"Math is the real world, okay it's everywhere, okay. Can I show you? You see how the petals spiral?

The number of petals in each row is the sum of the preceding two rows,

the Fibonacci Sequence. It's found in the structure of crystals and the spiral of galaxies and a nautilus shell.

Using a magnifying glass can help for closer examination.

What's more, the ratio between each number in the sequence to the one before it is approximately 1.61803, what the Greeks call the Golden Ratio. It shows up in the pyramids of Giza and the Parthenon at Athens, the dimensions of this card.

This week, for example, I had my boys look at the outside of a purple cabbage, a banana, an onion, an apple and a pomegranate.

We have done an activity like this a few years ago. We looked at a kiwi, an avocado and an artichoke. I picked those particular fruits because I was pretty sure they were not familiar with them and so had to think about what they might look like inside based solely on what they saw on the outside. I had them sketch the outside of the fruits so that they would pay close attention to their shape, size, texture and color. Then they sketched what they guessed the inside would look like. If the outside was tough, would that make the inside soft? Would their be cavities inside? Would the color be the same or different? If the color was the same, would the shade be different? What would the seeds look like? Would they be small or large? Hard or soft? I made sure that they knew that I did not expect them to get the correct answers, but that I just wanted them them wonder, to infer. Then we cut them open to see how they looked.

And they drew them again, noting all the things that they had wondered about before.

And it's based on a number we can find in a flower.

Math is nature's language... its method of communicating directly with us.

Everything is numbers."

- Charlie Epps, NUMB3RS

**Who was Fibonacci and what is the Fibonacci sequence?**

Leonardo of Pisa, (yes, the same Pisa know for its Leaning Tower) known as Fibonacci, was born about 1175 AD and was probably the greatest European mathematician of the middle ages. Leonardo grew up with a North African education under the Moors. He traveled extensively around the Mediterranean coast meeting with many merchants learning their systems of doing arithmetic. Europe was still using the Roman number system and Fibonacci realized the importance of the Hindu-Arabic number system with its ten digits, decimal point and a symbol for zero, and so was one of the first people to introduce it into Europe. Another important discovery which was named after him is called the Fibonacci Sequence. You can arrive at this sequence by beginning with number one add one to it, and you get two, now add these two numbers together and you get three, now add the last two numbers together and you get 5, and the next two added together is eight. Keep going, and you will see that the ratio that occurs after the number 3, is the number 1:618, which is the Golden Ratio.

This ratio was first discovered by the Greek mathematician Pythagoras. Later, an Athenian architect using the Golden Section in building design came up with Phi, the number 1.618. Fibonacci made the next leap when he published a book in 1202 called “Liber Abaci”. He introduced a math problem where a pair of rabbits were placed in a field with the provision that they could not escape or die. At the age of 1 month the female gives birth to 2 new rabbits (1 male, 1 female). The female rabbit does this each month for 1 year. How many rabbits would there be at the end of the year? The answer to this question contains a series of numbers (1, 1, 2, 3, 5, 8, 13, 21, 34, 55…..). I am sure you are getting used to this sequence by now.

**So, what does this have to do with mathematics for children?**

We have spent a lot of time playing with, looking at and discovering patterns in our math. This is because math is really the study of patterns, and looking at the patterns in math can inspire in children the respect and love that math deserves. I encourage you to spend as much time as you can inspiring your students to make, look at and make discoveries with patterns. This appreciation for patterns can overlap into nature study, science and art.

This week, for example, I had my boys look at the outside of a purple cabbage, a banana, an onion, an apple and a pomegranate.

And they drew them again, noting all the things that they had wondered about before.

We did this exercise again this week, using different fruits and vegetables. This time it was ones they were familiar with. Did it make a difference in their ability to accurately predict? Not necessarily, as people don't tend to take the time to notice details. And that was what I was asking them to do. I encourage them to describe the shape, texture and markings of each of them. Note the color. Can you make your sketch have the exact color of the fruit or vegetable? What is the texture like -smooth, bumpy, crackly like paper? Does it pull apart? Does it have spots or other markings? How many sections does it divide into?

Using a magnifying glass can help for closer examination.

Can they predict what each piece will look like when it is cut open? Will the color be the same on the inside as on the outside? Do you expect to find a few seeds or a great many or none at all?

Discuss the lines and the proportion and the different shades of color.

Cut it open and talk about what you discover. Look at the different patterns inside each piece. Have them sketch them, or make prints of them to look at in the future. Do you see spiraling patterns?

Sketch the patterns...

Cut it open and talk about what you discover. Look at the different patterns inside each piece. Have them sketch them, or make prints of them to look at in the future. Do you see spiraling patterns?

Sketch the patterns...

and perhaps watercolor them.

My youngest son became so interested he asked for other fruits and vegtables to examine, so we examined a cucumber and a tomato as well.

At another time you could have them examine a branch of leaves and ask them to describe whatever they notice. Do you see different shades of color? Do you see any places where there are different shades of color? Do you see any places where there are different thicknesses? Do these always occur in the same place? what do you notice about the spaces between the leaves? Are all the leaves the same size? How are they attached to the branch? Are they like your arm and are straight across or do they zig-zag? Do you see any diagonals? Are all the parts the same texture? The leaves on many plants are staggered in a spiral pattern to permit optimum exposure to sunlight. If we apply the Golden Ratio to a circle we can see how it is that this plant exhibits Fibonacci qualities. Cut the leaves off the branch. Can they sort them?

Or here, at

At another time you could have them examine a branch of leaves and ask them to describe whatever they notice. Do you see different shades of color? Do you see any places where there are different shades of color? Do you see any places where there are different thicknesses? Do these always occur in the same place? what do you notice about the spaces between the leaves? Are all the leaves the same size? How are they attached to the branch? Are they like your arm and are straight across or do they zig-zag? Do you see any diagonals? Are all the parts the same texture? The leaves on many plants are staggered in a spiral pattern to permit optimum exposure to sunlight. If we apply the Golden Ratio to a circle we can see how it is that this plant exhibits Fibonacci qualities. Cut the leaves off the branch. Can they sort them?

Make an effort to encourage them to notice patterns during nature study. Encourage them to make as accurate a sketch as they are capable, or take photographs to examine later for patterns.

In the case of tapered pine cones or pineapples, we see a double set of spirals – one going in a clockwise direction and one in the opposite direction. When these spirals are counted, the two sets are found to be adjacent Fibonacci numbers.Similarly, sunflowers have a Golden Spiral seed arrangement. You can see the beautiful pattern easily. This provides a biological advantage because it maximizes the number of seeds that can be packed into a seed head.

Notice the numbers within other flowers. Lillies and Iris tend to have 3 petals. Buttercup, Wild rose, Larkspur, and Columbine tend to have 5. Delphiniums have 8. Marigolds have 13. Aster, Black-eyed susan, and Chicory have 21. Count Daisy petals; do their petals count out as a Fibonacci number?

Another great activity involving art and Fibonacci Sequence can be found here at Notice the numbers within other flowers. Lillies and Iris tend to have 3 petals. Buttercup, Wild rose, Larkspur, and Columbine tend to have 5. Delphiniums have 8. Marigolds have 13. Aster, Black-eyed susan, and Chicory have 21. Count Daisy petals; do their petals count out as a Fibonacci number?

*Teach Kids Art.*

Or here, at

*Almost Unschoolers*.

I love the Fibonacci Sequence - science, and math are so much the poetry of the Maker - and absolutely awe inspiring to read.

ReplyDeletethis has always interested me...although I am HORRIBLE in the math department...if I would have been able to experience a multisensory approach, as you offer your children, I am almost positive I would have been able to grasp mathematical concepts without my palms sweating.....lovely work!

ReplyDeleteI didn't learn about the Fibonacci Sequence and the Golden Ratio until grad. school! You are a wealth of information and insight. This is an awesome post. I will have to refer back to it again and again in the future.

ReplyDeleteWhat a wonderful post, I totally love it!!! Thanks for linking it up!!! Have a fun week!!!

ReplyDelete