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Year 8 Spring 2 Lesson 7: Introducing the Fibonacci SequenceObjectiveTo uncover the Fibonacci sequence from a variety of starting points.AimThe Fibonacci sequence and related sequences are now a part of the GCSE syllabus. This lesson introduces the sequence from a variety of starting points.Doing?this investigation?students notice patterns and make and prove conjectures. It offers one of the best ways for learners to discover Fibonacci sequences for themselves. Whether or not they have already met these sequences it is important to ask whether the pattern will continue and, more importantly, why it will continue?ResourcesPowerpoint HYPERLINK "" Record sheet for Leonardo’s Leaps and Making a beeline for those who require the structure.Activity 1: Sheep TalkExplain the problem from slide 1.Sequences of words are formed as follows: The first word only contains the single letter A. To get the next word in the sequence change each A in the previous word into B and each B in the previous word into AB. Students write down the first ten words in this sequence on their whiteboards.They then count the number of A's in each word in the sequence., the number of B's in each word in the sequence and the total number of letters in each word in the sequence. They now have three sequences of numbers. For the proof of why this process gives the Fibonacci sequence you have to consider how the letters change from one word to the next and generate copies of themselves. This can lead to a lot of mathematical talk and it can give excellent practice in mathematical reasoning and communication.Students can each fill in their own tables and be asked to work in pairs to see what patterns they can spot. When they see a pattern (conjecture) they should be asked to use the Sheep Rule to write down some more words in the sequence and see if the pattern continues (test the conjecture). If they find it does then they should be asked to try to explain why the pattern comes out that way.Students can be asked to write down the rule for the sequence in words and then perhaps if they can use symbols to give the rule. Check for different notations offered by the class members or choose to discuss with the class why it is convenient to have an agreed (standard) notation. This could lead comfortably to using algebraic notation:(The Fibonacci sequence can be defined as the sequence?Fn, where?F0?and?F1?are?0?or?1?and?Fn = Fn?1+Fn?2.)But don’t worry of you do not get onto this.Questions for progressionDo you notice a pattern in the sequence of numbers?How many A's do you think there will be in the next word (that you have not written down yet)?Would you like to write down the next word in the list and see if you were right?Look at an A in a word. How many A's come from it in the next word and in the word after that?How many A's in a word come from an AB in the word before it?How do you find the number of A's in the tenth word? Why?Would you have to write the whole list of 20 words to find out how many A's there would be in the twentieth word?Ask similar questions for B'sActivity 2: Leonardo’s LeapsIntroduce the problem to the class.Show them the slide with 4 steps and say there are 5 ways of getting up the 5 steps. Can they find them all on their whiteboards?They then investigate how many ways there are to get up any number of steps.How many ways to get up 15 steps.Questions for progressionCan you investigate for different numbers of steps?Can you spot a pattern?Can you find how many ways to get up 15 steps?How can you communicate your results? Can you put your results into a table?Activity 3: If students finish activity 2 and have explained their work well, they can move onto this activity.Making a Bee LineInvestigate how many different ways there are to get to each number.Questions for progressionCan you investigate for different numbers?Can you spot a pattern?How can you communicate your results? Can you put your results into a table?Why does this problem give the same sequence as the other two problems?Acknowledgements:Problems from a selection from NRich. ................
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