this worked well in getting them to that point. Why not take a peek at the preview to see what you think? And if you choose to purchase the item, I would love your feedback.

I trust your students will find this graphic organizer easy to use as well as being a helpful aid in writing papers.

A Go Figure Debut for a High School English Teacher Who Is New!

Her Store Logo
During college, Literary Roses worked as a writing tutor at a university writing center. This experience solidified her desire to become a teacher. Currently, she is a high school English teacher and has been for ten years. She teaches Shakespeare, poetry, plays, writing, and a many other different literary works.

Literary Roses loves to interact with her students! In some students she sees a similarity to those stressed out college students she tutored who were striving to learn with difficulty. She is aware that not all students will grasp what comes so easily to others, and their struggling makes her want to work to help them connect with difficult concepts and skills. Even though teaching is very challenging, she states that she values her profession and believes in its worth.

Only $19.99

The resources in her Teachers Pay Teachers store reflect her passion for literature. All of her products are geared towards eleventh and twelfth graders. One of her 112 products is entitled Elie Wiesel’s Night: Common Core Curriculum Unit.

This unit contains many power points that answer the ten most frequent questions students have regarding the Holocaust, and includes graphs and photos to aid in comprehension. It also teaches concepts such as the dehumanization of Holocaust prisoners, the symbolism of the young Pipel, personification and irony with so much more discussed in this unit. It has 113 ratings with one person in particular saying,

"This is the most incredible, creative unit I have ever located on Night!
It is one of my favorite autobiographies to teach, and my students will benefit from these incredible resources and activities. I've been teaching English for over 20 years,
and I have to say this is the most extraordinary, first-rate novel unit I have ever seen.
I cannot thank you enough. :-)"

Free Resource

Literary Roses also has over 20 free resources in her store. Introduction to the Romantic Period is just one of them. It is an 11 slide power point that explains the Romantic Period to students. It includes the causes for the shift in ideas and the characteristics reflected in the literature. If you are a high school English teacher, you might want to download it.

In addition to teaching and creating resources, Literary Roses enjoys shopping, running, reading, and being with her church family. Her two children keep her quite busy, but she believes being a mom and a teacher are truly works of the heart! Why not take a few minutes and check out the quality resources in her store? 

Heart Rebuses

Hearts and Valentines Rebus Puzzles
My college students love to do rebus puzzles like the one on your right. Do you know what the picture represents? The clue I will give you is that it has to do with an expression that contains the word "heart". (See the answer at the end of this post.)

Since it is close to Valentine's Day, and college students don't have class Valentine's parties, I decided to create several rebus puzzles that represent familiar expressions that contain the word "heart". (e.g. "From the Bottom of My Heart" or "Cross My Heart") Each illustration uses a picture or symbol to represent a word or phrase. The students must use logic and reasoning skills to solve the 24 rebuses. (Yes, I ended up with 24!)

If you purchase this resource, all you have to do is copy the 12 pages of illustrations (two per page) using a color copier. If you do not have access to a color copier, you can enhance the hearts by hand coloring them or have a student help you color. (My grandkids are great at coloring!)

Each class period during the month of February, I put up two heart illustrations as a focus activity. However, you could place one or more up at one time or all of them up at the same time. As my college students enter the room, they try to figure out what heart expressions the two pictures represent. Sometimes they solve them immediately; other times it takes them a while.  But no matter how long it takes, my students find that it is fun and engaging, in addition to being a very challenging Valentine's Day activity!

*The answer to the above rebus is "A heart full of love."  Did you get it?


It's a Puzzling Situation!

One of my colleagues completed a Leadership Project with her ten students that I want to share with you. She had two ‘alike’ 100 piece puzzles. (The puzzles are fairly inexpensive at Walmart or Dollar General.) Kay took these two similar puzzles which had alike colors/pictures on them and mixed them up. She then separated them into two baggies, and put each baggie in one of the original two boxes.

The class numbered off, 1-2-1-2...and so on, and then separated into two groups. At first, the students thought this was going to be a race to see which group could complete their puzzle first; however, each group started at the same time, writing the starting time on the board. After that, Kay didn’t say a word, and answered no questions! She simply observed the students. The students tried asking her, "Hey we don’t have all the edges; these pieces don’t match; are these the right puzzles?" Something is wrong; what's up?"

Kay waited to see who would take the lead to combine the groups, and how they joined. She wondered, "Would they join peacefully? Would they gather and form one group; two new groups; work together, or divide again?"  As she continued to observe, she began to write names on the board of those who were positive and took leadership. She then wrote the time on the board when they commenced to form one group.

When they finished, she held a Socratic Seminar (an Avid strategy) about how they felt concerning the activity. One student, who did not want to join a group in the beginning, became so involved during the project that he actually was the leader in getting the groups together.  It was one of those fantastic teacher moments!

Kay's students learned quite a bit from the activity since in reality, this is how life, social, and work environments are. She pointed out that they may not have a project that is going well, but by joining together with another group, you can problem solve, gain assistance, and acquire more pieces to your puzzle to accomplish your project.

Since working together doesn't seem to be a skill that comes naturally, I plan to use this activity with my college freshman as they begin their final group projects. Plus, as you think about your class and are puzzled about how you can get your students to work well in cooperative groups, keep this activity in mind.  It might just put the pieces together for you.

Plant Mathematics and Fibonacii

Oak Leaves
We continue to look at Fibonacci numbers and how nature continually exhibits this pattern. As stated in my last post, this number pattern can be linked to ordinary things we see every day such as the branching in trees, the arrangement of leaves on a stem, the flowering of an artichoke, or the fruitlets of a pineapple. BUT were you aware that scores of plants, including the elm or linden trees, grow their leaves, twigs and branches placed exactly half way (1/2) around the stem from each other?

Similarly, plants, like beech trees, have leaves located 1/3 of a revolution around the stem from the previous leaves. In the same way, plants like the oak tree have leaves positioned at 2/5 of a rotation. Plants like the holly continue this pattern at 3/8, while larches (conifers) are next at 5/13. The sequence extends on and on. Looking at these fractions side by side (see below) do you see a number pattern in the numerators?


Likewise pay attention to the precision of a similar pattern in the denominators. Interestingly, in the numerators and denominators, if you add the two sequential numbers together, you create a Fibonacci series where all numbers in the series are the sum of the two preceding numbers.



Mathematicians recognize this unique pattern as the Fibonacci sequence. Since patterns such as this one are commonplace in botany as well as other areas of science, they are regularly studied so we can better understand the relationship between mathematics and our world. In my opinion, such mathematical precision and accuracy can only be the product of an intelligent Designer. "Through Him all things were made; without Him nothing was made that has been made." (John 1:3 - NIV)

 

Fibonacci Numbers Are Everywhere?

Handsome 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, many numbers in the Fibonacci sequence can be linked to ordinary things we see around us such as the branching in trees, the arrangement of leaves on a stem, the flowering of an artichoke, or the fruitlets of a pineapple. In addition, numerous claims of Fibonacci numbers are found in common sources such as the spirals of shells or the curve of waves.

Fibonacci numbers can also be seen in the arrangement of seeds on sunflower heads. If you look at the seed arrangement in the center, you'll observe what looks like spiral patterns curving left and right or clockwise and counter clockwise. Incredibly, if you count the spirals, the total will be a Fibonacci number. Divide the sunflower spirals into those pointed left and right, and you'll get two consecutive Fibonacci numbers. ­

Many other plants in nature also illustrate this sequence. For instance, buttercups have 5 petals; lilies have 3 petals; some delphiniums have 8; corn marigolds have 13 petals; some asters have 21 while daisies can be found with 34 or 55 or even 89 petals.

Pine cones clearly show the Fibonacci Spirals. On the right is a picture of an ordinary pine cone seen from its base where the stalk attaches to the tree. Can you see the two sets of spirals going left and right? How many are in each set?
Here are two 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?
I want to close this discussion with a cartoon. It is written by Bill Amend for his cartoon strip Fox Trot which appeared in the newspaper on February 8, 2009. Just think! Now that you know something about Fibonacci numbers, you can understand the humor in the cartoon.