![]() In the Important Ideas, we use a flow chart to show the two cases of infinite geometric sums. A similar argument can be made for the time it takes to complete the journey. As n (the number of legs) goes to infinity, S(n) (the total distance traveled) goes to 12. This is a great introduction to the idea of limits, though we save the formal notation for next chapter, opting instead to use the more informal arrows. Pause to consider how strange this concept is even for us who have been teaching this content for a while! One student said it nicely when she explained that “even though there are infinitely many halves, they only add up to a fixed amount because the amount being added is getting smaller and smaller”. Students are excited for the debrief because they are honestly searching for some resolution about this weird chicken! The key ideas to bring up are that even though it would require infinitely many legs of the journey to cross the road, the total distance (sum) is finite. This is a key distinction between infinite arithmetic sequences (and infinite geometric series with a common ratio greater than or equal to 1) and infinite geometric sequences with a common ratio less than 1. In question 7, students are able to reason that adding any amount of pause would make him not arrive, since an infinite amount of equal length pauses would take an infinite amount of time. ![]() Walk away when you pose these advancing questions and allow students to wrestle further. Ask students whether he will cross the road in 128 seconds or whether he will never cross the road, because it can’t be both. At this point many still think he won’t cross the road but this is an obvious paradox. Students first say that it will take him infinitely long, but upon further inspection they realize it should take him 128 seconds, since the second 6 feet should take him just as long as the first six feet. Still students weren’t all that convinced. Surely the marble doesn’t skip half-way marks?! I then bring up a marble that is rolling on the floor. I ask my students: “How do we make it anywhere then? How do we even make it to the door? Don’t I have to get half-way to the door first? And then don’t I have to make it half of the remaining distance? You’re telling me I’ll never make it to the door?!” The students are quick to say that humans don’t walk like Bernie does, and we step over the half-way mark. Almost every group concluded that since there is a horizontal asymptote at y=12, Bernie will never cross the road! Be ready to play devil’s advocate here. In question 3, students use a graphical approach to explore what happens to Bernie’s total distance. Check to make sure students are writing the nth term and not the nth sum in the table (how far Bernie walks on each leg, not how far he’s walked in total). Question 1 and 2 are pretty straightforward for students and provide an easy entry point to the lesson. īooks VIII and IX of Euclid's Elements analyzes geometric progressions (such as the powers of two, see the article for details) and give several of their properties.We often ask why the chicken crossed the road, but today we explore an even more interesting question: how did the chicken cross the road? Get ready for heated arguments and mind-boggled students as your class explores Bernie’s path across a 12 foot road. ![]() It is the only known record of a geometric progression from before the time of Babylonian mathematics. It has been suggested to be Sumerian, from the city of Shuruppak. The general form of a geometric sequence isĪ, a r, a r 2, a r 3, a r 4, … ,Ī clay tablet from the Early Dynastic Period in Mesopotamia, MS 3047, contains a geometric progression with base 3 and multiplier 1/2. is a geometric sequence with common ratio 1/2.Įxamples of a geometric sequence are powers r k of a fixed non-zero number r, such as 2 k and 3 k. is a geometric progression with common ratio 3. In mathematics, a geometric progression, also known as a geometric sequence, is a sequence of non-zero numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. ![]() The first block is a unit block and the dashed line represents the infinite sum of the sequence, a number that it will forever approach but never touch: 2, 3/2, and 4/3 respectively. Mathematical sequence of numbers Diagram illustrating three basic geometric sequences of the pattern 1( r n−1) up to 6 iterations deep.
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