Sequence Diagrams

I introduced “sequence diagrams” as a visual tool to help students think about sequences (alongside graphs, tables, and visual patterns).  Instead of modeling the use of this tool, I had the students explore it through a series of questions.

The sequence diagram below shows an arithmetic sequence where the second term is 5 and the thirteenth term is 49.

How much is each little arrow worth?  // Most students realized that counting the number of little arrows and “spreading out” the +44 equally to each one would do the trick. Formally we introduced this concept as slope. Though unconventional, I prefer to introduce slope in terms of a unit ratio, i.e. the amount that y changes when x increases by 1. Usually, slope is simply introduced as “the ratio between the change in y and the change in y”. But ratio has many different interpretations, and I believe introducing slope as a ratio is not enough – one should make a choice about which ratio interpretation to use when introducing slope.

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What’s the y-intercept of this function? // The nice thing of sequence diagrams is that boxes can be added and removed as suits its user. Students were quick to add boxes to the left. 

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What’s the value of box X?  // I gave my students hints in the form of two more questions: How many +4 arrows are there from box zero to box X?  How large is the big arrow?

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What happens if we change all the little arrows from +4 to x4?  // The shift to geometric sequences and exponential functions was very natural with a sequence diagram since many of the same patterns of reasoning carried over. Students were able to figure out a sensible expression for the value of the large arrow, thereby finding the explicit formula of an exponential function without the need for direct instruction.

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Reflection: Sequence diagrams aren’t all that different from tables. But for some reason the boxes and arrows helped the students reason about sequences better than cohorts of students in the past.

The inspiration for these sequence diagrams came from my recent experiences reading a book called “Visual Group Theory”.  In college, I never really understood abstract algebra. I could get through the tests OK but it never really stuck.  Visual Group Theory framed the foundations of abstract algebra in terms of Cayley graphs and everything made so much sense after that.

Like Cayley graphs, sequence diagrams encapsulate many concepts (sequence progressions, y-intercepts, slope, explicit formulas, recursive formulas) into a single, modular, visual package that can be shrunk, extended, and edited to suit different purposes. I hate forcing students to memorize formulas, like A_n = A_1 + (n-1)d or f(x) = ab^x. This is my attempt to avoid that.

How I Taught Domain and Range This Year

I’ve plotted a point, and now my x-collection is 2.

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Now what’s my x-collection? // -3

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What’s my x-collection now?  // Cool discussion on the nature of a line as an infinite collection of points. Students expressed answer as all the values between -3 and 2, including -3 and 2. I suggested the notation -3 << 2.

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How about now? // No change in the x-collection. But the “y-collection” (their words) changed.

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How about now? // Again, no change in the x-collection

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How about now?  // Students added 4 to the x-collection, alongside -3 << 2

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From there, students learned that “x-collection” is formally called domain. Students were able to extend their thinking to the “y-collection”, aka range without much assistance.

This lesson allowed me to change things up on the fly based on my read on the students. Too simple? Introduce open circles and strange twisty curves. Too complex? Stick with straight lines.

Piecewise functions made their first appearance immediately. This isn’t usually the case. But I think students were able to reason about them successfully because the lesson was grounded in the already familiar concept of plotting points. This is in contrast to some externally imposed technique such as “take a ruler and hold it parallel to the y-axis; move the ruler right until the graph stops”.