The Traveling Point

Locus of Points problems are not explicit in CCSS-M, but I think that they complement the G-GPE (Expressing Geometric Properties with Equations) domain nicely. I also like a question about the locus of points in a plane equidistant from two given points reminds students of the significance of the perpendicular bisector of a segment. And I am reminded by a phrase that resonated within me when I attended some Achieve the Core professional development a few months ago: The standards are the floor, not the ceiling.

The question:

And the results – in equation form:

And graph form. How can you tell that the points on the line are equidistant from the given points? What do you notice? How often do we ask a question like this without providing students the opportunity to connect the equation and the graph … by just asking for one or the other?

One of the Standards for Mathematical Practice is to use appropriate tools strategically.

One of my own mathematics professors shared this problem with me after observing students work on it in a classroom in Japan. I have posed it to my students the past 8 years or so, and it works nicely with our locus of points lesson.

I find that the problem is difficult for some student to visualize. In fact, most students immediately jump to the conclusion that the point moves along the following path.

Some students are successful when they just think about the problem (even though they some draw tentatively, in case they are wrong).

But others need additional tools to help them “see” the math.

Some groups were successful physically modeling the problem with an index card. One student traces the point while another student rotates the card.

Less than half of the students were successful when I sent out the first poll. But after using appropriate tools strategically, more were successful. One group needed targeted help, which I was able to give as the others in the class moved to the next problem.

Several years ago we began to use TI-Nspire to enhance the problem. For some students, seeing the rectangle rotate helps.

Students use the dynamic geometry capabilities of TI-Nspire to rotate the rectangle – and then some choose to draw in the locus of points that A travels.

The physical motion that they notice as they use the rotation tool is what the rest of the students need to be successful.

 

This year easing the hurry syndrome meant we didn’t get to explore the quarter circle in much detail. Last year, we rotated a quarter circle in the same manner.

The first results are from asking my students the length of the path that A travels after they interacted only with pencil and paper. I didn’t mark the correct answer at this point – or even show my students the results.

At this point, I showed them the following & asked if they wanted to change their answer to the poll. (Thanks to Jeff for creating this document.)

Students responded to this QP once they saw the object moving. You can see that more were successful. Again, I didn’t show students the results of their poll.

Next, I clicked “Show path” and let students watch.

Do you want to keep your response? Or change it based on what you’ve now seen? Students responded to Quick Poll after seeing the trace of the path. All students didn’t need the trace – but some needed it to be able to visualize – and ultimately all students were successful.

There is another great task that we didn’t have time to explore in class last year or this year. But just posing the question is enough for a few students.

What tools can help you build the locus of points that A travels? Or can you visualize how the string unwraps without tools?

And so the journey continues, using tools strategically to help students visualize the results and to explore and deepen understanding of concepts …

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