Key Stage 4 - The magic of maths

Notes for teachers

At a glance

This mathematics lesson is suitable for KS4 or more able KS3 students.

They explore numbers in nature and discover that Fibonacci numbers and spirals frequently appear. They will then use the sequence to discover and define patterns before proving why one of the patterns exists. Students will also use the video 'Give us a hand' to explore how mathematicians use mathematical modelling to explore the world around us.

Learning Outcomes

Students learn what the Fibonacci sequence is.

Students are able to find and describe patterns in sequences of numbers.

http://www.oxfordsparks.net/animations/give-us-hand

Each student will need

1 copy of the pages 1-2 of the pupil worksheet a non-permanent marker pen and a cloth to wipe it away calculator piece of squared paper ruler

You will also need

images from pages 3-5 of the pupil worksheet cut out, laminated and displayed around the room other natural objects that display Fibonacci numbers e.g. flowers, pine cones, romanesco cauliflower,

sunflower seed heads, lettuce heads

Possible Lesson Activities

1. Starter activity Ask the class what the connection is between a flower, the Milky Way and a shell. They will

probably conclude that it has something to do with maths! Tell them that in this lesson they will be looking for patterns and by the end of the lesson

they will have worked out the connection. Give each student a copy of page 1 of the pupil worksheet. Ask them to read through the

section Modelling with maths. Discuss with the class that mathematicians look for patterns in things including those found in nature. They can use these patterns to build models that allow them to make predictions to understand the world around them.

2. Paired activity: Finding patterns The students then work in pairs to complete the first task on the pupil worksheet. Place

the laminated images from pages 3-5 of the pupil worksheet around the room and/or place real examples of the objects for students to study. They should count the repeating objects, for example the petals on the flowers. They can use their pens to mark the images to make the counting easier, for example counting the spirals. L and M will take the longest time to count so you may wish to prepare a few copies of these. Go round and talk to the students as they do this and ask them if they can see any connection between the numbers. It will help to write them down in order.

Discuss with the class what they found: A: an apple has 5 seeds. B: there are 13 ridges on the top of a poppy seedcase. C: the flower has 8 petals. D: clover has 3 leaves. E: the flower has 5 petals and 5 stamens. F: hands have 5 digits, each finger has 3 joints and the thumb has 2. G: the flower has 13 petals.

http://www.oxfordsparks.net/animations/give-us-hand

H: the lettuce has 3 leaves coming from its stalk. I: the flowers have 21 petals. J: some leaves have 1 main vein, others have 3 and some have 5 or 7. K: the aloe vera has 5 spirals coming from its centre. L: the pine cone has 8 clockwise spirals and 13 anti-clockwise spirals.M: the sunflower head has 34 clockwise spirals and 55 anti-clockwise spirals.

Note: if you use your own objects, make sure they contain Fibonacci numbers! Some flowers have four petals (although this is not common). Also, the flowers of some species have different numbers of petals with the average being a Fibonacci number.

They should have a sequence which is part of the Fibonacci sequence. Expand on this to show: 1,1,2,3,5,8,13,21,34,55. Question them to see if they can work out the pattern in the sequence and ask them to continue the sequence for another 5 terms.

At this point you may wish to discuss why nature seems to choose numbers from this sequence. For further information you can refer to the video 'Doodling in math: spirals, Fibonacci, and being a plant (2 of 3)' from the weblinks below.

Give out page 2 of the pupil worksheet and ask students to continue to work in pairs to complete the next task. Here they are guided to look for other patterns in the sequence of the square numbers of the Fibonacci numbers. Some students may need hints to find them.

http://www.oxfordsparks.net/animations/give-us-hand

3. Class activity: The Fibonacci spiral Go though the answers to page 2 of the pupil worksheet.

Tell the class that they are now going to prove why 12 + 12 + 22 +32 +52 +82 = 8 x 13. Give each student a piece of squared paper and ask them to draw a box with sides of 1

square. Then, they should draw another box of the same size to its right. Now they have a rectangle showing 12 +12

. If possible, demonstrate how to do this on the board. Question the class - how can we show 22? They draw a square 2 x 2 at the top of their

rectangle. Then a 3 x 3 square to the left, a 5 x 5 square below and an 8 x 8 square to the right. It will look like the image below:

Ask pairs to discuss how this proves the sum above. Listen to their answers and then explain that they now have a rectangle with sides of 8 and 13 (area of 104) which is the sum of the square of the first 6 numbers in the Fibonacci sequence.

http://www.oxfordsparks.net/animations/give-us-hand

Now, demonstrate how they can draw a smooth line connecting the corners of each square in the order they were drawn to create a Fibonacci spiral. They can do this freehand or use a compass. Students may want to continue the pattern and draw a bigger spiral.

Explain that this spiral is a special shape which is often found in nature. Show them some examples using images from an internet search. Some examples are the nautilus shell, the Milky Way galaxy, the cochlea from the ear and hurricanes.

4. Plenary Show the students the video 'Give us a hand' which discusses chirality or handedness. We

get to meet Professor Alain Goriely from The University of Oxford who explains how he uses mathematical modelling to make predictions.

Ask the students if their Fibonacci spirals are right or left-handed. They will see that they spiral in an anti-clockwise direction which means they are right-handed. In fact, most spiral shaped objects in nature are right-handed including 90% of sea shells and even DNA - the helical molecule that codes for our characteristics.

Ask students to revisit the question you posed at the start of the lesson and write a brief answer.

Weblinks

www.khanacademy.org/math/recreational-math/vi-hart/spirals-fibonacci/v/doodling-in-math-class-spirals-fibonacci-and-being-a-plant-2-of-3

Doodling in math: spirals, Fibonacci, and being a plant. A series of videos looking at the Fibonacci numbers in plants.

www.youtube.com/watch?v=kkGeOWYOFoA

Nature by numbers. This short animation looks at where the Fibonacci numbers and spirals are found in nature.

http://www.oxfordsparks.net/animations/give-us-hand