Warm-up

 

If there were 40.3 million flights before Covid and only 19.6 million flights after Covid arrived, what is the decrease in flights between the pre and post numbers.

The Mathematics Of Tear Free Hair Brushing.

I ran across an article on this topic and was totally intrigued.  I have hair down to the bottom of my rear and sometimes it gets nicely tangled and over the years I've learned enough tricks to make it almost manageable to brush my hair without pulling tons out. It requires a good detangled which doesn't always do much, tons of conditioner, and some swearing on occasion. As we all know, you start combing from the ends up. So I had to read this article and share it with everyone based on the title.  It's nice to learn how math explains the way to make brushing hair easier.

A professor of applied math who used to try to detangle and comb his daughters hair noted that the topology, geometry, and mechanics involved in detangling posed some very interesting mathematical questions. The answers to these questions can also be applied to textile manufacturing, and certain chemical processes such as polymer processing.

This gentleman and two others recently published a paper explaining why the standard brushing method is the most effective way to detangle fibers. In order to look at the problem, they simplified it by simulating two helically entwined filaments rather than looking a whole head of hair. Using this model, they examined how the double helix is detangled via a single stiff tine that leaves two separate filaments and they measured the forces involved and the deformations connected with combing hair before simulating the results numerically.

They assigned a mathematical value to the short strokes that begin at the free end and move up towards the clamped end and end up removing the tangles as link density.  Link density refers to the amount of hair strands that are braided with each other. They also identified the optimal minimal length of the stroke, or the minimal length needed to detangle the hair.  Any shorter and the hair would never get detangled. Other people took this information and used it to create an algorithm that allows a robot to comb hair. 

The next step for these researchers is to look at the mechanics involved in brushing curlier hair and how it reacts to humidity and temperature. So they know have the mathematics to explain this method and it's interesting that the information could be utilized to teach a robot to comb hair effectively.  Let me know what you think, I'd love to hear.  Have a great day.

Fact Fluency Games Without Technology

 

I love having the students play games on line but I don't always have that option.  The internet at my school is so-so at best and if we have school wide testing, no one other than the testers can be on, otherwise they get kicked off quickly and frequently.  So I have several games I keep handy to have students play.  

One of my favorites is Bingo.  I use it to help students practice their multiplication and division facts, order of operations, fractions, and everything else but today I'm focusing on fact fluency.

I hand out blank bingo cards and have students write on either products if they practice multiplication or possible answers for division depending what I"m doing.  I let them chose the numbers they wanted.  Then I call out the problem like 7 x 8 and they have to come up with the answer of 56 and then cross out the answer.  I might go with 6 divided by 6 = and they have to come up with the answer of 1 and cross it out.  Yes some problems may have the same answers such as 6 x 6 or 9 x 4 so they might find the number is already crossed out but that has never been a problem.

Then there is checkers fact fluency where you place mathematical problems on all the white or all the black squares.  All pieces follow the normal rules for movement except the student must give the correct answer for the problem to land on that square. The winner is the one who has captured all the other pieces.

What about a match game with factors and answers on separate squares and you have to match them up.  For instance one card might have 8 and another card would have 4 x 2 and a third card would have 8 x 1. All three would match up because 8 equals the other two.  This could be done for any of the operations. A variation of this is to create a puzzle with answers and problems with squares.  The squares have a problem or answer around the sides so students match the problem with the answer on a different square. 

The photo shows how it might be set up for two squares. One is an edge piece and the other is a middle piece.  Students have to match the equation with the answer till it done.  It's like a jigsaw puzzle and requires some thought.

The last suggestion is for a fact search.  In this you create a grid filled with the numbers that make up factors so that students fill in the appropriate signs to make it true.  They might see 7. 8.  5. 6 and rewrite it as 7 x 8 = 56.  These are easy to find online such as this web site.  This is a nice way to have students practice it.  

This is just a few suggestions for things to do.  Let me know what you think, I'd love to hear.  Have a good day.

Practicing Fact Fluency Using Technology

 

In many school districts, students have not had proper math instruction over the past two years.  The students I work with have not had much math since the pandemic started due to being red.  They do not have access to the internet and their reading levels are quite low so they have trouble following any written instruction I include in take home packets.  So now that they are back in class, I'm slowly discovering their weaknesses.

I gave my students an assignment the other day that required them to read a verbal description of money and translate it into a written amount.  The description might be three dollar bills, six quarters, three dimes, and four nickels.  I discovered that half of them didn't know a nickel was worth five cents and about the same number couldn't multiply six times 25 or three times ten.  One of my students did have a strategy to account for that so she could do the answers.  She'd draw the number of coins and then skip count to get the answer.

Since my students do not know their multiplication tables, I have to include time in class for them to practice.  Usually, I incorporate it into the assignments but sometimes I have to look at ways for students to get a focused practice session.

One way is to have them use apps or web based programs for practice. The assignment should not be one in which they fill in the blank in ordered tables such as 2 x ___. = 2, 2 x ____ = 4 etc or 2 x 1 = ____, 2 x 2 = _____ etc because in the first they understand they have to write in the numbers 1,2,3..... in order.  In the second one, they can skip count to the end without really learning their facts.  Therefore, one should look for a program or app that has lets them solve problems that are not in order so they can't use a pattern to complete the assignment.

 One iPad app I've had students use is something called Sushi Monster which has them building fact fluency for both addition and multiplication.  It has seven levels for addition, five for multiplication and each level is made up of four rounds.  Each level targets 14 different numbers and if a student redoes a round because they didn't do well, or they need additional practice, the app uses new numbers so they can't write down the answers to use again. It also increases speed so they get faster at using their fact fluency.  The app is made by Houghton-Mifflin and best of all, it is free.  I have used it with both middle school and 9th graders effectively before.

Another free to download app is called Monster Math. This one is geared for grades one to five but it can be used for higher grades.  It allows students to practice all four operations in a game format and it is set up so students can practice multiple skills at once while meeting the common core standards.  The program uses stories and adaptive technology to set the expectations for where they are and not where they should be. In addition, it is set up with voice over narration so students can follow along. Another nice thing about this app is that it allows students to play with others if you want a more competitive game.   Although it is listed as free and can be used it does have a subscription option.

Then there is Prodigy games which is a web based online game site that is free. It is easy to sign up for and can be done at home or in school.  In addition, students can access it with or without a class code but if the student does it at home, they need permission from their parent to access it for more than one month.   The game site has students answer math questions to complete quests and earn in-game rewards and it is good for students up to the 8th grade but could easily be used in high school.

I'll be back next time with games that can be done in the classroom without technology.  I'm including nanotechnology based games because many schools do not have great internet including mine.  We have to quit using the internet if there are students are doing online testing otherwise people get kicked off.  Let me know what you think, I'd love to hear from you.  Have a great week.

Warm-up

 

If you need three pounds of tomatoes to make one quart of tomato sauce, how many quarts will you end up with if you harvest 327 pounds of tomatoes?

Warm-up

 If there are 3 globe sized tomatoes in a pound, how many tomatoes will be in 50 pounds?

Can You Get More Of The Pizza?

 

Is it possible to share a pizza so one person gets more of the pizza?  Let's see how it all works out.  We know that most pizza's are cut with an even number of pieces so if two people share a pizza, they will both end up with an equal number of slices. If the slices are of equal size, no problem.  If as in reality, the slices are not equal, then the person will go after the largest piece available to end up with the most.

Now to look at this from a more mathematical point of view since mathematics can explain the world.  When taking slices from a pizza, you either take the next one over - a shift - or you skip the empty spot - a jump.

If the person's strategy includes a number of jumps, then it is referred to as a j-jump strategy.  In addition, the pizza can be represented by a circular sequence with each piece being given a certain weight or the weight of P.  Now a player can have gain g if the strategy they use results in a certain number of pieces whose sum of weights is equal to g.

If a person does not have a jump strategy, there is only a 1/3 possibility of making it but if they have a one jump strategy, it jumps to a 7/16 possibility. If the person has a two jump strategy, it moves up to a 4/9 possibility.  Each jump increases the chances of coming out ahead. This is all based on an even number of pieces.

If there is an odd number of pieces, then it is a characteristic cycle instead of a circular cycle.  The characteristic cycle has an arc defined as a certain number of consecutive pieces with a specific length and weight. Furthermore, the arc of length defined as (n + 1)/2 is called a half circle. In the zero jump strategy, it has a potential of 1/3 as the lower bound.  This means it covers three half circles since the potential of v is the minimum number of weights of half circles covering v.  The upper bound is 100100100.

For a one jump strategy, the results depend on how and when the jump is made and could be a 1 if n is one,  4/9 if n is an odd number beginning at 15 or 1/2 for everything else,  a nice piecewise function.   It is best to use a one jump strategy when n is even or 9, 11, 13 or a two jump strategy for the odd numbers beginning at 15. 

The best two numbers of pieces the pizza should be cut into is 15 or 21 so one person gets more pieces than the other. I admit, it's been a while since I've had to read anything written in formal mathematical language so it took me a bit.  As a teacher, I don't spend much time reading things like this because I'm just trying to fill in missing foundational knowledge.  Let me know what you think, I'd love to hear.  Have a great day.

Forced To Make A Move In Games.

You are getting ready to play chess or a math game and you have the choice of going first or second.  Which should you choose even though everything in you is screaming to go first? 

In some games, going first provides only a small advantage but in other games, it is better to let your opponent go first. Either way, there are circumstances in some games where a person is forced to make a move.  That is what Zugswang is about.  Zugswang is a German word meaning "move compulsion" and was explored by some researchers.  One game to think about letting the other person go first is Chess.  

There are situations in chess where the person must make a move such as in the 1971 game between Bobby Fischer (white) and Mark Taimanov (black).  White had a bishop while black has a knight but white also has a pawn that can go all the way to the other side and become queen. In order to prevail, black must capture white's pawn by possibly sacrificing his knight but the best this leads to is a draw.  This is circumstance where he must move (Zugswang) either the knight or his king. In addition, white has the opportunity to make a move that doesn't lead to anything so that black must make another move.  In the end, White won since black had to make moves at various times.

In the game of Nim where two players are engaged in subtracting from a starting number selected by the second player.  In this game, players must subtract at least one but no more than one more than the tens digit of the number.  Player B must choose a starting number between 90 and 99 so that player A can begin subtracting. There are a few numbers that when chosen will automatically lead to player A being forced to make a move so they loose. If player B starts with any of the following numbers or gets the subtraction to any of these numbers - 93, 82, 72, 63, 55, 48, 42, 36, 31, 26, 22, 18, 15, 12, 9, 7, 5, 3, 1 player A will loose.  If on the other hand, player A is able to get the total to one of these numbers, player B will loose.  

Then there is Sim which is played on a hexagonal shaped playing board with lines going from all points across the center to the other points and along the outside.  There are two players, red and blue. The idea is to color in the lines across the middle or outside but they can't make three edges that form a triangle or they lose.  So picture if you will the hexagon is labeled A, B, C, D, E, F.  The first player starts by choosing AC, the blue EC and continuing on alternating until the red colors in the lines AC, AE, AF, AD, and AB and the blue has chosen lines EC, EB, ED, EF and it is blue's move.  No matter what move they make next, they will complete the third leg of the triangle and lose.  They are forced to make the move (Zugswang) and will lose.

On Friday, we'll look at this in terms of sharing a pizza so one person might be able to get more of the pizza than the other.  Let me know what you think, I'd love to hear.  Have fun with the idea of Zugswang and enjoy your day.

Difference Between Jamboard And Slides

 

As teachers, we have access to so many different tools.  Some are better for one type of teaching while others work on a different assignment.  I've often wondered how Jamboard and Slides differ in their potential uses so I'm taking a closer look. 

One of the biggest differences is that google slides offers a choice of templates when you hit the + or new button but Jamboard doesn't.  This isn't something to worry about because it is possible to find Jamboard templates available on the internet with a quick search.

As far as the background goes, slides allows you to set the background for one or for all the slides but Jamboard requires that you set the background for each frame so you have to set it one at a time. This is fine if you want different backgrounds such as plain versus coordinate plane but if you want them all any color other than white, you have to go in and set them.  This being said, Jamboard does have the ability to duplicate slides so if you want several that are identical, you can do this.

On the other hand, Jamboard appears to be the better tool to have students interact because it has sticky notes, shapes, markers, etc, all available to the students immediately.  In slides, you have to preset everything you might need.  In addition, students who use Jamboard have learned it they are expected to interact but slides is more about creating one slide to present their thoughts.

Two things that make Jamboard stand out over slides is the marker and the sticky notes. The Marker does a much better job than the scribble feature in Slides and the sticky notes allows the teacher or students to make notes off to the side.  Sticky notes are possible in slides but it is more difficult to do.

When it comes to adding images, it is about the same for both whether you upload, grab something from your drive, etc. However, slides allows students to explore the internet, your drive, photos from slides itself but Jamboard does not allow that.  Now as far as general features, slides offers more features than Jamboard which makes Jamboard easier for students to learn to use. In addition, slides allows you to have more slides over Jamboard because Jamboard seems to limit the number of frames to 20.  So if you have more than 20 students in your class, you would have to pair up students to work so everyone participates.

In addition, slides allows you to establish hyperlinks where as Jamboard does not. This makes it harder to send students to websites, documents, or other material. Slides allows the creator to link to a shared resource so much better than Jamboard.  Furthermore, in Jamboard, you can upload photos, use a URL, do a google image search, find something in your drive, or use your camera but it does not appear that you can import a video. It is possible to import videos into slides and the same goes for audio files. 

This is a quick look at the differences between Jamboard and slides.  I suggest you go to play with each one to see which you want to use and to get a better of idea of when to each one.  Have a great day and let me know what you think, I'd love to hear.

Happy Easter

 

Its Coming

 Its Coming!

Easter Based Math

Easter is rapidly approaching and many of our students look forward to the traditional celebrations. It's always nice to be able to incorporate math that is associated with the holiday.  Again, Yummy math to the rescue with some very nice activities that are not the standard worksheet based problem.

One activity I like has students compare giant chocolate easter bunnies.  In the exercise, students make a guess as to whether the bunny is solid or hallow and must explain how they determined the choice.  They also calculate price per pound, price per inch, and height of the bunny compared to the student, based on the information provided,  Students are also asked questions such as which is better price per inch or price per pound and asked to explain why they chose the answer they did.  The activity also looks at a Guinness Book of Records bunny and ends by having students write down observations on a huge chocolate egg. 

Another activity focuses on using food coloring to color eggs. The activity has a chart at the top with instructions on the number of drops needed to color a white cake, white frosting, and to dye Easter eggs.  It begins by asking students to make observations followed by questions on the number of drops required for each.  The questions require some real thought to answer as they are higher order.  

Then there is a wonderful activity on candy sold each year.  This one has students reading and interpreting data on a pie chart concerning the amount of candy sold in a year and the percent of candy sold for various holidays such as Valentines day. They are requested to use three different methods to find answers so they don't use a calculator.  They are asked to make conclusions based on the results.

We mustn't forget the activity on Peeps, those yellow, marshmallowy, coconut covered candies one associates with easter. In this activity, students work on estimating the number of Peeps sold for easter.  At the end of the lesson, the teacher provides the number so students can compare their estimates with an official one.  

Although many of these activities are said to be for upper elementary and middle school, they can be used all the way up to high school since they work on skills that are not as frequently covered as we'd like. In addition, these are a much better thing to do than to rely on those easter worksheets that use eggs, or ducks, or rabbits with assorted operations that are nothing more than a regular worksheet disguised for the holiday.

So now you've got an instant activity to celebrate Easter, and give students practice in what they need. Let me know what you think, I'd love to hear.  Have a great day and enjoy your Easter Weekend.

Equat|O (equatio) Extension.

 

If you've ever tried to include a mathematical equation, especially one with an exponent or root,  you know how hard it can be.  I have had to do it and ended up going back to early programming days of ^ to indicate it's a power or sqrt for square root because I couldn't find the symbol I wanted. 

If you haven't heard of it before, it is a chrome extension which allows you to construct creations, formulas, and so much more.  It is relatively easy to use and can be used with Google suites quite easily.  As noted Monday, if you are a teacher, you can download Equatio and upgrade to premium so you have access to all of it.  

It has an equation editor  which allows you to type in the equation you want.  It will make suggestions as you are typing things in based on the most commonly used equations and formulas and what you've done previously.  It also has a populated gallery of symbols, redone layouts, and common formulas.  This makes it easier to get the math done quicker.

In addition, it has a graphing editor which allows you to type in the equation and the editor graphs it on a four quadrant coordinate plane. Once it is graphed, it can be imported into Jamboard or other google application.  Another thing it offers is handwriting recognition. The handwriting recognition can be found on the tool.  It allows you to draw or write out the math and then translates it into print so it can be transported into the google app you want it in.  

There is another feature which makes it quite interesting.  The feature is called "Insert Mathspace". When you click on it, it opens a new tab one can use to insert math graphics such as shapes.  If you click on shapes, you'll find lots of math graphics such as fraction bars, standard geometric shapes, number lines, etc.  Then if you decide instead to use smart shapes, which allows you to create dynamic shapes so you can adjust the properties to make exactly what you want such as a number line, a grid, a protractor, coordinate plane, angle measure, etc.  

The smart shapes allow you to design drag and drop fractions to match  visual representations or matching one visual representation with an equivalent visual representation so students see fractions represented in more than one way. You could create analog clocks to match up with digital representations, or even have a protractor that can be used to measure unknown angles.

Equatio has so much to offer and the premium version is available for free to educators.  If you've never used it before, check it out in the Chrome store and have fun.  Let me know what you think, I'd love to hear.  Have a great day.

Revisiting Jamboard.

 

Many teachers use a smart board as part of their normal routine while asking students to use different parts of Google Suite to show their work.  There is a way to have students use Jamboard to show their work because it is an interactive whiteboard with so many features.  These features allow students to easily show their work. In addition, the teacher can post problems using the sticky note feature. Furthermore, since Jamboard allows the teacher to set up multiple pages with multiple problems. 

Lets look at how one can assign the work to students via google classroom, Jamboard, and google docs.  The first thing you want to do is to create the actual assignment in Jamboard.  Begin by going to Jamboard.google.com  so you can create the assignment.  Click on the plus sign in the bottom right hand corner.  The plus sign tells Jamboard it's a new assignment. Type in the title of the assignment in the upper left corner.  Add a sticky note by clicking on "sticky note" in the menu and type the problem on the sticky note before saving it.  If you need to resize the sticky note and move it to where you want it located.  If you want to provide immediate feedback, create another sticky note with the answer so students can check. When you've finished creating the assignment, close it up, and it will automatically save. Should you want a Jamboard with multiple problems, you can create multiple slides with different problems for students to work their way through.

Students can place their answers on Jamboard or they can write them in a google doc.  The advantage of using goole docs is one can type the answer in so it looks neater but this is not necessary.  If you want the answers in google docs, go to Docs.google.com and open up a blank document.  Place the document title in the upper left corner and then set it up as answer sheet before saving. 

Now to put it all together in google classroom.  Go into google classroom, and click on the classroom tab. Choose create assignment and assign it a title. First thing is to type out detailed instructions explaining to students they should show their work in Jamboard while placing their final answer in google docs.  Attach the Jamboard and google doc to the assignment but make sure you click make a copy for each student so each person has the problem to work.  Don't forget to hit the assign button so it is assigned.  

Unfortunately, most documents are not set up to do equations with variables, division, and other such symbols. This is where Equat|O comes in.  Equat|O is a Chrome extension which allows one to make equations that can be inserted in Jamboard.  After installing the extension, upgrade to the premium version which is free to educators.  What this does is that when you open Jamboard, it will now have the Equat|O icon at the top right hand corner of Chrome. When you click on the icon, the toolbar will open up at the bottom of the screen.  

It is suggested you go into the options part of Equat|O and select the xx-large size for math which creates a high resolution version so if it has to be resized, it doesn't look pixelated.  Go into Equat|O editor, type in your math problem, and when you are done, click "Copy math as...." an "image" and then paste it into the Jamboard slide.

I'll go into the math editor a bit more on another day.  So now you are ready to create a practice session using Jamboard, docs, and google classroom.  Let me know what you think, I'd love to hear.  Have a great dy.

 

Warm-up

 Stores sold 146 million pounds of candy for Easter.  If that is enough to give every man, woman, and child a half pound of candy, what is the population of the United States.

Warm-up

 

There are 180 million eggs used every Easter.  If each chicken layer 6 eggs in a week, how many chickens were needed to lay 180 million eggs?

Multiplication Chart In Middle School Or High School

 

Before COVID hit, I would have told you that students needed to have their multiplication tables memorized since we know it makes it easier for them to solve problems and learn the more complex problems but this year, I am teaching middle school students who have not had any real math since COVID hit.  The district switched to a pseudo block schedule so they only had to offer one semester of math for a year of credit but students still didn't get the full amount due to school having to go virtual every semester.

Since most of my students do not know their multiplication tables, I let them use a multiplication chart/table so they can look it up. I could let them use a calculator but the calculator does all the work and they have no way to connect factors with the final product.  I decided if they had to look factors up on a table, they would begin to see relationships between numbers and perhaps even connect skip counting with multiplication. 

Since we are working with fractions, I've also been teaching them to use the tables to learn to reduce fractions so they don't have to remember all the rules of divisibility. The thing about the rules of divisibility is simply that most students never learn them and as a teacher, I don't remember all the rules.  I tend to only remember 2, 3, 5, 6, and 10.  In addition, this also helps them learn about division and its relationship to multiplication.  It builds connections which helps 

Furthermore, making them use a multiplication chart/table means they are being exposed to it.  I remember hearing that when students practice something for 21 days, they will learn it, so I am hoping that as they use the charts every day, they will learn more multiplication facts by the time the semester is over.  When they use the charts to learn by practicing, they do not learn the facts by rote in order from one times one to twelve times twelve. 

Most students are really good at memorizing the first half of their facts from one times one to six times six but they don't always remember the larger ones like eight times nine. When they are exposed to all the facts in no particular order, I think they will know them better and they won't have to go through the facts to get to the one they want.

I honestly don't know how well this will work but what I do know is the students who have struggled with multiplication are spending actual time learning the process of finding common denominators, reducing fractions, changing from mixed numbers to improper fractions and back.  They are learning to read the tables/charts, becoming more familiar with those charts and spend more time doing the work.  I have seen students who began the semester exhibiting avoidance behaviors so they'd do no work to doing the problems using a whiteboard and marker while completing the assignment.

Due to COVID, students are getting behind and we have to adjust our thinking to giving students the tools they need to do the work while filling in the gaps in their learning.  This allows them to move forward and succeed.  I'd love to hear what you think about this.  Have a great day.

Paper Airplanes Lead To New Understanding Of Aerodynamics.

Imagine if you will, a room full of adults who are launching paper airplanes, watching them, before taking notes on the way they fly, how far they flew, and other such information.  Then you go back to your classroom to explain why it's important to do this.  Most of your students would look at you strangely because you've spent so much time telling them not to fly their own creations.  

Several scientists got together to answer the question on what is needed to create a good paper airplane and what is needed for it to have a smooth slide as it flies. In the process of answering the question, these scientists from New York University discovered that the aerodynamics involved in keeping a paper airplane level is different from those that keep a regular airplane stable.  In fact, up until this study, there has been no mathematical model to explain how the simplest versions fly.

Although it is quite easy to make a paper airplane, the aerodynamics involved in their flight is much more complex than expected.  They began with trying to identify what keeps a paper airplane gliding smoothly because a paper airplane has no engine and must use gravity and a good design to cross the distance.  Thus the scientists ended up looking at what factors control flight stability.

Consequently, the scientists built and flew paper airplanes with differing centers of mass to see how they controlled flight stability. To do this, they changed the amount of thin copper tape in the front of the plane so they changed the center of mass.   In addition, they dropped plates with varying weights  into a water tank and took both sets of results to create a new aerodynamic model.  It turns out paper airplanes have a center of pressure that conventional airplanes do not have. 

After it was done, they concluded that the center of mass must be in just the right place so the paper airplane is able to glide well.  If the mass was located on the wing or a little off center, the plane fluttered or tumbled.  If the mass was too close to an edge, it would dive downwards.  If they got the mass in just the perfect spot, the plane glided beautifully. 

Using the data from the flights and dropping plates, scientists created a mathematical model that formed the basis of a "flight simulator" because it was able to replicate the various flight paths. In addition, the modeling program explained why the paper airplane is able to glide so well.  It appears the shifting of the center pressure is due to the thin flat wings of the paper airplane and is responsible for the ability to glide.  It is hoped that these results will be used with drones and gliders in the future.

This is cool.  Let me know what you think, I'd love to hear.  Have a great day.

Where Else Do We Observe The Fibonacci sequence In Nature

 

Last time we saw how someone used the Fibonacci sequence to improve his solar energy design since it appears in tree in addition to flowers.  The Fibonacci sequence is one that is found all over the place in nature, so I thought I'd share where else we see it.

Some of the most recognizable representatives of the Fibonacci sequence in nature are in a spiral shape such as a certain snail shells and the nautilus but there are others. 

Let's start with flowers because individual flowers can display the Fibonacci sequence in a couple of ways.  First, with the number of petals for the lily with three, the buttercup with five, chicory with 21, or the daisy with 34.  Each petal is 0.618034 per rotation for maximum use of sunlight, just like with trees. The other way is with the centers or seed heads and the way they spiral such as in the sunflower. The seeds are produced in the center of the flower where they migrate outwards as they fill in the center. 

Another seed that shows this pattern are pinecones as they spiral out from the center.  Actually pinecones use two spirals going upwards in opposite directions such as 3 steps along the right and 5 steps along the left, meeting at the back.  Next time you are out, walking around pine trees, pick up a pinecone and check it out. In addition, the 0.618034 turn can be observed in certain aloe and other cactus plants because each segment is that much further on. The growth pattern spirals just go round and round and round.A beautiful example of the Fibonacci sequence is found in the romanesque broccoli where each spiral is made up of the sequence and tastes so good.  

Now for some not so common examples.  If you look at hurricanes on the weather channel it shows a beautiful spiral consistent with the Fibonacci sequence. Look at most spiral galaxies in space and note the pattern, it is perfect to match the sequence.  If you check the ovary of an angle fish or look at how cancer cells divide, you'll see the same spiral or map the spiral onto a chicken egg, it matches up beautifully 

Back to your nature walk. Look for a chameleon and check out how it's tail spirals, or the way an American millipede curls up or the way a Pangolin curls or even the way a fiddlehead fern spirals.  All of these follow the Fibonacci sequence. 

There are examples that are not nature. You can find examples in the art world with  the "Great Wave Off Kanagawa" by Katsushika Hokusai.  The Fibonacci sequence is found in the curl and wave in the painting but it is not the only piece of art this is true for.   Then there is the person who applied the Fibonacci sequence to the population density and land mass of Africa to show the population density matched the sequence around the landmass.  Finally, the spiral patterns found in fingerprints also follows the sequence. 

So basically everywhere in the world, we can find the Fibonacci sequence.  Have fun looking for it the next time you take a walk, check out art, look at your fingerprint but have fun.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

 If you have 8 pounds of raspberries and each pound has 3.25 cups of berries, how many pints of raspberry jam will you make if each jar requires 2 cups of berries?

Warm up

 

If there are 3.25 cups of raspberries per pound, how many cups of raspberries will you have?

New Place You Find The Fibonacci sequence!

 

If you've read this blog long enough or you've paid attention to articles, you know we see the Fibonacci sequence in nature, specifically flowers all the time.  Now someone has discovered that it exists in at least one other place.  Remember the Fibonacci sequence is the set of numbers beginning with 0, 1 and the next term is the sum of the previous two so it's 0,1,1,2,3,5,8, etc. The sequence was discovered by an Italian who was studying bunnies.

Believe it or not, a gentleman who is designing a solar energy project t, discovered a new Fibonacci sequence as he was designing shafts for.  Others have designed shafts or solar trees but none of them have been able to replicate the natural stability of structure and ability to absorb solar energy of real trees.  

As far as plants go, leaves gather energy from the sun so they want to have the maximum area available for absorbing it.  So the new leaves tend to begin growing just a bit further round the stem from the last one. It turns out the placement is based on the reciprocal of the golden ratio or about 61.8 percent around the stem from the last leaf.  We also know that the best way to approximate the golden ratio is by using the Fibonacci sequence.

Now, it is well known that the Fibonacci sequence is found in the leaves, and branches but this person noted that it appears in the way the tree itself grows.  Think about it.  The main trunk usually splits into two branches, one is smaller than the trunk and the other one is smaller than the trunk and the other branch so you have three different sized branches.  Each tree crotch splits in the same way.  He used Leonardo da Vinci's rule for area preservation which states that the total sum of the thickness of all the branches cannot be larger than the main trunk. 

To build it, he used standard sized aluminum and PVC stock pipes ranging from 1 to 4 inch diameters. He fit the pipes into three dimensionally printed connectors with three openings. These connectors helped the branches fit the rule of area preservation.  In addition, the openings had to be of three different sizes.  The largest is at the and the two branches are two different sizes, both smaller than the main trunk. He combined the connectors to create the final "tree".  

He decided that the tree would have one largest sized trunk -A, one of the next sized down - B, two of the third size - C, three of size -D, five for size -E, and so on.  So he built his artificial trees based on this pattern. So he used this pattern to create his "trees" for the energy. At the end, when he counted it all up, he discovered it matched the Fibonacci sequence. 

So this is a new application for this series.