![]() ![]() ![]() This will allow them to benefit from the sharing and justification of ideas, collaboration (mahi tahi) and peer learning (tuakana teina) providing opportunities for students to work in mixed groupings, pairs, as individuals, and with the whole class.using calculators to ease the cognitive load associated with making calculations with larger whole numbers, so students can focus more on pattern and less on finding answers.pacing the sequencing of patterns appropriately allowing time for students to discuss their predictions with others.using recording tools such as tables and branch diagrams (featured in the PowerPoints, and of your own creation in response to students’ needs) to support students’ search for patterns.This could include creating a family tree of direct relatives with class members physically acting out scenarios using people or materials. ![]() Once your family has a solid grasp on where to find the Fibonacci Sequence and where to see the Golden Ratio in action, do some collecting! Challenge your kids to see how many different natural objects they can find throughout the summer that show these relationships – they may be surprised to see how diverse their collection will become.The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to differentiate include: Each of the three short videos provides simple explanations and illustrations of how exactly the Fibonacci Sequence and Golden Ratio manifest in familiar places – and pairs this with some silly stuff, too! To help illustrate the Fibonacci Sequence and Golden Ratio for kids, check out Vi Hart’s three-part Fibonacci-inspired Doodling in Math series. (Additionally, testing out seed patterns based on non-phi numbers is a great way to experiment with the mathematical reasoning behind non-Fibonacci patterns!) When the place in which new seeds are located is determined by phi, then there is no extra space between seeds – the maximum number have been made to fit. In order for plants to have the best possible chance at survival, they need to pack in the maximum number of seeds that they can in the most efficient way possible. appears at an angle that is 1.618 rotations from the one before it – phi determines where each new seed or leaf will appear. Each new petal, seed, or seed-holding pocket in a cone grows in a place whose relationship to the one before it is determined by this special number. Directly related to this is the Golden Ratio’s determination of the exact placement of each of these petals, seeds, and seed pockets. The Fibonacci sequence describes the pattern in which the petals of flowers grow, the number of spirals of seeds that evergreen cones have, and the pattern in which flowers fit the most seeds possible into their centers. So what do these numbers mean to us as amateur mathematicians and naturalists? They help us to understand the deeper non-aesthetic meaning behind the patterns that we observe in nature. If you divide any two consecutive numbers in the sequence – 55 and 34, for example – the quotient will be phi. ![]() The ratio was originally discovered in order to describe the relationship between the longest and shortest sides of a rectangle thought to be the most beautiful to the human eye, but it is very closely related to the Fibonacci Sequence. Meanwhile, the Golden Ratio is a number called phi, which is equivalent to roughly 1.618. It is a series of numbers (1, 1, 2, 3, 5, 8, 13, 21, 34, 55, etc.) in which each number in the pattern is equal to the two numbers before it – for example, 1+1=2, 2+1=3, and 2+3=5. The Fibonacci Sequence and the Golden Ratio describe, in mathematical terms, the reasons for nature’s plethora of spiral patterns. Beneath the aesthetically pleasing shapes of petals, seeds, and branches are two fascinating mathematical concepts that explain nature’s tendency to expand in spirals. The spirals that appear all around us are no accident of nature – while they’re beautiful to look at, their purpose is much more important than vanity alone. Evergreen cones, heads of broccoli and cauliflower, and tree branches all display noticeable iterations of this spiraling pattern, too. Snail shells, too, show growth rings that become gradually larger as they spiral away from the shell’s center. Sunflowers, for example, seem to spiral their seeds from their centers in some sort of mathematical pattern. Nature is filled with patterns – spirals, in particular, are especially noticeable in species of plants and animals. The Fibonacci sequence describes the pattern in which flowers fit the most seeds possible into their centers. ![]()
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