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Start by drawing a point on a cartesian grid. The position of the point has been selected carefully.

The student at the back right of the class begins. They know nothing. They may either take a guess at what will happen next – or may ask for the next slide. There is no disgrace in choosing to be a scientist instead of making a wild speculation 😉

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The next student may now select to take a guess at what will happen next – or save face (at what will almost certainly be an incorrect guess) and just ask to go directly to the next slide.

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The third student now gets the same choice. For the first time there is enough information that maybe 10% of students might guess the right answer, but there is certainly not enough information to be too confident. The third student can also choose to just go to the next slide and save face…

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The fourth student may start to see some bits of a pattern – not enough to really understand what’s going on, but maybe good enough to make an educated guess and get it right 25% of the time.

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There are plenty of ideas that students may have. Take a break from asking individual students and ask the whole class for some ideas about the behaviour of this spiral.

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This is wrong. Skip immediately to the next slide.

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That’s better. What fraction of the class got it right? There is still no shame in being wrong. Scientists need to have courage to make hypotheses and then break them. Don’t be afraid of failure!

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Where do you think it will go next. By now some clear patterns are emerging. For example – the spiral is acting like a good spiral… always turning clockwise. That observation may not have been precisely stated, but it should be. What other precise observations can you make?

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No – that’s wrong.

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If you have not made an observation about the length of each successive line segment do so now. The next student should try to guess where the next line segment will be drawn.

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Take a vote about where the next line segment will end.

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What do you think of this spiral? Could some students with a strong artistic sense comment?

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I found this last segment a bit ugly. The curve was moving quite gracefully outward, but this turn just seemed too abrupt. Nevertheless it is correct. Nature is not always beautiful. Neither is this Babylonian Spiral.

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The next person should probably be able to precisely guess the correct length of the next line segment.

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The next phase of a scientist is to ask questions and to start going beyond just an understanding of how to create the spiral. Curiosity is key. What are you curious about here? This is not an easy thing to learn, so I will tell you in a few slides two of the burning questions that I had about this Spiral.

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One more guess…

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Let’s ask questions:

1) How does this behave like a spiral?

2) How does this not behave as a spiral?

3) Does it keep on moving away from the initial point? (To answer this question I calculated the square of the distance to the initial point.)

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So right away I see that the spiral behaves in a funny way. It is not slowly getting further and further away. It was actually further away after the eighth step than on subsequent steps. I don’t get the feeling that this will continue, but I am curious just how this Babylonian Spiral will behave.

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See what happens for yourselves. Scientific curiosity should drive you. If you do not have it, that’s okay, but you should learn the type of things that drive curiosity. It is a beautiful thing to learn and will make your life richer.

Download a printable sheet here.

PS. Don’t feel obliged to continue with the distance calculations unless that interests you. Finding out what happens to the spiral was certainly more interesting to me.

Babylonian Spiral

(MathPickle, 2015)

The Scientific Method is best taught in math class with mini universes where students must experiment and deduce the laws that govern.

The Babylonians knew a lot of right triangles, but as far as we know, they never proved the Pythagorean theorem. This Babylonian Spiral is a construct that uses right triangles, but I will leave it up to you to find the rules governing its construction. I don’t think you will need me to email a solution. Your students are bright enough.

Standards for Mathematical Practice

MathPickle puzzle and game designs engage a wide spectrum of student abilities while targeting the following Standards for Mathematical Practice:

MP1 Toughen up!

Students develop grit and resiliency in the face of nasty, thorny problems. It is the most sought after skill for our students.

MP2 Think abstractly!

Students take problems and reformat them mathematically. This is helpful because mathematics lets them use powerful operations like addition.

MP3 Work together!

Students discuss their strategies to collaboratively solve a problem and identify missteps in a failed solution. MathPickle recommends pairing up students for all its puzzles.

MP4 Model reality!

Students create a model that mimics the real world. Discoveries made by manipulating the model often hint at something in the real world.

MP5 Know the tools.

Students master the tools at their fingertips - whether it's a pencil or an online app. 

MP6 Be precise!

Students learn to communicate using precise terminology. MathPickle encourages students not only to use the precise terms of others, but to invent and rigorously define their own terms.

MP7 Be observant!

Students learn to identify patterns. This is one of the things that the human brain does very well. We sometimes even identify patterns that don't really exist 😉

MP8 Be lazy!?!

Students learn to seek for shortcuts. Why would you want to add the numbers one through a hundred if you can find an easier way to do it?


Please use MathPickle in your classrooms. If you have improvements to make, please contact me. I'll give you credit and kudos 😉 For a free poster of MathPickle's ideas on elementary math education go here.

Gordon Hamilton

(MMath, PhD)